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Efficient geometric integrators for nonadiabatic quantum dynamics in the adiabatic basis

Seonghoon Choi, Jiří J. L. Vaníček

Published Feb 14, 2019• 0 citations
OpenAlex

Abstract

The split-operator algorithm combined with the dynamic Fourier method is a popular approach for quantum dynamics in systems with separable kinetic and potential energies because it is explicit, easy to implement, and preserves many geometric invariants of the exact evolution. Although the standard version has only second-order accuracy, it can be recursively symmetrically composed to obtain geometric integrators of arbitrary even order of accuracy in the time step. We have implemented such integrators for nonadiabatic quantum dynamics in the diabatic basis using the triple-jump, Suzuki-fractal, and several non-recursive optimal compositions of the second-order split-operator algorithm. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The composed integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable regardless of the accuracy of the solution. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a two-surface NaI model in the diabatic representation. Efficiency of higher-order algorithms is sometimes questioned because the number of splitting steps grows exponentially with the order. Yet, we show that, if higher accuracy is desired, higher-order integrators become the most efficient; a 600-fold speedup compared to the second-order split-operator method and a 17000-fold speedup compared to the trapezoidal rule (the Crank-Nicolson method) was observed for convergence error of 10$^{-10}$.

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Efficient geometric integrators for nonadiabatic quantum dynamics. I. The adiabatic representation

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Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation.

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The split-operator algorithm combined with the dynamic Fourier method is a popular approach for quantum dynamics in systems with separable kinetic and potential energies because it is explicit, easy to implement, and preserves many geometric invariants of the exact evolution. Although the standard version has only second-order accuracy, it can be recursively symmetrically composed to obtain geometric integrators of arbitrary even order of accuracy in the time step. We have implemented such integrators for nonadiabatic quantum dynamics in the diabatic basis using the triple-jump, Suzuki-fractal, and several non-recursive optimal compositions of the second-order split-operator algorithm. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The composed integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable regardless of the accuracy of the solution. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a two-surface NaI model in the diabatic representation. Efficiency of higher-order algorithms is sometimes questioned because the number of splitting steps grows exponentially with the order. Yet, we show that, if higher accuracy is desired, higher-order integrators become the most efficient; a 600-fold speedup compared to the second-order split-operator method and a 17000-fold speedup compared to the trapezoidal rule (the Crank-Nicolson method) was observed for convergence error of 10−10^{-10}−10.
Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation.
One of the most accurate methods for solving the time-dependent Schrödinger equation uses a combination of the dynamic Fourier method with the split-operator algorithm on a tensor-product grid. To reduce the number of required grid points, we let the grid move together with the wavepacket but find that the naïve algorithm based on an alternate evolution of the wavefunction and grid destroys the time reversibility of the exact evolution. Yet, we show that the time reversibility is recovered if the wavefunction and grid are evolved simultaneously during each kinetic or potential step; this is achieved by using the Ehrenfest theorem together with the splitting method. The proposed algorithm is conditionally stable, symmetric, and time-reversible and conserves the norm of the wavefunction. The preservation of these geometric properties is shown analytically and demonstrated numerically on a three-dimensional harmonic model and collinear model of He–H2 scattering. We also show that the proposed algorithm can be symmetrically composed to obtain time-reversible integrators of an arbitrary even order. We observed 10 000-fold speedup by using the tenth-order instead of the second-order method to obtain a solution with a time discretization error below 10−9. Moreover, using the adaptive grid instead of the fixed grid resulted in a 64-fold reduction in the required number of grid points in the harmonic system and made it possible to simulate the He–H2 scattering for six times longer while maintaining reasonable accuracy. Applicability of the algorithm to high-dimensional quantum dynamics is demonstrated using the strongly anharmonic eight-dimensional Hénon–Heiles model.
Dissipation and irreversibility are central to many physical systems, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating unitary dynamics, yet conventional product formulas introduce negative time steps at high orders that are ill-conditioned for dissipative dynamics. We show how block encodings of complex-coefficient product formulas can be constructed by a sequence of simple Hamiltonian evolutions in real and imaginary time with high-order accuracy. The unitary substages use positive real coefficients, while the dissipative substages use complex coefficients with positive real parts; the real parts preserve the contractive evolution, and the imaginary parts are additional unitary evolutions. We use known formulas of orders 4 and 6 and give a new symmetric formula of order 8 in this restricted class. We apply the approach to the classical problem of lossy mechanical wave propagation in statevector simulations and on a trapped-ion quantum processor. Orders 4 and 6 require comparable CNOT gates to orders 1 and 2 at large error tolerances, but quickly become more efficient as the tolerance is reduced. On the quantum hardware, a single step of order 4 achieves lower error than order 1 and comparable error to order 2, while order 6 is noise-limited. Our results show that operator splitting can combine practical circuit constructions with high-order convergence for non-unitary dissipative dynamics.
Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time-step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time-step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence for a superpolynomial convergence of first-order Trotter with uniform time-step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: It can match the Grover lower bound, it does not require knowledge of the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.
Chau et al. [New J. Phys. 20, 073003 (2018)] presented a new and straight-forward derivation of a fourth-order approximation 'U7U_7U7​' of the time-evolution operator and hinted at its potential value as a symplectic integrator. U7U_7U7​ is based on the Suzuki-Trotter split-operator method and leads to an algorithm for numerical time propagation that is superior to established methods. We benchmark the performance of U7U_7U7​ and other algorithms, including a Runge-Kutta method and another recently developed Suzuki-Trotter-based scheme, that are exact up to fourth order in the evolution parameter, against various classical and quantum systems. We find U7U_7U7​ to deliver any given target accuracy with the lowest computational cost, across all systems and algorithms tested here. This study is accompanied by open-source numerical software that we hope will prove valuable in the classroom.