Lower Bounds for Unitary Property Testing with Proofs and Advice

2024-01-15

arXiv.org

AI Summary

This work investigates property testing of quantum black-box unitary operators, focusing on establishing lower bounds on the query complexity of testers equipped with quantum proofs and quantum advice. By establishing an equivalence between unitary property testing and unitary channel discrimination, and leveraging quantum query analysis, relativized SBQP/QMA(2) techniques, and a sample-to-query amplification theorem, we prove—first time—that QMA(2)/qpoly testers cannot improve the efficiency of testing several fundamental unitary properties. Our bound eliminates prior logarithmic factors, revealing an intrinsic limitation of unentangled quantum proofs in high-precision unitary testing. The results yield tight lower bounds for key problems including quantum phase estimation, entanglement entropy estimation, and Gibbs sampling. Furthermore, we construct relativized oracles that achieve strict separations: QMA(2) vs. SBQP, and QMA/qpoly vs. SBQP.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

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 $mathsf{QMA}(2) otsupset mathsf{SBQP}$ and $mathsf{QMA}/mathsf{qpoly} otsupset mathsf{SBQP}$. 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.

Problem

• Proving lower bounds for unitary property testing complexity
• Exploring limitations of quantum proofs and advice
• Enhancing lower bounds for quantum phase estimation

Innovation

• Connects unitary testing to channel discrimination
• Applies to testers with quantum proofs and advice
• Removes log factors in lower bounds