Time Evolution Of Quantum System Research Articles

Adiabatic Time Evolution of Highly Excited States

Adiabatic time evolution of quantum systems is a widely used tool with applications ranging from state preparation through simplifications of computations and topological transformations to optimization and quantum computing. Adiabatic time evolution generally works well for gapped ground states, but not for thermal states in the middle of the spectrum that lack a protecting energy gap. Here we show that quantum many-body scars—a particular type of highly excited states—are suitable for adiabatic time evolution despite the absence of a protecting energy gap. Considering two rather different models, namely a one-dimensional model constructed from tensor networks and a two-dimensional fractional quantum Hall model with anyons, we find that the quantum scars perform similarly to gapped ground states with respect to adiabatic dynamics when the required final adiabatic fidelity is around 0.99. The maximum speed at which the scar state of the one-dimensional model can be adiabatically transformed decreases as a power law with system size, as opposed to exponentially for both generic thermal and disorder-driven localized states. At constant and very low ramp speed, we find that the deviation of the fidelity from unity scales linearly with ramp speed for scar states, but quadratically for gapped ground states. The gapped ground states hence perform better when the required adiabatic fidelities are very high, such as 0.9999 and above. We identify two mechanisms for leakage out of the scar state and use them to explain our results. While manipulating a single, isolated ground state is common in quantum applications, adiabatic evolution of scar states provides the flexibility to manipulate an entire tower of ground-state-like states simultaneously in a single system. Published by the American Physical Society 2024

Near-term distributed quantum computation using mean-field corrections and auxiliary qubits

Distributed quantum computation is often proposed to increase the scalability of quantum hardware, as it reduces cooperative noise and requisite connectivity by sharing quantum information between distant quantum devices. However, such exchange of quantum information itself poses unique engineering challenges, requiring high gate fidelity and costly non-local operations. To mitigate this, we propose near-term distributed quantum computing, focusing on approximate approaches that involve limited information transfer and conservative entanglement production. We first devise an approximate distributed computing scheme for the time evolution of quantum systems split across any combination of classical and quantum devices. Our procedure harnesses mean-field corrections and auxiliary qubits to link two or more devices classically, optimally encoding the auxiliary qubits to both minimize short-time evolution error and extend the approximate scheme’s performance to longer evolution times. We then expand the scheme to include limited quantum information transfer through selective qubit shuffling or teleportation, broadening our method’s applicability and boosting its performance. Finally, we build upon these concepts to produce an approximate circuit-cutting technique for the fragmented pre-training of variational quantum algorithms. To characterize our technique, we introduce a non-linear perturbation theory that discerns the critical role of our mean-field corrections in optimization and may be suitable for analyzing other non-linear quantum techniques. This fragmented pre-training is remarkably successful, reducing algorithmic error by orders of magnitude while requiring fewer iterations.

Operator formalism for the propagation of paraxial electromagnetic fields

The operator formalism of the time evolution of quantum systems is applied to describe the propagation of light beams in homogeneous and inhomogeneous media, which are governed by the paraxial wave equation. This approach allowed us to address the important case of a Gaussian beam propagating in homogeneous and isotropic space, along with the depiction of relevant beam parameters, such as the beam waist, curvature radius and Gouy phase. By using this method, we show that a special class of operators can be used to generate arbitrary families of Gaussian beams when applied to the fundamental Gaussian mode. We exemplify with the reproduction of the well-known higher-order modes of the Hermite- and Laguerre–Gaussian beams, as well as with the generation of a new family of Gaussian modes, whose propagation properties could also be described. The present method provides a new venue to investigate physical phenomena in the realm of paraxial wave optics.

Quantum simulation of the Lindblad equation using a unitary decomposition of operators

Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm---utilizing a decomposition of nonunitary operators approach---that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.

Loschmidt amplitude spectrum in dynamical quantum phase transitions

Dynamical quantum phase transitions (DQPTs) are criticalities in the time evolution of quantum systems and their existence has been theoretically predicted and experimentally observed. However, how the system behaves in the vicinity of DQPT and its connection to physical observables remains an open question. In this work, we introduce the concept of the Loschmidt amplitude spectrum (LAS), which extends the Loscmidt amplitude - the detector of the transition - by considering the overlap of the initial state to all the eigenstates of the prequench Hamiltonian. By analysing the LAS in the integrable transverse-field Ising model, we find that the system undergoes a population redistribution in the momentum space across DQPT. In the quasiparticle picture, all the lower-half k modes are excited when the system is at DQPT. The LAS is also applicable to study the dynamics of non-integrable models where we have investigated the Ising model with next-nearest-neighbour interactions as an example. The time evolution of the system's magnetization is found to be connected to the products of the LAS and there exists a simultaneous overlap of the time-evolved state to pairs of eigenstates of the prequnech Hamiltonian that possess spin configurations of negative magnetization. Our findings provide a better understanding of the characteristics of the out-of-equilibrium system around DQPT.

A group-theoretic approach to the disentanglement of generalized squeezing operators

The disentangled form of unitary operators is an indispensable tool for physical applications such as the study of squeezing properties or the time evolution of quantum systems. Here we derive a closed form disentanglement for the most general element of group ISp(2,R), whose generating Lie algebra is obtained by joining the Heisenberg-Weyl algebra to su(1,1). We attain the disentanglement formula resorting to an extension of the Truax method and check our findings through an independent factorization approach, based on the use of displacement operators. As a result we obtain a new form of factorized squeezing operators, whose action on the light vacuum state is calculated.

Parallel time integration using Batched BLAS (Basic Linear Algebra Subprograms) routines

We present an approach for integrating the time evolution of quantum systems. We leverage the computation power of graphics processing units (GPUs) to perform the integration of all time steps in parallel. The performance boost is especially prominent for small to medium-sized quantum systems. The devised algorithm can largely be implemented using the recently-specified batched versions of the BLAS routines, and can therefore be easily ported to a variety of platforms. Our PARAllelized Matrix Exponentiation for Numerical Time evolution (PARAMENT) implementation runs on CUDA-enabled graphics processing units. Program summaryProgram Title: PARAMENTCPC Library link to program files:https://doi.org/10.17632/zy5v4xs89d.1Developer's repository link:https://github.com/parament-integrator/paramentLicensing provisions: Apache 2.0Programming language: C / CUDA / PythonNature of problem: Time-integration of the Schrödinger equation with a time-dependent Hamiltonian for quantum systems with a small Hilbert space but many time-steps.Solution method: A 4th order Magnus integrator, highly parallelized on a GPU, implemented using a small subset of BLAS functions for improved portability.

Hardware-efficient variational quantum algorithms for time evolution

Parameterized quantum circuits are a promising technology for achieving a quantum advantage. An important application is the variational simulation of time evolution of quantum systems. To make the most of quantum hardware, variational algorithms need to be as hardware-efficient as possible. Here we present alternatives to the time-dependent variational principle that are hardware-efficient and do not require matrix inversion. In relation to imaginary time evolution, our approach significantly reduces the hardware requirements. With regards to real time evolution, where high precision can be important, we present algorithms of systematically increasing accuracy and hardware requirements. We numerically analyze the performance of our algorithms using quantum Hamiltonians with local interactions.

A new look at quantal time evolution

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.

Alternative approach to time evolution of quantum systems

This study focuses on the temporal evolution of quantum systems based on epistemological and ontological arguments and reconsiders some fundamental concepts of quantum theory. Initially, the time dependent Schrödinger wave equation (TDSWE) is criticized for the lack of a potential energy term and thereafter, a novel time dependent momentum operator is defined which eventually leads to an alternative TDSWE, including a potential energy term that remains in harmony with current theory. The wave nature is then generally examined by comparing time evolutions of classical and Schrödinger waves, dissimilarities then carefully inspected and ultimately a novel second order TDSWE is suggested. Finally, time evolution of the probability density is discussed in light of this new approach, and additionally some further insights and conclusions are briefly outlined that conform with the standard theory.

Diagrammatic approach for analytical non-Markovian time evolution: Fermi's two-atom problem and causality in waveguide quantum electrodynamics

Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.

Time evolution in quantum systems: a closer look at student understanding

Time evolution of quantum systems has been shown to be one of the most difficult components of a typical undergraduate quantum mechanics course. In this work, we examine the current literature, and then take a closer look at the process that students use to determine how the quantum state of a spin-1/2 particle evolves with time. We divide the process of writing a time-dependent state into five elements and use these to both directly probe student understanding and guide our coding of student responses. We focus on three elements of this process, including knowledge of the Hamiltonian, the energy eigenstates and eigenvalues, and what basis should be used when writing the state as a function of time using the phase . Analysis of four exam questions given at three institutions suggests that knowledge of the energy eigenbasis and its importance for time evolution may be a weak point in student understanding.

Enhancing student visual understanding of the time evolution of quantum systems

A combined simulation tutorial helps students develop a correct visual understanding of the time evolution of the wave function and how this leads to the time evolution of the probability density.

The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.

Work drives time evolution

We propose the idea that time evolution of quantum systems is driven by work. The formalism presented here falls within the scope of a recently proposed theory of gravitating quantum matter where extractible work, and not energy, is responsible for gravitation. Our main assumption is that extractible work, and not the Hamiltonian, dictates dynamics. We find that expectation values of meaningful quantities, such as the occupation number, deviate from those predicted by standard quantum mechanics. The scope, applications and validity of this proposal are also discussed.

Derivation of the statistics of quantum measurements from the action of unitary dynamics

Quantum statistics is defined by Hilbert space products between the eigenstates associated with state preparation and measurement. The same Hilbert space products also describe the dynamics generated by a Hamiltonian when one of the states is an eigenstate of energy E and the other represents an observable B. In this paper, we investigate this relation between the observable time evolution of quantum systems and the coherence of Hilbert space products in detail. It is shown that the times of arrival for a specific value of B observed with states that have finite energy uncertainties can be used to derive the Hilbert space product between eigenstates of energy E and eigenstates of the dynamical variable B. In these Hilbert space products, quantum phases and interference effects appear in the form of an action that relates energy to time in the experimentally observable dynamics of localized states. Quantum effects emerge in the measurement statistics when the precise control of energy in quantum state preparation results in a coherent randomization of the dynamics, such that two different arrival times contribute to the quantum statistics of the same measurement outcome B. The non-classical features associated with quantum interference can thus be explained as a consequence of quantum dynamics and its role in state preparation and measurement, indicating that the apparent randomness of control described by the energy-time uncertainties is not merely a technical problem but rather originates from the fundamental nature of interactions between physical systems.

Dynamical memory effects in correlated quantum channels

Memory effects play a fundamental role in the study of the dynamics of open quantum systems. There exist two conceptually distinct notions of memory discussed for quantum channels in the literature. In quantum information theory quantum channels with memory are characterised by the existence of correlations between successive applications of the channel on a sequence of quantum systems. In open quantum systems theory memory effects arise dynamically during the time evolution of quantum systems, and define non-Markovian dynamics. Here we relate and combine these two different concepts of memory. In particular, we study the interplay between correlations between multiple uses of quantum channels and non-Markovianity as non-divisibility of the $t$-parametrized family of channels defining the dynamical map.

Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems

In our former contribution (Cruz et al., 2015), we have shown the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation. The formalism developed in the previous work is extended to effective approaches for the description of dissipative quantum systems. By means of simple examples we show the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents. We prove that the environmental parameter γ is strongly related with the sensitivity to the choice of initial conditions.

Application of generalized operator representation in the time evolution of quantum systems

We have systematically explored the application of generalized operator representation including P-, W-, and Husimi representation in the time evolution of quantum systems. In particular, by using the method of differentiation within an ordered product of operators, we give the normally and antinormally ordered forms of the integral kernels of Husimi operator representations and its corresponding formulations. By making use of the generalized operator representation, we transform exponentially complex operator equations into tractable phase---space equations. As a simple application, the time evolution equation of Husimi function in the amplitude dissipative channel is clearly obtained.

Quantum quench dynamics in the transverse field Ising model at nonzero temperatures

The recently discovered dynamical phase transition denotes non-analytic behavior in the real time evolution of quantum systems in the thermodynamic limit and has been shown to occur in different systems at zero temperature [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. In this paper we extend the analysis to non-zero temperature by studying a generalized form of the Loschmidt echo, the work distribution function, of a quantum quench in the transverse field Ising model. Although the quantitative behavior at non-zero temperatures still displays features derived from the zero temperature non-analyticities, it is shown that in this model dynamical phase transitions do not exist if $T>0$. This is a consequence of the system being initialized in a thermal state. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski relation can be exploited as a symmetry relation for a global quench or to obtain the change of the equilibrium free energy density.