A general formulation of time-optimal quantum control and optimality of singular protocols

Abstract

We present a general theoretical framework for finding the time-optimal unitary evolution of the quantum systems when the Hamiltonian is subject to arbitrary constraints. Quantum brachistochrone (QB) is such a framework based on the variational principle, whose drawback is that it only deals with equality constraints. While inequality constraints can be reduced to equality ones in some situations, they usually cannot, especially when a drift field, an uncontrollable part, is present in the Hamiltonian. We first develop a framework based on Pontryagin’s maximum principle (MP) in order to deal with inequality constraints as well. The new framework contains QB as a special case, and their detailed correspondence is given. Second, we address the problem of singular controls, which satisfy MP trivially so as to cause a trouble in determining the optimal protocol. To overcome this difficulty, we derive an additional necessary condition for a singular protocol to be optimal by applying the generalized Legendre–Clebsch condition. Third, we discuss general relations among the drift, the singular controls, and the inequality constraints. Finally, we demonstrate how our framework and results work in some examples. We also discuss the physical meaning of singular controls.

Time-optimal quantum transformations with bounded bandwidth

“…The problem discussed in this paper is an example of a brachistochrone problem [8,9,10,12,15,17]. For constrained closed quantum systems, such problems are typically reformulated as one or more time-local relations for the Hamiltonian using the calculus of variations [8,9,15,17]. However, such relations are generally of little help in determining the shortest possible transformation time.…”

“…We allow the Hamiltonian governing the transformation to be time-dependent, but we assume that its energy bandwidth is uniformly bounded. Such an assumption is in many cases physically justified [8,9,10,11,12,13,14,15,16,17].…”

Time-optimal quantum transformations with bounded bandwidth

“…The problem discussed in this paper is an example of a brachistochrone problem [8-10, 12, 15, 17]. For constrained closed quantum systems, such problems are typically reformulated as one or more time-local relations for the Hamiltonian using the calculus of variations [8,9,15,17]. However, such relations are generally of little help in determining the shortest possible transformation time.…”

From quantum speed limits to energy-efficient quantum gates

“…For a comprehensive review we refer to the literature [23][24][25]. In principle, studying the QSL has given rise to two fundamentally different questions [39], namely either to quantify the minimal time a quantum system needs to evolve between distinct states [40][41][42][43][44][45], or to bound the maximal rate with which quantum states can evolve [46][47][48][49]. While either version of the QSL can be exploited to assess the time cost and energy cost of a quantum computation [50][51][52], previous formulations are limited by the fact that the QSL has been considered as an inherent property of the evolving state -and not of the Hamiltonian generating the dynamics.…”