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

More documents

Recommendations

Info

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
Page 3 and 4: Decoherence in quantum computation
Page 5: S. BRAVYI AND A. KITAEV PHYSICAL RE
Page 9 and 10: S. BRAVYI AND A. KITAEV PHYSICAL RE
Page 19 and 20: PRL 103, 170504 (2009) PHYSICAL REV
Page 21 and 22: PRL 103, 170504 (2009) PHYSICAL REV
Page 23 and 24: VOLUME 86, NUMBER 22 P H Y S I C A
Page 25 and 26: VOLUME 86, NUMBER 22 P H Y S I C A
Page 27 and 28: LETTERSNATURE PHYSICS DOI: 10.1038/
Page 29: LETTERS‘parent’ Hamiltonian tha

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

Create successful ePaper yourself

Delete template?

Save as template?