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  • Open Access

Effective Compression of Quantum Braided Circuits Aided by ZX-Calculus

Michael Hanks1,*, Marta P. Estarellas1,*,†, William J. Munro2,1, and Kae Nemoto1

  • 1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
  • 2NTT Basic Research Laboratories & NTT Research Center for Theoretical Quantum Physics, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

  • *These authors contributed equally to this work.
  • †mpestarellas@nii.ac.jp
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Phys. Rev. X 10, 041030 – Published 11 November, 2020

DOI: https://doi.org/10.1103/PhysRevX.10.041030

Abstract

Mapping a quantum algorithm to any practical large-scale quantum computer will require a sequence of compilations and optimizations. At the level of fault-tolerant encoding, one likely requirement of this process is the translation into a topological circuit, for which braided circuits represent one candidate model. Given the large overhead associated with encoded circuits, it is paramount to reduce their size in terms of computation time and qubit number through circuit compression. While these optimizations have typically been performed in the language of three-dimensional diagrams, such a representation does not allow an efficient, general, and scalable approach to reduction or verification. We propose the use of the ZX-calculus as an intermediate language for braided circuit compression, demonstrating advantage by comparing results using this approach with those previously obtained for the compression of |A⟩ and |Y⟩ state distillation circuits. We then provide a benchmark of our method against a small set of Clifford+T circuits, yielding compression percentages of ∼77%. Our results suggest that the overheads of braided, defect-based circuits are comparable to those of their lattice-surgery counterparts, restoring the potential of this model for surface-code quantum computation.

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Popular Summary

The fast development of quantum technology has brought the field of quantum computing into an era where proof-of-principle devices have enough qubits to outpace a classical computer, but those few qubits are still quite noisy and therefore prone to error. To achieve a truly practical advantage over classical computers, quantum devices need many more physical qubits to encode a logical qubit so that errors can be corrected. To tackle this challenge, we present an efficient compression method that reduces the resources associated with arbitrary circuits designed to run in one common quantum computer architecture known as the braiding model.

For a language capable of optimizing the volume of quantum circuits as well as verifying their correctness, we propose using the “ZX-calculus,” a graphical language introduced as a way to describe mappings between qubits. This language is able to unify the different architectures of surface-code-based computation, allowing us to exploit a hybrid approach of compilation. Our method presents an advantage of a 40% compression rate with respect to previous reductions, yielding compression rates greater than 70% of the original volume. We also show that this hybrid approach surpasses the currently preferred model for compilation.

With such compression rates, we reignite the potential of the braiding architecture model and bring practical fault-tolerant computation one step closer. Our work opens a new venue of research in large-scale quantum computing found on the hybridization of surface-code-based models of compilation.

Article Text

References (56)

  1. D. Deutsch, Quantum Theory, the Church–Turing Principle and the Universal Quantum Computer, Proc. R. Soc. A 400, 97 (1985).
  2. P. W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM J. Comput. 26, 1484 (1997).
  3. L. 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) (Association for Computing Machinery Press, New York, 1996).
  4. F. Arute et al., Quantum Supremacy Using a Programmable Superconducting Processor, Nature (London) 574, 505 (2019).
  5. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface Codes: Towards Practical Large-Scale Quantum Computation, Phys. Rev. A 86, 032324 (2012).
  6. S. B. Bravyi and A. Yu. Kitaev, Quantum Codes on a Lattice with Boundary, arXiv:quant-ph/9811052.
  7. M. H. Freedman and D. A. Meyer, Projective Plane and Planar Quantum Codes, Found. Comput. Math. 1, 325 (2001).
  8. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological Quantum Memory, J. Math. Phys. (N.Y.) 43, 4452 (2002).
  9. A. G. Fowler and S. J. Devitt, A Bridge to Lower Overhead Quantum Computation, arXiv:1209.0510.
  10. C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, Surface Code Quantum Computing by Lattice Surgery, New J. Phys. 14, 123011 (2012).
  11. A. Paetznick and A. G. Fowler, Quantum Circuit Optimization by Topological Compaction in the Surface Code, arXiv:1304.2807.
  12. B. Coecke, Quantum Picturalism, Contemp. Phys. 51, 59 (2010).
  13. N. Chancellor, A. Kissinger, J. Roffe, S. Zohren, and D. Horsman, Graphical Structures for Design and Verification of Quantum Error Correction, arXiv:1611.08012.
  14. D. Litinski, A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery, Quantum 3, 128 (2019).
  15. T. M. Stace, S. D. Barrett, and A.C. Doherty, Thresholds for Topological Codes in the Presence of Loss, Phys. Rev. Lett. 102, 200501 (2009).
  16. T. M. Stace and S. D. Barrett, Error Correction and Degeneracy in Surface Codes Suffering Loss, Phys. Rev. A 81, 022317 (2010).
  17. S. D. Barrett and T. M. Stace, Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors, Phys. Rev. Lett. 105, 200502 (2010).
  18. H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Measurement-Based Quantum Computation, Nat. Phys. 5, 19 (2009).
  19. A. Paler, Design Methods for Reliable Quantum Circuits, Ph.D. thesis, Universität Passau, 2015.
  20. A. G. Fowler, A. M. Stephens, and P. Groszkowski, High-Threshold Universal Quantum Computation on the Surface Code, Phys. Rev. A 80, 052312 (2009).
  21. B. J. Brown, K. Laubscher, M. S. Kesselring, and J. R. Wootton, Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code, Phys. Rev. X 7, 021029 (2017).
  22. A. Paler, S. Devitt, K. Nemoto, and I. Polian, Synthesis of Topological Quantum Circuits, in Proceedings of the 2012 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH ’12) (Association for Computing Machinery Press, New York, 2012).
  23. Y. Ding, A. Holmes, A. Javadi-Abhari, D. Franklin, M. Martonosi, and F. Chong, Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures, in Proceedings of the 51st Annual IEEE/ACM International Symposium on Microarchitecture, 2018 (MICRO51) (IEEE, New York, 2018).
  24. A. Paler, A. G. Fowler, and R. Wille, Online Scheduled Execution of Quantum Circuits Protected by Surface Codes, Quantum Inf. Comput. 17, 1335 (2017).
  25. A. G. Fowler, Time-Optimal Quantum Computation, arXiv:1210.4626.
  26. Reversible Computation, edited by J. Krivine and J.-B. Stefani (Springer International Publishing, Switzerland, 2015).
  27. A. Paler and S. J. Devitt, Specification Format and a Verification Method of Fault-Tolerant Quantum Circuits, Phys. Rev. A 98, 022302 (2018).
  28. B. Coecke and R. Duncan, Interacting Quantum Observables, Automata, Languages and Programming (Springer, Berlin, 2008), pp. 298–310.
  29. B. Coecke and R. Dunca, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13, 043016 (2011).
  30. M. Backens, The ZX-Calculus is Complete for the Single-Qubit Clifford+T Group, in Proceedings of the 11th workshop on Quantum Physics and Logic (QPL 2014), EPTCS 172, pp. 293–303.
  31. B. Coecke and Q. Wang, ZX-Rules for 2-Qubit Clifford+T Quantum Circuits, arXiv:1804.05356.
  32. N. de Beaudrap and D. Horsman, The ZX Calculus Is a Language for Surface Code Lattice Surgery, Quantum 4, 218 (2020).
  33. C. Horsman, Quantum Picturalism for Topological Cluster-State Computing, New J. Phys. 13, 095011 (2011).
  34. A. Kissinger and J. van de Wetering, Reducing the Number of Non-Clifford Gates in Quantum Circuits, Phys. Rev. A 102, 022406 (2020).
  35. M. Backens, Making the Stabilizer ZX-Calculus Complete for Scalars, in Proceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015), Electronic Proceedings in Theoretical Computer Science, edited by C. Heunen, P. Selinger, and J. Vicary (Open Publishing Association, 2015), Vol. 195, pp. 17–32, 10.4204/EPTCS.195.2.
  36. Quantomatic/pyzx, https://github.com/Quantomatic/pyzx/tree/master/circuits.
  37. A. Kissinger and J. van de Wetering, pyzx: Large Scale Automated Diagrammatic Reasoning, in Electronic Proceedings in Theoretical Computer Science 318, (2020), pp. 229–241.
  38. R. Raussendorf, J. Harrington, and K. Goyal, Topological Fault-Tolerance in Cluster State Quantum Computation, New J. Phys. 9, 199 (2007).
  39. A. M. Meier, B. Eastin, and E. Knill, Magic-State Distillation with the Four-Qubit Code, arXiv:1204.4221.
  40. S. Bravyi and J. Haah, Magic-State Distillation with Low Overhead, Phys. Rev. A 86, 052329 (2012).
  41. C. Jones, Multilevel Distillation of Magic States for Quantum Computing, Phys. Rev. A 87, 042305 (2013).
  42. C. Jones, Low-Overhead Constructions for the Fault-Tolerant Toffoli Gate, Phys. Rev. A 87, 022328 (2013).
  43. C. Jones, Distillation Protocols for Fourier States in Quantum Computing, arXiv:1303.3066.
  44. B. Eastin, Distilling One-Qubit Magic States into Toffoli States, Phys. Rev. A 87, 032321 (2013).
  45. A. Paetznick and B. W. Reichardt, Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction, Phys. Rev. Lett. 111, 090505 (2013).

  46. It is worth emphasizing that these circuits do not represent the state of the art in terms of overall cost; we select them here primarily for the purpose of comparative reduction.

  47. S. Bravyi and A. Kitaev, Universal Quantum Computation with Ideal Clifford Gates and Noisy Ancillas, Phys. Rev. A 71, 022316 (2005).
  48. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels, Phys. Rev. Lett. 70, 1895 (1993).
  49. D. Gottesman and I. L. Chuang, Quantum Teleportation Is a Universal Computational Primitive, Nature (London) 402, 390 (1999).
  50. D. Gottesman and I. L. Chuang, Demonstrating the Viability of Universal Quantum Computation Using Teleportation and Single-Qubit Operations, Nature (London) 402, 390 (1999).
  51. A. G. Fowler and K. Goyal, Topological Cluster State Quantum Computing, Quantum Inf. Comput. 9, 721 (2009).
  52. C. Gidney and A. G. Fowler, Efficient Magic State Factories with a Catalyzed |CCZ⟩ to 2|T⟩ Transformation, Quantum 3, 135 (2019).

  53. In Fig. 1 of Ref. [52], the authors suggest outer bounding boxes for the CCZ factory and catalyzed 2−T circuit of 3×6×5.5=99 and 4×4×6.5=104, respectively, for a combined volume of 203. However, it is possible that some space may be saved when these components are packed or tiled within a larger circuit. To account for this, we use a highly conservative estimate for the volume, derived from Fig. 18 of Ref. [52]: we assign the CCZ factory a volume of 2×5×6=60 and the catalyzed 2−T circuit a volume of 3×4×5=60, for a combined total of 120.

  54. D. Litinski, Magic State Distillation: Not as Costly as You Think, Quantum 3, 205 (2019).
  55. A. G. Fowler and C. Gidney, Low Overhead Quantum Computation Using Lattice Surgery, arXiv:1808.06709.
  56. A. Paler (private communication).

Outline

Information

Phys. Rev. X 10, 041030– Published 11 November, 2020
  • Received 31 January 2020
  • Revised 23 May 2020
  • Accepted 1 September 2020
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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