Nearly tight Trotterization of interacting electrons

Yuan Su1,2, Hsin-Yuan Huang1,3, and Earl T. Campbell4

1Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA
2Google Research, Venice, CA 90291, USA
3Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA
4AWS Center for Quantum Computing, Pasadena, CA 91125, USA

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We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the $\eta$-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use $\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)n^{o(1)}$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when $n=\eta^{2-o(1)}$. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

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[1] Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proceedings of the 35th ACM Symposium on Theory of Computing, pages 20–29, 2003. 10.1145/​780542.780546. arXiv:quant-ph/​0301023.

[2] Dong An and Lin Lin. Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm, 2019. arXiv:1909.05500.

[3] Dong An, Di Fang, and Lin Lin. Time-dependent unbounded Hamiltonian simulation with vector norm scaling. Quantum, 5: 459, May 2021. ISSN 2521-327X. 10.22331/​q-2021-05-26-459. arXiv:2012.13105.

[4] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. 10.1126/​science.1113479. arXiv:quant-ph/​0604193.

[5] Ryan Babbush, Jarrod McClean, Dave Wecker, Alán Aspuru-Guzik, and Nathan Wiebe. Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation. Physical Review A, 91 (2): 022311, 2015. 10.1103/​PhysRevA.91.022311. arXiv:1410.8159.

[6] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials. Physical Review X, 8: 011044, Mar 2018. 10.1103/​PhysRevX.8.011044. arXiv:1706.00023.

[7] Ryan Babbush, Dominic W. Berry, Jarrod R. McClean, and Hartmut Neven. Quantum simulation of chemistry with sublinear scaling in basis size. npj Quantum Information, 5 (1): 92, Nov 2019. ISSN 2056-6387. 10.1038/​s41534-019-0199-y. arXiv:1807.09802.

[8] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin Chan. Quantum algorithms for quantum chemistry and quantum materials science. Chemical Reviews, 120 (22): 12685–12717, 2020. 10.1021/​acs.chemrev.9b00829. arXiv:2001.03685.

[9] Dominic W. Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47 (10): 105301, feb 2014. 10.1088/​1751-8113/​47/​10/​105301. arXiv:1010.2745.

[10] Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders. Efficient quantum algorithms for simulating sparse Hamiltonians. Communications in Mathematical Physics, 270 (2): 359–371, 2007. 10.1007/​s00220-006-0150-x. arXiv:quant-ph/​0508139.

[11] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Exponential improvement in precision for simulating sparse Hamiltonians. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 283–292, 2014. 10.1145/​2591796.2591854. arXiv:1312.1414.

[12] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating Hamiltonian dynamics with a truncated Taylor series. Physical Review Letters, 114 (9): 090502, 2015a. 10.1103/​PhysRevLett.114.090502. arXiv:1412.4687.

[13] Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science, pages 792–809, 2015b. 10.1109/​FOCS.2015.54. arXiv:1501.01715.

[14] Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush. Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization. Quantum, 3: 208, December 2019. ISSN 2521-327X. 10.22331/​q-2019-12-02-208. arXiv:1902.02134.

[15] Fernando G. S. L. Brandao and Krysta M. Svore. Quantum speed-ups for solving semidefinite programs. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science, pages 415–426, 2017. 10.1109/​FOCS.2017.45. arXiv:1609.05537.

[16] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. Strategies for solving the Fermi-Hubbard model on near-term quantum computers. Physical Review B, 102: 235122, Dec 2020. 10.1103/​PhysRevB.102.235122. arXiv:1912.06007.

[17] Zhenyu Cai. Resource estimation for quantum variational simulations of the Hubbard model. Physical Review Applied, 14: 014059, Jul 2020. 10.1103/​PhysRevApplied.14.014059. arXiv:1910.02719.

[18] Earl Campbell. Random compiler for fast Hamiltonian simulation. Physical Review Letters, 123: 070503, Aug 2019. 10.1103/​PhysRevLett.123.070503. arXiv:1811.08017.

[19] Earl T. Campbell. Early fault-tolerant simulations of the Hubbard model, 2020. arXiv:2012.09238.

[20] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119 (19): 10856–10915, 2019. 10.1021/​acs.chemrev.8b00803. arXiv:1812.09976.

[21] Chi-Fang Chen, Hsin-Yuan Huang, Richard Kueng, and Joel A. Tropp. Quantum simulation via randomized product formulas: Low gate complexity with accuracy guarantees, 2020. arXiv:2008.11751.

[22] Andrew M. Childs and Yuan Su. Nearly optimal lattice simulation by product formulas. Physical Review Letters, 123: 050503, Aug 2019. 10.1103/​PhysRevLett.123.050503. arXiv:1901.00564.

[23] Andrew M. Childs, Richard Cleve, Enrico Deotto, Edward Farhi, Sam Gutmann, and Daniel A. Spielman. Exponential algorithmic speedup by quantum walk. In Proceedings of the 35th ACM Symposium on Theory of Computing, pages 59–68, 2003. 10.1145/​780542.780552. arXiv:quant-ph/​0209131.

[24] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115 (38): 9456–9461, 2018. 10.1073/​pnas.1801723115. arXiv:0905.0887.

[25] Andrew M. Childs, Aaron Ostrander, and Yuan Su. Faster quantum simulation by randomization. Quantum, 3: 182, September 2019a. ISSN 2521-327X. 10.22331/​q-2019-09-02-182. arXiv:1805.08385.

[26] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. A theory of Trotter error, 2019b. arXiv:1912.08854.

[27] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. Theory of Trotter error with commutator scaling. Physical Review X, 11: 011020, Feb 2021. 10.1103/​PhysRevX.11.011020.

[28] Laura Clinton, Johannes Bausch, and Toby Cubitt. Hamiltonian simulation algorithms for near-term quantum hardware, 2020. arXiv:2003.06886.

[29] Andrew J. Ferris. Fourier transform for fermionic systems and the spectral tensor network. Physical Review Letters, 113: 010401, Jul 2014. 10.1103/​PhysRevLett.113.010401. arXiv:1310.7605.

[30] Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21 (6-7): 467–488, 1982. 10.1007/​BF02650179.

[31] Hrant Gharibyan, Masanori Hanada, Masazumi Honda, and Junyu Liu. Toward simulating superstring/​M-theory on a quantum computer, 2020. arXiv:2011.06573.

[32] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019. ISBN 978-1-4503-6705-9. 10.1145/​3313276.3316366. arXiv:1806.01838.

[33] Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low. Quantum algorithm for simulating real time evolution of lattice Hamiltonians. In Proceedings of the 59th IEEE Symposium on Foundations of Computer Science, pages 350–360, 2018. 10.1109/​FOCS.2018.00041. arXiv:1801.03922.

[34] Stuart Hadfield and Anargyros Papageorgiou. Divide and conquer approach to quantum Hamiltonian simulation. New Journal of Physics, 20 (4): 043003, 2018. 10.1088/​1367-2630/​aab1ef.

[35] Jad C. Halimeh, Haifeng Lang, Julius Mildenberger, Zhang Jiang, and Philipp Hauke. Gauge-symmetry protection using single-body terms, 2020. arXiv:2007.00668.

[36] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103 (15): 150502, 2009. 10.1103/​PhysRevLett.103.150502. arXiv:0811.3171.

[37] Trygve Helgaker, Poul Jørgensen, and Jeppe Olsen. Molecular electronic-structure theory. John Wiley & Sons, 2013. 10.1002/​9781119019572.

[38] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge university press, 2012. 10.1017/​CBO9781139020411.

[39] Stephen P. Jordan, Keith S. M. Lee, and John Preskill. Quantum algorithms for quantum field theories. Science, 336 (6085): 1130–1133, 2012. 10.1126/​science.1217069. arXiv:1111.3633.

[40] Stephen P. Jordan, Keith S. M. Lee, and John Preskill. Quantum algorithms for fermionic quantum field theories, 2014. arXiv:1404.7115.

[41] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. Quantum simulation of electronic structure with linear depth and connectivity. Physical Review Letters, 120 (11): 110501, 2018. 10.1103/​PhysRevLett.120.110501. arXiv:1711.04789.

[42] Ian D. Kivlichan, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, and Ryan Babbush. Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via Trotterization. Quantum, 4: 296, 2020. 10.22331/​q-2020-07-16-296. arXiv:1902.10673.

[43] Tomotaka Kuwahara, Álvaro M. Alhambra, and Anurag Anshu. Improved thermal area law and quasilinear time algorithm for quantum Gibbs states. Physical Review X, 11: 011047, Mar 2021. 10.1103/​PhysRevX.11.011047. arXiv:2007.11174.

[44] J. P. F. LeBlanc, Andrey E. Antipov, Federico Becca, Ireneusz W. Bulik, Garnet Kin-Lic Chan, Chia-Min Chung, Youjin Deng, Michel Ferrero, Thomas M. Henderson, Carlos A. Jiménez-Hoyos, E. Kozik, Xuan-Wen Liu, Andrew J. Millis, N. V. Prokof'ev, Mingpu Qin, Gustavo E. Scuseria, Hao Shi, B. V. Svistunov, Luca F. Tocchio, I. S. Tupitsyn, Steven R. White, Shiwei Zhang, Bo-Xiao Zheng, Zhenyue Zhu, and Emanuel Gull. Solutions of the two-dimensional hubbard model: Benchmarks and results from a wide range of numerical algorithms. Physical Review X, 5: 041041, Dec 2015. 10.1103/​PhysRevX.5.041041. arXiv:1505.02290.

[45] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through tensor hypercontraction, 2020. arXiv:2011.03494.

[46] Lin Lin and Yu Tong. Near-optimal ground state preparation. Quantum, 4: 372, December 2020a. ISSN 2521-327X. 10.22331/​q-2020-12-14-372. arXiv:2002.12508.

[47] Lin Lin and Yu Tong. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum, 4: 361, November 2020b. ISSN 2521-327X. 10.22331/​q-2020-11-11-361. arXiv:1910.14596.

[48] Norbert M. Linke, Sonika Johri, Caroline Figgatt, Kevin A. Landsman, Anne Y. Matsuura, and Christopher Monroe. Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer. Physical Review A, 98: 052334, Nov 2018. 10.1103/​PhysRevA.98.052334. arXiv:1712.08581.

[49] Yi-Xiang Liu, Jordan Hines, Zhi Li, Ashok Ajoy, and Paola Cappellaro. High-fidelity trotter formulas for digital quantum simulation. Physical Review A, 102: 010601, Jul 2020. 10.1103/​PhysRevA.102.010601. arXiv:1903.01654.

[50] Seth Lloyd. Universal quantum simulators. Science, pages 1073–1078, 1996. 10.1126/​science.273.5278.1073.

[51] Guang Hao Low and Isaac L. Chuang. Hamiltonian simulation by uniform spectral amplification, 2017a. arXiv:1707.05391.

[52] Guang Hao Low and Isaac L. Chuang. Optimal Hamiltonian simulation by quantum signal processing. Physical Review Letters, 118: 010501, 2017b. 10.1103/​PhysRevLett.118.010501. arXiv:1606.02685.

[53] Guang Hao Low and Isaac L. Chuang. Hamiltonian simulation by qubitization. Quantum, 3: 163, July 2019. 10.22331/​q-2019-07-12-163. arXiv:1610.06546.

[54] Guang Hao Low and Nathan Wiebe. Hamiltonian simulation in the interaction picture, 2018. arXiv:1805.00675.

[55] Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe. Well-conditioned multiproduct Hamiltonian simulation, 2019. arXiv:1907.11679.

[56] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum computational chemistry. Reviews of Modern Physics, 92 (1): 015003, 2020. 10.1103/​RevModPhys.92.015003. arXiv:1808.10402.

[57] Jarrod R. McClean, Ryan Babbush, Peter J. Love, and Alán Aspuru-Guzik. Exploiting locality in quantum computation for quantum chemistry. The Journal of Physical Chemistry Letters, 5 (24): 4368–4380, 2014. 10.1021/​jz501649m. arXiv:1407.7863.

[58] Richard Meister, Simon C. Benjamin, and Earl T. Campbell. Tailoring term truncations for electronic structure calculations using a linear combination of unitaries, 2020. arXiv:2007.11624.

[59] Mario Motta, Erika Ye, Jarrod R. McClean, Zhendong Li, Austin J. Minnich, Ryan Babbush, and Garnet Kin-Lic Chan. Low rank representations for quantum simulation of electronic structure. npj Quantum Information, 7 (1): 83, May 2021. ISSN 2056-6387. 10.1038/​s41534-021-00416-z. arXiv:1312.2579.

[60] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme. Quantum algorithms for fermionic simulations. Physical Review A, 64: 022319, Jul 2001. 10.1103/​PhysRevA.64.022319. arXiv:cond-mat/​0012334.

[61] Peter Otte. Boundedness properties of fermionic operators. Journal of Mathematical Physics, 51 (8): 083503, 2010. 10.1063/​1.3464264. arXiv:0911.4438.

[62] Yingkai Ouyang, David R. White, and Earl T. Campbell. Compilation by stochastic Hamiltonian sparsification. Quantum, 4: 235, 2020. 10.22331/​q-2020-02-27-235. arXiv:1910.06255.

[63] Tianyi Peng, Aram W. Harrow, Maris Ozols, and Xiaodi Wu. Simulating large quantum circuits on a small quantum computer. Physical Review Letters, 125: 150504, Oct 2020. 10.1103/​PhysRevLett.125.150504. arXiv:1904.00102.

[64] Michael E. Peskin and Daniel V. Schroeder. An introduction to quantum field theory. CRC press, 2018. 10.1201/​9780429503559.

[65] David Poulin, Matthew B. Hastings, Dave Wecker, Nathan Wiebe, Andrew C. Doherty, and Matthias Troyer. The Trotter step size required for accurate quantum simulation of quantum chemistry. Quantum Information and Computation, 15 (5-6): 361–384, 2015. arXiv:1406.4920.

[66] Google AI Quantum and collaborators. Observation of separated dynamics of charge and spin in the Fermi-Hubbard model, 2020. arXiv:2010.07965.

[67] Patrick Rall. Faster coherent quantum algorithms for phase, energy, and amplitude estimation, 2021. arXiv:2103.09717.

[68] Markus Reiher, Nathan Wiebe, Krysta M. Svore, Dave Wecker, and Matthias Troyer. Elucidating reaction mechanisms on quantum computers. Proceedings of the National Academy of Sciences, 114 (29): 7555–7560, 2017. 10.1073/​pnas.1619152114. arXiv:1605.03590.

[69] Burak Şahinoğlu and Rolando D. Somma. Hamiltonian simulation in the low energy subspace, 2020. arXiv:2006.02660.

[70] Nicolas P. D. Sawaya, Tim Menke, Thi Ha Kyaw, Sonika Johri, Alán Aspuru-Guzik, and Gian Giacomo Guerreschi. Resource-efficient digital quantum simulation of $d$-level systems for photonic, vibrational, and spin-$s$ Hamiltonians. npj Quantum Information, 6 (1): 49, Jun 2020. ISSN 2056-6387. 10.1038/​s41534-020-0278-0. arXiv:1909.12847.

[71] Jacob T. Seeley, Martin J. Richard, and Peter J. Love. The Bravyi-Kitaev transformation for quantum computation of electronic structure. The Journal of Chemical Physics, 137 (22): 224109, 2012. 10.1063/​1.4768229. arXiv:1208.5986.

[72] Alexander F. Shaw, Pavel Lougovski, Jesse R. Stryker, and Nathan Wiebe. Quantum algorithms for simulating the lattice Schwinger model. Quantum, 4: 306, August 2020. ISSN 2521-327X. 10.22331/​q-2020-08-10-306. arXiv:2002.11146.

[73] Rolando D. Somma. Quantum simulations of one dimensional quantum systems, 2015. arXiv:1503.06319.

[74] Rolando D. Somma. A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation. Journal of Mathematical Physics, 57: 062202, 2016. 10.1063/​1.4952761. arXiv:1512.03416.

[75] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush. Fault-tolerant quantum simulations of chemistry in first quantization, 2021. arXiv:2105.12767.

[76] Masuo Suzuki. Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics. Journal of Mathematical Physics, 26 (4): 601–612, 1985. 10.1063/​1.526596.

[77] Masuo Suzuki. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Physics Letters A, 146 (6): 319–323, 1990. 10.1016/​0375-9601(90)90962-N.

[78] Borzu Toloui and Peter J. Love. Quantum algorithms for quantum chemistry based on the sparsity of the CI-matrix, 2013. arXiv:1312.2579.

[79] Minh C. Tran, Andrew Y. Guo, Yuan Su, James R. Garrison, Zachary Eldredge, Michael Foss-Feig, Andrew M. Childs, and Alexey V. Gorshkov. Locality and digital quantum simulation of power-law interactions. Physical Review X, 9: 031006, Jul 2019. 10.1103/​PhysRevX.9.031006. arXiv:1808.05225.

[80] Minh C. Tran, Yuan Su, Daniel Carney, and Jacob M. Taylor. Faster digital quantum simulation by symmetry protection. PRX Quantum, 2: 010323, Feb 2021. 10.1103/​PRXQuantum.2.010323. arXiv:2006.16248.

[81] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. Quantum computing enhanced computational catalysis, 2020. arXiv:2007.14460.

[82] Kianna Wan and Isaac Kim. Fast digital methods for adiabatic state preparation, 2020. arXiv:2004.04164.

[83] Dave Wecker, Bela Bauer, Bryan K. Clark, Matthew B. Hastings, and Matthias Troyer. Gate count estimates for performing quantum chemistry on small quantum computers. Physical Review A, 90: 022305, Aug 2014. 10.1103/​PhysRevA.90.022305. arXiv:1312.1695.

[84] Dave Wecker, Matthew B Hastings, Nathan Wiebe, Bryan K Clark, Chetan Nayak, and Matthias Troyer. Solving strongly correlated electron models on a quantum computer. Physical Review A, 92 (6): 062318, 2015. 10.1103/​PhysRevA.92.062318. arXiv:1506.05135.

[85] James D. Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik. Simulation of electronic structure hamiltonians using quantum computers. Molecular Physics, 109 (5): 735–750, 2011. 10.1080/​00268976.2011.552441. arXiv:1001.3855.

[86] Gian Carlo Wick. The evaluation of the collision matrix. Physical Review, 80: 268–272, Oct 1950. 10.1103/​PhysRev.80.268.

[87] Shenglong Xu, Leonard Susskind, Yuan Su, and Brian Swingle. A sparse model of quantum holography, 2020. arXiv:2008.02303.

[88] Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin, Reinhard M Noack, Hao Shi, Steven R White, Shiwei Zhang, and Garnet Kin-Lic Chan. Stripe order in the underdoped region of the two-dimensional Hubbard model. Science, 358 (6367): 1155–1160, 2017. 10.1126/​science.aam7127. arXiv:1701.00054.

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