Photonic Boson Sampling in a Tunable Circuit The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Broome, M. A., A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White. “Photonic Boson Sampling in a Tunable Circuit.” Science 339, no. 6121 (February 15, 2013): 794–798. As Published http://dx.doi.org/10.1126/science.1231440 Publisher American Association for the Advancement of Science (AAAS) Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/85873 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ REPORTS matrices for this graph at momenta –p/4 and graphtobeO(n13g5), and the total evolution time 18. A. J. Hoffman et al., Phys.Rev.Lett.107, 053602 (2011). –p/2 are to be O(n12g5). The bounds we prove, although 19. J.K.Gamble,M.Friesen,D.Zhou,R.Joynt,S.N.Coppersmith, Phys.Rev.A81, 052313 (2010). 0 1 almost certainly not optimal, are sufficient to es- 00e−ip=4 20. J. Smith, Electron. Notes Discrete Math. 38, 795 (2011). tablish universality with only polynomial over- ć S ð−p=4Þ¼@ − A 21. Y. Omar, N. Paunkovi , Phys. Rev. A 74, 042304 (2006). switch 0 10 22. P. K. Pathak, G. S. Agarwal, Phys. Rev. A 75, 032351 (2007). −ip=4 head. Because it is also possible to efficiently e 00 simulate a multiparticle quantum walk of the type 23. Y. Lahini et al., Phys. Rev. A 86, 011603 (2012). we consider using a universal quantum com- 24. A. Schreiber et al., Science 336, 55 (2012). 0 1 25. A. Ahlbrecht et al., New J. Phys. 14, 073050 (2012). 10 0 puter, this model exactly captures the power of 26. P. P. Rohde, A. Schreiber, M. Štefaňák,I.Jex,C.Silberhorn, @ A quantum computation. New J. Phys. 13, 013001 (2011). Sswitchð−p=2Þ¼ 00−1 ð5Þ − 27. B.M.Terhal,D.P.DiVincenzo,Phys.Rev.A65, 032325 (2002). 0 10 References and Notes 28. S. Aaronson, A. Arkhipov, “The computational complexity 1. E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998). of linear optics,” Proceedings of the 43rd ACM The momentum switch has perfect transmission 2. D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, Symposium on Theory of Computing (2011), pp. 333–342. between vertices 1 and 3 at momentum –p/4 and “Quantum walks on graphs,” Proceedings of the 33rd 29. R. Ionicioiu, P. Zanardi, Phys. Rev. A 66, 050301 (2002). perfect transmission between vertices 2 and 3 at ACM Symposium on Theory of Computing (2001), pp. 50–59. 30. A. Mizel, D. A. Lidar, M. Mitchell, Phys. Rev. Lett. 99, momentum –p/2. Thus, the path a particle fol- 3. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, 070502 (2007). “ ” 31. A. M. Childs, D Gosset, J. Math. Phys. 53, 102207 lows through the switch depends on its momen- J. Watrous, One-dimensional quantum walks, Proceedings of the 33rd ACM Symposium on Theory of (2012). tum: A particle with momentum –p/2 follows the Computing (2001), pp. 37–49. 32. Materials and methods are available as supplementary double line in Fig. 3A, whereas a particle with 4. A. M. Childs et al., “Exponential algorithmic speedup by materials on Science Online. momentum –p/4 follows the single line. quantum walk,” Proceedings of the 35th ACM Symposium 33. M. Valiente, Phys. Rev. A 81, 042102 (2010). – 34. B. A. Blumer, M. S. Underwood, D. L. Feder, Phys. Rev. A The graph used to implement the Cq gate is on Theory of Computing (2003), pp. 59 68. 5. A. Ambainis, SIAM J. Comput. 37, 210 (2007). 84, 062302 (2011). shown in Fig. 3B [see section S4 of (32)forthe 6. E. Farhi, J. Goldstone, S. Gutmann, Theory of Computing numbers of vertices on each of the paths]. To see 4, 169 (2008). Acknowledgments: This work was supported in part by why this graph implements a Cq gate, consid- 7. A. 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Materials and Methods onto the vertical path as it passes through the first 14. L. Sansoni et al., Phys. Rev. Lett. 108, 010502 (2012). momentum switch and is routed to the right as it Supplementary Text 15. I. L. Chuang, Y. Yamamoto, Phys. Rev. A 52, 3489 (1995). Figs. S1 to S10 passes through the second switch. If both par- 16. G. K. Brennen, C. M. Caves, P. S. Jessen, I. H. Deutsch, References Phys. Rev. Lett. 82, 1060 (1999). ticles begin in the state j1in〉, they interact on the vertical path and the wave function acquires a 17. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, M. Greiner, 10 September 2012; accepted 17 December 2012 Nature 462, 74 (2009). 10.1126/science.1229957 phase eiq. To implement a circuit, the subgraphs rep- resenting circuit elements are connected by paths. Figure 4 depicts a graph corresponding to a sim- Photonic Boson Sampling ple two-qubit computation. Timing is important: Wave packets must meet on the vertical paths for interactions to occur. We achieve this by choosing in a Tunable Circuit the numbers of vertices on each of the segments Matthew A. Broome,1,2* Alessandro Fedrizzi,1,2 Saleh Rahimi-Keshari,2 Justin Dove,3 in the graph appropriately, taking into account 3 2 1,2 the different propagation speeds of the two wave Scott Aaronson, Timothy C. Ralph, Andrew G. White packets [see section S4 of (32)]. In section S3.1 of (32), we present a refinement of our scheme Quantum computers are unnecessary for exponentially efficient computation or simulation if the using planar graphs with maximum degree four. Extended Church-Turing thesis is correct. The thesis would be strongly contradicted by physical By analyzing the full (n +1)–particle inter- devices that efficiently perform tasks believed to be intractable for classical computers. Such a task acting many-body system, we prove that our al- is boson sampling: sampling the output distributions of n bosons scattered by some passive, linear gorithm performs the desired quantum computation unitary process. We tested the central premise of boson sampling, experimentally verifying that up to an error term that can be made arbitrarily three-photon scattering amplitudes are given by the permanents of submatrices generated from a small (32). Our analysis goes beyond the scat- unitary describing a six-mode integrated optical circuit. We find the protocol to be robust, working tering theory discussion presented above; we take even with the unavoidable effects of photon loss, non-ideal sources, and imperfect detection. into account the fact that both the wave packets Scaling this to large numbers of photons should be a much simpler task than building a universal and the graphs are finite. Specifically, we prove quantum computer. that by choosing the size of the wave packets, the number of vertices in the graph, and the total major motivation for scalable quan- puters are realistic physical devices, then the evolution time to be polynomial functions of both tum computing is Shor’s algorithm (1), Extended Church-Turing (ECT) thesis—that n and g, the error in simulating an n-qubit, g-gate Awhich enables the efficient factoring of any function efficiently computed on a realistic quantum circuit is bounded above by an arbi- large composite numbers into their constituent physical device can be efficiently computed trarily small constant [section S5 of (32)]. For primes. The presumed difficulty of this task is the on a probabilistic Turing machine—means that example, for the Bose-Hubbard model and for basis of the majority of today’s public-key en- a classical efficient factoring algorithm exists. the nearest-neighbor interaction model, we prove cryption schemes. It may be that scalable quan- Such an algorithm, long sought after, would that the error can be made arbitrarily small by tum computers are not realistic if, for example, enable us to break public-key cryptosystems choosing the size of the wave packets to be quantum mechanics breaks down for large num- such as RSA. A third possibility is that the ECT O(n12g4), the total number of vertices in the bers of qubits (2). If, however, quantum com- thesis itself is wrong. 794 15 FEBRUARY 2013 VOL 339 SCIENCE www.sciencemag.org REPORTS How do we answer this trilemma? As yet exponentially, whereas Bob’s experimental run- every input distributed to every output. The 6- there is no evidence that large-scale quantum time does not. It becomes intractable to verify input × 6-output modes of U are represented by computers are inherently impossible (that will Bob’s output against Alice’s, and, unlike for two orthogonal polarizations in 3 × 3 spatial modes need to be tested directly via experiment), and Shor’s algorithm, there is no known efficient of a fused-fiber beamsplitter (FBS), an intrin- there is no efficient classical factoring algorithm algorithm to verify the result (4).