Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Time-optimal quantum control via differential geometry

Xiaoting Wang, Michele Allegra, Kurt A. Jacobs, Seth Lloyd, Cosmo Lupo, Masoud Mohseni

Published Feb 20, 2017• 4 citations
Publisher

Abstract

Compared with many other methods which only give time sub-optimal designs, the quantum brachistochrone equation has a great potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far it has been of limited use, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. This identification allows us to design a numerical method that can efficiently solve the brachistochrone problem by first solving a family of geodesic equations.

DOI
10.1117/12.2256267

Related papers

Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems

Xiaoting Wang, Michele Allegra, Kurt A. Jacobs +3 more

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.

arXiv:1408.2465•Physical Review Letters•Apr 28, 2015•111 citations

Minimum-Time Quantum Control and the Quantum Brachistochrone Equation

Jing Yang, Adolfo del Campo

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.

arXiv:2204.12792•Apr 27, 2022•2 citations

Algorithm for solving unconstrained unitary quantum brachistochrone problems

Francesco Campaioli, William Sloan, Kavan Kishore Modi +1 more

The quantum brachistochrone problem (QBP) aims to find the optimal map steering a quantum system from a given initial state to a given final state in the minimum time, related to the quantum speed limit. An iterative method is introduced to solve the unconstrained, unitary QBP for mixed states. It could be interesting for quantum control applications.

arXiv:1906.02934•Physical Review A•Dec 20, 2019•21 citations

最速量子操作

陽介 奥平, Yosuke Okudaira

We present a general framework for finding the time-optimal quantum evolution inspired by the classical problem called the brachistochrone problem.At first, the time-optimal control problem for achieving a pure state evolution from a given initial state to a desired final state is formulated.We minimize the duration time for this transition by finding the optimal control for the Hamiltonian in an available set subject to a certain constraint dictated by physical conditions.Then, it will be extended to the unitary evolution which has greater relevance for quantum computation.This provides a systematic way to make an efficient unitary gate on a certain physical system.To demonstrate this framework, we choose the anisotropic Heisenberg model and find a general solution.Since it is not always possible to find a general solution for the fundamental equation for the time-optimal control, we introduce the simulated annealing method as a simple numerical method to show that our framework can be used for the system whose general solution cannot be found.The numerical analysis based on the Krotov iteration method is introduced and applied to the time-optimal quantum control problem to solve the problem efficiently.The optimal realization of the quantum Fourier transform for two or more qubit systems are derived.We consider two generalizations of the quantum brachistochrone for mixed state evolutions.One is the unitary evolution with ancilla system ended up with the final measurement on the ancilla.The other is the repetition of measurement or decoherence model which can be represented by the evolution with the master equation in the Lindblad form.Also, the relation between the fidelity to the target and the duration time is considerd.A.4 Quantum speed limit . . . . . . . . .

•Mar 1, 2008•0 citations

Quantum brachistochrone for multiple qubits

Ding Wang, Haowei Shi, Yueheng Lan

Abstract Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating possible redundant complexity involved in previous algorithms. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the ‘ground state’ solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.

New Journal of Physics•Aug 1, 2021•6 citations

Accelerating the computation of quantum brachistochrone

Ding Wang, Haowei Shi, Yueheng Lan

Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating redundant complexity involved in previous formulations. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the 'ground state' solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.

arXiv:2011.12629•Nov 25, 2020•0 citations
Compared with many other methods which only give time sub-optimal designs, the quantum brachistochrone equation has a great potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far it has been of limited use, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. This identification allows us to design a numerical method that can efficiently solve the brachistochrone problem by first solving a family of geodesic equations.
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.
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.
We present a general framework for finding the time-optimal quantum evolution inspired by the classical problem called the brachistochrone problem.At first, the time-optimal control problem for achieving a pure state evolution from a given initial state to a desired final state is formulated.We minimize the duration time for this transition by finding the optimal control for the Hamiltonian in an available set subject to a certain constraint dictated by physical conditions.Then, it will be extended to the unitary evolution which has greater relevance for quantum computation.This provides a systematic way to make an efficient unitary gate on a certain physical system.To demonstrate this framework, we choose the anisotropic Heisenberg model and find a general solution.Since it is not always possible to find a general solution for the fundamental equation for the time-optimal control, we introduce the simulated annealing method as a simple numerical method to show that our framework can be used for the system whose general solution cannot be found.The numerical analysis based on the Krotov iteration method is introduced and applied to the time-optimal quantum control problem to solve the problem efficiently.The optimal realization of the quantum Fourier transform for two or more qubit systems are derived.We consider two generalizations of the quantum brachistochrone for mixed state evolutions.One is the unitary evolution with ancilla system ended up with the final measurement on the ancilla.The other is the repetition of measurement or decoherence model which can be represented by the evolution with the master equation in the Lindblad form.Also, the relation between the fidelity to the target and the duration time is considerd.A.4 Quantum speed limit . . . . . . . . .
Abstract Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating possible redundant complexity involved in previous algorithms. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the ‘ground state’ solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.
Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating redundant complexity involved in previous formulations. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the 'ground state' solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.