Projected Least-Squares Quantum Process Tomogra-phy

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 d2, 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.

Automated summary

This paper proposes a new method called projected least squares (PLS) for quantum process tomography (QPT). The method involves computing the least-squares estimator of an unknown channel's Choi matrix and projecting it onto the convex set of Choi matrices. The authors provide closed form solutions for the estimators in various experimental setups and propose a novel two-step method for projecting the estimators onto the set of matrices representing physical quantum channels. They also provide concentration bounds, sampling complexities, and confidence regions for the error of the estimators. Numerical experiments demonstrate that PLS achieves high accuracy and computational tractability.