Physically motivated decompositions of single qutrit gates

TL;DR

This paper explores a physically motivated decomposition of single qutrit gates, providing a parameterization of U(3) matrices relevant for controlling superconducting qutrits, and discusses over-parameterization issues and optimal implementation paths.

Contribution

It introduces a decomposition of U(3) matrices based on exponential of diagonal and off-diagonal matrices, aiding experimental control of qutrits and analyzing parameterization redundancies.

Findings

Confirmed the decomposition can represent any U(3) element.

Identified over-parameterization issues with parameter ranges.

Determined the shortest implementation path for a qutrit gate.

Abstract

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.