Lower Bounds for Learning Quantum States with Single-Copy Measurements

TL;DR

This paper establishes fundamental lower bounds on the resources needed for quantum state learning using single-copy measurements, showing optimality of existing algorithms and limitations of adaptivity.

Contribution

It provides new lower bounds for quantum tomography and shadow tomography, confirming the optimality of Pauli tomography and showing that adaptivity offers no advantage with certain measurement constraints.

Findings

Optimality of Pauli tomography in sample complexity.

Lower bounds for learning rank r states with arbitrary and constant-outcome measurements.

Adaptivity does not improve efficiency with efficiently implementable single-copy measurements.

Abstract

We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown $d$-dimensional state. We first revisit a known lower bound due to Haah et al. (2017) on quantum tomography with accuracy $\epsilon$ in trace distance, when the measurements choices are independent of previously observed outcomes (i.e., they are nonadaptive). We give a succinct proof of this result. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes. In particular, this rigorously establishes the optimality of the folklore ``Pauli tomography" algorithm in terms of its sample complexity. We also derive novel bounds of $\Omega(r^2 d/\epsilon^2)$ and $\Omega(r^2 d^2/\epsilon^2)$ for learning rank $r$ states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of different measurements used by an algorithm. We extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of $\exp(O(d))$ measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy based on computing sample means of the given observables is optimal.