A search for quantum coin-flipping protocols using optimization techniques

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  (Formula presented.) for any given  (Formula presented.). However, no explicit description of these protocols is known; in fact, the smallest bias achieved by known explicit protocols is  (Formula presented.) (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  (Formula presented.) each of varying dimension which include the best known explicit protocol (with bias  (Formula presented.)). After checking over (Formula presented.) 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 (Formula presented.).