Distributional Property Testing in a Quantum World

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling

Likelihood-free hypothesis testing

Consider the problem of testing $Z \sim \mathbb P^{\otimes m}$ vs $Z \sim \mathbb Q^{\otimes m}$ from $m$ samples. Generally, to achieve a small error rate it is necessary and sufficient to have $m \asymp 1/\epsilon^2$, where $\epsilon$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions $\mathbb P$ and $\mathbb Q$ and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem, which we call likelihood-free (or simulation-based) hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ (which are a priori only known to belong to a large non-parametric family $\mathcal P$) is given through $n$ iid samples from each. We demostrate existence of a fundamental trade-off between $n$ and $m$ given by $nm \asymp n^2_\mathsf{GoF}(\epsilon,\mathcal P)$, where $n_\mathsf{GoF}$ is the minimax sample complexity of testing between the hypotheses $H_0: \mathbb P= \mathbb Q$ vs $H_1: \mathsf{TV}(\mathbb P,\mathbb Q) \ge \epsilon$. We show this for three non-parametric families $\cal P$: $\beta$-smooth densities over $[0,1]^d$, the Gaussian sequence model over a Sobolev ellipsoid, and the collection of distributions $\mathcal P$ on a large alphabet $[k]$ with pmfs bounded by $c/k$ for fixed $c$. The test that we propose (based on the $L^2$-distance statistic of Ingster) simultaneously achieves all points on the tradeoff curve for these families. In particular, when $m\gg 1/\epsilon^2$ our test requires the number of simulation samples $n$ to be orders of magnitude smaller than what is needed for density estimation with accuracy $\asymp \epsilon$ (under $\mathsf{TV}$). This demonstrates the possibility of testing without fully estimating the distributions.Comment: 48 pages, 1 figur