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arXiv:1104.1148

A method for exponential operator decomposition

Seçkin Şefi, Peter van Loock

Published Apr 6, 2011• 1 citation
arXivHTMLPDF

Abstract

Exponential operator decompositions are an important tool in many fields of physics, for example, in quantum control, quantum computation, or condensed matter physics. In this work, we present a method for obtaining such decompositions, which is efficient in terms of the required number of operators. Compared to existing schemes, our more direct approach is general, in the sense that it can be applied to various kinds of operators including nested commutation operators, and it is systematic.

DOI
10.48550/arxiv.1104.1148

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Exponential operator decompositions are an important tool in many fields of physics, for example, in quantum control, quantum computation, or condensed matter physics. In this work, we present a method for obtaining such decompositions, which is efficient in terms of the required number of operators. Compared to existing schemes, our more direct approach is general, in the sense that it can be applied to various kinds of operators including nested commutation operators, and it is systematic.
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 diagrammatic method is one of the most powerful tools of theoretical physics. It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory. The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams ( graphs ). Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs . There exist several kinds of diagrams. We will try to present them in a systematic way. In Sect. 20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure. This formalism is used extensively in classical statistical physics. It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables. We use the term “Gaussian integration” in a rather broad sense. Beside commuting “bosonic” variables, we also consider anti-commuting “fermionic” variables, where we use the Berezin integral with respect to a Gaussian weight. Even in the case of commuting variables, the “Gaussian integral” is not necessarily meant in the sense of measure theory. It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight. However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.
Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.