Projected Least-Squares Quantum Process Tomography

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 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.

Sample-Optimal Quantum Process Tomography with Non-Adaptive Incoherent Measurements

“…This Lemma shows that the diamond and 2 distances satisfy the same inequality with respect to the infinity norm between the Choi states when d in = d out = d. Since the algorithm of [Sur+22] approximates first the Choi state in the infinity norm, we obtain the same upper bound for the diamond distance. For general dimensions, we obtain the following complexity: Theorem 3.3.…”

“…In this section, we propose an upper bound on the complexity of the quantum process tomography problem. We generalize the algorithm proposed by [Sur+22] which is ancilla-free.…”

“…Theorem 3.1. [Sur+22] There is an ancilla-free process tomography algorithm that learns a quantum channel (of d in = d out = d) in the distance d 2 using only a number of measurements:…”

“…This result is stated in Theorem 2.1. Next, we show that the process tomography algorithm of [Sur+22] can be generalized to approximate an unknown quantum channel to within ε in the diamond norm (1) using a number of incoherent measurements Õ(d 3 in d 3 out /ε 2 ) (Theorem 3.3). For this, we relate the diamond norm between two quantum channels and the operator norm between their corresponding Choi states which improves on the usual inequality with the 1-norm: M ⋄ ≤ d in J M 1 (see e.g.…”

“…for learning in the diamond norm. [Sur+22] propose an algorithm for estimating the Choi state in the 2 norm that requires only Õ(d 4 /ε 2 ) ancilla-free incoherent measurements (when d in = d out = d). This article generalizes this result to the diamond norm and general input/output dimensions and shows that this algorithm is optimal up to a logarithmic factor.…”