Finsler metrics in quantum circuit optimization

Abstract

AbstractA brief review is given of possible Finsler metrics de ned on the SU (2 n )group tangent bundle which may be suitable for quantum circuit optimiza-tion. 1 INTRODUCTION In the Riemannian geometry of quantum computation, a Riemannian hasbeen exploited for the purpose of quantum circuit optimization [1]-[4]. Finslermetrics are a more general class of metrics of which Riemannian metrics area special case based on a quadratic form [5]-[8]. A Finsler need not bequadratic. In fact Bernard Riemann was the rst to consider a basedon a quartic form [9], but he settled on a quadratic form because of its greatersimplicity. Much later, Paul Finsler introduced Finsler geometry in his thesis[10]. The purpose of the present work is to brie‡y review possible Finsler metricsfor quantum circuit optimization [11],[1] . 2 QUANTUM CIRCUIT OPTIMIZATION For a manifold M with a so-called local metric [11], there is a function F (x;y )of coordinates x on M and y on the tangent space manifold TM . One requiresF (x;y ) = 0 for xed x, and F (x;y ) is vanishing for y = 0. F (x;y ) is positivelyhomogeneous of degree 1 in y , namely,F (x;y ) = F (x;y );  > 0; (1)and satis es the triangle inequality in the tangent space variable y , namely,F (x;y