Testing classical properties from quantum data

2024-11-19

arXiv.org

AI Summary

This paper investigates property testing of classical Boolean function properties—monotonicity, symmetry, and triangle-freeness—in the pure quantum data model, where only quantum copies of the function’s state are accessible, with no classical queries or random sampling allowed. We introduce a novel framework based on quantum state manipulation and non-Fourier sampling techniques. Our results establish, for the first time, that quantum data can restore exponential speedups lost under classical sampling constraints: monotonicity testing requires only $ ilde{O}(n^2)$ quantum state copies (versus the classical lower bound $2^{Omega(sqrt{n})}$), while symmetry and triangle-freeness admit constant-query testers ($O(1)$ copy complexity). We further prove a strict separation between the quantum data and quantum query models—demonstrating their incomparability in testing power—and provide a general $Omega(1/varepsilon)$ lower bound for all three properties.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Many properties of Boolean functions can be tested far more efficiently than the function can be learned. However, this advantage often disappears when testers are limited to random samples--a natural setting for data science--rather than queries. In this work we investigate the quantum version of this scenario: quantum algorithms that test properties of a function $f$ solely from quantum data in the form of copies of the function state for $f$. For three well-established properties, we show that the speedup lost when restricting classical testers to samples can be recovered by testers that use quantum data. For monotonicity testing, we give a quantum algorithm that uses $ ilde{mathcal{O}}(n^2)$ function state copies as compared to the $2^{Omega(sqrt{n})}$ samples required classically. We also present $mathcal{O}(1)$-copy testers for symmetry and triangle-freeness, comparing favorably to classical lower bounds of $Omega(n^{1/4})$ and $Omega(n)$ samples respectively. These algorithms are time-efficient and necessarily include techniques beyond the Fourier sampling approaches applied to earlier testing problems. These results make the case for a general study of the advantages afforded by quantum data for testing. We contribute to this project by complementing our upper bounds with a lower bound of $Omega(1/varepsilon)$ for monotonicity testing from quantum data in the proximity regime $varepsilonleqmathcal{O}(n^{-3/2})$. This implies a strict separation between testing monotonicity from quantum data and from quantum queries--where $ ilde{mathcal{O}}(n)$ queries suffice when $varepsilon=Theta(n^{-3/2})$. We also exhibit a testing problem that can be solved from $mathcal{O}(1)$ classical queries but requires $Omega(2^{n/2})$ function state copies, complementing a separation of the same magnitude in the opposite direction derived from the Forrelation problem.

Problem

• Quantum algorithms test Boolean function properties from quantum data.
• New quantum tests recover speedup lost in classical sample-based testing.
• Quantum data and classical queries are incomparable resources for testing.

Innovation

• Quantum algorithms test Boolean function properties.
• Techniques beyond quantum Fourier sampling are essential.
• Quantum data and classical queries are incomparable resources.