The Knowledge Complexity of Quantum Problems

Abstract: Foundational results in theoretical computer science have established that everything provable, is provable in zero knowledge. However, this assertion fundamentally assumes a classical interpretation of computation and many interesting physical statements that one can hope to prove are not characterized. In this work, we consider decision problems, where the problem instance itself is specified by a (pure) quantum state. We discuss several motivating examples for this notion and, as our main technical result, we show that every quantum problem that is provable with an interactive protocol, is also provable in zero-knowledge. Our protocol achieves unconditional soundness and computational zero-knowledge, under standard assumptions in cryptography. In addition, we show how our techniques yield a protocol for the Uhlmann transformation problem that achieves a meaningful notion of zero-knowledge, also in the presence of a malicious verifier.