Simulation of quantum circuits by low-rank stabilizer decompositions

Sergey Bravyi; Dan Browne; Padraic Calpin; Earl Campbell; David Gosset; Mark Howard

doi:10.22331/q-2019-09-02-181

Simulation of quantum circuits by low-rank stabilizer decompositions

Sergey Bravyi¹, Dan Browne², Padraic Calpin², Earl Campbell³, David Gosset^1,4, and Mark Howard³

¹IBM T.J. Watson Research Center, Yorktown Heights NY 10598
²Department of Physics and Astronomy, University College London, London, UK
³Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
⁴Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Canada

Published:	2019-09-02, volume 3, page 181
Eprint:	arXiv:1808.00128v2
Doi:	https://doi.org/10.22331/q-2019-09-02-181
Citation:	Quantum 3, 181 (2019).

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Abstract

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of $\it{stabilizer}$ $\textit{rank}$, which for a pure state $\psi$ is defined to be the smallest integer $\chi$ such that $\psi$ is a superposition of $\chi$ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of $\chi\sim10^6$ stabilizer states and sample from the full $n$-bit output distribution, improving on previous simulations which used $\sim 10^3$ stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 $T$ gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.

► BibTeX data

@article{Bravyi2019simulationofquantum,
  doi = {10.22331/q-2019-09-02-181},
  url = {https://doi.org/10.22331/q-2019-09-02-181},
  title = {Simulation of quantum circuits by low-rank stabilizer decompositions},
  author = {Bravyi, Sergey and Browne, Dan and Calpin, Padraic and Campbell, Earl and Gosset, David and Howard, Mark},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {181},
  month = sep,
  year = {2019}
}

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[49] Arman Sauliere, Guglielmo Lami, Corentin Boyer, Jacopo De Nardis, and Andrea De Luca, "Universality in the Anticoncentration of Noisy Quantum Circuits at Finite Depths", PRX Quantum 7 2, 020365 (2026).

[50] Yu-Ao Chen, Gilad Gour, Xin Wang, Lei Zhang, and Chenghong Zhu, "Physical implementability for reversible magic state manipulation", Communications Physics 8 1, 4 (2025).

[51] Matthew Sutcliffe and Aleks Kissinger, "Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus", Electronic Proceedings in Theoretical Computer Science 426, 247 (2025).

[52] Kaifeng Bu, Weichen Gu, and Arthur Jaffe, "Quantum higher-order Fourier analysis and the Clifford hierarchy", Proceedings of the National Academy of Sciences 122 45, e2515667122 (2025).

[53] Jovan Odavić, Tobias Haug, Gianpaolo Torre, Alioscia Hamma, Fabio Franchini, and Salvatore Marco Giampaolo, "Complexity of frustration: A new source of non-local non-stabilizerness", SciPost Physics 15 4, 131 (2023).

[54] Andi Gu, Lorenzo Leone, Soumik Ghosh, Jens Eisert, Susanne F. Yelin, and Yihui Quek, "Pseudomagic Quantum States", Physical Review Letters 132 21, 210602 (2024).

[55] Kaifeng Bu, Dax Enshan Koh, Lu Li, Qingxian Luo, and Yaobo Zhang, "Statistical complexity of quantum circuits", Physical Review A 105 6, 062431 (2022).

[56] Fu Shuangshuang, Li Xiaohui, and Luo Shunlong, "Dynamics of atomic magic in the Jaynes–Cummings model", Quantum Information Processing 22 1, 7 (2022).

[57] Markus Heinrich and David Gross, "Robustness of Magic and Symmetries of the Stabiliser Polytope", Quantum 3, 132 (2019).

[58] Hiroki Hamaguchi, Kou Hamada, and Nobuyuki Yoshioka, "Handbook for Quantifying Robustness of Magic", Quantum 8, 1461 (2024).

[59] Wenlong Sun and Yuanfeng Jin, "Leveraging magic resources for quantum channel capacity enhancement under stabilizer convolution", Physical Review A 112 5, 052412 (2025).

[60] Lingxuan Feng and Shunlong Luo, "Entanglement and Magic Correlations Generated by Discrete Beam‐Splitters", Annalen der Physik 537 10, e00288 (2025).

[61] Christopher Chamberland and Kyungjoo Noh, "Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits", npj Quantum Information 6 1, 91 (2020).

[62] Matthew Amy and Lucas Shigeru Stinchcombe, "Polynomial-Time Classical Simulation of Hidden Shift Circuits via Confluent Rewriting of Symbolic Sums", Quantum 9, 1926 (2025).

[63] Somnath Maity and Ryusuke Hamazaki, "Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit", Physical Review Research 8 1, 013324 (2026).

[64] Vsevolod I. Yashin and Maria A. Elovenkova, "Characterization of non-adaptive Clifford channels", Quantum Information Processing 24 3, 99 (2025).

[65] Alessio Cicero, Mohammad Ali Maleki, Muhammad Waqar Azhar, Anton Frisk Kockum, and Pedro Trancoso, "Simulation of Quantum Computers: Review and Acceleration Opportunities", ACM Transactions on Quantum Computing 7 1, 1 (2026).

[66] Ludovico Lami, Bartosz Regula, Ryuji Takagi, and Giovanni Ferrari, "Framework for resource quantification in infinite-dimensional general probabilistic theories", Physical Review A 103 3, 032424 (2021).

[67] Xingjian Lyu and Kaifeng Bu, "Displaced fermionic Gaussian states and their classical simulation", Journal of Physics A: Mathematical and Theoretical 58 29, 295301 (2025).

[68] Wira Azmoon Ahmad and Matthew Sutcliffe, "Dynamic T-decomposition for classical simulation of quantum circuits", International Journal of Modern Physics C 2543002 (2025).

[69] Xin Hong, Yuan Feng, Sanjiang Li, and Mingsheng Ying, Proceedings of the 41st IEEE/ACM International Conference on Computer-Aided Design 1 (2022) ISBN:9781450392174.

[70] Poetri Sonya Tarabunga and Tobias Haug, "Efficient mutual magic and magic capacity with matrix product states", SciPost Physics 19 4, 085 (2025).

[71] Lorenzo Leone and Lennart Bittel, "Stabilizer entropies are monotones for magic-state resource theory", Physical Review A 110 4, L040403 (2024).

[72] Patryk Lipka-Bartosik, Henrik Wilming, and Nelly H. Y. Ng, "Catalysis in quantum information theory", Reviews of Modern Physics 96 2, 025005 (2024).

[73] Lingxuan Feng and Shunlong Luo, "Optimal ququint diagonal gates for generating Wigner negativity", Physical Review A 113 4, 042434 (2026).

[74] Aleks Kissinger and John van de Wetering, "Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions", Quantum Science and Technology 7 4, 044001 (2022).

[75] Bartosz Regula, Ludovico Lami, and Mark M. Wilde, "Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols", Physical Review A 107 4, 042401 (2023).

[76] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[77] Kwok Ho Wan, Zhenghao Zhong, and Ainhoa Zapirain, "Simulating magic state cultivation with few Clifford terms", Quantum 10, 2134 (2026).

[78] Lingxuan Feng and Shunlong Luo, "From stabilizer states to SIC-POVM fiducial states", Theoretical and Mathematical Physics 213 3, 1747 (2022).

[79] Filipa C. R. Peres and Ernesto F. Galvão, "Quantum circuit compilation and hybrid computation using Pauli-based computation", Quantum 7, 1126 (2023).

[80] Ben Priestley and Petros Wallden, "A practically scalable approach to the closest vector problem for sieving via QAOA with fixed angles", Quantum Science and Technology 11 2, 025018 (2026).

[81] James R. Seddon and Earl T. Campbell, "Quantifying magic for multi-qubit operations", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 2227, 20190251 (2019).

[82] Yasunari Suzuki, Yoshiaki Kawase, Yuya Masumura, Yuria Hiraga, Masahiro Nakadai, Jiabao Chen, Ken M. Nakanishi, Kosuke Mitarai, Ryosuke Imai, Shiro Tamiya, Takahiro Yamamoto, Tennin Yan, Toru Kawakubo, Yuya O. Nakagawa, Yohei Ibe, Youyuan Zhang, Hirotsugu Yamashita, Hikaru Yoshimura, Akihiro Hayashi, and Keisuke Fujii, "Qulacs: a fast and versatile quantum circuit simulator for research purpose", Quantum 5, 559 (2021).

[83] Poetri Sonya Tarabunga and Emanuele Tirrito, "Magic transition in measurement-only circuits", npj Quantum Information 11 1, 166 (2025).

[84] Lingyun Wan, Jie Liu, Zhenyu Li, and Jinlong Yang, "Hybrid Hamiltonian Simulation for Excitation Dynamics", The Journal of Physical Chemistry Letters 15 45, 11234 (2024).

[85] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin, "Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications", PRX Quantum 7 1, 010201 (2026).

[86] Tobias Haug, Leandro Aolita, and M.S. Kim, "Probing quantum complexity via universal saturation of stabilizer entropies", Quantum 9, 1801 (2025).

[87] Florian Krötz, Xiao-Ting Michelle To, Korbinian Staudacher, and Dieter Kranzlmüller, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 106 (2025) ISBN:979-8-3315-5736-2.

[88] Ryuji Takagi and Bartosz Regula, "General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks", Physical Review X 9 3, 031053 (2019).

[89] Liyuan Chen, Roy J. Garcia, Kaifeng Bu, and Arthur Jaffe, "Magic of random matrix product states", Physical Review B 109 17, 174207 (2024).

[90] Saeed Mehraban and Mehrdad Tahmasbi, Proceedings of the 56th Annual ACM Symposium on Theory of Computing 608 (2024) ISBN:9798400703836.

[91] Santiago Cifuentes, Samson Wang, Thais L. Silva, Mario Berta, and Leandro Aolita, "Quantum Computational Complexity of Matrix Functions", PRX Quantum 7 2, 020364 (2026).

[92] Zejun Liu and Bryan K. Clark, "Classical simulability of Clifford+T circuits with Clifford-augmented matrix product states", Physical Review Research 8 2, 023116 (2026).

[93] Dominik Hangleiter, Marcin Kalinowski, Dolev Bluvstein, Madelyn Cain, Nishad Maskara, Xun Gao, Aleksander Kubica, Mikhail D. Lukin, and Michael J. Gullans, "Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes", PRX Quantum 6 2, 020338 (2025).

[94] Lu Li, Kaifeng Bu, Dax Enshan Koh, Arthur Jaffe, and Seth Lloyd, "Wasserstein complexity of quantum circuits", Journal of Physics A: Mathematical and Theoretical 58 26, 265302 (2025).

[95] Justin Provazza, Klaas Gunst, Huanchen Zhai, Garnet K.-L. Chan, Toru Shiozaki, Nicholas C. Rubin, and Alec F. White, "Fast Emulation of Fermionic Circuits with Matrix Product States", Journal of Chemical Theory and Computation 20 9, 3719 (2024).

[96] Kaifeng Bu and Dax Enshan Koh, "Efficient Classical Simulation of Clifford Circuits with Nonstabilizer Input States", Physical Review Letters 123 17, 170502 (2019).

[97] Neil Dowling, Kavan Modi, and Gregory A. L. White, "Bridging Entanglement and Magic Resources within Operator Space", Physical Review Letters 135 16, 160201 (2025).

[98] Guoming Chen, Qiang Chen, Shun Long, Weiheng Zhu, Zeduo Yuan, and Yilin Wu, "Quantum convolutional neural network for image classification", Pattern Analysis and Applications 26 2, 655 (2023).

[99] Jacob Biamonte, "Universal variational quantum computation", Physical Review A 103 3, L030401 (2021).

[100] Zi-Wen Liu and Andreas Winter, "Many-Body Quantum Magic", PRX Quantum 3 2, 020333 (2022).

[101] Jiayu He and Shuangshuang Fu, "Renormalization of magic and quantum phase transition in spin models", Quantum Information Processing 22 3, 161 (2023).

[102] Adrian Chapman and Steven T. Flammia, "Characterization of solvable spin models via graph invariants", Quantum 4, 278 (2020).

[103] Nicholas C. Rubin, Klaas Gunst, Alec White, Leon Freitag, Kyle Throssell, Garnet Kin-Lic Chan, Ryan Babbush, and Toru Shiozaki, "The Fermionic Quantum Emulator", Quantum 5, 568 (2021).

[104] Shalin Jose, Akshay Kannan Sairam, and Anil Shaji, "Efficient Classical Simulation of the DQC1 Circuit with Zero Discord", Quantum 9, 1895 (2025).

[105] Kaitlin N. Smith, Michael A. Perlin, Pranav Gokhale, Paige Frederick, David Owusu-Antwi, Richard Rines, Victory Omole, and Frederic Chong, Proceedings of the 50th Annual International Symposium on Computer Architecture 1 (2023) ISBN:9798400700958.

[106] Linmao Wang and Zhaoqi Wu, "Quantifying magic via quantum (α, β) Jensen–Shannon divergence", Communications in Theoretical Physics 78 5, 055103 (2026).

[107] Nikolaos Koukoulekidis, Hyukjoon Kwon, Hyejung H. Jee, David Jennings, and M. S. Kim, "Faster Born probability estimation via gate merging and frame optimisation", Quantum 6, 838 (2022).

[108] Yuxuan Du, Min-Hsiu Hsieh, and Dacheng Tao, "Efficient learning for linear properties of bounded-gate quantum circuits", Nature Communications 16 1, 3790 (2025).

[109] Roy J. Garcia, Kaifeng Bu, and Arthur Jaffe, "Resource theory of quantum scrambling", Proceedings of the National Academy of Sciences 120 17, e2217031120 (2023).

[110] Yu Wang, "Quantum Advantage via Efficient Postprocessing on Qudit Classical Shadow Tomography", Physical Review Letters 135 20, 200601 (2025).

[111] Jiace Sun, Lixue Cheng, and Shi-Xin Zhang, "Stabilizer ground states for simulating quantum many-body physics: theory, algorithms, and applications", Quantum 9, 1782 (2025).

[112] Stefan Hillmich, Igor L. Markov, and Robert Wille, 2020 57th ACM/IEEE Design Automation Conference (DAC) 1 (2020) ISBN:978-1-7281-1085-1.

[113] Alessandro Sinibaldi, Antonio Francesco Mello, Mario Collura, and Giuseppe Carleo, "Nonstabilizerness of neural quantum states", Physical Review Research 7 4, 043289 (2025).

[114] Angelo Russomanno, Gianluca Passarelli, Davide Rossini, and Procolo Lucignano, "Nonstabilizerness in the unitary and monitored quantum dynamics of XXZ-staggered and Sachdev-Ye-Kitaev models", Physical Review B 112 6, 064312 (2025).

[115] Ben Harper, Azar Nakhl, Thomas Quella, Martin Sevior, and Muhammad Usman, Proceedings of the Supercomputing Asia and International Conference on High Performance Computing in Asia Pacific Region 1 (2026) ISBN:9798400720673.

[116] Daniel Volya and Prabhat Mishra, Proceedings of the 28th Asia and South Pacific Design Automation Conference 216 (2023) ISBN:9781450397834.

[117] Jeffrey Marshall and Namit Anand, "Simulation of quantum optics by coherent state decomposition", Optica Quantum 1 2, 78 (2023).

[118] Lennart Bittel, Antonio A. Mele, Jens Eisert, and Lorenzo Leone, "PAC-learning of free-fermionic states is NP-hard", Quantum 9, 1665 (2025).

[119] Nikita A. Nemkov, Evgeniy O. Kiktenko, and Aleksey K. Fedorov, "Fourier expansion in variational quantum algorithms", Physical Review A 108 3, 032406 (2023).

[120] Benjamin Bichsel, Anouk Paradis, Maximilian Baader, and Martin Vechev, "Abstraqt: Analysis of Quantum Circuits via Abstract Stabilizer Simulation", Quantum 7, 1185 (2023).

[121] Otso Kinanen, Andrés D. Muñoz-Moller, Vlad Stirbu, Juan M. Murillo, and Tommi Mikkonen, "Toolchain for faster iterations in quantum software development", Computing 107 4, 99 (2025).

[122] Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang, Proceedings of the 56th Annual ACM Symposium on Theory of Computing 1352 (2024) ISBN:9798400703836.

[123] Tyler D. Ellison, Kohtaro Kato, Zi-Wen Liu, and Timothy H. Hsieh, "Symmetry-protected sign problem and magic in quantum phases of matter", Quantum 5, 612 (2021).

[124] Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandrà, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, Juan Atalaya, Ryan Babbush, Joseph C. Bardin, Rami Barends, Joao Basso, Andreas Bengtsson, Sergio Boixo, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Alan R. Derk, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Edward Farhi, Austin G. Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Julian Kelly, Seon Kim, Alexei Kitaev, Paul V. Klimov, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Michael Newman, Murphy Yuezhen Niu, Thomas E. O’Brien, Alex Opremcak, Eric Ostby, Balint Pato, Andre Petukhov, Nicholas Redd, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, Igor Aleiner, Kostyantyn Kechedzhi, Vadim Smelyanskiy, and Yu Chen, "Information scrambling in quantum circuits", Science 374 6574, 1479 (2021).

[125] Krzysztof Warmuz, Ernest Dokudowiec, Chandrashekar Radhakrishnan, and Tim Byrnes, "Magic Monotone for Faithful Detection of Nonstabilizerness in Mixed States", Physical Review Letters 135 1, 010203 (2025).

[126] Poulami Das, Swamit Tannu, Siddharth Dangwal, and Moinuddin Qureshi, MICRO-54: 54th Annual IEEE/ACM International Symposium on Microarchitecture 950 (2021) ISBN:9781450385572.

[127] Timothée Goubault de Brugière, Marc Baboulin, Benoît Valiron, and Cyril Allouche, Lecture Notes in Computer Science 11537, 3 (2019) ISBN:978-3-030-22740-1.

[128] James R. Seddon, Bartosz Regula, Hakop Pashayan, Yingkai Ouyang, and Earl T. Campbell, "Quantifying Quantum Speedups: Improved Classical Simulation From Tighter Magic Monotones", PRX Quantum 2 1, 010345 (2021).

[129] Benjamin Lovitz and Vincent Steffan, "New techniques for bounding stabilizer rank", Quantum 6, 692 (2022).

[130] Niel de Beaudrap and Steven Herbert, "Fast Stabiliser Simulation with Quadratic Form Expansions", Quantum 6, 803 (2022).

[131] Hyunjoon Shin, Seokhyeon Lee, Myeongjin Kwak, and Yongtae Kim, Proceedings of the 40th ACM International Conference on Supercomputing 1296 (2026) ISBN:9798400725227.

[132] Poetri Sonya Tarabunga, Martina Frau, Tobias Haug, Emanuele Tirrito, and Lorenzo Piroli, "A nonstabilizerness monotone from stabilizerness asymmetry", Quantum Science and Technology 10 4, 045026 (2025).

[133] Piotr Sierant, Jofre Vallès-Muns, and Artur Garcia-Saez, "Computing quantum magic of state vectors", Quantum 10, 2059 (2026).

[134] Wonjun Lee, Minki Hhan, Gil Young Cho, and Hyukjoon Kwon, "Shallow Quantum Circuit for Generating Extremely Low-Entangled Approximate State Designs", Physical Review Letters 136 15, 150603 (2026).

[135] Sabri Meyer, Francesco Scala, Francesco Tacchino, and Aurelien Lucchi, "Gradient scalability and Taylor surrogation of quantum cost landscapes", Physical Review Research 8 2, 023325 (2026).

[136] Beatrice Magni and Xhek Turkeshi, "Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits", Quantum 9, 1956 (2025).

[137] Amolak Ratan Kalra and Pulkit Sinha, "Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones", Quantum 10, 2179 (2026).

[138] Xingjian Zhang, Zhaokai Pan, and Guoding Liu, "Unconditional quantum magic advantage in shallow circuit computation", Nature Communications 15 1, 10513 (2024).

[139] Naga Dileep Varikuti, Soumik Bandyopadhyay, and Philipp Hauke, "Impact of Clifford operations on non-stabilizing power and quantum chaos", Quantum 10, 2017 (2026).

[140] Beatrice Magni, Alexios Christopoulos, Andrea De Luca, and Xhek Turkeshi, "Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudomagic States", Physical Review X 15 3, 031071 (2025).

[141] Alexis Ralli, Tim Weaving, and Peter J Love, "Noncontextual Pauli Hamiltonians", Journal of Physics A: Mathematical and Theoretical 58 33, 335301 (2025).

[142] Laszlo Gyongyosi and Sandor Imre, "Quantum circuit design for objective function maximization in gate-model quantum computers", Quantum Information Processing 18 7, 225 (2019).

[143] Lingxuan Feng and Shunlong Luo, "От стабилизирующих состояний к фидуциальным состояниям, задающим симметричную информационно-полную положительную операторнозначную меру", Теоретическая и математическая физика 213 3, 505 (2022).

[144] Oliver Hahn, Ryuji Takagi, Giulia Ferrini, and Hayata Yamasaki, "Classical simulation and quantum resource theory of non-Gaussian optics", Quantum 9, 1881 (2025).

[145] Gianluca Passarelli, Angelo Russomanno, and Procolo Lucignano, "Nonstabilizerness of a boundary time crystal", Physical Review A 111 6, 062417 (2025).

[146] Simone Cepollaro, Stefano Cusumano, Alioscia Hamma, Giorgio Lo Giudice, and Jovan Odavić, "Harvesting stabilizer entropy and nonlocality from a quantum field", Physical Review D 112 10, 105012 (2025).

[147] Alejandro Sopena, Max Hunter Gordon, Germán Sierra, and Esperanza López, "Simulating quench dynamics on a digital quantum computer with data-driven error mitigation", Quantum Science and Technology 6 4, 045003 (2021).

[148] Yasuhiro Kondo, Ryuhei Mori, and Ramis Movassagh, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) 1296 (2022) ISBN:978-1-6654-2055-6.

[149] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu, "Benchmarking one-shot distillation in general quantum resource theories", Physical Review A 101 6, 062315 (2020).

[150] Bartosz Regula, "Tight constraints on probabilistic convertibility of quantum states", Quantum 6, 817 (2022).

[151] Sergey Bravyi, David Gosset, and Yinchen Liu, "How to Simulate Quantum Measurement without Computing Marginals", Physical Review Letters 128 22, 220503 (2022).

[152] Ying-Jie 英杰 Qu 曲, Zhao 钊 Chen 陈, Wei-Jie 伟杰 Wang 王, and Hong-Yang 鸿洋 Ma 马, "Approximate error correction scheme for three-dimensional surface codes based reinforcement learning", Chinese Physics B 32 10, 100307 (2023).

[153] Yifei Huang and Peter Love, "Feynman-path-type simulation using stabilizer projector decomposition of unitaries", Physical Review A 103 2, 022428 (2021).

[154] Laura García-Álvarez, Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini, "Efficient simulatability of continuous-variable circuits with large Wigner negativity", Physical Review Research 2 4, 043322 (2020).

[155] Hammam Qassim, Hakop Pashayan, and David Gosset, "Improved upper bounds on the stabilizer rank of magic states", Quantum 5, 606 (2021).

[156] Deniz Karhan, Sedat Çimen, and Merve Güllü, 2025 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA) 1 (2025) ISBN:979-8-3315-3562-9.

[157] Neil Dowling, Pavel Kos, and Xhek Turkeshi, "Magic Resources of the Heisenberg Picture", Physical Review Letters 135 5, 050401 (2025).

[158] Salvatore Mandra, Jeffrey Marshall, Eleanor G. Rieffel, and Rupak Biswas, 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS) 99 (2021) ISBN:978-1-7281-8674-0.

[159] Xiaohui Li and Shunlong Luo, "Optimal diagonal qutrit gates for creating Wigner negativity", Physics Letters A 460, 128620 (2023).

[160] Yifan Zhang and Yuxuan Zhang, "Classical Simulability of Quantum Circuits with Shallow Magic Depth", PRX Quantum 6 1, 010337 (2025).

[161] Denis A. Kulikov, Vsevolod I. Yashin, Aleksey K. Fedorov, and Evgeniy O. Kiktenko, "Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations", Physical Review A 109 1, 012219 (2024).

[162] Muthukumaran Malarvel, K.P. Kaliyamurthie, and Antonidoss. A, 2025 IEEE First International Conference on Innovations in Engineering and Next-Generation Technologies for Sustainability (ICINVENTS) 1 (2025) ISBN:979-8-3315-5662-4.

[163] Beatriz Dias and Robert Koenig, "Classical simulation of non-Gaussian fermionic circuits", Quantum 8, 1350 (2024).

[164] Mark Koch, Richie Yeung, and Quanlong Wang, "Contraction of ZX diagrams with triangles via stabiliser decompositions", Physica Scripta 99 10, 105122 (2024).

[165] David Gosset, Daniel Grier, Alex Kerzner, and Luke Schaeffer, "Fast simulation of planar Clifford circuits", Quantum 8, 1251 (2024).

[166] Clement Charles, Erik J. Gustafson, Elizabeth Hardt, Florian Herren, Norman Hogan, Henry Lamm, Sara Starecheski, Ruth S. Van de Water, and Michael L. Wagman, "Simulating Z2 lattice gauge theory on a quantum computer", Physical Review E 109 1, 015307 (2024).

[167] Yu Luo, Fanxu Meng, and Youle Wang, "Epsilon measures of state-based quantum resource theory", Physical Review A 109 5, 052413 (2024).

[168] Jiaqing Jiang and Xin Wang, "Lower Bound for the T Count Via Unitary Stabilizer Nullity", Physical Review Applied 19 3, 034052 (2023).

[169] Hiroki Hamaguchi, Kou Hamada, Naoki Marumo, and Nobuyuki Yoshioka, "Faster computation of nonstabilizerness", Physical Review Applied 23 1, 014069 (2025).

[170] Arne Heimendahl, Markus Heinrich, and David Gross, "The axiomatic and the operational approaches to resource theories of magic do not coincide", Journal of Mathematical Physics 63 11, 112201 (2022).

[171] Junjie Chen, Yuxuan Yan, and You Zhou, "Magic of quantum hypergraph states", Quantum 8, 1351 (2024).

[172] Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang, "Agnostic Tomography of Stabilizer Product States", Quantum 10, 2027 (2026).

[173] Wenjing Hou, Zhanyuan Fu, Lei Li, Yiran Wang, and Yongming Li, "The convolution-based quantum Jensen-Shannon divergence and its application in quantifying magic resource", Physics Letters A 589, 131793 (2026).

[174] Gaurav Saxena and Gilad Gour, "Quantifying multiqubit magic channels with completely stabilizer-preserving operations", Physical Review A 106 4, 042422 (2022).

[175] Xiaohui Li and Shunlong Luo, "Optimality of T-gate for generating magic resource", Communications in Theoretical Physics 75 4, 045101 (2023).

[176] Arne Heimendahl, Felipe Montealegre-Mora, Frank Vallentin, and David Gross, "Stabilizer extent is not multiplicative", Quantum 5, 400 (2021).

[177] Oliver Hahn, Alessandro Ferraro, Lina Hultquist, Giulia Ferrini, and Laura García-Álvarez, "Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding", Physical Review Letters 128 21, 210502 (2022).

[178] Shuangshuang Fu, Xiaohui Li, and Shunlong Luo, "Detecting quantum phase transition via magic resource in the XY spin model", Physical Review A 106 6, 062405 (2022).

[179] Chengsi Mao, Changhao Yi, and Huangjun Zhu, "Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate", Physical Review Letters 134 16, 160801 (2025).

[180] Beatriz Dias and Robert König, "Classical simulation of non-Gaussian bosonic circuits", Physical Review A 110 4, 042402 (2024).

[181] Lei-Yi-Nan Liu, Su Yi, and Jian Cui, "Stabilizer Rényi entropy for translation-invariant matrix product states", Physical Review A 113 6, 062414 (2026).

[182] Snehasish Roy Chowdhury, Sahil Gopalkrishna Naik, Ananya Chakraborty, Ram Krishna Patra, Subhendu B. Ghosh, Pratik Ghosal, Manik Banik, and Ananda G. Maity, "Gottesman-Knill limit on one-way communication complexity: Tracing the quantum advantage down to magic resources", Physical Review A 113 4, 042607 (2026).

[183] Gianluca Passarelli, Procolo Lucignano, Davide Rossini, and Angelo Russomanno, "Chaos and magic in the dissipative quantum kicked top", Quantum 9, 1653 (2025).

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Simulation of quantum circuits by low-rank stabilizer decompositions

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