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 $$\tfrac{1}{\sqrt{2}} + \delta $$12+δ for any given $$\delta > 0$$δ>0. However, no explicit description of these protocols is known; in fact, the smallest bias achieved by known explicit protocols is $$1/4$$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 $$0.2499$$0.2499 each of varying dimension which include the best known explicit protocol (with bias $$1/4$$1/4). After checking over $$10^{16}$$1016 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 $$1/4$$1/4.