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Explicit quantum weak coin flipping protocols with arbitrarily small bias

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We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. A weak coin-flipping protocol has a bias ϵ if neither player can force the outcome towards their preferred value with probability more than 1/2+ϵ. While it is known that classically ϵ=1/2, Mochon showed in 2007 [arXiv:0711.4114] that quantumly weak coin flipping can be achieved with arbitrarily small bias, i.e. ϵ(k)=1/(4k+2) for arbitrarily large k, and he proposed an explicit protocol approaching bias 1/6. So far, the best known explicit protocol is the one by Arora, Roland and Weis, with ϵ(2)=1/10 (corresponding to k=2) [STOC'19, p. 205-216]. In the current work, we present the construction of protocols approaching arbitrarily close to zero bias, i.e. ϵ(k) for arbitrarily large k. We connect the algebraic properties of Mochon's assignments---at the heart of his proof of existence---with the geometric properties of the unitaries whose existence he proved. It is this connection that allows us to find these unitaries analytically.