Quantum property testing: A survey and one new result

Ron

Ashley Montanaro, Ronald de Wolf

The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow “far” from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. These arise for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.

Ashley Montanaro, Ronald de Wolf

The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow "far" from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.

Kean Chen, Qisheng Wang, Zhicheng Zhang

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.

Yoshifumi Inui, François Le Gall

Yoshifumi Inui, François Le Gall