Hamiltonian Dynamics Research Articles

Simulating open quantum systems with giant atoms

Abstract Open quantum many-body systems are of both fundamental and applicational interest. However, it remains an open challenge to simulate and solve such systems, both with state-of-the-art classical methods and with quantum-simulation protocols. To overcome this challenge, we introduce a simulator for open quantum many-body systems based on giant atoms, i.e., atoms (possibly artificial), that couple to a waveguide at multiple points, which can be wavelengths apart. We first show that a simulator consisting of two giant atoms can simulate the dynamics of two coupled qubits, where one qubit is subject to different drive amplitudes and dissipation rates. This simulation enables characterizing the quantum Zeno crossover in this model. We further show that by equipping the simulator with post-selection, it becomes possible to simulate the effective non-Hermitian Hamiltonian dynamics of the system and thereby characterize the transition from oscillatory to non-oscillatory dynamics due to varying dissipation rates. We demonstrate and analyze the robustness of these simulation results against noise affecting the giant atoms. Finally, we discuss and show how giant-atom-based simulators can be scaled up for digital-analog simulation of large open quantum many-body systems, e.g., generic dissipative spin models.

Generalized Conformable Hamiltonian Dynamics with Higher-Order Derivatives

Generalized Conformable Hamiltonian Dynamics with Higher-Order Derivatives

Designing open quantum systems with known steady states: Davies generators and beyond

We provide a systematic framework for constructing generic models of nonequilibrium quantum dynamics with a target stationary (mixed) state. Our framework identifies (almost) all combinations of Hamiltonian and dissipative dynamics that relax to a steady state of interest, generalizing the Davies’ generator for dissipative relaxation at finite temperature to nonequilibrium dynamics targeting arbitrary stationary states. We focus on Gibbs states of stabilizer Hamiltonians, identifying local Lindbladians compatible therewith by constraining the rates of dissipative and unitary processes. Moreover, given terms in the Lindbladian not compatible with the target state, our formalism identifies the operations – including syndrome measurements and local feedback – one must apply to correct these errors. Our methods also reveal new models of quantum dynamics: for example, we provide a “measurement-induced phase transition” in which measurable two-point functions exhibit critical (power-law) scaling with distance at a critical ratio of the transverse field and rate of measurement and feedback. Time-reversal symmetry – defined naturally within our formalism – can be broken both in effectively classical and intrinsically quantum ways. Our framework provides a systematic starting point for exploring the landscape of dynamical universality classes in open quantum systems, as well as identifying new protocols for quantum error correction.

Krylov complexity and Trotter transitions in unitary circuit dynamics

We investigate many-body dynamics where the evolution is governed by unitary circuits through the lens of “Krylov complexity,” a recently proposed measure of complexity and quantum chaos. We extend the formalism of Krylov complexity to unitary circuit dynamics and focus on Floquet circuits arising as the Trotter decomposition of Hamiltonian dynamics. For short Trotter steps the results from Hamiltonian dynamics are recovered, whereas a large Trotter step results in different universal behavior characterized by the existence of local : operators with vanishing autocorrelation functions, as exemplified in dual-unitary circuits. These operators exhibit maximal complexity growth, act as a memoryless bath for the dynamics, and can be directly probed in current quantum computing setups. These two regimes are separated by a crossover in chaotic systems. Conversely, we find that free integrable systems exhibit a nonanalytic transition between these different regimes, where maximally ergodic operators appear at a critical Trotter step. Published by the American Physical Society 2025

Generators of groups of Hamiltonian maps

AbstractWe prove that analytic Hamiltonian dynamics on tori, annuli, or Euclidean space can be approximated by a composition of nonlinear shear maps where each of the shears depends only on the position or only on the momentum.

The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

How well can quantum computers simulate classical dynamical systems? There is increasing effort in developing quantum algorithms to efficiently simulate dynamics beyond Hamiltonian simulation, but so far exact resource estimates are not known. In this work, we provide two significant contributions. First, we give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states – either the solution at a final time, or an encoding of the whole history within a time interval. Second, we show that the stability properties of a large class of classical dynamics allow their fast-forwarding, making their quantum simulation much more time-efficient. From this point of view, quantum Hamiltonian dynamics is a boundary case that does not allow this form of stability-induced fast-forwarding. In particular, we find that the history state can always be output with complexity O(T1/2) for any stable linear system. We present a range of asymptotic improvements over state-of-the-art in various regimes. We illustrate our results with a family of dynamics including linearized collisional plasma problems, coupled, damped, forced harmonic oscillators and dissipative nonlinear problems. In this case the scaling is quadratically improved, and leads to significant reductions in the query counts after inclusion of all relevant constant prefactors.

Optimized Trajectory Unraveling for Classical Simulation of Noisy Quantum Dynamics.

The dynamics of open quantum systems can be simulated by unraveling it into an ensemble of pure state trajectories undergoing nonunitary monitored evolution, which has recently been shown to undergo measurement-induced entanglement phase transition. Here, we show that, for an arbitrary decoherence channel, one can optimize the unraveling scheme to lower the threshold for entanglement phase transition, thereby enabling efficient classical simulation of the open dynamics for a broader range of decoherence rates. Taking noisy random unitary circuits as a paradigmatic example, we analytically derive the optimum unraveling basis that on average minimizes the threshold. Moreover, we present a heuristic algorithm that adaptively optimizes the unraveling basis for given noise channels, also significantly extending the simulatable regime. When applied to noisy Hamiltonian dynamics, the heuristic approach indeed extends the regime of efficient classical simulation based on matrix product states beyond conventional quantum trajectory methods. Finally, we assess the possibility of using a quasi-local unraveling, which involves multiple qubits and time steps, to efficiently simulate open systems with an arbitrarily small but finite decoherence rate.

On non-autonomous Hamiltonian dynamics, dual spaces, and kinetic lifts

On non-autonomous Hamiltonian dynamics, dual spaces, and kinetic lifts

Zigzag Path Connects Two Monte Carlo Samplers: Hamiltonian Counterpart to a Piecewise Deterministic Markov Process

Zigzag and other piecewise deterministic Markov process samplers have attracted significant interest for their non-reversibility and other appealing properties for Bayesian posterior computation. Hamiltonian Monte Carlo is another state-of-the-art sampler, exploiting fictitious momentum to guide Markov chains through complex target distributions. We establish an important connection between the zigzag sampler and a variant of Hamiltonian Monte Carlo based on Laplace-distributed momentum. The position and velocity component of the corresponding Hamiltonian dynamics travels along a zigzag path paralleling the Markovian zigzag process; however, the dynamics is non-Markovian in this position-velocity space as the momentum component encodes non-immediate pasts. This information is partially lost during a momentum refreshment step, in which we preserve its direction but resample magnitude. In the limit of increasingly frequent momentum refreshments, we prove that Hamiltonian zigzag converges strongly to its Markovian counterpart. This theoretical insight suggests that, when retaining full momentum information, Hamiltonian zigzag can better explore target distributions with highly correlated parameters by suppressing the diffusive behavior of Markovian zigzag. We corroborate this intuition by comparing performance of the two zigzag cousins on high-dimensional truncated multivariate Gaussians, including a 11,235-dimensional target arising from a Bayesian phylogenetic multivariate probit modeling of HIV virus data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity

We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024

Predicting arbitrary state properties from single Hamiltonian quench dynamics

Analog quantum simulation is an essential routine for quantum computing and plays a crucial role in studying quantum many-body physics. Typically, the quantum evolution of an analog simulator is largely determined by its physical characteristics, lacking the precise control or versatility of quantum gates. This limitation poses challenges in extracting physical properties on analog quantum simulators, an essential step of quantum simulations. To address this issue, we introduce the protocol, which uses a single quench Hamiltonian for estimating arbitrary state properties, eliminating the need for ancillary systems and random unitaries. Additionally, we derive the sample complexity of this protocol and show that it performs comparably to the classical shadow protocol. The Hamiltonian shadow protocol does not require sophisticated control and can be applied to a wide range of analog quantum simulators. We demonstrate its utility through numerical demonstrations with Rydberg atom arrays under realistic parameter settings. The new protocol significantly broadens the application of randomized measurements for analog quantum simulators without precise control and ancillary systems. Published by the American Physical Society 2024

Robustly learning the Hamiltonian dynamics of a superconducting quantum processor

Precise means of characterizing analog quantum simulators are key to developing quantum simulators capable of beyond-classical computations. Here, we precisely estimate the free Hamiltonian parameters of a superconducting-qubit analog quantum simulator from measured time-series data on up to 14 qubits. To achieve this, we develop a scalable Hamiltonian learning algorithm that is robust against state-preparation and measurement (SPAM) errors and yields tomographic information about those SPAM errors. The key subroutines are a novel super-resolution technique for frequency extraction from matrix time-series, tensorESPRIT, and constrained manifold optimization. Our learning results verify the Hamiltonian dynamics on a Sycamore processor up to sub-MHz accuracy, and allow us to construct a spatial implementation error map for a grid of 27 qubits. Our results constitute an accurate implementation of a dynamical quantum simulation that is precisely characterized using a new diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.

High-Fidelity Two-Qubit Gates between Fluxonium Qubits with a Resonator Coupler

We take a bottom-up first-principles approach to designing a two-qubit gate between fluxonium qubits for minimal error, speed, and control simplicity. Our proposed architecture consists of two fluxoniums coupled via a resonator. The use of a simple linear coupler has many practical benefits, including the possibility of material optimization for suppressing loss, reducing fabrication complexity, and increasing yield by circumventing the need for Josephson junctions. Crucially, a resonator-as-coupler approach also suggests a clear path to increased connectivity between fluxonium qubits, by reducing capacitive loading when the coupler has a high impedance. After performing analytic and numerical analyses of the circuit Hamiltonian and gate dynamics, we tune circuit parameters to destructively interfere sources of coherent error, revealing an efficient fourth-order scaling of coherent error with gate duration. For component properties from the literature, we predict an open-system average controlled-Z () gate infidelity of 1.86×10−4 in 70 ns. Published by the American Physical Society 2024

Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions

Compared with time-independent Hamiltonians, the dynamics of generic quantum Hamiltonians H(t) are complicated by the presence of time ordering in the evolution operator. In the context of digital quantum simulation, this difficulty prevents a direct adaptation of many time-independent simulation algorithms for time-dependent simulation. However, there exists a framework within the theory of dynamical systems that eliminates time ordering by adding a “clock” degree of freedom. In this work, we provide a computational framework, based on this reduction, for encoding time-dependent dynamics as time-independent systems. As a result, we make two advances in digital Hamiltonian simulation. First, we create a time-dependent simulation algorithm based on performing qubitization on the augmented clock system and, in doing so, provide the first qubitization-based approach to time-dependent Hamiltonians that goes beyond Trotterization of the ordered exponential. Second, we define a natural generalization of multiproduct formulas for time-ordered exponentials and then propose and analyze an algorithm based on these formulas. Unlike other algorithms of similar accuracy, the multiproduct approach achieves commutator scaling, meaning that this method outperforms existing methods for physically local time-dependent Hamiltonians with sufficient smoothness. Our work reduces the disparity between time-dependent and time-independent simulation and indicates a step toward optimal quantum simulation of time-dependent Hamiltonians. Published by the American Physical Society 2024

Learning Hamiltonian dynamics with reproducing kernel Hilbert spaces and random features

A method for learning Hamiltonian dynamics from a limited and noisy dataset is proposed. The method learns a Hamiltonian vector field on a reproducing kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in particular, odd Hamiltonian vector fields. This is done with a symplectic kernel, and it is shown how the kernel can be modified to an odd symplectic kernel to impose the odd symmetry. A random feature approximation is developed for the proposed odd kernel to reduce the problem size. The performance of the method is validated in simulations for three Hamiltonian systems. It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.

Constrained Hamiltonian dynamics for electrons in magnetic field and additional forces besides the Lorentz force acting on electrons

Abstract We consider the forces acting on electrons in magnetic field including the constraints and a condition arising from quantum mechanics. The force is calculated as the electron mass, m e , multiplied by the total time-derivative of the velocity field evaluated using the quantum mechanical many-electron wave function. The velocity field includes a term of the Berry connection from the many-body wave function; thereby, quantum mechanical effects are included. It is shown that additional important forces besides the Lorentz force exist; they include the gradient of the electron velocity field kinetic energy, the gradient of the chemical potential, and the ‘force’ for producing topologically protected loop currents. These additional forces are shown to be important in superconductivity, electric current in metallic wires, and charging of capacitors.

Tuning diagonal scale matrices for HMC

Three approaches for adaptively tuning diagonal scale matrices for HMC are discussed and compared. The common practice of scaling according to estimated marginal standard deviations is taken as a benchmark. Scaling according to the mean log-target gradient (ISG), and a scaling method targeting that the frequency of when the underlying Hamiltonian dynamics crosses the respective medians should be uniform across dimensions, are taken as alternatives. Numerical studies suggest that the ISG method leads in many cases to more efficient sampling than the benchmark, in particular in cases with strong correlations or non-linear dependencies. The ISG method is also easy to implement, computationally cheap and would be relatively simple to include in automatically tuned codes as an alternative to the benchmark practice.

From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review.

In this review work, we outline a conceptual path that, starting from the numerical investigation of the transition between weak chaos and strong chaos in Hamiltonian systems with many degrees of freedom, comes to highlight how, at the basis of equilibrium phase transitions, there must be major changes in the topology of submanifolds of the phase space of Hamiltonian systems that describe systems that exhibit phase transitions. In fact, the numerical investigation of Hamiltonian flows of a large number of degrees of freedom that undergo a thermodynamic phase transition has revealed peculiar dynamical signatures detected through the energy dependence of the largest Lyapunov exponent, that is, of the degree of chaoticity of the dynamics at the phase transition point. The geometrization of Hamiltonian flows in terms of geodesic flows on suitably defined Riemannian manifolds, used to explain the origin of deterministic chaos, combined with the investigation of the dynamical counterpart of phase transitions unveils peculiar geometrical changes of the mechanical manifolds in correspondence to the peculiar dynamical changes at the phase transition point. Then, it turns out that these peculiar geometrical changes are the effect of deeper topological changes of the configuration space hypersurfaces ∑v=VN-1(v) as well as of the manifolds {Mv=VN-1((-∞,v])}v∈R bounded by the ∑v. In other words, denoting by vc the critical value of the average potential energy density at which the phase transition takes place, the members of the family {∑v}v<vc are not diffeomorphic to those of the family {∑v}v>vc; additionally, the members of the family {Mv}v>vc are not diffeomorphic to those of {Mv}v>vc. The topological theory of the deep origin of phase transitions allows a unifying framework to tackle phase transitions that may or may not be due to a symmetry-breaking phenomenon (that is, with or without an order parameter) and to finite/small N systems.

BaHaMAs: a method for uncertainty quantification in geodetic time series and its application in short-term prediction of length of day

Some of the important geodetic time series used in various Earth science disciplines are provided without uncertainty estimates. This can affect the validity of conclusions based on such data. However, an efficient uncertainty quantification algorithm to tackle this problem is currently not available. Here we present a methodology to approximate the aleatoric uncertainty in time series, called Bayesian Hamiltonian Monte Carlo Autoencoders (BaHaMAs). BaHaMAs is based on three elements: (1) self-supervised autoencoders that learn the underlying structure of the time series, (2) Bayesian machine learning that accurately quantifies the data uncertainty, and (3) Monte Carlo sampling that follows the Hamiltonian dynamics. The method can be applied in various fields in the Earth sciences. As an example, we focus on Atmospheric and Oceanic Angular Momentum time series (AAM and OAM, respectively), which are typically provided without uncertainty information. We apply our methodology to 3-hourly AAM and OAM time series and quantify the uncertainty in the data from 1976 up to the end of 2022. Furthermore, since Length of Day (LOD) is a geodetic time series that is closely connected to AAM and OAM and its short-term prediction is important for various space-geodetic applications, we show that the use of the derived uncertainties alongside the time series of AAM and OAM improves the prediction performance of LOD on average by 17% for different time spans. Finally, a comparison with alternative uncertainty quantification baseline methods, i.e., variational autoencoders and deep ensembles, reveals that BaHaMAs is more accurate in quantifying uncertainty.Graphical

Intrinsic statistical regularity of topological charges revealed in dynamical disk model.

Identifying ordered structures hidden in the packings of particles is a common scientific question in multiple fields. In this work, we investigate the dynamical organizations of a large number of initially randomly packed repulsive particles confined on a disk under the Hamiltonian dynamics by the recently developed algorithm called the random batch method. This algorithm is specifically designed for reducing the computational complexity of long-range interacting particle systems. We highlight the revealed intrinsic statistical regularity of topological charges that is otherwise unattainable by the continuum analysis of particle density. We also identify distinct collective dynamics of the interacting particles under short- and long-range repulsive forces. This work shows the robustness and effectiveness of the concept of topological charge for characterizing the convoluted particle dynamics, and itdemonstrates the promising potential of the random batch method for exploring fundamental scientific questions arising in a variety of long-range interacting particle systems in soft matter physics and other relevant fields.