Compressed Sensing Shadow Tomography

TL;DR

This paper introduces a compressed sensing protocol for quantum shadow tomography that efficiently reconstructs time-dependent expectation values of many observables, significantly reducing measurement shots needed in quantum simulations.

Contribution

It combines classical shadow techniques with compressed sensing to reduce the number of measurements in reconstructing spectrally sparse quantum signals over time.

Findings

Reconstruction requires only O(s log^2 s log N) random timesteps for s-sparse signals.

Total measurement shots are reduced by up to a factor of N/s compared to naive methods.

Numerical experiments show effective reconstruction with fewer measurements in noisy quantum dynamics.

Abstract

Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix $S_{ij}=\text{Tr}(O_i \rho(t_j))$ for a collection of $M$ low-weight Pauli observables $\{O_i\}_{i=1}^M$ over $N$ timesteps $\{t_j\}_{j=1}^N$, while minimizing the total number of device shots. We propose a Compressed Sensing Shadow Tomography (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples $m\ll N$ timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-$N$ signal via standard $\ell_1$-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly $s$-sparse signals in a unitary transform basis, we show that $m=O \left(s\log^2 s \log N\right)$ random timesteps suffice (with high probability), leading to total-shot savings scaling as $\widetilde{\Theta}(N/s)$ (i.e., up to polylogarithmic factors) relative to collecting shadows at all $N$ timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.