Quantum annealing with manufactured spins

DROID: discrete-time simulation for ring-oscillator-based Ising design

Many combinatorial problems can be mapped to Ising machines, i.e., networks of coupled oscillators that settle to a minimum-energy ground state, from which the problem solution is inferred. This work proposes DROID, a novel event-driven method for simulating the evolution of a CMOS Ising machine to its ground state. The approach is accurate under general delay-phase relations that include the effects of the transistor nonlinearities and is computationally efficient. On a realistic-size all-to-all coupled ring oscillator array, DROID is nearly four orders of magnitude faster than a traditional HSPICE simulation and two orders of magnitude faster than a commercial fast SPICE solver in predicting the evolution of a coupled oscillator system and is demonstrated to attain a similar distribution of solutions as the hardware.

Near-ultrastrong nonlinear light-matter coupling in superconducting circuits

Light-matter interaction between an atom and an electromagnetic resonator is ubiquitous in quantum technologies. Although linear light-matter coupling g σ ̂ x ( a ̂ + a ̂ † ) can reach the ultrastrong regime g/ω > 10-1, nonlinear light-matter couplingχ 2 σ ̂ z a ̂ † a ̂ is typically perturbative and limited to χ/ω < 10-2. Nonlinear coupling has the advantage of commuting with the atomicσ ̂ z and photonica ̂ † a ̂ Hamiltonian, allowing for fundamental operations such as quantum-non-demolition measurement. Here, we use a superconducting circuit to demonstrate the experimental realization of near-ultrastrong χ/ω = (4.852 ± 0.006) × 10-2. We also show signatures of light-light nonlinear coupling ( χ a ̂ † a ̂ b ̂ † b ̂ ) and χ/2π = 580.3 ± 0.4 MHz matter-matter nonlinear coupling (χ 4 σ ̂ z , a σ ̂ z , b ), representing the largest reported ZZ interaction between two coherent qubits. Such advances in the nonlinear coupling strength of light, matter modes enable new physical regimes and could lead to orders of magnitude faster qubit readout and gates.

Scaling Advantage in Approximate Optimization with Quantum Annealing

Quantum annealing is a heuristic optimization algorithm that exploits quantum evolution to find low-energy states. Quantum annealers have scaled up in recent years to tackle increasingly larger and more highly connected discrete optimization and quantum simulation problems. Nevertheless, a computational quantum advantage in exact optimization using quantum annealing hardware has so far remained elusive. Here, we present evidence for a quantum annealing scaling advantage in approximate optimization. The advantage is relative to the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM). The setting is a family of 2D spin-glass problems with high-precision spin-spin interactions. To achieve this advantage, we implement quantum annealing correction (QAC): an embedding of a bit-flip error-correcting code with energy penalties that leverages the properties of the D-Wave Advantage quantum annealer to yield over 1,300 error-suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric for low-energy states. We demonstrate that, with QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low-energy states with an optimality gap of at least 1.0%. This amounts to the first demonstration of an algorithmic quantum speedup in approximate optimization.

Quantum Circuit and Mapping Algorithms for Wavepacket Dynamics: Case Study of Anharmonic Hydrogen Bonds in Protonated and Hydroxide Water Clusters

The accurate computational study of wavepacket nuclear dynamics is considered to be a classically intractable problem, particularly with increasing dimensions. Here, we present two algorithms that, in conjunction with other methods developed by us, may result in one set of contributions for performing quantum nuclear dynamics in arbitrary dimensions. For one of the two algorithms discussed here, we present a direct map between the Born-Oppenheimer Hamiltonian describing the nuclear wavepacket time evolution and the control parameters of a spin-lattice Hamiltonian that describes the dynamics of qubit states in an ion-trap quantum computer. This map is exact for three qubits, and when implemented, the dynamics of the spin states emulates those of the nuclear wavepacket in a continuous representation. However, this map becomes approximate as the number of qubits grows. In a second algorithm, we present a general quantum circuit decomposition formalism for such problems using a method called the Quantum Shannon Decomposition. This algorithm is more robust and is exact for any number of qubits at the cost of increased circuit complexity. The resultant circuit is implemented on IBM's quantum simulator (QASM) for 3-7 qubits, without using a noise model so as to test the intrinsic accuracy of the method. In both cases, the wavepacket dynamics is found to be in good agreement with the classical propagation result and the corresponding vibrational frequencies obtained from the wavepacket density time evolution are in agreement to within a few tenths of a wavenumber.

Variational Ground-State Quantum Adiabatic Theorem

We present a variational quantum adiabatic theorem, which states that, under certain assumptions, the adiabatic dynamics projected onto a variational manifold follow the instantaneous variational ground state. We focus on low-entanglement variational manifolds and target Hamiltonians with classical ground states. Despite highly entangled intermediate states along the exact adiabatic path, the variational evolution converges to the target ground state. We demonstrate this approach with several examples that align with our theoretical analysis.

Free-energy machine for combinatorial optimization

Finding optimal solutions to combinatorial optimization problems (COPs) is pivotal in both scientific and industrial domains. Considerable efforts have been invested on developing accelerated methods utilizing sophisticated models and advanced computational hardware. However, the challenge remains to achieve both high efficiency and broad generality in problem-solving. Here we propose a general method, free-energy machine (FEM), based on the ideas of free-energy minimization in statistical physics, combined with automatic differentiation and gradient-based optimization in machine learning. FEM flexibly addresses various COPs within a unified framework and efficiently leverages parallel computational devices such as graphics processing units. We benchmark FEM on diverse COPs including maximum cut, balanced minimum cut and maximum k-satisfiability, scaled to millions of variables, across synthetic and real-world instances. The findings indicate that FEM remarkably outperforms state-of-the-art algorithms tailored for individual COP in both efficiency and efficacy, demonstrating the potential of combining statistical physics and machine learning for broad applications.

Frustration Elimination and Excited State Search in Coherent Ising Machines

Frustration, that is, the impossibility of satisfying the energetic preferences between all spin pairs simultaneously, underlies the complexity of many fundamental properties in spin systems, including the computational difficulty in determining their ground states. Coherent Ising machines (CIMs) have been proposed as a promising analog computational approach to efficiently find different degenerate ground states of large and complex Ising models. However, CIMs also face challenges in solving frustrated Ising models: frustration not only reduces the probability of finding good solutions, but it also prohibits the leveraging of quantum effects in doing so. To circumvent these detrimental effects of frustration, we show how frustrated Ising models can be mapped to frustration-free CIM configurations by including ancillary modes and modifying the coupling protocol used in current CIM designs. Such frustration elimination may empower current CIMs to benefit from quantum effects in dealing with frustrated Ising models. In addition, these ancillary modes can also enable error detection and searching for excited states.

Enumeration Approach to Atom-to-Atom Mapping Accelerated by Ising Computing

Chemical reactions are regarded as transformations of chemical structures, and the question of which atoms in the reactants correspond to which atoms in the products has attracted chemists for a long time. Atom-to-atom mapping (AAM) is a procedure that establishes such correspondence(s) between the atoms of reactants and products in a chemical reaction. Currently, automatic AAM tools play a pivotal role in various chemoinformatics tasks. However, achieving accurate automatic AAM for complex or unknown reactions within a reasonable computation time remains a significant challenge due to the combinatorial nature of the problem and the difficulty in applying appropriate reaction rules. In this study, we propose a rule-free AAM algorithm, which enumerates all atom-to-atom correspondences that minimize the number of bond cleavages and formations during the reaction. To reduce the computational burden associated with the combinatorial optimization (i.e., minimizing bond changes), we introduce Ising computing, a computing paradigm that has gained significant attention for its efficiency in solving hard combinatorial optimization problems. We found that our Ising computing framework outperforms conventional combinatorial optimization algorithms in terms of computation times, making it feasible to solve the AAM problem without reaction rules in an acceptable time. Furthermore, our AAM algorithm successfully found the correct AAM solution for all problems in a benchmark data set. In contrast, conventional AAM algorithms based on chemical heuristics failed for several problems. Specifically, these algorithms either failed to find the optimal solution in terms of bond changes, or they identified only one optimal solution, which was incorrect when multiple optimal solutions exist. These results emphasize the importance of enumerating all optimal correspondences that minimize bond changes, which is effectively achieved by our Ising-computing framework.

Finding independent sets in large-scale graphs with a coherent Ising machine

Combinatorial optimizers using physical systems, or "Ising machines," have been intensively studied as a way of computation that is more efficient than current digital computers. Here, we demonstrate that a coherent Ising machine (CIM) based on degenerate optical parametric oscillator (DOPO) pulses can find solutions to large-scale maximum independent set problems, which is an essential optimization problem that is closely related to real-world optimization tasks. We implemented a stable external field for each Ising spin as an interaction between a DOPO pulse and a group of ferromagnetically coupled auxiliary pulses. Consequently, the CIM delivered independent sets for dense graphs with up to 40,000 nodes. We compared the performance of the CIM with that of optimized simulated annealing algorithm implemented on a digital computer and found that the CIM outperformed the central processing unit (CPU) in terms of the time needed to obtain good approximate solutions when the graph size exceeded several thousand nodes.

A fast quantum algorithm for solving partial differential equations

The numerical solution of partial differential equations (PDEs) is essential in computational physics. Over the past few decades, various quantum-based methods have been developed to formulate and solve PDEs. Solving PDEs incurs high-time complexity for high-dimensional real-world problems, and using traditional methods becomes practically inefficient. This paper presents a fast hybrid classical-quantum paradigm based on successive over-relaxation (SOR) to accelerate solving PDEs. Using the discretization method, this approach reduces the PDE solution to solving a system of linear equations, which is then addressed using the block SOR method. The block SOR method is employed to address qubit limitations, where the entire system of linear equations is decomposed into smaller subsystems. These subsystems are iteratively solved block-wise using Advantage quantum computers developed by D-Wave Systems, and the solutions are subsequently combined to obtain the overall solution. The performance of the proposed method is evaluated by solving the heat equation for a square plate with fixed boundary temperatures and comparing the results with the best existing method. Experimental results show that the proposed method can accelerate the solution of high-dimensional PDEs by using a limited number of qubits up to 2 times the existing method.

Implementing Arbitrary Ising Models with a Trapped-Ion Quantum Processor

A promising paradigm of quantum computing for achieving practical quantum advantages is quantum annealing or quantum approximate optimization algorithm, where the classical problems are encoded in Ising interactions. However, it is challenging to build a quantum system that can efficiently map any structured problems. Here, we present a trapped-ion quantum processor that can efficient encode arbitrary Ising models with all-to-all connectivity for up to four spins. We implement the spin-spin interactions by using the coupling of trapped ions to multiple collective motional modes and realize the programmability through phase modulation of the Raman laser beams that are individually addressed on ions. As an example, we realize several Ising models with different interaction connectivities, where the interactions can be ferromagnetic or antiferromagnetic. We confirm the programmed interaction geometry by observing the ground states of the corresponding models through quantum state tomography. Our experimental demonstrations serve as an important basis for realizing practical quantum advantages with trapped ions.

Cyclic quantum annealing: searching for deep low-energy states in 5000-qubit spin glass

Quantum computers promise a qualitative speedup in solving a broad spectrum of practical optimization problems. The latter can be mapped onto the task of finding low-energy states of spin glasses, which is known to be exceedingly difficult. Using D-Wave's 5000-qubit quantum processor, we demonstrate that a recently proposed iterative cyclic quantum annealing algorithm can find deep low-energy states in record time. We also find intricate structures in a low-energy landscape of spin glasses, such as a power-law distribution of connected clusters with a small surface energy. These observations offer guidance for further improvement of the optimization algorithms.

BEOL-Compatible 4F2 Oscillator Using Vertical InGaAs Biristor for Highly Scalable Monolithic 3D Ising Solver

Ising solvers are important for efficiently addressing non-deterministic polynomial-time (NP)-hard combinatorial optimization problems (COPs), where scalability and compactness are crucial for practical applications. In this study, an experimental demonstration of an oscillator-based Ising solver employing a highly scalable 4F2 InGaAs bi-stable resistor (biristor) is presented. It is first explored the oscillation behavior of the InGaAs biristor, establishing that classical Ising spins can be emulated using the sub-harmonic injection locking (SHIL) technique. Furthermore, capacitive and resistive coupling between two coupled InGaAs biristors is demonstrated, leading to out-of-phase and in-phase coupling, respectively. Employing this foundational technology, it is experimentally achieved a solution to the MaxCUT problem with the InGaAs biristor-based Ising solver, supplemented by simulation-based behavior evaluations. This emerging device architecture offers a viable pathway to surmount the scaling limitations faced by present hardware-based Ising solvers, representing a significant step forward in the development of efficient, scalable solutions for complex optimization challenges.

Quantum annealing for nearest neighbour compliance problem

Quantum Computing has emerged as a promising alternative, utilising quantum mechanics for faster computations. This paper explores the nearest neighbour compliance (NNC) Problem in Gate-based Quantum Computers, where quantum gates are constrained to operate on physically adjacent qubits. The NNC problem aims to optimise the insertion of SWAP-gates to ensure compliance with these constraints while minimising their count. This work introduces Quantum Annealing to tackle the NNC problem, proposing two Quadratic Unconstrained Optimisation Problem formulations. The formulations are tested on a contemporary Quantum Annealer, and their performance is compared with previous methods. It shows that the prospect of using Quantum Annealing is promising, however, the current state of the hardware makes that finding the embedding is the limiting factor.

Optimization Applications as Quantum Performance Benchmarks

Combinatorial optimization is anticipated to be one of the primary use cases for quantum computation in the coming years. The Quantum Approximate Optimization Algorithm and Quantum Annealing can potentially demonstrate significant run-time performance benefits over current state-of-the-art solutions. Inspired by existing methods to characterize classical optimization algorithms, we analyze the solution quality obtained by solving Max-cut problems using gate-model quantum devices and a quantum annealing device. This is used to guide the development of an advanced benchmarking framework for quantum computers designed to evaluate the trade-off between run-time execution performance and the solution quality for iterative hybrid quantum-classical applications. The framework generates performance profiles through compelling visualizations that show performance progression as a function of time for various problem sizes and illustrates algorithm limitations uncovered by the benchmarking approach. As an illustration, we explore the factors that influence quantum computing system throughput, using results obtained through execution on various quantum simulators and quantum hardware systems.

Ultrathin Ge-YF3 antireflective coating with 0.5 % reflectivity on high-index substrate for long-wavelength infrared cameras

Achieving long-wavelength infrared (LWIR) cameras with high sensitivity and shorter exposure times faces challenges due to series reflections from high-refractive index lenses within compact optical systems. However, designing effective antireflective coatings to maximize light throughput in these systems is complicated by the limited range of transparent materials available for the LWIR. This scarcity narrows the degrees of freedom in design, complicating the optimization process for a system that aims to minimize the number of physical layers and address the inherent large refractive mismatch from high-index lenses. In this study, we use discrete-to-continuous optimization to design a subwavelength-thick antireflective multilayer coating on high-refractive index Si substrate for LWIR cameras, where the coating consists of few (e.g., five) alternating stacks of high- and low-refractive-index thin films (e.g., Ge-YF3, Ge-ZnS, or ZnS-YF3). Discrete optimization efficiently reveals the configuration of physical layers through binary optimization supported by a machine learning model. Continuous optimization identifies the optimal thickness of each coating layer using the conventional gradient method. As a result, considering the responsivity of a LWIR camera, the discrete-to-continuous strategy finds the optimal design of a 2.3-μm-thick antireflective coating on Si substrate consisting of five physical layers based on the Ge-YF3 high-low index pair, showing an average reflectance of 0.54 % within the wavelength range of 8-13 μm. Moreover, conventional thin-film deposition (e.g., electron-beam evaporator) techniques successfully realize the designed structure, and Fourier-transform infrared spectroscopy (FTIR) and thermography confirm the high performance of the antireflective function.

Modeling routing problems in QUBO with application to ride-hailing

Many emerging commercial services are based on the sharing or pooling of resources for common use with the aim of reducing costs. Businesses such as delivery-, mobility-, or transport-as-a-service have become standard in many parts of the world, fulfilling on-demand requests for customers in live settings. However, it is known that many of these problems are NP-hard, and therefore both modeling and solving them accurately is a challenge. Here we focus on one such routing problem, the Ride Pooling Problem (RPP), where multiple customers can request on-demand pickups and drop-offs from shared vehicles within a fleet. The combinatorial optimization task is to optimally pool customer requests using the limited set of vehicles, akin to a small-scale flexible bus route. In this work, we propose a quadratic unconstrained binary optimization (QUBO) program and introduce efficient formulation methods for the RPP to be solved using metaheuristics, and specifically emerging quantum optimization algorithms.

Quantum Dynamical Interpretation of the Mean Strategy

The method of quantum dynamics is employed to investigate the mean strategy in the swarm intelligence algorithm. The physical significance of the population mean point is explained as the location where the optimal solution with the highest likelihood can be found once a quantum system has reached a ground state. Through the use of the double well function and the CEC2013 test suite, controlled experiments are conducted to perform a comprehensive performance analysis of the mean strategy. The empirical results indicate that implementing the mean strategy not only enhances solution diversity but also yields accurate, efficient, stable, and effective outcomes for finding the optimal solution.

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Solving the resource constrained project scheduling problem with quantum annealing

Quantum annealing emerges as a promising approach for tackling complex scheduling problems such as the resource-constrained project scheduling problem (RCPSP). This study represents the first application of quantum annealing to solve the RCPSP, analyzing 12 well-known mixed integer linear programming (MILP) formulations and converting the most qubit-efficient one into a quadratic unconstrained binary optimization (QUBO) model. We then solve this model using the D-wave advantage 6.3 quantum annealer, comparing its performance against classical computer solvers. Our results indicate significant potential, particularly for small to medium-sized instances. Further, we introduce time-to-target and Atos Q-score metrics to evaluate the effectiveness of quantum annealing and reverse quantum annealing. The paper also explores advanced quantum optimization techniques, such as customized anneal schedules, enhancing our understanding and application of quantum computing in operations research.

Scalable interconnection using a superconducting flux qubit

Superconducting quantum computers are rapidly reaching scales where bottlenecks to scaling arise from the practical aspects of the fabrication process. To improve quantum computer performance, implementation technology that guarantees the scalability of the number of qubits is essential. Increasing the degrees of freedom in routing by 2.5-dimensional implementation is important for realizing circuit scalability. We report an implementation technology to overcome the scaling bottlenecks using a reliable connection qubit with a demonstration of quantum annealing. The method comprises interconnection based on quantum annealing using a superconducting flux qubit, precise coupling status control, and flip-chip bonding. We perform experiments and simulations with a proof-of-concept demonstration of qubit coupling via interconnection using a flux qubit. The coupling status is strictly controllable by quantum annealing. A low-temperature flip-chip bonding technology is introduced for the 2.5-dimensional interconnection. The superconducting flux qubit, formed across two different chips via bumps, is demonstrated for the first time to show a state transition like that in a conventional qubit. The quantum annealing flux qubit and flip-chip bonding enable new interconnections between qubits. A perspective on the possibility of applying this technology to the connection between gate-type qubits is described.

The quantum transition of the two-dimensional Ising spin glass

Quantum annealers are commercial devices that aim to solve very hard computational problems1, typically those involving spin glasses2,3. Just as in metallurgic annealing, in which a ferrous metal is slowly cooled4, quantum annealers seek good solutions by slowly removing the transverse magnetic field at the lowest possible temperature. Removing the field diminishes the quantum fluctuations but forces the system to traverse the critical point that separates the disordered phase (at large fields) from the spin-glass phase (at small fields). A full understanding of this phase transition is still missing. A debated, crucial question regards the closing of the energy gap separating the ground state from the first excited state. All hopes of achieving an exponential speed-up, compared to classical computers, rest on the assumption that the gap will close algebraically with the number of spins5-9. However, renormalization group calculations predict instead that there is an infinite-randomness fixed point10. Here we solve this debate through extreme-scale numerical simulations, finding that both parties have grasped parts of the truth. Although the closing of the gap at the critical point is indeed super-algebraic, it remains algebraic if one restricts the symmetry of possible excitations. As this symmetry restriction is experimentally achievable (at least nominally), there is still hope for the quantum annealing paradigm11-13.

MEMS Oscillators-Network-Based Ising Machine with Grouping Method

Combinatorial optimization (CO) has a broad range of applications in various fields, including operations research, computer science, and artificial intelligence. However, many of these problems are classified as nondeterministic polynomial-time (NP)-complete or NP-hard problems, which are known for their computational complexity and cannot be solved in polynomial time on traditional digital computers. To address this challenge, continuous-time Ising machine solvers have been developed, utilizing different physical principles to map CO problems to ground state finding. However, most Ising machine prototypes operate at speeds comparable to digital hardware and rely on binarizing node states, resulting in increased system complexity and further limiting operating speed. To tackle these issues, a novel device-algorithm co-design method is proposed for fast sub-optimal solution finding with low hardware complexity. On the device side, a piezoelectric lithium niobate (LiNbO3) microelectromechanical system (MEMS) oscillator network-based Ising machine without second-harmonic injection locking (SHIL) is devised to solve Max-cut and graph coloring problems. The LiNbO3 oscillator operates at speeds greater than 9 GHz, making it one of the fastest oscillatory Ising machines. System-wise, an innovative grouping method is used that achieves a performance guarantee of 0.878 for Max-cut and 0.658 for graph coloring problems, which is comparable to Ising machines that utilize binarization.

Development of optimization method for truss structure by quantum annealing

In this study, we developed a new method of topology optimization for truss structures by quantum annealing. To perform quantum annealing analysis with real variables, representation of real numbers as a sum of random number combinations is employed. The nodal displacement is expressed with binary variables. The Hamiltonian H is formulated on the basis of the elastic strain energy and position energy of a truss structure. It is confirmed that truss deformation analysis is possible by quantum annealing. For the analysis of the optimization method for the truss structure, the cross-sectional area of the truss is expressed with binary variables. The iterative calculation for the changes in displacement and cross-sectional area leads to the optimal structure under the prescribed boundary conditions.

Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.

Ising model partition-function computation as a weighted counting problem

While the Ising model is most often used to understand physical phenomena, its natural connection to combinatorial reasoning also makes it one of the best models to probe complex systems in science and engineering. We bring a computational lens to the study of Ising models, where our computer-science perspective is twofold: On the one hand, we show that partition function computation (#Ising) can be reduced to weighted model counting (WMC). This enables us to take off-the-shelf model counters and apply them to #Ising. We show that one model counter (TensorOrder) outperforms state-of-the-art tools for #Ising on midsize and topologically unstructured instances, suggesting the tool would be a useful addition to a portfolio of partition function solvers. On the other hand, we consider the computational complexity of #Ising and relate it to the logic-based counting of constraint-satisfaction problems or #CSP. We show that known dichotomy results for #CSP give an easy proof of the hardness of #Ising and provide intuition on where the difficulty of #Ising comes from.

Quantum annealing of a frustrated magnet

Quantum annealing, which involves quantum tunnelling among possible solutions, has state-of-the-art applications not only in quickly finding the lowest-energy configuration of a complex system, but also in quantum computing. Here we report a single-crystal study of the frustrated magnet α-CoV2O6, consisting of a triangular arrangement of ferromagnetic Ising spin chains without evident structural disorder. We observe quantum annealing phenomena resulting from time-reversal symmetry breaking in a tiny transverse field. Below ~ 1 K, the system exhibits no indication of approaching the lowest-energy state for at least 15 hours in zero transverse field, but quickly converges towards that configuration with a nearly temperature-independent relaxation time of ~ 10 seconds in a transverse field of ~ 3.5 mK. Our many-body simulations show qualitative agreement with the experimental results, and suggest that a tiny transverse field can profoundly enhance quantum spin fluctuations, triggering rapid quantum annealing process from topological metastable Kosterlitz-Thouless phases, at low temperatures.

Energy-efficient superparamagnetic Ising machine and its application to traveling salesman problems

The growth of artificial intelligence leads to a computational burden in solving non-deterministic polynomial-time (NP)-hard problems. The Ising computer, which aims to solve NP-hard problems faces challenges such as high power consumption and limited scalability. Here, we experimentally present an Ising annealing computer based on 80 superparamagnetic tunnel junctions (SMTJs) with all-to-all connections, which solves a 70-city traveling salesman problem (TSP, 4761-node Ising problem). By taking advantage of the intrinsic randomness of SMTJs, implementing global annealing scheme, and using efficient algorithm, our SMTJ-based Ising annealer outperforms other Ising schemes in terms of power consumption and energy efficiency. Additionally, our approach provides a promising way to solve complex problems with limited hardware resources. Moreover, we propose a cross-bar array architecture for scalable integration using conventional magnetic random-access memories. Our results demonstrate that the SMTJ-based Ising computer with high energy efficiency, speed, and scalability is a strong candidate for future unconventional computing schemes.

Magnetic Flux Sensor Based on Spiking Neurons with Josephson Junctions

Josephson junctions (JJs) are superconductor-based devices used to build highly sensitive magnetic flux sensors called superconducting quantum interference devices (SQUIDs). These sensors may vary in design, being the radio frequency (RF) SQUID, direct current (DC) SQUID, and hybrid, such as D-SQUID. In addition, recently many of JJ's applications were found in spiking models of neurons exhibiting nearly biological behavior. In this study, we propose and investigate a new circuit model of a sensory neuron based on DC SQUID as part of the circuit. The dependence of the dynamics of the designed model on the external magnetic flux is demonstrated. The design of the circuit and derivation of the corresponding differential equations that describe the dynamics of the system are given. Numerical simulation is used for experimental evaluation. The experimental results confirm the applicability and good performance of the proposed magnetic-flux-sensitive neuron concept: the considered device can encode the magnetic flux in the form of neuronal dynamics with the linear section. Furthermore, some complex behavior was discovered in the model, namely the intermittent chaotic spiking and plateau bursting. The proposed design can be efficiently applied to developing the interfaces between circuitry and spiking neural networks. However, it should be noted that the proposed neuron design shares the main limitation of all the superconductor-based technologies, i.e., the need for a cryogenic and shielding system.

Parametric Frequency Divider Based Ising Machines

We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.

Effect of constraint relaxation on the minimum vertex cover problem in random graphs

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.

Controlling NMR spin systems for quantum computation

Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.

Efficiency optimization in quantum computing: balancing thermodynamics and computational performance

We investigate the computational efficiency and thermodynamic cost of the D-Wave quantum annealer under reverse-annealing with and without pausing. Our demonstration on the D-Wave 2000Q annealer shows that the combination of reverse-annealing and pausing leads to improved computational efficiency while minimizing the thermodynamic cost compared to reverse-annealing alone. Moreover, we find that the magnetic field has a positive impact on the performance of the quantum annealer during reverse-annealing but becomes detrimental when pausing is involved. Our results, which are reproducible, provide strategies for optimizing the performance and energy consumption of quantum annealing systems employing reverse-annealing protocols.

Native Approach to Controlled-Z Gates in Inductively Coupled Fluxonium Qubits

The fluxonium qubits have emerged as a promising platform for gate-based quantum information processing. However, their extraordinary protection against charge fluctuations comes at a cost: when coupled capacitively, the qubit-qubit interactions are restricted to XX interactions. Consequently, effective ZZ or XZ interactions are only constructed either by temporarily populating higher-energy states, or by exploiting perturbative effects under microwave driving. Instead, we propose and demonstrate an inductive coupling scheme, which offers a wide selection of native qubit-qubit interactions for fluxonium. In particular, we leverage a built-in, flux-controlled ZZ interaction to perform qubit entanglement. To combat the increased flux-noise-induced dephasing away from the flux-insensitive position, we use a continuous version of the dynamical decoupling scheme to perform noise filtering. Combining these, we demonstrate a 20 ns controlled-z gate with a mean fidelity of 99.53%. More than confirming the efficacy of our gate scheme, this high-fidelity result also reveals a promising but rarely explored parameter space uniquely suitable for gate operations between fluxonium qubits.

Quantum computing for chemistry and physics applications from a Monte Carlo perspective

This Perspective focuses on the several overlaps between quantum algorithms and Monte Carlo methods in the domains of physics and chemistry. We will analyze the challenges and possibilities of integrating established quantum Monte Carlo solutions into quantum algorithms. These include refined energy estimators, parameter optimization, real and imaginary-time dynamics, and variational circuits. Conversely, we will review new ideas for utilizing quantum hardware to accelerate the sampling in statistical classical models, with applications in physics, chemistry, optimization, and machine learning. This review aims to be accessible to both communities and intends to foster further algorithmic developments at the intersection of quantum computing and Monte Carlo methods. Most of the works discussed in this Perspective have emerged within the last two years, indicating a rapidly growing interest in this promising area of research.

Extensive antibody search with whole spectrum black-box optimization

In designing functional biological sequences with machine learning, the activity predictor tends to be inaccurate due to shortage of data. Top ranked sequences are thus unlikely to contain effective ones. This paper proposes to take prediction stability into account to provide domain experts with a reasonable list of sequences to choose from. In our approach, multiple prediction models are trained by subsampling the training set and the multi-objective optimization problem, where one objective is the average activity and the other is the standard deviation, is solved. The Pareto front represents a list of sequences with the whole spectrum of activity and stability. Using this method, we designed VHH (Variable domain of Heavy chain of Heavy chain) antibodies based on the dataset obtained from deep mutational screening. To solve multi-objective optimization, we employed our sequence design software MOQA that uses quantum annealing. By applying several selection criteria to 19,778 designed sequences, five sequences were selected for wet-lab validation. One sequence, 16 mutations away from the closest training sequence, was successfully expressed and found to possess desired binding specificity. Our whole spectrum approach provides a balanced way of dealing with the prediction uncertainty, and can possibly be applied to extensive search of functional sequences.

Spatial-photonic Ising machine by space-division multiplexing with physically tunable coefficients of a multi-component model

This paper proposes a space-division multiplexed spatial-photonic Ising machine (SDM-SPIM) that physically calculates the weighted sum of the Ising Hamiltonians for individual components in a multi-component model. Space-division multiplexing enables tuning a set of weight coefficients as an optical parameter and obtaining the desired Ising Hamiltonian at a time. We solved knapsack problems to verify the system's validity, demonstrating that optical parameters impact the search property. We also investigated a new dynamic coefficient search algorithm to enhance search performance. The SDM-SPIM would physically calculate the Hamiltonian and a part of the optimization with an electronics process.

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

Non-Markovian cost function for quantum error mitigation with Dirac Gamma matrices representation

This paper investigates the non-Markovian cost function in quantum error mitigation (QEM) and employs Dirac Gamma matrices to illustrate two-qubit operators, significant in relativistic quantum mechanics. Amid the focus on error reduction in noisy intermediate-scale quantum (NISQ) devices, understanding non-Markovian noise, commonly found in solid-state quantum computers, is crucial. We propose a non-Markovian model for quantum state evolution and a corresponding QEM cost function, using simple harmonic oscillators as a proxy for environmental noise. Owing to their shared algebraic structure with two-qubit gate operators, Gamma matrices allow for enhanced analysis and manipulation of these operators. We evaluate the fluctuations of the output quantum state across various input states for identity and SWAP gate operations, and by comparing our findings with ion-trap and superconducting quantum computing systems' experimental data, we derive essential QEM cost function parameters. Our findings indicate a direct relationship between the quantum system's coupling strength with its environment and the QEM cost function. The research highlights non-Markovian models' importance in understanding quantum state evolution and assessing experimental outcomes from NISQ devices.

Quantum-inspired encoding enhances stochastic sampling of soft matter systems

Quantum advantage in solving physical problems is still hard to assess due to hardware limitations. However, algorithms designed for quantum computers may engender transformative frameworks for modeling and simulating paradigmatically hard systems. Here, we show that the quadratic unconstrained binary optimization encoding enables tackling classical many-body systems that are challenging for conventional Monte Carlo. Specifically, in self-assembled melts of rigid lattice ring polymers, the combination of high density, chain stiffness, and topological constraints results in divergent autocorrelation times for real-space Monte Carlo. Our quantum-inspired encoding overcomes this problem and enables sampling melts of lattice rings with fixed curvature and compactness, unveiling counterintuitive topological effects. Tackling the same problems with the D-Wave quantum annealer leads to substantial performance improvements and advantageous scaling of sampling computational cost with the size of the self-assembled ring melts.

Nonnegative/Binary matrix factorization for image classification using quantum annealing

Classical computing has borne witness to the development of machine learning. The integration of quantum technology into this mix will lead to unimaginable benefits and be regarded as a giant leap forward in mankind's ability to compute. Demonstrating the benefits of this integration now becomes essential. With the advance of quantum computing, several machine-learning techniques have been proposed that use quantum annealing. In this study, we implement a matrix factorization method using quantum annealing for image classification and compare the performance with traditional machine-learning methods. Nonnegative/binary matrix factorization (NBMF) was originally introduced as a generative model, and we propose a multiclass classification model as an application. We extract the features of handwritten digit images using NBMF and apply them to solve the classification problem. Our findings show that when the amount of data, features, and epochs is small, the accuracy of models trained by NBMF is superior to classical machine-learning methods, such as neural networks. Moreover, we found that training models using a quantum annealing solver significantly reduces computation time. Under certain conditions, there is a benefit to using quantum annealing technology with machine learning.

A quantum-inspired probabilistic prime factorization based on virtually connected Boltzmann machine and probabilistic annealing

Probabilistic computing has been introduced to operate functional networks using a probabilistic bit (p-bit), broadening the computational abilities in non-deterministic polynomial searching operations. However, previous developments have focused on emulating the operation of quantum computers similarly, implementing every p-bit with large weight-sum matrix multiplication blocks and requiring tens of times more p-bits than semiprime bits. In addition, operations based on a conventional simulated annealing scheme required a large number of sampling operations, which deteriorated the performance of the Ising machines. Here we introduce a prime factorization machine with a virtually connected Boltzmann machine and probabilistic annealing method, which are designed to reduce the hardware complexity and number of sampling operations. From 10-bit to 64-bit prime factorizations were performed, and the machine offers up to 1.2 × 108 times improvement in the number of sampling operations compared with previous factorization machines, with a 22-fold smaller hardware resource.

Initial State Encoding via Reverse Quantum Annealing and H-Gain Features

Quantum annealing is a specialized type of quantum computation that aims to use quantum fluctuations in order to obtain global minimum solutions of combinatorial optimization problems. Programmable D-Wave quantum annealers are available as cloud computing resources, which allow users low-level access to quantum annealing control features. In this article, we are interested in improving the quality of the solutions returned by a quantum annealer by encoding an initial state into the annealing process. We explore twoD-Wave features that allow one toencode such an initialstate: the reverse annealing (RA) and theh-gain(HG)features.RAaimstorefineaknownsolutionfollowinganannealpathstartingwithaclassical state representing a good solution, going backward to a point where a transverse field is present, and then finishing the annealing process with a forward anneal. The HG feature allows one to put a time-dependent weighting scheme on linear (h ) biases of the Hamiltonian, and we demonstrate that this feature likewise can be used to bias the annealing to start from an initial state. We also consider a hybrid method consisting of a backward phase resembling RA and a forward phase using the HG initial state encoding. Importantly, we investigate the idea of iteratively applying RA and HG to a problem, with the goal of monotonically improving on an initial state that is not optimal. The HG encoding technique is evaluated on a variety of input problems including the edge-weighted maximum cut problem and the vertex-weighted maximum clique problem, demonstrating that the HG technique is a viable alternative to RA for some problems. We also investigate how the iterative procedures perform for both RA and HG initial state encodings on random whole-chip spin glasses with the native hardware connectivity of the D-Wave Chimera and Pegasus chips.

Low-Rank Combinatorial Optimization and Statistical Learning by Spatial Photonic Ising Machine

The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.

Single-Gate In-Transistor Readout of Current Superposition and Collapse Utilizing Quantum Tunneling and Ferroelectric Switching

In nanostructure assemblies, the superposition of current paths forms microscopic electric circuits, and different circuit networks produce varying results, particularly when utilized as transistor channels for computing applications. However, the intricate nature of assembly networks and the winding paths of commensurate currents hinder standard circuit modeling. Inspired by the quantum collapse of superposition states for information decoding in quantum circuits, the implementation of analogous current path collapse to facilitate the detection of microscopic circuits by modifying their network topology is explored. Here, the superposition and collapse of current paths in gate-all-around polysilicon nanosheet arrays are demonstrated to enrich the computational resources within transistors by engineering the channel length and quantity. Switching the ferroelectric polarization of Hf0.5 Zr0.5 O2 gate dielectric, which drives these transistors out-of-equilibrium, decodes the output polymorphism through circuit topological modifications. Furthermore, a protocol for the single-electron readout of ferroelectric polarization is presented with tailoring the channel coherence. The introduction of lateral path superposition results into intriguing metal-to-insulator transitions due to transient behavior of ferroelectric switching. This ability to adjust the current networks within transistors and their interaction with ferroelectric polarization in polycrystalline nanostructures lays the groundwork for generating diverse current characteristics as potential physical databases for optimization-based computing.

Optimality guarantees for crystal structure prediction

Crystalline materials enable essential technologies, and their properties are determined by their structures. Crystal structure prediction can thus play a central part in the design of new functional materials^1,2. Researchers have developed efficient heuristics to identify structural minima on the potential energy surface^3-5. Although these methods can often access all configurations in principle, there is no guarantee that the lowest energy structure has been found. Here we show that the structure of a crystalline material can be predicted with energy guarantees by an algorithm that finds all the unknown atomic positions within a unit cell by combining combinatorial and continuous optimization. We encode the combinatorial task of finding the lowest energy periodic allocation of all atoms on a lattice as a mathematical optimization problem of integer programming^6,7, enabling guaranteed identification of the global optimum using well-developed algorithms. A single subsequent local minimization of the resulting atom allocations then reaches the correct structures of key inorganic materials directly, proving their energetic optimality under clear assumptions. This formulation of crystal structure prediction establishes a connection to the theory of algorithms and provides the absolute energetic status of observed or predicted materials. It provides the ground truth for heuristic or data-driven structure prediction methods and is uniquely suitable for quantum annealers^8-10, opening a path to overcome the combinatorial explosion of atomic configurations.

Hybrid quantum-classical machine learning for generative chemistry and drug design

Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome with hybrid architectures combining quantum computers with deep classical networks. As the first step toward this goal, we built a compact discrete variational autoencoder (DVAE) with a Restricted Boltzmann Machine (RBM) of reduced size in its latent layer. The size of the proposed model was small enough to fit on a state-of-the-art D-Wave quantum annealer and allowed training on a subset of the ChEMBL dataset of biologically active compounds. Finally, we generated 2331 novel chemical structures with medicinal chemistry and synthetic accessibility properties in the ranges typical for molecules from ChEMBL. The presented results demonstrate the feasibility of using already existing or soon-to-be-available quantum computing devices as testbeds for future drug discovery applications.

Quantum Annealing Designs Nonhemolytic Antimicrobial Peptides in a Discrete Latent Space

Increasing the variety of antimicrobial peptides is crucial in meeting the global challenge of multi-drug-resistant bacterial pathogens. While several deep-learning-based peptide design pipelines are reported, they may not be optimal in data efficiency. High efficiency requires a well-compressed latent space, where optimization is likely to fail due to numerous local minima. We present a multi-objective peptide design pipeline based on a discrete latent space and D-Wave quantum annealer with the aim of solving the local minima problem. To achieve multi-objective optimization, multiple peptide properties are encoded into a score using non-dominated sorting. Our pipeline is applied to design therapeutic peptides that are antimicrobial and non-hemolytic at the same time. From 200 000 peptides designed by our pipeline, four peptides proceeded to wet-lab validation. Three of them showed high anti-microbial activity, and two are non-hemolytic. Our results demonstrate how quantum-based optimizers can be taken advantage of in real-world medical studies.

Extracting higher-conductivity designs for solid polymer electrolytes by quantum-inspired annealing

Data-driven optimal structure exploration has become a hot topic in materials for energy-related devices. However, this method is still challenging due to the insufficient prediction accuracy of material properties and large exploration space for candidate structures. We propose a data trend analysis system for materials using quantum-inspired annealing. Structure-property relationships are learned by a hybrid decision tree and quadratic regression algorithm. Then, ideal solutions to maximize the property are explored by a Fujitsu Digital Annealer, which is unique hardware that can quickly extract promising solutions from the ample search space. The system's validity is investigated with an experimental study examining solid polymer electrolytes as potential components for solid-state lithium-ion batteries. A new trithiocarbonate polymer electrolyte offers a conductivity of 10-6 S cm-1 at room temperature, even though it is in a glassy state. Molecular design through data science will enable accelerated exploration of functional materials for energy-related devices.

Quantum critical dynamics in a 5,000-qubit programmable spin glass

Experiments on disordered alloys1-3 suggest that spin glasses can be brought into low-energy states faster by annealing quantum fluctuations than by conventional thermal annealing. Owing to the importance of spin glasses as a paradigmatic computational testbed, reproducing this phenomenon in a programmable system has remained a central challenge in quantum optimization4-13. Here we achieve this goal by realizing quantum-critical spin-glass dynamics on thousands of qubits with a superconducting quantum annealer. We first demonstrate quantitative agreement between quantum annealing and time evolution of the Schrödinger equation in small spin glasses. We then measure dynamics in three-dimensional spin glasses on thousands of qubits, for which classical simulation of many-body quantum dynamics is intractable. We extract critical exponents that clearly distinguish quantum annealing from the slower stochastic dynamics of analogous Monte Carlo algorithms, providing both theoretical and experimental support for large-scale quantum simulation and a scaling advantage in energy optimization.