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arXiv:1804.01125 · Progress of Theoretical and Experimental Physics

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Mauro Bologna

Published Mar 28, 2019• 2 citations
arXivHTMLPDF

Abstract

This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine the two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate closed formulas that have several applications in solving problems ranging from linear differential equations and quantum mechanics to non-linear differential equations.

DOI
10.1093/ptep/ptz035

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This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine the two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate closed formulas that have several applications in solving problems ranging from linear differential equations and quantum mechanics to non-linear differential equations.
Abstract Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area.In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previouslypublished alternatives. They can also be used as basic methods in recursive procedures to increasethe order of approximation.We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.
Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.
We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.
We provide a recursive method for systematically constructing product formula approximations to exponentials of commutators, giving approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating many-body interaction Hamiltonians.