Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

Abstract

Quantum optimal control has numerous important applications ranging from pulses in magnetic resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.

Recovering models of open quantum systems from data via polynomial optimization: Towards globally convergent quantum system identification

“…discussed in detail the benefits of the polynomial optimization formulation and explained methods for obtaining global optima. Improved estimates of models of open quantum systems should, in turn, allow for improved pulse shaping for 2-qubit gates [35] and improved error cancellation [72].…”

“…Consider, for example, the uses of the estimates in quantum optimal control [33,34]. An imprecise estimate of the open quantum system may lead to poor quality control signals, which in turn lead to low-fidelity 2-qubit gates [35] or poor outcomes in the laser control [36] of chemical reactions. In contrast, the guarantees of global asymptotic convergence for the optimization over semi-algebraic sets make it possible to obtain the best possible estimates.…”

Exploiting term sparsity in Noncommutative Polynomial Optimization

“…d+1,j for all d, j hold for a given k ≥ 1. Then from ( 21), ( 29), (30) and by the induction hypothesis, we have…”

“…If the maximal chordal extension is used in (30), then the matrices in Π G Following from Theorem 3.6, we have the following two-level hierarchy of lower bounds for the optimum λ min (f, S) of (EQ 0 ): (33)…”

“…matrices, operators on a Hilbert space). The applications of NCPOPs include control theory and linear systems in engineering [37], quantum theory and quantum information science [14,32,30], matrix factorization ranks [15], machine learning [50,51] and so on, and new applications are emerging.…”