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The field of neural quantum states has recently experienced a tremendous progress, making them a competitive tool of computational quantum many-body physics. However, their largest achievements to date mostly concern interacting spin systems, while their utility for quantum chemistry remains yet to be demonstrated. Two main complications are the peaked structure of the molecular wave functions, which impedes sampling, and large number of terms in second quantised Hamiltonians, which hinders scaling to larger molecule sizes. In this paper we address these issues jointly and argue that the peaked structure might actually be key to drastically more efficient calculations. Specifically, we introduce a novel algorithm for autoregressive sampling without replacement and a procedure to calculate a computationally cheaper surrogate for the local energy. We complement them with a custom modification of the stochastic reconfiguration optimisation technique and a highly optimised GPU implementation. As a result, our calculations require substantially less resources and exhibit more than order of magnitude speedup compared to the previous works. On a single GPU we study molecules comprising up to 118 qubits and outperform the ``golden standard'' CCSD(T) benchmark in Hilbert spaces of $\sim 10^{15}$ Slater determinants, which is orders of magnitude larger than what was previously achieved. We believe that our work underscores the prospect of NQS for challenging quantum chemistry calculations and serves as a favourable ground for the future method development., Comment: Main text: 15 pages, 5 figures; Supplementary Material: 20 pages, 23 figures

The utility of a quantum computer depends heavily on the ability to reliably perform accurate quantum logic operations. For finding optimal control solutions, it is of particular interest to explore model-free approaches, since their quality is not constrained by the limited accuracy of theoretical models for the quantum processor - in contrast to many established gate implementation strategies. In this work, we utilize a continuous-control reinforcement learning algorithm to design entangling two-qubit gates for superconducting qubits; specifically, our agent constructs cross-resonance and CNOT gates without any prior information about the physical system. Using a simulated environment of fixed-frequency, fixed-coupling transmon qubits, we demonstrate the capability to generate novel pulse sequences that outperform the standard cross-resonance gates in both fidelity and gate duration, while maintaining a comparable susceptibility to stochastic unitary noise. We further showcase an augmentation in training and input information that allows our agent to adapt its pulse design abilities to drifting hardware characteristics, importantly with little to no additional optimization. Our results exhibit clearly the advantages of unbiased adaptive-feedback learning-based optimization methods for transmon gate design., Comment: 18 + 8 pages, 13 + 6 figures

Reconstructing Hamiltonians from local measurements is key to enabling reliable quantum simulation: both validating the implemented model, and identifying any left-over terms with sufficient precision is a problem of increasing importance. Here we propose a deep-learning-assisted variational algorithm for Hamiltonian reconstruction by pre-processing a dataset that is diagnosed to contain thermal measurements of local operators. We demonstrate the efficient and precise reconstruction of local Hamiltonians, while long-range interacting Hamiltonians are reconstructed approximately. Away from equilibrium, for periodically and random multipolar driven systems, we reconstruct the effective Hamiltonian widely used for Floquet engineering of metastable steady states. Moreover, our approach allows us to reconstruct an effective quasilocal Hamiltonian even in the heating regime beyond the validity of the prethermal plateau, where perturbative expansions fail., Comment: 13 pages, 5 figures

Spectral functions are central to link experimental probes to theoretical models in condensed matter physics. However, performing exact numerical calculations for interacting quantum matter has remained a key challenge especially beyond one spatial dimension. In this work, we develop a versatile approach using neural quantum states to obtain spectral properties based on simulations of the dynamics of excitations initially localized in real or momentum space. We apply this approach to compute the dynamical structure factor in the vicinity of quantum critical points (QCPs) of different two-dimensional quantum Ising models, including one that describes the complex density wave orders of Rydberg atom arrays. When combined with deep network architectures we find that our method reliably describes dynamical structure factors of arrays with up to $24\times24$ spins, including the diverging time scales at critical points. Our approach is broadly applicable to interacting quantum lattice models in two dimensions and consequently opens up a route to compute spectral properties of correlated quantum matter in yet inaccessible regimes., Comment: Published version

Neural quantum states are a new family of variational ans\"atze for quantum-many body wave functions with advantageous properties in the notoriously challenging case of two spatial dimensions. Since their introduction a wide variety of different network architectures has been employed to study paradigmatic models in quantum many-body physics with a particular focus on quantum spin models. Nonetheless, many questions remain about the effect that the choice of architecture has on the performance on a given task. In this work, we present a unified comparison of a selection of popular network architectures and symmetrization schemes employed for ground state searches of prototypical spin Hamiltonians, namely the two-dimensional transverse-field Ising model and the J1-J2 model. In the presence of a non-trivial sign structure of the ground states, we find that the details of symmetrization crucially influence the performance. We describe this effect in detail and discuss its consequences, especially for autoregressive models, as their direct sampling procedure is not compatible with the symmetrization procedure that we found to be optimal., Comment: 7 + 3 pages, 3 figures; Fixed references and a typo in table 1

The quantum Jarzynski equality and the Crooks relation are fundamental laws connecting equilibrium processes with nonequilibrium fluctuations. They are promising tools to benchmark quantum devices and measure free energy differences. While they are well established theoretically and also experimental realizations for few-body systems already exist, their experimental validity in the quantum many-body regime has not been observed so far. Here, we present results for nonequilibrium protocols in systems with up to sixteen interacting degrees of freedom obtained on trapped ion and superconducting qubit quantum computers, which test the quantum Jarzynski equality and the Crooks relation in the many-body regime. To achieve this, we overcome present-day limitations in the preparation of thermal ensembles and in the measurement of work distributions on noisy intermediate-scale quantum devices. We discuss the accuracy to which the Jarzynski equality holds on different quantum computing platforms subject to platform-specific errors. The analysis reveals the validity of Jarzynski's equality in a regime with energy dissipation, compensated for by a fast unitary drive. This provides new insights for analyzing errors in many-body quantum simulators., Comment: 21 pages, 14 figures

The introduction of Neural Quantum States (NQS) has recently given a new twist to variational Monte Carlo (VMC). The ability to systematically reduce the bias of the wave function ansatz renders the approach widely applicable. However, performant implementations are crucial to reach the numerical state of the art. Here, we present a Python codebase that supports arbitrary NQS architectures and model Hamiltonians. Additionally leveraging automatic differentiation, just-in-time compilation to accelerators, and distributed computing, it is designed to facilitate the composition of efficient NQS algorithms., Comment: 33 pages, 7 figures. Revised version. Code repository: https://github.com/markusschmitt/vmc_jax

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an {\it extended QKZM} taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators., Comment: 12 pages, 7 figures

We develop a variational approach to simulating the dynamics of open quantum many-body systems using deep autoregressive neural networks. The parameters of a compressed representation of a mixed quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. We illustrate our approach by solving the dissipative quantum Heisenberg model in one and two dimensions for up to 40 spins and by applying it to the simulation of confinement dynamics in the presence of dissipation., Comment: 7 + 5 pages, 3 figures

The vast complexity is a daunting property of generic quantum states that poses a significant challenge for theoretical treatment, especially in non-equilibrium 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., Comment: 6+6 pages, 3+6 figures

Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks, one for the amplitude and the other for the phase of the variational wavefunction. By introducing concrete mitigation strategies against inherent numerical instabilities in the stochastic reconfiguration algorithm we obtain a variational energy comparable to that reported recently with neural networks that incorporate knowledge about the physical system. Through a detailed analysis of the individual components of the algorithm, we conclude that the rugged nature of the energy landscape constitutes the major obstacle in finding a satisfactory approximation to the ground state wavefunction, and prevents learning the correct sign structure. In particular, we show that in the present setup the neural network expressivity and Monte Carlo sampling are not primary limiting factors.

Dynamics in correlated quantum matter is a hard problem, as its exact solution generally involves a computational effort that grows exponentially with the number of constituents. While a remarkable progress has been witnessed in recent years for one-dimensional systems, much less has been achieved for interacting quantum models in higher dimensions, since they incorporate an additional layer of complexity. In this work, we employ a variational method that allows for an efficient and controlled computation of the dynamics of quantum many-body systems in one and higher dimensions. The approach presented here introduces a variational class of wavefunctions based on complex networks of classical spins akin to artificial neural networks, which can be constructed in a controlled fashion. We provide a detailed prescription for such constructions and illustrate their performance by studying quantum quenches in one- and two-dimensional models. In particular, we investigate the nonequilibrium dynamics of a genuinely interacting two-dimensional lattice gauge theory, the quantum link model, for which we have recently shown -- employing the technique discussed thoroughly in this paper -- that it features disorder-free localization dynamics [P. Karpov et al., Phys. Rev. Lett. 126, 130401 (2021)]. The present work not only supplies a framework to address purely theoretical questions but also could be used to provide a theoretical description of experiments in quantum simulators, which have recently seen an increased effort targeting two-dimensional geometries. Importantly, our method can be applied to any quantum many-body system with a well-defined classical limit., Comment: 19 pages, 12 figures; version published in Physical Review B

By means of the discrete truncated Wigner approximation we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition these transitions separate a phase with nonvanishing magnetization along the ordering direction from a symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\propto 1/r^{\alpha}$ decaying algebraically as a function of distance $r$ and a two-dimensional system with short-range nearest-neighbour interactions. In the former case we identify dynamical phase transitions for $\alpha \lesssim 2$ and we extract the critical exponents from a data collapse of the steady state magnetization for up to 1200 lattice sites. We find identical exponents for $\alpha \lesssim 0.5$, suggesting that the dynamical transitions in this regime fall into the same universality class as the nonergodic mean-field limit. The two-dimensional Ising model is believed to be thermalizing, which we also confirm using exact diagonalization for small system sizes. Thus, the dynamical transition is expected to correspond to the thermal phase transition, which is consistent with our data upon comparing to equilibrium quantum Monte-Carlo simulations. We further test the accuracy of the discrete truncated Wigner approximation by comparing against numerically exact methods such as exact diagonalization, tensor network as well as artificial neural network states and we find good quantitative agreement on the accessible time scales. Finally, our work provides an additional contribution to the understanding of the range and the limitations of qualitative and quantitative applicability of the discrete truncated Wigner approximation., Comment: 16 pages, 19 figures, version published in Physical Review B

The efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter constitutes a key challenge for current computational methods. This holds in particular in the regime of two spatial dimensions, whose experimental exploration is currently pursued with strong efforts in quantum simulators. In this work we present a versatile and efficient machine learning inspired approach based on a recently introduced artificial neural network encoding of quantum many-body wave functions. We identify and resolve some key challenges for the simulation of time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, we study the dynamics of the paradigmatic two-dimensional transverse field Ising model, as recently also realized experimentally in systems of Rydberg atoms. Calculating the nonequilibrium real-time evolution across a broad range of parameters, we, for instance, observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached time scales are comparable to or exceed the capabilities of state-of-the-art tensor network methods., Comment: 6+10 pages, 2+9 figures

Studying the zeroes of partition functions in the space of complex control parameters allows to understand formally how critical behavior of a many-body system can arise in the thermodynamic limit despite various no-go theorems for finite systems. In this work we propose protocols that can be realized in quantum simulators to measure the location of complex partition function zeroes of classical Ising models. The protocols are solely based on the implementation of simple two-qubit gates, local spin rotations, and projective measurements along two orthogonal quantization axes. Besides presenting numerical simulations of the measurement outcomes for an exemplary classical model, we discuss the effect of projection noise and the feasibility of the implementation on state of the art platforms for quantum simulation., Comment: 8 pages, 4 figures

For the characterization of the dynamics in quantum many-body systems the question how information spreads and becomes distributed over the constituent degrees of freedom is of fundamental interest. The delocalization of information under many-body dynamics has been dubbed scrambling and out-of-time-order correlators were proposed to probe this behavior. In this work we investigate the time-evolution of tripartite information as a natural operator-independent measure of scrambling, which quantifies to which extent the initially localized information can only be recovered by global measurements. Studying the dynamics of quantum lattice models with tunable integrability breaking we demonstrate that in contrast to quadratic models generic interacting systems scramble information irrespective of the chosen partitioning of the Hilbert space, which justifies the characterization as scrambler. Without interactions the dynamics of tripartite information in momentum space reveals unambiguously the absence of scrambling., Comment: 9 pages, 7 figures. Major revision of the original version

The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows to identify a Lyapunov exponent $\lambda_L$. In particular, we find that $\lambda_L$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator., Comment: 6+7 pages, 3+7 figures

Irreversibility, despite being a necessary condition for thermalization, still lacks a sound understanding in the context of isolated quantum many-body systems. In this work we approach this question by studying the behavior of generic many-body systems under imperfect effective time reversal, where the imperfection is introduced as a perturbation of the many-body state at the point of time reversal. Based on numerical simulations of the full quantum dynamics we demonstrate that observable echos occurring in this setting decay exponentially with a rate that is independent of the perturbation; hence, the sensitivity to perturbations is intrinsic to the system meaning that the dynamics is effectively irreversible.

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.

We study the Hall conductance in a Floquet topological insulator in the long time limit after sudden switches of the driving amplitude. Based on a high frequency expansion of the effective Hamiltonian and the micromotion operator we demonstrate that the Hall conductance as function of the driving amplitude follows universal non-analytic laws close to phase transitions that are related to conic gap closing points, namely a logarithmic divergence for gapped initial states and jumps of a definite height for gapless initial states. This constitutes a generalization of the results known for the static systems to the driven case., Comment: 17 pages, 5 figures; added discussion of partially filled bands as initial states and effect of weak disorder

The question of thermalisation in closed quantum many-body systems has received a lot of attention in the past few years. An intimately related question is whether a closed quantum system shows irreversible dynamics. However, irreversibility and what we actually mean by this in a quantum many-body system with unitary dynamics has been explored very little. In this work we investigate the dynamics of the Ising model in a transverse magnetic field involving an imperfect effective time reversal. We propose a definition of irreversibility based on the echo peak decay of observables. Inducing the effective time reversal by different protocols we find algebraic decay of the echo peak heights or an ever persisting echo peak indicating that the dynamics in this model is well reversible.

We study the Hall conductance of a Chern insulator after a global quench of the Hamiltonian. The Hall conductance in the long time limit is obtained by applying the linear response theory to the diagonal ensemble. It is expressed as the integral of the Berry curvature weighted by the occupation number over the Brillouin zone. We identify a topologically driven nonequilibrium phase transition, which is indicated by the nonanalyticity of the Hall conductance as a function of the energy gap m_f in the post-quench Hamiltonian H_f. The topological invariant for the quenched state is the winding number of the Green's function W, which equals the Chern number for the ground state of H_f. In the limit that m_f goes to zero, the derivative of the Hall conductance with respect to m_f is proportional to ln(|m_f|), with the constant of proportionality being the ratio of the change of W at m_f = 0 to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators such as the Dirac model, the Haldane model, or the Kitaev honeycomb model in the fermionic basis., Comment: 18 pages, 8 figures

The notion of a dynamical quantum phase transition (DQPT) was recently introduced in [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)] as the non-analytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model are studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS-type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the non-degenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.