Exploiting overcompleteness of Platonic-solid POVMs for shadow estimation

Accurately estimating expectation values of observables from finite measurement shots is a bottleneck in quantum chemistry and variational quantum algorithms. While randomized Pauli measurements are a standard approach, this paper investigates whether using more symmetric, informationally overcomplete (OC) measurements—specifically those based on Platonic solids—can reduce estimation variance through improved classical post-processing.

The authors study POVMs defined by the vertices of the five Platonic solids (tetrahedron, cube, octahedron, icosahedron, and dodecahedron) on the Bloch sphere. They employ locally-optimal (k-LO) dual frames to reconstruct observables from measurement statistics, which allows for variance reduction by exploiting the redundancy of overcomplete measurements. Furthermore, they propose a pre-processing strategy that jointly optimizes the POVM orientation and effect weights using a classically tractable proxy state, represented as a Matrix Product State (MPS) with varying bond dimensions.

The study reveals that the optimal POVM geometry is not universal and depends on the system size and the locality of the dual frame. For smaller systems, there is a non-monotonic relationship between the number of POVM effects and estimation performance. Crucially, the authors demonstrate that optimizing the POVM on an MPS proxy state can outperform standard randomized Pauli measurements. However, this success is contingent on the MPS bond dimension being sufficient to capture the relevant correlations of the target Hamiltonian. When the proxy is too simple (e.g., a product state), the optimization may fail to generalize to the true state. Additionally, the authors find that optimizing only the local Pauli-basis probabilities (a subset of the full optimization) captures most of the performance gains, offering a computationally efficient heuristic for near-term hardware.