Dimension Independent and Computationally Efficient Shadow Tomography

TL;DR

This paper introduces a dimension-independent shadow tomography algorithm with optimal sample complexity and efficiency, enabling practical quantum state estimation with minimal memory and noise robustness.

Contribution

The paper presents a novel shadow tomography algorithm that is both dimension-independent and highly efficient, improving upon previous methods in sample complexity and resource usage.

Findings

Sample complexity is $\Theta(\sqrt{m}\log m/\epsilon^2)$, independent of quantum state dimension.

Algorithm is efficient in quantum memory, gate complexity, and classical computation.

Robust to measurement noise and implementable as a low-memory read-once quantum circuit.

Abstract

We describe a new shadow tomography algorithm that uses $n=\Theta(\sqrt{m}\log m/\epsilon^2)$ samples, for $m$ measurements and additive error $\epsilon$, which is independent of the dimension of the quantum state being learned. This stands in contrast to all previously known algorithms that improve upon the naive approach. The sample complexity also has optimal dependence on $\epsilon$. Additionally, this algorithm is efficient in various aspects, including quantum memory usage (possibly even $O(1)$), gate complexity, classical computation, and robustness to qubit measurement noise. It can also be implemented as a read-once quantum circuit with low quantum memory usage, i.e., it will hold only one copy of $\rho$ in memory, and discard it before asking for a new one, with the additional memory needed being $O(m\log n)$. Our approach builds on the idea of using noisy measurements, but instead of focusing on gentleness in trace distance, we focus on the \textit{gentleness in shadows}, i.e., we show that the noisy measurements do not significantly perturb the expected values.