Lower Bounds for Unitary Property Testing with Proofs and Advice

TL;DR

This work introduces a unified lower-bound framework for quantum unitary property testing built on the discrimination of unitary channels via the diamond distance. The authors prove that, even when a tester has access to unentangled quantum proofs and quantum advice ($ extsf{QMA}(2)/ extsf{qpoly}$), certain unitary-property-testing tasks require query counts matching Heisenberg-limited scaling, and they obtain tight or near-tight bounds for key problems such as quantum phase estimation, entanglement entropy testing, subset support verification, and quantum amplitude estimation. They also show quantum oracle separations demonstrating that high-precision black-box problems resist reduction in query complexity despite available proofs or advice. Relative to prior methods (e.g., Wang and Zhang), the technique eliminates suboptimal logarithmic factors and extends to broader advice/proofs models, offering a versatile tool for a wide range of unitary-testing and related problems. The results illuminate fundamental limits on leveraging quantum proofs and advice in black-box quantum tasks and open questions about extending tight bounds to more problems and precision regimes.

Abstract

In unitary property testing a quantum algorithm, also known as a tester, is given query access to a black-box unitary and has to decide whether it satisfies some property. We propose a new technique for proving lower bounds on the quantum query complexity of unitary property testing and related problems, which utilises its connection to unitary channel discrimination. The main advantage of this technique is that all obtained lower bounds hold for any $\mathsf{C}$-tester with $\mathsf{C} \subseteq \mathsf{QMA}(2)/\mathsf{qpoly}$, showing that even having access to both (unentangled) quantum proofs and advice does not help for many unitary property testing problems. We apply our technique to prove lower bounds for problems like quantum phase estimation, the entanglement entropy problem, quantum Gibbs sampling and more, removing all logarithmic factors in the lower bounds obtained by the sample-to-query lifting theorem of Wang and Zhang (2023). As a direct corollary, we show that there exist quantum oracles relative to which $\mathsf{QMA}(2) \not\supset \mathsf{SBQP}$ and $\mathsf{QMA}/\mathsf{qpoly} \not\supset \mathsf{SBQP}$. The former shows that, at least in a black-box way, having unentangled quantum proofs does not help in solving problems that require high precision.

Figures (1)

• Figure 1: Query complexity model for unitary property tester making $T$ queries to a unitary of interest $U$. The initial state is $\ket{\psi_\textup{init}} = \ket{0}^{\otimes \mathsf{poly}(n)} \ket{\psi_\text{input}}$ is allowed to consist of an input-independent part and an input-dependent part, depending on the class $\mathsf{C}$ of the $\mathsf{C}$-property tester. The unitary of interest $U$ can be accessed directly, through its inverse, controlled or controlled inverse, i.e. we have that $\tilde{U}^t \in \{U_i,U_i^\dagger, cU_i, cU_i^\dagger \}$ for all $t \in [T]$, $i \in \{1,2\}$.

Theorems & Definitions (15)

• Theorem 1: Diamond-norm lower bound for unitary channel discrimination
• Theorem 2
• Theorem 3
• Definition 1: $\mathsf{BQP}$-tester
• Definition 2: $\mathsf{QMA}(k)$-tester
• Definition 2a: $\mathsf{QCMA}$-tester
• Definition 3: $\mathsf{BQP}\slash\mathsf{qpoly}$-tester
• Definition 3a: $\mathsf{BQP}\slash\mathsf{poly}$-tester
• Definition 4: $\mathsf{QMA}(k)\slash\mathsf{qpoly}$-tester
• Definition 5: $\mathsf{SBQP}$-tester