Maximum Likelihood, Minimum Effort

TL;DR

This paper introduces an efficient algorithm for estimating the maximum likelihood quantum state from noisy measurement data, involving basis transformation and a novel method for projecting onto physical states.

Contribution

It presents a new computational approach combining basis change and a linear-time projection method to efficiently estimate quantum states from measurement data.

Findings

Algorithm operates in polynomial time, specifically O(d^4) or better for special cases.

The method effectively handles Gaussian noise in measurement outcomes.

Provides a practical solution for quantum state tomography with improved efficiency.

Abstract

We provide an efficient method for computing the maximum likelihood mixed quantum state (with density matrix $\rho$) given a set of measurement outcome in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix $\mu$ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst $O(d^4)$ for the basis change plus $O(d^3)$ for finding $\rho$ where $d$ is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only $O(d^3)$ as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.