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In the pursuit of scalable and energy-efficient neuromorphic devices, recent research has unveiled a novel category of spiking oscillators, termed "thermal neuristors." These devices function via thermal interactions among neighboring vanadium dioxide resistive memories, emulating biological neuronal behavior. Here, we show that the collective dynamical behavior of networks of these neurons showcases a rich phase structure, tunable by adjusting the thermal coupling and input voltage. Notably, we identify phases exhibiting long-range order that, however, does not arise from criticality, but rather from the time non-local response of the system. In addition, we show that these thermal neuristor arrays achieve high accuracy in image recognition and time series prediction through reservoir computing, without leveraging long-range order. Our findings highlight a crucial aspect of neuromorphic computing with possible implications on the functioning of the brain: criticality may not be necessary for the efficient performance of neuromorphic systems in certain computational tasks., Comment: 20 pages, 14 figures

Digital MemComputing machines (DMMs), which employ nonlinear dynamical systems with memory (time non-locality), have proven to be a robust and scalable unconventional computing approach for solving a wide variety of combinatorial optimization problems. However, most of the research so far has focused on the numerical simulations of the equations of motion of DMMs. This inevitably subjects time to discretization, which brings its own (numerical) issues that would be otherwise absent in actual physical systems operating in continuous time. Although hardware realizations of DMMs have been previously suggested, their implementation would require materials and devices that are not so easy to integrate with traditional electronics. Addressing this, our study introduces a novel hardware design for DMMs, utilizing readily available electronic components. This approach not only significantly boosts computational speed compared to current models but also exhibits remarkable robustness against additive noise. Crucially, it circumvents the limitations imposed by numerical noise, ensuring enhanced stability and reliability during extended operations. This paves a new path for tackling increasingly complex problems, leveraging the inherent advantages of DMMs in a more practical and accessible framework., Comment: 24 pages, 17 figures

While the complementary metal-oxide semiconductor (CMOS) technology is the mainstream for the hardware implementation of neural networks, we explore an alternative route based on a new class of spiking oscillators we call thermal neuristors, which operate and interact solely via thermal processes. Utilizing the insulator-to-metal transition in vanadium dioxide, we demonstrate a wide variety of reconfigurable electrical dynamics mirroring biological neurons. Notably, inhibitory functionality is achieved just in a single oxide device, and cascaded information flow is realized exclusively through thermal interactions. To elucidate the underlying mechanisms of the neuristors, a detailed theoretical model is developed, which accurately reflects the experimental results. This study establishes the foundation for scalable and energy-efficient thermal neural networks, fostering progress in brain-inspired computing.

While most neuromorphic systems are based on nanoscale electronic devices, nature relies on ions for energy-efficient information processing. Therefore, finding memristive nanofluidic devices is a milestone toward realizing electrolytic computers mimicking the brain down to its basic principles of operation. Here, we present a nanofluidic device designed for circuit scale in-memory processing that combines single-digit nanometric confinement and large entrance asymmetry. Our fabrication process is scalable while the device operates at the second timescale with a conductance ratio in the range 10-60. In-operando optical microscopy unveils the origin of memory, arising from the reversible formation of liquid blisters modulating the device conductance. The combination of features of these mechano-ionic memristive switches permits assembling logic circuits composed of two interactive devices and an ohmic resistor. These results open the way to design multi-component ionic machinery, such as nanofluidic neural networks, and implementing brain-inspired ionic computations.

Memcomputing is a novel computing paradigm beyond the von-Neumann one. Its digital version is designed for the efficient solution of combinatorial optimization problems, which emerge in various fields of science and technology. Previously, the performance of digital memcomputing machines (DMMs) was demonstrated using software simulations of their ordinary differential equations. Here, we present the first hardware realization of a DMM algorithm on a low-cost FPGA board. In this demonstration, we have implemented a Boolean satisfiability problem solver. To optimize the use of hardware resources, the algorithm was partially parallelized. The scalability of the present implementation is explored and our FPGA-based results are compared to those obtained using a python code running on a traditional (von-Neumann) computer, showing one to two orders of magnitude speed-up in time to solution. This initial small-scale implementation is projected to state-of-the-art FPGA boards anticipating further advantages of the hardware realization of DMMs over their software emulation.

MemComputing is a new model of computation that exploits the non-equilibrium property-we call 'memory'-of any physical system to respond to external perturbations by keeping track of how it has reacted at previous times. Its digital, scalable version maps a finite string of symbols into a finite string of symbols. In this paper, I will discuss some analogies of the operation of MemComputing machines-in general, and digital in particular-with a few physical properties of the biological brain. These analogies could be a source of inspiration to improve on the design of these machines. In turn, they could suggest new directions of study in (computational) neuroscience.

Digital memcomputing machines (DMMs) are a new class of computing machines that employ non-quantum dynamical systems with memory to solve combinatorial optimization problems. Here, we show that the time to solution (TTS) of DMMs follows an inverse Gaussian distribution, with the TTS self-averaging with increasing problem size, irrespective of the problem they solve. We provide both an analytical understanding of this phenomenon and numerical evidence by solving instances of the 3-SAT (satisfiability) problem. The self-averaging property of DMMs with problem size implies that they are increasingly insensitive to the detailed features of the instances they solve. This is in sharp contrast to traditional algorithms applied to the same problems, illustrating another advantage of this physics-based approach to computation., Comment: 9 pages, 13 figures

In the "Beyond Moore's Law" era, with increasing edge intelligence, domain-specific computing embracing unconventional approaches will become increasingly prevalent. At the same time, adopting a variety of nanotechnologies will offer benefits in energy cost, computational speed, reduced footprint, cyber resilience, and processing power. The time is ripe for a roadmap for unconventional computing with nanotechnologies to guide future research, and this collection aims to fill that need. The authors provide a comprehensive roadmap for neuromorphic computing using electron spins, memristive devices, two-dimensional nanomaterials, nanomagnets, and various dynamical systems. They also address other paradigms such as Ising machines, Bayesian inference engines, probabilistic computing with p-bits, processing in memory, quantum memories and algorithms, computing with skyrmions and spin waves, and brain-inspired computing for incremental learning and problem-solving in severely resource-constrained environments. These approaches have advantages over traditional Boolean computing based on von Neumann architecture. As the computational requirements for artificial intelligence grow 50 times faster than Moore's Law for electronics, more unconventional approaches to computing and signal processing will appear on the horizon, and this roadmap will help identify future needs and challenges. In a very fertile field, experts in the field aim to present some of the dominant and most promising technologies for unconventional computing that will be around for some time to come. Within a holistic approach, the goal is to provide pathways for solidifying the field and guiding future impactful discoveries., Comment: 80 pages accepted in Nano Futures

Inspired by the advancements in large language models based on transformers, we introduce the transformer quantum state (TQS): a versatile machine learning model for quantum many-body problems. In sharp contrast to Hamiltonian/task specific models, TQS can generate the entire phase diagram, predict field strengths with experimental measurements, and transfer such a knowledge to new systems it has never been trained on before, all within a single model. With specific tasks, fine-tuning the TQS produces accurate results with small computational cost. Versatile by design, TQS can be easily adapted to new tasks, thereby pointing towards a general-purpose model for various challenging quantum problems., Comment: 12 pages, 13 figures

Quantum computing employs some quantum phenomena to process information. It has been hailed as the future of computing but it is plagued by serious hurdles when it comes to its practical realization. MemComputing is a new paradigm that instead employs non-quantum dynamical systems and exploits time non-locality (memory) to compute. It can be efficiently emulated in software and its path towards hardware is more straightforward. I will discuss some analogies between these two computing paradigms, and the major differences that set them apart., Comment: 4 pages

The Peltier effect is the reverse phenomenon of the Seebeck effect, and has been observed experimentally in nanoscale junctions. However, despite its promising applications in local cooling of nanoelectronic devices, the role of strong electron correlations on such a phenomenon is still unclear. Here, by analyzing the thermoelectric properties of quantum impurity systems out of equilibrium, we unveil the essential role of electron-electron interactions and quantum resonant states in Peltier cooling, leading to the prediction of the Kondo cooling phenomenon. The existence of such Kondo cooling is validated by a reverse heat current and a lowered local temperature in a model junction. The discovery of this unconventional Peltier cooling offers a new approach toward nano-refrigeration, and highlights the unique role of strong electron correlations in nonequilibrium quantum systems., Comment: 6 pages, 4 figures

Voltage controlled magnetic anisotropy (VCMA) is a low-energy alternative to manipulate the ferromagnetic state, which has been recently considered also in antiferromagnets (AFMs). Here, we theoretically demonstrate that VCMA can be used to excite linear and parametric resonant modes in easy-axis AFMs with perpendicular anisotropy, thus opening the way for an efficient electrical control of the Neel vector, and for detection of high-frequency dynamics. Our work leads to two key results: (i) VCMA parametric pumping experiences the so-called exchange enhancement of the coupling efficiency and, thus, is 1-2 orders of magnitude more efficient than microwave magnetic fields or spin-orbit-torques, and (ii) it also allows for zero-field parametric resonance, which cannot be achieved by other parametric pumping mechanisms in AFMs with out-of-plane easy axis. Therefore, we demonstrate that the VCMA parametric pumping is the most promising method for coherent excitation and manipulation of AFM order in perpendicular easy-axis AFMs.

Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent even for extremely simple quantum states. Here, we propose to use mode-assisted training that provides global information via the modes of the NN distribution. Applied to quantum state tomography using restricted Boltzmann machines, this method improves the quality of reconstructed quantum states by orders of magnitude. The method is applicable to other types of NNs and may efficiently tackle problems previously unmanageable., Comment: 12 pages, 13 figures

Custodial symmetries are common in the Standard Model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory (memcircuit). In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spreads the state across the circuit. However, the resulting state is still protected against perturbations by the ensuing custodial chiral symmetry. These predictions can be verified experimentally and demonstrate the interplay between symmetry and memory., Comment: 11 pages, 11 figures

Information processing and storing by the same physical system is emerging as a promising alternative to traditional computing platforms. In turn, this requires the realization of elementary units whose memory content can be easily tuned and controlled. Here, we introduce a polariton-based quantum memristor where the memristive nature arises from the inter-cavity polariton exchange and is controlled by a time-varying atom-cavity detuning. A dynamical hysteresis is characterized by the fluctuations in the instantaneous polariton number, where the history information is encoded into a dynamical phase. Using a Lindblad master equation approach, we find that features of the quantum memristor dynamics, such as the area and circulation of the hysteresis loop, showcase a kind of "plasticity" controlled by quantum state initialization. This makes this quantum memristor very versatile for a wide range of applications, Comment: 9 pages, 4 figures

Surface plasmon enhanced processes and hot-carrier dynamics in plasmonic nanostructures are of great fundamental interest to reveal light-matter interactions at the nanoscale. Using plasmonic tunnel junctions as a platform supporting both electrically- and optically excited localized surface plasmons, we report a much greater (over 1000x) plasmonic light emission at upconverted photon energies under combined electro-optical excitation, compared with electrical or optical excitation separately. Two mechanisms compatible with the form of the observed spectra are interactions of plasmon-induced hot carriers and electronic anti-Stokes Raman scattering. Our measurement results are in excellent agreement with a theoretical model combining electro-optical generation of hot carriers through non-radiative plasmon excitation and hot-carrier relaxation. We also discuss the challenge of distinguishing relative contributions of hot carrier emission and the anti-Stokes electronic Raman process. This observed increase in above-threshold emission in plasmonic systems may open avenues in on-chip nanophotonic switching and hot carrier photocatalysis., Comment: 21 pages, 4 figures, plus 27 pages Supporting Info with 7 figures

It is believed that the $\pm J$ Ising spin-glass does not order at finite temperatures in dimension $d=2$. However, using a graphical representation and a contour argument, we prove rigorously the existence of a finite-temperature phase transition in $d\geq 2$ with $T_c \geq 0.4$. In the graphical representation, the low-temperature phase allows for the coexistence of multiple infinite clusters each with a rigidly aligned spin-overlap state. These clusters correlate negatively with each other, and are entropically stable without breaking any global symmetry. They can emerge in most graph structures and disorder measures., Comment: The proof of the finitude of zero-energy contours in Appendix D.2 contains a technical error, where we failed to notice that opening a red bond in any of the four quadrants can possibly restrict opening of any blue bonds in the contour. This means that the probability that the contour is zero-energy (no blue-bonds) cannot be bounded exponentially, thus the result does not immediately follow

The deep extension of the restricted Boltzmann machine (RBM), known as the deep Boltzmann machine (DBM), is an expressive family of machine learning models which can serve as compact representations of complex probability distributions. However, jointly training DBMs in the unsupervised setting has proven to be a formidable task. A recent technique we have proposed, called mode-assisted training, has shown great success in improving the unsupervised training of RBMs. Here, we show that the performance gains of the mode-assisted training are even more dramatic for DBMs. In fact, DBMs jointly trained with the mode-assisted algorithm can represent the same data set with orders of magnitude lower number of total parameters compared to state-of-the-art training procedures and even with respect to RBMs, provided a fan-in network topology is also introduced. This substantial saving in number of parameters makes this training method very appealing also for hardware implementations., Comment: 7 pages, 4 figures

Spin glasses are notoriously difficult to study both analytically and numerically due to the presence of frustration and metastability. Their highly non-convex landscapes require collective updates to explore efficiently. Currently, most state-of-the-art algorithms rely on stochastic spin clusters to perform non-local updates, but such "cluster algorithms" lack general efficiency. Here, we introduce a non-equilibrium approach for simulating spin glasses based on classical dynamics with memory. By simulating various classes of 3d spin glasses (Edwards-Anderson, partially-frustrated, and fully-frustrated models), we find that memory dynamically promotes critical spin clusters during time evolution, in a self-organizing manner. This facilitates an efficient exploration of the low-temperature phases of spin glasses., Comment: 21 pages, 9 figures

Digital memcomputing machines (DMMs) are a novel, non-Turing class of machines designed to solve combinatorial optimization problems. They can be physically realized with continuous-time, non-quantum dynamical systems with memory (time non-locality), whose ordinary differential equations (ODEs) can be numerically integrated on modern computers. Solutions of many hard problems have been reported by numerically integrating the ODEs of DMMs, showing substantial advantages over state-of-the-art solvers. To investigate the reasons behind the robustness and effectiveness of this method, we employ three explicit integration schemes (forward Euler, trapezoid and Runge-Kutta 4th order) with a constant time step, to solve 3-SAT instances with planted solutions. We show that, (i) even if most of the trajectories in the phase space are destroyed by numerical noise, the solution can still be achieved; (ii) the forward Euler method, although having the largest numerical error, solves the instances in the least amount of function evaluations; and (iii) when increasing the integration time step, the system undergoes a "solvable-unsolvable transition" at a critical threshold, which needs to decay at most as a power law with the problem size, to control the numerical errors. To explain these results, we model the dynamical behavior of DMMs as directed percolation of the state trajectory in the phase space in the presence of noise. This viewpoint clarifies the reasons behind their numerical robustness and provides an analytical understanding of the unsolvable-solvable transition. These results land further support to the usefulness of DMMs in the solution of hard combinatorial optimization problems., Comment: 12 pages, 11 figures

The end of Moore's law for CMOS technology has prompted the search for low-power computing alternatives, resulting in several promising proposals based on magnetic logic[1-8]. One approach aims at tailoring arrays of nanomagnetic islands in which the magnetostatic interactions constrain the equilibrium orientation of the magnetization to embed logical functionalities[9-12]. Despite the realization of several proofs of concepts of such nanomagnetic logic[13-15], it is still unclear what the advantages are compared to the widespread CMOS designs, due to their need for clocking[16, 17] and/or thermal annealing [18,19] for which fast convergence to the ground state is not guaranteed. In fact, it seems increasingly evident that "beyond CMOS" technology will require a fundamental rethinking of our computing paradigm[20]. In this respect, a type of terminal-agnostic logic was suggested[21], where a given gate is able to "self-organize" into its correct logical states, regardless of whether the signal is applied to the traditional input terminals, or the output terminals. Here, we introduce nanomagnetic self-organizing balanced logic gates, that employ stray-field coupled nanomagnetic islands to perform terminal-agnostic logic. We illustrate their capabilities by implementing reversible Boolean circuitry to solve a two-bit factorization problem via numerical modelling. In view of their design and mode of operation, we expect these systems to improve significantly over those suggested in Ref.[21], thus offering an alternative path to explore memcomputing, whose usefulness has already been demonstrated by solving a variety of hard combinatorial optimization problems[22].

Measuring local temperatures of open systems out of equilibrium is emerging as a novel approach to study the local thermodynamic properties of nanosystems. An operational protocol has been proposed to determine the local temperature by coupling a probe to the system and then minimizing the perturbation to a certain local observable of the probed system. In this paper, we first show that such a local temperature is unique for a single quantum impurity and the given local observable. We then extend this protocol to open systems consisting of multiple quantum impurities by proposing a local minimal perturbation condition (LMPC). The influence of quantum resonances on the local temperature is elucidated by both analytic and numerical results. In particular, we demonstrate that quantum resonances may give rise to strong oscillations of the local temperature along a multiimpurity chain under a thermal bias., Comment: 13 pages, 7 figures

Restricted Boltzmann machines (RBMs) are a powerful class of generative models, but their training requires computing a gradient that, unlike supervised backpropagation on typical loss functions, is notoriously difficult even to approximate. Here, we show that properly combining standard gradient updates with an off-gradient direction, constructed from samples of the RBM ground state (mode), improves their training dramatically over traditional gradient methods. This approach, which we call mode training, promotes faster training and stability, in addition to lower converged relative entropy (KL divergence). Along with the proofs of stability and convergence of this method, we also demonstrate its efficacy on synthetic datasets where we can compute KL divergences exactly, as well as on a larger machine learning standard, MNIST. The mode training we suggest is quite versatile, as it can be applied in conjunction with any given gradient method, and is easily extended to more general energy-based neural network structures such as deep, convolutional and unrestricted Boltzmann machines., Comment: 28 pages, 4 figures. Revision: Updated footnote format

Novel computational paradigms may provide the blueprint to help solving the time and energy limitations that we face with our modern computers, and provide solutions to complex problems more efficiently (with reduced time, power consumption and/or less device footprint) than is currently possible with standard approaches. Spintronics offers a promising basis for the development of efficient devices and unconventional operations for at least three main reasons: (i) the low-power requirements of spin-based devices, i.e., requiring no standby power for operation and the possibility to write information with small dynamic energy dissipation, (ii) the strong nonlinearity, time nonlocality, and/or stochasticity that spintronic devices can exhibit, and (iii) their compatibility with CMOS logic manufacturing processes. At the same time, the high endurance and speed of spintronic devices means that they can be rewritten or reconfigured frequently over the lifetime of a circuit, a feature that is essential in many emerging computing concepts. In this perspective, we will discuss how spintronics may aid in the realization of efficient devices primarily based on magnetic tunnel junctions and how those devices can impact in the development of three unconventional computing paradigms, namely, reservoir computing, probabilistic computing and memcomputing that in our opinion may be used to address some limitations of modern computers, providing a realistic path to intelligent hybrid CMOS-spintronic systems., Comment: 18 pages and 4 figures

The temperature of a physical system is operationally defined in physics as "that quantity which is measured by a thermometer" weakly coupled to, and at equilibrium with the system. This definition is unique only at global equilibrium in view of the zeroth law of thermodynamics: when the system and the thermometer have reached equilibrium, the "thermometer degrees of freedom" can be traced out and the temperature read by the thermometer can be uniquely assigned to the system. Unfortunately, such a procedure cannot be straightforwardly extended to a system out of equilibrium, where local excitations may be spatially inhomogeneous and the zeroth law of thermodynamics does not hold. With the advent of several experimental techniques that attempt to extract a single parameter characterizing the degree of local excitations of a (mesoscopic or nanoscale) system out of equilibrium, this issue is making a strong comeback to the forefront of research. In this paper, we will review the difficulties to define a unique temperature out of equilibrium, the majority of definitions that have been proposed so far, and discuss both their advantages and limitations. We will then examine a variety of experimental techniques developed for measuring the non-equilibrium local temperatures under various conditions. Finally we will discuss the physical implications of the notion of local temperature, and present the practical applications of such a concept in a variety of nanosystems out of equilibrium., Comment: Corrected version to Physics Reports

Many optimization problems can be cast into the maximum satisfiability (MAX-SAT) form, and many solvers have been developed for tackling such problems. To evaluate a MAX-SAT solver, it is convenient to generate hard MAX-SAT instances with known solutions. Here, we propose a method of generating weighted MAX-2-SAT instances inspired by the frustrated-loop algorithm used by the quantum annealing community. We extend the algorithm for instances of general bipartite couplings, with the associated optimization problem being the minimization of the restricted Boltzmann machine (RBM) energy over the nodal values, which is useful for effectively pre-training the RBM. The hardness of the generated instances can be tuned through a central parameter known as the frustration index. Two versions of the algorithm are presented: the random- and structured-loop algorithms. For the random-loop algorithm, we provide a thorough theoretical and empirical analysis on its mathematical properties from the perspective of frustration, and observe empirically a double phase transition behavior in the hardness scaling behavior driven by the frustration index. For the structured-loop algorithm, we show that it offers an improvement in hardness over the random-loop algorithm in the regime of high loop density, with the variation of hardness tunable through the concentration of frustrated weights., Comment: 38 pages, 9 figures

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent $\tau= 3/2$. We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent $\tau = 1.51 \pm 0.02$, in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution., Comment: 5 pages, 3 figures

Digital memcomputing machines (DMMs) are a class of computational machines designed to solve combinatorial optimization problems. A practical realization of DMMs can be accomplished via electrical circuits of highly non-linear, point-dissipative dynamical systems engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency ("logical defects") from the circuit. By employing a supersymmetric theory of dynamics, a DMM can be described by a cohomological field theory that allows for computation of certain topological matrix elements on instantons that have the mathematical meaning of intersection numbers on instantons. We discuss the "dynamical" meaning of these matrix elements, and argue that the number of elementary instantons needed to reach the solution cannot exceed the number of state variables of DMMs, which in turn can only grow at most polynomially with the size of the problem. These results shed further light on the relation between logic, dynamics and topology in digital memcomputing.

We introduce the notion, and develop the theory of local-noise spectroscopy (LNS) - a tool to study the properties of systems far from equilibrium by means of flux density correlations. As a test bed, we apply it to biased molecular junctions. This tool naturally extends those based on local fluxes, while providing complementary information on the system. As examples of the rich phenomenology that one can study with this approach, we show that LNS can be used to yield information on microscopic properties of bias-induced light emission in junctions, provide local resolution of intra-system interactions, and employed as a nano-thermometry tool. Although LNS may, at the moment, be difficult to realize experimentally, it can nonetheless be used as a powerful theoretical tool to infer a wide range of physical properties on a variety of systems of present interest., Comment: 7 pages, 4 figures

Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of clusters of variables, such solvers are able to navigate their state space and locate solutions more efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a non-convex landscape. Here, we give evidence that a benchmark derived from quantum annealing studies is solvable in polynomial time using digital memcomputing machines, which utilize a collection of dynamical components with memory to represent the structure of the underlying optimization problem. To illustrate the role of memory and clarify the structure of these solvers we propose a simple model of these machines that demonstrates the emergence of long-range order. This model, when applied to finding the ground state of the Ising frustrated-loop benchmarks, undergoes a transient phase of avalanches which can span the entire lattice and demonstrates a connection between long-range behavior and their probability of success. These results establish the advantages of computational approaches based on collective dynamics of continuous dynamical systems.

Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class efficiently, while requiring a reasonable compute time. However, although these algorithms have reached various degrees of success over the years, they still face considerable challenges when confronted with particularly hard problem instances, such as those of the MIPLIB 2010 library. In this work we propose a radically different non-algorithmic approach to ILP based on a novel physics-inspired computing paradigm: Memcomputing. This paradigm is based on digital (hence scalable) machines represented by appropriate electrical circuits with memory. These machines can be either built in hardware or, as we do here, their equations of motion can be efficiently simulated on our traditional computers. We first describe a new circuit architecture of memcomputing machines specifically designed to solve for the linear inequalities representing a general ILP problem. We call these self-organizing algebraic circuits, since they self-organize dynamically to satisfy the correct (algebraic) linear inequalities. We then show simulations of these machines using MATLAB running on a single core of a Xeon processor for several ILP benchmark problems taken from the MIPLIB 2010 library, and compare our results against a renowned commercial solver. We show that our approach is very efficient when dealing with these hard problems. In particular, we find within minutes feasible solutions for one of these hard problems (f2000 from MIPLIB 2010) whose feasibility, to the best of our knowledge, has remained unknown for the past eight years.

Memcomputing is a novel paradigm of computation that utilizes dynamical elements with memory to both store and process information on the same physical location. Its building blocks can be fabricated in hardware with standard electronic circuits, thus offering a path to its practical realization. In addition, since memcomputing is based on non-quantum elements, the equations of motion describing these machines can be simulated efficiently on standard computers. In fact, it was recently realized that memcomputing, and in particular its digital (hence scalable) version, when simulated on a classical machine provides a significant speed-up over state-of-the-art algorithms on a variety of non-convex problems. Here, we stress-test the capabilities of this approach on finding approximate solutions to hard combinatorial optimization problems. These fall into a class which is known to require exponentially growing resources in the worst cases, even to generate approximations. We recently showed that in a region where state of the art algorithms demonstrate this exponential growth, simulations of digital memcomputing machines performed using the Falcon$^\copyright$ simulator of MemComputing, Inc. only require time and memory resources that scale linearly. These results are extended in a stress-test up to $64\times10^6$ variables (corresponding to about 1 billion literals), namely the largest case that we could fit on a single node with 128 GB of DRAM. Since memcomputing can be applied to a wide variety of optimization problems, this stress test shows the considerable advantage of non-combinatorial, physics-inspired approaches over standard combinatorial ones.

The quantum variance of an observable is a fundamental quantity in quantum mechanics, and the variance provides additional information other than the average itself. By examining the relation between the particle-current variance $(\delta J)^2$ and the average current $J$ in both closed and open interacting fermionic systems, we show the emergence of a multi-valued Lissajous curve between $\delta J$ and $J$ due to interactions. As a closed system we considered the persistent current in a benzene-like lattice enclosing an effective magnetic flux and solved it by exact diagonalization. For the open system, the steady-state current flowing through a few lattice sites coupled to two particle reservoirs was investigated using a Lindblad equation. In both cases, interactions open a loop and change the topology of the corresponding $\delta J$-$J$ Lissajous curve, showing that this effect is model-independent. We finally discuss how the predicted phenomena can be observed in ultracold atoms, thus offering an alternative way of probing the dynamics of many-body systems., Comment: Main text: 5 pages, 3 figures. Supplemental information: 5 pages, 9 figures