Dimension Independent and Computationally Efficient Shadow Tomography

We describe a new shadow tomography algorithm that uses $n=\mathrm{\Theta }(\sqrt{m}\log m\mathrm{/}{\epsilon }^2)$ 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 $\mathit{\text{gentleness}}\mathit{\text{in}}\mathit{\text{shadows}}$, i.e., we show that the noisy measurements do not significantly perturb the expected values.