Quantum state tomography as a numerical optimization problem

Abstract

We present a framework that formulates the quest for the most efficient quantum state tomography measurement set as an optimization problem which can be solved numerically, where the optimization goal is the maximization of the information gain. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit-qutrit system, which could be realized e.g. by the N-14 nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e., projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for quantum state tomography, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.

Optimal Quantum State Tomography with Noisy Gates

“…One approach to look for an optimal solution is to use parallel searches with many well chosen starting points in order to explore well the space of potential optimal solutions. This approach has been used successfully for similar problems in [20,21]. We use Powell's method as a local search with a set of 500 diverse starting points, each at least 0.01 distance from each other using a Jaccard-based distance measure.…”

“…The problem of finding the optimal quorum of projection operators under noise is a non-convex continuous optimization problem with the derivatives of the function not easily obtained, and with multiple local maxima. From our previous work [20,21], we know that using a local optimizer (Powell's method) with well-chosen starting points performs very well. Here, we use the Powell's derivative free method started in parallel with multiple sufficiently diverse starting points in order to improve the exploration of the search space.…”

“…For a situation where one out of N qubits is measured, an optimal quorum [19] consists of projectors on socalled mutually unbiased subspaces. Numerically optimized QST measurement sets consisting of independent rank-1 projection operators [20] and projectors on half-dimensional subspaces in dimension six [21] have been obtained. In the first case, the numerical solution outperforms a set that constitutes projectors from a set of MUBs and in the latter case, the solution approximates mutually unbiased subspaces.…”