A List of Complexity Bounds for Property Testing by Quantum Sample-to-Query Lifting

TL;DR

This paper compiles 49 quantum complexity bounds for property testing, leveraging quantum sample-to-query lifting, including 41 new bounds and 18 near-optimal results, covering properties of distributions and quantum states.

Contribution

It provides a comprehensive list of new and existing quantum bounds for property testing using quantum sample-to-query lifting techniques.

Findings

41 new quantum complexity bounds for property testing.

18 bounds are near-optimal or optimal.

Includes bounds for properties like entropy and distribution closeness.

Abstract

Quantum sample-to-query lifting, a relation between quantum sample complexity and quantum query complexity presented in Wang and Zhang (SIAM J. Comput. 2025), was significantly strengthened by Tang, Wright, and Zhandry (2025) to the case of state-preparation oracles. In this paper, we compile a list of quantum lower and upper bounds for property testing that are obtained by quantum sample-to-query lifting. The problems of interest include testing properties of probability distributions and quantum states, such as entropy and closeness. This collection contains new results, as well as new proofs of known bounds. In total, we present 49 complexity bounds, where 41 are new and 18 are (near-)optimal.