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Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $\Lambda$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $\mu$s at distance-5 up to a million cycles, with a cycle time of 1.1 $\mu$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms., Comment: 10 pages, 4 figures, Supplementary Information

We compute the electrical conductivity of the strongly interacting medium in the presence of strong magnetic background fields, $eB=4,9~GeV^2$, and for different values of the temperature, both in the confined and in the deconfined Quark-Gluon Plasma (QGP) phase. The conductivity is obtained from the Euclidean lattice time correlator of the electrical current, computed on gauge configurations sampled from Monte-Carlo simulations of an improved staggered discretization of $N_f = 2+1$ QCD. We perform the inverse Laplace transform of the correlator adopting a recently-proposed version of the standard Backus--Gilbert procedure for the inversion. The results obtained in the QGP phase show a sizable enhancement of the conductivity in the direction parallel to the magnetic field, as well as a suppression in the direction orthogonal to it. Such enhancement could be attributed to the manifestation of the Chiral Magnetic Effect (CME): following this guess, we extract the behaviour of the relaxation time of this process, extrapolate it to the continuum limit and compare it to previous results, finding it lower than expected in the explored range of temperatures.

Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators. Unlocking the full potential of such systems toward this goal requires flexible initial state preparation, precise time evolution, and extensive probes for final state characterization. We present a quantum simulator comprising 69 superconducting qubits which supports both universal quantum gates and high-fidelity analog evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. Emulating a two-dimensional (2D) XY quantum magnet, we leverage a wide range of measurement techniques to study quantum states after ramps from an antiferromagnetic initial state. We observe signatures of the classical Kosterlitz-Thouless phase transition, as well as strong deviations from Kibble-Zurek scaling predictions attributed to the interplay between quantum and classical coarsening of the correlated domains. This interpretation is corroborated by injecting variable energy density into the initial state, which enables studying the effects of the eigenstate thermalization hypothesis (ETH) in targeted parts of the eigenspectrum. Finally, we digitally prepare the system in pairwise-entangled dimer states and image the transport of energy and vorticity during thermalization. These results establish the efficacy of superconducting analog-digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

$\require{mhchem}$The square-kagome lattice Heisenberg antiferromagnet is a highly frustrated Hamiltonian whose material realizations have been scarce. We theoretically investigate the recently synthesized $\ce{Na6Cu7BiO4(PO4)4Cl3}$ where a Cu$^{2+}$ spin-$1/2$ square-kagome lattice (with six site unit cell) is decorated by a seventh magnetic site alternatingly above and below the layers. The material does not show any sign of long-range magnetic order down to 50 mK despite a Curie-Weiss temperature of $-212$ K indicating a quantum paramagnetic phase. Our DFT energy mapping elicits a purely antiferromagnetic Hamiltonian that features longer range exchange interactions beyond the pure square-kagome model and, importantly, we find the seventh site to be strongly coupled to the plane. We combine two variational Monte Carlo approaches, pseudo-fermion/Majorana functional renormalization group and Schwinger-Boson mean field calculations to show that the complex Hamiltonian of $\ce{Na6Cu7BiO4(PO4)4Cl3}$ still features a nonmagnetic ground state. We explain how the seventh Cu$^{2+}$ site actually aids the stabilization of the disordered state. We predict static and dynamic spin structure factors to guide future neutron scattering experiments., Comment: 5 pages, 4 figures. Supplement: 7 pages, 11 figures v2: minor revisions in text and figure labels

Twisted bilayer graphene is an exciting platform for exploring correlated quantum phases, extremely tunable with respect to both the single-particle bands and the interaction profile of electrons. Here, we investigate the phase diagram of twisted bilayer graphene as described by an extended Hubbard model on the honeycomb lattice with two fermionic orbitals (valleys) per site. Besides the special extended {\it cluster interaction} $Q$, we incorporate the effect of gating through an onsite Hubbard-interaction $U$. Within Quantum Monte Carlo (QMC), we find valence-bond-solid, N\'eel-valley antiferromagnetic or charge-density wave phases. Further, we elucidate the competition of these phases by noticing that the cluster interaction induces an exotic constraint on the Hilbert space, which we dub {\it the cluster rule}, in analogy to the famous pyrochlore spin-ice rule. Formulating the perturbative Hamiltonian by projecting into the cluster-rule manifold, we perform exact diagonalization and construct the fixed-point states of the observed phases. Finally, we compute the local electron density patterns as signatures distinguishing these phases, which could be observed with scanning tunneling microscopy. Our work capitalizes on the notion of cluster constraints in the extended Hubbard model of twisted bilayer graphene, and suggests a scheme towards realization of several symmetry-breaking insulating phases in a twisted-bilayer graphene sheet., Comment: 10 pages, 6 figures

Quantum enhanced optimization of classical cost functions is a central theme of quantum computing due to its high potential value in science and technology. The variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are popular variational approaches that are considered the most viable solutions in the noisy-intermediate scale quantum (NISQ) era. Here, we study the scaling of the quantum resources, defined as the required number of circuit repetitions, to reach a fixed success probability as the problem size increases, focusing on the role played by measurement shot noise, which is unavoidable in realistic implementations. Simple and reproducible problem instances are addressed, namely, the ferromagnetic and disordered Ising chains. Our results show that: (i) VQE with the standard heuristic ansatz scales comparably to direct brute-force search when energy-based optimizers are employed. The performance improves at most quadratically using a gradient-based optimizer. (ii) When the parameters are optimized from random guesses, also the scaling of QAOA implies problematically long absolute runtimes for large problem sizes. (iii) QAOA becomes practical when supplemented with a physically-inspired initialization of the parameters. Our results suggest that hybrid quantum-classical algorithms should possibly avoid a brute force classical outer loop, but focus on smart parameters initialization., Comment: 15 pages, 11 figures

The continued development of novel many-body approaches to ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. Here we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide the most extensive curated dataset of variational calculations of many-body quantum systems to date, identifying cases where state-of-the-art numerical approaches show limited accuracy, and novel algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods towards quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible., Comment: 25 pages, 5 figures

Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work, we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational ansatz to represent infinite translation-invariant quantum states. We show numerically that this ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favourable scaling of the ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm., Comment: 19 pages, 14 figures

Topological quantum chemistry (TQC) is a successful framework for identifying (noninteracting) topological materials. Based on the symmetry eigenvalues of Bloch eigenstates at maximal momenta, which are attainable from first principles calculations, a band structure can either be classified as an atomic limit, in other words adiabatically connected to independent electronic orbitals on the respective crystal lattice, or it is topological. For interacting systems, there is no single-particle band structure and hence, the TQC machinery grinds to a halt. We develop a framework analogous to TQC, but employing $n$-particle Green's function to classify interacting systems. Fundamentally, we define a class of interacting reference states that generalize the notion of atomic limits, which we call Mott atomic limits, and are symmetry protected topological states. Our formalism allows to fully classify these reference states (with $n=2$), which can themselves represent symmetry protected topological states. We present a comprehensive classification of such states in one-dimension and provide numerical results on model systems. With this, we establish Mott atomic limit states as a generalization of the atomic limits to interacting systems.

Pinch point singularities, associated with flat band magnetic excitations, are tell-tale signatures of Coulomb spin liquids. While their properties in the presence of quantum fluctuations have been widely studied, the fate of the complementary non-analytic features -- shaped as half-moons and stars -- arising from adjacent shallow dispersive bands has remained unexplored. Here, we address this question for the spin $S=1/2$ Heisenberg antiferromagnet on the kagome lattice with second and third neighbor couplings, which allows one to tune the classical ground state from flat bands to being governed by shallow dispersive bands for intermediate coupling strengths. Employing the complementary strengths of variational Monte Carlo, pseudo-fermion functional renormalization group, and density-matrix renormalization group, we establish the quantum phase diagram. The U(1) Dirac spin liquid ground state of the nearest-neighbor antiferromagnet remains remarkably robust till intermediate coupling strengths when it transitions into a pinwheel valence bond crystal displaying signatures of half-moons in its structure factor. Our work thus identifies a microscopic setting that realizes one of the proximate orders of the Dirac spin liquid identified in a recent work [Song, Wang, Vishwanath, He, Nat. Commun. 10, 4254 (2019)]. For larger couplings, we obtain a collinear magnetically ordered ground state characterized by star-like patterns., Comment: 7 pages, 5 figures. Supplemental Material appended

The variational approach is a cornerstone of computational physics, considering both conventional and quantum computing computational platforms. The variational quantum eigensolver (VQE) algorithm aims to prepare the ground state of a Hamiltonian exploiting parametrized quantum circuits that may offer an advantage compared to classical trial states used, for instance, in quantum Monte Carlo or tensor network calculations. While traditionally, the main focus has been on developing better trial circuits, we show that the algorithm's success crucially depends on other parameters such as the learning rate, the number $N_s$ of measurements to estimate the gradient components, and the Hamiltonian gap $\Delta$. We first observe the existence of a finite $N_s$ value below which the optimization is impossible, and the energy variance resembles the behavior of the specific heat in second-order phase transitions. Secondly, when $N_s$ is above such threshold level, and learning is possible, we develop a phenomenological model that relates the fidelity of the state preparation with the optimization hyperparameters as well as $\Delta$. More specifically, we observe that the computational resources scale as $1/\Delta^2$, and we propose a symmetry-enhanced simulation protocol that should be used if the gap closes. We test our understanding on several instances of two-dimensional frustrated quantum magnets, which are believed to be the most promising candidates for near-term quantum advantage through variational quantum simulations., Comment: 12 pages, 13 figures

Information retrieval systems are usually measured by labeling the relevance of results corresponding to a sample of user queries. In practical search engines, such measurement needs to be performed continuously, such as daily or weekly. This creates a trade-off between (a) representativeness of query sample to current query traffic of the product; (b) labeling cost: if we keep the same query sample, results would be similar allowing us to reuse their labels; and (c) overfitting caused by continuous usage of same query sample. In this paper we explicitly formulate this tradeoff, propose two new variants -- Stable and Semi-stable -- to simple and weighted random sampling and show that they outperform existing approaches for the continuous usage settings, including monitoring/debugging search engine or comparing ranker candidates.

We introduce version 3 of NetKet, the machine learning toolbox for many-body quantum physics. NetKet is built around neural-network quantum states and provides efficient algorithms for their evaluation and optimization. This new version is built on top of JAX, a differentiable programming and accelerated linear algebra framework for the Python programming language. The most significant new feature is the possibility to define arbitrary neural network ans\"atze in pure Python code using the concise notation of machine-learning frameworks, which allows for just-in-time compilation as well as the implicit generation of gradients thanks to automatic differentiation. NetKet 3 also comes with support for GPU and TPU accelerators, advanced support for discrete symmetry groups, chunking to scale up to thousands of degrees of freedom, drivers for quantum dynamics applications, and improved modularity, allowing users to use only parts of the toolbox as a foundation for their own code., Comment: 55 pages, 5 figures. Accompanying code at https://github.com/netket/netket

Feed-forward neural networks are a novel class of variational wave functions for correlated many-body quantum systems. Here, we propose a specific neural network ansatz suitable for systems with real-valued wave functions. Its characteristic is to encode the all-important rugged sign structure of a quantum wave function in a convolutional neural network with discrete output. Its training is achieved through an evolutionary algorithm. We test our variational ansatz and training strategy on two spin-1/2 Heisenberg models, one on the two-dimensional square lattice and one on the three-dimensional pyrochlore lattice. In the former, our ansatz converges with high accuracy to the analytically known sign structures of ordered phases. In the latter, where such sign structures are a priory unknown, we obtain better variational energies than with other neural network states. Our results demonstrate the utility of discrete neural networks to solve quantum many-body problems.

We investigate the nature of the ground-state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the $shuriken$ lattice by complementary state-of-the-art numerical techniques, such as variational Monte Carlo (VMC) with versatile Gutzwiller-projected Jastrow wave functions, unconstrained multi-variable variational Monte Carlo (mVMC), and pseudo-fermion/Majorana functional renormalization group (PF/PM-FRG) methods. We establish the presence of a quantum paramagnetic ground state and investigate its nature, by classifying symmetric and chiral quantum spin liquids, and inspecting their instabilities towards competing valence-bond-crystal (VBC) orders. Our VMC analysis reveals that a VBC with a pinwheel structure emerges as the lowest-energy variational ground state, and it is obtained as an instability of the U(1) Dirac spin liquid. Analogous conclusions are drawn from mVMC calculations employing accurate BCS pairing states supplemented by symmetry projectors, which confirm the presence of pinwheel VBC order by a thorough analysis of dimer-dimer correlation functions. Our work highlights the nontrivial role of quantum fluctuations via the Gutzwiller projector in resolving the subtle interplay between competing orders., Comment: Main paper (7 pages, 5 figures, 1 table) + Supplemental Material

The spin-1/2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram. Besides hosting several ordered phases, the model is debated to possess a spin-liquid ground state when only nearest-neighbor antiferromagnetic interactions are present. Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and complementary variational techniques. Specifically, we employ a RVB-like many-variable Monte Carlo ansatz and convolutional neural network quantum states for (variational) calculations with up to $4\times 4^3$ and $4 \times 3^3$ spins, respectively. We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit. Our main results are (1) the determination of the phase transition between the putative spin-liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry-breaking tendencies. We find clear indications of spontaneously broken inversion and rotational symmetry, calling the scenario of a featureless quantum spin-liquid into question. Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets., Comment: 9 pages, 5 figures

We study the electromagnetic (e.m.) conductivity of QGP in a magnetic background by lattice simulations with $N_f = 2+1$ dynamical rooted staggered fermions at the physical point. We study the correlation functions of the e.m.~currents at $T=200,\,250$\,MeV and use the Tikhonov approach to extract the conductivity. This is found to rise with the magnetic field in the direction parallel to it and to decrease in the transverse direction, giving evidence for both the Chiral Magnetic Effect and the magnetoresistance phenomenon in QGP. We also estimate the chiral charge relaxation time in QGP., Comment: 9 pages, 7 figures

Neural quantum states (NQS) attract a lot of attention due to their potential to serve as a very expressive variational ansatz for quantum many-body systems. Here we study the main factors governing the applicability of NQS to frustrated magnets by training neural networks to approximate ground states of several moderately-sized Hamiltonians using the corresponding wavefunction structure on a small subset of the Hilbert space basis as training dataset. We notice that generalization quality, i.e. the ability to learn from a limited number of samples and correctly approximate the target state on the rest of the space, drops abruptly when frustration is increased. We also show that learning the sign structure is considerably more difficult than learning amplitudes. Finally, we conclude that the main issue to be addressed at this stage, in order to use the method of NQS for simulating realistic models, is that of generalization rather than expressibility., Comment: 9+6 pages, 5+7 figures, v3: added results for larger systems, improved discussion, added new references

In this paper we study the shear viscosity temperature dependence of $SU(3)$--gluodynamics within lattice simulation. To do so, we measure the correlation functions of energy-momentum tensor in the range of temperatures $T/T_c\in [0.9, 1.5]$. To extract the values of shear viscosity we used two approaches. The first one is to fit the lattice data with some physically motivated ansatz for the spectral function with unknown parameters and then determine shear viscosity. The second approach is to apply the Backus-Gilbert method which allows to extract shear viscosity from the lattice data nonparametrically. The results obtained within both approaches agree with each other. Our results allow us to conclude that within the temperature range $T/T_c \in [0.9, 1.5]$ SU(3)--gluodynamics reveals the properties of a strongly interacting system, which cannot be described perturbatively, and has the ratio $\eta/s$ close to the value ${1}/{4\pi}$ in $N = 4$ Supersymmetric Yang-Mills theory., Comment: Typos in references are corrected and the acknowledgements are modified