Learning Distributions over Quantum Measurement Outcomes

TL;DR

This paper introduces an efficient method for learning the probability distributions of quantum measurement outcomes with multiple outcomes, improving over previous shadow tomography techniques especially when measurements have more than two outcomes.

Contribution

We develop an online shadow tomography method for estimating distributions over unknown quantum measurements with multiple outcomes, achieving near-optimal sample complexity.

Findings

Sample complexity scales logarithmically with the number of measurements and system dimension.

The method is optimal in its dependence on the number of measurement outcomes.

A matching lower bound confirms the efficiency of our approach.

Abstract

Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of $2$-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal for the dependence on $K$.