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A search for quantum coin-flipping protocols using optimization techniques

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Abstract

Coin-flipping is a cryptographic task in which two physically separated, mistrustful parties wish to generate a fair coin-flip by communicating with each other. Chailloux and Kerenidis (2009) designed quantum protocols that guarantee coin-flips with near optimal bias away from uniform, even when one party deviates arbitrarily from the protocol. The probability of any outcome in these protocols is provably at most \(\tfrac{1}{\sqrt{2}} + \delta \) for any given \(\delta > 0\). However, no explicit description of these protocols is known; in fact, the smallest bias achieved by known explicit protocols is \(1/4\) (Ambainis 2001). We take a computational optimization approach, based mostly on convex optimization, to the search for simple and explicit quantum strong coin-flipping protocols. We present a search algorithm to identify protocols with low bias within a natural class, protocols based on bit-commitment (Nayak and Shor in Phys Rev A 67(1):012304, 2003). The techniques we develop enable a computational search for protocols given by a mesh over the corresponding parameter space. We conduct searches for four-round and six-round protocols with bias below \(0.2499\) each of varying dimension which include the best known explicit protocol (with bias \(1/4\)). After checking over \(10^{16}\) protocols, a task which would be infeasible using semidefinite programming alone, we conjecture that the smallest achievable bias within the family of protocols we consider is \(1/4\).

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Acknowledgments

We thank Andrew Childs, Michele Mosca, Peter Høyer, and John Watrous for their comments and suggestions. A.N.’s research was supported in part by NSERC Canada, CIFAR, an ERA (Ontario), QuantumWorks, and MITACS. A part of this work was completed at Perimeter Institute for Theoretical Physics. Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through MRI. J.S.’s research is supported by NSERC Canada, MITACS, ERA (Ontario), ANR Project ANR-09-JCJC-0067-01, and ERC Project QCC 306537. L.T.’s research is supported in part by Discovery Grants from NSERC. Research at the Centre for Quantum Technologies at the National University of Singapore is partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant “Random numbers from quantum processes,” (MOE2012-T3-1-009).

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada

    Ashwin Nayak

  2. Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore, 117543, Singapore

    Jamie Sikora

  3. MajuLab CNRS-UNS-NUS-NTU International Joint Research Unit, UMI, Singapore, 3654, Singapore

    Jamie Sikora

  4. Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada

    Levent Tunçel

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  1. Ashwin Nayak
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Correspondence to Jamie Sikora.

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Nayak, A., Sikora, J. & Tunçel, L. A search for quantum coin-flipping protocols using optimization techniques. Math. Program. 156, 581–613 (2016). https://doi.org/10.1007/s10107-015-0909-y

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