The complexity of antiferromagnetic interactions and 2D lattices
Pages 636 - 672
Abstract
Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMA-complete. We also prove StoqMA-completeness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMA-complete 2-local interactions remain QMA-complete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMA-complete with positive coefficients remain QMA-complete when restricted to a 2D triangular lattice.
References
[1]
F. Barahona. On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen., 15(10):3241, 1982.
[2]
E. Barouch and B. M. McCoy. Statistical Mechanics of the XY Model. II. Spin-Correlation Functions. Phys. Rev. A, 3:786-804, 1971.
[3]
J. Biamonte and P. Love. Realizable Hamiltonians for universal adiabatic quantum computers. Phys. Rev. A, 78:012352, 2008. arXiv:0704.1287.
[4]
A. Böttcher. Introduction to large truncated Toeplitz matrices. Universitext. Springer, New York, 1999.
[5]
S. Bravyi. Monte Carlo simulation of stoquastic Hamiltonians, February 2014. arXiv:1402.2295.
[6]
S. Bravyi, A. Bessen, and B. Terhal. Merlin-Arthur games and stoquastic complexity, 2006. quant-ph/0611021.
[7]
S. Bravyi, D. DiVincenzo, and D. Loss. Schrieffer-Wolff transformation for quantum many-body systems. Ann. of Phys., 326(10):2793-2826, 2011. arXiv:1105.0675.
[8]
S. Bravyi and M. Hastings. On complexity of the quantum Ising model, October 2014. arXiv:1410.0703.
[9]
Y. Cao and D. Nagaj. Perturbative gadgets without strong interactions, 2014. arXiv:1408.5881.
[10]
A. Childs, D. Gosset, and Z. Webb. The Bose-Hubbard model is QMA-complete. In Proc. 41st International Conference on Automata, Languages and Programming (ICALP'14), pages 308-319, 2014. arXiv:1311.3297.
[11]
A. Childs, D. Gosset, and Z. Webb. Complexity of the XY antiferromagnet at fixed magnetization, 2015. arXiv:1503.07083.
[12]
T. Cubitt and A. Montanaro. Complexity classification of local Hamiltonian problems. SIAM Journal on Computing, 45(2):268-316, 2016. arXiv:1311.3161.
[13]
P. Deift, A. Its, and I. Krasovsky. On the asymptotics of a Toeplitz determinant with singularities. In Random Matrix Theory, Interacting Particle Systems and Integrable Systems, pages 93-146. Cambridge University Press, 2014. arXiv:1206.1292.
[14]
T. Ehrhardt. A Status Report on the Asymptotic Behavior of Toeplitz Determinants with FisherHartwig Singularities. In A. Dijksma, M. A. Kaashoek, and A. C. M. Ran, editors, Recent Advances in Operator Theory, number 124 in Operator Theory: Advances and Applications, pages 217-241. Birkhäuser Basel, 2001.
[15]
M. Hastings and T. Koma. Spectral Gap and Exponential Decay of Correlations. Comm. Math. Phys., 265(3):781-804, 2006. math-ph/0507008.
[16]
J. Kempe, A. Kitaev, and Oded Regev. The Complexity of the Local Hamiltonian Problem. SIAM Journal of Computing, 35(5):1070-1097, 2004. quant-ph/0406180.
[17]
A. Yu Kitaev, A. Shen, and M. N. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. AMS, 2002.
[18]
E. Lieb, T. Schultz, and D. Mattis. Two soluble models of an antiferromagnetic chain. Ann. of Phys., 16(3):407-466, 1961.
[19]
B. M. McCoy. Spin Correlation Functions of the X - Y Model. Phys. Rev., 173:531-541, 1968.
[20]
Barry M. McCoy, Jacques H. H. Perk, and Robert E. Shrock. Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations. Nuclear Physics B, 220(3):269-282, 1983.
[21]
M. Metlitski. The XY Model in One Dimension, 2004. Student Project for course PHYS 503 at the University of Bristish Colombia.
[22]
R. Oliveira and B. Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Inf. Comput., 8(10):0900-0924, April 2005. quant-ph/0504050.
[23]
A. Ovchinnikov. Fisher-Hartwig conjecture and the correlators in XY spin chain. Physics Letters A, 366(4-5):357-362, 2007. math-ph/0509026.
[24]
C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
[25]
J. H. H. Perk and H. W. Capel. Time-dependent xx-correlation functions in the one-dimensional XY-model. Physica A: Statistical Mechanics and its Applications, 89(2):265-303, 1977.
[26]
T. Schaefer. The complexity of satisfiability problems. In Proc. 10th Annual ACM Symp. Theory of Computing, pages 216-226, 1978.
[27]
N. Schuch. Complexity of commuting Hamiltonians on a square lattice of qubits. Quantum Inf. Comput., 11(11&12):901-912, 2011. arXiv:1105.2843.
[28]
N. Schuch and F. Verstraete. Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory. Nature Physics, 5(10):732-735, 2009. arXiv:0712.0483.
- The complexity of antiferromagnetic interactions and 2D lattices
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Rinton Press, Incorporated
Paramus, NJ
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Published: 01 June 2017
Revised: 05 May 2017
Received: 18 December 2015
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