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European Control Conference (ECC)

A stochastic Lyapunov feedback technique for propagator generation of quantum systems on U(n)

H. B. Silveira, Paulo Sérgio Pereira da Silva, Pierre Rouchon

Published Jul 1, 2013• 2 citations
Publisher

Abstract

This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.

DOI
10.23919/ecc.2013.6669234

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This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.
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).
This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs u = (u1, …, um). The T-sampling stabilisation problem is introduced and solved: given any initial condition X0 and any goal state , find a control law u = u(X, t) such that for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to . This is achieved by the tracking of T-periodic reference trajectories of the quantum system that pass by using the framework of Coron’s return method. The T-periodic reference trajectories are generated by applying controls that are a sum of a finite number M of harmonics of sin (2πt/T), whose amplitudes are parameterised by a vector a. The main result establishes that, for M big enough, X(jT) exponentially converges towards for almost all fixed a, 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 parameter vector of control amplitudes a = aj at each t = jT, and keeping it fixed for t ∈ [jT, (j + 1)T). It is shown in the paper that X(jT) exponentially converges towards almost surely. Simulation results have indicated that the convergence speed of X(jT) may be significantly improved with such stochastic technique. This is illustrated in the generation of the C–NOT quantum logic gate on U(4).
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, where 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‾: [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 which is a right-translation of W‾(t), at least when dist(X(t0),X‾(t0)) is finite. Furthermore, it is shown that 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, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory X‾(t), at least when dist(X(t0),X‾(t0))≤c. When 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.
In the design of complex quantum systems like ion traps for quantum computing, it is usually desired to stabilize a particular system state or make the system state track a desired trajectory. Several control theoretical approaches based on feedback seem attractive to solve such problems. But the uncertain dynamics introduced by measurement on quantum systems makes the synthesis of feedback control laws very complicated. Although we have not explicitly modeled the change in system dynamics due to measurement (we have assumed weak measurements), this is a first step towards a more detailed analysis and closed-loop feedback design. Here, we present a Lyapunov-based control approach on the lines of that developed by Mirrahimi, Rouchon, Turnici (2005). The states are assumed to be obtained from weak measurements. The Lyapunov control technique has not been applied to realistic quantum systems so far. We have extended and applied the technique to two realistic physical systems - the quantum harmonic oscillator and the n-qubit system. We also propose to extend this concept to ion traps.
In this paper, a Lyapunov-based feedback control (LFC) is proposed for the state transfer of a two-level stochastic quantum system. The quantum state is described in the Cartesian coordinate system. An exponential function is selected as the Lyapunov function to ensure the stability of the system. The Lyapunov-based feedback control strategy is designed according to the stochastic Lyapunov stability theorem. Numerical simulations results show that the proposed LFC strategy has the ability to transfer the stochastic system from any initial mixed state to the desired eigenstate, and the LFC has high convergence rate and high state transfer fidelity.