Projected Least-Squares Quantum Process Tomography

TL;DR

This paper introduces a projected least-squares method for quantum process tomography that improves accuracy and efficiency by combining least-squares estimation with a novel projection technique, supported by rigorous theoretical bounds and numerical experiments.

Contribution

The paper presents a new PLS approach for QPT, including closed-form estimators, a two-step projection method, and theoretical analysis with practical demonstrations.

Findings

Bounds on Frobenius and trace-norm errors are linear in rank.

For low-rank channels, error rates improve by a factor of d^2.

Numerical experiments show competitive accuracy and computational efficiency.

Abstract

We propose and investigate a new method of quantum process tomography (QPT) which we call projected least squares (PLS). In short, PLS consists of first computing the least-squares estimator of the Choi matrix of an unknown channel, and subsequently projecting it onto the convex set of Choi matrices. We consider four experimental setups including direct QPT with Pauli eigenvectors as input and Pauli measurements, and ancilla-assisted QPT with mutually unbiased bases (MUB) measurements. In each case, we provide a closed form solution for the least-squares estimator of the Choi matrix. We propose a novel, two-step method for projecting these estimators onto the set of matrices representing physical quantum channels, and a fast numerical implementation in the form of the hyperplane intersection projection algorithm. We provide rigorous, non-asymptotic concentration bounds, sampling complexities and confidence regions for the Frobenius and trace-norm error of the estimators. For the Frobenius error, the bounds are linear in the rank of the Choi matrix, and for low ranks, they improve the error rates of the least squares estimator by a factor $d^2$, where $d$ is the system dimension. We illustrate the method with numerical experiments involving channels on systems with up to 7 qubits, and find that PLS has highly competitive accuracy and computational tractability.