Classical Commitments to Quantum States

TL;DR

This paper introduces a new classical commitment scheme for quantum states that is efficient, based on post-quantum assumptions, and enables classical verification of quantum information with applications to QMA and quantum PCPs.

Contribution

It constructs a classical commitment scheme for quantum states from post-quantum LWE and noisy trapdoor claw-free functions, improving efficiency and removing the need for obfuscation assumptions.

Findings

Constructed a succinct classical commitment scheme for quantum states.

Derived a classical succinct argument system for QMA under LWE.

Provided a method to convert quantum PCPs into succinct arguments.

Abstract

We define the notion of a classical commitment scheme to quantum states, which allows a quantum prover to compute a classical commitment to a quantum state, and later open each qubit of the state in either the standard or the Hadamard basis. Our notion is a strengthening of the measurement protocol from Mahadev (STOC 2018). We construct such a commitment scheme from the post-quantum Learning With Errors (LWE) assumption, and more generally from any noisy trapdoor claw-free function family that has the distributional strong adaptive hardcore bit property (a property that we define in this work). Our scheme is succinct in the sense that the running time of the verifier in the commitment phase depends only on the security parameter (independent of the size of the committed state), and its running time in the opening phase grows only with the number of qubits that are being opened (and the security parameter). As a corollary we obtain a classical succinct argument system for QMA under the post-quantum LWE assumption. Previously, this was only known assuming post-quantum secure indistinguishability obfuscation. As an additional corollary we obtain a generic way of converting any X/Z quantum PCP into a succinct argument system under the quantum hardness of LWE.