Quantum Lower Bounds by Sample-to-Query Lifting

TL;DR

The paper introduces a quantum sample-to-query lifting theorem that connects the sample complexity of quantum state testing to the query complexity of unitary property testing, providing a new information-theoretic route to quantum lower bounds. The main result yields a quadratic relation between quantum sample and query complexities, which is shown to be tight up to logarithmic factors. Leveraging this lifting, the authors derive near-tight quantum lower bounds for quantum Gibbs sampling, entanglement entropy, Hamiltonian simulation, phase estimation, amplitude estimation, and matrix spectrum testing, and they provide unified proofs for several previously known lower bounds. The work suggests a flexible, information-theoretic framework for lower bounds in quantum query complexity and points to several open questions, including potential classical analogs and improvements of logarithmic factors. Overall, the lifting method broadens the toolkit for establishing fundamental limits of quantum computation and offers new insights into the interplay between sampling and querying quantum data.

Abstract

The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998, J. ACM 2001), the adversary method by Ambainis (STOC 2000, J. Comput. Syst. Sci. 2002), and the compressed oracle method by Zhandry (CRYPTO 2019) have been shown to be powerful in proving quantum query lower bounds for a wide variety of problems. In this paper, we propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem, which is from an information theory perspective. Using this method, we obtain the following new results: 1. A quadratic relation between quantum sample and query complexities regarding quantum property testing, which is optimal and saturated by quantum state discrimination. Here, the sample complexity is measured given sample access to the quantum state to be tested, while the query complexity is measured given query access to an oracle that block-encodes the quantum state. 2. A matching lower bound $\widetilde Ω(β)$ for quantum Gibbs sampling at inverse temperature $β$, showing that the quantum Gibbs sampler by Gilyén, Su, Low, and Wiebe (STOC 2019) is optimal. 3. A new lower bound $\widetilde Ω(1/\sqrtΔ)$ for the entanglement entropy problem with gap $Δ$, which was recently studied by She and Yuen (ITCS 2023). 4. A series of quantum query lower bounds for matrix spectrum testing, based on the sample lower bounds for quantum state spectrum testing by O'Donnell and Wright (STOC 2015, Comm. Math. Phys. 2021). In addition, we also provide unified proofs for some known lower bounds that have been proven previously via different techniques, including those for phase/amplitude estimation and Hamiltonian simulation.

Figures (4)

• Figure 1: Commutative diagram for the form of lifting theorems, where $F$ is the set of all possible $f$'s and $F' = L\lparen F\rparen \coloneqq \{L\lparen f\rparen:f \in F\}$.
• Figure 2: $\mathsf{S}$-$\mathsf{Q}_\diamond$ relation for quantum state testing vs. $\mathsf{D}$-$\mathsf{Q}$ relation for Boolean functions. Here, a quantum state testing problem $\mathcal{P} = \lparen\mathcal{P}^{\textup{yes}}, \mathcal{P}^{\textup{no}}\rparen$ is expressed as a partial function $\mathcal{P} \colon \mathcal{D} \lparen\mathbb{C}^{N}\rparen\rightharpoonup\{0,1\}$, where $0$ and $1$ stand for yes and no instances, respectively.
• Figure 3: Diagram of reductions and relationships amongst our results.
• Figure 4: Tester for $\mathcal{P}$ based on $\mathcal{A}$.

Theorems & Definitions (87)

• Theorem 1.1: Quantum sample-to-query lifting, \ref{['thm:main']}
• proof : Proof sketch of \ref{['thm:lifting']}
• Remark 1.2
• Corollary 1.3: Query lower bound for quantum state discrimination
• Theorem 1.4: Lifting is tight, \ref{['thm:tightness']}
• proof : Proof sketch
• Remark 1.5
• Corollary 1.6: Optimality of quantum Gibbs sampling, \ref{['thm:gibbs']}
• proof : Proof sketch of \ref{['thm:gibbs-query-lower-bound']}
• Corollary 1.7: Impossibility of $O\lparen\sqrt{\beta}\rparen$-time quantum Gibbs sampling, \ref{['thm:sqrt-beta-gibbs']}