Two-qubit Stabilizer Circuits with Recovery II: Analysis

Authors Wim van Dam , Raymond Wong

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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

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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)


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.

Subject Classification

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


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  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:
  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:
  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:
  4. Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Phys. Rev. A, 86:052329, Nov 2012. URL:
  5. Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71:022316, Feb 2005. URL:
  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:
  7. Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, 2006. URL:
  8. Guillaume Duclos-Cianci and David Poulin. Reducing the quantum-computing overhead with complex gate distillation. Phys. Rev. A, 91:042315, Apr 2015. URL:
  9. Guillaume Duclos-Cianci and Krysta M. Svore. Distillation of nonstabilizer states for universal quantum computation. Phys. Rev. A, 88:042325, Oct 2013. URL:
  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:
  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:
  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:
  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:
  15. Andrew J. Landahl and Chris Cesare. Complex instruction set computing architecture for performing accurate quantum Z rotations with less magic, Feb 2013. URL:
  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:
  17. Ben Reichardt. Quantum universality by state distillation. Quantum Information and Computation, 9:1030-1052, 2009. URL:
  18. Neil J. Ross. Optimal ancilla-free Clifford+V approximation of z-rotations. Quantum Information and Computation, 15(11-12):932-950, 2015. URL:
  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:
  20. Peter Selinger. Efficient Clifford+T approximation of single-qubit operators. Quantum Information and Computation, 15(1-2):159-180, 2015. URL:
  21. Wim van Dam and Raymond Wong. Two-qubit Stabilizer Circuits with Recovery I: Existence, Mar 2018. URL: