The Knowledge Complexity of Quantum Problems

2025-10-08

AI Summary

This work investigates the feasibility of zero-knowledge proofs for quantum decision problems—specifically, promise problems defined by quantum states. Method: We present the first generic construction demonstrating that all quantum interactive verifiable problems admit zero-knowledge proof protocols. For the practically relevant Uhlmann transformation problem, we design an explicit protocol. Our construction integrates quantum information theory with classical cryptographic techniques under standard cryptographic assumptions, yielding a quantum interactive proof system with statistical soundness and computational zero-knowledge. Contribution/Results: The protocol achieves information-theoretic privacy against malicious verifiers and provides the first zero-knowledge characterization of broad quantum complexity classes—including QIP—thereby bridging quantum complexity theory and post-quantum cryptography. This establishes foundational tools for privacy-preserving verification of quantum computations and advances the theoretical understanding of quantum proofs in adversarial settings.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

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.

Problem

• Extends zero-knowledge proofs to quantum state decision problems
• Shows quantum provable problems are zero-knowledge provable
• Develops zero-knowledge protocol for Uhlmann transformation problem

Innovation

• Zero-knowledge proofs for quantum state problems
• Unconditional soundness with computational zero-knowledge guarantees
• Secure Uhlmann transformation protocol against malicious verifiers