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A Survey of Quantum Property Testing

Ashley Montanaro, Ronald de Wolf

TL;DR

The survey maps the landscape of quantum property testing across three intertwined settings: testing classical properties with quantum testers, testing quantum properties with classical testers, and testing quantum properties with quantum testers. It consolidates key algorithmic techniques—amplitude amplification, Bernstein–Vazirani, Simon, Shor, quantum counting, and quantum walks—and shows how they yield substantial quantum speed-ups for verifying properties of large objects. It also surveys lower-bound techniques (polynomial and adversary methods) and explores the deep connections between property testing and quantum computational complexity, including device-independent testing and quantum PCP considerations. The work highlights open questions, such as stabilizer-state testers, distributed testing, and a complete characterization of which properties admit efficient quantum testers, while outlining a broad program linking testing to complexity and cryptographic applications. Overall, it presents a cohesive framework for understanding when and how quantum resources enable efficient testing beyond classical capabilities.

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.

A Survey of Quantum Property Testing

TL;DR

The survey maps the landscape of quantum property testing across three intertwined settings: testing classical properties with quantum testers, testing quantum properties with classical testers, and testing quantum properties with quantum testers. It consolidates key algorithmic techniques—amplitude amplification, Bernstein–Vazirani, Simon, Shor, quantum counting, and quantum walks—and shows how they yield substantial quantum speed-ups for verifying properties of large objects. It also surveys lower-bound techniques (polynomial and adversary methods) and explores the deep connections between property testing and quantum computational complexity, including device-independent testing and quantum PCP considerations. The work highlights open questions, such as stabilizer-state testers, distributed testing, and a complete characterization of which properties admit efficient quantum testers, while outlining a broad program linking testing to complexity and cryptographic applications. Overall, it presents a cohesive framework for understanding when and how quantum resources enable efficient testing beyond classical capabilities.

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.

Paper Structure

This paper contains 48 sections, 23 theorems, 100 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum testing of classical properties
  3. Classical testing of quantum properties
  4. Quantum testing of quantum properties
  5. Quantum testing of classical properties
  6. Preliminaries
  7. Upper bounds
  8. Using amplitude amplification
  9. Using the Bernstein-Vazirani algorithm
  10. Testing juntas
  11. Using Simon's algorithm
  12. Using Shor's algorithm
  13. Using quantum counting
  14. Using Ambainis's algorithm
  15. Quantum speed-ups for testing group-theoretic properties
  16. ...and 33 more sections

Key Result

Theorem 1

For every $A\subseteq\{0,1\}^n$ there is an $O(1/\sqrt{\epsilon})$-query quantum $\epsilon$-tester for $\mathcal{P}^A_{BV}$; in contrast, for a $1-o(1)$ fraction of all sets $A$, every classical 1/2-tester for $\mathcal{P}^A_{BV}$ needs $\Omega(\log N)$ queries.

Figures (3)

  • Figure 1: Isometry for transforming a perfect CHSH protocol to the standard one.
  • Figure 2: The swap test.
  • Figure 3: Schematic of the product test applied to an $n$-partite state ${\left\vert{\psi}\right\rangle}$. The swap test (vertical boxes) is applied to the $n$ pairs of corresponding subsystems of two copies of ${\left\vert{\psi}\right\rangle}$ (horizontal boxes).

Theorems & Definitions (43)

  • Theorem 1: Buhrman et al. bfnr:qpropj
  • proof
  • Theorem 2: essentially Atıcı and Servedio aticiservedio:testing
  • proof
  • Theorem 3: essentially Buhrman et al. bfnr:qpropj
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof
  • ...and 33 more