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arXiv:2511.11508 · Physical Review A

Efficient computation of quantum time-optimal control

Andrei A. Stepanenko, Kseniia S. Chernova, Maxim A. Gorlach

quant-phcond-mat.mes-hallphysics.opticsSubmitted Nov 14, 2025• Announced Nov 17, 2025• 0 citations
v1 · Nov 14, 2025v2 · Aug 28, 2026
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Abstract

We present an approach to compute time-optimal control of a quantum system which combines quantum brachistochrone and Lax pair techniques and enables efficient investigation of large-scale quantum systems. We illustrate our method by finding the fastest way to transfer a single-particle excitation in a nearest-neighbor-coupled infinitely large qubit lattice with the fixed sum of squares of the couplings.

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6 pages, 3 figures, 8 pages of Supplementary Materials

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We present an approach to compute time-optimal control of a quantum system which combines quantum brachistochrone and Lax pair techniques and enables efficient investigation of large-scale quantum systems. We illustrate our method by finding the fastest way to transfer a single-particle excitation in a nearest-neighbor-coupled infinitely large qubit lattice with the fixed sum of squares of the couplings.
Recent progress in quantum technologies enables the fabrication of large-scale qubit arrays, but their optimal control remains challenging. In our work, we present a protocol for perfect quantum state transfer in a nearest-neighbor coupled chain of qubits, achieving maximal fidelity and minimal transfer time simultaneously. We apply the quantum brachistochrone method and obtain a protocol for the transfer of Fock and Bell states. Our solution reaches the fundamental quantum speed limit in the lattices with time-varying couplings.
Recent technological advances have allowed the fabrication of large arrays of coupled qubits, serving as prototypes for quantum processors. However, the optimal control of such systems remains notoriously challenging, limiting the potential of large-scale quantum systems. Here, we investigate a model problem of quantum state transfer in a large nearest-neighbor-coupled qubit array. We derive an optimal control strategy that simultaneously achieves maximal fidelity and minimal transfer time, reaching the quantum speed limit in a lattice with time-varying couplings.
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 . . . . . . . . .
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.
Decoherence is inevitable when manipulating quantum systems. It decreases the quality of quantum manipulations and thus is one of the main obstacles for large-scale quantum computation, where high-fidelity quantum gates are needed. Generally, the longer a gate operation is, the more decoherence-induced gate infidelity will be. Therefore, how to shorten the gate time becomes an urgent problem to be solved. To this end, time-optimal control based on solving the quantum brachistochrone equation is a straightforward solution. Here, based on time-optimal control, we propose a scheme to realize universal quantum gates on superconducting qubits in a two-dimensional square lattice configuration, and the two-qubit gate fidelity approaches 99.999.9%99.9. Meanwhile, we can further accelerate the ZZZ-axis gate considerably by adjusting the detuning of the external driving. Finally, in order to reduce the influence of the dephasing error, decoherence-free subspace encoding is also incorporated in our physical implementation. Therefore, we present a fast quantum scheme which is promising for large-scale quantum computation.