Testing classical properties from quantum data

• URL: http://arxiv.org/abs/2411.12730v1
• Date: Tue, 19 Nov 2024 18:52:55 GMT
• Title: Testing classical properties from quantum data
• Authors: Matthias C. Caro, Preksha Naik, Joseph Slote,
• Abstract summary: 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 $tildemathcalO(n2)$ function state copies as compared to the $2Omega(sqrtn)$ samples required classically. We also present $mathcalO(1)$-copy testers for symmetry and triangle-freeness, comparing favorably to classical lower bounds of $Omega(n1/4)$ and $Omega(n
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• 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 $\tilde{\mathcal{O}}(n^2)$ 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 $\varepsilon\leq\mathcal{O}(n^{-3/2})$. This implies a strict separation between testing monotonicity from quantum data and from quantum queries--where $\tilde{\mathcal{O}}(n)$ 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.

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