Quantum Estimation via Convex Optimization

Abstract

A number of problems in quantum estimation can be formulated as a convex optimization [1]. Applications include: maximum likelihood estimation, optimal experiment design, quantum state detection, and quantum metrology under instrumentation constraints. This talk will draw on the work I have been involved with, e.g., [2], [3], [4]. Our work in optimal quantum error correction [5, 6] is also relevant. Great benefit is derived using an error model which is specific to the system. Obtaining the errors from tomography is a logical route. How to do this, however, is an open question. The constraint is the form required by the standard error-correction model upon which the optimization is constructed. I will present some ideas on how to do the tomography in this context. • Maximum Likelihood (ML) quantum estimation problems are easily formed as log-convex optimization problems [1]. These include estimation of the state (density), estimation of the distribution of known input states, estimation of the OSR elements for quantum process tomography, and estimation of the coefficients of a preselected basis set of OSR elements. Estimation of Hamiltonian parameters, unfortunately, is not a convex optimization. Associated with these estimation problems, including Hamiltonian parameter estimation, is an optimal experiment design (OED), which is convex, and which can determine the system configurations to maximize the estimation accuracy [2]. Experiments have been performed In Ian Walmsley’s Group at Oxford using these methods [7, 8]. • Quantum state detection can be formulated as a convex optimization problem in the matrices of the POVM which characterize the measurement apparatus. Minimizing the error probability is a semidefinite program (SDP) [9]. Maximizing the posterior probability of detection is a quasiconvex optimization problem [3]. • Quantum metrology subject to instrumentation constraints can be cast as a convex optimization problem [4]. Focusing on the single parameter case, the optimization problem is a linear program (LP). The Fisher information from the LP solution for the constrained problem can be compared to what is possible with no constraints, the Quantum Fisher Information. This approach is easily extended to the multi-parameter case. • Quantum Error Correction (QEC) that is optimized with respect to the specific system at hand can reduce ancilla overhead while raising error thresholds for fault-tolerant operation [5, 6, 10, 11]. The problem is cast as a bi-convex optimization problem, iterating between encoding and recovery, each being an SDP. In [5] we introduced two new aspects of this approach: (i) we modified the objective functions to account for robustness, and (ii) posed the problem in an indirect form which can be solved via a sequence of constrained least-squares problems. This opens the way for solving extremely large problems in a reasonable time period both from offline models and online from measured data, i.e., tomography.