Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm

Jessica Lemieux1, Bettina Heim2, David Poulin1,3, Krysta Svore2, and Matthias Troyer2

1Département de Physique & Institut Quantique, Université de Sherbrooke, Québec, Canada
2Quantum Architecture and Computation Group, Microsoft Research, Redmond, WA 98052, USA
3Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations. We thus reformulate the quantum walk, circumventing its implementation altogether by closely following the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.

► BibTeX data

► References

[1] Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03, page 20, New York, New York, USA, 2003. ACM Press. ISBN 1581136749. 10.1145/​780542.780546.

[2] Andris Ambainis. Quantum walk algorithm for element distinctness. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 22–31, 2004. 10.1109/​focs.2004.54.

[3] Francisco Barahona. On the computational complexity of Ising spin glass models. Journal of Physics A: Mathematical and General, 15: 3241–3253, 1982. 10.1088/​0305-4470/​15/​10/​028.

[4] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. Divincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52 (5): 3457–3467, nov 1995. ISSN 10502947. 10.1103/​PhysRevA.52.3457.

[5] Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Physical Review A - Atomic, Molecular, and Optical Physics, 91 (5): 052317, may 2015. ISSN 10941622. 10.1103/​PhysRevA.91.052317.

[6] S. Boixo, E. Knill, and R. D. Somma. Fast quantum algorithms for traversing paths of eigenstates. may 2010. URL https:/​/​arxiv.org/​abs/​1005.3034.

[7] Sergio Boixo, Emanuel Knill, and Rolando Somma. Eigenpath traversal by phase randomization. Quantum Information and Computation, 9 (9&10): 0833, 2009. URL http:/​/​arxiv.org/​abs/​0903.1652.

[8] N Cody Jones, James D Whitfield, Peter L McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, and Yoshihisa Yamamoto. Faster quantum chemistry simulation on fault-tolerant quantum computers. New Journal of Physics, 14 (11): 115023, nov 2012. ISSN 1367-2630. 10.1088/​1367-2630/​14/​11/​115023.

[9] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Quantum Computation by Adiabatic Evolution. jan 2000. URL http:/​/​arxiv.org/​abs/​quant-ph/​0001106.

[10] Roy J. Glauber. Time-dependent statistics of the Ising model. Journal of Mathematical Physics, 4 (2): 294–307, feb 1963. ISSN 00222488. 10.1063/​1.1703954.

[11] Jeongwan Haah. Product Decomposition of Periodic Functions in Quantum Signal Processing. jun 2018. 10.22331/​q-2019-10-07-190.

[12] W K Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57 (1): 97–109, apr 1970. ISSN 0006-3444. 10.1093/​biomet/​57.1.97.

[13] Janus Collaboration, F. Belletti, M. Cotallo, A. Cruz, L. A. Fernández, A. Gordillo, M. Guidetti, A. Maiorano, F. Mantovani, E. Marinari, V. Martín-Mayor, A. Muñoz-Sudupe, D. Navarro, G. Parisi, S. Pérez-Gaviro, M. Rossi, J. J. Ruiz-Lorenzo, S. F. Schifano, D. Sciretti, A. Tarancón, R. Tripiccione, and J. L. Velasco. JANUS: an FPGA-based System for High Performance Scientific Computing. Computing in Science & Engineering, 11 (1): 48–58, 2009. 10.1109/​MCSE.2009.11.

[14] Janus Collaboration, M. Baity-Jesi, R. A. Banos, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, M. Guidetti, D. Iniguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Munoz Sudupe, D. Navarro, G. Parisi, M. Pivanti, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, P. Tellez, R. Tripiccione, and D. Yllanes. Reconfigurable computing for Monte Carlo simulations: results and prospects of the Janus project. The European Physical Journal Special Topics, 210 (33), 2012. 10.1140/​epjst/​e2012-01636-9.

[15] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220 (4598): 671–680, 1983. ISSN 00368075. 10.1126/​science.220.4598.671.

[16] A. Yu. Kitaev. Quantum measurements and the Abelian Stabilizer Problem. nov 1995. URL http:/​/​arxiv.org/​abs/​quant-ph/​9511026.

[17] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin. Resource estimate for quantum many-body ground state preparation on a quantum computer. 2020. URL https:/​/​arxiv.org/​abs/​2006.04650.

[18] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. oct 2016. 10.22331/​q-2019-07-12-163.

[19] Guang Hao Low and Isaac L. Chuang. Optimal Hamiltonian Simulation by Quantum Signal Processing. Physical Review Letters, 118 (1): 010501, jan 2017. ISSN 10797114. 10.1103/​PhysRevLett.118.010501.

[20] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. Methodology of resonant equiangular composite quantum gates. Physical Review X, 6 (4): 041067, dec 2016. ISSN 21603308. 10.1103/​PhysRevX.6.041067.

[21] F. Magniez, A. Nayak, J. Roland, and M. Santha. Search via quantum walk. SIAM Journal on Computing, 40: 142–164. 10.1137/​090745854.

[22] Chris Marriott and John Watrous. Quantum Arthur-Merlin games. In Computational Complexity, volume 14, pages 122–152. Springer, jun 2005. 10.1007/​s00037-005-0194-x.

[23] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21 (6): 1087–1092, jun 1953. ISSN 00219606. 10.1063/​1.1699114.

[24] Troels F. Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V. Isakov, David Wecker, John M. Martinis, Daniel A. Lidar, and Matthias Troyer. Defining and detecting quantum speedup. Science, 345 (6195): 420–424, jul 2014. ISSN 10959203. 10.1126/​science.1252319.

[25] Neil J Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of Z-rotations. Quantum Information and Computation, 16 (11&12): 0901, 2016. URL http:/​/​arxiv.org/​abs/​1403.2975.

[26] Terry Rudolph and Lov Grover. A 2 rebit gate universal for quantum computing. oct 2002. URL https:/​/​arxiv.org/​abs/​quant-ph/​0210187.

[27] R. D. Somma, S. Boixo, H. Barnum, and E. Knill. Quantum simulations of classical annealing processes. Physical Review Letters, 101 (13): 130504, sep 2008. ISSN 00319007. 10.1103/​PhysRevLett.101.130504.

[28] Mario Szegedy. Quantum speed-up of Markov Chain based algorithms. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 32–41, 2004. 10.1109/​focs.2004.53.

[29] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete. Quantum Metropolis sampling. Nature, 471 (7336): 87–90, mar 2011. ISSN 00280836. 10.1038/​nature09770.

[30] Marija Vucelja. Lifting—A nonreversible Markov chain Monte Carlo algorithm. American Journal of Physics, 84 (12): 958–968, dec 2016. ISSN 0002-9505. 10.1119/​1.4961596.

[31] Man-Hong Yung and Alán Aspuru-Guzik. A quantum-quantum Metropolis algorithm. Proceedings of the National Academy of Sciences of the United States of America, 109 (3): 754–9, jan 2012. ISSN 1091-6490. 10.1073/​pnas.1111758109.

Cited by

[1] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin, "Resource estimate for quantum many-body ground state preparation on a quantum computer", arXiv:2006.04650.

The above citations are from SAO/NASA ADS (last updated successfully 2020-07-14 02:00:07). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-07-14 02:00:06).