Time-Optimal Quantum Evolution

In optimal quantum-mechanical evolutions, motion can occur along nonpredetermined paths of shortest length in an optimal time. Alternatively, optimal evolutions can happen along predefined paths with no waste of energy resources and $100%$ speed efficiency. Unfortunately, realistic physical scenarios typically result in less-than-ideal evolutions. In this paper we study different families of suboptimal qubit Hamiltonians, both stationary and time varying, for which the so-called geodesic efficiency and the speed efficiency of the corresponding quantum evolutions are less than one. Furthermore, after proposing an alternative hybrid efficiency measure constructed out of the two previously mentioned efficiency quantifiers, we provide illustrative examples where the average departures from time optimality and $100%$ speed efficiency are globally captured over a limited time period. In particular, due to this hybrid measure, quantum evolutions are partitioned in four categories: geodesic unwasteful, nongeodesic unwasteful, geodesic wasteful, and nongeodesic wasteful. Finally, we discuss Hamiltonians specified by magnetic-field configurations, both stationary and nonstationary, yielding optimal hybrid efficiency, that it, both time optimality and $100%$ speed efficiency, over a finite time interval.In optimal quantum-mechanical evolutions, motion can occur along nonpredetermined paths of shortest length in an optimal time. Alternatively, optimal evolutions can happen along predefined paths with no waste of energy resources and $100%$ speed efficiency. Unfortunately, realistic physical scenarios typically result in less-than-ideal evolutions. In this paper we study different families of suboptimal qubit Hamiltonians, both stationary and time varying, for which the so-called geodesic efficiency and the speed efficiency of the corresponding quantum evolutions are less than one. Furthermore, after proposing an alternative hybrid efficiency measure constructed out of the two previously mentioned efficiency quantifiers, we provide illustrative examples where the average departures from time optimality and $100%$ speed efficiency are globally captured over a limited time period. In particular, due to this hybrid measure, quantum evolutions are partitioned in four categories: geodesic unwasteful, nongeodesic unwasteful, geodesic wasteful, and nongeodesic wasteful. Finally, we discuss Hamiltonians specified by magnetic-field configurations, both stationary and nonstationary, yielding optimal hybrid efficiency, that it, both time optimality and $100%$ speed efficiency, over a finite time interval.

arXiv:2408.14230Physical Review AFeb 25, 20257 citations

Time evolution of quantum systems

Institute of Physics eBooksNov 26, 20040 citations

Near-Optimal Quantum Time Evolution Circuits via Provably Convergent Compression

Erenay Karacan, Isabel Nha Minh Le, Matteo D'Anna +3 more

Variational compression can significantly lower implementation overheads for encoding the time evolution of Hamiltonians into quantum circuits. However, they usually lack global convergence guarantees and well-established scaling behavior. In this work, we provide a recipe for choosing the initial point of such variational optimizations that guarantees convergence to a quantum circuit with near-optimal gate complexity $\mathcal{O}\left( N \, t \, \text{polylog}(N \, t/ε) \right)$ for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For $N = 48$ sites, evolution time $t=0.1$ and infidelity $ε\approx1\%$ , the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups. Thereby, our recipe extends digital quantum simulators toward system sizes and geometries that are challenging for classical computation.Variational compression can significantly lower implementation overheads for encoding the time evolution of Hamiltonians into quantum circuits. However, they usually lack global convergence guarantees and well-established scaling behavior. In this work, we provide a recipe for choosing the initial point of such variational optimizations that guarantees convergence to a quantum circuit with near-optimal gate complexity $\mathcal{O}\left( N \, t \, \text{polylog}(N \, t/ε) \right)$ for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For $N = 48$ sites, evolution time $t=0.1$ and infidelity $ε\approx1\%$, the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups. Thereby, our recipe extends digital quantum simulators toward system sizes and geometries that are challenging for classical computation.

quant-pharXiv:2605.17067May 19, 20260 citations

Time-optimal unitary operations

A. Carlini, Akio Hosoya, Tatsuhiko Koike +1 more

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.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.

arXiv:quant-ph/0608039Physical Review AApr 10, 2007126 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 . . . . . . . . .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, 20080 citations

100%

speed efficiency. Unfortunately, realistic physical scenarios typically result in less-than-ideal evolutions. In this paper we study different families of suboptimal qubit Hamiltonians, both stationary and time varying, for which the so-called geodesic efficiency and the speed efficiency of the corresponding quantum evolutions are less than one. Furthermore, after proposing an alternative hybrid efficiency measure constructed out of the two previously mentioned efficiency quantifiers, we provide illustrative examples where the average departures from time optimality and

100%

speed efficiency are globally captured over a limited time period. In particular, due to this hybrid measure, quantum evolutions are partitioned in four categories: geodesic unwasteful, nongeodesic unwasteful, geodesic wasteful, and nongeodesic wasteful. Finally, we discuss Hamiltonians specified by magnetic-field configurations, both stationary and nonstationary, yielding optimal hybrid efficiency, that it, both time optimality and

100%

speed efficiency, over a finite time interval.

\mathcal{O}\left( N \, t \, \text{polylog}(N \, t/ε) \right)

for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For

N = 48

sites, evolution time

t=0.1

and infidelity

ε\approx1\%

, the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups. Thereby, our recipe extends digital quantum simulators toward system sizes and geometries that are challenging for classical computation.