Cartan Decomposition of SU(2^n), Constructive Controllability of Spin systems and Universal Quantum Computing

TL;DR

This paper presents a method for explicitly parameterizing any unitary operation on n qubits using Cartan decompositions of SU(2^n), facilitating the construction of quantum gates for universal quantum computing.

Contribution

It introduces a constructive approach based on Cartan decompositions to systematically design quantum gate sequences for n-qubit systems, emphasizing geometric insights.

Findings

Explicit parameterization of unitaries on n qubits

Pulse sequences for implementing arbitrary unitaries

Discussion on optimality of the control design

Abstract

In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.