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arXiv:2411.08644 · Optical Materials Express

Optimizing state transfer in a three-qubit array via quantum brachistochrone method [Invited]

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

Published Feb 13, 2025• 5 citations
arXivHTMLPDF

Abstract

The quantum brachistochrone method has recently emerged as a technique that allows one to implement the desired unitary evolution operator in a physical system within a minimal time. Here, we apply this approach to the problem of time-optimal quantum state transfer in an array of three qubits with time-varying nearest-neighbor couplings and analytically derive the fastest protocol.

DOI
10.1364/ome.547936

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The quantum brachistochrone method has recently emerged as a technique that allows one to implement the desired unitary evolution operator in a physical system within a minimal time. Here, we apply this approach to the problem of time-optimal quantum state transfer in an array of three qubits with time-varying nearest-neighbor couplings and analytically derive the fastest protocol.
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.
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 quantum speed limit for a single-particle excitation in a nearest-neighbor-coupled qubit lattice with switchable couplings and fixed sum of their squares. We obtain the solution for a finite system with up to 10 000 qubits and for the effectively infinite lattice closed into ring. As another application of our method, we derive the quantum speed limit for a lattice with time-independent 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 . . . . . . . . .
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.
Abstract We analytically investigate the role of entanglement in time-optimal state evolution as an application of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly coupled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entanglement plays a role for W and Greenberger–Horne–Zeilinger ( GHz ) initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.