Post-processing optimization and optimal bounds for non-adaptive shadow tomography

TL;DR

The paper tackles non-adaptive shadow tomography with informationally overcomplete POVMs by optimizing the classical post-processing stage to minimize estimator variance. It casts the reconstruction-coefficient choice as a convex minimax problem, solving for state-dependent optimal coefficients and then maximizing the resulting variance bound over all quantum states, with guaranteed convergence. This yields the tightest state-independent variance bound for a fixed measurement scheme and observable, often surpassing canonical estimators and, in structured targets, dramatically altering sampling-scaling behavior. The approach provides a practical, task-adaptive upgrade to shadow tomography that can substantially reduce sampling overhead and improve scalability in concrete quantum-information estimation tasks.

Abstract

Informationally overcomplete POVMs are known to outperform minimally complete measurements in many tomography and estimation tasks, and they also leave a purely classical freedom in shadow tomography: the same observable admits infinitely many unbiased linear reconstructions from identical measurement data. We formulate the choice of reconstruction coefficients as a convex minimax problem and give an algorithm with guaranteed convergence that returns the tightest state-independent variance bound achievable by post-processing for a fixed POVM and observable. Numerical examples show that the resulting estimators can dramatically reduce sampling complexity relative to standard (canonical) reconstructions, and can even improve the qualitative scaling with system size for structured noncommuting targets.

Figures (2)

• Figure 1: a) Post-processing optimization. The panel schematizes the shadow tomography procedure with optimized post-processing: the unknown state is sampled $K$ with a fixed POVM and the relative frequencies of outcomes $\{\,f_k\}_{k=1}^n$ are recorded; these frequencies are combined with decomposition coefficients $\{x_k^{\,\star}\}_{k=1}^n$ of the target observable in terms of the same POVM, which can be obtained through a classical convex optimization scheme. The resulting classical estimator thus approximates the actual expectation value $\ev{O}$. b) Optimized variance of product observables. The optimal upper bound on variance for tensors of single-qubit projectors constrained on the prime meridian of the Bloch sphere in terms of a single angle parameter $\theta\in[0,\pi]$ (see main text for details). The panel compares the scaling of the optimal variance (in log scale) with number of qubits for $\theta = \frac{s}{30}\cdot \frac{\pi}{2}$, where the index $s$ assumes all integer values $1\leq s\leq 30$: each run presents and increment of $\Delta \theta = \frac{\pi}{60}$. The scaling slope varies continuously between the sub-linear optimal case, for $\theta = 0 \wedge \frac{\pi}{2}$ (red line), and the exponential worst case scenario, for $\theta = \frac{\pi}{4}$, which coincides with the scaling obtained from the canonical coefficients (black dotted line).
• Figure 2: Optimized variance of sums of parametrized local Pauli observables The left panel a) shows the comparison of the optimal upper bound on variance (full line) with the canonical maximal variance (dotted line) in terms of the balance parameter $\theta$ for different number of qubits. The right panel b) shows the scaling of the square root of the optimal variance (in order to highlight the scaling) with the number of qubits for chosen fixed values of $\theta$. Black dotted line shows the scaling using canonical coefficients (the same for every value of $\theta$ considered).