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Dynamic quantum circuits (DQCs) incorporate mid-circuit measurements and gates conditioned on these measurement outcomes. DQCs can prepare certain long-range entangled states in constant depth, making them a promising route to preparing complex quantum states on devices with a limited coherence time. Almost all constructions of DQCs for state preparation have been formulated analytically, relying on special structure in the target states. In this work, we approach the problem of state preparation variationally, developing scalable tensor network algorithms which find high-fidelity DQC preparations for generic states. We apply our algorithms to critical states, random matrix product states, and subset states. We consider both DQCs with a fixed number of ancillae and those with an extensive number of ancillae. Even in the few ancillae regime, the DQCs discovered by our algorithms consistently prepare states with lower infidelity than a static quantum circuit of the same depth. Notably, we observe constant fidelity gains across system sizes and circuit depths. For DQCs with an extensive number of ancillae, we introduce scalable methods for decoding measurement outcomes, including a neural network decoder and a real-time decoding protocol. Our work demonstrates the power of an algorithmic approach to generating DQC circuits, broadening their scope of applications to new areas of quantum computing.

The ability to perform ab initio molecular dynamics simulations using potential energies calculated on quantum computers would allow virtually exact dynamics for chemical and biochemical systems, with substantial impacts on the fields of catalysis and biophysics. However, noisy hardware, the costs of computing gradients, and the number of qubits required to simulate large systems present major challenges to realizing the potential of dynamical simulations using quantum hardware. Here, we demonstrate that some of these issues can be mitigated by recent advances in machine learning. By combining transfer learning with techniques for building machine-learned potential energy surfaces, we propose a new path forward for molecular dynamics simulations on quantum hardware. We use transfer learning to reduce the number of energy evaluations that use quantum hardware by first training models on larger, less accurate classical datasets and then refining them on smaller, more accurate quantum datasets. We demonstrate this approach by training machine learning models to predict a molecule's potential energy using Behler-Parrinello neural networks. When successfully trained, the model enables energy gradient predictions necessary for dynamics simulations that cannot be readily obtained directly from quantum hardware. To reduce the quantum resources needed, the model is initially trained with data derived from low-cost techniques, such as Density Functional Theory, and subsequently refined with a smaller dataset obtained from the optimization of the Unitary Coupled Cluster ansatz. We show that this approach significantly reduces the size of the quantum training dataset while capturing the high accuracies needed for quantum chemistry simulations., Comment: 1- pages, 12 figures

Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose a novel and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one and two dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. Furthermore, we analyze the computational cost and provide both analytical and numerical evidences for it to be polynomial in system size., Comment: 6 pages, 4 figures + 7 pages, 5 figures

Variational Quantum Eigensolvers (VQE) are a promising approach for finding the classically intractable ground state of a Hamiltonian. The Unitary Block Optimization Scheme (UBOS) is a state-of-the-art VQE method which works by sweeping over gates and finding optimal parameters for each gate in the environment of other gates. UBOS improves the convergence time to the ground state by an order of magnitude over Stochastic Gradient Descent (SGD). It nonetheless suffers in both rate of convergence and final converged energies in the face of highly noisy expectation values coming from shot noise. Here we develop two classical post-processing techniques which improve UBOS especially when measurements have large noise. Using Gaussian Process Regression (GPR), we generate artificial augmented data using original data from the quantum computer to reduce the overall error when solving for the improved parameters. Using Double Robust Optimization plus Rejection (DROPR), we prevent outlying data which are atypically noisy from resulting in a particularly erroneous single optimization step thereby increasing robustness against noisy measurements. Combining these techniques further reduces the final relative error that UBOS reaches by a factor of three without adding additional quantum measurement or sampling overhead. This work further demonstrates that developing techniques which use classical resources to post-process quantum measurement results can significantly improve VQE algorithms., Comment: 18 pages, 11 figures

Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it was shown that the AKLT state -- a paradigmatic example of an MPS -- can be exactly prepared with an adaptive quantum circuit of constant-depth, an impossible feat with local unitary gates due to its nonzero correlation length [Smith et al., PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming optimal preparation protocols that rely on unitary circuits alone. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states., Comment: 25 pages, 5 figures

The ground state of second-quantized quantum chemistry Hamiltonians provides access to an important set of chemical properties. Wavefunctions based on ML architectures have shown promise in approximating these ground states in a variety of physical systems. In this work, we show how to achieve state-of-the-art energies for molecular Hamiltonians using the the neural network backflow wave-function. To accomplish this, we optimize this ansatz with a variant of the deterministic optimization scheme based on SCI introduced by [Li, et. al JCTC (2023)] which we find works better than standard MCMC sampling. For the molecules we studied, NNBF gives lower energy states than both CCSD and other neural network quantum states. We systematically explore the role of network size as well as optimization parameters in improving the energy. We find that while the number of hidden layers and determinants play a minor role in improving the energy, there is significant improvements in the energy from increasing the number of hidden units as well as the batch size used in optimization with the batch size playing a more important role.

High-coherence qubits, which can store and manipulate quantum states for long times with low error rates, are necessary building blocks for quantum computers. We propose a superconducting qubit architecture that uses a Floquet flux drive to modify the spectrum of a static fluxonium molecule. The computational eigenstates have two key properties: disjoint support to minimize bit flips, along with first- and second-order insensitivity to flux noise dephasing. The rates of the three main error types are estimated through numerical simulations, with predicted coherence times of approximately 50 ms in the computational subspace and erasure lifetimes of about 500 $\mu$s. We give a protocol for high-fidelity single qubit rotation gates via additional flux modulation on timescales of roughly 500 ns. Our results indicate that driven qubits are able to outperform some of their static counterparts., Comment: 15 pages, 8 figures. Fixed error in Figure 2(a); added citation

Among the variational wave functions for Fermionic Hamiltonians, neural network backflow (NNBF) and hidden fermion determinant states (HFDS) are two prominent classes to provide accurate approximations to the ground state. Here we develop a unifying view of fermionic neural quantum states casting them all in the framework of NNBF. NNBF wave-functions have configuration-dependent single-particle orbitals (SPO) which are parameterized by a neural network. We show that HFDS with $r$ hidden fermions can be written as a NNBF with an $r \times r$ determinant Jastrow and a restricted low-rank $r$ additive correction to the SPO. Furthermore, we show that in NNBF wave-functions, such determinant Jastrow's can generically be removed at the cost of further complicating the additive SPO correction increasing its rank by $r$. We numerically and analytically compare additive SPO corrections generated by the product of two matrices with inner dimension $r$. We find that larger $r$ wave-functions span a larger space and give evidence that simpler and more direct updates to the SPO's tend to be more expressive and better energetically. These suggest the standard NNBF approach is preferred amongst other related choices. Finally, we uncover that the row-selection used to select single-particle orbitals allows significant sign and amplitude modulation between nearby configurations and is partially responsible for the quality of NNBF and HFDS wave-functions., Comment: 19 pages, 11 figures

Confocal fluorescence microscopy is widely applied for the study of point-like emitters such as biomolecules, material defects, and quantum light sources. Confocal techniques offer increased optical resolution, dramatic fluorescence background rejection and sub-nanometer localization, useful in super-resolution imaging of fluorescent biomarkers, single-molecule tracking, or the characterization of quantum emitters. However, rapid, noise-robust automated 3D focusing on point-like emitters has been missing for confocal microscopes. Here, we introduce FiND (Focusing in Noisy Domain), an imaging-free, non-trained 3D focusing framework that requires no hardware add-ons or modifications. FiND achieves focusing for signal-to-noise ratios down to 1, with a few-shot operation for signal-to-noise ratios above 5. FiND enables unsupervised, large-scale focusing on a heterogeneous set of quantum emitters. Additionally, we demonstrate the potential of FiND for real-time 3D tracking by following the drift trajectory of a single NV center indefinitely with a positional precision of < 10 nm. Our results show that FiND is a useful focusing framework for the scalable analysis of point-like emitters in biology, material science, and quantum optics., Comment: 17 pages, 7 figures

Unitary t-designs are distributions on the unitary group whose first t moments appear maximally random. Previous work has established several upper bounds on the depths at which certain specific random quantum circuit ensembles approximate t-designs. Here we show that these bounds can be extended to any fixed architecture of Haar-random two-site gates. This is accomplished by relating the spectral gaps of such architectures to those of 1D brickwork architectures. Our bound depends on the details of the architecture only via the typical number of layers needed for a block of the circuit to form a connected graph over the sites. When this quantity is independent of width, the circuit forms an approximate t-design in linear depth. We also give an implicit bound for nondeterministic architectures in terms of properties of the corresponding distribution over fixed architectures., Comment: 29 pages, 8 figures

The variational quantum eigensolver (VQE) is a promising algorithm for demonstrating quantum advantage in the noisy intermediate-scale quantum (NISQ) era. However, optimizing VQE from random initial starting parameters is challenging due to a variety of issues including barren plateaus, optimization in the presence of noise, and slow convergence. While simulating quantum circuits classically is generically difficult, classical computing methods have been developed extensively, and powerful tools now exist to approximately simulate quantum circuits. This opens up various strategies that limit the amount of optimization that needs to be performed on quantum hardware. Here we present and benchmark an approach where we find good starting parameters for parameterized quantum circuits by classically simulating VQE by approximating the parameterized quantum circuit (PQC) as a matrix product state (MPS) with a limited bond dimension. Calling this approach the variational tensor network eigensolver (VTNE), we apply it to the 1D and 2D Fermi-Hubbard model with system sizes that use up to 32 qubits. We find that in 1D, VTNE can find parameters for PQC whose energy error is within 0.5% relative to the ground state. In 2D, the parameters that VTNE finds have significantly lower energy than their starting configurations, and we show that starting VQE from these parameters requires non-trivially fewer operations to come down to a given energy. The higher the bond dimension we use in VTNE, the less work needs to be done in VQE. By generating classically optimized parameters as the initialization for the quantum circuit one can alleviate many of the challenges that plague VQE on quantum computers., Comment: 10 pages

In this paper, we describe a tensor network simulation of a neutral atom quantum system under the presence of noise, while introducing a new purity-preserving truncation technique that compromises between the simplicity of the matrix product state and the positivity of the matrix product density operator. We apply this simulation to a near-optimized iteration of the quantum approximate optimization algorithm on a transverse field Ising model in order to investigate the influence of large system sizes on the performance of the algorithm. We find that while circuits with a large number of qubits fail more often under noise that depletes the qubit population, their outputs on a successful measurement are just as robust under Rydberg atom dissipation or qubit dephasing as smaller systems. However, such circuits might not perform as well under coherent multi-qubit errors such as Rydberg atom crosstalk. We also find that the optimized parameters are especially robust to noise, suggesting that a noisier quantum system can be used to find the optimal parameters before switching to a cleaner system for measurements of observables., Comment: 14 pages, 11 figures

We analyze functionals that characterize the distribution of eigenstates in Fock space through a tool derived from algebraic topology: persistent homology. Drawing on recent generalizations of the localization landscape applicable to mid-spectrum eigenstates, we introduce several novel persistent homology observables in the context of many-body localization that exhibit transitional behavior near the critical point. We demonstrate that the persistent homology approach to localization landscapes and, in general, functionals on the Fock space lattice offer insights into the structure of eigenstates unobtainable by traditional means.

The rise of automation and machine learning (ML) in electron microscopy has the potential to revolutionize materials research through autonomous data collection and processing. A significant challenge lies in developing ML models that rapidly generalize to large data sets under varying experimental conditions. We address this by employing a cycle generative adversarial network (CycleGAN) with a reciprocal space discriminator, which augments simulated data with realistic spatial frequency information. This allows the CycleGAN to generate images nearly indistinguishable from real data and provide labels for ML applications. We showcase our approach by training a fully convolutional network (FCN) to identify single atom defects in a 4.5 million atom data set, collected using automated acquisition in an aberration-corrected scanning transmission electron microscope (STEM). Our method produces adaptable FCNs that can adjust to dynamically changing experimental variables with minimal intervention, marking a crucial step towards fully autonomous harnessing of microscopy big data., Comment: 25 pages, 6 figures, 2 tables

We present a neural flow wavefunction, Gauge-Fermion FlowNet, and use it to simulate 2+1D lattice compact quantum electrodynamics with finite density dynamical fermions. The gauge field is represented by a neural network which parameterizes a discretized flow-based transformation of the amplitude while the fermionic sign structure is represented by a neural net backflow. This approach directly represents the $U(1)$ degree of freedom without any truncation, obeys Guass's law by construction, samples autoregressively avoiding any equilibration time, and variationally simulates Gauge-Fermion systems with sign problems accurately. In this model, we investigate confinement and string breaking phenomena in different fermion density and hopping regimes. We study the phase transition from the charge crystal phase to the vacuum phase at zero density, and observe the phase seperation and the net charge penetration blocking effect under magnetic interaction at finite density. In addition, we investigate a magnetic phase transition due to the competition effect between the kinetic energy of fermions and the magnetic energy of the gauge field. With our method, we further note potential differences on the order of the phase transitions between a continuous $U(1)$ system and one with finite truncation. Our state-of-the-art neural network approach opens up new possibilities to study different gauge theories coupled to dynamical matter in higher dimensions.

Gauge Theory plays a crucial role in many areas in science, including high energy physics, condensed matter physics and quantum information science. In quantum simulations of lattice gauge theory, an important step is to construct a wave function that obeys gauge symmetry. In this paper, we have developed gauge equivariant neural network wave function techniques for simulating continuous-variable quantum lattice gauge theories in the Hamiltonian formulation. We have applied the gauge equivariant neural network approach to find the ground state of 2+1-dimensional lattice gauge theory with U(1) gauge group using variational Monte Carlo. We have benchmarked our approach against the state-of-the-art complex Gaussian wave functions, demonstrating improved performance in the strong coupling regime and comparable results in the weak coupling regime.

We probe the bulk geometry of the Wegner Wilson Flow (WWF) in the context of many-body localization, by addressing efficiency and bulk entanglement growth measures through approximating upper bounds on the boundary entanglement entropy. We connect these upper bounds to the Fubini-Study metric and clarify how a central quantity, the information fluctuation complexity, distinguishes bulk unitary rotation from entanglement production. We also give a short new proof of the small incremental entangling theorem in the absence of ancillas, achieving a dimension-independent, universal factor of $c= 2$.

Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects., Comment: 19 pages, 9 figures

Quantum process tomography is a powerful tool for understanding quantum channels and characterizing properties of quantum devices. Inspired by recent advances using classical shadows in quantum state tomography [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020).], we have developed ShadowQPT, a classical shadow method for quantum process tomography. We introduce two related formulations with and without ancilla qubits. ShadowQPT stochastically reconstructs the Choi matrix of the device allowing for an a-posteri classical evaluation of the device on arbitrary inputs with respect to arbitrary outputs. Using shadows we then show how to compute overlaps, generate all $k$-weight reduced processes, and perform reconstruction via Hamiltonian learning. These latter two tasks are efficient for large systems as the number of quantum measurements needed scales only logarithmically with the number of qubits. A number of additional approximations and improvements are developed including the use of a pair-factorized Clifford shadow and a series of post-processing techniques which significantly enhance the accuracy for recovering the quantum channel. We have implemented ShadowQPT using both Pauli and Clifford measurements on the IonQ trapped ion quantum computer for quantum processes up to $n=4$ qubits and achieved good performance., Comment: Revised final version

Solving for quantum ground states is important for understanding the properties of quantum many-body systems, and quantum computers are potentially well-suited for solving for quantum ground states. Recent work has presented a nearly optimal scheme that prepares ground states on a quantum computer for completely generic Hamiltonians, whose query complexity scales as $\delta^{-1}$, i.e. inversely with their normalized gap. Here we consider instead the ground state preparation problem restricted to a special subset of Hamiltonians, which includes those which we term "nearly-frustration-free": the class of Hamiltonians for which the ground state energy of their block-encoded and hence normalized Hamiltonian $\alpha^{-1}H$ is within $\delta^y$ of -1, where $\delta$ is the spectral gap of $\alpha^{-1}H$ and $0 \leq y \leq 1$. For this subclass, we describe an algorithm whose dependence on the gap is asymptotically better, scaling as $\delta^{y/2-1}$, and show that this new dependence is optimal up to factors of $\log \delta$. In addition, we give examples of physically motivated Hamiltonians which live in this subclass. Finally, we describe an extension of this method which allows the preparation of excited states both for generic Hamiltonians as well as, at a similar speedup as the ground state case, for those which are nearly frustration-free., Comment: 13 pages, 1 figure

The variational quantum eigensolver (VQE) combines classical and quantum resources in order simulate classically intractable quantum states. Amongst other variables, successful VQE depends on the choice of variational ansatz for a problem Hamiltonian. We give a detailed description of a quantum circuit version of the 2D isometric tensor network (isoTNS) ansatz which we call qisoTNS. We benchmark the performance of qisoTNS on two different 2D spin 1/2 Hamiltonians. We find that the ansatz has several advantages. It is qubit efficient with the number of qubits allowing for access to some exponentially large bond-dimension tensors at polynomial quantum cost. In addition, the ansatz is robust to the barren plateau problem due emergent layerwise training. We further explore the effect of noise on the efficacy of the ansatz. Overall, we find that qisoTNS is a suitable variational ansatz for 2D Hamiltonians with local interactions., Comment: 9 pages, 9 figures

Entanglement is a key quantum phenomena and understanding transitions between phases of matter with different entanglement properties are an interesting probe of quantum mechanics. We numerically study a model of a 2D tensor network proposed to have an entanglement entropy transition first considered by Vasseur et al.[Phys. Rev. B 100, 134203 (2019)]. We find that by varying the bond dimension of the tensors in the network we can observe a transition between an area and volume phase with a logarithmic critical point around $D\approx 2$. We further characterize the critical behavior measuring a critical exponent using entanglement entropy and the tripartite quantum mutual information, observe a crossover from a `nearly pure' to entangled area law phase using the the distributions of the entanglement entropy and find a cubic decay of the pairwise mutual information at the transition. We further consider the dependence of these observables for different R\'enyi entropy. This work helps further validate and characterize random tensor networks as a paradigmatic examples of an entanglement transition.

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr\"{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

A topological $\theta$-term in gauge theories, including quantum chromodynamics in 3+1 dimensions, gives rise to a sign problem that makes classical Monte Carlo simulations impractical. Quantum simulations are not subject to such sign problems and are a promising approach to studying these theories in the future. In the near term, it is interesting to study simpler models that retain some of the physical phenomena of interest and their implementation on quantum hardware. For example, dimensionally-reducing gauge theories on small spatial tori produces quantum mechanical models which, despite being relatively simple to solve, retain interesting vacuum and symmetry structures from the parent gauge theories. Here we consider quantum mechanical particle-on-a-circle models, related by dimensional reduction to the 1+1d Schwinger model, that possess a $\theta$-term and realize an 't Hooft anomaly or global inconsistency at $\theta = \pi$. These models also exhibit the related phenomena of spontaneous symmetry breaking and instanton-anti-instanton interference in real time. We propose an experimental scheme for the real-time simulation of a particle on a circle with a $\theta$-term and a $\mathbb{Z}_n$ potential using a synthetic dimension encoded in a Rydberg atom. Simulating the Rydberg atom with realistic experimental parameters, we demonstrate that the essential physics can be well-captured by the experiment, with expected behavior in the tunneling rate as a function of $\theta$. Similar phenomena and observables can also arise in more complex quantum mechanical models connected to higher-dimensional nonabelian gauge theories by dimensional reduction., Comment: 25 pages, 11 figures; minor changes and refs added, version accepted for publication in PRD

We generally expect quantum systems to thermalize and satisfy the eigenstate thermalization hypothesis (ETH), which states that finite energy density eigenstates are thermal. However, some systems, such as many-body localized systems and systems with quantum many-body scars, violate ETH and have high-energy athermal eigenstates. In systems with scars, most eigenstates thermalize, but a few atypical scar states do not. Scar states can give rise to a periodic revival when time-evolving particular initial product states, which can be detected experimentally. Recently, a family of spin Hamiltonians was found with magnetically ordered 3-colored eigenstates that are quantum many-body scars [Lee et al. Phys. Rev. B 101, 241111(2020)]. These models can be realized in any lattice that can be tiled by triangles, such as the triangular or kagome lattices, and have been shown to have close connections to the physics of quantum spin liquids in the Heisenberg kagome antiferromagnet. In this work, we introduce a generalized family of $n$-colored Hamiltonians with "spiral colored" eigenstates made from $n$-spin motifs such as polygons or polyhedra. We show how these models can be realized in many different lattice geometries and provide numerical evidence that they can exhibit quantum many-body scars with periodic revivals that can be observed by time-evolving simple product states. The simple structure of these Hamiltonians makes them promising candidates for future experimental studies of quantum many-body scars., Comment: 10 pages, 11 figures, 1 table

Variational quantum algorithms are a promising hybrid framework for solving chemistry and physics problems with broad applicability to optimization as well. They are particularly well suited for noisy intermediate scale quantum (NISQ) computers. In this paper, we describe the unitary block optimization scheme (UBOS) and apply it to two variational quantum algorithms: the variational quantum eigensolver (VQE) and variational time evolution. The goal of VQE is to optimize a classically intractable parameterized quantum wave function to target a physical state of a Hamiltonian or solve an optimization problem. UBOS is an alternative to other VQE optimization schemes with a number of advantages including fast convergence, less sensitivity to barren plateaus, the ability to tunnel through some local minima and no hyperparameters to tune. We additionally describe how UBOS applies to real and imaginary time-evolution (TUBOS)., Comment: 17 pages, 11 figures

Symmetries such as gauge invariance and anyonic symmetry play a crucial role in quantum many-body physics. We develop a general approach to constructing gauge invariant or anyonic symmetric autoregressive neural network quantum states, including a wide range of architectures such as Transformer and recurrent neural network (RNN), for quantum lattice models. These networks can be efficiently sampled and explicitly obey gauge symmetries or anyonic constraint. We prove that our methods can provide exact representation for the ground and excited states of the 2D and 3D toric codes, and the X-cube fracton model. We variationally optimize our symmetry incorporated autoregressive neural networks for ground states as well as real-time dynamics for a variety of models. We simulate the dynamics and the ground states of the quantum link model of $\text{U(1)}$ lattice gauge theory, obtain the phase diagram for the 2D $\mathbb{Z}_2$ gauge theory, determine the phase transition and the central charge of the $\text{SU(2)}_3$ anyonic chain, and also compute the ground state energy of the SU(2) invariant Heisenberg spin chain. Our approach provides powerful tools for exploring condensed matter physics, high energy physics and quantum information science.

Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Zd gauge group on different geometries. Focusing on the special case of Z2 gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wavefunction for the Z2 theory away from the exactly solvable limit, and to demonstrate the confining/deconfining phase transition of the Wilson loop order parameter.

The theory of open quantum systems lays the foundations for a substantial part of modern research in quantum science and engineering. Rooted in the dimensionality of their extended Hilbert spaces, the high computational complexity of simulating open quantum systems calls for the development of strategies to approximate their dynamics. In this paper, we present an approach for tackling open quantum system dynamics. Using an exact probabilistic formulation of quantum physics based on positive operator-valued measure (POVM), we compactly represent quantum states with autoregressive transformer neural networks; such networks bring significant algorithmic flexibility due to efficient exact sampling and tractable density. We further introduce the concept of String States to partially restore the symmetry of the autoregressive transformer neural network and improve the description of local correlations. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator using a forward-backward trapezoid method and find the steady state via a variational formulation. Our approach is benchmarked on prototypical one and two-dimensional systems, finding results which closely track the exact solution and achieve higher accuracy than alternative approaches based on using Markov chain Monte Carlo to sample restricted Boltzmann machines. Our work provides general methods for understanding quantum dynamics in various contexts, as well as techniques for solving high-dimensional probabilistic differential equations in classical setups.

Matrix product states (MPS) and `dressed' ground states of quadratic mean fields (e.g. Gutzwiller projected Slater Determinants) are both important classes of variational wave-functions. This latter class has played important roles in understanding superconductivity and quantum spin-liquids. We present a novel method to obtain both the finite and infinite MPS (iMPS) representation of the ground state of an arbitrary fermionic quadratic mean-field Hamiltonian, (which in the simplest case is a Slater determinant and in the most general case is a Pfaffian). We also show how to represent products of such states (e.g. determinants times Pfaffians). From this representation one can project to single occupancy and evaluate the entanglement spectra after Gutzwiller projection. We then obtain the MPS and iMPS representation of Gutzwiller projected mean-field states that arise from the variational slave-fermion approach to the $S=1$ Bilinear-Biquadratic (BLBQ) quantum spin chain. To accomplish this, we develop an approach to orthogonalize degenerate iMPS to find all the states in the degenerate ground-state manifold. We find the energies of the MPS and iMPS states match the variational energies closely indicating the method is accurate and there is minimal loss due to truncation error. We then present the first exploration of the entanglement spectra of projected slave-fermion states exploring their qualitative features and finding good qualitative agreement with the respective exact ground state spectra found from DMRG.

Quantum control, which refers to the active manipulation of physical systems described by the laws of quantum mechanics, constitutes an essential ingredient for the development of quantum technology. Here we apply Differentiable Programming (DP) and Natural Evolution Strategies (NES) to the optimal transport of Majorana zero modes in superconducting nanowires, a key element to the success of Majorana-based topological quantum computation. We formulate the motion control of Majorana zero modes as an optimization problem for which we propose a new categorization of four different regimes with respect to the critical velocity of the system and the total transport time. In addition to correctly recovering the anticipated smooth protocols in the adiabatic regime, our algorithms uncover efficient but strikingly counter-intuitive motion strategies in the non-adiabatic regime. The emergent picture reveals a simple but high fidelity strategy that makes use of pulse-like jumps at the beginning and the end of the protocol with a period of constant velocity in between the jumps, which we dub the jump-move-jump protocol. We provide a transparent semi-analytical picture, which uses the sudden approximation and a reformulation of the Majorana motion in a moving frame, to illuminate the key characteristics of the jump-move-jump control strategy. We verify that the jump-move-jump protocol remains robust against the presence of interactions or disorder, and corroborate its high efficacy on a realistic proximity coupled nanowire model. Our results demonstrate that machine learning for quantum control can be applied efficiently to quantum many-body dynamical systems with performance levels that make it relevant to the realization of large-scale quantum technology., Comment: Revised manuscript: Addition of results on proximity coupled semiconducting nanowire, disorder and interactions. Extra discussions and new conclusion. 18 pages, 8 figures + appendix (12 pages, 8 figures)

Closed, interacting, quantum systems have the potential to transition to a many-body localized (MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failing to thermalize. In this work we study the distribution of correlations throughout the ergodic-MBL phase diagram. We find the typical correlations in the MBL phase decay as a stretched exponential with range $r$ eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as $e^{-A\sqrt{r}}$, a decay that is reminiscent of the random singlet phase. While the standard deviation of the $\log(QMI)$ has a range dependence, the $\log(QMI)$ converges to a range-invariant distribution on all other moments (i.e., the skewness and higher) at the transition. The universal nature of these distributions provides distinct phenomenology of the transition different from both the ergodic and MBL phenomenologies. In addition to the typical correlations, we study the extreme correlations in the system, finding that the probability of strong long-range correlations is maximal at the transition, suggesting the proliferation of resonances there. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, finding that this probability is non-zero deep in the MBL phase but vanishes at moderate disorder well above the transition., Comment: Main paper: 9 pages, 10 figures. Appendix: 11 pages, 11 figures

The Density Matrix Renormalization Group (DMRG) algorithm is a powerful tool for solving eigenvalue problems to model quantum systems. DMRG relies on tensor contractions and dense linear algebra to compute properties of condensed matter physics systems. However, its efficient parallel implementation is challenging due to limited concurrency, large memory footprint, and tensor sparsity. We mitigate these problems by implementing two new parallel approaches that handle block sparsity arising in DMRG, via Cyclops, a distributed memory tensor contraction library. We benchmark their performance on two physical systems using the Blue Waters and Stampede2 supercomputers. Our DMRG performance is improved by up to 5.9X in runtime and 99X in processing rate over ITensor, at roughly comparable computational resource use. This enables higher accuracy calculations via larger tensors for quantum state approximation. We demonstrate that despite having limited concurrency, DMRG is weakly scalable with the use of efficient parallel tensor contraction mechanisms.

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of $\ell$-bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only $\pm 1$ eigenvalues). While MBL and $\ell$-bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary $\ell$-bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the $\ell$-bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality $\ell$-bits at large disorder strength and rapid qualitative changes in the distributions of $\ell$-bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models which further validates the evidence of MBL phenomenology in the other two and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality., Comment: main paper: 7 pages, 4 figures; supplement: 29 pages, 24 figures; added figures to supplement and made minor changes to main paper

An interacting quantum system can transition from an ergodic to a many-body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian. Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in the MBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstates of the disordered Heisenberg model as the disorder strength, $W$, is varied adiabatically. We focus on the probability that two `colliding' eigenstates hybridize as a function of both the range $R$ at which they differ as well as the strength of their hybridization. We find, in the MBL phase, that the probability of a colliding eigenstate hybridizing strongly at range $R$ decays as $Pr(R)\propto \exp [-R/\eta]$, with a length scale $\eta(W) = 1 / (B \log(W / W_c) )$ which diverges at the critical disorder strength $W_c$. This leads to range-invariance at the transition, suggesting the formation of resonating cat states at all ranges. This range invariance does not survive to the ergodic phase, where hybridization is exponentially more likely at large range, a fact that can be understood with simple combinatorial arguments. In fact, compensating for these combinatorial effects allows us to define an additional correlation length $\xi$ in the MBL phase which is in excellent agreement with previous works and which takes the critical value $1 / \log(2)$ at the transition, found in previous works to destabilize the MBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-level collisions of eigenstates close in energy.

The Kondo-Heinsberg chain is an interesting model of a strongly correlated system which has a broad superconducting state with pair-density wave (PDW) order. Some of us have recently proposed that this PDW state is a symmetry-protected topological (SPT) state, and the gapped spin sector of the model supports Majorana zero modes. In this work, we reexamine this problem using a combination of numeric and analytic methods. In extensive density matrix renormalization group calculations, we find no evidence of a topological ground state degeneracy or the previously proposed Majorana zero modes in the PDW phase of this model. This result motivated us to reexamine the original arguments for the existence of the Majorana zero modes. A careful analysis of the effective continuum field theory of the model shows that the Hilbert space of the spin sector of the theory does not contain any single Majorana fermion excitations. This analysis shows that the PDW state of the doped 1D Kondo-Heisenberg model is not an SPT with Majorana zero modes., Comment: 13 pages, 8 figures

We study the promising idea of using dipolar molecular systems as analog quantum simulators for quantum link models, which are discrete versions of lattice gauge theories. In a quantum link model the link variables have a finite number of degrees of freedom and discrete values. We construct the effective Hamiltonian of a system of dipolar molecules with electric dipole-dipole interactions, where we use the tunable parameters of the system to match it to the target Hamiltonian describing a U(1) quantum link model in 1+1 dimensions., Comment: Proceedings of the 37th International Symposium on Lattice Field Theory - Lattice 2019, Wuhan (China)

2D materials offer an ideal platform to study the strain fields induced by individual atomic defects, yet challenges associated with radiation damage have so-far limited electron microscopy methods to probe these atomic-scale strain fields. Here, we demonstrate an approach to probe single-atom defects with sub-picometer precision in a monolayer 2D transition metal dichalcogenide, WSe$_{2-2x}$Te$_{2x}$. We utilize deep learning to mine large datasets of aberration-corrected scanning transmission electron microscopy images to locate and classify point defects. By combining hundreds of images of nominally identical defects, we generate high signal-to-noise class-averages which allow us to measure 2D atomic coordinates with up to 0.3 pm precision. Our methods reveal that Se vacancies introduce complex, oscillating strain fields in the WSe$_{2-2x}$Te$_{2x}$ lattice which cannot be explained by continuum elastic theory. These results indicate the potential impact of computer vision for the development of high-precision electron microscopy methods for beam-sensitive materials., Comment: All authors are from University of Illinois at Urbana-Champaign. 41 pages, 5 figures, 6 supplementary figures, and supplementary information

Gauge theories appear broadly in physics, ranging from the standard model of particle physics to long-wavelength descriptions of topological systems in condensed matter. However, systems with sign problems are largely inaccessible to classical computations and also beyond the current limitations of digital quantum hardware. In this work, we develop an analog approach to simulating gauge theories with an experimental setup that employs dipolar spins (molecules or Rydberg atoms). We consider molecules fixed in space and interacting through dipole-dipole interactions, avoiding the need for itinerant degrees of freedom. Each molecule represents either a site or gauge degree of freedom, and Gauss law is preserved by a direct and programmatic tuning of positions and internal state energies. This approach can be regarded as a form of analog systems programming and charts a path forward for near-term quantum simulation. As a first step, we numerically validate this scheme in a small-system study of U(1) quantum link models in (1+1) dimensions with link spin S = 1/2 and S = 1 and illustrate how dynamical phenomena such as string inversion and string breaking could be observed in near-term experiments. Our work brings together methods from atomic and molecular physics, condensed matter physics, high-energy physics, and quantum information science for the study of nonperturbative processes in gauge theories.

The fundamental question of how to best simulate quantum systems using conventional computational resources lies at the forefront of condensed matter and quantum computation. It impacts both our understanding of quantum materials and our ability to emulate quantum circuits. Here we present an exact formulation of quantum dynamics via factorized generalized measurements which maps quantum states to probability distributions with the advantage that local unitary dynamics and quantum channels map to local quasi-stochastic matrices. This representation provides a general framework for using state-of-the-art probabilistic models in machine learning for the simulation of quantum many-body dynamics. Using this framework, we have developed a practical algorithm to simulate quantum circuits with the Transformer, a powerful ansatz responsible for the most recent breakthroughs in natural language processing. We demonstrate our approach by simulating circuits which build GHZ and linear graph states of up to 60 qubits, as well as a variational quantum eigensolver circuit for preparing the ground state of the transverse field Ising model on six qubits. Our methodology constitutes a modern machine learning approach to the simulation of quantum physics with applicability both to quantum circuits as well as other quantum many-body systems., Comment: 12 pages, 11 figures including supplementary materials

Symmetry is at the heart of modern physics. Phases of matter are classified by symmetry breaking, topological phases are characterized by non-local symmetries, and point group symmetries are critical to our understanding of crystalline materials. Symmetries could then be used as a criterion to engineer quantum systems with targeted properties. Toward that end, we have developed a novel approach, the symmetric Hamiltonian construction (SHC), that takes as input symmetries, specified by integrals of motion or discrete symmetry transformations, and produces as output all local Hamiltonians consistent with these symmetries (see github.com/ClarkResearchGroup/qosy for our open-source code). This approach builds on the slow operator method [PRE 92, 012128]. We use our new approach to construct new Hamiltonians for topological phases of matter. Topological phases of matter are exotic quantum phases with potential applications in quantum computation. In this work, we focus on two types of topological phases of matter: superconductors with Majorana zero modes and $Z_2$ quantum spin liquids. In our first application of the SHC approach, we analytically construct a large and highly tunable class of superconducting Hamiltonians with Majorana zero modes with a given targeted spatial distribution. This result lays the foundation for potential new experimental routes to realizing Majorana fermions. In our second application, we find new $Z_2$ spin liquid Hamiltonians on the square and kagome lattices. These new Hamiltonians are not sums of commuting operators nor frustration-free and, when perturbed appropriately (in a way that preserves their $Z_2$ spin liquid behavior), exhibit level-spacing statistics that suggest non-integrability. This result demonstrates how our approach can automatically generate new spin liquid Hamiltonians with interesting properties not often seen in solvable models., Comment: 26 pages, 11 figures, 2 tables

Tensor networks are a powerful formalism for transforming one set of degrees of freedom to another. They have been heavily used in analyzing the geometry of bulk/boundary correspondence in conformal field theories. Here we develop a tensor-network version of the Wilson-Wegner Renormalization Group Flow equations to efficiently generate a unitary tensor network which diagonalizes many-body localized Hamiltonians. Treating this unitary tensor network as a bulk geometry, we find this emergent geometry corresponds to the shredded horizon picture: the circumference of the network shrinks exponentially with distance into the bulk, with spatially distant points being largely disconnected., Comment: 4+8 pages; 5+12 figures

Quantum Monte Carlo simulations of quantum many body systems are plagued by the Fermion sign problem. The computational complexity of simulating Fermions scales exponentially in the projection time $\beta$ and system size. The sign problem is basis dependent and an improved basis, for fixed errors, lead to exponentially quicker simulations. We show how to use sign-free quantum Monte Carlo simulations to optimize over the choice of basis on large two-dimensional systems. We numerically illustrate these techniques decreasing the `badness' of the sign problem by optimizing over single-particle basis rotations on one and two-dimensional Hubbard systems. We find a generic rotation which improves the average sign of the Hubbard model for a wide range of $U$ and densities for $L \times 4$ systems. In one example improvement, the average sign (and hence simulation cost at fixed accuracy) for the $16\times 4$ Hubbard model at $U/t=4$ and $n=0.75$ increases by $\exp\left[8.64(6)\beta\right]$. For typical projection times of $\beta\gtrapprox 100$, this accelerates such simulation by many orders of magnitude.

Representing a target quantum state by a compact, efficient variational wave-function is an important approach to the quantum many-body problem. In this approach, the main challenges include the design of a suitable variational ansatz and optimization of its parameters. In this work, we address both of these challenges. First, we define the variational class of Computational Graph States (CGS) which gives a uniform framework for describing all computable variational ansatz. Secondly, we develop a novel optimization scheme, supervised wave-function optimization (SWO), which systematically improves the optimized wave-function by drawing on ideas from supervised learning. While SWO can be used independently of CGS, utilizing them together provides a flexible framework for the rapid design, prototyping and optimization of variational wave-functions. We demonstrate CGS and SWO by optimizing for the ground state wave-function of 1D and 2D Heisenberg models on nine different variational architectures including architectures not previously used to represent quantum many-body wave-functions and find they are energetically competitive to other approaches. One interesting application of this architectural exploration is that we show that fully convolution neural network wave-functions can be optimized for one system size and, using identical parameters, produce accurate energies for a range of system sizes. We expect these methods to increase the rate of discovery of novel variational ansatz and bring further insights to the quantum many body problem., Comment: 10 + 4 pages; 8 + 3 figures

Motivated by the recent discovery of a macroscopically degenerate exactly solvable point of the spin-$1/2$ $XXZ$ model for $J_z/J=-1/2$ on the kagome lattice [H. J. Changlani et al. Phys. Rev. Lett 120, 117202 (2018)] -- a result that holds for arbitrary magnetization -- we develop an exact mapping between its exact quantum three-coloring wavefunctions and the characteristic localized and topological magnons. This map, involving resonating two-color loops, is developed to represent exact many-body ground state wavefunctions for special high magnetizations. Using this map we show that these exact ground state solutions are valid for any $J_z/J \geq -1/2$. This demonstrates the equivalence of the ground-state wavefunction of the Ising, Heisenberg and $XY$ regimes all the way to the $J_z/J=-1/2$ point for these high magnetization sectors. In the hardcore bosonic language, this means that a certain class of exact many-body solutions, previously argued to hold for purely repulsive interactions ($J_z \geq 0$), actually hold for attractive interactions as well, up to a critical interaction strength. For the case of zero magnetization, where the ground state is not exactly known, we perform density matrix renormalization group calculations. Based on the calculation of the ground state energy and measurement of order parameters, we provide evidence for a lack of any qualitative change in the ground state on finite clusters in the Ising ($J_z \gg J$), Heisenberg ($J_z=J$) and $XY$ ($J_z=0$) regimes, continuing adiabatically to the vicinity of the macroscopically degenerate $J_z/J=-1/2$ point. These findings offer a framework for recent results in the literature, and also suggest that the $J_z/J=-1/2$ point is an unconventional quantum critical point whose vicinity may contain the key to resolving the spin-$1/2$ kagome problem., Comment: 10 pages, 10 figures

Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this paper, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman, adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to find the optimal transformation. NNB directly dresses a mean-field state, can be systematically improved and directly alters the sign structure of the wave-function. It generalizes the standard backflow which we show how to explicitly represent as a NNB. We benchmark the NNB on a Hubbard model at intermediate doping finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.

We investigate the classical phase diagram of the stuffed honeycomb Heisenberg lattice, which consists of a honeycomb lattice with a superimposed triangular lattice formed by sites at the center of each hexagon. This lattice encompasses and interpolates between the honeycomb, triangular and dice lattices, preserving the hexagonal symmetry while expanding the phase space for potential spin liquids. We use a combination of iterative minimization, classical Monte Carlo and analytical techniques to determine the complete ground state phase diagram. It is quite rich, with a variety of non-coplanar and non-collinear phases not found in the previously studied limits. In particular, our analysis reveals the triangular lattice critical point to be a multicritical point with two new phases vanishing via second order transitions at the critical point. We analyze these phases within linear spin wave theory and discuss consequences for the S = 1/2 spin liquid., Comment: 18 pages, 23 figures

A prime characterization of many-body localized (MBL) systems is the entanglement of their eigenstates; in contrast to the typical ergodic phase whose eigenstates are volume law, MBL eigenstates obey an area law. In this work, we show that a spin-disordered Hubbard model has both a large number of area-law eigenstates as well as a large number of eigenstates whose entanglement scales logarithmically with system size (log-law). This model, then, is a microscopic Hamiltonian which is neither ergodic nor many-body localized. We establish these results through a combination of analytic arguments based on the eta-pairing operators combined with a numerical analysis of eigenstates. In addition, we describe and simulate a dynamic time evolution approach starting from product states through which one can separately probe the area law and log-law eigenstates in this system., Comment: 12 pages, 18 figures

QMCPACK is an open source quantum Monte Carlo package for ab-initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses Slater-Jastrow type trial wave functions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliary field quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit (CPU) and graphical processing unit (GPU) systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://www.qmcpack.org .

Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output. However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians. We introduce an alternative computational "inverse method," the Eigenstate-to-Hamiltonian Construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems. EHC takes as input a wave function $|\psi_T\rangle$ and produces as output Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix. EHC is widely applicable to a number of models and in this work we consider seven different examples. Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model., Comment: 13 pages, 7 figures, 1 table; new example in results section; updated supplement; additional references; other minor changes