Time-evolved States Research Articles

Quantum topological data analysis via the estimation of the density of states

We develop a quantum topological data analysis (QTDA) protocol based on the estimation of the density of states (DOS) of the combinatorial Laplacian. Computing topological features of graphs and simplicial complexes is crucial for analyzing data sets and building explainable artificial intelligence solutions. This task becomes computationally hard for simplicial complexes with over 60 vertices and high-degree topological features due to a combinatorial scaling. We propose to approach the task by embedding underlying hypergraphs as effective quantum Hamiltonians and evaluating their density of states from the time evolution. Specifically, we compose propagators as quantum circuits using the Cartan decomposition of effective Hamiltonians and sample overlaps of time-evolved states using multifidelity protocols. Next we develop various postprocessing routines and implement a Fourier-like transform to recover the rank (and kernel) of Hamiltonians. This enables us to estimate the Betti numbers, revealing the topological features of simplicial complexes. We test our protocol on noiseless and noisy quantum simulators and run examples on IBM quantum processors. We observe the resilience of the proposed QTDA approach to real-hardware noise even in the absence of error mitigation, showing the promise for near-term device implementations and highlighting the utility of global DOS-based estimators. Published by the American Physical Society 2024

Measurable Krylov spaces and eigenenergy count in quantum state dynamics

In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements.

Closed-loop designed open-loop control of quantum systems: An error analysis

Quantum Lyapunov control, an important class of quantum control methods, aims at generating converging dynamics guided by Lyapunov-based theoretical tools. However, unlike the case of classical systems, disturbance caused by quantum measurement hinders direct and exact realization of the theoretical feedback dynamics designed with Lyapunov theory. Regarding this issue, the idea of closed-loop designed open-loop control has been mentioned in literature, which means to design the closed-loop dynamics theoretically, simulate the closed-loop system, generate control pulses based on simulation and apply them to the real plant in an open-loop fashion. Based on bilinear quantum control model, we analyze in this article the error, i.e., difference between the theoretical and real systems’ time-evolved states, incurred by the procedures of closed-loop designed open-loop control. It is proved that the error at an arbitrary time converges to a unitary transformation of initialization error as the number of simulation grids between 0 and that time tends to infinity. Moreover, it is found that once the simulation accuracy reaches a certain level, adopting more accurate (thus possibly more expensive) numerical simulation methods does not efficiently improve convergence. We also present an upper bound on the error norm and an example to illustrate our results.

Quench dynamics of interacting bosons: generalized coherent states versus multi-mode Glauber states

Multi-mode Glauber coherent states (MMGS) as well as Bloch states with zero quasi-momentum, which are a special case of generalized coherent states (GCS), are frequently used to describe condensed phases of bosonic many-body systems. The difference of two-point correlators of MMGS and GCS vanishes in the thermodynamic limit. Using the established expansion of MMGS in terms of GCS, we derive a Fourier-type relation between the (auto-)correlation functions of the two different time-evolved states. This relation reveals that the (auto-)correlation and thus the dynamical free-energy density for the two cases are still different, even in the thermodynamic limit, due to the lack of the U(1) symmetry of the MMGS. Analytic results for the deep lattice model of interacting bosons for increasing filling factors show multiple sharp structures in the dynamical free energy-density of increasing complexity. These are explained using the evolution of Husimi functions in phase space.

Understanding dynamical phase transitions in a spinor Bose–Einstein condensate via quantum and semiclassical analyses

Phase transitions in nonequilibrium dynamics of quantum many-body system, known as dynamical phase transitions (DPTs), play an important role for understanding various dynamical phenomena observed in different branches of physics. In general, there are two types of DPTs, the first one is characterized by distinct evolutionary behaviors of a physical observable, while the second one is marked by the vanishing overlap between the time-evolved and initial states. Here, we focus on exploring such DPTs from both quantum and semiclassical perspectives in a spinor Bose–Einstein condensate (BEC), an ideal platform for investigating nonequilibrium dynamics. Utilizing the sudden quench process, we demonstrate that the system exhibits both types of DPTs as the control parameter is quenched through the critical value, referring to as the critical quenching. We show analytically how to determine the critical quenching via the semiclassical approach and carry out a detailed examination of both semiclassical and quantum signatures of DPTs. In particular, we reveal that the occurrence of DPTs is triggered by the separatrix in the underlying semiclassical system. Our findings offer deeper insights into the properties of DPTs and verify the usefulness of semiclassical analysis for studying DPTs in quantum systems with well-defined semiclassical limit.

Quantum Entanglement and State-Transference in Fenna–Matthews–Olson Complexes: A Post-Experimental Simulation Analysis in the Computational Biology Domain

Fenna-Mathews-Olson complexes participate in the photosynthetic process of Sulfur Green Bacteria. These biological subsystems exhibit quantum features which possibly are responsible for their high efficiency; the latter may comprise multipartite entanglement and the apparent tunnelling of the initial quantum state. At first, to study these aspects, a multidisciplinary approach including experimental biology, spectroscopy, physics, and math modelling is required. Then, a global computer modelling analysis is achieved in the computational biology domain. The current work implements the Hierarchical Equations of Motion to numerically solve the open quantum system problem regarding this complex. The time-evolved states obtained with this method are then analysed under several measures of entanglement, some of them already proposed in the literature. However, for the first time, the maximum overlap with respect to the closest separable state is employed. This authentic multipartite entanglement measure provides information on the correlations, not only based on the system bipartitions as in the usual analysis. Our study has led us to note a different view of FMO multipartite entanglement as tiny contributions to the global entanglement suggested by other more basic measurements. Additionally, in another related trend, the initial state, considered as a Förster Resonance Energy Transfer, is tracked using a novel approach, considering how it could be followed under the fidelity measure on all possible permutations of the FMO subsystems through its dynamical evolution by observing the tunnelling in the most probable locations. Both analyses demanded significant computational work, making for a clear example of the complexity required in computational biology.

Performance evaluation of invariant-based inverse engineering by quantum speed limits

Quantum speed limits for two time-evolved states are introduced and applied to overlap between true dynamics and approximate dynamics. In particular, we point out that the present idea is suitable for invariant-based inverse engineering, i.e., the worst case performance of invariant-based inverse engineering can be evaluated by using a designed time-evolved state. As demonstrations, we apply the present method to stimulated Raman adiabatic passage and quantum annealing, and then we find that the present idea brings valuable insight to control scheduling.

From observations to complexity of quantum states via unsupervised learning

The vast complexity is a daunting property of generic quantum states that poses a significant challenge for theoretical treatment, especially in nonequilibrium setups. Therefore, it is vital to recognize states which are locally less complex and thus describable with (classical) effective theories. We use unsupervised learning with autoencoder neural networks to detect the local complexity of time-evolved states by determining the minimal number of parameters needed to reproduce local observations. The latter can be used as a probe of thermalization, to assign the local complexity of density matrices in open setups, and for the reconstruction of underlying Hamiltonian operators. Our approach is an ideal diagnostics tool for data obtained from (noisy) quantum simulators because it requires only practically accessible local observations.

Quantum dynamics simulations beyond the coherence time on noisy intermediate-scale quantum hardware by variational Trotter compression

We demonstrate a postquench dynamics simulation of a Heisenberg model on present-day IBM quantum hardware that extends beyond the coherence time of the device. This is achieved using a hybrid quantum-classical algorithm that propagates a state using Trotter evolution and then performs a classical optimization that effectively compresses the time-evolved state into a variational form. When iterated, this procedure enables simulations to arbitrary times with an error controlled by the compression fidelity and a fixed Trotter step size. We show how to measure the required cost function, the overlap between the time-evolved and variational states, on present-day hardware, making use of several error mitigation methods. In addition to carrying out simulations on real hardware, we investigate the performance and scaling behavior of the algorithm with noiseless and noisy classical simulations. We find the main bottleneck in going to larger system sizes to be the difficulty of carrying out the optimization of the noisy cost function.

Real-Time Evolution for Ultracompact Hamiltonian Eigenstates on Quantum Hardware

In this work we present a detailed analysis of variational quantum phase estimation (VQPE), a method based on real-time evolution for ground- and excited-state estimation on near-term hardware. We derive the theoretical ground on which the approach stands, and demonstrate that it provides one of the most compact variational expansions to date for solving strongly correlated Hamiltonians, when starting from an appropriate reference state. At the center of VQPE lies a set of equations, with a simple geometrical interpretation, which provides conditions for the time evolution grid in order to decouple eigenstates out of the set of time-evolved expansion states, and connects the method to the classical filter-diagonalization algorithm. Furthermore, we introduce what we call the unitary formulation of VQPE, in which the number of matrix elements that need to be measured scales linearly with the number of expansion states, and we provide an analysis of the effects of noise that substantially improves previous considerations. The unitary formulation allows for a direct comparison to iterative phase estimation. Our results mark VQPE as both a natural and highly efficient quantum algorithm for ground- and excited-state calculations of general many-body systems. We demonstrate a hardware implementation of VQPE for the transverse field Ising model. Furthermore, we illustrate its power on a paradigmatic example of strong correlation (Cr2 in the def2-SVP basis set), and show that it is possible to reach chemical accuracy with as few as approximately 50 time steps.4 MoreReceived 5 April 2021Revised 17 February 2022Accepted 1 March 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020323Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum computationQuantum simulationCondensed Matter, Materials & Applied PhysicsQuantum Information

GPU-accelerated solutions of the nonlinear Schrödinger equation for simulating 2D spinor BECs

As a first approximation beyond linearity, the nonlinear Schrödinger equation (NLSE) reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally complex. In this paper, we showcase a code development approach, demonstrating how computational time can be significantly reduced with readily available graphics processing unit (GPU) hardware and a straightforward code migration using open-source libraries. This process shows how CPU computations with power-law scaling in computation time with grid size can be made linear using GPUs. As a specific case study, we investigate the Gross-Pitaevskii equation, a specific version of the nonlinear Schrödinger model, as it describes in two dimensions a trapped, interacting, two-component Bose-Einstein condensate (BEC) subject to a spatially dependent interspin coupling, resulting in an analog to a spin-Hall system. This computational approach lets us probe high-resolution spatial features – revealing an interaction-dependent phase transition – all in a reasonable amount of time. Our computational approach is particularly relevant for research groups looking to easily accelerate straightforward numerical simulation of physical phenomena. Program summaryProgram Title:spinor-gpeCPC Library link to program files:https://doi.org/10.17632/sn6gjp99td.1Developer's repository link:https://github.com/ultracoldYEG/spinor-gpeLicensing provisions: MITProgramming language: PythonNature of problem: Calculate the ground- and time-evolved states to the quasi-2D, pseudospinor (two-component) nonlinear Schrodinger equation, with additional Zeeman interaction and momentum-dependent, interspin coupling.Solution method:spinor-gpe is a high-level, object-oriented Python package built on Numpy and PyTorch. It propagates the pseudospinor NLSE using a time-splitting spectral method. This package accelerates solutions using NVIDIA hardware and PyTorch's cuFFT libraries and tensor functionality.Additional comments including restrictions and unusual features: The NVIDIA CUDA backend of PyTorch is not supported on Mac computers. To run this code on a Mac system, the dependency installation will need to exclude the CUDA toolkit.

General interaction quenches in a Luttinger liquid

We discuss a general interaction quench in a Luttinger liquid described by a paired bosonic Hamiltonian. By employing su(1, 1) Lie algebra, the post-quench time-evolved wavefunctions are obtained analytically, from which the time evolution of the entanglement in momentum space can be investigated. We note that depending on the choice of Bogoliubov quasiparticles, the expressions of wavefunctions, which describe time-evolved paired states, can take different forms. The correspondence between the largest entanglement eigenvalue in momentum space and the wavefunction overlap in quench dynamics is discussed, which generalizes the results of Dóra et al (2016, Phys. Rev. Lett. 117, 010 603). A numerical demonstration on an XXZ lattice model is presented via the exact diagonalization method.

The Jaynes\u2013Cummings model of a two-level atom in a single-mode para-Bose cavity field

The coherent states in the parity deformed analog of standard boson Glauber coherent states are generated, which admit a resolution of unity with a positive measure. The quantum-mechanical nature of the light field of these para-Bose states is studied, and it is found that para-Bose order plays an important role in the nonclassical behaviors including photon antibunching, sub-Poissonian statistics, signal-to-quantum noise ratio, quadrature squeezing effect, and multi-peaked number distribution. Furthermore, we consider the Jaynes-Cummings model of a two-level atom in a para-Bose cavity field with the initial states of the excited and Glauber coherent ones when the atom makes one-photon transitions, and obtain exact energy spectrum and eigenstates of the deformed model. Nonclassical properties of the time-evolved para-Bose atom-field states are exhibited through evaluating the fidelity, evolution of atomic inversion, level damping, and von Neumann entropy. It is shown that the evolution time and the para-Bose order control these properties.

Entanglement of nitrogen-vacancy-center ensembles with initial squeezing

In this paper, we investigate entanglement of an experimental system of two nitrogen-vacancy-center ensembles which are initially squeezed under the one-axis twisting Hamiltonian. We take into account three scenarios in which initial squeezing and entanglement are mediated by phonons or photons: (a) the phonon-squeezed photon-entangled scenario, (b) the phonon-squeezed phonon-entangled scenario, and (c) the photon-squeezed photon-entangled scenario. For our investigation, we employ the Tavis-Cummings model, which includes dissipative decoherence of the collective spin ensemble, and analyze the system both for a relatively small number of spins and in the limit of a large number of spins using the approach of a quantum master equation. Although evidence in the literature on idealized coupled oscillator systems and coupled quantum kicked tops suggests that initial squeezing can enhance entanglement, we find that, in the realistic system studied in this paper, initial squeezing can improve entanglement overall when the field mode interacts in a particular manner with the two spin ensembles. Our analysis using the Holstein-Primakoff transformation and Wigner characteristic function in the rotating frame of reference shows that the entanglement enhancement is a subtle consequence of the way in which the dissipative decoherence rotates the state of the collective spin ensemble such that enhancement depends on the time-evolved rotated states between the presence and absence of initial squeezing.

Quantum Power Method by a Superposition of Time-Evolved States

We propose a quantum-classical hybrid algorithm of the power method, here dubbed as the quantum power method, to evaluate H^n|ψ⟩ with quantum computers, where n is a non-negative integer, H^ is a time-independent Hamiltonian of interest, and |ψ⟩ is a quantum state. We show that the number of gates required for approximating H^n scales linearly in the power and the number of qubits, making it a promising application for near-term quantum computers. Using numerical simulation, we show that the power method can control systematic errors in approximating the Hamiltonian power H^n for n as large as 100. As an application, we combine our method with a multireference Krylov-subspace-diagonalization scheme to show how one can improve the estimation of ground-state energies and the ground-state fidelities found using a variational-quantum-eigensolver scheme. Finally, we outline other applications of the quantum power method, including several moment-based methods. We numerically demonstrate the connected-moment expansion for the imaginary-time evolution and compare the results with the multireference Krylov-subspace diagonalization.16 MoreReceived 11 August 2020Revised 19 November 2020Accepted 7 January 2021DOI:https://doi.org/10.1103/PRXQuantum.2.010333Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum computationQuantum gatesQuantum simulationQuantum Information

Quench dynamics and zero-energy modes: The case of the Creutz model

In most lattice models, the closing of a band gap typically occurs at high-symmetry points in the Brillouin zone. Differently, in the Creutz model $-$ describing a system of spinless fermions hopping on a two-leg ladder pierced by a magnetic field $-$ the gap closing at the quantum phase transition between the two topologically nontrivial phases of the model can be moved by tuning the hopping amplitudes. We take advantage of this property to examine the nonequilibrium dynamics of the model after a sudden quench of the magnetic flux through the plaquettes of the ladder. For a quench to one of the equilibrium quantum critical points we find that the revival period of the Loschmidt echo $-$ measuring the overlap between initial and time-evolved states $-$ is controlled by the gap closing zero-energy modes. In particular, and contrary to expectations, the revival period of the Loschmidt echo for a finite ladder does not scale linearly with size but exhibits jumps determined by the presence or absence of zero-energy modes. We further investigate the conditions for the appearance of dynamical quantum phase transitions in the model and find that, for a quench {\em to} an equilibrium critical point, such transitions occur only for ladders of sizes which host zero-energy modes. Exploiting concepts from quantum thermodynamics, we show that the average work and the irreversible work per lattice site exhibit a weak dependence on the size of the system after a quench {\em across} an equilibrium critical point, suggesting that quenching into a different phase induces effective correlations among the particles.

Dynamics of entanglement and state-space trajectories followed by a system of four-qubit in the presence of random telegraph noise: common environment (CE) versus independent environments (IEs)

The paper investigates the dynamics of entanglement and explores some geometrical characteristics of the trajectories in state space, in four-qubit Greenberger-Horne-Zeilinger (GHZ) - and W-type states, coupled to common and independent classical random telegraph noise (RTN) sources. It is shown from numerical simulations that: (i) the dynamics of entanglement depends drastically not only on the input configuration of the qubits and the presence or absence of memory effects, but also on whether the qubits are coupled to the RTN in a CE or IEs; (ii) a considerable amount of entanglement can be indefinitely trapped when the qubits are embedded in a CE; (iii) the CE configuration preserves better the entanglement initially shared between the qubits than the IEs configuration, however, for W-type states, there is a period of time and/or certain values of the purity for which, the opposite can be found. Thanks to the results obtained in our earlier works on three-qubit models, we are able to conclude that entanglement becomes more robustly protected from decay when the number of qubits of the system increases. Finally, we find that the trajectories in state space of the system quantified by the quantum Jensen Shannon divergence (QJSD) between the time-evolved states of the qubits and some reference states may be curvilinear or chaotic.

Dynamical Quantum Phase Transitions in Spin Chains with Long-Range Interactions: Merging Different Concepts of Nonequilibrium Criticality.

We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.

Quantum dynamics in transverse-field Ising models from classical networks

The efficient representation of quantum many-body states with classical resources is a key challenge in quantum many-body theory. In this work we analytically construct classical networks for the description of the quantum dynamics in transverse-field Ising models that can be solved efficiently using Monte Carlo techniques. Our perturbative construction encodes time-evolved quantum states of spin-1/2 systems in a network of classical spins with local couplings and can be directly generalized to other spin systems and higher spins. Using this construction we compute the transient dynamics in one, two, and three dimensions including local observables, entanglement production, and Loschmidt amplitudes using Monte Carlo algorithms and demonstrate the accuracy of this approach by comparisons to exact results. We include a mapping to equivalent artificial neural networks, which were recently introduced to provide a universal structure for classical network wave functions.

Loschmidt Echo Revivals: Critical and Noncritical.

A quantum phase transition is generally thought to imprint distinctive characteristics on the nonequilibrium dynamics of a closed quantum system. Specifically, the Loschmidt echo after a sudden quench to a quantum critical point-measuring the time dependence of the overlap between initial and time-evolved states-is expected to exhibit an accelerated relaxation followed by periodic revivals. We here introduce a new exactly solvable model, the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a counterexample. A parallell analysis of the quench dynamics of the three-site spin-interacting XY model allows us to pinpoint the conditions under which a periodic Loschmidt revival actually appears.