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View all- Gharibian S(2024)Guest Column: The 7 faces of quantum NPACM SIGACT News10.1145/3639528.363953554:4(54-91)Online publication date: 3-Jan-2024
Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians
Abstract
We examine the problem of determining whether a multiqubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding 3-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy; namely, we present an algorithm which decides, in a number of arithmetic operations over $\mathbb{R}$ which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.
References
[1]
D. Aharonov and A. B. Grilo, Stoquastic PCP vs. randomness, in IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), 2019, pp. 1000--1023, https://doi.org/10.1109/FOCS.2019.00065.
[2]
D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput., 37 (2007), pp. 166--194, https://doi.org/10.1137/S0097539705447323.
[3]
T. Albash, Role of nonstoquastic catalysts in quantum adiabatic optimization, Phys. Rev. A, 99 (2019), 042334, https://doi.org/10.1103/PhysRevA.99.042334.
[4]
T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Modern Phys., 90 (2018), 015002, https://doi.org/10.1103/RevModPhys.90.015002.
[5]
J. Bausch and E. Crosson, Analysis and limitations of modified circuit-to-Hamiltonian constructions, Quantum, 2 (2018), p. 94, https://doi.org/10.22331/q-2018-09-19-94.
[6]
A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994, https://doi.org/10.1137/1.9781611971262.
[7]
J. D. Biamonte and P. J. Love, Realizable Hamiltonians for universal adiabatic quantum computers, Phys. Rev. A, 78 (2008), 012352, https://doi.org/10.1103/PhysRevA.78.012352.
[8]
R. F. Bishop and D. J. J. Farnell, Marshall-Peierls sign rules, the quantum Monte Carlo method, and frustration, in Recent Progress in Many-Body Theories, World Scientific, 2000, pp. 457--460, https://doi.org/10.1142/9789812792754_0052.
[9]
S. Bravyi, A. J. Bessen, and B. M. Terhal, Merlin-Arthur Games and Stoquastic Complexity, preprint, http://arXiv.org/abs/quant-ph/0611021, 2006.
[10]
S. Bravyi, D. P. Divincenzo, R. Oliveira, and B. M. Terhal, The complexity of stoquastic local Hamiltonian problems, Quantum Inf. Comput., 8 (2008), p. 361--385, https://dl.acm.org/doi/10.5555/2011772.2011773.
[11]
S. Bravyi and B. Terhal, Complexity of stoquastic frustration-free Hamiltonians, SIAM J. Comput., 39 (2009), pp. 1462--1485, https://doi.org/10.1137/08072689X.
[12]
J. Bringewatt, W. Dorland, S. P. Jordan, and A. Mink, Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians, Phys. Rev. A, 97 (2018), 022323, https://doi.org/10.1103/PhysRevA.97.022323.
[13]
S. Chandrasekharan, Fermion bag approach to lattice field theories, Phys. Rev. D, 82 (2010), 025007, https://doi.org/10.1103/PhysRevD.82.025007.
[14]
T. Cubitt and A. Montanaro, Complexity classification of local Hamiltonian problems, SIAM J. Comput., 45 (2016), pp. 268--316, https://doi.org/10.1137/140998287.
[15]
D. Hangleiter, I. Roth, D. Nagaj, and J. Eisert, Easing the Monte Carlo sign problem, Sci. Adv., 6 (2020), eabb8341, https://doi.org/10.1126/sciadv.abb8341.
[16]
M. B. Hastings, Obstructions to classically simulating the quantum adiabatic algorithm, Quantum Inf. Comput., 13 (2013), p. 1038--1076, https://dl.acm.org/doi/10.5555/2535639.2535647.
[17]
M. B. Hastings, How quantum are non-negative wavefunctions?, J. Math. Phys., 57 (2016), 015210, https://doi.org/10.1063/1.4936216.
[18]
N. Hatano and M. Suzuki, Representation basis in quantum Monte Carlo calculations and the negative-sign problem, Phys. Lett. A, 163 (1992), pp. 246--249, https://doi.org/10.1016/0375-9601(92)91006-D.
[19]
I. Hen, Resolution of the sign problem for a frustrated triplet of spins, Phys. Rev. E, 99 (2019), 033306, https://doi.org/10.1103/PhysRevE.99.033306.
[20]
I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput., 10 (1981), pp. 718--720, https://doi.org/10.1137/0210055.
[21]
L. Hormozi, E. W. Brown, G. Carleo, and M. Troyer, Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass, Phys. Rev. B, 95 (2017), 184416, https://doi.org/10.1103/PhysRevB.95.184416.
[22]
M. Ioannou, Stoquastic Hamiltonians and the Monte Carlo sign problem, Ph.D thesis, LMU Munich, 2019; supervision by Prof. B. M. Terhal.
[23]
M. Jarret, S. P. Jordan, and B. Lackey, Adiabatic optimization versus diffusion Monte Carlo methods, Phys. Rev. A, 94 (2016), 042318, https://doi.org/10.1103/PhysRevA.94.042318.
[24]
J. Klassen and B. M. Terhal, Two-local qubit Hamiltonians: When are they stoquastic?, Quantum, 3 (2019), p. 139, https://doi.org/10.22331/q-2019-05-06-139.
[25]
D. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, 2005.
[26]
E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Sign problem in the numerical simulation of many-electron systems, Phys. Rev. B, 41 (1990), pp. 9301--9307, https://doi.org/10.1103/PhysRevB.41.9301.
[27]
M. Marvian, D. A. Lidar, and I. Hen, On the computational complexity of curing non-stoquastic Hamiltonians, Nature Commun., (2019), https://www.nature.com/articles/s41467-019-09501-6.
[28]
H. Nishimori and K. Takada, Exponential enhancement of the efficiency of quantum annealing by non-stoquastic Hamiltonians, Frontiers ICT, 4 (2017), 2, https://doi.org/10.3389/fict.2017.00002.
[29]
I. Ozfidan et al., Demonstration of a nonstoquastic Hamiltonian in coupled superconducting flux qubits, Phys. Rev. Appl., 13 (2020), 034037, https://doi.org/10.1103/PhysRevApplied.13.034037.
[30]
Z. Ringel and D. L. Kovrizhin, Quantized gravitational responses, the sign problem, and quantum complexity, Sci. Adv., 3 (2017), e1701758, https://doi.org/10.1126/sciadv.1701758.
[31]
M. Suzuki, Quantum Monte Carlo Methods in Condensed Matter Physics, World Scientific, 1993, https://doi.org/10.1142/2262.
[32]
G. Torlai, J. Carrasquilla, M. T. Fishman, R. G. Melko, and M. P. A. Fisher, Wave-function positivization via automatic differentiation, Phys. Rev. Res., 2 (2020), 032060, https://doi.org/10.1103/PhysRevResearch.2.032060.
[33]
M. Troyer, Boulder School 2010: Computational and Conceptual Approaches to Quantum Many-Body Systems, Lecture notes, Yale University, 2010.
[34]
W. Vinci and D. A. Lidar, Non-stoquastic Hamiltonians in quantum annealing via geometric phases, npj Quant. Inf., 3 (2017), 38, https://www.nature.com/articles/s41534-017-0037-z.
[35]
P. Werner, A. Comanac, L. de' Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impurity models, Phys. Rev. Lett., 97 (2006), 076405, https://doi.org/10.1103/PhysRevLett.97.076405.
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Published: 01 January 2020
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