Robert Raussendorf Statement and Readings - Pacific Institute of ...

Decoherence in quantum computation - foe or friend?Robert Raussendorf, University of British ColumbiaAbstract: Decoherence is detrimental to quantum computation because it makes the computation“noisy”. Or is it? Upon closer inspection, it turns out that decoherence can both compromize andhelp realize quantum computation. Which of the two applies does very much depend on thedecoherence model considered.I will start out by proving the expected, namely that decoherence, for a certain (justifiable) class ofdecoherence models, does indeed compromise quantum computation. In this regard, I will reviewa result of Bravyi and Kitaev [1b]/ van Dam and Howard [1a] demonstrating an upper bound tothe error threshold for fault-tolerant quantum computation. The significance of this upper boundis that no method of error correction, however clever, can put the quantum computation back ontrack if the decoherence level per elementary gate operation is above the threshold value.In the second part of my introduction, I will discuss two computational models [2], [3] that usedecoherent dynamics to realize quantum computation. In the case of [2], the computation is drivenby local projective measurements on a highly entangled quantum state. Therein, the entanglementof the initial quantum state is progressively destroyed as the computation proceeds. Thus, entanglementis a resource for this computational model. In the second case, Ref. [3], universal quantumcomputation is implemented in a dissipative quantum system whose evolution is governed by timeindependentand local couplings to the environment. Due to the purely dissipative nature of theprocess, this way of doing quantum computation exhibits some inherent robustness and defies someof the DiVincenzo criteria for quantum computation.Suggested Reading:[1a] Sergey Bravyi and Alexei Kitaev, Phys. Rev. A 71, 022316 (2005).[1b] Wim van Dam and Mark Howard, Phys. Rev. Lett. 103, 170504 (2009).[2] R. Raussendorf and H.J Briegel, Phys. Rev. Lett. 86, 5188 (2001).[3] F. Verstraete, M. Wolf and J.I. Cirac, Nature Physics 5, 633 - 636 (2009).Remark: The results of refs. [1a] and [1b] are very closely related. For background reading, Irecommend Ref. [1b] over [1a] because it is shorter. In my introduction, I will discuss [1a] (firstpart only), however, because the result therein is better suited for graphical display.1

PHYSICAL REVIEW A 71, 022316 2005Universal quantum computation with ideal Clifford gates and noisy ancillasSergey Bravyi* and Alexei Kitaev †Institute for Quantum Information, California Institute of Technology, Pasadena, 91125 California, USAReceived 6 May 2004; published 22 February 2005We consider a model of quantum computation in which the set of elementary operations is limited toClifford unitaries, the creation of the state 0, and qubit measurement in the computational basis. In addition,we allow the creation of a one-qubit ancilla in a mixed state , which should be regarded as a parameter of themodel. Our goal is to determine for which universal quantum computation UQC can be efficiently simulated.To answer this question, we construct purification protocols that consume several copies of andproduce a single output qubit with higher polarization. The protocols allow one to increase the polarizationonly along certain “magic” directions. If the polarization of along a magic direction exceeds a threshold valueabout 65%, the purification asymptotically yields a pure state, which we call a magic state. We show that theClifford group operations combined with magic states preparation are sufficient for UQC. The connection ofour results with the Gottesman-Knill theorem is discussed.DOI: 10.1103/PhysRevA.71.022316PACS numbers: 03.67.Lx, 03.67.PpI. INTRODUCTION AND SUMMARYThe theory of fault-tolerant quantum computation definesan important number called the error threshold. If the physicalerror rate is less than the threshold value , it is possibleto stabilize computation by transforming the quantum circuitinto a fault-tolerant form where errors can be detected andeliminated. However, if the error rate is above the threshold,then errors begin to accumulate, which results in rapid decoherenceand renders the output of the computation useless.The actual value of depends on the error correction schemeand the error model. Unfortunately, this number seems to berather small for all known schemes. Estimates vary from10 −6 see Ref. 1 to 10 −4 see Refs. 2–4, which is hardlyachievable with the present technology.In principle, one can envision a situation in which qubitsdo not decohere, and a subset of the elementary gates isrealized exactly due to special properties of the physical system.This scenario could be realized experimentally usingspin, electron, or other many-body systems with topologicallyordered ground states. Excitations in two-dimensionaltopologically ordered systems are anyons—quasiparticleswith unusual statistics described by nontrivial representationsof the braid group. If we have sufficient control ofanyons, i.e., are able to move them around each other, fusethem, and distinguish between different particle types, thenwe can realize some set of unitary operators and measurementsexactly. This set may or may not be computationallyuniversal. While the universality can be achieved with sufficientlynontrivial types of anyons 5–8, more realistic systemsoffer only decoherence protection and an incomplete setof topological gates. See Refs. 9,10 about non-Abeliananyons in quantum Hall systems and Refs. 11,12 abouttopological orders in Josephson junction arrays. Nevertheless,universal computation is possible if we introduce some*Email address: serg@cs.caltech.edu† Email address: kitaev@iqi.caltech.eduadditional operations e.g., measurements by Aharonov-Bohm interference 13 or some gates that are not related totopology at all. Of course, these nontopological operationscannot be implemented exactly and thus are prone to errors.In this situation, the threshold error rate may becomesignificantly larger than the values given above because weneed to correct only errors of certain special type and weintroduce a smaller amount of error in the correction stage.The main purpose of the present paper is to illustrate thisstatement by a particular computational model.The model is built upon the Clifford group—the group ofunitary operators that map the group of Pauli operators toitself under conjugation. The set of elementary operations isdivided into two parts: O=O ideal O faulty . Operations fromO ideal are assumed to be perfect. We list these operationsbelow:i prepare a qubit in the state 0;ii apply unitary operators from the Clifford group;iii measure an eigenvalue of a Pauli operator x , y ,or z on any qubit.Here we mean nondestructive projective measurement.We also assume that no errors occur between the operations.It is well known that these operations are not sufficient foruniversal quantum computation UQC unless a quantumcomputer can be efficiently simulated on a classical computer.More specifically, the Gottesman-Knill theorem statesthat by operations from O ideal one can only obtain quantumstates of a very special form called stabilizer states. Such astate can be specified as an intersection of eigenspaces ofpairwise commuting Pauli operators, which are referred to asstabilizers. Using the stabilizer formalism, one can easilysimulate the evolution of the state and the statistics of measurementson a classical probabilistic computer see Ref.14 or a textbook 15 for more details.The set O faulty describes faulty operations. In our model, itconsists of just one operation: prepare an ancillary qubit in amixed state . The state should be regarded as a parameterof the model. From the physical point of view, is mixeddue to imperfections of the preparation procedure entanglementof the ancilla with the environment, thermal fluctua-1050-2947/2005/712/02231614/$23.00022316-1©2005 The American Physical Society

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 2005tions, etc.. An essential requirement is that by preparing nqubits we obtain the state n , i.e., all ancillary qubits areindependent. The independence assumption is similar to theuncorrelated errors model in the standard fault-tolerant computationtheory.Our motivation for including all Clifford group gates intoO ideal relies mostly on the recent progress in the fault-tolerantimplementation of such gates. For instance, using a concatenatedstabilizer code with good error correcting propertiesto encode each qubit and applying gates transversally so thaterrors do not propagate inside code blocks one can implementClifford gates with an arbitrary high precision, see Ref.16. However, these nearly perfect gates act on encodedqubits. To establish a correspondence with our model, oneneeds to prepare an encoded ancilla in the state . It can bedone using the schemes for fault-tolerant encoding of an arbitraryknown one-qubit state described by Knill in Ref. 17.In the more recent paper 18 Knill constructed a scheme offault-tolerant quantum computation which combines i theteleported computing and error correction technique by Gottesmanand Chuang 19; ii the method of purification ofCSS states by Dür and Briegel 20; and iii the magic statesdistillation algorithms described in the present paper. As wasargued in Ref. 18, this scheme is likely to yield a muchhigher value for the threshold it may be up to 1%.Unfortunately, ideal implementation of the Clifford groupcannot be currently achieved in any realistic physical systemwith a topological order. What universality classes of anyonsallow one to implement all Clifford group gates but do notallow one to simulate UQC is an interesting open problem.To fully utilize the potential of our model, we allow adaptivecomputation. It means that a description of an operationto be performed at step t may be a function of all measurementoutcomes at steps 1,…,t−1. For even greater generality,the dependence may be probabilistic. This assumptiondoes not actually strengthen the model since tossing a faircoin can be simulated using O ideal At this point, we need tobe careful because the proper choice of operations should notonly be defined mathematically—it should be computed bysome efficient algorithm. In all protocols described below,the algorithms will actually be very simple. Let us point outthat dropping the computational complexity restriction stillleaves a nontrivial problem: can we prepare an arbitrary multiqubitpure state with any given fidelity using only operationsfrom the basis O?The main question that we address in this paper is asfollows: For which density matrices can one efficientlysimulate universal quantum computation by adaptive computationin the basis O?It will be convenient to use the Bloch sphere representationof one-qubit states: = 1 2 I + x x + y y + z z .The vector x , y , z will be referred to as the polarizationvector of . Let us first consider the subset of states satisfying x + y + z 1.This inequality says that the vector x , y , z lies inside theoctahedron O with vertices ±1, 0, 0, 0, ±1, 0, 0, 0, ±1,FIG. 1. Left: the Bloch sphere and the octahedron O. Right: theoctahedron O projected on the x−y plane. The magic states correspondto the intersections of the symmetry axes of O with the Blochsphere. The empty and filled circles represent T-type and H-typemagic states, respectively.see Fig. 1. The six vertices of O represent the six eigenstatesof the Pauli operators x , y , and z . We can prepare thesestates by operations from O ideal only. Since is a convexlinear combination probabilistic mixture of these states, wecan prepare by operations from O ideal and by tossing a coinwith suitable weights. Thus we can rephrase the Gottesman-Knill theorem in the following way.Theorem 1. Suppose the polarization vector x , y , z ofthe state belongs to the convex hull of ±1, 0, 0, 0, ±1, 0,0, 0, ±1. Then any adaptive computation in the basis O canbe efficiently simulated on a classical probabilistic computer.This observation leads naturally to the following question:is it true that UQC can be efficiently simulated whenever lies in the exterior of the octahedron O? In an attempt toprovide at least a partial answer, we prove the universalityfor a large set of states. Specifically, we construct two particularschemes of UQC simulation based on a method whichwe call magic states distillation. Let us start by defining themagic states.Definition 1. Consider pure states H,TC 2 such thatandTT = 1 2I + 1 3 x + y + z ,HH = 1 2I + 1 2 x + z .The images of T and H under the action of one-qubitClifford operators are called magic states of T type and Htype, respectively.This notation is chosen since H and T are eigenvectorsof certain Clifford group operators: the Hadamard gate H andthe operator usually denoted T, see Eq. 7. Denote the onequbitClifford group by C 1 . Overall, there are 8 magic statesof T type, UT,UC 1 up to a phase and 12 states of Htype, UH,UC 1 , see Fig. 1. Clearly, the polarization vectorsof magic states are in one-to-one correspondence withrotational symmetry axes of the octahedron O H-type statescorrespond to 180° rotations and T-type states correspond to120° rotations. The role of magic states in our constructionis twofold. First, adaptive computation in the basis O idealtogether with the preparation of magic states of either typeallows one to simulate UQC see Sec. III. Second, by adap-022316-2

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…tive computation in the basis O ideal one can “purify” imperfectmagic states. It is a rather surprising coincidence thatone and the same state can comprise both of these properties,and that is the reason why we call them magic states.More exactly, a magic state distillation procedure yieldsone copy of a magic state with any desired fidelity fromseveral copies of the state , provided that the initial fidelitybetween and the magic state to be distilled is large enough.In the course of distillation, we use only operations from theset O ideal . By constructing two particular distillationschemes, for T-type and H-type magic states, respectively,we prove the following theorems.Theorem 2. Let F T be the maximum fidelity between and a T-type magic state, i.e.,F T = maxUC 1 TU † UT.Adaptive computation in the basis O=O ideal allows oneto simulate universal quantum computation wheneverF T F T = +21 1 3 0.910.71/2Theorem 3. Let F H be the maximum fidelity between and an H-type magic state,F H = maxUC 1 HU † UH.Adaptive computation in the basis O=O ideal allows oneto simulate universal quantum computation wheneverF H F H 0.927.The quantities F T and F H have the meaning of thresholdfidelity since our distillation schemes increase the polarizationof , converging to a magic state as long as the inequalitiesF T F T or F H F H are fulfilled. If they are notfulfilled, the process converges to the maximally mixed state.The conditions stated in the theorems can also be understoodin terms of the polarization vector x , y , z . Indeed, let usassociate a “magic direction” with each of the magic states.Then Theorems 2 and 3 say that the distillation is possible ifthere is a T direction such that the projection of the vector x , y , z onto that T direction exceeds the threshold valueof 2F T 2 −10.655, or if the projection on some of the Hdirections is greater than 2F H 2 −10.718.Let us remark that, although the proposed distillationschemes are probably not optimal, the threshold fidelities F Tand F H cannot be improved significantly. Indeed, it is easy tocheck that the octahedron O corresponding to probabilisticmixtures of stabilizer states can be defined aswhereF T * =O = :F T F T * ,121 + 1 31/2 0.888.It means that F T * is a lower bound on the threshold fidelity F Tfor any protocol distilling T-type magic states. Thus any potentialimprovement to Theorem 2 may only decrease F Tfrom 0.910 down to F T * =0.888. From a practical perspective,the difference between these two numbers is not important.On the other hand, such an improvement would be ofgreat theoretical interest. Indeed, if Theorem 2 with F T replacedby F T * is true, it would imply that the Gottesman-Knilltheorem provides necessary and sufficient conditions for theclassical simulation, and that a transition from classical touniversal quantum behavior occurs at the boundary of theoctahedron O. This kind of transition has been discussed incontext of a general error model 21. Our model is simpler,which gives hope for sharper results.By the same argument, one can show that the quantitydefF * H =maxOPHYSICAL REVIEW A 71, 022316 2005 HH = 1 21 + 1 21/2 0.924is a lower bound on the threshold fidelity F H for any protocoldistilling H-type magic states.A similar approach to UQC simulation was suggested inRef. 22, where Clifford group operations were used to distillthe entangled three-qubit state 000+001+010+100, which is necessary for the realization of the Toffoligate.The rest of the paper is organized as follows. Section IIcontains some well-known facts about the Clifford group andstabilizer formalism, which will be used throughout the paper.In Sec. III we prove that magic states together withoperations from O ideal are sufficient for UQC. In Sec. IVideal magic are substituted by faulty ones and the error ratethat our simulation algorithm can tolerate is estimated. InSec. V we describe a distillation protocol for T-type magicstates. This protocol is based on the well-known five-qubitquantum code. In Sec. VI a distillation protocol for H-typemagic states is constructed. It is based on a certain CSSstabilizer code that encodes one qubit into 15 and admits anontrivial automorphism 23. Specifically, the bitwise applicationof a certain non-Clifford unitary operator preserves thecode subspace and effects the same operator on the encodedqubit. We conclude with a brief summary and a discussion ofopen problems.II. CLIFFORD GROUP, STABILIZERS, AND SYNDROMEMEASUREMENTSLet C n denote the n-qubit Clifford group. Recall that it is afinite subgroup of U2 n generated by the Hadamard gate Happlied to any qubit, the phase-shift gate K applied to anyqubit, and the controlled-not gate x which may be appliedto any pair qubits,H = 1 2 1 11 − 1, K = 1 00 i, x = I 00 x.The Pauli operators x , y , z belong to C 1 , for instance, z=K 2 and x =HK 2 H. The Pauli group PnC n is generatedby the Pauli operators acting on n qubits. It is known 24that the Clifford group C n augmented by scalar unitary operatorse i I coincides with the normalizer of Pn in the uni-1022316-3

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 2005tary group U2 n . Hermitian elements of the Pauli group areof particular importance for quantum error correction theory;they are referred to as stabilizers. These are operators of theform± 1 ¯ n, j 0,x,y,z,where 0 =I. Let us denote by Sn the set of all n-qubitstabilizers:Sn = S Pn : S † = S.For any two stabilizers S 1 ,S 2 we have S 1 S 2 = ±S 2 S 1 and S 12=S 2 2 =I. It is known that for any set of pairwise commutingstabilizers S 1 ,…,S k Sn there exists a unitary operator VC n such thatVS j V † = z j,j = 1,…,k,where z j denotes the operator z applied to the jth qubit,e.g., z 1= z I ¯ I.These properties of the Clifford group allow us to introducea very useful computational procedure which can berealized by operations from O ideal . Specifically, we can performa joint nondestructive eigenvalue measurement for anyset of pairwise commuting stabilizers S 1 ,…,S k Sn. Theoutcome of such a measurement is a sequence of eigenvalues= 1 ,…, k , j = ±1, which is usually called a syndrome.For any given outcome, the quantum state is acted upon bythe projectork1 = j=1 2 I + jS j .Now, let us consider a computation that begins with anarbitrary state and consists of operations from O ideal . It isclear that we can defer all Clifford operations until the veryend if we replace the Pauli measurements by general syndromemeasurements. Thus the most general transformationthat can be realized by O ideal is an adaptive syndrome measurement,meaning that the choice of the stabilizer S j to bemeasured next depends on the previously measured values of 1 ,…, j−1 . In general, this dependence may involve cointossing. Without loss of generality one can assume that S jcommutes with all previously measured stabilizersS 1 ,…,S j−1 for all possible values of 1 ,…, j−1 and cointossing outcomes. Adaptive syndrome measurement hasbeen used in Ref. 25 to distill entangled states of a bipartitesystem by local operations.III. UNIVERSAL QUANTUM COMPUTATION WITHMAGIC STATESIn this section, we show that operations from O ideal aresufficient for universal quantum computation if a supply ofideal magic states is also available. First, consider a onequbitstateA = 2 −1/2 0 + e i 1and suppose that is not a multiple of /2. We now describea procedure that implements the phase shift gate2022316-4e i = 1 00 e i by consuming several copies of A and using only operationsfrom O ideal .Let =a0+b1 be the unknown initial state whichshould be acted on by e i . Prepare the state 0 = A and measure the stabilizer S 1 = z z . Note that bothoutcomes of this measurement appear with probability 1/2.If the outcome is “+1”, we are left with the state 1 + = a0,0 + be i 1,1.In the case of “−1” outcome, the resulting state is 1 − = ae i 0,1 + b1,0.Let us apply the gate x 1,2 the first qubit is the controlone. The above two states are mapped to 2 + = x 1,2 1 + = a0 + be i 1 0, 2 − = x 1,2 1 − = ae i 0 + b1 1.Now the second qubit can be discarded, and we are left withthe state a0+be ±i 1, depending upon the measured eigenvalue.Thus the net effect of this circuit is the application ofa unitary operator that is chosen randomly between e i and e −i and we know which of the two possibilities hasoccurred.Applying the circuit repeatedly, we effect the transformationse ip 1 , e ip 2 ,… for some integers p 1 , p 2 ,… whichobey the random-walk statistics. It is well known that such arandom walk visits each integer with the probability 1. Itmeans that sooner or later we will get p k =1 and thus realizethe desired operator e i . The probability that we will needmore than N steps to succeed can be estimated as cN −1/2 forsome constant c0. Note also that if is a rational multipleof 2, we actually have a random walk on a cyclic group Z q .In this case, the probability that we will need more than Nsteps decreases exponentially with N.The magic state H can be explicitly written in the standardbasis asH = cos 80 + sin 81.Note that HKH=e i/8 A −/4 . So if we are able to preparethe state H, we can realize the operator e −i/4 . It doesnot belong to the Clifford group. Moreover, the subgroup ofU2 generated by e −i/4 and C 1 is dense in U2. 1 Thusthe operators from C 1 and C 2 together with e −i/4 constitutea universal basis for quantum computation.The magic state T can be explicitly written in the standardbasis:1 Recall that the action of the Clifford group C 1 on the set ofoperators ± x , ± y , ± z coincides with the action of rotational symmetrygroup of a cube on the set of unit vectors ±e x , ±e y , ±e z ,respectively.3

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…T = cos 0 + e i/4 sin 1, cos2 = 1 3.Let us prepare an initial state 0 =T T and measure thestabilizer S 1 = z z . The outcome +1 appears with probabilityp + =cos 4 +sin 4 =2/3. If the outcome is −1, we discardthe reduced state and try again, using a fresh pair ofmagic states. On average, we need three copies of the Tstate to get the outcome +1. The reduced state correspondingto the outcome +1 is 1 = cos 0,0 + i sin 1,1, = 12 .Let us apply the gate x 1,2 and discard the secondqubit. We arrive at the state 2 = cos 0 + i sin 1.Next apply the Hadamard gate H: 3 = H 2 = 2 −1/2 e i 0 + e −2i 1 = A −/6 .We can use this state as described above to realize the operatore −i/6 . It is easy to check that Clifford operatorstogether with e −i/6 constitute a universal set of unitarygates.Thus we have proved that the sets of operationsO ideal H and O ideal T are sufficient for universalquantum computation.IV. ERROR ANALYSISTo establish a connection between the simulation algorithmsdescribed in Sec. III and the universality theoremsstated in the introduction we have to substitute ideal magicstates by faulty ones. Before doing that let us discuss theideal case in more detail. Suppose that a quantum circuit tobe simulated uses a gate basis in which the only non-Cliffordgate is the phase shift e −i/4 or e −i/6 . One can applythe algorithm of Sec. III to simulate each non-Clifford gateindependently. To avoid fluctuations in the number of magicstates consumed at each round, let us set a limit of K magicstates per round, where K is a parameter to be chosen later.As was pointed out in Sec. III, the probability for some particularsimulation round to “run out of budget” scales asexp−K for some constant 0. If at least one simulationround runs out of budget, we declare a failure and the wholesimulation must be aborted. Denote the total number of non-Clifford gates in the circuit by L. The probability p a for thewhole simulation to be aborted can be estimated asp a 1 − 1 − exp− K L L exp− K 1,provided that L exp−K1. We will assumeK −1 ln L,so the abort probability can be neglected.Each time the algorithm requests an ideal magic state, itactually receives a slightly nonideal one. Such nearly perfectmagic states must be prepared using the distillation methods4described in Secs. V and VI. Let us estimate an affordableerror rate out for distilled magic states. Since there are Lnon-Clifford gates in the circuit, one can tolerate an errorrate of the order 1/L in implementation of these gates. 2 Eachnon-Clifford gate requires Kln L magic states. Thus thewhole simulation is reliable enough if one chooses out 1/L ln L.What are the resources needed to distill one copy of amagic state with the error rate out ? To be more specific, letus talk about H-type states. It will be shown in Sec. VI thatthe number n of raw undistilled ancillas needed to distillone copy of the H magic state with an error rate not exceeding out scales asn ln1/ out , = log 3 15 2.5,see Eq. 39. Taking out from Eq. 5, one getsn ln L .Since the whole simulation requires KLL ln L copies ofthe distilled H state, we needN Lln L +1raw ancillas overall.Summarizing, the simulation theorems stated in the introductionfollow from the following results the last one willbe proved later:i the circuits described in Sec. III allow one to simulateUQC with the sets of operations O ideal H andO ideal T;ii these circuits work reliably enough if the states Hand T are slightly noisy, provided that the error rate doesnot exceed out 1/L ln L;iii a magic state having an error rate out can be preparedfrom copies of the raw ancillary state using the distillationschemes provided that F T F T or F H F H .The distillation requires resources that are polynomial inln L.V. DISTILLATION OF T-TYPE MAGIC STATESSuppose we are given n copies of a state , and our goalis to distill one copy of the magic state T. The polarizationvector of can be brought into the positive octant of theBloch space by a Clifford group operator, so we can assumethat x , y , z 0.In this case, the fidelity between and T is the largest oneamong all T-type magic states, i.e.,A related quantity,PHYSICAL REVIEW A 71, 022316 2005F T = TT.2 This fault tolerance does not require any redundancy in theimplementation of the circuit e.g., the use of concatenated codes.It is achived automatically because in the worst case the error probabilityaccumulates linearly in the number of gates. In our modelonly non-Clifford gates are faulty.5022316-5

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 2005 = 1 − TT = 1 21 − 1 3 x + y + z ,will be called the initial error probability. By definition, 01/2.The output of the distillation algorithm will be some onequbitmixed state out . To quantify the proximity between outand T, let us define a final error probability: out = 1 − T out T.It will be certain function of n and . The asymptotic behaviorof this function for n→ reveals the existence of athreshold error probability, 0 = 1 21 − 3 7 0.173,TT 0 = e +i/3 T 0 , TT 1 = e −i/3 T 1 ,T 0,1 T 0,1 = 1 2I ± 1 3 x + y + z .defNote that T 0 = T and T 1 = y HT 0 are T-type magicstates.Let us apply a dephasing transformation,D = 1 3 + TT† + T † T,to each copy of the state . The transformation D can berealized by applying one of the operators I,T,T −1 chosenwith probability 1/3 each. Since9such that for 0 the function out n, converges to zero.We will see that for small ,we haveDT 0 T 1 = DT 1 T 0 = 0, out n, 5 n , = 1/log 2 30 0.2. 6D = 1 − T 0 T 0 + T 1 T 1 .10On the other hand, if 0 , the output state converges to themaximally mixed state, i.e., lim n→ out n,=1/2.Before coming to a detailed description of the distillationalgorithm, let us outline the basic ideas involved in its construction.The algorithm recursively iterates an elementarydistillation subroutine that transforms five copies of an imperfectmagic state into one copy having a smaller errorprobability. This elementary subroutine involves a syndromemeasurement for certain commuting stabilizers S 1 ,S 2 ,S 3 ,S 4S5. If the measured syndrome 1 , 2 , 3 , 4 is nontrivial j =−1 for some j, the distillation attempt fails andthe reduced state is discarded. If the measured syndrome istrivial j =1 for all j, the distillation attempt is successful.Applying a decoding transformation a certain Clifford operatorto the reduced state, we transform it to a single-qubitstate. This qubit is the output of the subroutine.Our construction is similar to concatenated codes used inmany fault-tolerant quantum computation techniques, but itdiffers from them in two respects. First, we do not need tocorrect errors—it suffices only to detect them. Once an errorhas been detected, we simply discard the reduced state, sinceit does not contain any valuable information. This allows usto achieve higher threshold error probability. Second, we donot use quantum codes in the way for which they were originallydesigned: in our scheme, the syndrome is measured ona product state.The state T is an eigenstate for the unitary operatorT = e i/4 KH = ei/4 2 1 1i − i C 1 . 7Note that T acts on the Pauli operators as follows: 3T x T † = z , T z T † = y , T y T † = x . 8We will denote its eigenstates by T 0 and T 1 , so thatWe will assume that the dephasing transformation is appliedat the very first step of the distillation, so has the form 10.Thus the initial state for the elementary distillation subroutineis in = 5 =x0,1 5 x 1 − 5−x T x T x ,11where x=x 1 ,…,x 5 is a binary string, x is the number of1’s in x, anddefT x = T x1 ¯ T x5 .The stabilizers S 1 ,…,S 4 to be measured on the state incorrespond to the famous five-qubit code, see Refs. 26,27.They are defined as follows:S 1 = x z z x I,S 2 = I x z z x ,S 3 = x I x z z ,S 4 = z x I x z .12This code has a cyclic symmetry, which becomes explicit ifwe introduce an auxiliary stabilizer, S 5 =S 1 S 2 S 3 S 4 = z z x I x . Let L be the two-dimensional code subspacespecified by the conditions S j =, j=1,…, 4, and bethe orthogonal projector onto L:4 = 1 I + S j .16 j=1It was pointed out in Ref. 16 that the operators133 The operator denoted by T in Ref. 16 does not coincide withour T. They are related by the substitution T→e −i/4 T † though.andXˆ = x 5 , Ŷ = y 5 , Ẑ = z 5 ,022316-6

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…PHYSICAL REVIEW A 71, 022316 2005Tˆ = T 514commute with , thus preserving the code subspace. Moreover,Xˆ ,Ŷ ,Ẑ obey the same algebraic relations as one-qubitPauli operators, e.g., Xˆ Ŷ =iẐ. Let us choose a basis in L suchthat Xˆ ,Ŷ, and Ẑ become logical Pauli operators x , y , and z , respectively. How does the operator Tˆ act in this basis?From Eq. 8 we immediately getTˆ Xˆ Tˆ † = Ẑ, Tˆ ẐTˆ † = Ŷ, Tˆ ŶTˆ † = Xˆ .Therefore Tˆ coincides with the logical operator T up to anoverall phase factor. This factor is fixed by the condition thatthe logical T has eigenvalues e ±i/3 .Let us find the eigenvectors of Tˆ that belong to L. Considertwo particular states from L, namelyT 1 L = 6T00000 , and T 0 L = 6T11111 .In the Appendix we show thatT 00000 T 00000 = T 11111 T 11111 = 1 6 ,15so that the states T 0 L and T 1 L are normalized. Taking intoaccount that Tˆ ,=0 and thatwe getTˆ T x = e i/35−2x T x for all x 0,1 5 ,Tˆ T 1 L = 6Tˆ T00000 = 6Tˆ T00000 = e −i/3 T 1 L .Analogously, one can check thatTˆ T 0 L = e +i/3 T 0 L .16It follows that Tˆ is exactly the logical operator T, includingthe overall phase, and T 0 L and T 1 L are the logical states T 0 and T 1 up to some phase factors, which are not importantfor us. Therefore we haveT L 0,1 T L 0,1 = 2I 1 ± 1 Xˆ + Ŷ + Ẑ. 317Now we are in a position to describe the syndrome measurementperformed on the state in . The unnormalized reducedstate corresponding to the trivial syndrome is as follows: s = in =x0,1 5 x 1 − 5−x T x T x ,18see Eq. 11. The probability for the trivial syndrome to beobserved isp s = Tr s .Note that the state T x is an eigenvector of Tˆ for any x0,1 5 . But we know that the restriction of Tˆ on L haseigenvalues e ±i/3 . At the same time, Eq. 16 implies thatTˆ T x = − T x whenever x=1 or x=4. This eigenvalue equation is not acontradiction only ifT x = 0 for x = 1,4.This equality can be interpreted as an error correction property.Indeed, the initial state in is a mixture of the desiredstate T 00000 and unwanted states T x with x0. We caninterpret the number of “1” components in x as a number oferrors. Once the trivial syndrome has been measured, we canbe sure that either no errors or at least two errors have occurred.Such error correction, however, is not directly relatedto the minimal distance of the code.It follows from Eq. 16 that for x=2, 3 one hasTˆ T x =e ±i/3 T x , so that T x must be proportional toone of the states T 0 L , T 1 L . Our observations can be summarizedas follows:T x =6−1/2 T L 1 , if x = 0,0, if x = 1,a x T L 0 , if x = 2,b x T L 191 , if x = 3,0, if x = 4,6 −1/2 T L 0 , if x = 5.Here the coefficients a x ,b x depend upon x in some way. Theoutput state 18 can now be written as s =2 1 6 5 + 2 1 − 3 a x T 0 L T L 0 x:x=2+ 1 6 1 − 5 + 3 1 − 2 x:x=3b x 2 T 1 L T 1 L . 20To exclude the unknown coefficients a x and b x , we can usethe identityT 0 L T 0 L + T 1 L T 1 L = =x0,1 5 T x T x .Substituting Eq. 19 into this identity, we get a x 2 = b x 2 = 5x:x=2 x:x=36 .So the final expression for the output state s is as follows: s = 5 + 5 2 1 − 3T L6 0 T L 0 + 1 − 5 + 5 3 1 − 26T 1 L T 1 L .21Accordingly, the probability to observe the trivial syndromeisp s = 5 + 5 2 1 − 3 + 5 3 1 − 2 + 1 − 5. 226A decoding transformaion for the five-qubit code is a unitaryoperator VC 5 such that022316-7

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 2005FIG. 2. The final error probability out and the probability p s tomeasure the trivial syndrome as functions of the initial error probability for the T-type states distillation.VL = C 2 0,0,0,0.In other words, V maps the stabilizers S j , j=2, 3, 4, 5 to z j. The logical operators Xˆ ,Ŷ ,Ẑ are mapped to the Paulioperators x , y , z acting on the first qubit. From Eq. 17we infer thatVT L0,1 = T 0,1 0,0,0,0maybe up to some phase. The decoding should be followedby an additional operator A= y HC 1 , which swaps thestates T 0 and T 1 note that for small the state s is closeto T 1 L , while our goal is to distill T 0 . After that we get anormalized output statewhere out = 1 − out T 0 T 0 + out T 1 T 1 , out =t 5 + 5t 21 + 5t 2 + 5t 3 + t 5, t = 1 − . 23The plot of the function out is shown on Fig. 2. Itindicates that the equation out = has only one nontrivialsolution, = 0 0.173. The exact value is 0 = 1 21 − 3 7.If 0 , we can recursively iterate the elementary distillationsubroutine to produce as good an approximation to thestate T 0 as we wish. On the other hand, if 0 , the distillationsubroutine increases the error probability and iterationsconverge to the maximally mixed state. Thus 0 is athreshold error probability for our scheme. The correspondingthreshold polarization is 1−2 0 = 3/70.655. For a sufficientlysmall , one can use the approximation out 5 2 .The probability p s = p s to measure the trivial syndromedecreases monotonically from 1/6 for =0 to 1/16 for =1/2, see Fig. 2. In the asymptotic regime where is small,we can use the approximation p s p s 0=1/6.Now the construction of the whole distillation scheme isstraightforward. We start from n1 copies of the state =1−T 0 T 0 +T 1 T 1 . Let us split these states intogroups containing five states each and apply the elementarydistillation subroutine described above to each group independently.In some of these groups the distillation attemptfails, and the outputs of such groups must be discarded. Theaverage number of “successful” groups is obviously p s n/5n/30 if is small. Neglecting the fluctuations ofthis quantity, we can say that our scheme provides a constantyield r=1/30 of output states that are characterized by theerror probability out 5 2 . Therefore we can obtain r 2 nstates with out 5 3 4 , r 3 n states with out 5 7 8 , and so on.We have created a hierarchy of states with n states on thefirst level and four or fewer states on the last level. Let k bethe number of levels in this hierarchy and out the error probabilitycharacterizing the states on the last level. Up to smallfluctuations, the numbers n,k, out , and are related by thefollowing obvious equations:Their solution yields Eq. 6. out 1 5 52k , r k n 1. 24VI. DISTILLATION OF H-TYPE MAGIC STATESA distillation scheme for H-type magic states also worksby recursive iteration of a certain elementary distillation subroutinebased on a syndrome measurement for a suitable stabilizercode. Let us start with introducing some relevant codingtheory constructions, which reveal an unusual symmetryof this code and explain why it is particularly useful forH-type magic states distillation.Let F 2 n be the n-dimensional binary linear space and A bea one-qubit operator such that A 2 =I. With any binary vectoru=u 1 ,…,u n F 2 n we associate the n-qubit operatorAu = A u 1 A u 2 ¯ A u n.Let u,v= ni=1 u i v i mod 2 denote the standard binary innerproduct. If LF n 2 is a linear subspace, we denote by L theset of vectors which are orthogonal to L. The Hammingweight of a binary vector u is denoted by u. Finally, u·vF n 2 designates the bitwise product of u and v, i.e., u·v i=u i v i .A systematic way of constructing stabilizer codes wassuggested by Calderbank, Shor, and Steane, see Refs.28,29. Codes that can be described in this way will bereferred to as standard CSS codes. In addition, we consider022316-8

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…their images under an arbitrary unitary transformation VU2 applied to every qubit. Such “rotated” codes will becalled CSS codes.Definition 2. Consider a pair of one-qubit Hermitian operatorsA,B such thatA 2 = B 2 = I,AB = − BA,and a pair of binary vector spaces L A ,L B F 2 n , such thatu,v = 0 for all u L A ,v L B .A quantum code CSSA,L A ;B,L B is a decompositionC 2 n = H,,* *L A L Bwhere the subspace H, is defined by the conditionsAu = − 1 u ,Bv = − 1 v 25for all uL A and vL B . The linear functionals and arereferred to as A syndrome and B syndrome, respectively. Thesubspace H0,0 corresponding to the trivial syndromes ==0 is called the code subspace.The subspaces H, are well defined since the operatorsAu and Bv commute for any uL A and vL B :AuBv = − 1 u,v BvAu = BvAu.The number of logical qubits in a CSS code isk = log 2 dim H0,0 = n − dim L A − dim L B .Logical operators preserving the subspaces H, can bechosen asAu : u L B /L A and Bv : v L A /L B .By definition, L A L B and L B L A , so the factor spacesare well defined. In the case where A and B are Pauli operators,we get a standard CSS code. Generally, A=V z V †and B=V x V † for some unitary operator VSU2, so anarbitrary CSS code can be mapped to a standard one by asuitable bitwise rotation. By a syndrome measurement for aCSS code we mean a projective measurement associatedwith the decomposition 25.Consider a CSS code such that some of the operatorsAu, Bv do not belong to the Pauli group Pn. Let us posethis question: can one perform a syndrome measurement forthis code by operations from O ideal only? It may seem thatthe answer is no, because by definition of O ideal one cannotmeasure an eigenvalue of an operator unless it belongs to thePauli group. Surprisingly, this naive answer is wrong. Indeed,imagine that we have measured part of the operatorsAu, Bv namely, those that belong to the Pauli group.Now we may restrict the remaining operators to the subspacecorresponding to the obtained measurement outcomes. Itmay happen that the restriction of some unmeasured operatorAu, which does not belong to the Pauli group, coincideswith the restriction of some other operator Ãũ Pn. Ifthis is the case, we can safely measure Ãũ instead of Au.The 15-qubit code that we use for the distillation is actuallythe simplest to our knowledge CSS code exhibiting thisPHYSICAL REVIEW A 71, 022316 2005strange behavior. We now come to an explicit description ofthis code.Consider a function f of four Boolean variables. Denoteby fF 2 15 the table of all values of f except f0000. Thetable is considered as a binary vector, i.e.,f = „f0001, f0010, f0011,…, f1111….Let L 1 be the set of all vectors f, where f is a linear functionsatisfying f0=0. In other words, L 1 is the linear subspacespanned by the four vectors x j , j=1, 2, 3, 4 where x jis an indicator function for the jth input bit:L 1 = linear spanx 1 ,x 2 ,x 3 ,x 4 .Let also L 2 be the set of all vectors f, where f is a polynomialof degree at most 2 satisfying f0=0. In other words,L 2 is the linear subspace spanned by the four vectors x j andthe six vectors x i x j :L 2 = linear spanx 1 ,x 2 ,x 3 ,x 4 ,x 1 x 2 ,x 1 x 3 ,x 1 x 4 ,x 2 x 3 ,x 2 x 4 ,x 3 x 4 .26The definition of L 1 and L 2 resembles the definition of puncturedReed-Muller codes of order 1 and 2, respectively, seeRef. 30. Note also that L 1 is the dual space for the 15-bitHamming code. The relevant properties of the subspaces L jare stated in the following lemma.Lemma 1.1 For any uL 1 one has u0mod 8.2 For any vL 2 one has v0mod 2.3 Let 1 be the unit vector 1, 1,…, 1, 1. Then L 1=L 2 1 and L 2 =L 1 1.4 For any vectors u,vL 1 one has u·v0mod 4.5 For any vectors uL 1 and vL 2 one has u·v0mod 4.Proof.1 Any linear function f on F 2 4 satisfying f0=0 takesvalue 1 exactly eight times if f 0 or zero times if f =0.2 All basis vectors of L 2 have weight equal to 8 thevectors x i or 4 the vectors x i x j . By linearity, all elementsof L 2 have even weight.3 One can easily check that all basis vectors of L 1 areorthogonal to all basis vectors of L 2 , therefore L 1 L 2 ,L 2 L 1 . Besides, we have already proved that 1L 1 and1L 2 . Now the statement follows from dimension counting,since dim L 1 =4 and dim L 2 =10.4 Without loss of generality we may assume that u0and v0. If u=v, the statement has been already proved, seeproperty 1. If uv, then u=f, v=g for some linearlyindependent linear functions f and g. We can introduce newcoordinates y 1 ,y 2 ,y 3 ,y 4 on F 2 4 such that y 1 = fx and y 2=gx. Now u·v=y 1 y 2 =4.5 Let uL 1 and vL 2 . Since L 2 =L 1 1, there aretwo possibilities: vL 1 and v=1+w for some wL 1 . Thefirst case has been already considered. In the second case wehave022316-9

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 200515u · v = u j 1 − w j = u − u · w.j=1It follows from properties 1 and 4 that u·v0mod 4. Now consider the one-qubit Hermitian operatorA = 1 2 x + y =0e −i/4e +i/4 0 = e −1/4 K x ,where K is the phase shift gate, see Eq. 1. By definition, Abelongs to the Clifford group C 1 . One can easily check thatA 2 =I and A z =− z A, so the code CSS z ,L 2 ;A,L 1 is welldefined. We claim that its code subspace coincides with thecode subspace of a certain stabilizer code.Lemma 2. Consider the decompositionC 2 15 = L 2* H,,*L 1associated with the code CSS z ,L 2 ;A,L 1 and the decompositionC 2 15 = L 2* G,,*L 1associated with the stabilizer code CSS z ,L 2 ; x ,L 1 . Forany syndrome L 1 * one hasH0, = G0,.Moreover, for any L 2 * there exists some wF 2 15 such thatfor any L 1*H, = AwG0,.27This Lemma provides a strategy to measure a syndromeof the code CSS z ,L 2 ;A,L 1 by operations from O ideal .Specifically, we measure i.e., the z part of the syndromefirst, compute w=w, apply Aw † , measure using thestabilizers x x j , and apply Aw.Proof of the lemma. Consider an auxiliary subspace,H = L 1*H0, = G0,,*L 1corresponding to the trivial z syndrome for both CSS codes.Each state H0 can be represented as = c v v,vL 2where c v are some complex amplitudes and v=v 1 ,…,v 15 are vectors of the standard basis. Let us showthatAu = x u for any H, u L 1 .To this end, we represent A as x e i/4 K † . For any uL 1 andvL 2 we haveAuv = x ue i/4u−i/2u·v v = x uv,because u0mod 8 and u·v0mod 4 see Lemma 1,parts 1 and 5.Since for any uL 1 the operators Au and x u act onH in the same way, their eigenspaces must coincide, i.e.,H0,=G0, for any L 1 * .Let us now consider the subspace H, for arbitraryL 2 * , L 1 * . By definition, is a linear functional onL 2 F 2 15 ; we can extend it to a linear functional on F 2 15 , i.e.,represent it in the form v=w,v for some wF 2 15 . Thenfor any H,, vL 2 , and uL 1 we have z vAw † = − 1 w,v Aw † z v = Aw † ,AuAw † = Aw † Au = − 1 v Aw † as z and A anticommute, hence Aw † H0,. ThusH, = AwH0, = AwG0,.Lemma 2 is closely related to an interesting property ofthe stabilizer code CSS z ,L 2 ; x ,L 1 , namely the existenceof a non-Clifford automorphism 23. Consider a one-qubitunitary operator W such thatW z W † = z and W x W † = A.It is defined up to an overall phase and obviously does notbelong to the Clifford group C 1 . However, the bitwise applicationof W, i.e., the operator W 15 , preserves the code subspaceG0,0. Indeed, W 15 G0,0 corresponds to the trivialsyndrome of the codeCSSW z W † ,L 2 ;W x W † ,L 1 = CSS z ,L 2 ;A,L 1 .Thus W 15 G0,0=H0,0. But H0,0=G0,0 due to thelemma.Now we are in a position to describe the distillationscheme and to estimate its threshold and yield. Suppose weare given 15 copies of the state , and our goal is to distillone copy of an H-type magic state. We will actually distillthe state,A 0 = 1 20 + e i 4 1 = e i 8 HK † H.Note that A 0 is an eigenstate of the operator A; specifically,AA 0 =A 0 . Let us also introduce the stateA 1 = z A 0 ,which satisfies AA 1 =−A 1 . Since the Clifford group C 1 actstransitively on the set of H-type magic states, we can assumethat the fidelity between and A 0 is the maximum oneamong all H-type magic states, so that022316-10

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…F H = A0 A 0 .As in Sec. V we define the initial error probability = 1 − F H 2 = A 1 A 1 .Applying the dephasing transformationD = 1 2 + AA† to each copy of , we can guarantee that is diagonal in theA 0 ,A 1 basis, i.e., = D = 1 − A 0 A 0 + A 1 A 1 .Since AC 1 , the dephasing transformation can be realizedby operations from O ideal . Thus our initial state is in = 15 = u 1 − 15−u A u A u ,15uF 228where A u =A u0 ¯ A u15 .According to the remark following the formulation ofLemma 2, we can measure the syndrome , of the codeCSS z ,L 2 ;A,L 1 by operations from O ideal only. Let us followthis scheme, omitting the very last step. So, we beginwith the state in , measure , compute w=w, applyAw † , and measure . We consider the distillation attemptsuccessful if =0. The measured value of is not importantat this stage. In fact, for any L 2 * the unnormalized postmeasurementstate is s = Aw † in Aw = in .In this equation is the projector onto the code subspaceH0,0=G0,0, i.e., = z A for z = 1L 2 vL 2 z v, A = 1L 1 uL 1Au.Let us compute the state s = in . SinceAuA w = − 1 u,w A w , z vA w = A w+v ,29one can easily see that A A w =A w if wL 1 , otherwise A A w =0. On the other hand, z A w does not vanish anddepends only on the coset of L 2 that contains w. There areonly two such cosets in L 1 because L 1 =L 2 1, seeLemma 1, and the corresponding projected states areA 0 L = L2 z A 0¯0 =A 1 L = L2 z A 1¯1 =1 L2 vL 2A v ,1 L2 vL 2A v+1 .30The states A L0,1 form an orthonormal basis of the code subspace.The projections of A w for wL 1 onto the code subspaceare given by these formulas:A w =1 L2 A 0 L if w L 2 ,A w =PHYSICAL REVIEW A 71, 022316 20051 L2 A 1 L if w L 2 + 1.Now the unnormalized final state s = in can be expandedas s1L 2 vL 21 − 15−v v A 0 L A 0 L + 1L 2 vL 2 15−v 1 − v A 1 L A 1 L .The distillation succeeds with probabilityp s = L 2 Tr s = 15−v 1 − v .vL 1The factor L 2 reflects the number of possible values of ,which all give rise to the same state s .To complete the distillation procedure, we need to apply adecoding transformation that would map the twodimensionalsubspace H0,0C 2 15 onto the Hilbertspace of one qubit. Recall that H0,0=G0,0 is the codesubspace of the stabilizer code CSS z ,L 2 ; x ,L 1 . Its logicalPauli operators can be chosen asXˆ = x 15 , Ŷ = y 15 , Ẑ = − z 15 .It is easy to see that Xˆ ,Ŷ ,Ẑ obey the correct algebraic relationsand preserve the code subspace. The decoding can berealized as a Clifford operator VC 15 that maps Xˆ ,Ŷ ,Ẑ tothe Pauli operators x , y , z acting on the first qubit. Theremaining 14 qubits become unentangled with the first one,so we can safely disregard them. Let us show that the logicalstate A 0 L is transformed into A 0 up to some phase. Forthis, it suffices to check that A 0 L Xˆ A 0 L =A 0 x A 0 ,A 0 L ŶA 0 L =A 0 y A 0 , and A 0 L ẐA 0 L =A 0 z A 0 . Verifyingthese identities becomes a straightforward task if we representA 0 L in the standard basis:A L 0 = L 2 1/2 2 −15/2 e i/4u uuL 2= 2 −5/2 uL 1u + e −i/4 u + 1.To summarize, the distillation subroutine consists of thefollowing steps.1 Measure eigenvalues of the Pauli operators z x j , z x j x k for j,k=1,2,3,4. The outcomes determine the zsyndrome, L 2 * .2 Find w=wF 2 15 such that w,v=v for any vL 2 .3 Apply the correcting operator Aw † .4 Measure eigenvalues of the operators x x j . Theoutcomes determine the A syndrome, L 1 * .5 Declare failure if 0, otherwise proceed to the nextstep.6 Apply the decoding transformation, which takes the022316-11

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 2005code subspace to the Hilbert space of one qubit.The subroutine succeeds with probabilityp s = 15−v 1 − v .vL 131In the case of success, it produces the normalized outputstate out = 1 − out A 0 A 0 + out A 1 A 1 32characterized by the error probability out = p s−1vL 2 15−v 1 − v .33The sums in Eqs. 31 and 33 are special forms of socalledweight enumerators. The weight enumerator of a subspaceLF 2 n is a homogeneous polynomial of degree n intwo variables, namelyIn this notation,W L x,y = x n−u y u .uLp s = W L1,1 − , out = W L 2,1 − W L1,1 − .The MacWilliams identity 30 relates the weight enumeratorof L to that of L :W L x,y = 1L W Lx + y,x − y.Applying this identity and taking into account that L 2 =L 1 1 and that u0mod 2 for any uL 1 see Lemma 1,we getp s = 1 16 W L 11,1 − 2, out = 1 21 − W L 11 − 2,1W L1 1,1 − 2.34The weight enumerator of the subspace L 1 is particularlysimple:W L1 x,y = x 15 + 15x 7 y 8 .Substituting this expression into Eq. 34, we arrive at thefollowing formulas:1 + 151 − 28p s = , 3516 out = 1 − 151 − 27 + 151 − 2 8 − 1 − 2 1521 + 151 − 2 8 . 36The function out is plotted in Fig. 3. Solving the equation out = numerically, we find the threshold error probability: 0 0.141.37FIG. 3. The final error probability out for the H-type statesdistillation.Let us examine the asymptotic properties of this scheme.For small the distillation subroutine succeeds with probabilityclose to 1, therefore the yield is close to 1/15. Theoutput error probability is out 35 3 .38Now suppose that the subroutine is applied recursively. Fromn copies of the state with a given , we distill one copy ofthe magic state A 0 with the final error probability out n, 1 35 353 k , 15 k n,where k is the number of recursion levels here we neglectthe fluctuations in the number of successful distillation attempts.Solving these equation, we obtain the relation out n, 35n , = 1/log 3 15 0.4. 39It characterizes the efficiency of the distillation scheme.VII. CONCLUSION AND SOME OPEN PROBLEMSWe have studied a simplified model of fault-tolerant quantumcomputation in which operations from the Cliffordgroup are realized exactly, whereas decoherence occurs onlyduring the preparation of nontrivial ancillary states. Themodel is fully characterized by a one-qubit density matrix describing these states. It is shown that a good strategy forsimulating universal quantum computation in this model is“magic states distillation.” By constructing two particulardistillation schemes we find a threshold polarization of above which the simulation is possible.The most exciting open problem is to understand the computationalpower of the model in the region of parameters1 x + y + z 3/ 7 which corresponds to FT * F T F T , see Sec. I. In this region, the distillation scheme basedon the five-quit code does not work, while the Gottesman-Knill theorem does not yet allow the classical simulation.One possibility is that a transition from classical to universalquantum behavior occurs on the octahedron boundary, x + y + z =1.022316-12

UNIVERSAL QUANTUM COMPUTATION WITH IDEAL…To prove the existence of such a transition, one it sufficesto construct a T-type states distillation scheme having thethreshold fidelity F T * . A systematic way of constructing suchschemes is to replace the five-qubit by a GF4-linear stabilizercode. A nice property of these codes is that the bitwiseapplication of the operator T preserves the code subspace andacts on the encoded qubit as T, see Ref. 31 for more details.One can check that the error-correcting effect described inSec. V takes place for an arbitrary GF4-linear stabilizercode, provided that the number of qubits is n=6k−1 for anyinteger k. Unfortunately, numerical simulations we performedfor some codes with n=11 and n=17 indicate thatthe threshold fidelity increases as the number of qubits increases.So it may well be the case that the five-qubit code isthe best GF4-linear code as far as the distillation is concerned.From the experimental point of view, an exciting openproblem is to design a physical system in which reliablestorage of quantum information and its processing by Cliffordgroup operations is possible. Since our simulationscheme tolerates strong decoherence on the ancilla preparationstage, such a system would be a good candidate for apractical quantum computer.ACKNOWLEDGMENTSWe thank Mikhail Vyalyi for bringing to our attentionmany useful facts about the Clifford group. This work hasbeen supported in part by the National Science Foundationunder Grant No. EIA-0086038.APPENDIXThe purpose of this section is to prove Eq. 15. Let usintroduce this notation:Tˆ 0 = T 00000 and Tˆ 1 = T 11111 .Consider the set S + 5S5 consisting of all possible tensorproducts of the Pauli operators x , y , z on five qubitsclearly, S + 5=4 5 =S5/2 since elements of S5 mayhave a plus or minus sign. For each gS + 5 let g0,5 be the number of qubits on which g acts nontriviallye.g., x x y I I=3. We haveTˆ 0Tˆ 0 = 1 2 5gS + 5 1g 3Now let us expand the formula 13 for the projector .Denote by G P5 the Abelian group generated by the stabilizersS 1 ,S 2 ,S 3 ,S 4 . It consists of 16 elements. Repeatedlyconjugating the stabilizer S 1 by the operator Tˆ =T 5 , we getthree elements of G:S 1 = x z z x I,g.S 1 S 3 S 4 = z y y z I,S 3 S 4 = y x x y I.Due to the cyclic symmetry mentioned in Sec. V, the 15cyclic permutations of these elements also belong to G; togetherwith the identity operator they exhaust the group G.Thus GS + 5, and we have = 1 h.16 hGTaking into account that Trgh=2 5 g,h for any g,hS + 5,we getTˆ 0Tˆ 0 = 1 2 9 PHYSICAL REVIEW A 71, 022316 2005hG gS + 53 −g/2 Trgh = 1 3 −g/2 = 1 16 gG 6 .Similar calculations show that Tˆ 1Tˆ 1= 1 6 .1 E. Knill, R. Laflamme, and W. Zurek, Science 279, 3421998.2 C. Zalka, e-print quant-ph/9612028.3 A. Steane, Phys. Rev. Lett. 78, 2252 1997.4 E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math.Phys. 43, 4452 2002.5 A. Kitaev, Ann. Phys. N.Y. 303, 2 2003.6 M. Freedman, M. Larsen, and Z. Wang, e-print quant-ph/0001108.7 M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Bull., NewSer., Am. Math. Soc. 40, 31 2002.8 C. Mochon, Phys. Rev. A 69, 032306 2004.9 G. Moore and N. Read, Nucl. Phys. B 360, 362 1991.10 C. Nayak and F. Wilczek, Nucl. Phys. B 479, 529 1996.11 B. Doucot and J. Vidal, Phys. Rev. Lett. 88, 227005 2001.12 M. Feigel’man and L. Ioffe, Phys. Rev. B 66, 224503 2002.13 J. Preskill, e-print quant-ph/9712048.14 D. Gottesman, Ph.D. thesis, Caltech, Pasadena, 1997, URLhttp://arxiv.org/abs/quant-ph/9705052.15 M. Nielsen and I. Chuang, Quantum Computation and QuantumInformation Cambridge University Press, Cambridge, England,2000.16 D. Gottesman, Phys. Rev. A 57, 127 1998.17 E. Knill, e-print quant-ph/0402171.18 E. Knill, e-print quant-ph/0404104.19 D. Gottesman and I. Chuang, Nature London 402, 3901999.20 W. Dur and H. Briegel, Phys. Rev. Lett. 90, 067901 2003.21 D. Aharonov, e-print quant-ph/9602019.22 E. Dennis, Phys. Rev. A 63, 052314 2001.23 E. Knill, R. Laflamme, and W. Zurek, e-print quant-ph/9610011.022316-13

S. BRAVYI AND A. KITAEV PHYSICAL REVIEW A 71, 022316 200524 A. Calderbank, E. Rains, P. Shor, and N. Sloane, Phys. Rev.Lett. 78, 405 1997.25 A. Ambainis and D. Gottesman, e-print quant-ph/0310097.26 C. Bennett, D. DiVincenzo, J. Smolin, and W. Wootters, Phys.Rev. A 54, 3824 1996.27 R. Laflamme, C. Miquel, J. Paz, and W. Zurek, Phys. Rev.Lett. 77, 198 1996.28 A. Calderbank and P. Shor, Phys. Rev. A 54, 1098 1996.29 A. Steane, Proc. R. Soc. London, Ser. A 452, 2551 1996.30 F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam, 1981.31 A. Calderbank, E. Rains, P. Shor, and N. Sloane, e-print quantph/9608006.022316-14

PRL 103, 170504 (2009) PHYSICAL REVIEW LETTERSweek ending23 OCTOBER 2009Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer OperationsWim van Dam*Department of Computer Science, University of California, Santa Barbara, California 93106, USAand Department of Physics, University of California, Santa Barbara, California 93106, USAMark Howard †Department of Physics, University of California, Santa Barbara, California 93106, USA(Received 21 July 2009; published 23 October 2009)We study how much noise can be tolerated by a universal gate set before it loses its quantumcomputationalpower. Specifically we look at circuits with perfect stabilizer operations in addition toimperfect nonstabilizer gates. We prove that for all unitary single-qubit gates there exists a tightdepolarizing noise threshold that determines whether the gate enables universal quantum computationor if the gate can be simulated by a mixture of Clifford gates. This exact threshold is determined by theClifford polytope spanned by the 24 single-qubit Clifford gates. The result is in contrast to the situationwherein nonstabilizer qubit states are used; the thresholds in that case are not currently known to be tight.DOI: 10.1103/PhysRevLett.103.170504Introduction.—A way to study the resources needed foruniversal quantum computation (UQC) is to analyze thetransition from a system that can provide UQC to one thatis classically efficiently simulable. A particularly usefulexample of a classically simulable system is given by thestabilizer operations, which are made by a combination ofpreparation of j0i states, unitary Clifford gates, measurementsin the fj0i; j1ig basis, and classical control determinedby the measurement outcomes. The Gottesman-Knill theorem tells us that stabilizer operations can beefficiently simulated classically (see, for example, [1],Theorem 10.7), while it also known that the addition ofany other one-qubit gate outside the Clifford group willenable the system to perform UQC. This fact provides uswith a framework for testing tolerance to noise—one canexamine how noisy this additional non-Clifford gate can bebefore it becomes classically simulable itself. If the non-Clifford operation has become a probabilistic combinationof Clifford gates due to the noise, then we know that we areunequivocally in the classical computational regime. Thenoise rate where the extra gate becomes simulable (whereit enters the ‘‘Clifford polytope’’ [2]) is thus an upperbound for fault tolerance. If the converse is true—i.e., ifany operation outside the Clifford polytope enables UQC,then the threshold is tight. In this Letter we show that forsingle-qubit gates undergoing depolarizing such a tightnoise threshold does indeed apply. We will do so byproving that any depolarized gate that lies outside theClifford polytope of single-qubit operations, in combinationwith noiseless stabilizing operations, allows for UQC.This result should be contrasted to the situation for nonstabilizerqubit states where the thresholds in that case arenot currently known to be tight. In fact, a recent result byCampbell and Browne [3] states that achieving tightthresholds for all nonstabilizer qubit states is impossibleif the number of copies of the resource state must be finite.PACS numbers: 03.67.Lx, 03.67.PpWe will consistently assume that Clifford gates can beimplemented perfectly, motivated by the fact that thesegates can be implemented fault tolerantly by applyingthem transversally and to encoded states [4–7]. The faulttolerantimplementation of Clifford gates naturally carrieswith it a threshold of its own, independent of the kind wediscuss in this Letter. The current model is particularlyrelevant to the so-called Pfaffian quantum Hall state intopological quantum computation [8,9], the two-qubitClifford group (but only the Clifford group) can be implementedusing braiding making these operations naturallyfault tolerant. The additional resource required to performUQC will likely be highly noisy, and so we can see theparallels with our model.We will begin by listing a couple of previously knownresults in this area. Next, we will discuss the connectionbetween the geometry of the Clifford polytope and stabilizermeasurements, and show that tightness of a magicstatedistillation procedure for single-qubit states automaticallyensures tight thresholds for non-Clifford gates undergoingany kind of unital noise. Finally, we show thatcurrently known magic-state distillation techniques aresufficient to prove tight thresholds for a non-Clifford gateundergoing depolarizing noise.Previously known results.—The idea of using perfectstabilizer operations in conjunction with imperfect nonstabilizerstates to perform UQC originates with Knill [7].Shortly after, Bravyi and Kitaev [10] showed that mostnonstabilizer qubit states (when sufficiently many copiesare available) can be purified (‘‘distilled’’), using onlystabilizer operations, towards a pure nonstabilizer state (a‘‘magic state’’). Since a universal gate set can be createdfrom perfect stabilizer operations and a supply of magicstates [10], we see that allowing access to a supply ofappropriate (possibly impure) nonstabilizer qubit ancillaspromotes the power of stabilizer operations from classi-0031-9007=09=103(17)=170504(4) 170504-1 Ó 2009 The American Physical Society

PRL 103, 170504 (2009) PHYSICAL REVIEW LETTERSweek ending23 OCTOBER 2009cally simulable to UQC. The conditions on the ancillaryqubits to enable UQC is that they are sufficiently close tobeing one of the 20 so-called magic states that lie on thesurface of the Bloch sphere. The two classes of magic state(see Fig. 1) are the jHi type and jTi type, where all jHitype states can be derived by applying a Clifford operationptoffiffiffisome canonical representative jHi ¼ðj0iþe i=4 j1iÞ=2 , and similarly for jTi-type states likep jTi¼cosð#Þj0iþe i=4 sinð#Þj1i, where cosð2#Þ ¼1= ffiffiffi3 . The routines usedin [10] were unable to distill qubit states just outside theedges and faces of the octahedron of Fig. 1. Reichardt [11]subsequently proposed an improved routine that closed thegap in the jHi direction (along the edges of the octahedron).Virmani et al. [12] suggested using the convex hullof Clifford operations in order to find gates’ robustness tovarious types of noise. In particular, they considered gatesthat are diagonal in the computational basis. Plenio andVirmani [13] subsequently extended this idea by analyzingcases where noise was allowed to affect the stabilizer operationstoo. Buhrman et al. [2] used a similar idea (thatnoise causes non-Clifford gates to eventually become ableto be implemented via Clifford gates only) to find the non-Clifford gate that is most resistant to depolarizing noise—a=8 rotation about the Z axis (or the same gate modulosome Clifford operation). Reichardt [14] showed that thisparticular gate enabled UQC right up to its threshold noiserate (about 45%), as well as considering in detail the processof reducing multiqubit states to single-qubit states usingpostselected stabilizer operations. Our current resulthere generalizes this tightness result to all possible singlequbitgates.Preliminaries and notation.—Let us parameterize anarbitrary single-qubit SU(2) gate as followsUð; ; Þ ¼ ei cosðÞ e i sinðÞe i sinðÞ e i cosðÞ: (1)The representation of this rotation in SO(3) is denoted byFIG. 1 (color online). Magic states and the octahedron: Someof the single-qubit magic states: jHi type states are designatedwith black arrows, jTi type states with white arrows. Theoctahedron defined by jxjþjyjþjzj 1 depicts the singlequbitstates that can be created by stabilizer operations.Reichardt [11] showed that distillation techniques work rightup to the edges of the octahedron (i.e., tight in the jHi direction).Current distillation techniques are unable to distill states justoutside the faces of the octahedron (i.e., not tight in the jTidirection).170504-2Rð; ; Þ. Implementing a rotation R while suffering depolarizingnoise (with noise rate p), means that this noisyoperation is represented by the rescaling M ¼ð1 pÞR, afact that we will need later.Often we will apply the unitary Uð; ; Þ to one half ofan entangled Bell pair, ji ¼p 1 ffiffi2ðj00iþj11iÞ, yielding ¼ðI UÞjihjðI UÞ y : (2)If we use the two-qubit Pauli operators as a basis for thedensity matrix then we can find the 16 real coefficientsc ij ¼ Trðð i j ÞÞ so that ¼ 1 4Xcij ð i j Þ; i;j 2fI; X; Y; Zg: (3)Since we have applied a local unitary to a maximallyentangled state, the coefficients (c IX , c IY , c IZ , c XI , c YI ,c ZI ) are always zero. Comparing the 9 coefficientsfc XX ;c XY ; ...;c ZZ g one can see that these are the same asthe entries of the SO(3) matrix Rð; ; Þ. More precisely,01c XX c YX c ZXRð; ; Þ ¼@c XY c YY c ZYA; (4)c XZ c YZ c ZZwhere the c ij are obviously also functions of (, , ).If we represent the 24 single-qubit Clifford operations asSO(3) matrices, then they are simply signed permutationmatrices with unit determinant (they are a matrix representationof the elements of the chiral octahedral symmetrygroup or, equivalently, the symmetry group S 4 ). We labelthese operations C i and so the convex hull of the C i (the socalledClifford polytope) is given by X 24P ¼ p i C i with p i 0 and X24p i ¼ 1 : (5)i¼1Geometrically, the Clifford polytope is a closed polyhedronin R 9 that has 24 vertices (each vertex representingone of the C i ). This polytope can also be defined by thebounding inequalities of its 120 facets. The concise descriptionof these facets used by Buhrman et al. [2]isgivenby the setF ¼fC i FC j ji; j 2f1; ...; 24g;F 2fA; A T ;Bgg; (6)where0 10 11 0 00 1 0A ¼ @ 1 0 0A and B ¼ @ 1 0 1 A: (7)1 0 01 0 1At times we will have reason to refer to different subsets ofthe set of facets F so we use the obvious notation F ¼F A [ F AT [ F B . It is useful to note that all the facetsderived from A comprise a single column with 1 entriesand zeros elsewhere, and similarly for the row facetsderived from A T ; hence, jF A j¼jF A Tj¼3 2 3 ¼ 24.There are jF B j¼72 ‘‘B-type’’ facets, which can be constructedas follows: (i) Pick one position in a 3 3 matrixF, e.g., row i and column j and put 1 there (9 2 ¼ 18i¼1

PRL 103, 170504 (2009) PHYSICAL REVIEW LETTERSweek ending23 OCTOBER 2009choices), (ii) Fill the remaining entries not in row i orcolumn j with 1 such that detðFÞ ¼ 2 (4 choices).To determine whether or not an operation M is inside theClifford polytope P we take the elementwise inner product(or Frobenius inner product) between M and the facets F 2F of the polytopeM F ¼ X3i;j¼1M i;j F i;j ¼ TrðM T FÞ: (8)Using the above notation, a 3 3 matrix M is inside thepolytope P if and only if for all F 2 F we have M F 1.Interpreting the facets of the Clifford polytope.—Ourproof will involve applying some non-Clifford gate toone half of a Bell Pair [as in Eq. (2)] and then postselectingon the outcomes of various stabilizer measurements. Aftersome further stabilizer operations, this measurement ultimatelyhas the effect of taking our two-qubit state to asingle-qubit state 0 (times some stabilizer state that we donot care about), which we then distill using magic-statedistillation (see [14] for a more general discussion of thesekinds of techniques). For example, performing a YX measurementon and postselecting on a ‘‘þ1’’ outcome (i.e.,projecting with ¼ 1 2ðII þ YXÞ) leads to a single-qubitstate 0 with a Bloch vector given by~rð 0 Þ¼ 0; c XZ c ZY;c II þ c YXc XY þ c ZZ: (9)c II þ c YXThe form of this vector means that it lies in the YZ plane(see Fig. 1), where we know that distillation techniqueswork right up to the edge jyjþjzj ¼1 of the octahedron.We can check if ~r is outside the octahedron by simplycomparing the L 1 norm of ~r with 1. Rearranged, thecondition k~rk 1 > 1 for being outside the octahedron isjc XZ c ZY jþj ðc XY þ c ZZ Þj > jc II þ c YX j: (10)Given the correspondence between the coefficients c ij andthe elements of R [see Eq. (4)] we can rewrite the abovecondition (dropping the absolute value operators) as a facetinequality010 1 0R F>1 where F ¼ @ 1 0 1 A: (11)1 0 1This facet is a legitimate ‘‘B-type’’ facet and a littlethought shows that, had we applied the single-qubit Paulioperations [as SO(3) rotations] X, Y or Z to ~rð 0 Þ above, wewould arrive at three other ‘‘B-type’’ facets0 10 1 0@ 1 0 1A;1 0 10 10 1 0@ 1 0 1A;1 0 10@10 1 01 0 1 A; (12)1 0 1respectively. Note that all four facet inequalities combinedcould be simplified to the form Eq. (10) above. We omit thedetails, but it is straightforward to show that all 72‘‘B-type’’ facets correspond to postselecting on some(weight two) Pauli operator, and possibly performing asingle-qubit Pauli rotation on the resulting 0 .170504-3It is somewhat more straightforward to see the geometricalinterpretation of the ‘‘A ðTÞ -type’’ facets. For example,the canonical A given in Eq. (7), merely returns the sum ofthe elements of the Bloch vector ~r, arising from a rotationapplied to the X ‘‘þ1’’ eigenstate.0 1R A ¼ X31r i where ~r ¼ R@0 A: (13)i¼10In general, an operation M having an inner product greaterthan one with some ‘‘A-type’’ facet simply means that M,applied to some initial vector corresponding to a stabilizerstate, brings that vector to a final position outside theoctahedron.The preceding discussion shows us that if magic-statedistillation were possible everywhere outside the octahedron,then every unital operation outside the Clifford polytopewould be distillable—either straightforwardly or byusing postselection, depending on which facet it violated.Using current (not tight) distillation routines however, wewould be unable to deal with some operations violating an‘‘A-type’’ facet by a fairly small amount. In the nextsection we show that, for depolarizing noise, any noisyrotation violating an ‘‘A-type’’ facet also violates a‘‘B-type’’ facet. Since ‘‘B-type’’ facets correspond to jHistate distillation, the results we obtain are tight.Tight threshold for depolarizing noise.—The claim weshall prove is that, anytime a matrix M ¼ð1 pÞR, representinga depolarized rotation, is outside some ‘‘A-type’’facet then there exists a ‘‘B-type’’ facet that M also liesoutside. In fact, we will prove the slightly stronger statementthat for all R 2 SOð3Þ8A 2 F A [ F A T; 9B 2 F B such that R ðB AÞ0:(14)To simplify the proof we will repeatedly make use of thesymmetries of the problem [see Eq. (6)]. We will pick acanonical ‘‘A-type’’ facet and assume that this gives thelargest inner product with R of all the F 2 F A . If there wasa larger inner product with some F 2 F A T, then we couldjust relabel R T as R. We can assume that the facet with onesin the first column [the A in Eq. (7)] gives the biggest innerproduct since Tr½R T ðC i AC j ÞŠ ¼ TrðC j R T C i AÞ¼Tr½ðC k RC l Þ T AŠ, which shows that the inner product of Rwith a different F 2 F A is the same as the inner productbetween a different (but related via Clifford operations)rotation and the canonical ‘‘A-type’’ facet.The proof will hinge on an entry of R, outside of the firstcolumn, being larger than the rest of the elements outsidethe first column. As such, let us define 12 matrices closelyrelated to A and call them A 0 i0 1 0 10 11 10 110A 0 1 ¼ @ 1 0 0A A 0 2 ¼ @ 100A A 0 12 ¼ @ 100A1 0 0 100101(15)

PRL 103, 170504 (2009) PHYSICAL REVIEW LETTERSweek ending23 OCTOBER 2009such that the index i of the largest inner product R A 0 i tellsus the sign and location of the largest magnitude elementoutside the first column. Once again, symmetry allows usto assume that A 0 1 yields the largest inner product becausethe rest of the A 0 i can be derived from A 0 1 via Cliffordrotations801 < 001fA 0 i g¼ @ 100A:010jA 0 10 191 0 0@ 0 0 1Aj 2f1;2;3g=kk 2f1;2;3;4g; :0 10(16)801 0 1 0< þ þ þþR 2 @ þ þ A; @ þ þ A; @ þ:þ þ þ þ þ þ þFor a matrix R to be an SO(3) rotation there are constraintson the signs of the elements R i;j ; i.e., there are eightchoices for the first column, 6 choices for the secondcolumn and 2 for the third. Given that A is the maximumfacet for R, we have fixed the signs positively in the firstcolumn, reducing the number of types of rotation to 6 2 ¼ 12. Since A 0 1 gives the maximum inner product with Rof all A 0 i we have that R 1;2 < 0, which reduces the numberof rotation types further to 3 2 ¼ 6. It can be shown thatR 1;2 having larger magnitude than the rest of the elementsR i;j ði 2f1; 2; 3g;j2f2; 3gÞ restricts the type of rotationfurther to one the following four typesþ1 0 19þ þ =A; @ þ A; : (17)þ þ þ þThis should not be surprising if one considers that R 1;2 ¼ðR 2;1 R 3;3 R 2;3 R 3;1 Þ because of the structure of SO(3)matrices, and the sign patterns listed above ensure jR 1;2 j isas large as possible.We claim that the B 2 F B of Eq. (9) will suffice toprove the desired inequality R ðB AÞ 0, which readsin matrix form0 101þ 1 1 0@ þ A@0 0 1 A 0: (18)þ þ 0 0 1Using the relevant entries of R we define a pair of 2 vectors~u and ~v as ~u ¼ðR 1;1 ;R 1;2 Þ, ~v ¼ðR 2;3 ;R 3;3 Þ so that we canrewrite the above inequality Eq. (17) ask ~vk 1 k~uk 1 0: (19)The L 2 normalization of all the rows and columns of therotation matrix R means that ~u and ~v have the same L 2norm. With reference to Fig. 2, it should be clear thatbecause ~u has an L 1 norm at least as big as that of ~v(because jR 1;2 jjR 2;3 j; jR 3;3 j), it holds that the L 1 norm of~v is automatically at least as large as the L 1 norm of ~u, asdesired.Summary.—We showed that for any unitary one-qubitgate undergoing depolarizing noise with rate p it holds thatFIG. 2. Proof of Eq. (19): For any pair of two vectors ~u and ~vwith the same L 2 norm, the vector with greater L 1 norm hassmaller L 1 norm. A vector pointing towards the black dot hassimultaneously minimal L 1 norm and maximal L 1 norm.if its SO(3) representation M ¼ð1 pÞR lies outside theClifford polytope P , then it must be the case that there is afacet B 2 F B such that M B>1. In turn, this means thatif this noisy gate is applied to a Bell pair ji ¼p 1 ffiffi2ðj00iþj11iÞ and an appropriate stabilizer measurement is performed,then, conditionally on the outcome of the measurement,one obtains a state that can be transformed usingClifford gates into a single-qubit state with jyjþjzj > 1 inthe Bloch ball representation. By the result of Reichardt[11] such states enable stabilizing operations to performuniversal quantum computation.This material is based upon work supported by theNational Science Foundation under Grant No. 0917244.*vandam@cs.ucsb.edu† mhoward@physics.ucsb.edu[1] Michasel A. Nielsen and Isaac L. Chuang, QuantumComputation and Quantum Information (CambridgeUniversity Press, Cambridge, England, 2000).[2] H. Buhrman, R. Cleve, M. Laurent, N. Linden, A.Schrijver, and F. Unger, Annual IEEE Symposium onFoundations of Computer Science (2006), p 411.[3] E. T. Campbell and D. E. Browne, arXiv:0908.0836v2.[4] A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098(1996).[5] A. Steane, Proc. R. Soc. A 452, 2551 (1996).[6] D. Gottesman, Phys. Rev. A 57, 127 (1998).[7] E. H. Knill, arXiv:quant-ph/0402171v1.[8] M. Freedman, C. Nayak, and K. Walker, Phys. Rev. B 73,245307 (2006).[9] L. S. Georgiev, Phys. Rev. B 74, 235112 (2006).[10] S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).[11] Ben W. Reichardt, Quant. Info. Proc. 4, 251 (2005).[12] S. Virmani, S. F. Huelga, and M. B. Plenio, Phys. Rev. A71, 042328 (2005).[13] M. B. Plenio and S. Virmani, arXiv:0810.4340.[14] B. W. Reichardt, arXiv:quant-ph/0608085v1.170504-4

VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001A One-Way Quantum ComputerRobert Raussendorf and Hans J. BriegelTheoretische Physik, Ludwig-Maximilians-Universität München, Germany(Received 25 October 2000)We present a scheme of quantum computation that consists entirely of one-qubit measurements on aparticular class of entangled states, the cluster states. The measurements are used to imprint a quantumlogic circuit on the state, thereby destroying its entanglement at the same time. Cluster states are thusone-way quantum computers and the measurements form the program.DOI: 10.1103/PhysRevLett.86.5188A quantum computer promises efficient processing ofcertain computational tasks that are intractable with classicalcomputer technology [1]. While basic principles of aquantum computer have been demonstrated in the laboratory[2], scalability of these systems to a large number ofqubits [3], essential for practical applications such as theShor algorithm, represents a formidable challenge. Mostof the current experiments are designed to implement sequencesof highly controlled interactions between selectedparticles (qubits), thereby following models of a quantumcomputer as a (sequential) network of quantum logicgates [4,5].Here we propose a different model of a scalable quantumcomputer. In our model, the entire resource for thequantum computation is provided initially in the form ofa specific entangled state (a so-called cluster state [6]) ofa large number of qubits. Information is then written ontothe cluster, processed, and read out from the cluster byone-particle measurements only. The entangled state ofthe cluster thereby serves as a universal “substrate” for anyquantum computation. Cluster states can be created efficientlyin any system with a quantum Ising-type interaction(at very low temperatures) between two-state particles ina lattice configuration.We consider two- and three-dimensional arrays ofqubits that interact via an Ising-type next-neighbor interaction[6] described by a Hamiltonian H int gt 3P 11sza12s a0 za,a 0 2 2 2 1 4 gt P a,a 0 sza s a0 z [7] whosestrength gt can be controlled externally. A possiblerealization of such a system is discussed below. A qubit atsite a can be in two states j0 a j0 z,a or j1 a j1 z,a ,the eigenstates of the Pauli phase flip operator szasza ji a 21 i ji a . These two states form the computationalbasis. Each qubit can equally be in an arbitrarysuperposition state aj0 1 bj1, jaj 2 1 jbj 2 1. Forour purpose, we initially prepare all qubits in the superpositionj1 j0 1 j1 p 2, an eigenstate of thePauli spin flip operator s x s x j6 6j6. H int isthen switched on for an appropriately chosen finite timeinterval T, where R T0 dt gt p, by which a unitarytransformation S is realized. Since H int acts uniformly onthe lattice, entire clusters of neighboring particles becomeentangled in one single step. The quantum state jF C ,PACS numbers: 03.67.Lx, 03.65.Udthe state of a cluster C of neighboring qubits, which isthereby created provides in advance all entanglement thatis involved in the subsequent quantum computation. It hasbeen shown [6] that the cluster state jF C is characterizedby a set of eigenvalue equationss axOa 0 [ngbhas a0 z jF C 6jF C , (1)where ngbha specifies the sites of all qubits that interactwith the qubit at site a [ C . The eigenvalues are determinedby the distribution of the qubits on the lattice.The equations (1) are central for the proposed computationscheme. As an example, a measurement on an individualqubit of a cluster has a random outcome. On the otherhand, Eqs. (1) imply that any two qubits at sites a, a 0 [ Ccan be projected into a Bell state by measuring a subset ofthe other qubits in the cluster. This property will be used todefine quantum channels that allow us to propagate quantuminformation through a cluster.We show that a cluster state jF C can be used as a substrateon which any quantum circuit can be imprinted byone-qubit measurements. In Fig. 1 this scheme is illustrated.For simplicity, we assume that in a certain regionof the lattice each site is occupied by a qubit. This requirementis not essential as will be explained below [see(d)]. In the first step of the computation, a subset ofqubits is measured in the basis of s z which effectivelyremoves them. In Fig. 1 these qubits are denoted by “ Ø.”information flowquantum gateFIG. 1. Quantum computation by measuring two-state particleson a lattice. Before the measurements the qubits are in thecluster state jF C of (1). Circles Ø symbolize measurements ofs z , vertical arrows are measurements of s x , while tilted arrowsrefer to measurements in the x-y plane.5188 0031-90070186(22)5188(4)$15.00 © 2001 The American Physical Society

VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001The state jF C is thereby projected into a tensor productjm C nN ≠ j ˜F N consisting of the state jm C nN ofall measured particles (subset C nN ) on one side and anentangled state j ˜F N of yet unmeasured particles (subsetN , C ), on the other side. These unmeasured particlesdefine a “network” N corresponding to the shaded structurein Fig. 1. The state j ˜F N of the network is relatedto a cluster state jF N on N by a local unitary transformationwhich depends on the set of measurement resultsm. More specifically, j ˜F N satisfies Eqs. (1)—with Creplaced by the subcluster N —except for a possible differencein the sign factors, which are determined by themeasurement results m.To process quantum information with this network, itsuffices to measure its particles in a certain order and in acertain basis. Quantum information is thereby propagatedhorizontally through the cluster by measuring the qubitson the wire while qubits on vertical connections are usedto realize two-bit quantum gates. The basis in which acertain qubit is measured depends in general on the resultsof preceding measurements. The processing is finishedonce all qubits except the last one on each wire have beenmeasured. At this point, the results of previous measurementsdetermine in which basis these “output” qubits needto be measured for the final readout. We note that, in theentire process, only one-qubit measurements are required.The amount of entanglement therefore decreases with everymeasurement [8] and all entanglement involved in theprocess is provided by the initial resource, the cluster state.This is different from the scheme of Ref. [11], which usesBell measurements (capable of producing entanglement)to realize quantum gates.In the following, we show that any quantum logic circuitcan be implemented on a cluster state. The purpose of thisis twofold. First, it serves as an illustration of how to implementa particular quantum circuit in practice. Second,in showing that any quantum circuit can be implementedon a sufficiently large cluster we demonstrate the universalityof the proposed scheme. For pedagogical reasons wefirst explain a scheme with one essential modification withrespect to the proposed scheme: before the entanglementoperation S, certain qubits are selected as input qubits andthe input information is written onto them, while the remainingqubits are prepared in j1. This step weakens thescheme since it affects the character of the cluster state asa genuine resource. It can, however, be avoided [see (e)].Points (a) to (c) are concerned with the basic elements of aquantum circuit, quantum gates, and wires, point (d) withthe composition of gates to circuits.(a) Information propagation in a wire for qubits. A qubitcan be teleported from one site of a cluster to any othersite. In particular, consider a chain of an odd number ofqubits 1 to n prepared in the state jc in 1 ≠ j1 2 ≠ · · · ≠j1 n and subsequently entangled by S. The state that wasoriginally encoded in qubit 1, jc in , is now delocalizedand can be transferred to site n by performing s x measurements(basis j1 j j0 x,j , j2 j j1 x,j ) at qubitsites j 1, . . . , n 2 1 with measurement outcomes s j [0, 1. The resulting state is js 1 x,1 ≠ · · · ≠ js n21 x,n21 ≠jc out n . The output state jc out is related to the input statejc in by a unitary transformation U S [ 1, s x , s z , s x s z which depends on the outcomes of the s x measurementsat sites 1 to n 2 1. A similar argument can be given for aneven number of qubits. The effect of U S can be accountedfor at the end of a computation as shown below [see (d)].It is noteworthy that not all classical information gainedby the s x measurements needs to be stored to identify thetransformation U S . Instead, U S is determined by the valuesof only two classical bits which are updated with everymeasurement.(b) An arbitrary rotation U R [ SU2 can be achievedin a chain of five qubits. Consider a rotation in itsEuler representation U R j, h, z U x zU z hU x j,where U x a exp2ia s x2 ,U za exp2ia s z2 . Initially,the first qubit is in some state jc in , whichis to be rotated, and the other qubits are in j1;i.e., their common state reads jC 1,...,5 jc in 1 ≠j1 2 ≠ j1 3 ≠ j1 4 ≠ j1 5 . After the five qubits areentangled by S they are in the state SjC 1,...,5 12jc in 1 j0 2 j2 3 j0 4 j2 5 2 12jc in 1 j0 2 j1 3 j1 4 j1 5 212jcin 1 j1 2 j1 3 j0 4 j2 5 1 12jcin 1 j1 2 j2 3 j1 4 j1 5 ,where jcin s z jc in . Now, the state jc in can be rotatedby measuring qubits 1 to 4, while it is teleported to site 5 atthe same time. The qubits 1, . . . , 4 are measured in appropriatelychosen bases B j a j j0 j1e ia jj1 p2 j, j0 j2e ia jj1 p2 jwhereby the measurement outcomes s j [ 0, 1 for j 1, . . . , 4 are obtained. Here, s j 0 means that qubit j isprojected into the first state of B j a j . The resultingstate is js 1 a1 ,1 ≠ js 2 a2 ,2 ≠ js 3 a3 ,3 ≠ js 4 a4 ,4 ≠ jc out 5with jc out Ujc in . For the choice a 1 0 (measurings x of qubit 1) the rotation U has the form U s s 21s 4x s s 11s 3z U R 21 s111 a 2 , 21 s 2 a 3, 21 s 11s 3a 4 . Insummary, the procedure to implement an arbitraryrotation U R j, h, z , specified by its Euler anglesj, h, z is (i) measure qubit 1 in B 1 0; (ii) measurequbit 2 in B 2 21 s111 j; (iii) measure qubit 3 inB 3 21 s 2 h; (iv) measure qubit 4 in B 4 21 s 11s 3z .In this way the rotation UR 0 is realized: URj, 0 h, z s s 21s 4x s s 11s 3s s 21s 4x s s 11s 3zz U R j, h, z . The extra rotation U S can be accounted for at the end of the computation,as is described below in (d).(c) To perform the gate CNOTc, t in ! t out j0 cc 0j ≠ 1 t in!t out 1 j1 cc 1j ≠ s t in!t out x between a controlqubit c and a target qubit t, four qubits, arrangedas depicted Fig. 2a, are required. During the actionof the gate, the target qubit t is transferred from t into t out . The following procedure has to be implemented.Let qubit 4 be the control qubit. First, the stateji 1 z,1 ≠ ji 4 z,4 ≠ j1 2 ≠ j1 3 is prepared and then theentanglement operation S is performed. Second, s x ofqubits 1 and 2 is measured. The measurement results5189

VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001(a)target in1 2 34controltarget out(b)target incontrol intarget outcontrol outFIG. 2. Realization of a CNOT gate by one-particle measurements.See text.s j [ 0, 1 correspond to projections of the qubits j intojs j x,j , j 1, 2. The quantum state created by this procedureis js 1 x,1 ≠ js 2 x,2 ≠ U 34S ji 4 z,4 ≠ ji 1 1 i 4 mod2 z,3 ,where U 34S s 3s 111. The input state is thusz s 3s 2xs 4s 1zacted upon by the CNOT and successive s x and s z rotationsU 34S , depending on the measurement results s 1 , s 2 .These unwanted extra rotations can again be accountedfor as described in (d). For practical purposes it is moreconvenient if the control qubit is, as the target qubit,transferred to another site during the action of the gate.When a CNOT is combined with other gates to form aquantum circuit it will be used in the form shown inFig. 2b.To explain the working principle of the CNOT gate we,for simplicity, refer to the minimal implementation withfour qubits. The minimal CNOT can be viewed as a wirefrom qubit 1 to qubit 3 with an additional qubit, No. 4,attached. From the eigenvalue equations (1) it can now bederived that, if qubit 4 is in an eigenstate ji 4 z,4 of s z , thenthe value of i 4 [ 0, 1 determines whether a unit wire ora spin flip s x (modulo the same correction U 3S for bothvalues of i 4 ) is being implemented. In other words, onces x of qubits 1 and 2 have been measured, the value i 4 ofqubit 4 controls whether the target qubit is flipped or not.(d) Quantum circuits. The gates described —the CNOTand arbitrary one-qubit rotations — form a universal set[5]. In the implementation of a quantum circuit on a clusterstate the site of every output qubit of a gate overlapswith the site of an input qubit of a subsequent gate. Becauseof this, the entire entanglement operation can beperformed at the beginning. To see this, compare thefollowing two strategies. Given a quantum circuit implementedon a network N of qubits which is dividedinto two consecutive circuits, circuit 1 is implemented onnetwork N 1 and circuit 2 is implemented on networkN 2 , and N N 1 < N 2 . There is an overlap O N 1 > N 2 which contains the sites of the output qubitsof circuit 1 (these are identical to the sites of the inputqubits of circuit 2). The sites of the readout qubits form aset R , N 2 . Strategy (i) consists of the following steps:(1) write input and entangle all qubits on N ; (2) measurequbits [ N nR to implement the circuit. Strategy(ii) consists of (1) write input and entangle the qubits onN 1 , (2) measure the qubits in N 1 nO . This implementsthe circuit on N 1 and writes the intermediate output toO ; (3) entangle the qubits on N 2 ; (4) measure all qubitsin N 2 nR. Steps 3 and 4 implement the circuit 2 on N 2 .The measurements on N 1 nO commute with the entanglementoperation restricted to N 2 , since they act on differentsubsets of particles. Therefore the two strategies aremathematically equivalent and yield the same results. Itis therefore consistent to entangle in a single step at thebeginning and perform all measurements afterwards.Two further points should be addressed in connectionwith circuits. First, the randomness of the measurementresults does not jeopardize the function of the circuit.Depending on the measurement results, extra rotationss x and s z act on the output qubit of every implementedgate. By use of the relations U R j, h, z szs s xs0 szs s x s0 U R 21 s j, 21 s0 h, 21 s z , and CNOTc, ts ts tzs cs cz s ts0 txs cs0 cx s ts tz s cs c1s t c1sz s 0 ts0 txs cs0 cx CNOTc, t,these extra rotations can be pulled through the network toact upon the output state. There they can be accountedfor by adjusting the measurement basis for the finalreadout. The above relations imply that for a rotationU R j, h, z—different from the CNOT gate—the accumulatedextra rotations U S at the input side of U R need tobe determined before the measurement bases that realizeU R can be specified. This introduces a partial temporalordering of the measurements on the whole cluster.Second, quantum circuits can also be implemented onirregular clusters. In that case, qubits may be missingwhich are required for the standard implementation of thecircuit. This can be compensated by a large flexibility inshape of the gates and wires. The components can be bentand stretched to fit to the cluster structure as long as thetopology of the circuit implementation does not change.Irregular clusters are found in lattices with a finite siteoccupation probability 0 , p , 1. In such a situation,the possibility of universal quantum computation isclosely linked to the phenomenon of percolation. For pabove a certain critical value p c , which depends on thedimension of the lattice, an infinitely extended clusterexists that may be used as the carrier of the quantumcircuit. In two dimensions, for example, exactly onesuch cluster C exists. Suppose this cluster is dividedinto two subclusters C 1 and C 2 by a one-dimensional cutO C 1 > C 2 . It can be shown, e.g., by using Russo’sformula [12] from percolation theory that, for any cut O ,jO j `. Therefore there is no upper bound, in principle,to the “capacity” of the cluster, i.e., to the number ofqubits that can be processed across such a cut.(e) Full scheme. It is important to note that the stepof writing the input information onto the qubits beforethe cluster is entangled was introduced only for pedagogicalreasons. For illustration of this point considera chain of five qubits in the state Sj1 1 ≠ j1 2 ≠ · · · ≠j1 5 . Clearly, there is no local information on any of thequbits. However, by measuring qubits 1 to 4 along suitabledirections, qubit 5 can be projected into any desired state(modulo U S ). What is used here is the knowledge that the5190

VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001resource has been prepared with qubit 1 in the state j1 1before the entanglement operation. By the four measurements,this qubit is then rotated as described in (b). Inorder to use qubit 5 for further processing, the five-qubitchain considered here should, of course, be part of a largercluster such that particle 5 is still entangled with the remainingnetwork, after particles 1 to 4 have been measured.The method of preparing the input state remains thesame, in this case, as explained in (d). In a similar mannerany desired input state can be prepared if the rotationsare replaced by a circuit preceding the proper circuit forcomputation. In summary, no input information needs tobe written to the qubits before they are entangled. Clusterstates are thus a genuine resource for quantum computationvia measurements only.For a cluster of a given finite size, the number of computationalsteps may be too large to fit on the cluster. In thiscase, the computation can be split into consecutive parts,for each of which there is sufficient space on the cluster.The modified procedure consists then of repeatedly (re)entanglingthe cluster and imprinting the actual part of thecircuit —by measuring all of the lattice qubits exceptthe ones carrying the intermediate quantum output — untilthe whole calculation is performed. This procedure hasalso the virtue that qubits involved in the later part of acalculation need not be protected from decoherence for along time while the calculation is still being performed ata remote place of the cluster. Standard error-correctiontechniques [13,14] may then be used on each part of thecircuit to stabilize the computation against decoherence.A possible implementation of such a quantum computeruses neutral atoms stored in periodic micropotentials[15–18] where Ising-type interactions can be realizedby controlled collisions between atoms in neighboringpotential wells [16,18]. This system combines smalldecoherence rates with a high scalability. The questionof scalability is linked to the percolation phenomenon,as mentioned earlier. For a site occupation probabilityabove the percolation threshold, there exists a clusterwhich is bounded in size only by the trap dimensions.For optical lattices in three dimensions, single-atom siteoccupation with a filling factor of 0.44 has been reported[19] which is significantly above the percolation thresholdof 0.31 [20]. As in other proposed implementations forquantum computing, the addressability of single qubitsin the lattice is, however, still a problem. (For recentprogress, see Ref. [21]). Recently, it has also been shownthat implementations based on arrays of capacitively coupledquantum dots may be used to realize an Ising-typeinteraction [22].In conclusion, we have described a new scheme ofquantum computation that consists entirely of one-qubitmeasurements on a particular class of entangled states,the cluster states. The measurements are used to imprint aquantum circuit on the state, thereby destroying its entanglementat the same time. Cluster states are thus one-wayquantum computers and the measurements form theprogram.We thank D. E. Browne, D. P. DiVincenzo, A. Schenzle,and H. Wagner for helpful discussions. This work wassupported by the Deutsche Forschungsgemeinschaft.[1] C. H. Bennett and D. P. DiVincenzo, Nature (London) 404,247 (2000).[2] See Ref. [1] for a recent review.[3] J. I. Cirac and P. Zoller, Nature (London) 404, 579 (2000).[4] D. Deutsch, Proc. R. Soc. London 425, 73 (1989).[5] A. Barenco et al., Phys. Rev. A 52, 3457 (1995).[6] H.-J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910(2001).[7] The second Hamiltonian is of the standard Ising form. Thesymbol “” means that the states generated from a giveninitial state, under the action of these Hamiltonians, areidentical up to a local rotation on certain qubits. We usethe first Hamiltonian to make the computational schememore transparent. The conclusions drawn in the paper are,however, the same for both Hamiltonians.[8] By the “amount of entanglement” contained in the resource,we mean any measure that satisfies the criteria ofan entanglement monotone [9]. For cluster states, the entanglementcan be calculated, e.g., in terms of the Schmidtmeasure of Ref. [10].[9] G. Vidal, J. Mod. Opt. 47, 355 (2000).[10] J. Eisert and H.-J. Briegel, quant-ph/0007081.[11] D. Gottesman and I. L. Chuang, Nature (London) 402, 390(1999).[12] See, e.g., G. Grimmett, Percolation (Springer-Verlag, NewYork, 1989).[13] A. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098(1996).[14] A. Steane, Phys. Rev. Lett. 77, 793 (1996).[15] G. K. Brennen et al., Phys. Rev. Lett. 82, 1060 (1999).[16] D. Jaksch et al., Phys. Rev. Lett. 82, 1975 (1999).[17] T. Calarco et al., Phys. Rev. A 61, 022304 (2000).[18] H.-J. Briegel et al., J. Mod. Opt. 47, 415 (2000).[19] M. T. DePue et al., Phys. Rev. Lett. 82, 2262 (1999).[20] J. M. Ziman, Models of Disorder (Cambridge UniversityPress, Cambridge, United Kingdom, 1979).[21] R. Scheunemann et al., Phys. Rev. A 62, 051801(R) (2000).[22] T. Tanamoto, quant-ph/0009030.5191

LETTERSPUBLISHED ONLINE: 20 JULY 2009 | DOI: 10.1038/NPHYS1342Quantum computation and quantum-stateengineering driven by dissipationFrank Verstraete 1 *, Michael M. Wolf 2 and J. Ignacio Cirac 3 *The strongest adversary in quantum information science isdecoherence, which arises owing to the coupling of a systemwith its environment 1 . The induced dissipation tends to destroyand wash out the interesting quantum effects that give riseto the power of quantum computation 2 , cryptography 2 andsimulation 3 . Whereas such a statement is true for manyforms of dissipation, we show here that dissipation can alsohave exactly the opposite effect: it can be a fully fledgedresource for universal quantum computation without anycoherent dynamics needed to complement it. The coupling tothe environment drives the system to a steady state wherethe outcome of the computation is encoded. In a similarvein, we show that dissipation can be used to engineer alarge variety of strongly correlated states in steady state,including all stabilizer codes, matrix product states 4 , and theirgeneralization to higher dimensions 5 .The situation we have in mind is shown in Fig. 1. A quantumsystem composed of N particles (such as qubits) is organized inspace according to a particular geometry (in the figure, a onedimensionallattice). Neighbouring systems are coupled to somelocal environments, which are dissipative in nature and tend todrive the system to a steady state. Our idea is to engineer thosecouplings, so that the environments drive the system to a desiredfinal state. The coupling to the environment will be static, so that thedesired state is obtained after some time without having to activelycontrol the system. Note that the role of the environments is todissipate (or, more precisely, evacuate) the entropy of the system,and by choosing the couplings appropriately we can use this effectto drive our system.We will show first how to design the interactions withthe environment to implement universal quantum computation.This new method, which we refer to as dissipative quantumcomputation (DQC), defies some of the standard criteria forquantum computation because it requires neither state preparation,nor unitary dynamics 6 . However, it is nevertheless as powerful asstandard quantum computation. Then we will show that dissipationcan be engineered 7 to prepare ground states of frustration-freeHamiltonians. Those include matrix product states 4,8,9 (MPSs) andprojected entangled pair states 5,9 (PEPSs), such as graph states 10and Kitaev 11 and Levin–Wen 12 topological codes. Both DQC anddissipative state engineering (DSE) are robust in the sense that,given the dissipative nature of the process, the system is driventowards its steady state independent of the initial state and henceof eventual perturbations along the way.Here, we will concentrate first on DQC, showing how givenany quantum circuit one can construct a locally acting masterequation for which the steady state is unique, encodes the outcomeof the circuit and is reached in polynomial time (with respect tothe one corresponding to the circuit). Then we will show howto construct dissipative processes that drive the system to theground state of any frustration-free Hamiltonian. In the Methodssection, we will prove that MPS (ref. 9) and certain kinds ofPEPS (ref. 9) can be efficiently prepared using this method, andin Supplementary Information we will give details of the proofs.In this letter we will not consider specific physical set-ups whereour ideas can be implemented. Nevertheless, the Methods sectionwill provide a universal way of engineering the master equationsrequired for DQC and DSE, which can be easily adapted to currentexperiments 13 based on, for example, atoms in optical lattices 14or trapped ions 15 . Thus, we expect that our predictions may beexperimentally tested in the near future.Let us start with DQC by considering N qubits in a line and aquantum circuit specified by a sequence of nearest-neighbour qubitoperations {U t } T t=1 . We define |ψ t 〉 := U t U t−1 ...U 1 |0〉 1 ⊗...|0〉 N , sothat |ψ T 〉 is the final state after the computation. Our goal is to finda master equation ˙ρ =L(ρ) with a Liouvillian in Lindblad form 16L(ρ) = ∑ kL k ρL † k − 1 2{L†kL k ,ρ } +(1)where the L k acts locally and has a steady state, ρ 0 : (1) that is unique;(2) that can be reached in a time poly(T); (3) such that ψ T can beextracted from it in a time poly(T). As in Feynman’s constructionof a quantum simulator 3 , we consider another auxiliary registerwith states {|t〉} T t=0, which will represent the time. We choosethe Lindblad operatorsL i = |0〉 i 〈1|⊗|0〉 t 〈0|L t = U t ⊗|t +1〉〈t|+U †t⊗|t〉〈t +1|where i = 1,...,N and t = 0,...,T. It is clear that the L terms actlocally except for the interaction with the extra register, which canbe made local as well. Furthermore,ρ 0 = 1 ∑|ψ t 〉〈ψ t |⊗|t〉〈t|T +1tis a steady state, that is, L(ρ 0 )=0. Given such a state, the result of theactual quantum computation can be read out with probability 1/Tby measuring the time register. In Supplementary Information, weshow that ρ 0 is the unique steady state and that the Liouvillian hasa spectral gap = π 2 /(2T +3) 2 . This means indeed that the steadystate will be reached in polynomial time in T. Note that this gap isindependent of N as well as of the actual quantum computation thatis carried out (that is, independent of the U t ). It is also shown thatthe same gap is retained if the clock register is encoded in the unary1 Fakultät für Physik, Universität Wien, 1090 Wien, Austria, 2 Niels Bohr Institute, 2100 Copenhagen, Denmark, 3 Max-Planck-Institut für Quantenoptik,85748 Garching, Germany. *e-mail: fverstraete@gmail.com; ignacio.cirac@mpq.mpg.de.NATURE PHYSICS | VOL 5 | SEPTEMBER 2009 | www.nature.com/naturephysics 633© 2009 Macmillan Publishers Limited. All rights reserved.

LETTERSNATURE PHYSICS DOI: 10.1038/NPHYS1342Figure 1 | Schematic representation of the set-up. We consider a collection of N quantum particles, locally coupled to a set of environments. Thecouplings are engineered in such a way that the system reaches the desired state in the long-time limit.way proposed by Kitaev and co-workers 17 , making the Lindbladoperators strictly local. A sketch of the proof is as follows. First, wedo a similarity transformation on L that replaces all gates U i with theidentity gates, showing that its spectrum is independent of the actualquantum computation. Second, another similarity transformationis done that makes L Hermitian and block-diagonal. Each block canthen be diagonalized exactly leading to the claimed gap.In some sense, the present formalism can be seen as a robustway of doing adiabatic quantum computation 18 (errors do notaccumulate and the path does not have to be engineered carefully)and implementing quantum random walks 19 , and it might thereforebe easier to tackle interesting open questions, such as the quantumprobabilistically-checkable-proofs theorem, in this setting 20 . Inaddition, it seems that the dissipative way of preparing ground statesis more natural than to use adiabatic time evolution, as nature itselfprepares them by cooling.Let us now turn to DSE and consider again a quantum systemwith N particles on a lattice in any dimension. We are interested inground states Ψ, of HamiltoniansH = ∑ λthat are frustration-free, meaning that Ψ minimizes the energy ofeach H λ individually, and local in the sense that H λ acts non-triviallyonly on a small set λ ⊂ {1,...,N } of sites (for example, nearestneighbours). We can assume the terms H λ to be projectors andwe will denote the orthogonal projectors by P λ = 1 − H λ . States Ψof the considered form are, for example, all PEPS (including MPSand stabilizer states 21 ).We will consider discrete time evolution generated by a tracepreservingcompletely positive map instead of a master equation.These two approaches are basically equivalent 22 as every localcompletely positive map T can be associated with a local Liouvillianthrough L(ρ) = N [T (ρ)−ρ], which leads to the same fixed pointsand spectrum. We choose completely positive maps of the formT (ρ) = ∑ λH λ[]p λ P λ ρP λ + 1 m∑U λ,i H λ ρH λ U † λ,imwhere the p λ terms are probabilities and U λ,1 ,...,U λ,m is a set ofunitaries acting non-trivially only within region λ. They effectivelyrotate part of the high-energy space (with support of H λ ) to thezero-energy space, so that tr[T (ρ)Ψ] ≥ tr[ρΨ] increases. As forLiouvillians (1), we could similarly take L λ,i = U i H λ , or the onesassociated with the completely positive map.We show now that for every frustration-free Hamiltonian,the completely positive map in equation (2) converges to theground-state space if we choose the unitaries U λ,i to be completelydepolarizing, that is, T (ρ) ∝ ∑ λ P λρP λ + 1 λ ⊗ tr λ [H λ ρ]/tr[1 λ ].For ease of notation, we will explain the proof for the case of ai=1(2)one-dimensional ring with nearest-neighbour interactions labelledby the first site λ = 1,...,N . Assume ρ is such that its expectationvalue with respect to the projector Ψ onto the ground-state spaceof H is non-increasing under applications of T , that is, in particulartr[ρΨ] = tr[T N (ρ)Ψ]. Expressing this in the Heisenberg picture inwhich T ∗ (Ψ) = Ψ + ∑ λ H λtr λ (Ψ)/(d 2 N ), we get[]1N∑ N∏tr[ρΨ] ≥ tr[ρΨ]+(d 2 N ) tr ( )ρ Hλ+µ tr N λ+µ (Ψ)νN≥ tr[ρΨ]+(d 2 N ) tr[ρH]Nµ=1 λ=1where the first inequality comes from discarding (positive) terms inthe sum and the second one is due to bounding all partial tracesof H λ from below by the respective smallest eigenvalue ν. Notethat the latter is strictly positive unless H has a product state asthe ground state (in which case the statement becomes trivial).Hence, we must have tr[ρH] = 0; that is, ρ is a ground state of H.It is easily seen that the same argument applies for more generalinteractions on arbitrary lattices.Once we have shown that the steady state after the applicationof the completely positive map lies within the desired subspace(the ground-state space of the frustration-free Hamiltonian), thenext question to be addressed is how efficient the process is. Thisdepends on the spectral gap, δ, of the completely positive map (or,equivalently, of the corresponding Liouvillian), as the time to reachthe steady state, τ = O(1/δ). Thus, the above procedure will beefficient as long as the gap vanishes only polynomially with thenumber of systems, N . Similarly to what occurs with many-bodyHamiltonians, the determination of such a gap is, in general, verycomplicated. For a wide range of interesting models, however, itcan be proved that this gap scales favourably. This is the case forall MPS as well as for a rich subfamily of PEPS that includes allstabilizer states (such as Kitaev’s toric code 11 and the Levin–Wenstates 12 ). In the Methods section, we characterize such a subfamilyof states, and in Supplementary Information we give the technicalproofs of our statements. Here, we will qualitatively explain howour method works efficiently for some families of states. For thatwe note that the action of the completely positive map (2) canbe interpreted as randomly choosing a region λ (according to p λ ,which we may set equal to 1/N ), then measuring P λ and applyinga correction according to the unitaries if the outcome was negative.We denote by R n the set of regions λ where ϕ satisfies the conditionH λ |ϕ〉 = 0. If we measure now in one of those regions, we willobviously obtain a positive result, and thus R n will remain thesame. If we measure in another region, we may have a positive ornegative result, something that may change the set R n . By imposingcertain conditions on the operators H λ and U λ,i , we can make surethat in each step R n cannot be reduced and that the probability of634 NATURE PHYSICS | VOL 5 | SEPTEMBER 2009 | www.nature.com/naturephysics© 2009 Macmillan Publishers Limited. All rights reserved.

NATURE PHYSICS DOI: 10.1038/NPHYS1342being enlarged is non-vanishing. This automatically ensures thatthe τ scales only polynomially with the number of systems. Inone dimension, however, one can get rid of all those restrictionsand show that any MPS can be prepared in a time that also scalesfavourably with N . The fact that all MPS states can be preparedwith our method, together with the results reported in refs 23, 24,automatically implies the existence of phase transitions driven bydissipation in the following sense. By changing the parameters ofthe operators H λ appearing in the completely positive map (2), wechange the steady state of that map. It is possible to choose modelsfor which that state changes abruptly at some particular value ofthat parameter in such a way that the correlation length divergesand an order parameter appears (an example can be found in theSupplementary Information).We have investigated the computational power of purelydissipative processes, and proved that it is equivalent to that ofthe quantum circuit model of quantum computation. We havealso shown that dissipative dynamics can be used to create groundstates (such as MPS or PEPS) of frustration-free Hamiltonians ofstrongly correlated quantum spin systems. We believe that thesenew methods can be experimentally tested using atoms or ions withcurrent set-ups (see the Methods section).Let us stress that we have been concerned here with a proofof-principledemonstration that dissipation provides us with analternative way of carrying out quantum computations or stateengineering. We believe, however, that much more efficient andpractical schemes can be developed and adapted to specificimplementations. We also think that these results open upsome interesting questions that deserve further investigation: forexample, how the use of fault-tolerant computations can makeour scheme more robust, or how one can design translationallyinvariant completely positive maps that prepare MPS moreefficiently, or the importance and generality of the set of commutingHamiltonians (see the Methods section), which is intimatelyconnected to the fixed points of the renormalization grouptransformations on PEPS (as it happens with MPS; ref. 25).Furthermore, the model of DQC might well lead to the constructionof new quantum algorithms, as, for example, quantum randomwalks can more easily be formulated within this context. Finally,other ideas related to this work can be easily addressed using themethods introduced; for example, thermal states of commutingHamiltonians can be engineered using DSE because the Metropolisway of sampling over classical spin configurations can be adoptedto the case of commuting operators. Similar techniques could beapplied to free fermionic and bosonic systems, and, more generally,it should be possible to devise DSE schemes converging to theground or thermal states of frustrated Hamiltonians by combiningunitary and dissipative dynamics.Note added. Concurrently with the submission of this paper,refs 26 and 27 appeared in which a similar quantum-reservoirengineering was used to prepare many-body states and nonequilibriumquantum phases.MethodsEngineering dissipation. Here we show how to engineer the local dissipation thatgives rise to the master equations (1) and completely positive maps (2). They arecomposed of local terms, involving few particles (typically two), so that we just haveto show how to implement those. To simplify the exposition, we will treat thoseparticles as a single one and assume that one has full control over its dynamics (forexample, one can apply arbitrary gates).Let us start with the completely positive maps. It is clear that by applying aquantum gate to the particle and a ‘fresh’ ancilla and then tracing the ancilla onecan generate any physical action (that is, completely positive map) on the system.Furthermore, by repeating the same process with short time intervals one cansubject the system to an arbitrary time-independent master equation. This lastprocess may not be efficient. An alternative way works as follows. Let us assumethat the ancilla is a qubit interacting with a reservoir such that it fulfils a masterLETTERSequation with Liouville operator L a = √ Γ σ − , where σ − = |0〉〈1|. Now, we couplethe ancilla to the system with a Hamiltonian H = (σ − L † + σ † −L). In the limitΓ ≫ , one can adiabatically eliminate the level |1〉 of the ancilla 28 by applyingsecond-order perturbation theory to the Liouvillian (albeit for non-Hermitianoperators). In this way we obtain an effective master equation for ρ describingthe system alone, with Liouville operator / √ Γ L. By using several ancillaswith Hamiltonians H = (σ − L i + σ † −L † i ) and following the same procedure weobtain the desired master equation. Although we have not specified here a physicalsystem, one could use atoms. In that case, the ancilla could be an atom itself with|0〉 and |1〉 an electronic ground and excited level, respectively, so that spontaneousemission gives rise to the dissipation. The coupling to the system (other atoms)could be achieved using standard ideas used in the implementation of quantumcomputation using those systems 13 .Efficient state preparation. We have shown that it is possible to engineerdissipative processes that prepare ground states of frustration-free Hamiltonians insteady state. In the proof, the time for this preparation scales as N N , which may bean issue for experiments with large number of particles. Here we give much moreefficient methods for certain classes of frustration-free Hamiltonians.We consider first frustration-free Hamiltonians for which [H λ ,H µ ] = 0 andshow that, under certain conditions, the corresponding ground states can beprepared in a time that scales only polynomially with the number of particles. Thecorresponding set of ground states contains important families, such as stabilizerstates (for example, cluster states and topological codes), or certain kinds of PEPS,namely, those that have (commuting) parent Hamiltonians with the injectivitycondition (as defined in refs 8, 29). Note that there was no known way of efficientpreparation for the latter.Loosely speaking, we will consider two classes of Hamiltonians.(1) Hamiltonians for which all excitations can be locally annihilated. In this case thetime of convergence scales as τ = O(logN ). (2) Interactions where excitations haveto be moved along the lattice before they can annihilate and τ =O(N logN ).To see how the first case can occur notice that, when iterating T , thecorrection on λ does not change the outcome of previous measurements onneighbouring regions because∀λ ≠ λ ′ : [U λ,i ,H λ ′ ] = 0 (3)In fact, this can always be achieved by regrouping the regions into larger oneshaving an interior I(λ) ⊂ λ on which only H λ acts non-trivially and letting theU λ,i solely act on I(λ). Denote by q the largest probability for obtaining twice anegative measurement outcome on the same region λ. The energy tr[HT M (ρ)]after M applications of T decreases then as N (1−(1−q)/N ) M such that it takesO((N logN )/(1−q)) steps to converge to a ground state. The relaxation time of thecorresponding Liouvillian is thus τ = O(logN 1/1−q ). Clearly, this is a reasonablebound only if q

LETTERS‘parent’ Hamiltonian that has a gap. Denoting by ρ the reduced density operatorcorresponding to particles k and k + 1, H k and P k = 1 − H k will denote theprojectors onto its kernel and range, respectively. Note that tr(P k ) = D 2 . Wetake N = 2 n for simplicity, but this is clearly not necessary. We construct thechannel T in several steps. We first define a channel acting on two neighbouringparticles k,k +1, as followsR r,c (X) := P k XP k + P kD 2 tr(H kX)Here, k = 2 r−1 (2c −1), where r = 1,...,n and c = 1,...,2 n−r . The action of thesemaps has a tree structure, where the index r indicates the row in the tree, whereas cdoes it for the column. Now we define recursively,S r,c := (1−ɛ r )(S r−1,2c +S r−1,2c+1 )+ɛ r R r,c2Here, r = 2,...,n, c = 1,...,2 n−r , S 1,c := R 1,c and ɛ r+1 = 1/M r , where M = CN 2and C ≫ 1 (see Supplementary Information). Note that S r,1 acts on the first 2 rparticles, S r,2 on the next 2 r and so on. We finally defineT := (1−ɛ n+1 )S n,1 +ɛ n+1 R n,2 (4)In the Supplementary Information, we show that this map achieves the fixed point(up to an exponentially small error in C) in a time O(N log 2 (N ) ). The intuitionbehind the completely positive map (4) is that the channels S 1,c , which are the onesthat most often applied, project the state of every second nearest neighbour ontothe right subspace. Then S 2,c do the same with half of the pairs that have not beenprojected. Then S 3,c does the same on half of the rest, and so on.Received 11 March 2008; accepted 18 June 2009; published online20 July 2009References1. Aliferis, P., Gottesman, D. & Preskill, J. Quantum accuracy threshold forconcatenated distance-3 codes. Quant. Inf. Comput. 6, 97–165 (2006).2. Nielsen, M. A. & Chuang, I. L. Quantum Computation and QuantumInformation (Cambridge Univ. Press, 2000).3. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21,467–488 (1982).4. 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Kraus, B. et al. Preparation of entangled states by quantum Markov processes.Phys. Rev. A 78, 042307 (2008).28. Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G. Atom-PhotonInteractions (Wiley, 1992).29. Perez-Garcia, D., Verstraete, F., Cirac, J. I. & Wolf, M. M. PEPS as uniqueground states of local Hamiltonians. Quant. Inf. Comput. 8, 0650–0663 (2008).AcknowledgementsWe thank D. Perez-Garcia for discussions and acknowledge financial support bythe EU projects QUEVADIS, SCALA, the FWF, QUANTOP, FNU, SFB FoQuS, theDFG, Forschungsgruppe 635, the Munich Center for Advanced Photonics (MAP)and Caixa Manresa.Author contributionsAll authors have contributed equally to this paper.Additional informationSupplementary information accompanies this paper on www.nature.com/naturephysics.Reprints and permissions information is available online at http://npg.nature.com/reprintsandpermissions. Correspondence and requests for materials should beaddressed to F.V. or J.I.C.636 NATURE PHYSICS | VOL 5 | SEPTEMBER 2009 | www.nature.com/naturephysics© 2009 Macmillan Publishers Limited. All rights reserved.