Two-local qubit Hamiltonians: when are they stoquastic?

Joel Klassen; Barbara M. Terhal

doi:10.22331/q-2019-05-06-139

Two-local qubit Hamiltonians: when are they stoquastic?

Joel Klassen¹ and Barbara M. Terhal^1,2

¹QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands
²Institute for Theoretical Nanoelectronics, Forschungszentrum Juelich, D-52425 Juelich, Germany

Published:	2019-05-06, volume 3, page 139
Eprint:	arXiv:1806.05405v3
Scirate:	https://scirate.com/arxiv/1806.05405v3
Doi:	https://doi.org/10.22331/q-2019-05-06-139
Citation:	Quantum 3, 139 (2019).

We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a concrete example where using unitaries beyond Clifford rotations is required in order to decide stoquasticity. We report on simple results for $n$-qubit Hamiltonians with identical 2-local terms on bipartite graphs. Our most significant result is that we give an efficient algorithm to determine whether an arbitrary $n$-qubit XYZ Heisenberg Hamiltonian is stoquastic by local basis changes.

► BibTeX data

@article{Klassen2019twolocalqubit,
  doi = {10.22331/q-2019-05-06-139},
  url = {https://doi.org/10.22331/q-2019-05-06-139},
  title = {Two-local qubit {H}amiltonians: when are they stoquastic?},
  author = {Klassen, Joel and Terhal, Barbara M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {139},
  month = may,
  year = {2019}
}

► References

[1] S. Bravyi, D. P. DiVincenzo, R. I. Oliveira, and B. M. Terhal, ``The Complexity of Stoquastic Local Hamiltonian Problems,'' Quantum Information and Computation 8 no. 5, (2008) 0361-0385 , arXiv:0606140 [quant-ph].
https://doi.org/10.26421/QIC8.5
arXiv:0606140

[2] N. J. Cerf and O. C. Martin, ``Projection Monte Carlo methods : an algorithmic analysis,'' International Journal of Modern Physics C 6 no. 5, (1995) 693-723.
https://doi.org/10.1142/S0129183195000587

[3] S. Sorella and L. Capriotti, ``Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem,'' Physical Review B - Condensed Matter and Materials Physics 61 no. 4, (2000) 2599-2612, arXiv:9902211 [cond-mat].
https://doi.org/10.1103/PhysRevB.61.2599
arXiv:9902211

[4] S. Bravyi, ``Monte Carlo Simulation of Stoquastic Hamiltonians,'' Quantum Information and Computation 15 no. 13/14, (2015) 1122-1140, arXiv:1402.2295.
https://doi.org/10.26421/QIC15.13-14
arXiv:1402.2295

[5] S. Wessel, ``Monte Carlo Simulations of Quantum Spin Models Institute for Theoretical Solid State Physics,'' in Autumn School on Correlated Electrons. 2013. https://www.cond-mat.de/events/correl13/manuscripts/.
https://www.cond-mat.de/events/correl13/manuscripts/

[6] E. Crosson, Classical and Quantum Computation in Ground States and Beyond. PhD thesis, University of Washington, 2015. http://hdl.handle.net/1773/34128.
http://hdl.handle.net/1773/34128

[7] S. Bravyi and D. Gosset, ``Polynomial-Time Classical Simulation of Quantum Ferromagnets,'' Physical Review Letters 119 no. 10, (2017) , arXiv:1612.05602.
https://doi.org/10.1103/PhysRevLett.119.100503
arXiv:1612.05602

[8] T. Albash and D. A. Lidar, ``Adiabatic quantum computation,'' Reviews of Modern Physics 90 no. 1, (2018) 015002, arXiv:1611.04471 [quant-ph].
https://doi.org/10.1103/RevModPhys.90.015002
arXiv:1611.04471

[9] S. Bravyi and B. Terhal, ``Complexity of stoquastic frustration-free Hamiltonians,'' SIAM J. Comput. 39 no. 4, (2009) 1642, arXiv:0806.1746.
https://doi.org/10.1137/08072689X
arXiv:0806.1746

[10] M. B. Hastings and M. H. Freedman, ``Obstructions To Classically Simulating The Quantum Adiabatic Algorithm,'' Quantum Information and Computation 13 no.11/12, (2013) 1038-1076 arXiv:1302.5733.
https://doi.org/10.26421/QIC13.11-12
arXiv:1302.5733

[11] J. Bringewatt, W. Dorland, S. P. Jordan, and A. Mink, ``Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians,'' Phys. Rev. A 97 no. 2, (Feb., 2018) 022323, arXiv:1709.03971 [quant-ph].
https://doi.org/10.1103/PhysRevA.97.022323
arXiv:1709.03971

[12] D. Kafri, C. Quintana, Y. Chen, A. Shabani, J. M. Martinis, and H. Neven, ``Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime,'' Physical Review A 95 no. 5, (May, 2017) 052333, arXiv:1606.08382.
https://doi.org/10.1103/PhysRevA.95.052333
arXiv:1606.08382

[13] L. Hormozi, E. W. Brown, G. Carleo, and M. Troyer, ``Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass,'' Phys. Rev. B 95 no. 18, (May, 2017) 184416, arXiv:1609.06558 [quant-ph].
https://doi.org/10.1103/PhysRevB.95.184416
arXiv:1609.06558

[14] G. Samach, ``Tunable XX-Coupling Between High Coherence Flux Qubits,'' in APS March Meeting 2018. 2018. https://meetings.aps.org/Meeting/MAR18/Session/L33.13.
https://meetings.aps.org/Meeting/MAR18/Session/L33.13

[15] V. I. Iglovikov, E. Khatami, and R. T. Scalettar, ``Geometry dependence of the sign problem in quantum Monte Carlo simulations,'' Phys. Rev. B 92 (Jul, 2015) 045110. https://link.aps.org/doi/10.1103/PhysRevB.92.045110.
https://doi.org/10.1103/PhysRevB.92.045110

[16] C. Wu and S.-C. Zhang, ``Sufficient condition for absence of the sign problem in the fermionic quantum monte carlo algorithm,'' Phys. Rev. B 71 (Apr, 2005) 155115.
https://doi.org/10.1103/PhysRevB.71.155115

[17] Z.-X. Li and H. Yao, ``Sign-Problem-Free Fermionic Quantum Monte Carlo: Developments and Applications,'' arXiv e-prints (May, 2018) arXiv:1805.08219, arXiv:1805.08219 [cond-mat.str-el].
https://doi.org/10.1146/annurev-conmatphys-033117-054307
arXiv:1805.08219

[18] R. F. Bishop and D. J. J. Farnell, ``Marshall-Peierls sign rules, the quantum monte carlo method, and frustration,'' International Journal of Modern Physics B 15 no. 10n11, (2001) 1736-1739.
https://doi.org/10.1142/9789812792754_0052

[19] M. Marvian, D. A. Lidar, and I. Hen, ``On the Computational Complexity of Curing non-stoquastic Hamiltonians,'' Nature Communications 10 no. 1, (2019) 1571 , arXiv:1802.03408.
https://doi.org/10.1038/s41467-019-09501-6
arXiv:1802.03408

[20] B. M. Terhal, ``The Power and Use of Stoquastic Hamiltonians,'' in Adiabatic Quantum Computing Conference. 2017. https://www.youtube.com/watch?v=4dK30QExF4M.
https://www.youtube.com/watch?v=4dK30QExF4M

[21] T. Cubitt, A. Montanaro, and S. Piddock, ``Universal Quantum Hamiltonians,'' National Academy of Sciences 115 no. 38 (2018) 9497-9502 , arXiv:1701.05182.
https://doi.org/10.1073/pnas.1804949115%20
arXiv:1701.05182

[22] S. Bravyi and M. Hastings, ``On complexity of the quantum Ising model,'' Communications in Mathematical Physics 349 no. 1 (2017) 1-45 , arXiv:1410.0703 [quant-ph].
https://doi.org/10.1007/s00220-016-2787-4
arXiv:1410.0703

[23] D. Grier and L. Schaeffer, ``The Classification of Stabilizer Operations over Qubits,'' ArXiv e-prints (Mar., 2016) , arXiv:1603.03999 [quant-ph].
arXiv:1603.03999

[24] Y. Makhlin, ``Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations,'' Quantum Information Processing 1 no. 4, (2002) 243-252, arXiv:0002045 [quant-ph].
https://doi.org/10.1023/A:1022144002391
arXiv:0002045

[25] N. Linden, S. Popescu, and A. Sudbery, ``Nonlocal Parameters for Multiparticle Density Matrices,'' Physical Review Letters 83 no. 2, (1999) 243-247, arXiv:9801076 [quant-ph].
https://doi.org/10.1103/PhysRevLett.83.243
arXiv:9801076

[26] M. Grassl, M. Rötteler, and T. Beth, ``Computing local invariants of quantum-bit systems,'' Phys. Rev. A 58 (Sept., 1998) 1833-1839, quant-ph/9712040.
https://doi.org/10.1103/PhysRevA.58.1833
arXiv:quant-ph/9712040

[27] R. A. Bertlmann and P. Krammer, ``Bloch vectors for qudits,'' Journal of Physics A: Mathematical and Theoretical 41 no. 23, (2008) , arXiv:0806.1174.
https://doi.org/10.1088/1751-8113/41/23/235303
arXiv:0806.1174

[28] T. F. Gonzalez, ``Clustering to minimize the maximum intercluster distance,'' Theoretical Computer Science 38 (1985) 293-306.
https://doi.org/10.1016/0304-3975(85)90224-5

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[3] Andrew J. Kerman, "Superconducting qubit circuit emulation of a vector spin-1/2", arXiv:1810.01352.

[4] Dorit Aharonov and Alex B. Grilo, "Stoquastic PCP vs. Randomness", arXiv:1901.05270.

[5] Dominik Hangleiter, Ingo Roth, Daniel Nagaj, and Jens Eisert, "Easing the Monte Carlo sign problem", arXiv:1906.02309.

[6] I. Ozfidan, C. Deng, A. Y. Smirnov, T. Lanting, R. Harris, L. Swenson, J. Whittaker, F. Altomare, M. Babcock, C. Baron, A. J. Berkley, K. Boothby, H. Christiani, P. Bunyk, C. Enderud, B. Evert, M. Hager, A. Hajda, J. Hilton, S. Huang, E. Hoskinson, M. W. Johnson, K. Jooya, E. Ladizinsky, N. Ladizinsky, R. Li, A. MacDonald, D. Marsden, G. Marsden, T. Medina, R. Molavi, R. Neufeld, M. Nissen, M. Norouzpour, T. Oh, I. Pavlov, I. Perminov, G. Poulin-Lamarre, M. Reis, T. Prescott, C. Rich, Y. Sato, G. Sterling, N. Tsai, M. Volkmann, W. Wilkinson, J. Yao, and M. H. Amin, "Demonstration of nonstoquastic Hamiltonian in coupled superconducting flux qubits", arXiv:1903.06139.

The above citations are from Crossref's cited-by service (last updated 2019-07-15 18:11:56) and SAO/NASA ADS (last updated 2019-07-15 18:11:57). The list may be incomplete as not all publishers provide suitable and complete citation data.

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