Hidden variable model for quantum computation with magic states on qudits of any dimension

Michael Zurel1,2, Cihan Okay3, Robert Raussendorf2,4, and Arne Heimendahl5

1Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada
2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, Canada
3Department of Mathematics, Bilkent University, Ankara, Turkey
4Institute of Theoretical Physics, Leibniz University Hannover, Hannover, Germany
5Department of Mathematics and Computer Science, University of Cologne, Cologne, Germany

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Abstract

It was recently shown that a hidden variable model can be constructed for universal quantum computation with magic states on qubits. Here we show that this result can be extended, and a hidden variable model can be defined for quantum computation with magic states on qudits with any Hilbert space dimension. This model leads to a classical simulation algorithm for universal quantum computation.

Multimedia: Michael Zurel, University of British Columbia | “Hidden Variable Model for Quantum Computation with Magic States on Any Number of Qudits of Any Dimension

We present a description of universal quantum computation via a probabilistic hidden variable model. This extends a previous result which applied only to qubits (Hilbert space dimension 2) to quantum computations on systems of any dimension. This model gives us a classical simulation algorithm for universal quantum computation that proceeds by sampling from the probability distributions that define the model. This model extends Gross' Wigner function and the qudit stabilizer formalism.

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► References

[1] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis, ``Quantum supremacy using a programmable superconducting processor'' Nature 574, 505-510 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-1666-5
arXiv:1910.11333

[2] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O’Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, and Adam Zalcman, ``Hartree-Fock on a superconducting qubit quantum computer'' Science 369, 1084–1089 (2020).
https:/​/​doi.org/​10.1126/​science.abb9811
arXiv:2004.04174

[3] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Andreas Bengtsson, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Yu-An Chen, Ben Chiaro, Roberto Collins, Stephen J. Cotton, William Courtney, Sean Demura, Alan Derk, Andrew Dunsworth, Daniel Eppens, Thomas Eckl, Catherine Erickson, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Jonathan A. Gross, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William Huggins, Lev B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Michael Marthaler, Orion Martin, John M. Martinis, Anika Marusczyk, Sam McArdle, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Carlos Mejuto-Zaera, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Eric Ostby, Bálint Pató, Andre Petukhov, Harald Putterman, Chris Quintana, Jan-Michael Reiner, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Peter Schmitteckert, Marco Szalay, Norm M. Tubman, Amit Vainsencher, Theodore White, Nicolas Vogt, Z. Jamie Yao, Ping Yeh, Adam Zalcman, and Sebastian Zanker, ``Observation of separated dynamics of charge and spin in the Fermi-Hubbard model'' (2020).
arXiv:2010.07965

[4] Matthew P. Harrigan, Kevin J. Sung, Matthew Neeley, Kevin J. Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Alan Ho, Sabrina Hong, Trent Huang, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Martin Leib, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Charles Neill, Florian Neukart, Murphy Yuezhen Niu, Thomas E. O’Brien, Bryan O’Gorman, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Andrea Skolik, Vadim Smelyanskiy, Doug Strain, Michael Streif, Marco Szalay, Amit Vainsencher, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Leo Zhou, Hartmut Neven, Dave Bacon, Erik Lucero, Edward Farhi, and Ryan Babbush, ``Quantum approximate optimization of non-planar graph problems on a planar superconducting processor'' Nature Physics 17, 332–336 (2021).
https:/​/​doi.org/​10.1038/​s41567-020-01105-y
arXiv:2004.04197

[5] Petar Jurcevic, Ali Javadi-Abhari, Lev S Bishop, Isaac Lauer, Daniela F Bogorin, Markus Brink, Lauren Capelluto, Oktay Günlük, Toshinari Itoko, Naoki Kanazawa, Abhinav Kandala, George A Keefe, Kevin Krsulich, William Landers, Eric P Lewandowski, Douglas T McClure, Giacomo Nannicini, Adinath Narasgond, Hasan M Nayfeh, Emily Pritchett, Mary Beth Rothwell, Srikanth Srinivasan, Neereja Sundaresan, Cindy Wang, Ken X Wei, Christopher J Wood, Jeng-Bang Yau, Eric J Zhang, Oliver E Dial, Jerry M Chow, and Jay M Gambetta, ``Demonstration of quantum volume 64 on a superconducting quantum computing system'' Quantum Science and Technology 6, 025020 (2021).
https:/​/​doi.org/​10.1088/​2058-9565/​abe519
arXiv:2008.08571

[6] Andrew D. King, Juan Carrasquilla, Jack Raymond, Isil Ozfidan, Evgeny Andriyash, Andrew Berkley, Mauricio Reis, Trevor Lanting, Richard Harris, Fabio Altomare, Kelly Boothby, Paul I. Bunyk, Colin Enderud, Alexandre Fréchette, Emile Hoskinson, Nicolas Ladizinsky, Travis Oh, Gabriel Poulin-Lamarre, Christopher Rich, Yuki Sato, Anatoly Yu. Smirnov, Loren J. Swenson, Mark H. Volkmann, Jed Whittaker, Jason Yao, Eric Ladizinsky, Mark W. Johnson, Jeremy Hilton, and Mohammad H. Amin, ``Observation of topological phenomena in a programmable lattice of 1,800 qubits'' Nature 560, 456–460 (2018).
https:/​/​doi.org/​10.1038/​s41586-018-0410-x
arXiv:1803.02047

[7] K. Wright, K. M. Beck, S. Debnath, J. M. Amini, Y. Nam, N. Grzesiak, J.-S. Chen, N. C. Pisenti, M. Chmielewski, C. Collins, K. M. Hudek, J. Mizrahi, J. D. Wong-Campos, S. Allen, J. Apisdorf, P. Solomon, M. Williams, A. M. Ducore, A. Blinov, S. M. Kreikemeier, V. Chaplin, M. Keesan, C. Monroe, and J. Kim, ``Benchmarking an 11-qubit quantum computer'' Nature Communications 10, 5464 (2019).
https:/​/​doi.org/​10.1038/​s41467-019-13534-2
arXiv:1903.08181

[8] Yunseong Nam, Jwo-Sy Chen, Neal C. Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R. Brown, Stewart Allen, Jason M. Amini, Joel Apisdorf, Kristin M. Beck, Aleksey Blinov, Vandiver Chaplin, Mika Chmielewski, Coleman Collins, Shantanu Debnath, Kai M. Hudek, Andrew M. Ducore, Matthew Keesan, Sarah M. Kreikemeier, Jonathan Mizrahi, Phil Solomon, Mike Williams, Jaime David Wong-Campos, David Moehring, Christopher Monroe, and Jungsang Kim, ``Ground-state energy estimation of the water molecule on a trapped-ion quantum computer'' npj Quantum Information 6, 33 (2020).
https:/​/​doi.org/​10.1038/​s41534-020-0259-3
arXiv:1902.10171

[9] Sonika Johri, Shantanu Debnath, Avinash Mocherla, Alexandros Singh, Anupam Prakash, Jungsang Kim, and Iordanis Kerenidis, ``Nearest centroid classification on a trapped ion quantum computer'' npj Quantum Information 7, 122 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00456-5
arXiv:2012.04145

[10] Manuel S. Rudolph, Ntwali Bashige Toussaint, Amara Katabarwa, Sonika Johri, Borja Peropadre, and Alejandro Perdomo-Ortiz, ``Generation of High-Resolution Handwritten Digits with an Ion-Trap Quantum Computer'' Physical Review X 12, 031010 (2022).
https:/​/​doi.org/​10.1103/​PhysRevX.12.031010
arXiv:2012.03924

[11] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Phys. Rev. A 64, 012310 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.012310

[12] Sergey Bravyiand Alexei Kitaev ``Universal quantum computation with ideal Clifford gates and noisy ancillas'' Physical Review A 71, 022316 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.022316

[13] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot, ``Roads towards fault-tolerant universal quantum computation'' Nature 549, 172–179 (2017).
https:/​/​doi.org/​10.1038/​nature23460
arXiv:1612.07330

[14] Daniel Gottesman ``The Heisenberg representation of quantum computers'' Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics 32–43 (1999).

[15] Scott Aaronsonand Daniel Gottesman ``Improved simulation of stabilizer circuits'' Physical Review A 70, 052328 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.70.052328

[16] E. Wigner ``On the Quantum Correction For Thermodynamic Equilibrium'' Physical Review 40, 749–759 (1932).
https:/​/​doi.org/​10.1103/​PhysRev.40.749

[17] R.L. Hudson ``When is the Wigner quasi-probability density non-negative?'' Reports on Mathematical Physics 6, 249–252 (1974).
https:/​/​doi.org/​10.1016/​0034-4877(74)90007-X

[18] Anatole Kenfackand Karol yczkowski ``Negativity of the Wigner function as an indicator of non-classicality'' Journal of Optics B: Quantum and Semiclassical Optics 6, 396–404 (2004).
https:/​/​doi.org/​10.1088/​1464-4266/​6/​10/​003

[19] William K Wootters ``A Wigner-function formulation of finite-state quantum mechanics'' Annals of Physics 176, 1–21 (1987).
https:/​/​doi.org/​10.1016/​0003-4916(87)90176-X

[20] Kathleen S. Gibbons, Matthew J. Hoffman, and William K. Wootters, ``Discrete phase space based on finite fields'' Physical Review A 70, 062101 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.70.062101

[21] Ernesto F. Galvão ``Discrete Wigner functions and quantum computational speedup'' Physical Review A 71, 042302 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.042302

[22] Cecilia Cormick, Ernesto F. Galvão, Daniel Gottesman, Juan Pablo Paz, and Arthur O. Pittenger, ``Classicality in discrete Wigner functions'' Physical Review A 73, 012301 (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.73.012301

[23] David Gross ``Hudson’s theorem for finite-dimensional quantum systems'' Journal of Mathematical Physics 47, 122107 (2006).
https:/​/​doi.org/​10.1063/​1.2393152

[24] David Gross ``Non-negative Wigner functions in prime dimensions'' Applied Physics B 86, 367–370 (2006).
https:/​/​doi.org/​10.1007/​s00340-006-2510-9

[25] David Gross ``Computational power of quantum many-body states and some results on discrete phase spaces'' thesis (2008) https:/​/​www.thp.uni-koeln.de/​gross/​files/​diss.pdf.
https:/​/​www.thp.uni-koeln.de/​gross/​files/​diss.pdf

[26] R. L. Stratonovich ``On Distributions in Representation Space'' Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki 31, 1012–1020 (1956) [English translation: Soviet Physics JETP, 4.6:891–898 (1957)].
http:/​/​jetp.ras.ru/​cgi-bin/​dn/​e_004_06_0891.pdf

[27] C. Brifand A. Mann ``A general theory of phase-space quasiprobability distributions'' Journal of Physics A: Mathematical and General 31, L9–L17 (1998).
https:/​/​doi.org/​10.1088/​0305-4470/​31/​1/​002

[28] Robert Raussendorf, Cihan Okay, Michael Zurel, and Polina Feldmann, ``The role of cohomology in quantum computation with magic states'' Quantum 7, 979 (2023).
https:/​/​doi.org/​10.22331/​q-2023-04-13-979
arXiv:2110.11631

[29] Victor Veitch, Christopher Ferrie, David Gross, and Joseph Emerson, ``Negative quasi-probability as a resource for quantum computation'' New Journal of Physics 14, 113011 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​11/​113011
arXiv:1201.1256

[30] Michael Zurel ``Hidden variable models and classical simulation algorithms for quantum computation with magic states on qubits'' thesis (2020).
https:/​/​doi.org/​10.14288/​1.0394790

[31] Hakop Pashayan, Joel J. Wallman, and Stephen D. Bartlett, ``Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities'' Physical Review Letters 115 (2015).
https:/​/​doi.org/​10.1103/​physrevlett.115.070501
arXiv:1503.07525

[32] Robert W. Spekkens ``Negativity and Contextuality are Equivalent Notions of Nonclassicality'' Physical Review Letters 101, 020401 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.020401
arXiv:0710.5549

[33] Nicolas Delfosse, Cihan Okay, Juan Bermejo-Vega, Dan E Browne, and Robert Raussendorf, ``Equivalence between contextuality and negativity of the Wigner function for qudits'' New Journal of Physics 19, 123024 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8fe3
arXiv:1610.07093

[34] Robert Raussendorf ``Contextuality in measurement-based quantum computation'' Physical Review A 88 (2013).
https:/​/​doi.org/​10.1103/​physreva.88.022322
arXiv:0907.5449

[35] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson, ``Contextuality supplies the “magic” for quantum computation'' Nature 510, 351–355 (2014).
https:/​/​doi.org/​10.1038/​nature13460
arXiv:1401.4174

[36] Farid Shahandeh ``Quantum computational advantage implies contextuality'' (2021).
arXiv:2112.00024

[37] Huangjun Zhu ``Permutation Symmetry Determines the Discrete Wigner Function'' Physical Review Letters 116, 040501 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.040501
arXiv:1504.03773

[38] Angela Karanjai, Joel J. Wallman, and Stephen D. Bartlett, ``Contextuality bounds the efficiency of classical simulation of quantum processes'' (2018).
arXiv:1802.07744

[39] David Schmid, Haoxing Du, John H. Selby, and Matthew F. Pusey, ``Uniqueness of Noncontextual Models for Stabilizer Subtheories'' Physical Review Letters 129, 120403 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.129.120403
arXiv:2101.06263

[40] Nicolas Delfosse, Philippe Allard Guerin, Jacob Bian, and Robert Raussendorf, ``Wigner Function Negativity and Contextuality in Quantum Computation on Rebits'' Physical Review X 5, 021003 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.021003
arXiv:1409.5170

[41] Juan Bermejo-Vega, Nicolas Delfosse, Dan E. Browne, Cihan Okay, and Robert Raussendorf, ``Contextuality as a Resource for Models of Quantum Computation with Qubits'' Physical Review Letters 119 (2017).
https:/​/​doi.org/​10.1103/​physrevlett.119.120505
arXiv:1610.08529

[42] Robert Raussendorf, Dan E. Browne, Nicolas Delfosse, Cihan Okay, and Juan Bermejo-Vega, ``Contextuality and Wigner-function negativity in qubit quantum computation'' Physical Review A 95 (2017).
https:/​/​doi.org/​10.1103/​physreva.95.052334
arXiv:1511.08506

[43] Mark Howardand Earl Campbell ``Application of a Resource Theory for Magic States to Fault-Tolerant Quantum Computing'' Physical Review Letters 118 (2017).
https:/​/​doi.org/​10.1103/​physrevlett.118.090501
arXiv:1609.07488

[44] Lucas Kociaand Peter Love ``Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states'' Physical Review A 96, 062134 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.062134
arXiv:1705.08869

[45] Robert Raussendorf, Juani Bermejo-Vega, Emily Tyhurst, Cihan Okay, and Michael Zurel, ``Phase-space-simulation method for quantum computation with magic states on qubits'' Physical Review A 101, 012350 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.012350
arXiv:1905.05374

[46] Michael Zurel, Cihan Okay, and Robert Raussendorf, ``Hidden Variable Model for Universal Quantum Computation with Magic States on Qubits'' Physical Review Letters 125, 260404 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.260404
arXiv:2004.01992

[47] Cihan Okay, Michael Zurel, and Robert Raussendorf, ``On the extremal points of the $\Lambda$-polytopes and classical simulation of quantum computation with magic states'' Quantum Information & Computation 21 (2021).
https:/​/​doi.org/​10.26421/​QIC21.13-14-2
arXiv:2104.05822

[48] Daniel Gottesman ``Fault-Tolerant Quantum Computation with Higher-Dimensional Systems'' Chaos, Solitons & Fractals 10, 1749–1758 (1999).
https:/​/​doi.org/​10.1016/​s0960-0779(98)00218-5

[49] Sergey Bravyi, Graeme Smith, and John A Smolin, ``Trading classical and quantum computational resources'' Physical Review Letters 6, 021043 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.021043
arXiv:1506.01396

[50] Cihan Okay, Sam Roberts, Stephen D. Bartlett, and Robert Raussendorf, ``Topological proofs of contextuality in quantum mechanics'' Quantum Information and Computation 17, 1135–1166 (2017).
https:/​/​doi.org/​10.26421/​QIC17.13-14-5
arXiv:1701.01888

[51] Gunter M. Ziegler ``Lectures on polytopes'' Springer-Verlag (1995).

[52] John B DeBrotaand Blake C Stacey ``Discrete Wigner functions from informationally complete quantum measurements'' Phys. Rev. A 102, 032221 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.032221
arXiv:1912.07554

[53] E.G. Beltramettiand S. Bugajski ``The Bell phenomenon in classical frameworks'' Journal of Physics A: Mathematical and General 29, 247 (1996).
https:/​/​doi.org/​10.1088/​0305-4470/​29/​2/​005

[54] Vašek Chvátal ``Linear Programming'' W.H. Freeman Company (1983).

[55] Arne Heimendahl ``The Stabilizer Polytope and Contextuality for Qubit Systems'' thesis (2019) http:/​/​www.mi.uni-koeln.de/​opt/​wp-content/​uploads/​2020/​07/​MT_Arne_Heimendahl.pdf.
http:/​/​www.mi.uni-koeln.de/​opt/​wp-content/​uploads/​2020/​07/​MT_Arne_Heimendahl.pdf

[56] Vlad Gheorghiu ``Standard form of qudit stabilizer groups'' Physics Letters A 378, 505–509 (2014).
https:/​/​doi.org/​10.1016/​j.physleta.2013.12.009
arXiv:1101.1519

[57] Chuangxun Cheng ``A character theory for projective representations of finite groups'' Linear Algebra and its Applications 469, 230–242 (2015).
https:/​/​doi.org/​10.1016/​j.laa.2014.11.027

[58] Ben W. Reichardt ``Quantum universality by state distillation'' Quantum Information and Computation 9, 1030–1052 (2009).
https:/​/​doi.org/​10.26421/​QIC9.11-12-7

[59] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson, ``The resource theory of stabilizer quantum computation'' New Journal of Physics 16, 013009 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​1/​013009
arXiv:1307.7171

[60] William M. Kirbyand Peter J. Love ``Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers'' Physical Review Letters 123 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.123.200501
arXiv:1904.02260

[61] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' thesis (1997) arXiv:quant-ph/​9705052.
arXiv:9705052

[62] Stanislaw Szarekand Guillaume Aubrun ``Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory'' American Mathematical Society (2017).

[63] Richard A. Brualdi ``Introductory Combinatorics'' Pearson Education International (2012).

Cited by

[1] Robert Raussendorf, Cihan Okay, Michael Zurel, and Polina Feldmann, "The role of cohomology in quantum computation with magic states", arXiv:2110.11631, (2021).

[2] Michael Zurel, Lawrence Z. Cohen, and Robert Raussendorf, "Simulation of quantum computation with magic states via Jordan-Wigner transformations", arXiv:2307.16034, (2023).

[3] Michael Zurel, Cihan Okay, and Robert Raussendorf, "Simulating quantum computation: how many "bits" for "it"?", arXiv:2305.17287, (2023).

[4] William F. Braasch and William K. Wootters, "A quantum prediction as a collection of epistemically restricted classical predictions", Quantum 6, 659 (2022).

[5] 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).

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