Classical Annealing Research Articles

Anomaly detection speed-up by quantum restricted Boltzmann machines

Quantum machine learning promises to revolutionize traditional machine learning by efficiently addressing hard tasks for classical computation. While claims of quantum speed-up have been announced for gate-based quantum computers and photon-based boson samplers, demonstration of an advantage by adiabatic quantum annealers (AQAs) is open. Here we quantify the computational cost and the performance of restricted Boltzmann machines (RBMs), a widely investigated machine learning model, by classical and quantum annealing. Despite the lower computational complexity of the quantum RBM being lost due to physical implementation overheads, a quantum speed-up may arise as a reduction by orders of magnitude of the computational time. By employing real-world cybersecurity datasets, we observe that the negative phase on sufficiently challenging tasks is computed up to 64 times faster by AQAs during the exploitation phase. Therefore, although a quantum speed-up highly depends on the problem’s characteristics, it emerges in existing hardware on real-world data.

Diabatic quantum and classical annealing of the Sherrington-Kirkpatrick model

Quantum annealing is a contender to solve combinatorial optimization problems based on quantum dynamics. While significant efforts have been undertaken to investigate the quality of the solutions and the required run times, much less attention has been paid to understanding the dynamics of quantum annealing and the process leading to the solution during the sweep itself. In this comprehensive study, we investigate various aspects of the quantum annealing dynamics using different approaches. We perform quantum annealing, simulated quantum annealing, and classical annealing on several hundred instances of the Sherrington-Kirkpatrick model with intermediate system sizes up to 22 spins using numerical simulations. We observe qualitative differences between the quantum and classical methods, in particular at intermediate times, where a peak in the fidelity, also known as diabatic bump, appears for hard instances. Furthermore, we investigate the two-point correlation functions, which feature differences at intermediate times as well. At short times, however, the methods are similar again, which can be explained by relating the short-time expansion of quantum annealing to a high-temperature expansion, thus allowing one in principle to find the classical solution already at short times, albeit at prohibitive sampling cost.

Supply chain logistics with quantum and classical annealing algorithms

Noisy intermediate-scale quantum (NISQ) hardware is almost universally incompatible with full-scale optimization problems of practical importance which can have many variables and unwieldy objective functions. As a consequence, there is a growing body of literature that tests quantum algorithms on miniaturized versions of problems that arise in an operations research setting. Rather than taking this approach, we investigate a problem of substantial commercial value, multi-truck vehicle routing for supply chain logistics, at the scale used by a corporation in their operations. Such a problem is too complex to be fully embedded on any near-term quantum hardware or simulator; we avoid confronting this challenge by taking a hybrid workflow approach: we iteratively assign routes for trucks by generating a new binary optimization problem instance one truck at a time. Each instance has sim 2500 quadratic binary variables, putting it in a range that is feasible for NISQ quantum computing, especially quantum annealing hardware. We test our methods using simulated annealing and the D-Wave Hybrid solver as a place-holder in wait of quantum hardware developments. After feeding the vehicle routes suggested by these runs into a highly realistic classical supply chain simulation, we find excellent performance for the full supply chain. Our work gives a set of techniques that can be adopted in contexts beyond vehicle routing to apply NISQ devices in a hybrid fashion to large-scale problems of commercial interest.

Supplementing recurrent neural networks with annealing to solve combinatorial optimization problems

Combinatorial optimization problems can be solved by heuristic algorithms such as simulated annealing (SA) which aims to find the optimal solution within a large search space through thermal fluctuations. This algorithm generates new solutions through Markov-chain Monte Carlo techniques which can result in severe limitations, such as slow convergence and a tendency to stay within the same local search space at small temperatures. To overcome these shortcomings, we use the variational classical annealing (VCA) framework that combines autoregressive recurrent neural networks (RNNs) with traditional annealing to sample solutions that are uncorrelated. In this paper, we demonstrate the potential of using VCA as an approach to solving real-world optimization problems. We explore VCA’s performance in comparison with SA at solving three popular optimization problems: the maximum cut problem (Max-Cut), the nurse scheduling problem (NSP), and the traveling salesman problem (TSP). For all three problems, we find that VCA outperforms SA on average in the asymptotic limit by one or more orders of magnitude in terms of relative error. Interestingly, we reach large system sizes of up to 256 cities for the TSP. We also conclude that in the best case scenario, VCA can serve as a great alternative when SA fails to find the optimal solution.

Integration of Simulated Quantum Annealing in Parallel Tempering and Population Annealing for Heterogeneous-Profile QUBO Exploration

Simulated Quantum Annealing (SQA) is a heuristic algorithm which can solve Quadratic Unconstrained Binary Optimization (QUBO) problems by emulating the exploration of the solution space done by a quantum annealer. It mimics the quantum superposition and tunnelling effects through a set of correlated replicas of the spins system representing the problem to be solved and performing Monte Carlo steps. However, the effectiveness of SQA over a classical algorithm strictly depends on the cost/energy profile of the target problem. In fact, quantum annealing only performs well in exploring functions with high and narrow peaks, while classical annealing is better in overcoming flat and wide energy-profile barriers. Unfortunately, real-world problems have a heterogeneous solution space and the probability of success of each solver depends on the size of the energy profile region compatible with its exploration mechanism. Therefore, significant advantages could be obtained by exploiting hybrid solvers, which combine SQA and classical algorithms. This work proposes four new quantum-classical algorithms: Simulated Quantum Parallel Tempering (SQPT), Simulated Quantum Population Annealing (SQPA), Simulated Quantum Parallel Tempering - Population Annealing v1 (SQPTPA1) and Simulated Quantum Parallel Tempering - Population Annealing v2 (SQPTPA2). They are obtained by combining SQA, Parallel Tempering (PT), and Population Annealing (PA). Their results are compared with those provided by SQA, considering benchmark QUBO problems, characterized by different profiles. Even though this work is preliminary, the obtained results are encouraging and prove hybrid solvers’ potential in solving a generic optimization problem.

Magnetic properties evolution and crystallization behaviour of vacuum- and air-long-term-annealed rapidly quenched Fe80.3Co5Cu0.7B14 alloy.

This work aims to investigate the isothermal crystallization behaviour, crystal structure and magnetic properties evolution of long-term (up to 300h) low temperature (210 and 260°C) vacuum- and air-annealed Fe80.3Co5Cu0.7B14alloy. Before the α-Fe(Co) phase crystallization, the primary relaxation process has been identified at a temperature range up to 340°C. The relaxation process performed under 210°C for 300h did not initiate the crystallization process. However, the topological and compositional short-range rearrangements improved magnetic properties remarkably. Annealing 150h at 260°C helps to deliver enough energy to stabilize the glassy state and initiate the crystallization process fully. Structural and magnetic properties evolution of 150h annealing at 260°C corresponds to the evolution presented during isochronal 20min annealing at 310°C. Magnetic properties Bs = 1.75-1.79T, Hc < 20 A/m and P10/50are similar to those for 20min of annealing at 310°C. Comparison of core power losses from up to 400kHz frequency dependences of long-term low temperature annealed alloy with 20min classical annealing at 310°C shown that presented here long-term annealing is energetically insufficient to bring the glassy state system into the same low level of core power losses efficiency.

Hardening of Cobalt Ferrite Nanoparticles by Local Crystal Strain Release: Implications for Rare Earth Free Magnets.

In this work, we demonstrate that the reduction of the local internal stress by a low-temperature solvent-mediated thermal treatment is an effective post-treatment tool for magnetic hardening of chemically synthesized nanoparticles. As a case study, we used nonstoichiometric cobalt ferrite particles of an average size of 32(8) nm synthesized by thermal decomposition, which were further subjected to solvent-mediated annealing at variable temperatures between 150 and 320 °C in an inert atmosphere. The postsynthesis treatment produces a 50% increase of the coercive field, without affecting neither the remanence ratio nor the spontaneous magnetization. As a consequence, the energy product and the magnetic energy storage capability, key features for applications as permanent magnets and magnetic hyperthermia, can be increased by ca. 70%. A deep structural, morphological, chemical, and magnetic characterization reveals that the mechanism governing the coercive field improvement is the reduction of the concomitant internal stresses induced by the low-temperature annealing postsynthesis treatment. Furthermore, we show that the medium where the mild annealing process occurs is essential to control the final properties of the nanoparticles because the classical annealing procedure (T > 350 °C) performed on a dried powder does not allow the release of the lattice stress, leading to the reduction of the initial coercive field. The strategy here proposed, therefore, constitutes a method to improve the magnetic properties of nanoparticles, which can be particularly appealing for those materials, as is the case of cobalt ferrite, currently investigated as building blocks for the development of rare-earth free permanent magnets.

Degeneracy engineering for classical and quantum annealing: A case study of sparse linear regression in collider physics

Classical and quantum annealing are computing paradigms that have been proposed to solve a wide range of optimization problems. In this paper, we aim to enhance the performance of annealing algorithms by introducing the technique of degeneracy engineering, through which the relative degeneracy of the ground state is increased by modifying a subset of terms in the objective Hamiltonian. We illustrate this novel approach by applying it to the example of $\ell_0$-norm regularization for sparse linear regression, which is in general an NP-hard optimization problem. Specifically, we show how to cast $\ell_0$-norm regularization as a quadratic unconstrained binary optimization (QUBO) problem, suitable for implementation on annealing platforms. As a case study, we apply this QUBO formulation to energy flow polynomials in high-energy collider physics, finding that degeneracy engineering substantially improves the annealing performance. Our results motivate the application of degeneracy engineering to a variety of regularized optimization problems.

Quantum and classical annealing in a continuous space with multiple local minima

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.

Perturbed ferromagnetic chain: Tunable test of hardness in the transverse-field Ising model

Quantum annealing in the transverse-field Ising model (TFIM) with open-system dynamics is known to use thermally assisted tunneling to drive computation. However, it is still subject to debate whether quantum systems in the presence of decoherence are more useful than those using classical dynamics to drive computation. We contribute to this debate by introducing the perturbed ferromagnetic chain (PFC), a chain of frustrated subsystems where the degree of frustration scales inversely with the perturbation introduced by a tunable parameter. This gives us an easily embeddable gadget whereby problem hardness can be tuned for systems of constant size. We outline the properties of the PFC and compare classical spin-vector Monte Carlo (SVMC) variants with the adiabatic quantum master equation. We demonstrate that SVMC methods get trapped in the exponentially large first-excited-state manifold when solving this frustrated problem, whereas evolution using quantum dynamics remains in the lowest energy eigenstates. This results in significant differences in ground-state probability when using either classical or quantum annealing dynamics in the TFIM.

The effect of annealing under acid or alkaline environment on the physicochemical and functional properties of wheat starch

The annealing of wheat starch was performed in dilute acid and alkaline media at 40 °C and 45 °C. To highlight the effect of these processes on wheat starch, the scanning electron microscopy and the optic microscopy under normal and polarized light, the Wide-Angle X-ray Scattering patterns and the colour parameters were analysed without revealing significant changes in comparison to the native wheat starch and samples submitted to the classical annealing process in pure water. Most striking changes induced were observed upon additional hydrothermal treatment above wheat starch gelatinization temperatures and when analysing the water-binding capacity and the solubility, the starch granules swelling kinetics, the pasting behaviour of starch-water dispersion as well as the flow properties of gels recovered. It is likely that the complementary changes induced by these variants of annealing are meanly related to the restructuration of the amorphous phase inside granules that modify irreversibly their swelling capacity as well as their susceptibility to collapse upon further hydrothermal treatments. Although the magnitude of the changes induced by these processes seems to depend on the incubation temperature, the same trends were observed when compared the results from the two incubation temperatures applied. Coupling the annealing process with incubation in diluted acid or alkaline media can be used as effective way to expand the possibilities of using wheat starches in various industries.

Entropic barriers as a reason for hardness in both classical and quantum algorithms

We study both classical and quantum algorithms to solve a hard optimization problem, namely 3-XORSAT on 3-regular random graphs. By introducing a new quasi-greedy algorithm that is not allowed to jump over large energy barriers, we show that the problem hardness is mainly due to entropic barriers. We study, both analytically and numerically, several optimization algorithms, finding that entropic barriers affect in a similar way classical local algorithms and quantum annealing. For the adiabatic algorithm, the difficulty we identify is distinct from that of tunneling under large barriers, but does, nonetheless, give rise to exponential running (annealing) times.

Optimizing the magnetic properties of hard and soft materials for producing exchange spring permanent magnets

The exploitation of the exchange coupling between hard and soft magnetic materials has been proposed for enhancing the magnetic performances of rare-earth free permanent magnets, with the aim of extending their use to all applications where moderate energy product (35–100 kJ m−3) is required. Strontium hexaferrite (SFO)/spinel ferrite composites seem particularly promising to achieve this target, although the conditions to maximize the effect while using techniques easily scalable to industrial production have not yet been identified. Within this framework, the optimization of the structural, chemical, and magnetic properties of the two moieties before the coupling procedure is crucial to enhance the energy product of the final composite. Here we report the syntheses of both nanometric SFO with high coercivity (ca. 525 kA m−1) and quasi-bulk saturation magnetization (68 Am2 kg−1) and a series of nanosized zinc-doped ferrite (Zn x Fe3−x O4, 0.0 ⩽ x ⩽ 0.4) through cheap, easily scalable and eco-friendly approaches. The structural and chemical stability of the two magnetic phases as a function of temperature were investigated up to 1100 °C, with the aim of finding the best compromise between preservation of the nanometric scale and magnetic properties. A very high-magnetization (106 Am2 kg−1) ferrite was obtained by annealing Zn0.3Fe2.7O4 nanopowder at the highest investigated temperature. A preliminary attempt at coupling the two phases, starting from a mixture of the nanopowders, was performed through a classic annealing process in the temperature range 500 °C–1100 °C. The adopted procedure allowed for obtaining an exchange coupled composite at 1100 °C where the two phases are intimately and homogeneously mixed, with micrometric (0.3–5 μm) and nanometric (up to 50 nm) spinel ferrite particles. Despite these promising results, no enhancement of the energy product was found, highlighting the need for further experimental efforts to improve the coupling procedure.

Optimizing the Modified Lam Annealing Schedule

Simulated annealing is a metaheuristic commonly used for combinatorial optimization in many industrial applications. Its runtime behavior is controlled by an algorithmic component known as the annealing schedule. The classic annealing schedules have control parameters that must be set o

Adiabatic evolution on a spatial-photonic Ising machine

Combinatorial optimization problems are crucial for widespread applications but remain difficult to solve on a large scale with conventional hardware. Novel optical platforms, known as coherent or photonic Ising machines, are attracting considerable attention as accelerators on optimization tasks formulable as Ising models. Annealing is a well-known technique based on adiabatic evolution for finding optimal solutions in classical and quantum systems made by atoms, electrons, or photons. Although various Ising machines employ annealing in some form, adiabatic computing on optical settings has been only partially investigated. Here, we realize the adiabatic evolution of frustrated Ising models with 100 spins programmed by spatial light modulation. We use holographic and optical control to change the spin couplings adiabatically, and exploit experimental noise to explore the energy landscape. Annealing enhances the convergence to the Ising ground state and allows to find the problem solution with probability close to unity. Our results demonstrate a photonic scheme for combinatorial optimization in analogy with adiabatic quantum algorithms and classical annealing methods but enforced by optical vector-matrix multiplications and scalable photonic technology.

Impact of nanosecond laser energy density on the C40-TiSi2 formation and C54-TiSi2 transformation temperature

The formation of Ti based contacts in new image sensor complementary metal–oxide–semiconductor technologies is limited by the requirement of a low thermal budget. The objectives of these new 3D-technologies are to promote ohmic, low resistance, repeatable, and reliable contacts by keeping the process temperature as low as possible. In this work, ultraviolet-nanosecond laser annealing was performed before classical rapid thermal annealing (RTA) to promote the formation at lower RTA temperatures of the low resistivity C54-TiSi2 phase. The laser energy density was varied from 0.30 to 1.00 J/cm² with three pulses in order to form the C40-TiSi2 phase and finally to obtain the C54-TiSi2 phase by a subsequent RTA at low temperatures. The formed Ti-silicides were characterized by four-point probe measurements, x-ray diffraction, transmission electron microscopy, and atom probe tomography. A threshold in the laser energy density for the formation of the C40-TiSi2 is observed at an energy density of 0.85 J/cm² for the targeted TiN/Ti stack on blanket wafers. The C40-TiSi2 formation by laser annealing prior to RTA enables to reduce the formation temperature of the C54-TiSi2 phase by 150 °C in comparison to a single RTA applied after the Ti/TiN deposition. This specific phase sequence is only observed for a laser energy density close to 0.85 J/cm². At higher energy densities, the presence of C49-TiSi2 or a mixture of C49-TiSi2 and C54-TiSi2 is observed. The underlying mechanisms of the phase sequence and formation are discussed in detail.

Water Annealing of TiO2 Nanotubes for Photocatalysis Revisited

AbstractOver the past years, so‐called water annealing of TiO2 nanotube (NT) arrays has attracted considerable attention as a tool to enhance the nanotubes’ photocatalytic properties. The present work investigates the effect of different water‐annealing treatments on TiO2 nanotubes and compares it to thermally annealed tubes. We find that the most effective water annealing is water immersion of TiO2 NT arrays at 70 °C for several days. The two main effects are i) a partial crystallization of the tubes to anatase, and ii) an etching of the tube walls that leads to a roughening of the walls and, thus, to a higher surface area. Under the best conditions, water annealing is almost as effective in providing enhanced photocatalytic AO7 degradation as a classic thermal annealing of the tubes. We, however, find that water annealed tubes – due to their only partial crystallization – do not perform well when used as photoelectrodes. For photocatalysis, most efficient is a combination of water annealing, followed by thermal annealing, which provides both a larger surface area and a high crystallinity.

Optimization of quantum computing models inspired by D-wave quantum annealing

With the slow progress of universal quantum computers, studies on the feasibility of optimization by a dedicated and quantum-annealing-based annealer are important. The quantum principle is expected to utilize the quantum tunneling effects to find the optimal solutions for the exponential-level problems while classical annealing may be affected by the initializations. This study constructs a new Quantum-Inspired Annealing (QIA) framework to explore the potentials of quantum annealing for solving Ising model with comparisons to the classical one. Through various configurations of the 1D Ising model, the new framework can achieve ground state, corresponding to the optimum of classical problems, with higher probability up to 28% versus classical counterpart (22% in case). This condition not only reveals the potential of quantum annealing for solving the Ising-like Hamiltonian, but also contributes to an improved understanding and use of the quantum annealer for various applications in the future.

Activation of α-Fe2 O3 for Photoelectrochemical Water Splitting Strongly Enhanced by Low Temperature Annealing in Low Oxygen Containing Ambient.

Photoelectrochemical (PEC) water splitting is a promising method for the conversion of solar energy into chemical energy stored in the form of hydrogen. Nanostructured hematite (α‐Fe2O3) is one of the most attractive materials for a highly efficient charge carrier generation and collection due to its large specific surface area and the short minority carrier diffusion length. In the present work, the PEC water splitting performance of nanostructured α‐Fe2O3 is investigated which was prepared by anodization followed by annealing in a low oxygen ambient (0.03 % O2 in Ar). It was found that low oxygen annealing can activate a significant PEC response of α‐Fe2O3 even at a low temperature of 400 °C and provide an excellent PEC performance compared with classic air annealing. The photocurrent of the α‐Fe2O3 annealed in the low oxygen at 1.5 V vs. RHE results as 0.5 mA cm−2, being 20 times higher than that of annealing in air. The obtained results show that the α‐Fe2O3 annealed in low oxygen contains beneficial defects and promotes the transport of holes; it can be attributed to the improvement of conductivity due to the introduction of suitable oxygen vacancies in the α‐Fe2O3. Additionally, we demonstrate the photocurrent of α‐Fe2O3 annealed in low oxygen ambient can be further enhanced by Zn‐Co LDH, which is a co‐catalyst of oxygen evolution reaction. This indicates low oxygen annealing generates a promising method to obtain an excellent PEC water splitting performance from α‐Fe2O3 photoanodes.

Obstacles to quantum annealing in a planar embedding of XORSAT

We introduce a planar embedding of the k-regular k-XORSAT problem, in which solutions are encoded in the ground state of a classical statistical mechanics model of reversible logic gates arranged on a square grid and acting on bits that represent the Boolean variables of the problem. The special feature of this embedding is that the resulting model lacks a finite-temperature phase transition, thus bypassing the first-order thermodynamic transition known to occur in the random graph representation of XORSAT. In spite of this attractive feature, the thermal relaxation into the ground state displays remarkably slow glassy behavior. The question addressed in this paper is whether this planar embedding can afford an efficient path to solution of k-regular k-XORSAT via quantum adiabatic annealing. We first show that our model bypasses an avoided level crossing and consequent exponentially small gap in the limit of small transverse fields. We then present quantum Monte Carlo results for our embedding of the k-regular k-XORSAT that strongly support a picture in which second-order and first-order transitions develop at a finite transverse field for k = 2 and k = 3, respectively. This translates into power-law and exponential dependences in the scaling of energy gaps with system size, corresponding to times-to-solution which are, respectively, polynomial and exponential in the number of variables. We conclude that neither classical nor quantum annealing can efficiently solve our reformulation of XORSAT, even though the original problem can be solved in polynomial time by Gaussian elimination.