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arXiv:quant-ph/0010100

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

Navin Khaneja, Steffen J. Glaser

Published Oct 29, 2000• 31 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.quant-ph/0010100

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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.
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The control of spin chains represents a very interesting problem from the point of view of quantum computation. The problem consists in defining a procedure to obtain any possible evolution operator of the spin chain by means of in external magnetic field. The set of possible evolution operators of the system corresponds to the unitary group SU(2/sup N/) (where N is the number of atoms in the chain) and the interactions involved can he set to correspond to elements in the corresponding Lie algebra. As a consequence, the whole problem can be formulated in Lie algebraic terms and the design issues be reduced to a suitable decomposition of the group elements. The goal of this paper is to introduce a Cartan decomposition for the unitary group based on a minimal switching decomposition of the special orthogonal group. We analyze its implications from the point of view of time optimality in the construction of a program as a sequence of quantum gates.
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This chapter discusses in detail the case of two-level systems and the associated Lie group. It gives an example of a general decomposition of SU(n), namely the decomposition into planar rotation, which is quite simple and can be obtained by straightforward computations. This decomposition gives a direct way to parametrize the special unitary group. The chapter presents the basics of Cartan decompositions of semisimple Lie groups. It gives some examples of applications of Lie group decompositions and introduces some notions of Lie group and Lie algebra theory. The chapter discusses Cartan decompositions of Lie groups. Cartan decompositions appear to be ubiquitous in quantum control and dynamics and in quantum information theory.