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Nanomagnetic artificial spin-systems are ideal candidates for neuromorphic hardware. Their passive memory, state-dependent dynamics and nonlinear GHz spin-wave response provide powerful computation. However, any single physical reservoir must trade-off between performance metrics including nonlinearity and memory-capacity, with the compromise typically hard-coded. Here, we present three artificial spin-systems and show how tuning system geometry and dynamics defines computing performance. We engineer networks where each node is a high-dimensional physical reservoir, implementing parallel, deep and multilayer physical neural network architectures. This solves the issue of physical reservoir performance compromise, allowing a small suite of synergistic physical systems to address diverse tasks and provide a broad range of reprogrammable computationally-distinct configurations. These networks outperform any single reservoir across a broad taskset. Crucially, we move beyond reservoir computing to present a method for reconfigurably programming inter-layer network connections, enabling on-demand task optimised performance.

Significance Technological improvement is the most important cause of long-term economic growth. We study the effects of technology improvement in the setting of a production network, in which each producer buys input goods and converts them to other goods, selling the product to households or other producers. We show how this network amplifies the effects of technological improvements as they propagate along chains of production. Longer production chains for an industry bias it toward faster price reduction, and longer production chains for a country bias it toward faster growth. These predictions are in good agreement with data and improve with the passage of time, demonstrating a key influence of production chains in price change and output growth over the long term., Technological improvement is the most important cause of long-term economic growth. In standard growth models, technology is treated in the aggregate, but an economy can also be viewed as a network in which producers buy goods, convert them to new goods, and sell the production to households or other producers. We develop predictions for how this network amplifies the effects of technological improvements as they propagate along chains of production, showing that longer production chains for an industry bias it toward faster price reduction and that longer production chains for a country bias it toward faster growth. These predictions are in good agreement with data from the World Input Output Database and improve with the passage of time. The results show that production chains play a major role in shaping the long-term evolution of prices, output growth, and structural change.

Recent studies in magnetic nanolithography show that a variety of complex magnetic states emerge as a function of a single magnetic island's aspect ratio. We propose a model which, in addition to fitting experiments, predicts magnetic states with continuous symmetry at particular aspect ratios and reveals a duality between vortex and vertex states. Our model then opens new means of engineering novel types of artificial spin systems, and their application to complex magnetic textures in devices and computing., Comment: 3 pages + epsilon + 18 supplementary material

In the last decade, nanoscale resistive devices with memory have been the subject of intense study because of their possible use in brain-inspired computing. However, operational endurance is one of the limiting factors in the adoption of such technology. For this reason, we discuss the emergence of current-induced memristance in magnetic materials, known for their durability. We show analytically and numerically that a single ferromagnetic layer can possess GHz memristance, due to a combination of two factors: a current-induced transfer of angular momentum (Zhang-Li torque) and the anisotropic magnetoresistance (AMR). We term the resulting effect the anisotropic magneto-memristance (AMM). We connect the AMM to the topology of the magnetization state, within a simple model of a 1-dimensional annulus-shaped magnetic layer, confirming the analytical results with micromagnetic simulations for permalloy. Our results open a new path towards the realization of single-layer magnetic memristive devices operating at GHz frequencies., 20 pages + supplementary material; 31 pages in total; 11 figures; Introduction: https://www.youtube.com/watch?v=WbUTtJq8FfM&feature=youtu.be

We introduce a Lyapunov function for the dynamics of memristive circuits, and compare the effectiveness of memristors in minimizing the function to widely used optimization software. We study in particular three classes of problems which can be directly embedded in a circuit topology, and show that memristors effectively attempt at (quickly) extremizing these functionals.

Strongly interacting artificial spin systems are moving beyond mimicking naturally occurring materials to emerge as versatile functional platforms, from reconfigurable magnonics to neuromorphic computing. Typically, artificial spin systems comprise nanomagnets with a single magnetization texture: collinear macrospins or chiral vortices. By tuning nanoarray dimensions we have achieved macrospin-vortex bistability and demonstrated a four-state metamaterial spin system, the 'artificial spin-vortex ice' (ASVI). ASVI can host Ising-like macrospins with strong ice-like vertex interactions and weakly coupled vortices with low stray dipolar field. Vortices and macrospins exhibit starkly differing spin-wave spectra with analogue mode amplitude control and mode frequency shifts of Δf = 3.8 GHz. The enhanced bitextural microstate space gives rise to emergent physical memory phenomena, with ratchet-like vortex injection and history-dependent non-linear fading memory when driven through global magnetic field cycles. We employed spin-wave microstate fingerprinting for rapid, scalable readout of vortex and macrospin populations, and leveraged this for spin-wave reservoir computation. ASVI performs non-linear mapping transformations of diverse input and target signals in addition to chaotic time-series forecasting.

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices $H=\langle H\rangle +\delta H$, with $\langle H\rangle$ rank-$1$ and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement., Comment: 20 pages, 4 figures. Clarifications and appendices added

We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref.[1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs, Entanglement, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CP-maps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones., Comment: Submission to SciPost

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. Via the mapping of spin ices to the Ising model, we clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the 1D Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such technique can be used also to obtain an upper bound, we find that in all the examples we studied but one, another upper bound based on Schrijver inequality is tighter., Comment: 12 pages double column; result unchanged, paper clarified

Strongly-interacting artificial spin systems are moving beyond mimicking naturally-occurring materials to emerge as versatile functional platforms, from reconfigurable magnonics to neuromorphic computing. Typically artificial spin systems comprise nanomagnets with a single magnetisation texture: collinear macrospins or chiral vortices. By tuning nanoarray dimensions we achieve macrospin/vortex bistability and demonstrate a four-state metamaterial spin-system 'Artificial Spin-Vortex Ice' (ASVI). ASVI can host Ising-like macrospins with strong ice-like vertex interactions, and weakly-coupled vortices with low stray dipolar-field. Vortices and macrospins exhibit starkly-differing spin-wave spectra with analogue-style mode-amplitude control and mode-frequency shifts of df = 3.8 GHz. The enhanced bi-textural microstate space gives rise to emergent physical memory phenomena, with ratchet-like vortex training and history-dependent nonlinear fading memory when driven through global field cycles. We employ spin-wave microstate fingerprinting for rapid, scaleable readout of vortex and macrospin populations and leverage this for spin-wave reservoir computation. ASVI performs linear and non-linear mapping transformations of diverse input signals as well as chaotic time-series forecasting. Energy costs of machine learning are spiralling unsustainably, developing low-energy neuromorphic computation hardware such as ASVI is crucial to achieving a zero-carbon computational future.

Quantum nano-devices are fundamental systems in quantum thermodynamics that have been the subject of profound interest in recent years. Among these, quantum batteries play a very important role. In this paper we lay down a theory of random quantum batteries and provide a systematic way of computing the average work and work fluctuations in such devices by investigating their typical behavior. We show that the performance of random quantum batteries exhibits typicality and depends only on the spectral properties of the time evolving operator, the initial state and the measuring Hamiltonian. At given revival times a random quantum battery features a quantum advantage over classical random batteries. Our method is particularly apt to be used both for exactly solvable models like the Jaynes-Cummings model or in perturbation theory, e.g., systems subject to harmonic perturbations. We also study the setting of quantum adiabatic random batteries., Comment: 7 pages double column + 14 supplementary material single column; minor corrections; Accepted in Phys Rev Research

We study the role of coherence in closed and open quantum batteries. We obtain upper bounds to the work performed or energy exchanged by both closed and open quantum batteries in terms of coherence. Specifically, we show that the energy storage can be bounded by the Hilbert-Schmidt coherence of the density matrix in the spectral basis of the unitary operator that encodes the evolution of the battery. We also show that an analogous bound can be obtained in terms of the battery's Hamiltonian coherence in the basis of the unitary operator by evaluating their commutator. We apply these bounds to a 4-state quantum system and the anisotropic XY Ising model in the closed system case, and the Spin-Boson model in the open case., Comment: 14 pages double column, 9 pages appendix; revised manuscript with citations and new plot; results unchanged

Reservoir computing is a machine learning paradigm that uses a high-dimensional dynamical system, or \emph{reservoir}, to approximate and predict time series data. The scale, speed and power usage of reservoir computers could be enhanced by constructing reservoirs out of electronic circuits, and several experimental studies have demonstrated promise in this direction. However, designing quality reservoirs requires a precise understanding of how such circuits process and store information. We analyze the feasibility and optimal design of electronic reservoirs that include both linear elements (resistors, inductors, and capacitors) and nonlinear memory elements called memristors. We provide analytic results regarding the feasibility of these reservoirs, and give a systematic characterization of their computational properties by examining the types of input-output relationships that they can approximate. This allows us to design reservoirs with optimal properties. By introducing measures of the total linear and nonlinear computational capacities of the reservoir, we are able to design electronic circuits whose total computational capacity scales extensively with the system size. Our electronic reservoirs can match or exceed the performance of conventional "echo state network" reservoirs in a form that may be directly implemented in hardware., Comment: 11 pages double column + supplementary material; title changed, results clarified; to appear in PRE

We present a baseline stochastic framework for assessing inter-sectorial relationships in a generic economy. We show that - irrespective of the specific features of the technology matrix for a given country or a particular year - the Leontief multipliers (and any upstreamness/downstreamness indicator computed from the Leontief inverse matrix) follow a universal pattern, which we characterize analytically. We formulate a universal benchmark to assess the structural inter-dependence of sectors in a generic economy. Several empirical results on World Input-Output Database (WIOD, 2013 Release) are presented that corroborate our findings.

Systems of interacting nanomagnets known as artificial spin ice1–4 have allowed the design, realization and study of geometrically frustrated exotic collective states5–10 that are absent in natural magnets. We have experimentally measured11,12 the thermally induced moment fluctuations in the Shakti geometry of artificial spin ice. We show that its disordered moment configuration is a topological phase described by an emergent dimer-cover model13 with excitations that can be characterized as topologically charged defects. Examination of the low-energy dynamics of the system confirms that these effective topological charges have long lifetimes associated with their topological protection, that is, they can be created and annihilated only as charge pairs with opposite sign and are kinetically constrained. This manifestation of classical topological order14–19 demonstrates that geometrical design in nanomagnetic systems can lead to emergent, topologically protected kinetics that can limit pathways to equilibration and ergodicity. Experiments on the Shakti geometry of artificial spin ice show that its low-energy excitations are topologically protected, and that an emergent classical topological order influences the ergodicity and equilibration of this nanomagnetic system.