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

TL;DR

This paper introduces a convex optimization approach to optimize post-processing in shadow tomography, significantly reducing sampling complexity and improving scaling for structured quantum states.

Contribution

It formulates the reconstruction coefficient selection as a convex minimax problem and provides an algorithm with guaranteed convergence for optimal variance bounds.

Findings

Optimized estimators reduce sampling complexity.

Improved scaling with system size for structured targets.

Numerical results demonstrate significant variance reduction.

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.