An Exponential Advantage for Adaptive Tomography of Structured States under Pauli Basis Measurements

TL;DR

This paper demonstrates that adaptive measurement strategies can significantly reduce the number of copies needed for quantum state tomography of structured states under local Pauli measurements, compared to non-adaptive methods.

Contribution

It provides a concrete example where adaptivity reduces sample complexity from exponential to polynomial in quantum state tomography.

Findings

Adaptive strategies achieve polynomial copy complexity for certain structured states.

Non-adaptive strategies require exponential copies in the worst case.

The result highlights a regime where adaptivity offers a provable advantage in quantum tomography.

Abstract

Broad claims about whether adaptivity helps in quantum state tomography can be misleading unless the state family, measurement architecture, and error metric are specified carefully. We study a restricted but physically important regime: single-copy quantum state tomography under local Pauli basis measurements, where the allowed measurement settings are tensor-product measurement operators built from local single-qubit Pauli operators, and performance is measured in trace distance with high probability in a minimax sense over a known structured family. We construct an explicit discrete prefix/tree family of states for which adaptive measurement selection achieves polynomial copy complexity, while every non-adaptive design requires exponentially many copies in the worst case. The adaptive upper bound comes from stagewise prefix recovery using hierarchical breadcrumb information revealed by partial prefix matches. The non-adaptive lower bound is based on a rare-prefix mechanism: every fixed design under-samples some deep prefix subset, and outside that subset the competing hypotheses induce identical one-shot laws, so only an exponentially small fraction of the measurement budget contributes to the KL divergence between the full data distributions. The result isolates a concrete regime in which adaptivity provably changes the sample-complexity scaling under the experimentally common local Pauli measurement architecture.