Quantum State Search with Classical Interactive Advice

Abstract

We study whether a quantum verifier can find a hidden quantum state by exchanging classical messages with an unbounded prover who knows the state. The verifier has access to a reflection oracle about the target. For a Haar-random $n$-qubit target, we prove that achieving constant average fidelity with the target requires $(q+1)^2(B+1)=\Omega(2^n)$, where $q$ is the number of verifier queries and $B$ is the total prover-to-verifier communication. The bound allows arbitrary classical interaction and unrestricted computation between queries. Our proof extends the lower bound of Aaronson and Kuperberg (ToC'07) from a single classical proof to an arbitrary classical interaction. As a result, we obtain a quantum-oracle separation of QMA from QCIP. In particular, we give a unitary oracle $\mathcal{U}$ such that there exists a language $L\in\mathrm{QMA}^{\mathcal{U}}$ but $L\notin\mathrm{QCIP}^{\mathcal{U}}$. Our separation holds even for UniqueQMA, the restriction of QMA to verification with a unique quantum witness, which is known to be strictly weaker than QMA in the quantum black-box setting. We further give an oracle construction for a notion known as QMA--QCIP[2] gap problem. This gives average case hardness for a QMA language under classical access to a QCIP[2] decision oracle and is used in establishing negative results on witness encryption and quantum indistinguishability obfuscation.