Quantum Gate Generation by T-Sampling Stabilization

TL;DR

This paper introduces a T-sampling stabilization method for quantum systems on U(n), enabling the generation of arbitrary quantum gates through explicit feedback laws and a stochastic approach that improves convergence speed.

Contribution

It develops a novel T-sampling stabilization framework for quantum control using Coron's Return Method and introduces a stochastic control law that enhances convergence speed.

Findings

Exponential convergence of quantum states to target gates with explicit feedback laws.

Stochastic amplitude selection improves convergence speed in quantum gate generation.

Successful simulation of C--NOT gate on U(4) demonstrating the method's effectiveness.

Abstract

This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs. The T-sampling stabilization problem is introduced and solved: given any initial condition X(0) any goal state Xg find a feedback law such that X(jT) converges to Xg as the integer j tends to infinity. The purpose is to generate arbitrary quantum gates corresponding to Xg. This is achieved by the tracking of T-periodic reference trajectories that pass by Xg using the framework of Coron's Return Method. The T-periodic reference trajectories are generated by applying open-loop controls that are a sum of a finite number M of odd T-periodic function whose amplitudes are parameterized by a vector. The main result establishes that, for M big enough, X(jT) exponentially converges towards Xg for almost all fixed amplitude vectors, with explicit and completely constructive control laws. This paper also establishes a stochastic version of this deterministic control law. The key idea is to randomly choose a different amplitude vector at each sampling time jT, and keeping it fixed until the next sampling time (j+1)T. It is also shown that X(jT) exponentially converges towards Xg almost surely. Simulation results have indicated that the convergence speed may be significantly improved with such stochastic technique. This is illustrated in the generation of the C--NOT quantum logic gate on U(4).