We show that every language in QMA admits a classical-verifier, quantum-prover zero-knowledge argument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in QMA, with a classical verifier, introduced by Mahadev (FOCS 2018).
We show that every language in QMA admits a classical-verifier, quantum-prover zero-knowledge argument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in QMA, with a classical verifier, introduced by Mahadev (FOCS 2018).
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration.
With the rapid development of quantum computers, proofs of quantumness have recently become an interesting research direction. However, in current schemes for proofs of quantumness, quantum provers face the risk of being maliciously exploited by classical verifiers. Through malicious strategies in interaction with quantum provers, classical verifiers could solve some instances of hard problems that arise from the specific scheme in use. This is due to the lack of formalization that prevents malicious verifiers from extracting useful information in proofs of quantumness. To address this issue, we formalize zero-knowledge proofs of quantumness. Intuitively, the zero-knowledge property necessitates that the information gained by the classical verifier from interactions with the quantum prover should not surpass what can be simulated using a simulated classical prover interacting with the same verifier. As a result, the new zero-knowledge notion can prevent a malicious verifier from exploiting quantum advantage. We find that the classical zero-knowledge proof is sufficient to compile some existing proofs of quantumness schemes into zero-knowledge proofs of quantumness schemes. It appears to be more general to require zero-knowledge proof on the verifier side instead of the prover side. This helps to regulate the verifier's behavior from malicious to be honest-but-curious. As a result, both parties will play not only one role in the proofs of quantumness but also the dual role in the classical zero-knowledge proof. Specifically, Shor's factoring-based scheme and the learning with errors-based scheme in [Brakerski et al., FOCS, 2018] can be transformed into zero-knowledge proofs of quantumness by requiring an extractable non-interactive zero-knowledge argument on the verifier side. Zero-knowledge proofs of quantumness can thus be viewed as an enhanced security notion for proofs of quantumness.
We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.
A quantum verifier decides membership in QMA using a (single) quantum witness state. This witness may be arbitrary, and no requirement is placed on it beyond its acceptance probability. A protocol claiming to classically verify QMA should offer the same guarantee: any witness that would convince a quantum verifier should also suffice to convince a classical verifier. Unfortunately, existing classical verification of quantum computation (CVQC) protocols behave quite differently from the original QMA verifier: they require many copies of the witness and destroy every one, effectively demanding that the prover solve the hard problem of witness duplication before verification can begin.
We build a protocol to classically verify QMA languages that uses a single copy of the prover's QMA witness. Whenever the original QMA verifier accepts that witness with overwhelming probability, our protocol has negligible completeness and soundness errors and returns the witness negligibly disturbed after interaction with an honest verifier. The soundness of our CVQC is based on the post-quantum Learning With Errors (LWE) assumption.
As an intermediate step of independent interest, we introduce witness-preserving classical arguments for NP which enable provers that start out with a superposition over NP witnesses to convince a verifier without disturbing their superposition.
In a proof of knowledge (PoK), a verifier becomes convinced that a prover possesses privileged information. In combination with zero-knowledge proof systems, PoKs play an important role in security protocols such as in digital signatures and authentication schemes, as they enable a prover to demonstrate possession of certain information (such as a private key or a credential), without revealing it. A PoK is formally defined via the existence of an extractor, which is capable of reconstructing the key information that makes a verifier accept, given oracle access to any accepting prover. We extend this concept to the setting of a single classical verifier and multiple quantum provers and present the first statistical zero-knowledge (ZK) PoK proof system for problems in QMA. To achieve this, we establish the PoK property for the ZK protocol of Broadbent, Mehta, and Zhao (TQC 2024), which applies to the local Hamiltonian problem. More specifically, we construct an extractor which, given oracle access to a provers' strategy that leads to high acceptance probability, is able to reconstruct the ground state of a local Hamiltonian. Our result can be seen as a new form of self-testing, where, in addition to certifying a pre-shared entangled state, the verifier also certifies that provers have access to a quantum system, in particular, a ground state; this indicates a new level of verification for a proof of quantumness.