Bibliography

[1]    Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70(5):052328, 2004.

[2]    S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS), pages 415–425, 2004. arXiv:quant-ph/0402130.

[3]    Leonard M Adleman, Jonathan DeMarrais, and Ming-Deh A Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524–1540, 1997.

[4]    Dorit Aharonov. A simple proof that Toffoli and Hadamard are quantum universal. arXiv preprint quant-ph/0301040, 2003.

[5]    Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 176–188, 1997.

[6]    Dorit Aharonov and Tomer Naveh. Quantum np-a survey. arXiv preprint quant-ph/0210077, 2002.

[7]    Gernot Alber, Thomas Beth, Michał Horodecki, Paweł Horodecki, Ryszard Horodecki, Martin Rötteler, Harald Weinfurter, Reinhard Werner, Anton Zeilinger, Thomas Beth, et al. Quantum algorithms: Applicable algebra and quantum physics. Quantum information: an introduction to basic theoretical concepts and experiments, pages 96–150, 2001.

[8]    Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. arXiv preprint quant-ph/0504218, 2005.

[9]    A. Ambainis. New developments in quantum algorithms. arXiv:1006.4014, 2010.

[10]    M. Amy, D. Maslov, M. Mosca, and M. Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 32(6):818–830, 6 2013.

[11]    Matthew Amy. Formal methods in quantum circuit design. PhD thesis, University of Waterloo, 2019.

[12]    Matthew Amy, Jianxin Chen, and Neil J Ross. A finite presentation of CNOT-dihedral operators. In Bob Coecke and Aleks Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science. Open Publishing Association, 2018.

[13]    Matthew Amy, Matthew Crawford, Andrew N Glaudell, Melissa L Macasieb, Samuel S Mendelson, and Neil J Ross. Catalytic embeddings of quantum circuits. arXiv preprint arXiv:2305.07720, 2023.

[14]     Matthew Amy, Dmitri Maslov, and Michele Mosca. Polynomial-time T-depth optimization of Clifford+ T circuits via matroid partitioning. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 33(10):1476–1489, 2014.

[15]    Matthew Amy, Dmitri Maslov, Michele Mosca, and Martin Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 32(6):818–830, 2013.

[16]    Matthew Amy and Michele Mosca. T-count optimization and Reed-Muller codes. Transactions on Information Theory, 2019.

[17]    Matthew Amy and Neil J. Ross. Phase-state duality in reversible circuit design. Phys. Rev. A, 104:052602, Nov 2021.

[18]    Sanjeev Arora, László Babai, Jacques Stern, and Z Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. Journal of Computer and System Sciences, 54(2):317–331, 1997.

[19]    Miriam Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16(9):093021, 2014.

[20]    Miriam Backens. The zx-calculus is complete for the single-qubit clifford+t group. In Bob Coecke, Ichiro Hasuo, and Prakash Panangaden, editors, Proceedings of the 11th workshop on Quantum Physics and Logic, volume 172 of Electronic Proceedings in Theoretical Computer Science, pages 293–303. Open Publishing Association, 2014.

[21]    Miriam Backens and Aleks Kissinger. ZH: A complete graphical calculus for quantum computations involving classical non-linearity. In Peter Selinger and Giulio Chiribella, editors, Proceedings of the 15th International Conference on Quantum Physics and Logic, Halifax, Canada, 3-7th June 2018, volume 287 of Electronic Proceedings in Theoretical Computer Science, pages 18–34. Open Publishing Association, 2019.

[22]    Miriam Backens, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. Completeness of the ZH-calculus. Compositionality, 5, 7 2023.

[23]    Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, and John van de Wetering. There and back again: A circuit extraction tale. Quantum, 5:421, 3 2021.

[24]    Miriam Backens, Simon Perdrix, and Quanlong Wang. A simplified stabilizer zx-calculus. In Proceedings of the 13th International Conference on Quantum Physics and Logic, 2016. arXiv:1602.04744.

[25]    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Physical Review A, 52:3457–3467, 1995.

[26]    P. Benioff. The computer as a physical system: A microscopic quantum mechanical hamiltonian model of computers as represented by turing machines. Journal of Statistical Physics, 22:563–591, 1980.

[27]    Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996.

[28]    E. Berlekamp, R. McEliece, and H. van Tilborg. On the inherent intractability of certain coding problems (corresp.). IEEE Transactions on Information Theory, 24(3):384–386, 1978.

[29]    Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. SIAM Journal on Computing, 26(5):1411–1473, 1997.

[30]    Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating hamiltonian dynamics with a truncated taylor series. Phys. Rev. Lett., 114:090502, Mar 2015.

[31]    Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Interacting bialgebras are Frobenius. In International Conference on Foundations of Software Science and Computation Structures, pages 351–365. Springer, 2014.

[32]    Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum, 3:181, 9 2019.

[33]    Sergey Bravyi and David Gosset. Improved classical simulation of quantum circuits dominated by clifford gates. Phys. Rev. Lett., 116:250501, Jun 2016.

[34]    Sergey Bravyi, David Gosset, and Yinchen Liu. How to simulate quantum measurement without computing marginals. Phys. Rev. Lett., 128:220503, Jun 2022.

[35]    Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86:052329, Nov 2012.

[36]    Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71(2):022316, 2005.

[37]    Sergey Bravyi, Graeme Smith, and John A Smolin. Trading classical and quantum computational resources. Physical Review X, 6(2):021043, 2016.

[38]    Sergey B Bravyi and A Yu Kitaev. Quantum codes on a lattice with boundary. arXiv preprint quant-ph/9811052, 1998.

[39]    Hans J Briegel, David E Browne, Wolfgang Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5(1):19–26, 2009.

[40]    Hans J Briegel and Robert Raussendorf. Persistent entanglement in arrays of interacting particles. Physical Review Letters, 86(5):910, 2001.

[41]    A. Broadbent, J. Fitzsimons, and E. Kashefi. Universal Blind Quantum Computation, pages 517–526. Annual Symposium on Foundations of Computer Science. IEEE Computer Society, 2009.

[42]    Anne Broadbent and Elham Kashefi. Parallelizing quantum circuits. Theoretical Computer Science, 410(26):2489–2510, 2009.

[43]    Daniel E. Browne, Elham Kashefi, Mehdi Mhalla, and Simon Perdrix. Generalized flow and determinism in measurement-based quantum computation. New Journal of Physics, 9(8):250, 2007.

[44]    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthogonal geometry. Phys. Rev. Lett., 78:405–408, Jan 1997.

[45]    A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54:1098–1105, Aug 1996.

[46]    Earl Campbell. The Smallest Interesting Colour Code. https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/.

[47]    Earl Campbell. Random compiler for fast hamiltonian simulation. Phys. Rev. Lett., 123:070503, Aug 2019.

[48]    Titouan Carette. When Only Topology Matters. arXiv preprint arXiv:2102.03178, 2021.

[49]    Titouan Carette, Dominic Horsman, and Simon Perdrix. SZX-Calculus: Scalable Graphical Quantum Reasoning. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), volume 138 of Leibniz International Proceedings in Informatics (LIPIcs), pages 55:1–55:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.

[50]    Titouan Carette and Emmanuel Jeandel. A Recipe for Quantum Graphical Languages. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 118:1–118:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl–Leibniz-Zentrum für Informatik.

[51]    Nicholas Chancellor, Aleks Kissinger, Joschka Roffe, Stefan Zohren, and Dominic Horsman. Graphical structures for design and verification of quantum error correction. arXiv preprint arXiv:1611.08012, 2016.

[52]    A. Chi-Chih Yao. Quantum circuit complexity. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 352–361, November 1993.

[53]    B. Coecke and A. Kissinger. The compositional structure of multipartite quantum entanglement. In Automata, Languages and Programming, Lecture Notes in Computer Science, pages 297–308. Springer, 2010. arXiv:1002.2540.

[54]    B. Coecke, A. Kissinger, A. Merry, and S. Roy. The ghz/w-calculus contains rational arithmetic. Electronic Proceedings in Theoretical Computer Science, 52:34–48, 2010.

[55]    B. Coecke and D. Pavlovic. Quantum measurements without sums. In G. Chen, L. Kauffman, and S. Lamonaco, editors, Mathematics of Quantum Computing and Technology, pages 567–604. Taylor and Francis, 2007. arXiv:quant-ph/0608035.

[56]    B. Coecke, D. Pavlović, and J. Vicary. A new description of orthogonal bases. Mathematical Structures in Computer Science, to appear, 23:555–567, 2013. arXiv:quant-ph/0810.1037.

[57]    Bob Coecke and Ross Duncan. Interacting quantum observables. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, 2008.

[58]    Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13:043016, 2011.

[59]    Bob Coecke, Ross Duncan, Aleks Kissinger, and Quanlong Wang. Strong Complementarity and Non-locality in Categorical Quantum Mechanics. In 2012 27th Annual IEEE Symposium on Logic in Computer Science, pages 245–254, 2012.

[60]    Bob Coecke and Stefano Gogioso. Quantum in Pictures. 2023.

[61]    Bob Coecke and Aleks Kissinger. The compositional structure of multipartite quantum entanglement. In Automata, Languages and Programming, Lecture Notes in Computer Science, pages 297–308. Springer, 2010.

[62]    Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017.

[63]    Bob Coecke and Quanlong Wang. ZX-rules for 2-qubit Clifford+ T quantum circuits. In International Conference on Reversible Computation, pages 144–161. Springer, 2018.

[64]    Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah. Phase Gadget Synthesis for Shallow Circuits. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 213–228. Open Publishing Association, 2020.

[65]    Andrew Cross, Ali Javadi-Abhari, Thomas Alexander, Niel De Beaudrap, Lev S Bishop, Steven Heidel, Colm A Ryan, Prasahnt Sivarajah, John Smolin, Jay M Gambetta, et al. Openqasm 3: A broader and deeper quantum assembly language. ACM Transactions on Quantum Computing, 3(3):1–50, 2022.

[66]    Andrew W Cross, Lev S Bishop, John A Smolin, and Jay M Gambetta. Open quantum assembly language. Preprint, 2017.

[67]    Shawn X. Cui, Daniel Gottesman, and Anirudh Krishna. Diagonal gates in the Clifford hierarchy. Physical Review A, 95(1):012329, 2017.

[68]    Alexander M Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T Hann, Michael J Kastoryano, Emil T Khabiboulline, Aleksander Kubica, et al. Quantum algorithms: A survey of applications and end-to-end complexities. arXiv preprint arXiv:2310.03011, 2023.

[69]    V. Danos and E. Kashefi. Determinism in the one-way model. Physical Review A, 74(052310), 2006.

[70]    Vincent Danos, Elham Kashefi, and Prakash Panangaden. The measurement calculus. Journal of the ACM (JACM), 54(2):8–es, 2007.

[71]    Vincent Danos, Elham Kashefi, Prakash Panangaden, and Simon Perdrix. Extended measurement calculus. Semantic techniques in quantum computation, pages 235–310, 2009.

[72]    Christopher M Dawson, Andrew P Hines, Duncan Mortimer, Henry L Haselgrove, Michael A Nielsen, and Tobias J Osborne. Quantum computing and polynomial equations over the finite field z2. Quantum Information & Computation, 5(2):102–112, 2005.

[73]    Christopher M Dawson and Michael A Nielsen. The solovay-kitaev algorithm. Preprint, 2005.

[74]    Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020), volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1–11:23, Dagstuhl, Germany, 2020. Schloss Dagstuhl–Leibniz-Zentrum für Informatik.

[75]    Niel de Beaudrap, Xiaoning Bian, and Quanlong Wang. Techniques to Reduce π4-Parity-Phase Circuits, Motivated by the ZX Calculus. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 131–149. Open Publishing Association, 2020.

[76]    Niel de Beaudrap, Ross Duncan, Dominic Horsman, and Simon Perdrix. Pauli Fusion: a Computational Model to Realise Quantum Transformations from ZX Terms. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 85–105. Open Publishing Association, 2020.

[77]    Niel de Beaudrap and Dominic Horsman. The ZX calculus is a language for surface code lattice surgery. Quantum, 4, 2020.

[78]    Timothée Goubault de Brugière, Marc Baboulin, Benoît Valiron, Simon Martiel, and Cyril Allouche. Quantum cnot circuits synthesis for nisq architectures using the syndrome decoding problem. In Reversible Computation: 12th International Conference, RC 2020, Oslo, Norway, July 9-10, 2020, Proceedings 12, pages 189–205. Springer, 2020.

[79]    Jeroen Dehaene and Bart De Moor. Clifford group, stabilizer states, and linear and quadratic operations over GF(2). Physical Review A, 68(4):042318, 2003.

[80]    Nicolas Delfosse. Hierarchical decoding to reduce hardware requirements for quantum computing. arXiv preprint arXiv:2001.11427, 2020.

[81]    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452–4505, 2002.

[82]    D. Deutsch. Quantum computational networks. Proceedings of the Royal Society of London, 425, 1989.

[83]    D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, 1992.

[84]    David Deutsch. Quantum theory, the church–turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97–117, 1985.

[85]    Olivia Di Matteo and Michele Mosca. Parallelizing quantum circuit synthesis. Quantum Science and Technology, 1(1):015003, 2016.

[86]    David P DiVincenzo. Two-bit gates are universal for quantum computation. Physical Review A, 51(2):1015, 1995.

[87]    Ross Duncan and Kevin Dunne. Interacting Frobenius Algebras are Hopf. In 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1–10. IEEE, 2016.

[88]    Ross Duncan, Aleks Kissinger, Simon Perdrix, and John van de Wetering. Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus. Quantum, 4:279, 6 2020.

[89]    Ross Duncan and Maxime Lucas. Verifying the Steane code with Quantomatic. In Bob Coecke and Matty Hoban, editors, Proceedings of the 10th International Workshop on Quantum Physics and Logic, Castelldefels (Barcelona), Spain, 17th to 19th July 2013, volume 171 of Electronic Proceedings in Theoretical Computer Science, pages 33–49. Open Publishing Association, 2014.

[90]    Ross Duncan and Simon Perdrix. Graph states and the necessity of Euler decomposition. Mathematical Theory and Computational Practice, pages 167–177, 2009.

[91]    Ross Duncan and Simon Perdrix. Rewriting measurement-based quantum computations with generalised flow. In International Colloquium on Automata, Languages, and Programming, pages 285–296. Springer, 2010.

[92]    Ross Duncan and Simon Perdrix. Pivoting makes the zx-calculus complete for real stabilizers. In QPL 2013-10th Workshop on Quantum Physics and Logic, 2013.

[93]    Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17:449–467, 1965.

[94]    Matthew B Elliott, Bryan Eastin, and Carlton M Caves. Graphical description of the action of Clifford operators on stabilizer states. Physical Review A, 77(4):042307, 2008.

[95]     Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Stadler, Vlad Negnevitsky, Martin Ringbauer, Philipp Schindler, Hans J Briegel, et al. Entangling logical qubits with lattice surgery. Nature, 589(7841):220–224, 2021.

[96]    R. P. Feynman. Simulating physics with computers. International journal of theoretical physics, 21:467–488, 1982.

[97]    Lance Fortnow and John Rogers. Complexity limitations on quantum computation. Journal of Computer and System Sciences, 59(2):240–252, 1999.

[98]    Austin G Fowler and Craig Gidney. Low overhead quantum computation using lattice surgery. arXiv preprint arXiv:1808.06709, 2018.

[99]    Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012.

[100]    Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Phys. Rev. A, 80:052312, Nov 2009.

[101]    Liam Garvie and Ross Duncan. Verifying the Smallest Interesting Colour Code with Quantomatic. In Bob Coecke and Aleks Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 147–163. Open Publishing Association, 2018.

[102]    Craig Gidney. Constructing Large Controlled Nots. Blogpost, June 2015. https://algassert.com/circuits/2015/06/05/Constructing-Large-Controlled-Nots.html.

[103]    Craig Gidney. Halving the cost of quantum addition. Quantum, 2:74, 6 2018.

[104]    Craig Gidney and Martin Ekerå. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum, 5:433, 4 2021.

[105]    Craig Gidney and Austin G. Fowler. Efficient magic state factories with a catalyzed |CCZ to 2|T transformation. Quantum, 3:135, 4 2019.

[106]    Craig Gidney and N Cody Jones. A cccz gate performed with 6 t gates. arXiv preprint arXiv:2106.11513, 2021.

[107]    Brett Giles and Peter Selinger. Exact synthesis of multiqubit Clifford+T circuits. Physical Review A, 87(3):032332, 2013.

[108]    Mercedes Gimeno-Segovia, Pete Shadbolt, Dan E. Browne, and Terry Rudolph. From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation. Physical Review Letters, 115(2):020502, 2015.

[109]    Marek Gluza. Double-bracket quantum algorithms for diagonalization. Quantum, 8:1316, 4 2024.

[110]    Stefano Gogioso. A Diagrammatic Approach to Quantum Dynamics. In Markus Roggenbach and Ana Sokolova, editors, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019), volume 139 of Leibniz International Proceedings in Informatics (LIPIcs), pages 19:1–19:23, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.

[111]    Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum hamming bound. Physical Review A, 54(3):1862, 1996.

[112]    Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint quant-ph / 9705052, 1997.

[113]    Daniel Gottesman. The heisenberg representation of quantum computers. arXiv preprint quant-ph/9807006, 1998.

[114]    Daniel Gottesman. Surviving as a quantum computer in a classical world. Textbook manuscript preprint, 2024.

[115]    Daniel Gottesman and Isaac L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402(6760):390–393, 1999.

[116]    Timothèe Goubault de Brugière. Methods for optimizing the synthesis of quantum circuits. PhD thesis, Université Paris-Saclay, 2020.

[117]    Alexander S Green, Peter LeFanu Lumsdaine, Neil J Ross, Peter Selinger, and Benoît Valiron. Quipper: a scalable quantum programming language. In Proceedings of the 34th ACM SIGPLAN conference on Programming language design and implementation, pages 333–342, 2013.

[118]    Seth Greylyn. Generators and relations for the group u([12,i]). Master’s thesis, Dalhousie University, 2014.

[119]    David Gross and Maarten Van den Nest. The lu-lc conjecture, diagonal local operations and quadratic forms over gf(2). Quantum Info. Comput., 8(3):263–281, mar 2008.

[120]    Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC ’96, pages 212–219, New York, NY, USA, 1996. ACM.

[121]    A. Hadzihasanovic. A diagrammatic axiomatisation for qubit entanglement. In Proceedings of the 30th Annual IEEE Symposium on Logic in Computer Science (LICS), 2015. arXiv:1501.07082.

[122]    Amar Hadzihasanovic. A diagrammatic axiomatisation for qubit entanglement. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 573–584. IEEE, 2015.

[123]    Amar Hadzihasanovic, Kang Feng Ng, and Quanlong Wang. Two complete axiomatisations of pure-state qubit quantum computing. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’18, pages 502–511, New York, NY, USA, 2018. ACM.

[124]    Brian C Hall and Brian C Hall. Lie groups, Lie algebras, and representations. Springer, 2013.

[125]    Chris Heunen and Jamie Vicary. Categories for Quantum Theory: an introduction. Oxford University Press, USA, 2020.

[126]    Luke E Heyfron and Earl T Campbell. An efficient quantum compiler that reduces T count. Quantum Science and Technology, 4(015004), 2018.

[127]    Oscar Higgott. Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching. 3(3), 6 2022.

[128]    Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell. Improved decoding of circuit noise and fragile boundaries of tailored surface codes. Phys. Rev. X, 13:031007, Jul 2023.

[129]    Clare Horsman. Quantum picturalism for topological cluster-state computing. New Journal of Physics, 13(9):095011, 2011.

[130]    Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery. New Journal of Physics, 14(12):123011, 2012.

[131]    Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett., 118:090501, Mar 2017.

[132]    Min-Hsiu Hsieh and François Le Gall. Np-hardness of decoding quantum error-correction codes. Physical Review A—Atomic, Molecular, and Optical Physics, 83(5):052331, 2011.

[133]    IBM. Qiskit. https://www.ibm.com/quantum/qiskit.

[134]    Antonio deMarti iOlius, Josu Etxezarreta Martinez, Patricio Fuentes, and Pedro M. Crespo. Performance enhancement of surface codes via recursive minimum-weight perfect-match decoding. Phys. Rev. A, 108:022401, Aug 2023.

[135]    Emmanuel Jeandel, Simon Perdrix, and Margarita Veshchezerova. Addition and Differentiation of ZX-diagrams. Logical Methods in Computer Science, Volume 20, Issue 2, 5 2024.

[136]    Emmanuel Jeandel, Simon Perdrix, and Renaud Vilmart. A Complete Axiomatisation of the ZX-calculus for Clifford+T Quantum Mechanics. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 559–568. ACM, 2018.

[137]    Cody Jones. Low-overhead constructions for the fault-tolerant Toffoli gate. Physical Review A, 87(2):022328, 2013.

[138]    Stephen Jordan. The quantum algorithms zoo, 2022. https://quantumalgorithmzoo.org.

[139]    R. Jozsa. Quantum algorithms and the fourier transform. In Proceedings of the Santa Barbarba Conference on Coherence and Decoherence, Proceedings of the Royal Society of London, 1997.

[140]    Phillip Kaye, Raymond Laflamme, and Michele Mosca. An introduction to quantum computing. Oxford University Press, 2007.

[141]    Gregory M Kelly and Miguel L Laplaza. Coherence for compact closed categories. Journal of pure and applied algebra, 19:193–213, 1980.

[142]    Aleks Kissinger. Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code). arXiv preprint arXiv:2204.14038, 2022.

[143]    Aleks Kissinger and Arianne Meijer-van de Griend. Cnot circuit extraction for topologically-constrained quantum memories. Quant. Inf. Comput., 20(arXiv: 1904.00633):581–596, 2020.

[144]    Aleks Kissinger, Neil J. Ross, and John van de Wetering. Catalysing Completeness and Universality. arXiv preprint arXiv:2404.09915, 2024.

[145]    Aleks Kissinger and John van de Wetering. Universal MBQC with generalised parity-phase interactions and Pauli measurements. Quantum, 3:134, 4 2019.

[146]    Aleks Kissinger and John van de Wetering. PyZX: Large Scale Automated Diagrammatic Reasoning. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 229–241. Open Publishing Association, 2020.

[147]    Aleks Kissinger and John van de Wetering. Reducing the number of non-Clifford gates in quantum circuits. Physical Review A, 102:022406, 8 2020.

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

[149]    Aleks Kissinger and John van de Wetering. Scalable Spider Nests (...Or How to Graphically Grok Transversal Non-Clifford Gates). In Alejandro Díaz-Caro and Vladimir Zamdzhiev, editors, Proceedings of the 21st International Conference on Quantum Physics and Logic, Buenos Aires, Argentina, July 15-19, 2024, volume 406 of Electronic Proceedings in Theoretical Computer Science, pages 79–95. Open Publishing Association, 2024.

[150]    Aleks Kissinger, John van de Wetering, and Renaud Vilmart. Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions. In François Le Gall and Tomoyuki Morimae, editors, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1–5:13, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.

[151]    Aleks Kissinger and Vladimir Zamdzhiev. Quantomatic: A proof assistant for diagrammatic reasoning. In International Conference on Automated Deduction, pages 326–336. Springer, 2015.

[152]    A Yu Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6):1191, 1997.

[153]    A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of physics, 303(1):2–30, 2003.

[154]    Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit. Shorter quantum circuits via single-qubit gate approximation. Quantum, 7:1208, 12 2023.

[155]    Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434(7029):39–44, 2005.

[156]    Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55:900–911, Feb 1997.

[157]    Emanuel Knill, Raymond Laflamme, and Wojciech H Zurek. Resilient quantum computation. Science, 279(5349):342–345, 1998.

[158]    Anton Kotzig. Eulerian lines in finite 4-valent graphs and their transformations. In Colloqium on Graph Theory Tihany 1966, pages 219–230. Academic Press, 1968.

[159]    Stach Kuijpers, John van de Wetering, and Aleks Kissinger. Graphical fourier theory and the cost of quantum addition. Preprint, 2019.

[160]    S. Lack. Composing PROPs. Theory and Applications of Categories, 13:147–163, 2004.

[161]    Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77:198–201, Jul 1996.

[162]    R. Landauer. Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5(3):183–191, 1961.

[163]    Louis Lemonnier, John van de Wetering, and Aleks Kissinger. Hypergraph simplification: Linking the path-sum approach to the ZH-calculus. arXiv preprint arXiv:2003.13564, 2020.

[164]    Daniel Litinski. A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery. Quantum, 3:128, 3 2019.

[165]    Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013.

[166]    Saunders MacLane. Categorical algebra. Bulletin of the American Mathematical Society, 71(1):40–106, 1965.

[167]    F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes, volume 16 of North-Holland Mathematical Library. North Holland Publishing Co., 1977.

[168]    Y. I. Manin. Vychislimoe i nevychislimoe. Sovetskoye Radio, 1980.

[169]    Ketan Markov, Igor Patel, and John Hayes. Optimal synthesis of linear reversible circuits. Quantum Information and Computation, 8(3&4):0282–0294, 2008.

[170]    Dmitri Maslov and Martin Roetteler. Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations. IEEE Transactions on Information Theory, 64(7):4729–4738, 2018.

[171]    Dmitri Maslov and Ben Zindorf. Depth optimization of cz, cnot, and clifford circuits. IEEE Transactions on Quantum Engineering, 3:1–8, 2022.

[172]    Adam M. Meier, Bryan Eastin, and Emanuel Knill. Magic-state distillation with the four-qubit code. Quantum Info. Comput., 13(3–4):195–209, mar 2013.

[173]     Arianne Meijer-van de Griend and Ross Duncan. Architecture-aware synthesis of phase polynomials for nisq devices. In Stefano Gogioso and Matty Hoban, editors, Proceedings 19th International Conference on Quantum Physics and Logic, Wolfson College, Oxford, UK, 27 June - 1 July 2022, volume 394 of Electronic Proceedings in Theoretical Computer Science, pages 116–140. Open Publishing Association, 2023.

[174]    N David Mermin. Quantum computer science: an introduction. Cambridge University Press, 2007.

[175]    Giulia Meuli, Mathias Soeken, Earl Campbell, Martin Roetteler, and Giovanni de Micheli. The role of multiplicative complexity in compiling low t-count oracle circuits. In 2019 IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pages 1–8, 2019.

[176]    Giulia Meuli, Mathias Soeken, Martin Roetteler, Nikolaj Bjorner, and Giovanni De Micheli. Reversible pebbling game for quantum memory management. In 2019 Design, Automation & Test in Europe Conference & Exhibition (DATE), pages 288–291. IEEE, 2019.

[177]    Giulia Meuli, Mathias Soeken, Martin Roetteler, and Giovanni De Micheli. Ros: Resource-constrained oracle synthesis for quantum computers. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 119–130. Open Publishing Association, 2020.

[178]    Mehdi Mhalla and Simon Perdrix. Graph States, Pivot Minor, and Universality of (X,Z)-measurements. International Journal of Unconventional Computing, 9(1-2):153–171, 2013.

[179]    Microsoft. Announcing the Microsoft Quantum Development Kit. https://azure.microsoft.com/en-us/blog/quantum/2017/12/11/announcing-microsoft-quantum-development-kit/.

[180]    Jacob Miller and Akimasa Miyake. Hierarchy of universal entanglement in 2D measurement-based quantum computation. npj Quantum Information, 2:16036, 2016.

[181]    Jisho Miyazaki, Michal Hajdušek, and Mio Murao. Analysis of the trade-off between spatial and temporal resources for measurement-based quantum computation. Physical Review A, 91(5):052302, 2015.

[182]     A. Montanaro. Quantum algorithms: an overview. arXiv:1511.04206, 2015.

[183]    Mauro E. S. Morales, Pedro C. S. Costa, Daniel K. Burgarth, Yuval R. Sanders, and Dominic W. Berry. Greatly improved higher-order product formulae for quantum simulation, October 2022. arXiv:2210.15817 [quant-ph].

[184]    Sam Morley-Short, Sara Bartolucci, Mercedes Gimeno-Segovia, Pete Shadbolt, Hugo Cable, and Terry Rudolph. Physical-depth architectural requirements for generating universal photonic cluster states. Quantum Science and Technology, 3(1):015005, 2017.

[185]    Michele Mosca. Quantum algorithms. arXiv preprint arXiv:0808.0369, 2008.

[186]    Beatrice Nash, Vlad Gheorghiu, and Michele Mosca. Quantum circuit optimizations for NISQ architectures. Quantum Science and Technology, 5(2):025010, 2020.

[187]    Kang Feng Ng and Quanlong Wang. A universal completion of the ZX-calculus. Preprint, 2017.

[188]    M. A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge university press, 2010.

[189]    Bernhard Ömer. Procedural quantum programming. In AIP Conference Proceedings, volume 627, pages 276–285. American Institute of Physics, 2002.

[190]    Hakop Pashayan, Joel J. Wallman, and Stephen D. Bartlett. Estimating outcome probabilities of quantum circuits using quasiprobabilities. Phys. Rev. Lett., 115:070501, Aug 2015.

[191]    Simon Perdrix and Quanlong Wang. Supplementarity is necessary for quantum diagram reasoning. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of Leibniz International Proceedings in Informatics (LIPIcs), pages 76:1–76:14, Krakow, Poland, 2016.

[192]    Boldizsár Poór. A unique normal form for prime-dimensional qudit Clifford ZX-calculus. Master’s thesis, University of Oxford, 2022.

[193]    John Preskill. Quantum computation lecture notes, 2015. http://theory.caltech.edu/~preskill/ph229/.

[194]    Hammam Qassim, Hakop Pashayan, and David Gosset. Improved upper bounds on the stabilizer rank of magic states. Quantum, 5:606, 12 2021.

[195]    Quantinuum. TKET. https://www.quantinuum.com/developers/tket.

[196]    R. Raussendorf and H. J. Briegel. A one-way quantum computer. Physical Review Letters, 86:5188, 2001.

[197]    R. Raussendorf, J. Harrington, and K. Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9:199, 2007.

[198]    Robert Raussendorf and Hans J. Briegel. Computational model underlying the one-way quantum computer. Quantum Info. Comput., 2(6):443–486, oct 2002.

[199]    Robert Raussendorf, Dan E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Physical Review A, 68(2):22312, 2003.

[200]    Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett., 98:190504, May 2007.

[201]    Neil J. Ross and Peter Selinger. Optimal ancilla-free clifford+t approximation of z-rotations. Quantum Information and Computation, 16(11–12):901–953, 9 2016.

[202]    Francisco JR Ruiz, Tuomas Laakkonen, Johannes Bausch, Matej Balog, Mohammadamin Barekatain, Francisco JH Heras, Alexander Novikov, Nathan Fitzpatrick, Bernardino Romera-Paredes, John van de Wetering, Alhussein Fawzi, Konstantinos Meichanetzidis, and Pushmeet Kohli. Quantum Circuit Optimization with AlphaTensor. arXiv preprint arXiv:2402.14396, 2024.

[203]    Christian Schröder de Witt and Vladimir Zamdzhiev. The ZX-calculus is incomplete for quantum mechanics. In Bob Coecke, Ichiro Hasuo, and Prakash Panangaden, editors, Proceedings of the 11th workshop on Quantum Physics and Logic, Kyoto, Japan, 4-6th June 2014, volume 172 of Electronic Proceedings in Theoretical Computer Science, pages 285–292. Open Publishing Association, 2014.

[204]    J. Schwinger. Unitary operator bases. Proceedings of the National Academy of Sciences of the U.S.A., 46:570–579, 1960.

[205]    Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289–355. Springer, 2010.

[206]    Peter Selinger. Quantum circuits of T-depth one. Physical Review A, 87(4):042302, 2013.

[207]    Peter Selinger. Efficient Clifford+T Approximation of Single-Qubit Operators. Quantum Info. Comput., 15(1–2):159–180, jan 2015.

[208]    Yaoyun Shi. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Preprint, 2002.

[209]    P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 124–134. IEEE, 1994.

[210]    P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997.

[211]    Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R2493–R2496, Oct 1995.

[212]    Peter W Shor. Fault-tolerant quantum computation. In Proceedings of 37th conference on foundations of computer science, pages 56–65. IEEE, 1996.

[213]    Will Simmons. Relating Measurement Patterns to Circuits via Pauli Flow. In Chris Heunen and Miriam Backens, editors, Proceedings 18th International Conference on Quantum Physics and Logic, Gdansk, Poland, and online, 7-11 June 2021, volume 343 of Electronic Proceedings in Theoretical Computer Science, pages 50–101. Open Publishing Association, 2021.

[214]    Luka Skoric, Dan E Browne, Kenton M Barnes, Neil I Gillespie, and Earl T Campbell. Parallel window decoding enables scalable fault tolerant quantum computation. Nature Communications, 14(1):7040, 2023.

[215]    Anders Sørensen and Klaus Mølmer. Quantum computation with ions in thermal motion. Physical review letters, 82(9):1971, 1999.

[216]    A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77:793–797, Jul 1996.

[217]    A. M. Steane. Simple quantum error-correcting codes. Phys. Rev. A, 54:4741–4751, Dec 1996.

[218]    Andrew M Steane. Active stabilization, quantum computation, and quantum state synthesis. Physical Review Letters, 78(11):2252, 1997.

[219]    Masuo Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32(2):400–407, 1991.

[220]    Yuki Takeuchi, Tomoyuki Morimae, and Masahito Hayashi. Quantum computational universality of hypergraph states with pauli-x and z basis measurements. Scientific reports, 9(1):1–14, 2019.

[221]    Hale F Trotter. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4):545–551, 1959.

[222]    John van de Wetering. ZX-calculus for the working quantum computer scientist. arXiv preprint arXiv:2012.13966, 2020.

[223]    John van de Wetering and Matthew Amy. Optimising quantum circuits is generally hard. arXiv preprint arXiv:2310.05958, 2024.

[224]    John van de Wetering, Richie Yeung, Tuomas Laakkonen, and Aleks Kissinger. Optimal compilation of parametrised quantum circuits. arXiv preprint arXiv:2401.12877, 2024.

[225]    M. Van den Nest, J. Dehaene, and B. De Moor. Graphical description of the action of local Clifford transformations on graph states. Physical Review A, 69(2):9422, 2004.

[226]    Maarten Van Den Nest. Classical simulation of quantum computation, the gottesman-knill theorem, and slightly beyond. Quantum Info. Comput., 10(3):258–271, mar 2010.

[227]    Maarten Van den Nest, Jeroen Dehaene, and Bart De Moor. Graphical description of the action of local Clifford transformations on graph states. Physical Review A, 69(2):022316, 2004.

[228]    Vivien Vandaele. Lower t-count with faster algorithms. arXiv preprint arXiv:2407.08695, 2024.

[229]    Renaud Vilmart. A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1–10, 2019.

[230]    Suhas Vittal, Poulami Das, and Moinuddin Qureshi. Astrea: Accurate quantum error-decoding via practical minimum-weight perfect-matching. In Proceedings of the 50th Annual International Symposium on Computer Architecture, ISCA ’23, New York, NY, USA, 2023. Association for Computing Machinery.

[231]    David S. Wang, Austin G. Fowler, and Lloyd C. L. Hollenberg. Surface code quantum computing with error rates over 1 Phys. Rev. A, 83:020302, Feb 2011.

[232]    Quanlong Wang, Richie Yeung, and Mark Koch. Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning. arXiv preprint arXiv:2201.13250, 2022.

[233]    John Watrous. Quantum computation lecture notes, 2006. https://cs.uwaterloo.ca/~watrous/QC-notes/.

[234]    John Watrous. The theory of quantum information. Cambridge university press, 2018.

[235]    Mark A Webster, Armanda O Quintavalle, and Stephen D Bartlett. Transversal diagonal logical operators for stabiliser codes. New Journal of Physics, 25(10):103018, oct 2023.

[236]    Tzu-Chieh Wei, Ian Affleck, and Robert Raussendorf. The 2d aklt state is a universal quantum computational resource. Bulletin of the American Physical Society, 56, 2011.

[237]    Xanadu. PennyLane. https://pennylane.ai/.

[238]    Xinlan Zhou, Debbie W. Leung, and Isaac L. Chuang. Methodology for quantum logic gate construction. Phys. Rev. A, 62:052316, Oct 2000.