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Physical Review A

Time-optimal control based on path distance

Lin-Feng Du, Li‐Guo Qin, Qin Guo, Xue-xiang Xu, Jie‐Hui Huang

Published Aug 13, 2025• 0 citations
Publisher

Abstract

The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation ${\ensuremath{\omega}}_{H}=\frac{1}{2}({\ensuremath{\lambda}}_{max}\ensuremath{-}{\ensuremath{\lambda}}_{min})$, which is independent of the quantum state. Given an initial state ${\ensuremath{\rho}}_{0}$ in a quantum unitary system, the minimum time required to evolve it to a target state ${\ensuremath{\rho}}_{\ensuremath{\tau}}$ is ${\ensuremath{\tau}}_{min}=\frac{{\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})}{{\ensuremath{\omega}}_{H}}$, where ${\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})$ is the path distance contained in the shortest evolution from ${\ensuremath{\rho}}_{0}$ to ${\ensuremath{\rho}}_{\ensuremath{\tau}}$. This lower bound of the evolution time is tight and achievable in a finite-energy system.

DOI
10.1103/mv26-qytj

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The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation \ensuremathωH=12(\ensuremathλmax\ensuremath−\ensuremathλmin){\ensuremath{\omega}}_{H}=\frac{1}{2}({\ensuremath{\lambda}}_{max}\ensuremath{-}{\ensuremath{\lambda}}_{min})\ensuremathωH​=21​(\ensuremathλmax​\ensuremath−\ensuremathλmin​), which is independent of the quantum state. Given an initial state \ensuremathρ0{\ensuremath{\rho}}_{0}\ensuremathρ0​ in a quantum unitary system, the minimum time required to evolve it to a target state \ensuremathρ\ensuremathτ{\ensuremath{\rho}}_{\ensuremath{\tau}}\ensuremathρ\ensuremathτ​ is \ensuremathτmin=Lmin(\ensuremathρ0,\ensuremathρ\ensuremathτ)\ensuremathωH{\ensuremath{\tau}}_{min}=\frac{{\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})}{{\ensuremath{\omega}}_{H}}\ensuremathτmin​=\ensuremathωH​Lmin​(\ensuremathρ0​,\ensuremathρ\ensuremathτ​)​, where Lmin(\ensuremathρ0,\ensuremathρ\ensuremathτ){\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})Lmin​(\ensuremathρ0​,\ensuremathρ\ensuremathτ​) is the path distance contained in the shortest evolution from \ensuremathρ0{\ensuremath{\rho}}_{0}\ensuremathρ0​ to \ensuremathρ\ensuremathτ{\ensuremath{\rho}}_{\ensuremath{\tau}}\ensuremathρ\ensuremathτ​. This lower bound of the evolution time is tight and achievable in a finite-energy system.
For any pair of quantum states, an initial state |I and a final quantum state |F , in a Hilbert space, there are many Hamiltonians H under which |I evolves into |F .Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E max and E min , is held fixed.We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ .For Hermitian Hamiltonians, τ has a nonzero lower bound.However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle.The minimum value of τ can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |I to the vector |F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short.The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole.This result may have applications in quantum computing.
The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian H\mathcal HH (Def. 6.1), and the initial state of the system is the eigenstate of H\mathcal HH corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation i d∣ψ(t)⟩dt=H(t)∣ψ(t)⟩i\,\frac{\mathrm{d} | \psi(t) \rangle}{\mathrm{d}t} = \mathcal{H}(t) | \psi(t) \rangleidtd∣ψ(t)⟩​=H(t)∣ψ(t)⟩, and H(t)\mathcal{H}(t)H(t) is slowly changing. If, for any value of the time parameter ttt, there is always a gap between the lowest eigenvalue of H(t)\mathcal{H}(t)H(t) and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian H(t)\mathcal{H}(t)H(t) varies from an initial (time-independent) Hamiltonian HI\mathcal{H}_{\mathrm{I}}HI​ to a final Hamiltonian HF\mathcal{H}_{\mathrm{F}}HF​. If the change happens slowly, and the initial state of the system is an eigenstate of HI\mathcal{H}_{\mathrm{I}}HI​ with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian HF\mathcal{H}_{\mathrm{F}}HF​. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly HI\mathcal{H}_{\mathrm{I}}HI​ should be changed into HF\mathcal{H}_{\mathrm{F}}HF​. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.
I report a tight upper bound of the maximum speed of evolution from one quantum state $\ensuremath{\rho}$ to another ${\ensuremath{\rho}}^{\ensuremath{'}}$ with fidelity $F(\ensuremath{\rho},{\ensuremath{\rho}}^{\ensuremath{'}})$ less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state DE\mathcal{D}EDE, one may interpret DE\mathcal{D}EDE as a meaningful measure of the maximum information-processing capability of a system.
We introduce a metric in a geometric way for measuring the distance between two quantum states and derive the instantaneous angular velocity of a quantum state's dynamical evolution, which has two components: $\mathrm{\ensuremath{\Delta}}{E}_{t}/\ensuremath{\hbar}$ in the longitudinal direction and ${\overline{E}}_{t}/\ensuremath{\hbar}$ in the latitudinal direction. By applying the instantaneous angular velocities in the dynamical evolution governed by a time-independent Hamiltonian, the fastest way for a pure state to evolve to a target state is discussed. As a consequence, we generalize the Mandelstam-Tamm and Margolus-Levitin bounds without using the Heisenberg uncertainty relation and point out the conditions for saturating the two bounds. The fastest dynamical evolution of a quantum system with a particular time-dependent Hamiltonian is also briefly discussed. The quantum features in the dynamical evolution are revealed through phase change and lifetime of the initial state.
Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent Hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation between optimal times found in quantum control and the QSL bound, in the (relevant) time-dependent regime, by discussing the ubiquitous two-level Landau-Zener-type Hamiltonian.
Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent Hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation between optimal times found in quantum control and the QSL bound, in the (relevant) time-dependent regime, by discussing the ubiquitous two-level Landau-Zener–type Hamiltonian.