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Physical Review E

Concatenated toolkit for quantum optimal control wave-function propagation

Michael Hsieh, HERSCHEL A. RABITZ

Published Mar 17, 2008• 7 citations
Publisher

Abstract

Numerical propagation of the Schrödinger equation is the bottleneck in many quantum optimal control computations. For a quantum system of N states with an electric-field-dipole interaction, the use of a propagation toolkit introduced in a prior work yields an O(N) reduction in floating-point operations per wave function propagation. A concatenation scheme for the toolkit method is introduced, and a scaling analysis shows a significant additional reduction in computational cost. The method exploits the fact that the same sequences of discretized control field values are often repeated many times in a control simulation. The concatenated toolkit is benchmarked against the standard toolkit in a numerical simulation.

DOI
10.1103/physreve.77.037701

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Numerical propagation of the Schrödinger equation is the bottleneck in many quantum optimal control computations. For a quantum system of N states with an electric-field-dipole interaction, the use of a propagation toolkit introduced in a prior work yields an O(N) reduction in floating-point operations per wave function propagation. A concatenation scheme for the toolkit method is introduced, and a scaling analysis shows a significant additional reduction in computational cost. The method exploits the fact that the same sequences of discretized control field values are often repeated many times in a control simulation. The concatenated toolkit is benchmarked against the standard toolkit in a numerical simulation.
A toolkit of time-propagation operators, to be stored and recalled as needed, is incorporated into the algorithms for the optimal control of quantum systems. Typically, the control field ε(t) revisits the same values many times during the full time evolution. This repetition may be utilized to enhance efficiency through a convenient toolkit of propagators where the propagators are computed and stored only at a small number of discrete electric-field values in the dynamic range εmin⩽ε(t)⩽εmax. At each time step of the controlled evolution a specific member of the pre-calculated toolkit is selected as dictated by the local control field value. The toolkit can reduce the cost of control field design by a factor scaling as ∼N for quantum systems described in a basis set of N states. Optimal control with the toolkit is demonstrated for systems up to dimension N=30.
Recently the 'toolkit' discretization introduced to accelerate the numerical resolution of the time-dependent Schrödinger equation arising in quantum optimal control problems demonstrated good results on a large range of models. However, when coupling this class of methods with the so-called monotonically convergent algorithms, numerical instabilities affect the convergence of the discretized scheme. We present an adaptation of the 'toolkit' method which preserves the monotonicity of the procedure. The theoretical properties of the new algorithm are illustrated by numerical simulations.
Abstract Quantum optimal control theory is becoming increasingly crucial as quantum devices become more precise, but the need to quickly optimize these systems classically remains a significant bottleneck in their operation. Here we present a new theoretical quantum control framework for much faster optimization than the state of the art by replacing standard time propagation with a product of short-time propagators, each calculated using the Magnus expansion. The derived formulas for exact series terms and their gradients, based on earlier approximate integrals in a simulation setting, allow us to subsume the high cost of calculating commutators and integrals as an initial overhead. This provides an order of magnitude speedup for quantum control optimization.
Optimal control theory provides a general means for designing controls to manipulate quantum phenomena. Traditional implementation requires solving coupled nonlinear equations to obtain the optimal control solution, whereas this work introduces a combinatorial quantum control (CQC) algorithm to avoid this complexity. The CQC technique uses a predetermined toolkit of small time step propagators in conjunction with combinatorial optimization to identify a proper sequence for the toolkit members. Results indicate that the CQC technique exhibits invariance of search effort to the number of system states and very favorable scaling upon comparison to a standard gradient algorithm, taking into consideration that CQC is easily parallelizable.
In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time TTT, where the system is governed by a time-dependent Schrödinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schrödinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.
Accurate derivatives are important for locally traversing and converging efficiently in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that the computational feasibility of meeting this accuracy requirement depends on the choice of propagation scheme and problem representation. Even when exact propagation is sufficiently cheap it is, perhaps surprisingly, much more efficient to optimize the (appropriately) approximate propagators: Approximations in the dynamics are traded off for significant complexity reductions in the exact derivative calculations. Importantly, past the initial analytical considerations, only standard numerical techniques are explicitly required with straightforward application to realistic systems. These results are numerically verified for two concrete problems of increasing Hilbert space dimensionality. The best schemes obtain unit fidelity to machine precision, whereas the results for other schemes are separated consistently by orders of magnitude in computation time and in worst-case ten orders of magnitude in achievable fidelity. Since these gaps continually increase with system size and complexity, this methodology allows numerically efficient optimization of very high-dimensional dynamics, e.g., in many-body contexts, operating in the high-fidelity regime.