Proofs

Abstract

We're going to start by beating a retreat from QuantumLand, back onto the safe territory of computational complexity. In particular, we're going to see how, in the 1980s and 1990s, computational complexity theory reinvented the millennia-old concept of mathematical proof – making it probabilistic, interactive, and cryptographic. But then, having fashioned our new pruning-hooks (proving-hooks?), we're going to return to QuantumLand and reap the harvest. In particular, I’ll show you why, if you could see the entire trajectory of a hidden variable, then you could efficiently solve any problem that admits a “statistical zero-knowledge proof protocol,” including problems like Graph Isomorphism for which no efficient quantum algorithm is yet known. What is a proof? Historically, mathematicians have had two very different notions of “proof.” The first is that a proof is something that induces in the audience (or at least the prover!) an intuitive sense of certainty that the result is correct. In this view, a proof is an inner transformative experience – a way for your soul to make contact with the eternal verities of Platonic heaven.