Automatic design of Hamiltonians

Kiryl Pakrouski

Department of Physics, Princeton University, Princeton, NJ 08544, USA

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We formulate an optimization problem of Hamiltonian design based on the variational principle. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a weighted sum of relevant Hamiltonian properties specifying thereby the search query. Using fractional quantum Hall effect as a test system we illustrate how the framework can be used to determine a generating Hamiltonian of a finite-size model wavefunction (Moore-Read Pfaffian and Read-Rezayi states), find optimal conditions for an experiment or "extrapolate" given wavefunctions in a certain universality class from smaller to larger system sizes. We also discuss how the search for approximate generating Hamiltonians may be used to find simpler and more realistic models implementing the given exotic phase of matter by experimentally accessible interaction terms.

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[1] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko. Quantum boltzmann machine. Phys. Rev. X, 8: 021050, May 2018. 10.1103/​PhysRevX.8.021050. URL https:/​/​​doi/​10.1103/​PhysRevX.8.021050.

[2] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Phys. Rev. Lett., 110: 084101, Feb 2013. 10.1103/​PhysRevLett.110.084101. URL https:/​/​​doi/​10.1103/​PhysRevLett.110.084101.

[3] Eyal Bairey, Itai Arad, and Netanel H. Lindner. Learning a local hamiltonian from local measurements. Phys. Rev. Lett., 122: 020504, Jan 2019. 10.1103/​PhysRevLett.122.020504. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.020504.

[4] D.M. Basko, I.L. Aleiner, and B.L. Altshuler. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of Physics, 321 (5): 1126 – 1205, 2006. ISSN 0003-4916. https:/​/​​10.1016/​j.aop.2005.11.014. URL http:/​/​​science/​article/​pii/​S0003491605002630.

[5] Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research (JMLR), 18 (153): 1–43, 2018. URL http:/​/​​papers/​v18/​17-468.html.

[6] DAVID CEPERLEY and BERNI ALDER. Quantum monte carlo. Science, 231 (4738): 555–560, 1986. ISSN 0036-8075. 10.1126/​science.231.4738.555. URL https:/​/​​content/​231/​4738/​555.

[7] Eli Chertkov and Bryan K. Clark. Computational inverse method for constructing spaces of quantum models from wave functions. Phys. Rev. X, 8: 031029, Jul 2018. 10.1103/​PhysRevX.8.031029. URL https:/​/​​doi/​10.1103/​PhysRevX.8.031029.

[8] Marcus P. da Silva, Olivier Landon-Cardinal, and David Poulin. Practical characterization of quantum devices without tomography. Phys. Rev. Lett., 107: 210404, Nov 2011. 10.1103/​PhysRevLett.107.210404. URL https:/​/​​doi/​10.1103/​PhysRevLett.107.210404.

[9] J. M. Deutsch. Quantum statistical mechanics in a closed system. Phys. Rev. A, 43: 2046–2049, Feb 1991. 10.1103/​PhysRevA.43.2046. URL https:/​/​​doi/​10.1103/​PhysRevA.43.2046.

[10] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014. https:/​/​​abs/​1411.4028.

[11] R. Fletcher and C. M. Reeves. Function minimization by conjugate gradients. The Computer Journal, 7 (2): 149–154, 01 1964. ISSN 0010-4620. 10.1093/​comjnl/​7.2.149. URL https:/​/​​10.1093/​comjnl/​7.2.149.

[12] Hiroyuki Fujita, Yuya O. Nakagawa, Sho Sugiura, and Masaki Oshikawa. Construction of hamiltonians by supervised learning of energy and entanglement spectra. Phys. Rev. B, 97: 075114, Feb 2018. 10.1103/​PhysRevB.97.075114. URL https:/​/​​doi/​10.1103/​PhysRevB.97.075114.

[13] James R. Garrison and Tarun Grover. Does a single eigenstate encode the full hamiltonian? Phys. Rev. X, 8: 021026, Apr 2018. 10.1103/​PhysRevX.8.021026. URL https:/​/​​doi/​10.1103/​PhysRevX.8.021026.

[14] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov. Interacting electrons in disordered wires: Anderson localization and low-$t$ transport. Phys. Rev. Lett., 95: 206603, Nov 2005. 10.1103/​PhysRevLett.95.206603. URL https:/​/​​doi/​10.1103/​PhysRevLett.95.206603.

[15] Christopher E Granade, Christopher Ferrie, Nathan Wiebe, and D G Cory. Robust online hamiltonian learning. New Journal of Physics, 14 (10): 103013, oct 2012. 10.1088/​1367-2630/​14/​10/​103013. URL https:/​/​​10.1088.

[16] Martin Greiter, Xiao-Gang Wen, and Frank Wilczek. Paired hall state at half filling. Phys. Rev. Lett., 66: 3205–3208, Jun 1991. 10.1103/​PhysRevLett.66.3205. URL https:/​/​​doi/​10.1103/​PhysRevLett.66.3205.

[17] Martin Greiter, X.G. Wen, and Frank Wilczek. Paired hall states. Nuclear Physics B, 374 (3): 567 – 614, 1992. ISSN 0550-3213. https:/​/​​10.1016/​0550-3213(92)90401-V. URL http:/​/​​science/​article/​pii/​055032139290401V.

[18] Martin Greiter, Vera Schnells, and Ronny Thomale. Method to identify parent hamiltonians for trial states. Phys. Rev. B, 98: 081113(R), Aug 2018. 10.1103/​PhysRevB.98.081113. URL https:/​/​​doi/​10.1103/​PhysRevB.98.081113.

[19] F. D. M. Haldane. Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states. Phys. Rev. Lett., 51: 605–608, Aug 1983. 10.1103/​PhysRevLett.51.605. URL https:/​/​​doi/​10.1103/​PhysRevLett.51.605.

[20] Magnus R. Hestenes and Eduard Stiefel. Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards, 49: 409–436 (1953), 1952. ISSN 0160-1741. 10.6028/​jres.049.044.

[21] William Hutzel, John J. McCord, P. T. Raum, Ben Stern, Hao Wang, V. W. Scarola, and Michael R. Peterson. Particle-hole-symmetric model for a paired fractional quantum hall state in a half-filled landau level. Phys. Rev. B, 99: 045126, Jan 2019. 10.1103/​PhysRevB.99.045126. URL https:/​/​​doi/​10.1103/​PhysRevB.99.045126.

[22] Jainendra K. Jain. Composite Fermions. Cambridge University Press, 2007. 10.1017/​CBO9780511607561.

[23] H.J. Kappen. Learning quantum models from quantum or classical data. 2018. https:/​/​​abs/​1803.11278.

[24] J. Kiefer. Sequential minimax search for a maximum. Proc. Amer. Math. Soc., 4: 502–506, 1953. 10.1090/​S0002-9939-1953-0055639-3. URL http:/​/​​journals/​proc/​1953-004-03/​S0002-9939-1953-0055639-3.

[25] Mária Kieferová and Nathan Wiebe. Tomography and generative training with quantum boltzmann machines. Phys. Rev. A, 96: 062327, Dec 2017. 10.1103/​PhysRevA.96.062327. URL https:/​/​​doi/​10.1103/​PhysRevA.96.062327.

[26] Bartosz Kuśmierz and Arkadiusz Wójs. Emergence of jack ground states from two-body pseudopotentials in fractional quantum hall systems. Phys. Rev. B, 97: 245125, Jun 2018. 10.1103/​PhysRevB.97.245125. URL https:/​/​​doi/​10.1103/​PhysRevB.97.245125.

[27] Bartosz Kuśmierz, Arkadiusz Wójs, and G. J. Sreejith. Mean-field approximations for short-range four-body interactions at ${\nu}=\frac{3}{5}$. Phys. Rev. B, 99: 235141, Jun 2019. 10.1103/​PhysRevB.99.235141. URL https:/​/​​doi/​10.1103/​PhysRevB.99.235141.

[28] Hai-Jun Liao, Jin-Guo Liu, Lei Wang, and Tao Xiang. Differentiable programming tensor networks. Phys. Rev. X, 9: 031041, Sep 2019. 10.1103/​PhysRevX.9.031041. URL https:/​/​​doi/​10.1103/​PhysRevX.9.031041.

[29] Gregory Moore and Nicholas Read. Nonabelions in the fractional quantum hall effect. Nuclear Physics B, 360 (2): 362 – 396, 1991. ISSN 0550-3213. https:/​/​​10.1016/​0550-3213(91)90407-O. URL http:/​/​​science/​article/​pii/​055032139190407O.

[30] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. Non-abelian anyons and topological quantum computation. Rev. Mod. Phys., 80: 1083–1159, Sep 2008. 10.1103/​RevModPhys.80.1083. URL https:/​/​​doi/​10.1103/​RevModPhys.80.1083.

[31] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117 – 158, 2014. ISSN 0003-4916. https:/​/​​10.1016/​j.aop.2014.06.013. URL http:/​/​​science/​article/​pii/​S0003491614001596.

[32] Kiryl Pakrouski, Michael R. Peterson, Thierry Jolicoeur, Vito W. Scarola, Chetan Nayak, and Matthias Troyer. Phase diagram of the ${\nu}=5/​2$ fractional quantum hall effect: Effects of landau-level mixing and nonzero width. Phys. Rev. X, 5: 021004, Apr 2015. 10.1103/​PhysRevX.5.021004. URL https:/​/​​doi/​10.1103/​PhysRevX.5.021004.

[33] Kiryl Pakrouski, Matthias Troyer, Yang-Le Wu, Sankar Das Sarma, and Michael R. Peterson. Enigmatic 12/​5 fractional quantum hall effect. Phys. Rev. B, 94: 075108, Aug 2016. 10.1103/​PhysRevB.94.075108. URL https:/​/​​doi/​10.1103/​PhysRevB.94.075108.

[34] Arijeet Pal and David A. Huse. Many-body localization phase transition. Phys. Rev. B, 82: 174411, Nov 2010. 10.1103/​PhysRevB.82.174411. URL https:/​/​​doi/​10.1103/​PhysRevB.82.174411.

[35] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, 2014. 10.1038/​ncomms5213. URL https:/​/​​10.1038/​ncomms5213.

[36] Michael R. Peterson and Chetan Nayak. More realistic hamiltonians for the fractional quantum hall regime in gaas and graphene. Phys. Rev. B, 87: 245129, Jun 2013. 10.1103/​PhysRevB.87.245129. URL https:/​/​​doi/​10.1103/​PhysRevB.87.245129.

[37] Michael R. Peterson, Kwon Park, and S. Das Sarma. Spontaneous particle-hole symmetry breaking in the ${\nu}=5/​2$ fractional quantum hall effect. Phys. Rev. Lett., 101: 156803, Oct 2008. 10.1103/​PhysRevLett.101.156803. URL https:/​/​​doi/​10.1103/​PhysRevLett.101.156803.

[38] E. Polak and G. Ribiere. Note sur la convergence de méthodes de directions conjuguées. Revue française dínformatique et de recherche opérationnelle. Serie rouge, 3: 35–43, 1969. 10.1051/​m2an/​196903r100351. URL http:/​/​​article/​M2AN_1969__3_1_35_0.pdf.

[39] M. J. D. Powell. Nonconvex minimization calculations and the conjugate gradient method. In David F. Griffiths, editor, Numerical Analysis, pages 122–141, Berlin, Heidelberg, 1984. Springer Berlin Heidelberg. ISBN 978-3-540-38881-4. 10.1007/​BFb0099521.

[40] Richard E. Prange and Steven M. Girvin. The Quantum Hall Effect. Springer, New York, NY, 1987. https:/​/​​10.1007/​978-1-4684-0499-9.

[41] Xiao-Liang Qi and Daniel Ranard. Determining a local Hamiltonian from a single eigenstate. Quantum, 3: 159, July 2019. ISSN 2521-327X. 10.22331/​q-2019-07-08-159. URL https:/​/​​10.22331/​q-2019-07-08-159.

[42] N. Read and E. Rezayi. Beyond paired quantum hall states: Parafermions and incompressible states in the first excited landau level. Phys. Rev. B, 59: 8084–8092, Mar 1999. 10.1103/​PhysRevB.59.8084. URL https:/​/​​doi/​10.1103/​PhysRevB.59.8084.

[43] E. H. Rezayi and F. D. M. Haldane. Incompressible paired hall state, stripe order, and the composite fermion liquid phase in half-filled landau levels. Phys. Rev. Lett., 84: 4685–4688, May 2000. 10.1103/​PhysRevLett.84.4685. URL https:/​/​​doi/​10.1103/​PhysRevLett.84.4685.

[44] E. H. Rezayi and N. Read. Non-abelian quantized hall states of electrons at filling factors 12/​5 and 13/​5 in the first excited landau level. Phys. Rev. B, 79: 075306, Feb 2009. 10.1103/​PhysRevB.79.075306. URL https:/​/​​doi/​10.1103/​PhysRevB.79.075306.

[45] Edward H. Rezayi. Landau level mixing and the ground state of the ${\nu}=5/​2$ quantum hall effect. Phys. Rev. Lett., 119: 026801, Jul 2017. 10.1103/​PhysRevLett.119.026801. URL https:/​/​​doi/​10.1103/​PhysRevLett.119.026801.

[46] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. Thermalization and its mechanism for generic isolated quantum systems. Nature, 452 (7189): 854–858, 2008. 10.1038/​nature06838. URL https:/​/​​10.1038/​nature06838.

[47] A. Shabani, M. Mohseni, S. Lloyd, R. L. Kosut, and H. Rabitz. Estimation of many-body quantum hamiltonians via compressive sensing. Phys. Rev. A, 84: 012107, Jul 2011. 10.1103/​PhysRevA.84.012107. URL https:/​/​​doi/​10.1103/​PhysRevA.84.012107.

[48] Steven H. Simon and Edward H. Rezayi. Landau level mixing in the perturbative limit. Phys. Rev. B, 87: 155426, Apr 2013. 10.1103/​PhysRevB.87.155426. URL https:/​/​​doi/​10.1103/​PhysRevB.87.155426.

[49] Steven H. Simon, E. H. Rezayi, and Nigel R. Cooper. Pseudopotentials for multiparticle interactions in the quantum hall regime. Phys. Rev. B, 75: 195306, May 2007. 10.1103/​PhysRevB.75.195306. URL https:/​/​​doi/​10.1103/​PhysRevB.75.195306.

[50] I. Sodemann and A. H. MacDonald. Landau level mixing and the fractional quantum hall effect. Phys. Rev. B, 87: 245425, Jun 2013. 10.1103/​PhysRevB.87.245425. URL https:/​/​​doi/​10.1103/​PhysRevB.87.245425.

[51] Dam Thanh Son. Is the composite fermion a dirac particle? Phys. Rev. X, 5: 031027, Sep 2015. 10.1103/​PhysRevX.5.031027. URL https:/​/​​doi/​10.1103/​PhysRevX.5.031027.

[52] Mark Srednicki. Chaos and quantum thermalization. Phys. Rev. E, 50: 888–901, Aug 1994. 10.1103/​PhysRevE.50.888. URL https:/​/​​doi/​10.1103/​PhysRevE.50.888.

[53] G. J. Sreejith, Yuhe Zhang, and J. K. Jain. Surprising robustness of particle-hole symmetry for composite-fermion liquids. Phys. Rev. B, 96: 125149, Sep 2017. 10.1103/​PhysRevB.96.125149. URL https:/​/​​doi/​10.1103/​PhysRevB.96.125149.

[54] Brian Swingle and Isaac H. Kim. Reconstructing quantum states from local data. Phys. Rev. Lett., 113: 260501, Dec 2014. 10.1103/​PhysRevLett.113.260501. URL https:/​/​​doi/​10.1103/​PhysRevLett.113.260501.

[55] Ryo Tamura and Koji Hukushima. Method for estimating spin-spin interactions from magnetization curves. Phys. Rev. B, 95: 064407, Feb 2017. 10.1103/​PhysRevB.95.064407. URL https:/​/​​doi/​10.1103/​PhysRevB.95.064407.

[56] X. Turkeshi, T. Mendes-Santos, G. Giudici, and M. Dalmonte. Entanglement-guided search for parent hamiltonians. Phys. Rev. Lett., 122: 150606, Apr 2019. 10.1103/​PhysRevLett.122.150606. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.150606.

[57] Agnes Valenti, Evert van Nieuwenburg, Sebastian Huber, and Eliska Greplova. Hamiltonian learning for quantum error correction. Phys. Rev. Research, 1: 033092, Nov 2019. 10.1103/​PhysRevResearch.1.033092. URL https:/​/​​doi/​10.1103/​PhysRevResearch.1.033092.

[58] V. Vedral. The role of relative entropy in quantum information theory. Rev. Mod. Phys., 74: 197–234, Mar 2002. 10.1103/​RevModPhys.74.197. URL https:/​/​​doi/​10.1103/​RevModPhys.74.197.

[59] Alexander Weiße and Holger Fehske. Exact Diagonalization Techniques, pages 529–544. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008. ISBN 978-3-540-74686-7. 10.1007/​978-3-540-74686-7_18. URL https:/​/​​10.1007/​978-3-540-74686-7_18.

[60] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69: 2863–2866, Nov 1992. 10.1103/​PhysRevLett.69.2863. URL https:/​/​​doi/​10.1103/​PhysRevLett.69.2863.

[61] W. Zhu, S. S. Gong, F. D. M. Haldane, and D. N. Sheng. Fractional quantum hall states at ${\nu}=13/​5$ and $12/​5$ and their non-abelian nature. Phys. Rev. Lett., 115: 126805, Sep 2015. 10.1103/​PhysRevLett.115.126805. URL https:/​/​​doi/​10.1103/​PhysRevLett.115.126805.

[62] W. Zhu, Zhoushen Huang, and Yin-Chen He. Reconstructing entanglement hamiltonian via entanglement eigenstates. Phys. Rev. B, 99: 235109, Jun 2019. 10.1103/​PhysRevB.99.235109. URL https:/​/​​doi/​10.1103/​PhysRevB.99.235109.

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