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Two-qubit Stabilizer Circuits with Recovery II: Analysis

Authors Wim van Dam , Raymond Wong

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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • stabilizer circuit
  • recovery circuit
  • magic state

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Abstract

We study stabilizer circuits that use non-stabilizer qubits and Z-measurements to produce other non-stabilizer qubits. These productions are successful when the correct measurement outcome occurs, but when the opposite outcome is observed, the non-stabilizer input qubit is potentially destroyed. In preceding work [arXiv:1803.06081 (2018)] we introduced protocols able to recreate the expensive non-stabilizer input qubit when the two-qubit stabilizer circuit has an unsuccessful measurement outcome. Such protocols potentially allow a deep computation to recover from such failed measurements without the need to repeat the whole prior computation. Possible complications arise when the recovery protocol itself suffers from a failed measurement. To deal with this, we need to use nested recovery protocols. Here we give a precise analysis of the potential advantage of such recovery protocols as we examine its optimal nesting depth. We show that if the expensive input qubit has cost d, then typically a depth O(log d) recovery protocol is optimal, while a certain special case has optimal depth O(sqrt{d}). We also show that the recovery protocol can achieve a cost reduction by a factor of at most two over circuits that do not use recovery.

Cite As Get BibTex

Wim van Dam and Raymond Wong. Two-qubit Stabilizer Circuits with Recovery II: Analysis. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.TQC.2018.8

Author Details

Wim van Dam
  • Department of Computer Science, Department of Physics, University of California, Santa Barbara, CA, USA
Raymond Wong
  • Department of Computer Science, University of California, Santa Barbara, CA, USA

Funding

This material is based upon work supported by the National Science Foundation under Grants No. 0917244 and 1719118.

References

  1. Alex Bocharov, Yuri Gurevich, and Krysta M. Svore. Efficient decomposition of single-qubit gates into V basis circuits. Phys. Rev. A, 88:012313, Jul 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.012313.
  2. Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient synthesis of probabilistic quantum circuits with fallback. Phys. Rev. A, 91:052317, May 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.052317.
  3. Alex Bocharov, Martin Roetteler, and Krysta M. Svore. Efficient Synthesis of Universal Repeat-Until-Success Quantum Circuits. Phys. Rev. Lett., 114:080502, Feb 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.114.080502.
  4. Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86:052329, Nov 2012. URL: http://dx.doi.org/10.1103/PhysRevA.86.052329.
  5. Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71:022316, Feb 2005. URL: http://dx.doi.org/10.1103/PhysRevA.71.022316.
  6. Earl T Campbell and Joe O’Gorman. An efficient magic state approach to small angle rotations. Quantum Science and Technology, 1(1):015007, 2016. URL: http://stacks.iop.org/2058-9565/1/i=1/a=015007.
  7. Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, 2006. URL: https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf.
  8. Guillaume Duclos-Cianci and David Poulin. Reducing the quantum-computing overhead with complex gate distillation. Phys. Rev. A, 91:042315, Apr 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.042315.
  9. Guillaume Duclos-Cianci and Krysta M. Svore. Distillation of nonstabilizer states for universal quantum computation. Phys. Rev. A, 88:042325, Oct 2013. URL: http://dx.doi.org/10.1103/PhysRevA.88.042325.
  10. 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. URL: http://dx.doi.org/10.1103/PhysRevA.86.032324.
  11. Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker. Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count. Quantum, 1:31, Oct 2017. URL: http://dx.doi.org/10.22331/q-2017-10-03-31.
  12. N Cody Jones, James D Whitfield, Peter L McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, and Yoshihisa Yamamoto. Faster quantum chemistry simulation on fault-tolerant quantum computers. New Journal of Physics, 14(11):115023, 2012. URL: http://stacks.iop.org/1367-2630/14/i=11/a=115023.
  13. J.G. Kemény and J.L. Snell. Finite markov chains. University series in undergraduate mathematics. Springer-Verlag New York, 1976. Google Scholar
  14. Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates. Quantum Information and Computation, 13(7-8):607-630, 2013. URL: http://arxiv.org/abs/1206.5236.
  15. Andrew J. Landahl and Chris Cesare. Complex instruction set computing architecture for performing accurate quantum Z rotations with less magic, Feb 2013. URL: http://arxiv.org/abs/1302.3240.
  16. Adam Meier, Bryan Eastin, and Emanuel Knill. Magic-state distillation with the four-qubit code. Quantum Information and Computation, 13:195-209, 2013. URL: http://arxiv.org/abs/1204.4221.
  17. Ben Reichardt. Quantum universality by state distillation. Quantum Information and Computation, 9:1030-1052, 2009. URL: http://arxiv.org/abs/quant-ph/0608085v2.
  18. Neil J. Ross. Optimal ancilla-free Clifford+V approximation of z-rotations. Quantum Information and Computation, 15(11-12):932-950, 2015. URL: http://arxiv.org/abs/1409.4355.
  19. Neil J. Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of z-rotations. Quantum Information and Computation, 16(11-12):901-953, 2016. URL: http://www.rintonpress.com/xxqic16/qic-16-1112/0901-0953.pdf.
  20. Peter Selinger. Efficient Clifford+T approximation of single-qubit operators. Quantum Information and Computation, 15(1-2):159-180, 2015. URL: http://arxiv.org/abs/1212.6253.
  21. Wim van Dam and Raymond Wong. Two-qubit Stabilizer Circuits with Recovery I: Existence, Mar 2018. URL: http://arxiv.org/abs/1803.06081.
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