We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. First, we answer a longstanding open question by showing that the Consistency of Local Density Matrices problem is QMA-complete under Karp reductions. We also show for the first time a commit-and-open computational zero-knowledge proof system for all of QMA as a quantum analogue of a “sigma” protocol. We then define a Proof of Quantum Knowledge, which guarantees that a prover is effectively in possession of a quantum witness in an interactive proof, and show that our zero-knowledge proof system satisfies this definition. Finally, we show that our proof system can be used to establish that QMA has a quantum non-interactive zero-knowledge proof system in the secret parameters setting. Our main technique consists in developing locally simulatable proofs for all of QMA: this is an encoding of a QMA witness such that it can be efficiently verified by probing only five qubits and, furthermore, the reduced density matrix of any five-qubit subsystem can be computed in polynomial time and is independent of the witness. This construction follows the techniques of Grilo, Slofstra, and Yuen [FOCS 2019].

Publication

QMA-Hardness of Consistency of Local Density Matrices with Applications to Quantum Zero-Knowledge

We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. First, we answer a longstanding open question by showing that the Consistency of Local Density Matrices problem is QMA-complete under Karp reductions. We also show for the first time a commit-and-open computational zero-knowledge proof system for all of QMA as a quantum analogue of a “sigma” protocol. We then define a Proof of Quantum Knowledge, which guarantees that a prover is effectively in possession of a quantum witness in an interactive proof, and show that our zero-knowledge proof system satisfies this definition. Finally, we show that our proof system can be used to establish that QMA has a quantum non-interactive zero-knowledge proof system in the secret parameters setting. Our main technique consists in developing locally simulatable proofs for all of QMA: this is an encoding of a QMA witness such that it can be efficiently verified by probing only five qubits and, furthermore, the reduced density matrix of any five-qubit subsystem can be computed in polynomial time and is independent of the witness. This construction follows the techniques of Grilo, Slofstra, and Yuen [FOCS 2019].