A search for quantum coin-flipping protocols using optimization techniques

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 1 √ 2 +δ for any given δ > 0. However, no explicit description of these protocols is known, and the number of rounds in the protocols tends to infinity as δ goes to 0. 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, 2003). To make this search computationally feasible, we further restrict to commitment states à la Mochon (2005). An analysis of the resulting protocols via semidefinite programs (SDPs) unveils a simple structure. For example, we show that the SDPs reduce to second-order cone programs. We devise novel cheating strategies in the protocol by restricting the semidefinite programs and use the strategies to prune the search. The techniques we develop enable a computational search for protocols given by a mesh over the corresponding parameter space. The protocols have up to six rounds of communication, with messages of varying dimension and include the best known explicit protocol (with bias 1/4). We conduct two kinds of search: one for protocols ∗Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo. Address: 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada. Email: [email protected]. †Laboratoire d’Informatique Algorithmique: Fondements et Applications, Université Paris Diderot. Address: 5 rue Thomas-Mann 75205 Paris cedex 13, France. Email: [email protected]. ‡Department of Combinatorics and Optimization, University of Waterloo. Address: 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada. Email: [email protected].