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arXiv:1607.04054

Pulse Width Modulation for Simulating Time-Dependent Quantum Systems

Qiming Chen, Re-Bing Wu

Published Jul 14, 2016• 2 citations
arXivHTMLPDF

Abstract

This paper proposes a numerical method for solving time-dependent Schrodinger equations with finite spectral bandwidth, which applies to both periodic and non-periodic cases. We introduce the concept of Pulse Width Modulation (PWM), which is broadly used in classical control engineering, to simulate time-dependent quantum dynamics by switching between a finite number of constant Hamiltonians. The switching timings can be programmed to improve the precision and save computational resources. The effectiveness of the PWM method is demonstrated by numerical simulations, which shows that it is faster and more robust than the standard piecewise constant control systems. Moreover, we also propose the realization of PWM method for potential implementation in experimental systems.

DOI
10.48550/arxiv.1607.04054

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This paper proposes a numerical method for solving time-dependent Schrodinger equations with finite spectral bandwidth, which applies to both periodic and non-periodic cases. We introduce the concept of Pulse Width Modulation (PWM), which is broadly used in classical control engineering, to simulate time-dependent quantum dynamics by switching between a finite number of constant Hamiltonians. The switching timings can be programmed to improve the precision and save computational resources. The effectiveness of the PWM method is demonstrated by numerical simulations, which shows that it is faster and more robust than the standard piecewise constant control systems. Moreover, we also propose the realization of PWM method for potential implementation in experimental systems.
Illuminated by the Pulse Width Modulation (PWM) technology in classical control engineering, we propose the PWM approximation which transforms continuous and bang-bang control into each other. This method works by squeezing the approximation error into high-frequency components above a prescribed cutoff frequency ΩΩΩ, which can be tailored by modulating the switch time of the bang-bang control. Because the quantum dynamics under bang-bang control can be efficiently calculated, PWM allows us to speed up quantum control design particularly in large dimensional systems with small number of control variables. Moreover, PWM paves the way to implement various bang-bang control protocols in the literature with an arbitrary waveform, which is more desirable in laboratory.
This paper focuses on accelerating quantum optimal control design for complex quantum systems. Based on our previous work [{arXiv:1607.04054}], we combine Pulse Width Modulation (PWM) and gradient descent algorithm into solving quantum optimal control problems, which shows distinct improvement of computational efficiency in various cases. To further apply this algorithm to potential experiments, we also propose the smooth realization of the optimized control solution, e.g. using Gaussian pulse train to replace rectangular pulses. Based on the experimental data of the D-Norleucine molecule, we numerically find optimal control functions in 333-qubit and 666-qubit systems, and demonstrate its efficiency advantage compared with basic GRAPE algorithm.
In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.
We present a general framework for promoting first-order pulse sequences in quantum simulation to higher-order sequences that maintain robustness in the presence of finite pulse-width effects. Our approach maps a given first-order pulse sequence to a first-order Trotter formula, applies higher-order Trotter-formula constructions, and then compiles the resulting evolution back into physically implementable finite-width pulses via dynamically corrected gates. The resulting sequences achieve arbitrarily high-order error scaling with respect to the control cycle time of the underlying first-order sequence while maintaining robustness to finite pulse-width effects. The framework also enables the use of multi-product formulas for more efficient constructions. We apply the framework to several physically motivated quantum-simulation tasks and numerically verify the predicted error scalings.
We describe a computational method for simulating time-dependent quantum mechanical systems interacting with external fields. In this method, the Schrödinger equation is solved by expanding the wave function in the basis set of the unperturbed Hamiltonian. The expansion yields a set of coupled ordinary differential equations for the expansion coefficients. This coupled channel method can be applied to many time-dependent problems. We apply the coupled channel method to a particle in a box interacting with a laser pulse. We show that if only two states are involved, the method leads naturally to the solutions for Rabi flopping, and that the system exhibits Rabi flopping behavior even with realistic, nonmonochromatic laser pulses. We also discuss instances where simulations could improve the understanding or address misconceptions of beginning students in quantum mechanics.
A fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time-dependent Schrodinger equation. This method is a real-space-real-time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method, and an operator named adhesive operator. This method conserves the norm of the wave function, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.