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While many statistical properties of deep random quantum circuits can be deduced, often rigorously and other times heuristically, by an approximation to global Haar-random unitaries, the statistics of constant-depth random quantum circuits are generally less well-understood due to a lack of amenable tools and techniques. We circumvent this barrier by considering a related constant-time Brownian circuit model which shares many similarities with constant-depth random quantum circuits but crucially allows for direct calculations of higher order moments of its output distribution. Using mean-field (large-n) techniques, we fully characterize the output distributions of Brownian circuits at shallow depths and show that they follow a Porter-Thomas distribution, just like in the case of deep circuits, but with a truncated Hilbert space. The access to higher order moments allows for studying the expected and typical Linear Cross-entropy (XEB) benchmark scores achieved by an ideal quantum computer versus the state-of-the-art classical spoofers for shallow Brownian circuits. We discover that for these circuits, while the quantum computer typically scores within a constant factor of the expected value, the classical spoofer suffers from an exponentially larger variance. Numerical evidence suggests that the same phenomenon also occurs in constant-depth discrete random quantum circuits, like those defined over the all-to-all architecture. We conjecture that the same phenomenon is also true for random brickwork circuits in high enough spatial dimension., Comment: 15+25 pages, 4 figures

Solar irradiance forecasts can be dynamic and unreliable due to changing weather conditions. Near the Arctic circle, this also translates into a distinct set of further challenges. This work is forecasting solar irradiance with Norwegian data using variations of Long-Short-Term Memory units (LSTMs). In order to gain more trustworthiness of results, the probabilistic approaches Quantile Regression (QR) and Maximum Likelihood (MLE) are optimized on top of the LSTMs, providing measures of uncertainty for the results. MLE is further extended by using a Johnson's SU distribution, a Johnson's SB distribution, and a Weibull distribution in addition to a normal Gaussian to model parameters. Contrary to a Gaussian, Weibull, Johnson's SU and Johnson's SB can return skewed distributions, enabling it to fit the non-normal solar irradiance distribution more optimally. The LSTMs are compared against each other, a simple Multi-layer Perceptron (MLP), and a smart-persistence estimator. The proposed LSTMs are found to be more accurate than smart persistence and the MLP for a multi-horizon, day-ahead (36 hours) forecast. The deterministic LSTM showed better root mean squared error (RMSE), but worse mean absolute error (MAE) than a MLE with Johnson's SB distribution. Probabilistic uncertainty estimation is shown to fit relatively well across the distribution of observed irradiance. While QR shows better uncertainty estimation calibration, MLE with Johnson's SB, Johnson's SU, or Gaussian show better performance in the other metrics employed. Optimizing and comparing the models against each other reveals a seemingly inherent trade-off between point-prediction and uncertainty estimation calibration., Comment: 8 pages, 5 figures. To be published in the 2024 IEEE Conference Photovoltaic Specialists (PVSC) proceedings

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, $N$, of fermions goes to infinity. For SYK, the distance scales as $N^{1/2}$, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. $N^{.99}$, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size $N$. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity., Comment: 81 pages, 22 figures, revised and substantially expanded version to be published in Quantum, main results are unchanged and new results added

In a fast scrambling many-body quantum system, information is spread and entanglement is built up on a timescale that grows logarithmically with the system size. This is of fundamental interest in understanding the dynamics of many-body systems, as well as in efficiently producing entangled resource states and error-correcting codes. In this work, we identify a dynamical transition marking the onset of scrambling in quantum circuits with different levels of long-range connectivity. In particular, we show that as a function of the interaction range for circuits of different structures, the tripartite mutual information exhibits a scaling collapse around a critical point between two clearly defined regimes of different dynamical behaviour. We study this transition analytically in a related long-range Brownian circuit model and show how the transition can be mapped onto the statistical mechanics of a long-range Ising model in a particular region of parameter space. This mapping predicts mean-field critical exponents $\nu = -1/(1+s_c)$, which are consistent with the critical exponents extracted from Clifford circuit numerics. In addition to systems with conventional power-law interactions, we identify the same phenomenon in deterministic, sparse circuits that can be realised in experiments with neutral atom arrays., Comment: 19 pages, 9 figures

With an ever-increasing number of sensors in modern society, spatio-temporal time series forecasting has become a de facto tool to make informed decisions about the future. Most spatio-temporal forecasting models typically comprise distinct components that learn spatial and temporal dependencies. A common methodology employs some Graph Neural Network (GNN) to capture relations between spatial locations, while another network, such as a recurrent neural network (RNN), learns temporal correlations. By representing every recorded sample as its own node in a graph, rather than all measurements for a particular location as a single node, temporal and spatial information is encoded in a similar manner. In this setting, GNNs can now directly learn both temporal and spatial dependencies, jointly, while also alleviating the need for additional temporal networks. Furthermore, the framework does not require aligned measurements along the temporal dimension, meaning that it also naturally facilitates irregular time series, different sampling frequencies or missing data, without the need for data imputation. To evaluate the proposed methodology, we consider wind speed forecasting as a case study, where our proposed framework outperformed other spatio-temporal models using GNNs with either Transformer or LSTM networks as temporal update functions., Comment: Pre-print

Holography has taught us that spacetime is emergent and its properties depend on the entanglement structure of the dual theory. In this paper, we describe how changes in the entanglement due to a local projective measurement (LPM) on a subregion $A$ of the boundary theory modify the bulk dual spacetime. We find that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk spacetimes dual to the complementary unmeasured region $A^c$ that are cut off by end-of-the-world branes. Using a bulk calculation in $AdS_3$ and tensor network models of holography, we show that the portions of the bulk geometry that are preserved after the measurement depend on the size of $A$ and the state we project onto. The post-measurement bulk dual to $A^c$ includes regions that were originally part of the entanglement wedge of $A$ prior to measurement. This suggests that LPMs performed on a boundary subregion $A$ teleport part of the bulk information originally encoded in $A$ into the complementary region $A^c$. In semiclassical holography an arbitrary amount of bulk information can be teleported in this way, while in tensor network models the teleported information is upper-bounded by the amount of entanglement shared between $A$ and $A^c$ due to finite-$N$ effects. When $A$ is the union of two disjoint subregions, the measurement triggers an entangled/disentangled phase transition between the remaining two unmeasured subregions, corresponding to a connected/disconnected phase transition in the bulk description. Our results shed new light on the effects of measurement on the entanglement structure of holographic theories and give insight on how bulk information can be manipulated from the boundary theory. They could also be extended to more general quantum systems and tested experimentally, and represent a first step towards a holographic description of measurement-induced phase transitions., Comment: 60 pages + appendices, 27 figures

This study focuses on multi-step spatio-temporal wind speed forecasting for the Norwegian continental shelf. The study aims to leverage spatial dependencies through the relative physical location of different measurement stations to improve local wind forecasts. Our multi-step forecasting models produce either 10-minute, 1- or 4-hour forecasts, with 10-minute resolution, meaning that the models produce more informative time series for predicted future trends. A graph neural network (GNN) architecture was used to extract spatial dependencies, with different update functions to learn temporal correlations. These update functions were implemented using different neural network architectures. One such architecture, the Transformer, has become increasingly popular for sequence modelling in recent years. Various alterations have been proposed to better facilitate time series forecasting, of which this study focused on the Informer, LogSparse Transformer and Autoformer. This is the first time the LogSparse Transformer and Autoformer have been applied to wind forecasting and the first time any of these or the Informer have been formulated in a spatio-temporal setting for wind forecasting. By comparing against spatio-temporal Long Short-Term Memory (LSTM) and Multi-Layer Perceptron (MLP) models, the study showed that the models using the altered Transformer architectures as update functions in GNNs were able to outperform these. Furthermore, we propose the Fast Fourier Transformer (FFTransformer), which is a novel Transformer architecture based on signal decomposition and consists of two separate streams that analyse the trend and periodic components separately. The FFTransformer and Autoformer were found to achieve superior results for the 10-minute and 1-hour ahead forecasts, with the FFTransformer significantly outperforming all other models for the 4-hour ahead forecasts., Comment: (Please cite this version): https://doi.org/10.1016/j.apenergy.2022.120565 Code Available at https://github.com/LarsBentsen/FFTransformer

We calculate the frame potential for Brownian clusters of $N$ spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the $k$th frame potential comes within $\epsilon$ of the Haar value after a time of order $t \sim k N + k \log k + \log \epsilon^{-1}$. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary $k$-design after a time of order $t \sim k N$. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a $k$-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable., Comment: 20.5 pages, 5 figures

We investigate the emergence of different effective geometries in stochastic Clifford circuits with sparse coupling. By changing the probability distribution for choosing two-site gates as a function of distance, we generate sparse interactions that either decay or grow with distance as a function of a single tunable parameter. Tuning this parameter reveals three distinct regimes of geometry for the spreading of correlations and growth of entanglement in the system. We observe linear geometry for short-range interactions, treelike geometry on a sparse coupling graph for long-range interactions, and an intermediate fast scrambling regime at the crossover point between the linear and treelike geometries. This transition in geometry is revealed in calculations of the subsystem entanglement entropy and tripartite mutual information. We also study emergent lightcones that govern these effective geometries by teleporting a single qubit of information from an input qubit to an output qubit. These tools help to analyze distinct geometries arising in dynamics and correlation spreading in quantum many-body systems., Comment: 19 pages, 11 figures

With the increased penetration of wind energy into the power grid, it has become increasingly important to be able to predict the expected power production for larger wind farms. Deep learning (DL) models can learn complex patterns in the data and have found wide success in predicting wake losses and expected power production. This paper proposes a modular framework for attention-based graph neural networks (GNN), where attention can be applied to any desired component of a graph block. The results show that the model significantly outperforms a multilayer perceptron (MLP) and a bidirectional LSTM (BLSTM) model, while delivering performance on-par with a vanilla GNN model. Moreover, we argue that the proposed graph attention architecture can easily adapt to different applications by offering flexibility into the desired attention operations to be used, which might depend on the specific application. Through analysis of the attention weights, it was showed that employing attention-based GNNs can provide insights into what the models learn. In particular, the attention networks seemed to realise turbine dependencies that aligned with some physical intuition about wake losses., Comment: 12 pages, 7 figures, 1 appendix, in review process

Measurement-induced phase transitions arise due to a competition between the scrambling of quantum information in a many-body system and local measurements. In this work we investigate these transitions in different classes of fast scramblers, systems that scramble quantum information as quickly as is conjectured to be possible -- on a timescale proportional to the logarithm of the system size. In particular, we consider sets of deterministic sparse couplings that naturally interpolate between local circuits that slowly scramble information and highly nonlocal circuits that achieve the fast-scrambling limit. We find that circuits featuring sparse nonlocal interactions are able to withstand substantially higher rates of local measurement than circuits with only local interactions, even at comparable gate depths. We also study the quantum error-correcting codes that support the volume-law entangled phase and find that our maximally nonlocal circuits yield codes with nearly extensive contiguous code distance. Use of these sparse, deterministic circuits opens pathways towards the design of noise-resilient quantum circuits and error correcting codes in current and future quantum devices with minimum gate numbers., Comment: 11 pages, 6 figures (plus 11 page supplement with 11 figures)

We develop solvable models of large-$N$ hybrid quantum circuits on qubits and fermions with long-range power-law interactions and continuous local monitoring, which provide analytical access to the entanglement phase diagram and error-correcting properties of many-body entangled non-equilibrium states generated by such dynamics. In one dimension, the long-range coupling is irrelevant for $\alpha>3/2$, where $\alpha$ is the power-law exponent, and the models exhibit a conventional measurement-induced phase transition between volume- and area-law entangled phases. For $1/2<\alpha<3/2$ the long-range coupling becomes relevant, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for $\alpha<1$ the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases, indicating that the phase realizes a quantum error correcting code whose code distance scales as $L^{2-2\alpha}$. While the entanglement phase diagram is the same for both the interacting qubit and fermionic hybrid Brownian circuits, we find that long-range free-fermionic circuits exhibit a distinct phase diagram with two different fractal entangled phases., Comment: 8+12 pages, 6 figures, v2 close to published version

Competition between unitary dynamics that scrambles quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we study this phenomenon in an analytically tractable all-to-all Brownian hybrid circuit model composed of qubits. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables which are distinguished by the averaging procedure, and for concreteness we focus on a particular purity quantity for which the averaging is particularly simple. We represent the purity as a path integral coupling four replicas with twisted boundary conditions. Saddle-point analysis reveals a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in 0+1 dimensions. In addition to studying the purity of the entire system, we study subsystem purities and relate these results to manifestations of quantum error correction in the model. We also comment on the experimental feasibility for simulating this averaged purity, and corroborate our results with exact diagonalization for modest system sizes., Comment: 32 pages, 16 figures

Fast scramblers are dynamical quantum systems that produce many-body entanglement on a timescale that grows logarithmically with the system size $N$. We propose and investigate a family of deterministic, fast scrambling quantum circuits realizable in near-term experiments with arrays of neutral atoms. We show that three experimental tools -- nearest-neighbour Rydberg interactions, global single-qubit rotations, and shuffling operations facilitated by an auxiliary tweezer array -- are sufficient to generate nonlocal interaction graphs capable of scrambling quantum information using only $O(\log N)$ parallel applications of nearest-neighbor gates. These tools enable direct experimental access to fast scrambling dynamics in a highly controlled and programmable way, and can be harnessed to produce highly entangled states with varied applications., Comment: 7 pages with 4 figures (plus 7-page supplement with 4 figures); updated to match published version

We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and $L^2$ estimates for related oscillatory integrals., Comment: 31 pages. Reorganized the proof of Proposition 4. Revised version incorporating referee's suggestions. To appear in The Journal of Geometric Analysis

Using an ensemble of atoms in an optical cavity, we engineer a family of nonlocal Heisenberg Hamiltonians with continuously tunable anisotropy of the spin-spin couplings. We thus gain access to a rich phase diagram, including a paramagnetic-to-ferromagnetic Ising phase transition that manifests as a diverging magnetic susceptibility at the critical point. The susceptibility displays a symmetry between Ising interactions and XY (spin-exchange) interactions of the opposite sign, which is indicative of the spatially extended atomic system behaving as a single collective spin. Images of the magnetization dynamics show that spin-exchange interactions protect the coherence of the collective spin, even against inhomogeneous fields that completely dephase the non-interacting and Ising systems. Our results underscore prospects for harnessing spin-exchange interactions to enhance the robustness of spin squeezing protocols.

We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities on the cone. We also construct a Sobolev space adapted to translations on the Heisenberg group to which these averaging operators map all $L^p$ functions boundedly., Comment: 32 pages, 2 figures. Added references, added detail to proof of Corollary 1.1, result unchanged. Made additional changes reflecting anonymous referee's comments

We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on non-local interactions which couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, non-random couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system., Comment: 6 pages, 4 figures (plus 7-page supplement with 4 figures)

We show that a class of random all-to-all spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram due to the pairwise separable nature of the couplings. By controlling the experimental parameters, one can tune between integrable and chaotic dynamics on the one hand, and between classical and quantum regimes on the other hand. For two special values of a spin-anisotropy parameter, the model exhibits rational-Gaudin type integrability and it is characterized by an extensive set of spin-bilinear integrals of motion, independent of the spin size. More generically, we find a novel integrable structure with conserved charges that are not purely bilinear. Instead, they develop `dressing tails' of higher-body terms, reminiscent of the dressed local integrals of motion found in Many-Body Localized phases. Surprisingly, this new type of integrable dynamics found in finite-size spin-1/2 systems disappears in the large-$S$ limit, giving way to classical chaos. We identify parameter regimes for characterizing these different dynamical behaviors in realistic experiments, in light of the limitations set by cavity dissipation., Comment: v3: Published version. Included more data and updated the figures accordingly

We report direct observations of photon-mediated spin-exchange interactions in an atomic ensemble. Interactions extending over a distance of 500 microns are generated within a cloud of cold rubidium atoms coupled to a single mode of light in an optical resonator. We characterize the system via quench dynamics and imaging of the local magnetization, verifying the coherence of the interactions and demonstrating optical control of their strength and sign. Furthermore, by initializing the spin-1 system in the mF = 0 Zeeman state, we observe correlated pair creation in the mF = +/- 1 states, a process analogous to spontaneous parametric down-conversion and to spin mixing in Bose-Einstein condensates. Our work opens new opportunities in quantum simulation with long-range interactions and in entanglement-enhanced metrology.

Given a quantum many-body system with few-body interactions, how rapidly can quantum information be hidden during time evolution? The fast scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N. We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using Lieb-Robinson bounds, generalized Sachdev-Ye-Kitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: we construct random few-body circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes, and suggest methods to experimentally study scrambling with as many as 100 sparsely-connected quantum degrees of freedom., Comment: 11+24 pages; 1+3 figures. v2: minor revisions, added references. v3: many revisions to improve presentation

The strong light-matter coupling attainable in optical cavities enables the generation of highly squeezed states of atomic ensembles. It was shown in [Phys. Rev. A 66, 022314 (2002)] how an effective one-axis twisting Hamiltonian can be realized in a cavity setup. Here, we extend this work and show how an effective two-axis twisting Hamiltonian can be realized in a similar cavity setup. We compare the two schemes in order to characterize their advantages. In the absence of decoherence, the two-axis Hamiltonian leads to more squeezing than the one-axis Hamiltonian. If limited by decoherence from spontaneous emission and cavity decay, we find roughly the same level of squeezing for the two schemes scaling as (NC)^(1/2) where C is the single atom cooperativity and N is the total number of atoms. When compared to an ideal squeezing operation, we find that for specific initial states, a dissipative version of the one-axis scheme attains higher fidelity than the unitary one-axis scheme or the two-axis scheme. However, the unitary one-axis and two-axis schemes perform better for general initial states., Comment: 13 pages, 6 figures