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Quantum gate generation for systems with drift in U(n) using Lyapunov-LaSalle techniques

Authors: H.B. Silveira, P.S.Pereira da Silva, P. Rouchon, International Journal of Control, International Journal of Control, Vol 89 No 12, pp. 2466-2481, Apr 8 2016 DOI: 10.1080/00207179.2016.1161830
This paper considers right-invariant and controllable quantum systems with m inputs u = (u1,… , um) and state X(t) evolving on the unitary Lie group U(n). The ε-steering problem is introduced and solved for systems with drift: given any initial condition X0 at the initial time instant t0 ≥ 0, any goal state Xgoal U(n) and ε > 0, find a control law such that dist(X(t0+t‾),Xgoal)≤ϵ dist(X(t0+t-),Xgoal)≤ϵ, where t‾>0 t‾>0 is big enough and dist(X1, X2) is a convenient right-invariant notion of distance between two elements X1, X2 U(n). The purpose is to approximately generate arbitrary quantum gates corresponding to Xgoal. This is achieved by solving a tracking problem for a special kind of reference trajectories W‾‾
W‾: [t0, ∞) U(n), which are here called c-universal reference trajectories. It is shown that, for this special kind of trajectories, the tracking problem can be solved up to an error ε for any reference trajectory X‾‾(t)=W‾‾(t)R X‾(t)=W‾(t)R
which is a right-translation of W‾‾(t) W‾(t), at least when dist(X(t0),X‾‾(t0)) dist(X(t0),X‾(t0))
is finite. Furthermore, it is shown that dist(X(t),X‾‾(t)) dist(X(t),X‾(t))
converges uniformly exponentially to zero in the sense that the rate of convergence is independent of t0, R and X0. The approach considered here for showing such convergence is a generalisation of the results of a previous paper of the authors, which is mainly based on the central ideas of Jurdjevic and Quinn and Coron's return method. Taking a right-translation R such that X‾‾(t0+t‾)=Xgoal
X‾(t0+t‾)=Xgoal, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory X‾‾(t)X‾(t), at least when dist(X(t0),X‾‾(t0))≤c dist(X(t0),X‾(t0))≤c. When dist(X(t0),X‾‾(t0))≥c
dist(X(t0),X‾(t0))≥c, it is shown that the ε-steering problem can be globally solved in a two-iteration procedure. The underlying algorithmic complexity to get the steering control is essentially equivalent to the numerical integration of the Cauchy problem governing X(t). A numerical example considering a Toffoli quantum gate on U(8) for a chain of three coupled qubits that are controlled only locally is presented.
author = {P.S.Pereira da Silva H.B. Silveira, P. Rouchon},
title = {Quantum gate generation for systems with drift in U(n) using Lyapunov-LaSalle techniques},
journal = {International Journal of Control, International Journal of Control},
volume = {89},
number = {12},
pages = {2466-2481},
year = {2016},