Error regions in quantum state tomography: computational complexity caused by geometry of quantum states

TL;DR

This paper demonstrates that determining optimal quantum state uncertainty regions is computationally hard (NP-hard), highlighting a fundamental trade-off between accuracy and efficiency in quantum state tomography.

Contribution

The paper formalizes the complexity of quantum uncertainty quantification, proving NP-hardness for both frequentist and Bayesian approaches, and discusses implications for practical algorithms.

Findings

Quantum state uncertainty regions are NP-hard to compute.

There exists a trade-off between optimality and computational efficiency.

The results apply to both frequentist and Bayesian quantum statistics.

Abstract

The outcomes of quantum mechanical experiments are inherently random. It is therefore necessary to develop stringent methods for quantifying the degree of statistical uncertainty about the results of quantum experiments. For the particularly relevant task of quantum state estimation, it has been shown that a significant reduction in uncertainty can be achieved by taking the positivity of quantum states into account. However -- the large number of partial results and heuristics notwithstanding -- no efficient general algorithm is known that produces an optimal uncertainty region from experimental data and the prior constraint of positivity. Here, we make this problem precise and show that the general case is NP-hard. Our result leaves room for the existence of efficient approximate solutions, and therefore does not yet imply that the practical task of quantum uncertainty quantification is intractable. However, it does show that there exists a non-trivial trade-off between optimality and computational efficiency for error regions. We prove two versions of the result: One for frequentist and one for Bayesian statistics.