Compressed Sensing Shadow Tomography

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 $S_{ij}=\text{Tr}(O_i ρ(t_j))$ 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 $\widetildeΘ(N/s)$ (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.

Figures (9)

• Figure 1: Schematic of the CSST data-processing pipeline. We sample $m\ll N$ timesteps uniformly at random (red markers) and, at each sampled time, collect local classical-shadow snapshots that enable simultaneous estimation of the $M$ target low-weight Pauli expectation values $\{\Tr(O_i\rho(t_j))\}_{i=1}^M$. For each observable $O_i$, the resulting time-subsampled signal is then reconstructed on the full length-$N$ grid using Fourier/DCT-domain compressed sensing.
• Figure 2: $\langle O(t)\rangle$ over time for four selected observables of the $2\times3$ noisy Heisenberg model defined in \ref{['eq:heis_ham', 'eq:lme']}. We plot the ground truth from exact simulation, the baseline estimation using only shadow tomography, and the CSST reconstructions at $\alpha^*$ from each observable's $\alpha$ sweep. Remarkably, the subsampled CSST reconstruction---which uses only 60% of the available data---is significantly closer to the ground truth data than the baseline ST estimation.
• Figure 3: Baseline shadow-tomography noise levels and transform-domain compressibility for our noisy $2\times 3$ Heisenberg model. Left: Baseline ST RMSE [\ref{['eq:RMSE']}] as a function of snapshots per timestep $N_{\mathrm{ST}}$, grouped by Pauli weight $w\in\{1,2,3,4\}$. Markers show empirical RMSEs averaged over all weight-$w$ Pauli strings; solid curves show the corresponding finite-sample upper bounds from the Pauli-shadow analysis using Bernstein's inequality [\ref{['eq:bern']}]. Right: Weight-sector-averaged truncation RMSE of the exact (noise-free) Pauli traces under the orthonormal DCT-II: for each trace we keep the $s$ largest-magnitude DCT coefficients, invert the transform, and compute the resulting RMSE. The dashed guideline indicates an approximate $\propto s^{-2}$ decay over the power-law regime ($s\gtrsim 10^2$), highlighting strong DCT-domain compressibility.
• Figure 4: CSST reconstruction error versus LASSO regularization (same noisy $2\times 3$ Heisenberg model as in \ref{['fig:setup']}). Each panel corresponds to a Pauli-weight sector $w\in\{1,2,3,4\}$ and reports RMSE averaged over all weight-$w$ Pauli strings. Curves show the CSST reconstruction RMSE [\ref{['eq:RMSE']}] as a function of the regularization parameter $\alpha$ in \ref{['eq:lasso']}, for several choices of the number of randomly sampled timesteps $m$ (curve labels), at fixed snapshots $N_{\mathrm{ST}}=7437$ per sampled timestep. The dashed reference line indicates the average baseline ST RMSE using all $N$ timesteps without CS. Black annotations indicate $m^*$, the smallest sampled-timestep value for which the best-achieved RMSE (minimized over $\alpha$) beats the baseline ST line.
• Figure 5: Optimal regularization parameters for CSST. For each Pauli observable $O_i$ and each sampled-timestep budget $m$, we define the optimal value $\alpha_i^*$ as the value of $\alpha$ that minimizes the reconstruction RMSE over the $\alpha$-sweep shown in \ref{['fig:cserrs_sind']}. Lines show the mean of $\alpha_i^*$ within each Pauli-weight sector $w$ as a function of $m$, and shading shows one standard deviation across observables in that sector (same model and snapshot budgets as in \ref{['fig:cserrs_sind']}). The stabilization (variance reduction) of $\alpha^*$ at larger $m$ suggests that (for fixed $w$) a single near-optimal regularization choice can work well across many observables. Left: $\alpha_i^*$ for the $2\times3$ Heisenberg model. Right: $\alpha_i^*$ for the $2\times3$ TFIM model with initial state $\ket{\text{GHZ}}$ (see \ref{['app:tfim']} for the model Hamiltonian and additional data). Note the similarity with respect to the Heisenberg model, highlighting that the $\alpha_i^*$ are primarily a function of the noise only and not the intrinsic details of the chosen model. Thus, an $\alpha^*$ which works well on average for a given Pauli weight from one model's dataset should be applicable to other models as well. The sharp jump which occurs at low $m$ is due to the corresponding curves' minima in \ref{['fig:cserrs_sind']} moving from around $10^{-7}$ to $10^{-4}$.

Theorems & Definitions (5)

• Theorem 3.1: Local Classical Shadows huangPredictingManyProperties2020
• Definition 3.1: Restricted Isometry Property candesDecodingLinearProgramming2005
• Theorem 3.2: Compressed sensing candesStableSignalRecovery2005candesIntroductionCompressiveSampling2008
• Theorem 3.3: Fourier Compressed Sensing havivRestrictedIsometryProperty2015
• Theorem 3.4: End-to-end guarantee assuming exact sparsity