The Knowledge Complexity of Quantum Problems
Giulio Malavolta
TL;DR
The paper advances quantum zero-knowledge by showing that every problem provable in quantum interactive proof systems with quantum-state problem instances can be proven in zero-knowledge under standard cryptographic assumptions, notably EFI pairs. It introduces and analyzes the computational classes pQIPzk and p/mQMAzk, provides a conditional main theorem pQIP ⊆ pQIPzk, and develops explicit protocols via De Finetti reductions and round-collapsing compilers, including improvements for the Uhlmann transformation. The results illuminate how secure quantum computation, quantum MACs, and quantum state commitments can be orchestrated to preserve privacy while certifying quantum properties, with both honest-verifier and malicious-verifier guarantees under EFI assumptions. The work also delineates limits for mixed-state settings, illustrating impossibility results under LWE-based assumptions, and outlines pathways toward reactive secure computation for streaming/verifiable quantum tasks, showcasing practical avenues for quantum cryptography and state-property certification. Overall, the findings provide a structured framework for computational zero-knowledge in quantum contexts, with implications for quantum cryptography, state property testing, and quantum-information science.
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.