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International Journal of Control

Quantum gate generation by T-sampling stabilisation

Hector Bessa Silveira, Paulo Sérgio Pereira da Silva, Pierre Rouchon

Published Dec 12, 2013• 5 citations
Publisher

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 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).

DOI
10.1080/00207179.2013.873951

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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 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 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.
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.
A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let X¯ℓ−1(t) be the trajectory that is obtained in step ℓ−1, with X¯ℓ−1(t)=I. In step ℓ, X¯ℓ−1 is right-translated in order to displace X¯ℓ−1(Tf) to Xgoal. The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating X¯ℓ(T), and so on. A proof of the exponential convergence of X¯ℓ−1(Tf) to zero is provided for Tf large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time T∗. It presents excellent results for Tf=T∗. The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.
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.
In a previous work, the authors have introduced and treated the periodic motion planning problem for right-invariant driftless systems that evolve on SU(n), which in turn corresponds to an approximate motion planning control problem. Although the only assumption was the controllability of the system, the control laws are not explicitly known in general. By using analogous Lyapunov-like control techniques that were developed in that work, this paper complements its results and proves that if the system satisfies a more restrictive form of controllability, then the control laws are given by explicit and analytical expressions. Simulation results are presented for n = 4 and the generation of the C-NOT (Controlled-NOT) quantum logic gate.