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arXiv:physics/0312005

Positive Decompositions of Exponential Operators

Siu A. Chin

Published Dec 1, 2003• 0 citations
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Abstract

The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schr\\"odinger equation, it is immaterial whether the decomposition coefficients are positive or negative. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions which respect the time-irreversibility of the diffusion kernel can yield practical algorithms. These positive time step, or forward decompositions, are a highly restrictive class of factorization algorithms. This work proves a fundamental theorem: in order for a $2n$th order forward algorithm to become $(2n+2)$th order, one must include a new, higher order commutator in the decomposition process. Since these higher order commuatators are highly complex, it seems difficult to produce practical forward algorithms beyond fourth order. This proof generalize the Sheng-Suzuki theorem for the case of $n=1$. In particular, this work shows that it is not possible to have a sixth order forward algorithm by including just the $[V,[T,V]]$ commutator.

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The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schr\"odinger equation, it is immaterial whether the decomposition coefficients are positive or negative. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions which respect the time-irreversibility of the diffusion kernel can yield practical algorithms. These positive time step, or forward decompositions, are a highly restrictive class of factorization algorithms. This work proves a fundamental theorem: in order for a 2n2n2nth order forward algorithm to become (2n+2)(2n+2)(2n+2)th order, one must include a new, higher order commutator in the decomposition process. Since these higher order commuatators are highly complex, it seems difficult to produce practical forward algorithms beyond fourth order. This proof generalize the Sheng-Suzuki theorem for the case of n=1n=1n=1. In particular, this work shows that it is not possible to have a sixth order forward algorithm by including just the [V,[T,V]][V,[T,V]][V,[T,V]] commutator.
We propose a high order numerical decomposition of exponentials of Hermitian operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M Suzuki, implementing it for complex coefficients. We outline a convenient fourth-order formula which can be written compactly for an arbitrary number of non-commuting terms in the Hamiltonian and which is superior to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for a sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.
We propose a high order numerical decomposition of exponentials of hermitean operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M. Suzuki, however implementing it for complex coefficients. We outline a convenient fourth order formula which can be written compactly for arbitrary number of noncommuting terms in the Hamiltonian and which is superiour to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.
Abstract Suzuki–Trotter decompositions of exponential operators like exp ( H t ) are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators H = ∑ k A k , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators A 1 , 2 can be applied to such generic Suzuki–Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order n ⩽ 4 is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order n ⩽ 8 . Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of 10 −4 . Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.
The evolution operator of a quantum system in a time-dependent potential is factorized in unitary exponential operators at order 6. This expression is derived with the time-ordering method. It is compared with lower-order factorizations on several simple one-dimensional examples. Better accuracies are reached at sixth order for a given time step than at lower orders. Due to a significant increase of computation duration per time step, the sixth-order approximation is mainly useful when high accuracies are required.
The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher-order splitting will result in a higher-order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Green's function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically gives rise to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second-order forward propagators can have fourth-order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second-order propagators and their minimal correctors.