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ZX-Rules for 2-Qubit Clifford+T Quantum Circuits

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Reversible Computation (RC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11106))

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Abstract

ZX-calculus is a high-level graphical formalism for qubit computation. In this paper we give the ZX-rules that enable one to derive all equations between 2-qubit Clifford+T quantum circuits. Our rule set is only a small extension of the rules of stabiliser ZX-calculus, and substantially less than those needed for the recently achieved universal completeness. One of our rules is new, and we expect it to also have other utilities.

These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that ZX-calculus provides a more convenient arena for quantum circuit rewriting than restricting oneself to circuit equations. The reason for this is that ZX-calculus is not constrained by a fixed unitary gate set for performing intermediate computations.

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References

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

  2. Backens, M.: The ZX-calculus is complete for stabilizer quantum mechanics. New J. Phys. 16, 093021 (2014). arXiv:1307.7025

    Article  Google Scholar 

  3. Backens, M., Nabi Duman, A.: A complete graphical calculus for Spekkens’ toy bit theory. Found. Phys. (2015). arXiv:1411.1618

  4. Backens, M.: The ZX-calculus is complete for the single-qubit Clifford+T group. In: Coecke, B., Hasuo, I., Panangaden, P. (eds.) Proceedings of the 11th workshop on Quantum Physics and Logic. Electronic Proceedings in Theoretical Computer Science, vol. 172, pp. 293–303. Open Publishing Association (2014)

    Google Scholar 

  5. Backens, M., Perdrix, S., Wang, Q.: Towards a minimal stabilizer ZX-calculus. arXiv preprint arXiv:1709.08903 (2017)

  6. de Beaudrap, N., Horsman, D.: The ZX calculus is a language for surface code lattice surgery. arXiv preprint arXiv:1704.08670 (2017)

  7. Chancellor, N., Kissinger, A., Roffe, J., Zohren, S., Horsman, D.: Graphical structures for design and verification of quantum error correction. arXiv preprint arXiv:1611.08012 (2016)

  8. Coecke, B.: Quantum picturalism. Contemp. Phys. 51, 59–83 (2009). arXiv:0908.1787

    Article  Google Scholar 

  9. Coecke, B., Duncan, R.: Interacting quantum observables. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 298–310. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_25

    Chapter  Google Scholar 

  10. Coecke, B., Duncan, R.: Interacting quantum observables: categorical algebra and diagrammatics. New J. Phys. 13, 043016 (2011). arXiv:0906.4725

    Article  MathSciNet  Google Scholar 

  11. Coecke, B., Duncan, R., Kissinger, A., Wang, Q.: Strong complementarity and non-locality in categorical quantum mechanics. In: Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science (LICS) (2012). arXiv:1203.4988

  12. Coecke, B., Duncan, R., Kissinger, A., Wang, Q.: Generalised compositional theories and diagrammatic reasoning. In: Chiribella, G., Spekkens, R.W. (eds.) Quantum Theory: Informational Foundations and Foils. FTP, vol. 181, pp. 309–366. Springer, Dordrecht (2016). https://doi.org/10.1007/978-94-017-7303-4_10. arXiv:1203.4988

    Chapter  Google Scholar 

  13. Coecke, B., Kissinger, A.: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

  14. Coecke, B., Duncan, R.: Tutorial: graphical calculus for quantum circuits. In: Glück, R., Yokoyama, T. (eds.) RC 2012. LNCS, vol. 7581, pp. 1–13. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36315-3_1

    Chapter  Google Scholar 

  15. Coecke, B., Wang, Q.: ZX-rules for 2-qubit Clifford+T quantum circuits. arXiv preprint arXiv:1804.05356 (2018)

  16. Duncan, R., Perdrix, S.: Graph states and the necessity of Euler decomposition. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 167–177. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03073-4_18

    Chapter  MATH  Google Scholar 

  17. Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with generalised flow. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 285–296. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14162-1_24

    Chapter  Google Scholar 

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

  19. Hadzihasanovic, A.: The algebra of entanglement and the geometry of composition. Ph.D. thesis, University of Oxford (2017)

    Google Scholar 

  20. Horsman, C.: Quantum picturalism for topological cluster-state computing. New J. Phys. 13, 095011 (2011). arXiv:1101.4722

    Article  Google Scholar 

  21. Jeandel, E., Perdrix, S., Vilmart, R.: A complete axiomatisation of the ZX-calculus for Clifford+ T quantum mechanics. arXiv preprint arXiv:1705.11151 (2017)

  22. Jeandel, E., Perdrix, S., Vilmart, R.: Diagrammatic reasoning beyond Clifford+ T quantum mechanics. arXiv preprint arXiv:1801.10142 (2018)

  23. Kissinger, A., Quick, D.: Tensors, !-graphs, and non-commutative quantum structures. New Gener. Comput. 34(1–2), 87–123 (2016)

    Article  Google Scholar 

  24. Kissinger, A., Zamdzhiev, V.: Quantomatic: a proof assistant for diagrammatic reasoning. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 326–336. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21401-6_22

    Chapter  Google Scholar 

  25. Ng, K.F., Wang, Q.: A universal completion of the ZX-calculus. arXiv preprint arXiv:1706.09877 (2017)

  26. Selinger, P., Bian, X.: Relations for Clifford+T operators on two qubits (2015). Talk. https://www.mathstat.dal.ca/~xbian/talks/

  27. Schröder de Witt, C., Zamdzhiev, V.: The ZX calculus is incomplete for quantum mechanics (2014). arXiv:1404.3633

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Acknowledgements

This work was sponsored by Cambridge Quantum Computing Inc. for which we are grateful. QW also thanks Kang Feng Ng for useful discussions.

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Authors and Affiliations

  1. University of Oxford, Oxford, UK

    Bob Coecke & Quanlong Wang

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  1. Bob Coecke
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  2. Quanlong Wang
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Correspondence to Quanlong Wang .

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Editors and Affiliations

  1. University of Turku, Turku, Finland

    Jarkko Kari

  2. University of Leicester, Leicester, United Kingdom

    Irek Ulidowski

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Coecke, B., Wang, Q. (2018). ZX-Rules for 2-Qubit Clifford+T Quantum Circuits. In: Kari, J., Ulidowski, I. (eds) Reversible Computation. RC 2018. Lecture Notes in Computer Science(), vol 11106. Springer, Cham. https://doi.org/10.1007/978-3-319-99498-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-99498-7_10

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  • Online ISBN: 978-3-319-99498-7

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