A fixed point algorithm for improving fidelity of quantum gates^*

¹ Polytechnic School – PTC, University of São Paulo (USP), São Paulo, SP, Brazil.
² Centre Automatique et Systèmes, Mines ParisTech, Paris, France.
³ Departamento de Automação e Sistemas (DAS), Universidade Federal de Santa Catarina (UFSC), Florianópolis, SC, Brazil.

^** Corresponding author: paulo@lac.usp.br

Received: 11 November 2019
Accepted: 18 August 2020

Abstract

This work considers the problem of quantum gate generation for controllable quantum systems with drift. It is assumed that an approximate solution called seed is pre-computed by some known algorithm. This work presents a method, called Fixed-Point Algorithm (FPA) that is able to improve arbitrarily the fidelity of the given seed. When the infidelity of the seed is small enough and the approximate solution is attractive in the context of a tracking control problem (that is verified with probability one, in some sense), the Banach Fixed-Point Theorem allows to prove the exponential convergence of the FPA. Even when the FPA does not converge, several iterated applications of the FPA may produce the desired fidelity. The FPA produces only small corrections in the control pulses and preserves the original bandwidth of the seed. The computational effort of each step of the FPA corresponds to the one of the numerical integration of a stabilized closed loop system. A piecewise-constant and a smooth numerical implementations are developed. Several numerical experiments with a N-qubit system illustrates the effectiveness of the method in several different applications including the conversion of piecewise-constant control pulses into smooth ones and the reduction of their bandwidth.