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Physical Review A

Shortest evolution path between two mixed states and its realization

Jie‐Hui Huang, Shan-Shan Dong, Guang-Long Chen, Nanrun Zhou, Fu-Yao Liu, Li‐Guo Qin

Published Apr 8, 2024• 4 citations
Publisher

Abstract

In a quantum unitary system, an initial state may have different paths to evolve to a target state, and the only requirement is that the initial and target states share the same eigenvalue spectrum. We focus on the evolution between two nondegenerate mixed states in this paper, and investigate the shortest evolution path between them. By minimizing the path distance contained in the unitary operator connecting the initial and target states over a series of phases, the shortest evolution path could be figured out. This minimum path distance has an analytical form in the single-qubit dynamical system, and its solution in the three- or higher-dimensional dynamical system could be obtained numerically. Based on the unitary operator associated with the shortest evolution path, a general form for the Hamiltonian to realize it is presented. Here we present another way to study quantum optimal control, which is based on the path distance between the initial state and its evolution state, rather than the state distance between them.

DOI
10.1103/physreva.109.042405

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In a quantum unitary system, an initial state may have different paths to evolve to a target state, and the only requirement is that the initial and target states share the same eigenvalue spectrum. We focus on the evolution between two nondegenerate mixed states in this paper, and investigate the shortest evolution path between them. By minimizing the path distance contained in the unitary operator connecting the initial and target states over a series of phases, the shortest evolution path could be figured out. This minimum path distance has an analytical form in the single-qubit dynamical system, and its solution in the three- or higher-dimensional dynamical system could be obtained numerically. Based on the unitary operator associated with the shortest evolution path, a general form for the Hamiltonian to realize it is presented. Here we present another way to study quantum optimal control, which is based on the path distance between the initial state and its evolution state, rather than the state distance between them.
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.
The Hamiltonian and evolution time of a quantum unitary system completely determine a unitary operator, which turns an initial state ${\ensuremath{\rho}}_{0}$ into its time-evolved state ${\ensuremath{\rho}}_{t}$ and may also turn another state ${\ensuremath{\sigma}}_{0}$ into its time-evolved state ${\ensuremath{\sigma}}_{t}$. Although the two pairs of states $({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{t})$ and $({\ensuremath{\sigma}}_{0},{\ensuremath{\sigma}}_{t})$ are governed by the same Hamiltonian and experience equal evolution periods, their state distances are not necessarily equal to each other. Specifically, the evolution operator contains information about the evolution other than that reflected by two specific states connected through it. As a matter of fact, two pairs of states with different state distances may have equal path distances. Here we propose a geometric measure for the path distance contained in a quantum unitary evolution, which depends only on the unitary operator itself and is independent of any pair of states connected through it. This path distance meets the basic requirements for a good measure of distance and is valid during the whole evolution process, no matter how long the evolution time is. This path distance is bounded by the path length of the evolution from above and by the Bures angle between the two states connected through it from below. It has potential applications in quantum information processing, such as providing another way to study the optimal control between two quantum states. We finally present the form of Hamiltonian to realize a unitary operator on demand.
Abstract In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.
Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini–Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.
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 . . . . . . . . .
The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g. a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.