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The Brachistochrone Problem: From Euler to Quantum

Carlos Hermans

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

In this thesis I take a look at various aspects of the brachistochrone problem. In the first half I consider the classical brachistochrone problem, without friction, after which I generalize to also consider drag; friction proportional to the squared speed. I do this numerically, with the shooting method, and following some papers, which Euler wrote. Then I will look at how Euler's results compare to modern results and discuss how they are different. In the second half I consider the quantum brachistochrone problem, following a paper by A. Carlini, and afterwards I look at a related problem, where there is not enough time to change an electron spin between two predetermined states. This problem has been numerically solved in an example.

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Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpoint due to the boundary effect. We present the general solution to the full quantum brachistochrone equation and discuss its main features. Using it, we prove that the speed of evolution under constraints is reduced with respect to the unrestricted case. In addition, we find that solving the quantum brachistochrone equation is closely connected to solving the dynamics of the Lagrange multipliers, which is in general governed by nonlinear differential equations. Their numerical integration allows generating time-extremal trajectories. Furthermore, when the restricted operators form a closed subalgebra, the Lagrange multipliers become constant and the optimal Hamiltonian takes a concise form. The new class of analytically solvable models for the quantum brachistochrone problem opens up the possibility of applying it to many-body quantum systems, exploring notions related to geometry such as quantum speed limits, and advancing significantly the quantum state and gate preparation for quantum information processing.

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Andrzej Frydryszak, V. M. Tkachuk

We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.

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In this thesis I take a look at various aspects of the brachistochrone problem. In the first half I consider the classical brachistochrone problem, without friction, after which I generalize to also consider drag; friction proportional to the squared speed. I do this numerically, with the shooting method, and following some papers, which Euler wrote. Then I will look at how Euler's results compare to modern results and discuss how they are different. In the second half I consider the quantum brachistochrone problem, following a paper by A. Carlini, and afterwards I look at a related problem, where there is not enough time to change an electron spin between two predetermined states. This problem has been numerically solved in an example.
Minimum-time quantum control protocols can be obtained from the quantum brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev. Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly applied the variational calculus with fixed boundary conditions. We argue that the genuine quantum brachistochrone problem involves a variational problem with a movable endpoint, contrary to the classical brachistochrone problem. This formulation not only simplifies the derivation of the quantum brachistochrone equation but introduces an additional constraint at the endpoint due to the boundary effect. We present the general solution to the full quantum brachistochrone equation and discuss its main features. Using it, we prove that the speed of evolution under constraints is reduced with respect to the unrestricted case. In addition, we find that solving the quantum brachistochrone equation is closely connected to solving the dynamics of the Lagrange multipliers, which is in general governed by nonlinear differential equations. Their numerical integration allows generating time-extremal trajectories. Furthermore, when the restricted operators form a closed subalgebra, the Lagrange multipliers become constant and the optimal Hamiltonian takes a concise form. The new class of analytically solvable models for the quantum brachistochrone problem opens up the possibility of applying it to many-body quantum systems, exploring notions related to geometry such as quantum speed limits, and advancing significantly the quantum state and gate preparation for quantum information processing.
The quantum brachistochrone problem -- to find the time-optimal transition between given initial and final quantum states -- is investigated in this bachelor's thesis. First the quantum equivalent of distance (the Fubini-Study metric) is formulated, using geometry of spheres. Together with constraints, it is used to create the quantum action. Functional derivatives are then taken of said action to find equations of motion (eoms). For the unconstrained case (excluding finite energy), these eoms are solved in closed form, whilst general to be solved formulae are obtained for the constrained case. The latter is used to explicitly solve an example quantum brachistochrone problem: a spin-1/2 particle in a controllable magnetic field constrained to an x-y plane. Finally, the link with quantum computing is illustrated through the time-optimization of unitary transformations that the quantum brachistochrone yields us.
Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for a wide range of quantum control problems. So far, this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators. We expect this brachistochrone-geodesic connection to have broad applications, as it opens up minimal-time control to the tools of geometry. As one such application, we use it to obtain a fast numerical method to solve the brachistochrone problem, and apply this method to two examples demonstrating its power.
The quantum brachistochrone problem addresses the fundamental challenge of achieving the quantum speed limit in applications aiming to realize a given unitary operation in a quantum system. Specifically, it looks into optimization of the transformation of quantum states through controlled Hamiltonians, which form a small subset in the space of the system's observables. Here we introduce a broad family of completely integrable brachistochrone protocols, which arise from a judicious choice of the control Hamiltonian subset. Furthermore, we demonstrate how the inherent stability of the completely integrable protocols makes them numerically tractable and therefore practicable as opposed to their nonintegrable counterparts.
We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.