Efficient classical simulation of noisy random quantum circuits in one dimension
1Department of Physics, Yale University, New Haven, Connecticut 06520, USA
2AWS Center for Quantum Computing, Pasadena, CA, 91125, USA
3Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
4Department of Computer Science, University of Chicago, Chicago, Illinois 60637, USA
| Published: | 2020-09-11, volume 4, page 318 |
| Eprint: | arXiv:2003.13163v3 |
| Doi: | https://doi.org/10.22331/q-2020-09-11-318 |
| Citation: | Quantum 4, 318 (2020). |
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Abstract
Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.
Popular summary
Specifically, we study 1D noisy random circuit sampling as a simple model for exploring the adverse effects of the realistic gate errors. The key takeaway from our work is that in noisy settings, there exists a characteristic system size, determined solely by the gate error rate, above which adding more qubits does not bring about an exponential growth of the computing power of a noisy quantum system. That is, quality limits the utility of quantity. In particular, we demonstrate that matrix product operators are able to describe the dynamics of a 1D noisy system in a reliable and compressed way.
Our work provides a framework for assessing the utility of near-term quantum computing technologies based on noisy intermediate-scale quantum (NISQ) devices. Going beyond the chain architecture and investigating more general settings (e.g., planar architecture) would be a fruitful future research direction.
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► References
[1] P. W. Shor, ``Algorithms for quantum computation: discrete logarithms and factoring,'' in Proceedings 35th Annual Symposium on Foundations of Computer Science (1994) pp. 124–134.
https://doi.org/10.1109/SFCS.1994.365700
[2] S. Lloyd, ``Universal quantum simulators,'' Science 273, 1073–1078 (1996).
https://doi.org/10.1126/science.273.5278.1073
[3] P. W. Shor, ``Fault-tolerant quantum computation,'' in Proceedings of 37th Conference on Foundations of Computer Science (1996) pp. 56–65.
https://doi.org/10.1109/SFCS.1996.548464
[4] D. Gottesman, ``Fault-tolerant quantum computation with local gates,'' Journal of Modern Optics 47, 333–345 (2000).
https://doi.org/10.1080/09500340008244046
[5] S. Bravyi and A. Kitaev, ``Universal quantum computation with ideal clifford gates and noisy ancillas,'' Phys. Rev. A 71, 022316 (2005).
https://doi.org/10.1103/PhysRevA.71.022316
[6] E. Knill, ``Quantum computing with realistically noisy devices,'' Nature 434, 39–44 (2005).
https://doi.org/10.1038/nature03350
[7] D. Gottesman, ``An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation,'' arXiv e-prints , arXiv:0904.2557 (2009), arXiv:0904.2557 [quant-ph].
arXiv:0904.2557
[8] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, ``Surface codes: Towards practical large-scale quantum computation,'' Phys. Rev. A 86, 032324 (2012a).
https://doi.org/10.1103/PhysRevA.86.032324
[9] A. G. Fowler, A. C. Whiteside, A. L. McInnes, and A. Rabbani, ``Topological code autotune,'' Phys. Rev. X 2, 041003 (2012b).
https://doi.org/10.1103/PhysRevX.2.041003
[10] S. Bravyi and J. Haah, ``Magic-state distillation with low overhead,'' Phys. Rev. A 86, 052329 (2012).
https://doi.org/10.1103/PhysRevA.86.052329
[11] C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, ``Surface code quantum computing by lattice surgery,'' New Journal of Physics 14, 123011 (2012).
https://doi.org/10.1088/1367-2630/14/12/123011
[12] J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, ``Magic state distillation with low space overhead and optimal asymptotic input count,'' Quantum 1, 31 (2017).
https://doi.org/10.22331/q-2017-10-03-31
[13] C. Chamberland and M. E. Beverland, ``Flag fault-tolerant error correction with arbitrary distance codes,'' Quantum 2, 53 (2018).
https://doi.org/10.22331/q-2018-02-08-53
[14] R. Chao and B. W. Reichardt, ``Quantum error correction with only two extra qubits,'' Phys. Rev. Lett. 121, 050502 (2018a).
https://doi.org/10.1103/PhysRevLett.121.050502
[15] R. Chao and B. W. Reichardt, ``Fault-tolerant quantum computation with few qubits,'' npj Quantum Information 4, 42 (2018b).
https://doi.org/10.1038/s41534-018-0085-z
[16] C. Chamberland and A. W. Cross, ``Fault-tolerant magic state preparation with flag qubits,'' Quantum 3, 143 (2019).
https://doi.org/10.22331/q-2019-05-20-143
[17] D. Litinski, ``Magic State Distillation: Not as Costly as You Think,'' Quantum 3, 205 (2019).
https://doi.org/10.22331/q-2019-12-02-205
[18] R. Chao and B. W. Reichardt, ``Flag fault-tolerant error correction for any stabilizer code,'' PRX Quantum 1, 010302 (2020).
https://doi.org/10.1103/PRXQuantum.1.010302
[19] C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, ``Topological and subsystem codes on low-degree graphs with flag qubits,'' Phys. Rev. X 10, 011022 (2020a).
https://doi.org/10.1103/PhysRevX.10.011022
[20] C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, ``Triangular color codes on trivalent graphs with flag qubits,'' New Journal of Physics 22, 023019 (2020b).
https://doi.org/10.1088/1367-2630/ab68fd
[21] P. Das, C. A. Pattison, S. Manne, D. Carmean, K. Svore, M. Qureshi, and N. Delfosse, ``A Scalable Decoder Micro-architecture for Fault-Tolerant Quantum Computing,'' arXiv e-prints , arXiv:2001.06598 (2020), arXiv:2001.06598 [quant-ph].
arXiv:2001.06598
[22] N. Delfosse, ``Hierarchical decoding to reduce hardware requirements for quantum computing,'' arXiv e-prints , arXiv:2001.11427 (2020), arXiv:2001.11427 [quant-ph].
arXiv:2001.11427
[23] N. Delfosse, B. W. Reichardt, and K. M. Svore, ``Beyond single-shot fault-tolerant quantum error correction,'' arXiv e-prints , arXiv:2002.05180 (2020), arXiv:2002.05180 [quant-ph].
arXiv:2002.05180
[24] C. Chamberland and K. Noh, ``Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits,'' arXiv e-prints , arXiv:2003.03049 (2020), arXiv:2003.03049 [quant-ph].
arXiv:2003.03049
[25] J. Preskill, ``Quantum Computing in the NISQ era and beyond,'' Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79
[26] M. J. Bremner, R. Jozsa, and D. J. Shepherd, ``Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy,'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, 459–472 (2011).
https://doi.org/10.1098/rspa.2010.0301
[27] S. Aaronson and A. Arkhipov, ``The computational complexity of linear optics,'' in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC ’11 (Association for Computing Machinery, New York, NY, USA, 2011) p. 333–342.
https://doi.org/10.1145/1993636.1993682
[28] C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, ``Gaussian boson sampling,'' Phys. Rev. Lett. 119, 170501 (2017).
https://doi.org/10.1103/PhysRevLett.119.170501
[29] B. Fefferman and C. Umans, ``On the Power of Quantum Fourier Sampling,'' in 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 61, edited by A. Broadbent (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016) pp. 1:1–1:19.
https://doi.org/10.4230/LIPIcs.TQC.2016.1
[30] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, ``Characterizing quantum supremacy in near-term devices,'' Nature Physics 14, 595–600 (2018).
https://doi.org/10.1038/s41567-018-0124-x
[31] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, ``Photonic boson sampling in a tunable circuit,'' Science 339, 794–798 (2013).
https://doi.org/10.1126/science.1231440
[32] J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, ``Boson sampling on a photonic chip,'' Science 339, 798–801 (2013).
https://doi.org/10.1126/science.1231692
[33] M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, ``Experimental boson sampling,'' Nature Photonics 7, 540–544 (2013).
https://doi.org/10.1038/nphoton.2013.102
[34] A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ``Integrated multimode interferometers with arbitrary designs for photonic boson sampling,'' Nature Photonics 7, 545–549 (2013).
https://doi.org/10.1038/nphoton.2013.112
[35] A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro, and A. Laing, ``Classical boson sampling algorithms with superior performance to near-term experiments,'' Nature Physics 13, 1153–1157 (2017).
https://doi.org/10.1038/nphys4270
[36] P. Clifford and R. Clifford, ``The classical complexity of boson sampling,'' in Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 146–155.
https://doi.org/10.1137/1.9781611975031.10
[37] J. Renema, V. Shchesnovich, and R. Garcia-Patron, ``Classical simulability of noisy boson sampling,'' arXiv e-prints , arXiv:1809.01953 (2018), arXiv:1809.01953 [quant-ph].
arXiv:1809.01953
[38] R. García-Patrón, J. J. Renema, and V. Shchesnovich, ``Simulating boson sampling in lossy architectures,'' Quantum 3, 169 (2019).
https://doi.org/10.22331/q-2019-08-05-169
[39] A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, ``On the complexity and verification of quantum random circuit sampling,'' Nature Physics 15, 159–163 (2019).
https://doi.org/10.1038/s41567-018-0318-2
[40] S. Aaronson and L. Chen, ``Complexity-Theoretic Foundations of Quantum Supremacy Experiments,'' in 32nd Computational Complexity Conference (CCC 2017), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 79, edited by R. O'Donnell (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017) pp. 22:1–22:67.
https://doi.org/10.4230/LIPIcs.CCC.2017.22
[41] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, ``Quantum supremacy using a programmable superconducting processor,'' Nature 574, 505–510 (2019).
https://doi.org/10.1038/s41586-019-1666-5
[42] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, and R. Wisnieff, ``Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits,'' arXiv e-prints , arXiv:1910.09534 (2019), arXiv:1910.09534 [quant-ph].
arXiv:1910.09534
[43] R. Movassagh, ``Efficient unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling,'' arXiv e-prints , arXiv:1810.04681 (2018), arXiv:1810.04681 [quant-ph].
arXiv:1810.04681
[44] R. Movassagh, ``Quantum supremacy and random circuits,'' arXiv e-prints , arXiv:1909.06210 (2019), arXiv:1909.06210 [quant-ph].
arXiv:1909.06210
[45] G. Vidal, ``Efficient classical simulation of slightly entangled quantum computations,'' Phys. Rev. Lett. 91, 147902 (2003).
https://doi.org/10.1103/PhysRevLett.91.147902
[46] Y. Zhou, E. M. Stoudenmire, and X. Waintal, ``What limits the simulation of quantum computers?'' arXiv e-prints , arXiv:2002.07730 (2020), arXiv:2002.07730 [quant-ph].
arXiv:2002.07730
[47] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, ``Matrix product density operators: Simulation of finite-temperature and dissipative systems,'' Phys. Rev. Lett. 93, 207204 (2004).
https://doi.org/10.1103/PhysRevLett.93.207204
[48] M. Zwolak and G. Vidal, ``Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm,'' Phys. Rev. Lett. 93, 207205 (2004).
https://doi.org/10.1103/PhysRevLett.93.207205
[49] M.-D. Choi, ``Completely positive linear maps on complex matrices,'' Linear Algebra and its Applications 10, 285 – 290 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0
[50] J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. Laflamme, ``Symmetrized characterization of noisy quantum processes,'' Science 317, 1893–1896 (2007).
https://doi.org/10.1126/science.1145699
[51] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, ``Concentrating partial entanglement by local operations,'' Phys. Rev. A 53, 2046–2052 (1996a).
https://doi.org/10.1103/PhysRevA.53.2046
[52] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, ``Mixed-state entanglement and quantum error correction,'' Phys. Rev. A 54, 3824–3851 (1996b).
https://doi.org/10.1103/PhysRevA.54.3824
[53] B. M. Terhal, M. Horodecki, D. W. Leung, and D. P. DiVincenzo, ``The entanglement of purification,'' Journal of Mathematical Physics 43, 4286–4298 (2002), https://doi.org/10.1063/1.1498001.
https://doi.org/10.1063/1.1498001
arXiv:https://doi.org/10.1063/1.1498001
[54] J. Guth Jarkovský, A. Molnár, N. Schuch, and J. I. Cirac, ``Efficient description of many-body systems with matrix product density operators,'' PRX Quantum 1, 010304 (2020).
https://doi.org/10.1103/PRXQuantum.1.010304
[55] T. Prosen and I. Pižorn, ``Operator space entanglement entropy in a transverse ising chain,'' Phys. Rev. A 76, 032316 (2007).
https://doi.org/10.1103/PhysRevA.76.032316
[56] T. Prosen and M. Žnidarič, ``Matrix product simulations of non-equilibrium steady states of quantum spin chains,'' Journal of Statistical Mechanics: Theory and Experiment 2009, P02035 (2009).
https://doi.org/10.1088/1742-5468/2009/02/p02035
[57] S. Xu and B. Swingle, ``Accessing scrambling using matrix product operators,'' Nature Physics 16, 199–204 (2020).
https://doi.org/10.1038/s41567-019-0712-4
[58] P. Zanardi, ``Entanglement of quantum evolutions,'' Phys. Rev. A 63, 040304 (2001).
https://doi.org/10.1103/PhysRevA.63.040304
[59] D. Aharonov and M. Ben-Or, ``Polynomial simulations of decohered quantum computers,'' in Proceedings of 37th Conference on Foundations of Computer Science (1996) pp. 46–55.
https://doi.org/10.1109/SFCS.1996.548463
[60] A. W. Harrow and M. A. Nielsen, ``Robustness of quantum gates in the presence of noise,'' Phys. Rev. A 68, 012308 (2003).
https://doi.org/10.1103/PhysRevA.68.012308
[61] D. N. Page, ``Average entropy of a subsystem,'' Phys. Rev. Lett. 71, 1291–1294 (1993).
https://doi.org/10.1103/PhysRevLett.71.1291
[62] S. K. Foong and S. Kanno, ``Proof of page's conjecture on the average entropy of a subsystem,'' Phys. Rev. Lett. 72, 1148–1151 (1994).
https://doi.org/10.1103/PhysRevLett.72.1148
[63] J. Sánchez-Ruiz, ``Simple proof of page's conjecture on the average entropy of a subsystem,'' Phys. Rev. E 52, 5653–5655 (1995).
https://doi.org/10.1103/PhysRevE.52.5653
[64] S. Sen, ``Average entropy of a quantum subsystem,'' Phys. Rev. Lett. 77, 1–3 (1996).
https://doi.org/10.1103/PhysRevLett.77.1
[65] A. Dang, Distributed Matrix Product State Simulations of Large-Scale Quantum Circuits, Master's thesis, The University of Melbourne (2017).
[66] F. Verstraete and J. I. Cirac, ``Matrix product states represent ground states faithfully,'' Phys. Rev. B 73, 094423 (2006).
https://doi.org/10.1103/PhysRevB.73.094423
[67] G. Vidal, ``Class of quantum many-body states that can be efficiently simulated,'' Phys. Rev. Lett. 101, 110501 (2008).
https://doi.org/10.1103/PhysRevLett.101.110501
[68] Y. Li, X. Chen, and M. P. A. Fisher, ``Quantum zeno effect and the many-body entanglement transition,'' Phys. Rev. B 98, 205136 (2018).
https://doi.org/10.1103/PhysRevB.98.205136
[69] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, ``Unitary-projective entanglement dynamics,'' Phys. Rev. B 99, 224307 (2019).
https://doi.org/10.1103/PhysRevB.99.224307
[70] B. Skinner, J. Ruhman, and A. Nahum, ``Measurement-induced phase transitions in the dynamics of entanglement,'' Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009
[71] Y. Li, X. Chen, and M. P. A. Fisher, ``Measurement-driven entanglement transition in hybrid quantum circuits,'' Phys. Rev. B 100, 134306 (2019).
https://doi.org/10.1103/PhysRevB.100.134306
[72] M. Szyniszewski, A. Romito, and H. Schomerus, ``Entanglement transition from variable-strength weak measurements,'' Phys. Rev. B 100, 064204 (2019).
https://doi.org/10.1103/PhysRevB.100.064204
[73] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, ``Quantum error correction in scrambling dynamics and measurement-induced phase transition,'' Phys. Rev. Lett. 125, 030505 (2020).
https://doi.org/10.1103/PhysRevLett.125.030505
[74] M. J. Gullans and D. A. Huse, ``Dynamical purification phase transition induced by quantum measurements,'' arXiv e-prints , arXiv:1905.05195 (2019), arXiv:1905.05195 [quant-ph].
arXiv:1905.05195
[75] M. J. Gullans and D. A. Huse, ``Scalable probes of measurement-induced criticality,'' Phys. Rev. Lett. 125, 070606 (2020).
https://doi.org/10.1103/PhysRevLett.125.070606
[76] A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, ``Critical properties of the measurement-induced transition in random quantum circuits,'' Phys. Rev. B 101, 060301 (2020).
https://doi.org/10.1103/PhysRevB.101.060301
[77] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, ``Self-Organized Error Correction in Random Unitary Circuits with Measurement,'' arXiv e-prints , arXiv:2002.12385 (2020), arXiv:2002.12385 [cond-mat.stat-mech].
arXiv:2002.12385
[78] Y. Bao, S. Choi, and E. Altman, ``Theory of the phase transition in random unitary circuits with measurements,'' Phys. Rev. B 101, 104301 (2020).
https://doi.org/10.1103/PhysRevB.101.104301
[79] C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, ``Measurement-induced criticality in random quantum circuits,'' Phys. Rev. B 101, 104302 (2020).
https://doi.org/10.1103/PhysRevB.101.104302
[80] A. Bera and S. S. Roy, ``Growth of genuine multipartite entanglement in random unitary circuits,'' arXiv e-prints , arXiv:2003.12546 (2020), arXiv:2003.12546 [quant-ph].
arXiv:2003.12546
[81] Y. Li, X. Chen, A. W. W. Ludwig, and M. P. A. Fisher, ``Conformal invariance and quantum non-locality in hybrid quantum circuits,'' arXiv e-prints , arXiv:2003.12721 (2020), arXiv:2003.12721 [quant-ph].
arXiv:2003.12721
[82] D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology (1997).
https://doi.org/10.7907/rzr7-dt72
[83] D. Gottesman, ``The Heisenberg Representation of Quantum Computers,'' arXiv e-prints , quant-ph/9807006 (1998), arXiv:quant-ph/9807006 [quant-ph].
arXiv:quant-ph/9807006
[84] S. Aaronson and D. Gottesman, ``Improved simulation of stabilizer circuits,'' Phys. Rev. A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328
[85] L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, ``Exact dynamics in dual-unitary quantum circuits,'' Phys. Rev. B 101, 094304 (2020).
https://doi.org/10.1103/PhysRevB.101.094304
[86] D. Aharonov, ``Quantum to classical phase transition in noisy quantum computers,'' Phys. Rev. A 62, 062311 (2000).
https://doi.org/10.1103/PhysRevA.62.062311
[87] M. J. Bremner, A. Montanaro, and D. J. Shepherd, ``Achieving quantum supremacy with sparse and noisy commuting quantum computations,'' Quantum 1, 8 (2017).
https://doi.org/10.22331/q-2017-04-25-8
[88] M.-H. Yung and X. Gao, ``Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise?'' arXiv e-prints , arXiv:1706.08913 (2017), arXiv:1706.08913 [quant-ph].
arXiv:1706.08913
[89] X. Gao and L. Duan, ``Efficient classical simulation of noisy quantum computation,'' arXiv e-prints , arXiv:1810.03176 (2018), arXiv:1810.03176 [quant-ph].
arXiv:1810.03176
[90] S. Boixo, V. N. Smelyanskiy, and H. Neven, ``Fourier analysis of sampling from noisy chaotic quantum circuits,'' arXiv e-prints , arXiv:1708.01875 (2017), arXiv:1708.01875 [quant-ph].
arXiv:1708.01875
[91] F. Verstraete and J. I. Cirac, ``Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions,'' arXiv e-prints , cond-mat/0407066 (2004), arXiv:cond-mat/0407066 [cond-mat.str-el].
arXiv:cond-mat/0407066
[92] F. Verstraete and J. I. Cirac, ``Valence-bond states for quantum computation,'' Phys. Rev. A 70, 060302 (2004).
https://doi.org/10.1103/PhysRevA.70.060302
[93] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, ``Computational complexity of projected entangled pair states,'' Phys. Rev. Lett. 98, 140506 (2007).
https://doi.org/10.1103/PhysRevLett.98.140506
[94] J. Haferkamp, D. Hangleiter, J. Eisert, and M. Gluza, ``Contracting projected entangled pair states is average-case hard,'' Phys. Rev. Research 2, 013010 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013010
[95] J. Napp, R. L. La Placa, A. M. Dalzell, F. G. S. L. Brandao, and A. W. Harrow, ``Efficient classical simulation of random shallow 2D quantum circuits,'' arXiv e-prints , arXiv:2001.00021 (2019), arXiv:2001.00021 [quant-ph].
arXiv:2001.00021
[96] U. Schollwöck, ``The density-matrix renormalization group in the age of matrix product states,'' Annals of Physics 326, 96 – 192 (2011), january 2011 Special Issue.
https://doi.org/10.1016/j.aop.2010.09.012
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[1] James Allen, Matthew Otten, Stephen K. Gray, and Bryan K. Clark, "Simulating large one-dimensional neutral-atom quantum systems", Physical Review A 112 2, 022609 (2025).
[2] Minzhao Liu, Changhun Oh, Junyu Liu, Liang Jiang, and Yuri Alexeev, "Simulating lossy Gaussian boson sampling with matrix-product operators", Physical Review A 108 5, 052604 (2023).
[3] Meng Zhang, Chao Wang, Shaojun Dong, Hao Zhang, Yongjian Han, and Lixin He, "Entanglement entropy scaling of noisy random quantum circuits in two dimensions", Physical Review A 106 5, 052430 (2022).
[4] Changhun Oh, Minzhao Liu, Yuri Alexeev, Bill Fefferman, and Liang Jiang, "Classical algorithm for simulating experimental Gaussian boson sampling", Nature Physics 20 9, 1461 (2024).
[5] Sahar Atallah, Michael Garn, Sania Jevtic, Yukuan Tao, and Shashank Virmani, "Efficient classical simulation of cluster state quantum circuits with alternative inputs", Quantum 8, 1243 (2024).
[6] Thomas Schuster, Chao Yin, Xun Gao, and Norman Y. Yao, "A Polynomial-Time Classical Algorithm for Noisy Quantum Circuits", Physical Review X 15 4, 041018 (2025).
[7] Wei-guo Ma, Yun-Hao Shi, Kai Xu, and Heng Fan, "Tomography-assisted noisy quantum circuit simulator using matrix product density operators", Physical Review A 110 3, 032604 (2024).
[8] Elmer V. H. Doggen, Yuval Gefen, Igor V. Gornyi, Alexander D. Mirlin, and Dmitry G. Polyakov, "Evolution of many-body systems under ancilla quantum measurements", Physical Review B 107 21, 214203 (2023).
[9] Mingyu Huang, Ji Guan, Wang Fang, and Mingsheng Ying, 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE) 1 (2024) ISBN:978-3-9819263-8-5.
[10] Y S Teo, S U Shringarpure, S Cho, and H Jeong, "Linear-optical protocols for mitigating and suppressing noise in bosonic systems", Quantum Science and Technology 10 3, 035003 (2025).
[11] Daniel K. Mark, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi, "Benchmarking Quantum Simulators Using Ergodic Quantum Dynamics", Physical Review Letters 131 11, 110601 (2023).
[12] Mikheil Tsitsishvili, Dario Poletti, Marcello Dalmonte, and Giuliano Chiriacò, "Measurement induced transitions in non-Markovian free fermion ladders", SciPost Physics Core 7 1, 011 (2024).
[13] Zhuo Chen, Yimu Bao, and Soonwon Choi, "Optimized Trajectory Unraveling for Classical Simulation of Noisy Quantum Dynamics", Physical Review Letters 133 23, 230403 (2024).
[14] Sofía González-García, Shengqi Sang, Timothy H. Hsieh, Sergio Boixo, Guifré Vidal, Andrew C. Potter, and Romain Vasseur, "Random insights into the complexity of two-dimensional tensor network calculations", Physical Review B 109 23, 235102 (2024).
[15] Alberto Boretti, "Technical, economic, and societal risks in the progress of artificial intelligence driven quantum technologies", Discover Artificial Intelligence 4 1, 67 (2024).
[16] Aleksandr Berezutskii, Minzhao Liu, Atithi Acharya, Roman Ellerbrock, Johnnie Gray, Reza Haghshenas, Zichang He, Abid Khan, Viacheslav Kuzmin, Dmitry Lyakh, Danylo Lykov, Salvatore Mandrà, Christopher Mansell, Alexey Melnikov, Artem Melnikov, Vladimir Mironov, Dmitry Morozov, Florian Neukart, Alberto Nocera, Michael A. Perlin, Michael Perelshtein, Matthew Steinberg, Ruslan Shaydulin, Benjamin Villalonga, Markus Pflitsch, Marco Pistoia, Valerii Vinokur, and Yuri Alexeev, "Tensor networks for quantum computing", Nature Reviews Physics 7 10, 581 (2025).
[17] Song Cheng, Chenfeng Cao, Chao Zhang, Yongxiang Liu, Shi-Yao Hou, Pengxiang Xu, and Bei Zeng, "Simulating noisy quantum circuits with matrix product density operators", Physical Review Research 3 2, 023005 (2021).
[18] Ivan H. Deutsch, "Harnessing the Power of the Second Quantum Revolution", PRX Quantum 1 2, 020101 (2020).
[19] Hidetaka Manabe, Yasunari Suzuki, and Andrew S Darmawan, "Efficient simulation of leakage errors in quantum error correcting codes using tensor network methods", New Journal of Physics 27 11, 114512 (2025).
[20] Oliver Lunt, Jonas Richter, and Arijeet Pal, Quantum Science and Technology 251 (2022) ISBN:978-3-031-03997-3.
[21] Zhi Li, Shengqi Sang, and Timothy H. Hsieh, "Entanglement dynamics of noisy random circuits", Physical Review B 107 1, 014307 (2023).
[22] Su-un Lee, Changhun Oh, Yat Wong, Senrui Chen, and Liang Jiang, "Universal Spreading of Conditional Mutual Information in Noisy Random Circuits", Physical Review Letters 133 20, 200402 (2024).
[23] I.A. Aloisio, G.A.L. White, C.D. Hill, and K. Modi, "Sampling Complexity of Open Quantum Systems", PRX Quantum 4 2, 020310 (2023).
[24] Rosario Fazio, Jonathan Keeling, Leonardo Mazza, and Marco Schirò, "Many-body open quantum systems", SciPost Physics Lecture Notes 99 (2025).
[25] Cupjin Huang, Fang Zhang, Michael Newman, Xiaotong Ni, Dawei Ding, Junjie Cai, Xun Gao, Tenghui Wang, Feng Wu, Gengyan Zhang, Hsiang-Sheng Ku, Zhengxiong Tian, Junyin Wu, Haihong Xu, Huanjun Yu, Bo Yuan, Mario Szegedy, Yaoyun Shi, Hui-Hai Zhao, Chunqing Deng, and Jianxin Chen, "Efficient parallelization of tensor network contraction for simulating quantum computation", Nature Computational Science 1 9, 578 (2021).
[26] Mingyu Huang, Ji Guan, Wang Fang, and Mingsheng Ying, "Approximation Methods for Simulation and Equivalence Checking of Noisy Quantum Circuits", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 45 6, 2679 (2026).
[27] Hai Wang, Jue Nan, Tao Zhang, Xingze Qiu, Wenlan Chen, and Xiaopeng Li, "Kernel function based quantum algorithms for finite temperature quantum simulation", Physical Review B 108 8, 085102 (2023).
[28] Yuchen Guo and Shuo Yang, "Quantum state tomography with locally purified density operators and local measurements", Communications Physics 7 1, 322 (2024).
[29] En-Jui Kuo, Brayden Ware, Peter Lunts, Mohammad Hafezi, and Christopher David White, "Energy diffusion in weakly interacting chains with fermionic dissipation assisted operator evolution", Physical Review B 110 7, 075149 (2024).
[30] Zhihui Zhu, Joseph M. Lukens, and Brian T. Kirby, "On the connection between least squares, regularization, and classical shadows", Quantum 8, 1455 (2024).
[31] Matteo Votto, Marko Ljubotina, Cécilia Lancien, J. Ignacio Cirac, Peter Zoller, Maksym Serbyn, Lorenzo Piroli, and Benoît Vermersch, "Learning Mixed Quantum States in Large-Scale Experiments", Physical Review Letters 136 9, 090801 (2026).
[32] D. Wellnitz, G. Preisser, V. Alba, J. Dubail, and J. Schachenmayer, "Rise and Fall, and Slow Rise Again, of Operator Entanglement under Dephasing", Physical Review Letters 129 17, 170401 (2022).
[33] Hansveer Singh, Romain Vasseur, Andrew C. Potter, and Sarang Gopalakrishnan, "Mixed-state learnability transitions in monitored noisy quantum dynamics", Physical Review B 113 5, 054305 (2026).
[34] Eli Chertkov, Zihan Cheng, Andrew C. Potter, Sarang Gopalakrishnan, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Alex Hall, Aaron Hankin, Mitchell Matheny, Tanner Mengle, David Hayes, Brian Neyenhuis, Russell Stutz, and Michael Foss-Feig, "Characterizing a non-equilibrium phase transition on a quantum computer", Nature Physics 19 12, 1799 (2023).
[35] Michael Kolodrubetz, "Optimality of Lindblad unfolding in measurement phase transitions", Physical Review B 107 14, L140301 (2023).
[36] Yoshiki Sunada, Shingo Kono, Jesper Ilves, Takanori Sugiyama, Yasunari Suzuki, Tsuyoshi Okubo, Shuhei Tamate, Yutaka Tabuchi, and Yasunobu Nakamura, "Efficient Tomography of Microwave Photonic Cluster States", PRX Quantum 7 1, 010323 (2026).
[37] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti, "Continuous-time dynamics and error scaling of noisy highly entangling quantum circuits", Physical Review A 104 6, 062407 (2021).
[38] Thomas Ayral, "Classical and quantum algorithms for many-body problems", Comptes Rendus. Physique 26 G1, 25 (2025).
[39] Maurits S. J. Tepaske and David J. Luitz, "Compressed quantum error mitigation", Physical Review B 107 20, L201114 (2023).
[40] Yuguo Shao, Fuchuan Wei, Song Cheng, and Zhengwei Liu, "Simulating Noisy Variational Quantum Algorithms: A Polynomial Approach", Physical Review Letters 133 12, 120603 (2024).
[41] J. Alexander Jacoby, David A. Huse, and Sarang Gopalakrishnan, "Spectral gaps of local quantum channels in the weak-dissipation limit", Physical Review B 111 10, 104303 (2025).
[42] Yuchen Guo and Shuo Yang, "Efficient quantum circuit compilation for near-term quantum advantage", EPJ Quantum Technology 12 1, 69 (2025).
[43] Shuo Liu, Shao-Kai Jian, and Shi-Xin Zhang, "Noisy monitored quantum circuits", Journal of Physics: Condensed Matter 38 20, 203001 (2026).
[44] Yaodong Li, Yijian Zou, Paolo Glorioso, Ehud Altman, and Matthew P. A. Fisher, "Cross Entropy Benchmark for Measurement-Induced Phase Transitions", Physical Review Letters 130 22, 220404 (2023).
[45] Yuchen Guo and Shuo Yang, "Quantum Error Mitigation via Matrix Product Operators", PRX Quantum 3 4, 040313 (2022).
[46] Avner Bensoussan, Elena Chachkarova, Karine Even-Mendoza, Sophie Fortz, and Connor Lenihan, "AccelerQ: Accelerating Quantum Eigensolvers with Machine Learning on Quantum Simulators", Proceedings of the ACM on Programming Languages 9 OOPSLA2, 2279 (2025).
[47] Tam’si Ley, Anna Ouskova Leonteva, Johannes Schachenmayer, and Pierre Collet, Lecture Notes in Computer Science 13927, 64 (2023) ISBN:978-3-031-44354-1.
[48] Yuxuan Yan, Zhenyu Du, Junjie Chen, and Xiongfeng Ma, "Limitations of noisy quantum devices in computing and entangling power", npj Quantum Information 11 1, 188 (2025).
[49] Tibor Rakovszky, C. W. von Keyserlingk, and Frank Pollmann, "Dissipation-assisted operator evolution method for capturing hydrodynamic transport", Physical Review B 105 7, 075131 (2022).
[50] Zhi-Yuan Wei, Jon Nelson, Joel Rajakumar, Esther Cruz, Alexey V. Gorshkov, Michael J. Gullans, and Daniel Malz, "Measurement-Induced Entanglement in Noisy 2D Random Circuits", PRX Quantum 7 3, 033023 (2026).
[51] Shuvro Chowdhury, Kerem Y. Çamsari, and Supriyo Datta, "Emulating Quantum Circuits With Generalized Ising Machines", IEEE Access 11, 116944 (2023).
[52] Beatriz C. Dias, Domagoj Perković, Masudul Haque, Pedro Ribeiro, and Paul A. McClarty, "Quantum noise as a symmetry-breaking field", Physical Review B 108 6, L060302 (2023).
[53] Yuchen Guo, Jian-Hao Zhang, Hao-Ran Zhang, Shuo Yang, and Zhen Bi, "Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States", Physical Review X 15 2, 021060 (2025).
[54] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).
[55] Barbara Collignon, "A graph-based quantum-inspired approximation of the Feynman path integral", Academia Quantum 3 2(2026).
[56] GUO Yuchen and YANG Shuo, "Frontiers of tensor networks and their applications in quantum many-body physics", SCIENTIA SINICA Physica, Mechanica & Astronomica 56 8, 280011 (2026).
[57] Sara Murciano, Jérôme Dubail, and Pasquale Calabrese, "More on symmetry resolved operator entanglement", Journal of Physics A: Mathematical and Theoretical 57 14, 145002 (2024).
[58] A. Zabalo, M. J. Gullans, J. H. Wilson, R. Vasseur, A. W. W. Ludwig, S. Gopalakrishnan, David A. Huse, and J. H. Pixley, "Operator Scaling Dimensions and Multifractality at Measurement-Induced Transitions", Physical Review Letters 128 5, 050602 (2022).
[59] Zixin Huang and Cosmo Lupo, "Error filtration for quantum sensing via interferometry", Physical Review A 110 5, 052431 (2024).
[60] Weixiao Sun, Fuchuan Wei, Yuguo Shao, and Zhaohui Wei, "Sudden death of quantum advantage in correlation generations", Science Advances 10 47, eadr5002 (2024).
[61] Changhun Oh, Kyungjoo Noh, Bill Fefferman, and Liang Jiang, "Classical simulation of lossy boson sampling using matrix product operators", Physical Review A 104 2, 022407 (2021).
[62] Marcel Niedermeier, Jose L. Lado, and Christian Flindt, "Simulating the quantum Fourier transform, Grover's algorithm, and the quantum counting algorithm with limited entanglement using tensor networks", Physical Review Research 6 3, 033325 (2024).
[63] Zhen Qin, Casey Jameson, Zhexuan Gong, Michael B. Wakin, and Zhihui Zhu, "Quantum State Tomography for Matrix Product Density Operators", IEEE Transactions on Information Theory 70 7, 5030 (2024).
[64] Yuta Kikuchi, Conor Mc Keever, Luuk Coopmans, Michael Lubasch, and Marcello Benedetti, "Realization of quantum signal processing on a noisy quantum computer", npj Quantum Information 9 1, 93 (2023).
[65] Tian-Hua Yang and Dmitry A. Abanin, "Integrals of motion as slow modes in dissipative many-body operator dynamics", Physical Review B 112 9, 094304 (2025).
[66] Anupam Mitra, Tameem Albash, Philip Daniel Blocher, Jun Takahashi, Akimasa Miyake, Grant Biedermann, and Ivan H Deutsch, "Macroproperties vs. microstates in the classical simulation of critical phenomena in quench dynamics of 1D Ising models", New Journal of Physics 27 1, 013026 (2025).
[67] Weitang Li, Jonathan Allcock, Lixue Cheng, Shi-Xin Zhang, Yu-Qin Chen, Jonathan P. Mailoa, Zhigang Shuai, and Shengyu Zhang, "TenCirChem: An Efficient Quantum Computational Chemistry Package for the NISQ Era", Journal of Chemical Theory and Computation 19 13, 3966 (2023).
[68] Zhen Qin and Zhihui Zhu, "Quantum state tomography for tensor networks in two dimensions", Physical Review A 113 2, 022414 (2026).
[69] Minzhao Liu, Junyu Liu, Yuri Alexeev, and Liang Jiang, "Estimating the randomness of quantum circuit ensembles up to 50 qubits", npj Quantum Information 8 1, 137 (2022).
[70] Changhun Oh, Liang Jiang, and Bill Fefferman, "Spoofing Cross-Entropy Measure in Boson Sampling", Physical Review Letters 131 1, 010401 (2023).
[71] Yi-Ting Chen, Collin Farquhar, and Robert M. Parrish, "Low-rank density-matrix evolution for noisy quantum circuits", npj Quantum Information 7 1, 61 (2021).
[72] J. Helsen, I. Roth, E. Onorati, A.H. Werner, and J. Eisert, "General Framework for Randomized Benchmarking", PRX Quantum 3 2, 020357 (2022).
[73] Seongwook Shin, Yong Siah Teo, and Hyunseok Jeong, "Dequantizing quantum machine learning models using tensor networks", Physical Review Research 6 2, 023218 (2024).
[74] Baptiste Anselme Martin, Thomas Ayral, François Jamet, Marko J. Rančić, and Pascal Simon, "Combining matrix product states and noisy quantum computers for quantum simulation", Physical Review A 109 6, 062437 (2024).
[75] Yuchen Guo and Shuo Yang, "Locally purified density operators for noisy quantum circuits", Chinese Physics Letters 41 12, 120302 (2024).
[76] Zhen Qin, Joseph M. Lukens, Brian T. Kirby, and Zhihui Zhu, "Enhancing quantum state reconstruction with structured classical shadows", npj Quantum Information 11 1, 147 (2025).
[77] M. Szyniszewski, A. Romito, and H. Schomerus, "Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements", Physical Review Letters 125 21, 210602 (2020).
[78] Jordi Tura, "Imperfections Lower the Simulation Cost of Quantum Computers", Physics 13, 183 (2020).
[79] Alexander Zlokapa, Benjamin Villalonga, Sergio Boixo, and Daniel A. Lidar, "Boundaries of quantum supremacy via random circuit sampling", npj Quantum Information 9 1, 36 (2023).
[80] Izabella Lovas, Utkarsh Agrawal, and Sagar Vijay, "Quantum Coding Transitions in the Presence of Boundary Dissipation", PRX Quantum 5 3, 030327 (2024).
[81] Florian Brökemeier, S. Momme Hengstenberg, James W. T. Keeble, Caroline E. P. Robin, Federico Rocco, and Martin J. Savage, "Quantum magic and multipartite entanglement in the structure of nuclei", Physical Review C 111 3, 034317 (2025).
[82] Meng Zhang, Chao Wang, and Yongjian Han, "Noisy Random Quantum Circuit Sampling and its Classical Simulation", Advanced Quantum Technologies 6 7, 2300030 (2023).
[83] Guillermo Preisser, David Wellnitz, Thomas Botzung, and Johannes Schachenmayer, "Comparing bipartite entropy growth in open-system matrix-product simulation methods", Physical Review A 108 1, 012616 (2023).
[84] Oskar Słowik, Piotr Dulian, and Adam Sawicki, "Quantum circuit overhead", Physical Review A 114 1, 012454 (2026).
[85] Zihan Cheng and Matteo Ippoliti, "Efficient Sampling of Noisy Shallow Circuits Via Monitored Unraveling", PRX Quantum 4 4, 040326 (2023).
[86] Giuliano Chiriacò, Mikheil Tsitsishvili, Dario Poletti, Rosario Fazio, and Marcello Dalmonte, "Diagrammatic method for many-body non-Markovian dynamics: Memory effects and entanglement transitions", Physical Review B 108 7, 075151 (2023).
[87] Takato Yoshimura and Lucas Sá, "Robustness of quantum chaos and anomalous relaxation in open quantum circuits", Nature Communications 15 1, 9808 (2024).
[88] Markus Schmitt and Zala Lenarčič, "From observations to complexity of quantum states via unsupervised learning", Physical Review B 106 4, L041110 (2022).
[89] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay, "Random Quantum Circuits", Annual Review of Condensed Matter Physics 14 1, 335 (2023).
[90] Xiao Huang, Guanhua Chen, and Yao Yao, "Advantage of utilizing nonlocal magic resource in Haar-random circuits", Journal of Physics A: Mathematical and Theoretical 59 2, 025302 (2026).
[91] J. A. Montañez-Barrera, Michael R. von Spakovsky, Cesar E. Damian Ascencio, and Sergio Cano-Andrade, "Decoherence predictions in a superconducting quantum processor using the steepest-entropy-ascent quantum thermodynamics framework", Physical Review A 106 3, 032426 (2022).
[92] Anbang Wang, Jingning Zhang, and Ying Li, "Error-mitigated deep-circuit quantum simulation of open systems: Steady state and relaxation rate problems", Physical Review Research 4 4, 043140 (2022).
[93] Qi Zhang and Guang-Ming Zhang, "Noise-Induced Entanglement Transition in One-Dimensional Random Quantum Circuits", Chinese Physics Letters 39 5, 050302 (2022).
[94] Dmitry I. Lyakh, Thien Nguyen, Daniel Claudino, Eugene Dumitrescu, and Alexander J. McCaskey, "ExaTN: Scalable GPU-Accelerated High-Performance Processing of General Tensor Networks at Exascale", Frontiers in Applied Mathematics and Statistics 8, 838601 (2022).
[95] Aniket Rath, Vittorio Vitale, Sara Murciano, Matteo Votto, Jérôme Dubail, Richard Kueng, Cyril Branciard, Pasquale Calabrese, and Benoît Vermersch, "Entanglement Barrier and its Symmetry Resolution: Theory and Experimental Observation", PRX Quantum 4 1, 010318 (2023).
[96] Boaz Barak, Chi-Ning Chou, and Xun Gao, "Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits", arXiv:2005.02421, (2020).
[97] Benjamin Villalonga, Murphy Yuezhen Niu, Li Li, Hartmut Neven, John C. Platt, Vadim N. Smelyanskiy, and Sergio Boixo, "Efficient approximation of experimental Gaussian boson sampling", arXiv:2109.11525, (2021).
[98] Anthony P. Thompson, Arie Soeteman, Chris Cade, and Ido Niesen, "Non-zero noise extrapolation: accurately simulating noisy quantum circuits with tensor networks", arXiv:2501.13237, (2025).
[99] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Fault-tolerant quantum speedup from constant depth quantum circuits", arXiv:2005.11539, (2020).
[100] Zhi-Yuan Wei, Jon Nelson, Joel Rajakumar, Esther Cruz, Alexey V. Gorshkov, Michael J. Gullans, and Daniel Malz, "Measurement-induced entanglement in noisy 2D random circuits", arXiv:2510.12743, (2025).
[101] Avner Bensoussan, Elena Chachkarova, Karine Even-Mendoza, Sophie Fortz, and Connor Lenihan, "Accelerating Quantum Eigensolver Algorithms With Machine Learning", arXiv:2409.13587, (2024).
[102] Maxime Oliva, "An entanglement-aware quantum computer simulation algorithm", arXiv:2307.16870, (2023).
[103] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Fault-tolerant quantum speedup from constant depth quantum circuits", Physical Review Research 2 3, 033444 (2020).
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