Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi1,2 and Daniel Ranard1

1Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305 USA
2Institute for Advanced Study, Princeton NJ 08540 USA

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We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``$\textit{k}$-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

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[1] J. R. Garrison and T. Grover, ``Does a single eigenstate encode the full hamiltonian d?,'' Physical Review X, vol. 8, no. 2, p. 021026, 2018. https:/​/​​10.1103/​PhysRevX.8.021026.

[2] J. M. Deutsch, ``Quantum statistical mechanics in a closed system,'' Physical Review A, vol. 43, no. 4, p. 2046, 1991. https:/​/​​10.1103/​PhysRevA.43.2046.

[3] M. Srednicki, ``Chaos and quantum thermalization,'' Physical Review E, vol. 50, no. 2, p. 888, 1994. https:/​/​​10.1103/​PhysRevE.50.888.

[4] H. L. Haselgrove, M. A. Nielsen, and T. J. Osborne, ``Quantum states far from the energy eigenstates of any local hamiltonian,'' Physical review letters, vol. 91, no. 21, p. 210401, 2003. https:/​/​​10.1103/​PhysRevLett.91.210401.

[5] J. Chen, Z. Ji, Z. Wei, and B. Zeng, ``Correlations in excited states of local hamiltonians,'' Phys. Rev. A, vol. 85, p. 040303, Apr 2012. https:/​/​​10.1103/​PhysRevA.85.040303.

[6] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, ``Efficient quantum state tomography,'' Nature communications, vol. 1, p. 149, 2010. https:/​/​​10.1038/​ncomms1147.

[7] B. Swingle and I. H. Kim, ``Reconstructing quantum states from local data,'' Physical review letters, vol. 113, no. 26, p. 260501, 2014. https:/​/​​10.1103/​PhysRevLett.113.260501.

[8] V. Zauner, D. Draxler, L. Vanderstraeten, J. Haegeman, and F. Verstraete, ``Symmetry breaking and the geometry of reduced density matrices,'' New Journal of Physics, vol. 18, no. 11, p. 113033, 2016. https:/​/​​10.1088/​1367-2630/​18/​11/​113033.

[9] H. Kim, M. C. Bañuls, J. I. Cirac, M. B. Hastings, and D. A. Huse, ``Slowest local operators in quantum spin chains,'' Physical Review E, vol. 92, no. 1, p. 012128, 2015. https:/​/​​10.1103/​PhysRevE.92.012128.

[10] T. O'Brien, D. A. Abanin, G. Vidal, and Z. Papić, ``Explicit construction of local conserved operators in disordered many-body systems,'' Physical Review B, vol. 94, no. 14, p. 144208, 2016. https:/​/​​10.1103/​PhysRevB.94.144208.

[11] P. Zanardi, P. Giorda, and M. Cozzini, ``Information-theoretic differential geometry of quantum phase transitions,'' Physical review letters, vol. 99, no. 10, p. 100603, 2007. https:/​/​​10.1103/​PhysRevLett.99.100603.

[12] C. Davis and W. M. Kahan, ``The rotation of eigenvectors by a perturbation. iii,'' SIAM Journal on Numerical Analysis, vol. 7, no. 1, pp. 1–46, 1970. https:/​/​​10.1137/​0707001.

[13] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, ``Approximating gibbs states of local hamiltonians efficiently with projected entangled pair states,'' Physical review b, vol. 91, no. 4, p. 045138, 2015. https:/​/​​10.1103/​PhysRevB.91.045138.

[14] J. Cho, ``Correlations in quantum spin systems from the boundary effect,'' New Journal of Physics, vol. 17, no. 5, p. 053021, 2015. https:/​/​​10.1088/​1367-2630/​17/​5/​053021.

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[1] Shi-Yao Hou, Ningping Cao, Sirui Lu, Yi Shen, Yiu-Tung Poon, and Bei Zeng, "Determining system Hamiltonian from eigenstate measurements without correlation functions", New Journal of Physics 22 8, 083088 (2020).

[2] Zhi Li, Liujun Zou, and Timothy H. Hsieh, "Hamiltonian Tomography via Quantum Quench", Physical Review Letters 124 16, 160502 (2020).

[3] Vedran Dunjko, "Inside quantum black boxes", Nature Physics 17 8, 880 (2021).

[4] Eli Chertkov and Bryan K. Clark, "Motif magnetism and quantum many-body scars", Physical Review B 104 10, 104410 (2021).

[5] Xhek Turkeshi and Marcello Dalmonte, "Parent Hamiltonian reconstruction of Jastrow-Gutzwiller wavefunctions", SciPost Physics 8 3, 042 (2020).

[6] Tasio Gonzalez-Raya, Rodrigo Asensio-Perea, Ana Martin, Lucas C. Céleri, Mikel Sanz, Pavel Lougovski, and Eugene F. Dumitrescu, "Digital-Analog Quantum Simulations Using the Cross-Resonance Effect", PRX Quantum 2 2, 020328 (2021).

[7] Cécile Monthus, "Construction of many-body-localized models where all the eigenstates are matrix-product-states", Journal of Statistical Mechanics: Theory and Experiment 2020 8, 083301 (2020).

[8] Liangyu Che, Chao Wei, Yulei Huang, Dafa Zhao, Shunzhong Xue, Xinfang Nie, Jun Li, Dawei Lu, and Tao Xin, "Learning quantum Hamiltonians from single-qubit measurements", Physical Review Research 3 2, 023246 (2021).

[9] Eugene F. Dumitrescu and Pavel Lougovski, "Hamiltonian assignment for open quantum systems", Physical Review Research 2 3, 033251 (2020).

[10] Pavan Hosur, "Polynomial-time algorithm for studying physical observables in chaotic eigenstates", Physical Review B 103 19, 195159 (2021).

[11] Agnes Valenti, Evert van Nieuwenburg, Sebastian Huber, and Eliska Greplova, "Hamiltonian learning for quantum error correction", Physical Review Research 1 3, 033092 (2019).

[12] Eyal Bairey, Chu Guo, Dario Poletti, Netanel H Lindner, and Itai Arad, "Learning the dynamics of open quantum systems from their steady states", New Journal of Physics 22 3, 032001 (2020).

[13] Kiryl Pakrouski, "Automatic design of Hamiltonians", Quantum 4, 315 (2020).

[14] Nicholas O'Dea, Fiona Burnell, Anushya Chandran, and Vedika Khemani, "From tunnels to towers: Quantum scars from Lie algebras and q -deformed Lie algebras", Physical Review Research 2 4, 043305 (2020).

[15] Maxime Dupont, Nicolas Macé, and Nicolas Laflorencie, "From eigenstate to Hamiltonian: Prospects for ergodicity and localization", Physical Review B 100 13, 134201 (2019).

[16] Chenfeng Cao, Shi-Yao Hou, Ningping Cao, and Bei Zeng, "Supervised learning in Hamiltonian reconstruction from local measurements on eigenstates", Journal of Physics: Condensed Matter 33 6, 064002 (2020).

[17] Christian Kokail, Rick van Bijnen, Andreas Elben, Benoît Vermersch, and Peter Zoller, "Entanglement Hamiltonian tomography in quantum simulation", Nature Physics 17 8, 936 (2021).

[18] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller, "Theoretical and Experimental Perspectives of Quantum Verification", PRX Quantum 2 1, 010102 (2021).

[19] Zhoushen Huang, Aashish Clerk, and Ivar Martin, "Nondispersing Wave Packets in Lattice Floquet Systems", Physical Review Letters 126 10, 100601 (2021).

[20] Michael Matty, Yi Zhang, T. Senthil, and Eun-Ah Kim, "Entanglement clustering for ground-stateable quantum many-body states", Physical Review Research 3 2, 023212 (2021).

[21] Eli Chertkov, Benjamin Villalonga, and Bryan K. Clark, "Engineering topological models with a general-purpose symmetry-to-Hamiltonian approach", Physical Review Research 2 2, 023348 (2020).

[22] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar, "Sample-efficient learning of interacting quantum systems", Nature Physics 17 8, 931 (2021).

[23] Davide Rattacaso, Gianluca Passarelli, Antonio Mezzacapo, Procolo Lucignano, and Rosario Fazio, "Optimal parent Hamiltonians for time-dependent states", Physical Review A 104 2, 022611 (2021).

[24] W. Zhu, Zhoushen Huang, Yin-Chen He, and Xueda Wen, "Entanglement Hamiltonian of Many-Body Dynamics in Strongly Correlated Systems", Physical Review Letters 124 10, 100605 (2020).

[25] Enrico M. Brehm, Diptarka Das, and Shouvik Datta, "Probing thermality beyond the diagonal", Physical Review D 98 12, 126015 (2018).

[26] Eyal Bairey, Itai Arad, and Netanel H. Lindner, "Learning a Local Hamiltonian from Local Measurements", arXiv:1807.04564, Physical Review Letters 122 2, 020504 (2019).

[27] Eli Chertkov and Bryan K. Clark, "Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions", Physical Review X 8 3, 031029 (2018).

[28] W. Zhu, Zhoushen Huang, and Yin-Chen He, "Reconstructing entanglement Hamiltonian via entanglement eigenstates", Physical Review B 99 23, 235109 (2019).

[29] Martin Greiter, Vera Schnells, and Ronny Thomale, "Method to identify parent Hamiltonians for trial states", Physical Review B 98 8, 081113 (2018).

[30] X. Turkeshi, T. Mendes-Santos, G. Giudici, and M. Dalmonte, "Entanglement-Guided Search for Parent Hamiltonians", Physical Review Letters 122 15, 150606 (2019).

[31] Maxime Dupont and Nicolas Laflorencie, "Many-body localization as a large family of localized ground states", arXiv:1807.01313, Physical Review B 99 2, 020202 (2019).

[32] G. J. Sreejith, M. Fremling, Gun Sang Jeon, and J. K. Jain, "Search for exact local Hamiltonians for general fractional quantum Hall states", Physical Review B 98 23, 235139 (2018).

[33] Keith R. Fratus and Syrian V. Truong, "Does a Single Eigenstate of a Hamiltonian Encode the Critical Behaviour of its Finite-Temperature Phase Transition?", arXiv:1810.11092.

[34] Xu-Yang Hou, Ziwen Huang, Hao Guo, Yan He, and Chih-Chun Chien, "BCS thermal vacuum of fermionic superfluids and its perturbation theory", Scientific Reports 8, 11995 (2018).

[35] Vinit Kumar Singh and Jung Hoon Han, "Statistical recovery of the classical spin Hamiltonian", arXiv:1807.04884.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-20 00:24:33) and SAO/NASA ADS (last updated successfully 2021-10-20 00:24:35). The list may be incomplete as not all publishers provide suitable and complete citation data.