A Survey of Quantum Property Testing

TL;DR

This survey reviews recent advances in quantum property testing, covering quantum algorithms for classical and quantum objects, and discussing bounds, efficiencies, and open problems in the field.

Contribution

It provides a comprehensive overview of quantum property testing, highlighting known results, efficiencies over classical methods, and open questions in the domain.

Findings

Quantum testers can be exponentially more efficient than classical testers for certain properties.

Quantum property testing bounds are established for states and operations.

Connections to quantum information theory are explored.

Abstract

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.