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k1 and k2 (byconservationofenergyandmomentum). The effect of the interaction is to change the global phase of the wave function after scattering (relative to the case with no interaction). This phase can be calculated given the two momenta, the interaction U, and the particle type [section S2 of (32)]. In our scheme, we use k1 = –p/2 and k 2 = p/4, and we write e iq for the phase acquired at these momenta. This picture of the two-particle scattering process holds even on a long (but finite) path and with finite wave packets [section S5 of (32)]. We now describe our scheme for universal computation by multiparticle quantum walk. We encode n logical qubits into the locations of n indistinguishable particles on a graph of size poly(n)[seesectionS3.2of(32) for a refinement of our scheme using distinguishable particles]. In addition to these n particles, we use another particle to encode an ancilla qubit that facilitates two-qubit gates. We call the original n qubits computational qubits, and we call the ancilla qubit the mediator qubit. Time evolution of a simple initial state with the Hamiltonian corresponding to a suitably chosen graph G implements a quantum circuit on this encoded data. The quantum circuit to be simulated (on the n computational qubits) is given as a product of gates from a universal set consisting of singlequbit gates along with the two-qubit controlled phase gate CP = diag(1,1,1,–i). A controlled phase between computational qubits i and j can be expanded as the following sequence of gates also acting on the mediator qubit m. CPijjai,bj,0m〉 ¼ CNOTimCPjmCNOTimjai,bj,0m〉 ¼ HmCP 2 im HmCPjmHmCP 2 im Hmjai,bj,0m〉 Writing the controlled phase gate in this way, we transform the given n-qubit circuit into an A 1 2 3 = 1 2 3 Fig. 3. (A) Momentum switch. (B) Cq gate. 01, in 11, in 02, in 12, in 0m, in 1m, in (n+1)–qubit circuit with only single-qubit gates on computational qubits, Hadamard gates H on the mediator qubit, and controlled phase gates between the mediator qubit and arbitrary computational qubits. We represent each of the n+1 qubits using a dual-rail encoding with two paths that run through the graph (Fig. 1B). The encoded state j0〉 has a particle moving along the top path, whereas the encoded state j1〉 has a particle moving along the bottom path. The particle moves as a wave packet with momentum near k. Forthencom putational qubits we choose k = –p/4, and for the mediator qubit we choose k = –p/2 (for concreteness). To implement single-qubit unitaries, we design the graph so that the particles scatter through a series of small subgraphs while remaining far apart. When the particles are all far from each other, the interaction term in the Hamiltonian is negligible and the n+1 wave packets propagate independently [our analysis in section S4 of (32) accounts for the error in this approximation]. In this case, the multiparticle quantum walk can be viewed as a single-particle quantum walk for each of the particles. To apply a one-qubit unitary U to an encoded qubit, we insert an associated graph G % into the paths representing the qubit as follows. We attach two long “input” paths and two long “output” paths to four suitably chosen vertices of G % so that the S-matrix at the relevant momentum has the form 0 U′ S ¼ ð4Þ U 0 A particle incident on the input paths with the relevant momentum transmits perfectly to the output paths, with amplitudes determined by the 2 by 2 unitary matrix U. Graphs G % that implement a phase gate and a basis-changing gate at momentum –p/4 are B 0c,in 1c,in 0m,in 1m,in 0c,out 1c,out 0m,out 1m,out showninFig.2,AandB,respectively(7). The input and output paths are attached to the vertices denoted by open circles. The S-matrices at momentum –p/4 for these graphs are of the formgiveninEq.4,withlower-left2by2 submatrices. Uphase ¼ e−ip=4 0 0 1 Ubasis ¼ − i pffiffi 2 1 −i −i 1 These two gates allow us to approximate arbitrary single-qubit unitaries on each of the n computational qubits. The Hadamard gate is the only single-qubit gate we apply to the mediator qubit. We implement this gate using the graph in Fig. 2C, which has S-matrix at momentum k = –p/2 of the form in Eq. 4, with lower-left submatrix UH ¼ − eip=4 pffiffi 2 1 1 1 −1 which is the Hadamard gate up to an irrelevant global phase (34). To implement a controlled phase gate between the mediator qubit and a computational qubit, we use a subgraph that routes two particles toward each other and causes them to interact on a long path for a short time. After scattering, the system returns to a state where the particles are all far apart. We route a computational particle and the mediator particle toward each other along a long path only when the two associated qubits are in state j11〉. This allows us to implement the two-qubit gate Cq = diag(1,1,1,e iq ), where e iq is the two-particle scattering phase. For some models, such as the Bose-Hubbard model with U ¼ 2 þ ffiffi p iq 2,wehaveCq = CP because e = –i [sectionS2of(31)]. For nearest-neighbor inter- pffiffiffi actions with fermions, the choiceU ¼ −2 − gives e iq = i,soCP=(Cq) 3 . Although tuning the interaction strength makes the CP gate easier to implement, almost any interaction between indistinguishable particles allows for universal computation. We can approximate the required CP gate by repeating the Cq gate a times, where e iaq ≈ –i (which is possible for most values of q, assuming q is known). We route particles using a subgraph we call the momentum switch (Fig. 3A). The S- 01, out 11, out 02, out 12, out REPORTS 0m, out 1m, out Fig. 4. Schematic depiction of a graph simulating B2CP1,2T1 on two qubits for the Bose-Hubbard model with interaction strength U = 2 + ffiffiffi p 2. The dotted lines represent paths, and the single-qubit unitary gates represent their corresponding subgraphs. www.sciencemag.org SCIENCE VOL 339 15 FEBRUARY 2013 793 2 on February 14, 2013 www.sciencemag.org Downloaded from
REPORTS 794 matrices for this graph at momenta –p/4 and –p/2 are Sswitchð−p=4Þ ¼ 0 0 e−ip=4 0 −1 0 e−ip=4 0 @ 1 A 0 0 0 1 Sswitchð−p=2Þ ¼@ 0 0 0 1 0 −1 A ð5Þ 0 −1 0 The momentum switch has perfect transmission between vertices 1 and 3 at momentum –p/4 and perfect transmission between vertices 2 and 3 at momentum –p/2. Thus, the path a particle follows through the switch depends on its momentum: A particle with momentum –p/2 follows the double line in Fig. 3A, whereas a particle with momentum –p/4 follows the single line. The graph used to implement the Cq gate is shown in Fig. 3B [see section S4 of (32) forthe numbers of vertices on each of the paths]. To see why this graph implements a Cq gate, consider the movement of two particles as they pass through the graph. If either particle begins in the state j0in〉, it travels along a path to the output without interacting with the second particle. When either particle begins in the state j1in〉,itisrouted onto the vertical path as it passes through the first momentum switch and is routed to the right as it passes through the second switch. If both particles begin in the state j1in〉, they interact on the vertical path and the wave function acquires a phase e iq . To implement a circuit, the subgraphs representing circuit elements are connected by paths. Figure 4 depicts a graph corresponding to a simple two-qubit computation. Timing is important: Wave packets must meet on the vertical paths for interactions to occur. We achieve this by choosing the numbers of vertices on each of the segments in the graph appropriately, taking into account the different propagation speeds of the two wave packets [see section S4 of (32)]. In section S3.1 of (32), we present a refinement of our scheme using planar graphs with maximum degree four. By analyzing the full (n +1)–particle interacting many-body system, we prove that our algorithm performs the desired quantum computation up to an error term that can be made arbitrarily small (32). Our analysis goes beyond the scattering theory discussion presented above; we take into account the fact that both the wave packets and the graphs are finite. Specifically, we prove that by choosing the size of the wave packets, the number of vertices in the graph, and the total evolution time to be polynomial functions of both n and g, the error in simulating an n-qubit, g-gate quantum circuit is bounded above by an arbitrarily small constant [section S5 of (32)]. For example, for the Bose-Hubbard model and for the nearest-neighbor interaction model, we prove that the error can be made arbitrarily small by choosing the size of the wave packets to be O(n 12 g 4 ), the total number of vertices in the graphtobeO(n 13 g 5 ), and the total evolution time to be O(n 12 g 5 ). The bounds we prove, although almost certainly not optimal, are sufficient to establish universality with only polynomial overhead. Because it is also possible to efficiently simulate a multiparticle quantum walk of the type we consider using a universal quantum computer, this model exactly captures the power of quantum computation. References and Notes 1. E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998). 2. D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, “Quantum walks on graphs,” Proceedings of the 33rd ACM Symposium on Theory of Computing (2001), pp. 50–59. 3. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, “One-dimensional quantum walks,” Proceedings of the 33rd ACM Symposium on Theory of Computing (2001), pp. 37–49. 4. A. M. Childs et al., “Exponential algorithmic speedup by quantum walk,” Proceedings of the 35th ACM Symposium on Theory of Computing (2003), pp. 59–68. 5. A. Ambainis, SIAM J. Comput. 37, 210 (2007). 6. E. Farhi, J. Goldstone, S. Gutmann, Theory of Computing 4, 169 (2008). 7. A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009). 8. B. Do et al., J. Opt. Soc. Am. B 22, 499 (2005). 9. M. Karski et al., Science 325, 174 (2009). 10. H. B. Perets et al., Phys. Rev. Lett. 100, 170506 (2008). 11. Y. Bromberg, Y. Lahini, R. Morandotti, Y. Silberberg, Phys. Rev. Lett. 102, 253904 (2009). 12. A. Peruzzo et al., Science 329, 1500 (2010). 13. J. O. Owens et al., New J. Phys. 13, 075003 (2011). 14. L. Sansoni et al., Phys. Rev. Lett. 108, 010502 (2012). 15. I. L. Chuang, Y. Yamamoto, Phys. Rev. A 52, 3489 (1995). 16. G. K. Brennen, C. M. Caves, P. S. Jessen, I. H. Deutsch, Phys. Rev. Lett. 82, 1060 (1999). 17. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, M. Greiner, Nature 462, 74 (2009). Photonic Boson Sampling in a Tunable Circuit 18. A. J. Hoffman et al., Phys.Rev.Lett.107, 053602 (2011). 19. J.K.Gamble,M.Friesen,D.Zhou,R.Joynt,S.N.Coppersmith, Phys.Rev.A81, 052313 (2010). 20. J. Smith, Electron. Notes Discrete Math. 38, 795 (2011). 21. Y. Omar, N. Paunković, Phys. Rev. A 74, 042304 (2006). 22. P. K. Pathak, G. S. Agarwal, Phys. Rev. A 75, 032351 (2007). 23. Y. Lahini et al., Phys. Rev. A 86, 011603 (2012). 24. A. Schreiber et al., Science 336, 55 (2012). 25. A. Ahlbrecht et al., New J. Phys. 14, 073050 (2012). 26. P. P. Rohde, A. Schreiber, M. Štefaňák,I.Jex,C.Silberhorn, New J. Phys. 13, 013001 (2011). 27. B.M.Terhal,D.P.DiVincenzo,Phys.Rev.A65, 032325 (2002). 28. S. Aaronson, A. Arkhipov, “The computational complexity of linear optics,” Proceedings of the 43rd ACM Symposium on Theory of Computing (2011), pp. 333–342. 29. R. Ionicioiu, P. Zanardi, Phys. Rev. A 66, 050301 (2002). 30. A. Mizel, D. A. Lidar, M. Mitchell, Phys. Rev. Lett. 99, 070502 (2007). 31. A. M. Childs, D Gosset, J. Math. Phys. 53, 102207 (2012). 32. Materials and methods are available as supplementary materials on Science Online. 33. M. Valiente, Phys. Rev. A 81, 042102 (2010). 34. B. A. Blumer, M. S. Underwood, D. L. Feder, Phys. Rev. A 84, 062302 (2011). Acknowledgments: This work was supported in part by MITACS; Natural Sciences and Engineering Research Council of Canada; the Ontario Ministry of Research and Innovation; the Ontario Ministry of Training, Colleges, and Universities; and the U.S. Army Research Office. Supplementary Materials www.sciencemag.org/cgi/content/full/339/6121/791/DC1 Materials and Methods Supplementary Text Figs. S1 to S10 References 10 September 2012; accepted 17 December 2012 10.1126/science.1229957 Matthew A. Broome, 1,2 * Alessandro Fedrizzi, 1,2 Saleh Rahimi-Keshari, 2 Justin Dove, 3 Scott Aaronson, 3 Timothy C. Ralph, 2 Andrew G. White 1,2 Quantum computers are unnecessary for exponentially efficient computation or simulation if the Extended Church-Turing thesis is correct. The thesis would be strongly contradicted by physical devices that efficiently perform tasks believed to be intractable for classical computers. Such a task is boson sampling: sampling the output distributions of n bosons scattered by some passive, linear unitary process. We tested the central premise of boson sampling, experimentally verifying that three-photon scattering amplitudes are given by the permanents of submatrices generated from a unitary describing a six-mode integrated optical circuit. We find the protocol to be robust, working even with the unavoidable effects of photon loss, non-ideal sources, and imperfect detection. Scaling this to large numbers of photons should be a much simpler task than building a universal quantum computer. Amajor motivation for scalable quantum computing is Shor’s algorithm (1), which enables the efficient factoring of large composite numbers into their constituent primes. The presumed difficulty of this task is the basis of the majority of today’s public-key encryption schemes. It may be that scalable quantum computers are not realistic if, for example, quantum mechanics breaks down for large numbers of qubits (2). If, however, quantum com- 15 FEBRUARY 2013 VOL 339 SCIENCE www.sciencemag.org puters are realistic physical devices, then the Extended Church-Turing (ECT) thesis—that any function efficiently computed on a realistic physical device can be efficiently computed on a probabilistic Turing machine—means that a classical efficient factoring algorithm exists. Such an algorithm, long sought after, would enable us to break public-key cryptosystems such as RSA. A third possibility is that the ECT thesis itself is wrong. Downloaded from www.sciencemag.org on February 14, 2013
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