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Time Optimal Quantum Operartion for Mixed States(Micro-Macro Duality in Quantum Analysis)

A. Carlini, Akio Hosoya, Tatsuhiko Koike, Yosuke Okudaira

Published Jul 1, 2007• 0 citations
OpenAlex

Abstract

We formulate a variational principle for finding the time.optimalquantum evolution of mixed states subject to a master equation, when the Hamiltonian $H$ and the Lindblad operators $L_{f}$ are subject to certain constraints.We show that the problem can be reduced to solving first a fun- damental equation (the "quanbum brachistochrone") for $H(t)$ , which can be written down once the $\infty nstraints$ are specified, and then solving the constraints and the master equation for the $L_{f}(t)s$ and the density operator $\rho(t)$ .As an application of our fomalism, we $an\Phi ticM$ solve a simple one $qu$ , bit model where the optimal Lindblad operators correspond either to a measurement or to a decoherence process by the environment.

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We formulate a variational principle for finding the time.optimalquantum evolution of mixed states subject to a master equation, when the Hamiltonian HHH and the Lindblad operators LfL_{f}Lf​ are subject to certain constraints.We show that the problem can be reduced to solving first a fun- damental equation (the "quanbum brachistochrone") for H(t)H(t)H(t) , which can be written down once the ∞nstraints\infty nstraints∞nstraints are specified, and then solving the constraints and the master equation for the Lf(t)sL_{f}(t)sLf​(t)s and the density operator ρ(t)\rho(t)ρ(t) .As an application of our fomalism, we anΦticMan\Phi ticManΦticM solve a simple one quququ , bit model where the optimal Lindblad operators correspond either to a measurement or to a decoherence process by the environment.
We present a general formalism based on the variational principle for finding the time-optimal quantum evolution of mixed states governed by a master equation, when the Hamiltonian and the Lindblad operators are subject to certain constraints. The problem may be reduced to solving first a fundamental equation, which can be written down once the constraints are specified, for the Hamiltonian and then solving the constraints and the master equation for the Lindblad and the density operators. As an application of our formalism, we study a simple one-qubit model, where the optimal Lindblad operators can be simulated by a tunable coupling with an ancillary qubit.
In a series of previous works we formulated a variational principle to find out the time-optimal evolution for a quantum system with a given set of initial and final states.Here we show how to obtain the shortest duration time T(f) to reach the target state via quantum operations within a given fidelity larger than a specified value f ≤\leq≤ 1.
We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.
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.
Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.