A quantum-inspired method and apparatus for integer optimization problems

Abstract

A method for solving an integer optimization problem, comprising: defining a local representation of integer variables of the integer optimization problem as a set of qudits,; setting an state ansatz for the state of the qudits; setting a cost function as a function of the state representative of technical objectives and constraints; setting an optimization algorithm to minimize the cost function, and executing a quantum-inspired optimization method including: initializing control variables, evaluating the expression for the cost function, executing the defined optimization algorithm, conditionally rounding the current state of the qudits to a classical form, evaluating the cost function on the current state of the qudits, returning the solution found if the optimization method has converged or predetermined ending criteria have been met or repeating the optimization method with updated control variables to yield a tangible technical solution. Also, a computer program, a data processing apparatus or system, and a computer-readable medium.

Description

[0001] A QUANTUM-INSPIRED METHOD AND APPARATUS FOR INTEGER OPTIMIZATION PROBLEMS

[0002] BACKGROUND

[0003] Integer optimization problems are ubiquitous in various fields. From computer science to operation research or engineering, many problems can be formulated or casted as an instance of an integer optimization problem. An integer optimization problem involves finding the optimal solution of a given objective function subject to a set of constraints. The objective function is a function that we want to maximize or minimize, and the constraints are the conditions that the solution must satisfy. The solution to an integer optimization problem is a set of integer values that satisfy the constraints and optimize the objective function.

[0004] Examples of applications of integer optimization problems include: in computer science, the knapsack problem, the traveling salesman problem, or the job scheduling problem; in operation research, the transportation problem, the assignment problem, or the network flow problem; in engineering, the optimal design of a system, the optimal allocation of resources, or the optimal control of a system.

[0005] Among the integer optimization problems, the graph coloring problem emerges as one subset of the integer optimization problems. The graph coloring problem is a particular integer optimization problem that involves assigning colors to the vertices of a graph, under the constraint that no two adjacent vertices have the same color. The objective of the graph coloring problem is to minimize the number of colors used to color the vertices of the graph.

[0006] The graph coloring problem is a well-known NP-hard problem, and it has many applications in various fields, such as scheduling, register allocation, and frequency assignment.

[0007] Due to the relevance of these methods, multiple algorithms have been developed to solve graph coloring problems. The most common ones, such as brute-force, dynamic programming, greedy coloring, and contraction algorithms, have been widely used to solve graph coloring problems. However, these algorithms have limitations in terms of scalability and efficiency, especially when dealing with large graphs.

[0008] To account for large graphs, parallel, distributed, and decentralized algorithms have been proposed to solve graph coloring problems. These algorithms are designed to exploit the parallelism and distributed nature of the graph coloring problem to improve the efficiency and scalability of the algorithms. However, these algorithms are often complex and require specialized hardware or software to implement.

[0009] Other algorithms, such as metaheuristic algorithms, have been proposed to solve graph coloring problems. Metaheuristic algorithms are optimization algorithms that are inspired by natural phenomena or physical processes. These algorithms are designed to find good solutions, i.e., according to one or more criteria to be fulfilled, to optimization problems in a reasonable amount of time. Examples of metaheuristic algorithms include genetic algorithms, simulated annealing, ant colony optimization, and particle swarm optimization. However, these algorithms are often stochastic and require a large number of iterations to converge to a good solution.

[0010] To avoid these problems, some algorithms have emerged to solve graph coloring problems (GCPs), with a sub-category of relevance being quantum-inspired algorithms. These algorithms exploit the principles of quantum mechanics to solve optimization problems. Among the existing proposals, most of them involve casting the GCP as a Quadratic Unconstrained Binary Optimization (QUBO) problem, which can be solved using quantum annealing or quantum- inspired algorithms. Examples of these algorithms include Bowles et al. (EP4123480A1), which proposes a new quantum-inspired algorithm to solve QUBO problems. The algorithm, called Local Quantum Annealing, is based on the principles of quantum mechanics, and uses a quantum- inspired cost function to find a (non-necessarily optimal) solution to a QUBO problem by approximating the global solution as a combination of local solutions. Since the algorithm constrains the quantum state to be a product state during the quantum annealing, it is designed to be scalable and efficient, and it can be implemented on classical hardware, e.g., on one or more classical processing units, one or more graphical processing units, etc.

[0011] However, for integer programming this requires treating the integer variables as binary variables, which can lead to a significant increase in the number of variables and constraints. Additionally, these algorithms require the inclusion of penalty terms to enforce the integer constraints, which can complicate the optimization process. Furthermore, these algorithms require that the cost function be defined as a Hamiltonian quadratic function, which can be difficult to define, or even non-existent, for some problems. Moreover, the locality ansatz may be too restrictive for the solution methods proposed, as it limits heavily the proportion of the configuration space that can be explored with it, potentially leading to suboptimal solutions.

[0012] SUMMARY

[0013] The present disclosure relates to methods, devices, and systems for quantum-inspired integer optimization, which do not showcase the limitations of known quantum inspired algorithms listed above.

[0014] According to this disclosure, a qudit is defined as a quantum system with a finite number of levels, where the levels are binary or greater. Examples of these include, but are not limited to, qubits or qutrits. The methods disclosed herein are based on the use of qudits to represent the integer variables of an integer optimization problem. In some preferred embodiments, such qudits are represented using spherical coordinates, which allow for a natural representation of the integer variables. Other representations of the qudits, in accordance with some other embodiments, include, but are not limited to, the Cartesian representation, the Bloch sphere representation, the Wigner representation, or the Fock representation.

[0015] According to this disclosure, a state is defined as a given configuration of the quantum system. In some embodiments, the state is represented using the local basis of the qudits. In some embodiments, the state is represented as a single vector in the Hilbert space of the qudits. In some embodiments, the state is represented as a density matrix in the Hilbert space of the qudits. In some embodiments, the state is represented as a wave function in the Hilbert space of the qudits. In some embodiments, the state is represented as a probability distribution in the Hilbert space of the qudits. In some embodiments, the state is represented as a tensor network in the Hilbert space of the qudits. These modes of representation allow for the representation of quantum effects, such as entanglement, during the exploration of the configuration space. Examples of these tensor networks include, but are not limited to, matrix product states, projected entangled pair states, tree tensor networks, or quantum states parametrized through a neural network.

[0016] According to this disclosure, a cost function is defined as a function of the state that represents the objective function of the integer optimization problem. In some embodiments, the cost function is defined as a continuous function of the state, which allows for an efficient optimization process. In some embodiments, the cost function is written in terms of the angular momentum operator acting on the qudits, which allows for a more natural representation of the cost function in the embodiments where the integer variables are represented using n- dimensional spherical coordinates. In some embodiments, the cost function is written as a Hamiltonian operator acting on the qudits. In some embodiments, the cost function has different interaction strengths between different qudits. In some embodiments, the cost function comprises a map from a representation of, e.g., a state of the quantum system to a value system which represents the quality of the proposed solution to the optimisation problem.

[0017] In some embodiments, a penalty term is included in the cost function. In some embodiments, the penalty term is used to enforce the integer constraints of the integer optimization problem. In some embodiments, the penalty term is used to explore and / or extend the exploration of the solution space. In some embodiments, the penalty term is written as a function of the angular momentum operator acting on the qudits. In some embodiments, the penalty term is written as a Hamiltonian operator acting on the qudits.

[0018] In some embodiments, a random potential is added to the cost function. In some embodiments, the random potential is used to explore and / or extend the exploration of the solution space, by avoiding local minima. In some embodiments, the random potential is written as a function of the angular momentum operator acting on the qudits. In some embodiments, the random potential is written as a Hamiltonian operator acting on the qudits. In some embodiments, the random potential is a function of the time-step of the optimization process.

[0019] According to this disclosure, an optimization process is defined as a process of finding the minimum or maximum of the cost function. In some embodiments, the optimization process is performed using a quantum-inspired algorithm. Examples of quantum-inspired algorithms include, but are not limited to, quantum annealing, quantum-inspired annealing, quantum-inspired optimization, or a quantum-inspired variational algorithm. In some embodiments, the optimization process is performed using a classical algorithm. Examples of classical algorithms include, but are not limited to, gradient descent, adaptive movement estimation (ADAM), simulated annealing, a genetic algorithm, or particle swarm optimization. In some embodiments, the optimization process is performed using a hybrid quantum-classical algorithm. Examples of hybrid quantum- classical algorithms include, but are not limited to, quantum-inspired optimization with classical post-processing, quantum-inspired optimization with classical pre-processing, density-matrix renormalization group ground state search (DMRG), imaginary time evolution for optimizing the state, and / or quantum-inspired optimization with classical optimization.

[0020] According to this disclosure, a hyperparameter is defined as a parameter that is not related to the target optimization problem at hand, but that can be tuned to improve the performance of the optimization process. Examples of hyperparameters include, but are not limited to, the learning rate of the optimization process, the number of iterations of the optimization process, the temperature of the optimization process, or the strength of the penalty term.

[0021] In some embodiments, the cost function is a function of a given set of hyperparameters. In some embodiments, these hyperparameters comprise the time-step of the optimization process. In some embodiments, the hyperparameters themselves are optimized using a hyperparameter optimization process. Examples of hyperparameter optimization processes include, but are not limited to, grid search, random search, Bayesian optimization, reinforcement learning, or a genetic algorithm. In some embodiments, the hyperparameter optimization process belongs to the same family as the optimization process.

[0022] According to this disclosure, a method for solving an integer optimization problem is disclosed. The method comprises: defining a cost function as a function of qudits, where the cost function represents the objective function of the integer optimization problem; defining a penalty term as a function of the qudits, where the penalty term enforces the integer constraints of the integer optimization problem; defining a random potential as a function of the qudits, where the random potential promotes the exploration of the solution space; and performing an optimization process to find the minimum of the cost function.

[0023] In some embodiments, performing the optimization process comprises projecting the qudit system into the local basis of the qudits, thus providing a candidate solution for the integer optimization problem. In some embodiments, the projection is carried out at least once during the optimization process. In some embodiments, the projection is carried out at random stages of the optimization process. In some embodiments, the projection is carried out at fixed times of the optimization process. In some embodiments, the projection is carried out at the end of the optimization process. In some embodiments, the projection is carried out at every time-step of the optimization process. In some embodiments, the methods, devices, and systems disclosed herein are implemented on classical hardware. In some embodiments, the methods, devices, and systems disclosed herein are implemented on quantum hardware. In some embodiments, the methods, devices, and systems disclosed herein are implemented on hybrid quantum-classical hardware.

[0024] It is an object of this disclosure that the methods, devices, and systems disclosed herein provide a more efficient and scalable solution to integer optimization problems, compared to known quantum-inspired algorithms.

[0025] It is a purpose of this disclosure to provide improved quantum-inspired methods and apparatuses for integer optimization problems. For example, in manufacturing, the disclosed methods can determine an optimal job schedule by representing jobs as qudits and machine assignments as qudit states, minimizing total production time. In communication networks, the methods can optimize bandwidth allocation by representing data flows as qudits and allocated bandwidth segments as qudit states, thereby maximizing network throughput under capacity constraints. In power systems, the optimal dispatch of generation units can be determined by representing generators as qudits and their output levels as qudit states, minimizing generation costs while satisfying demand and grid stability constraints. Examples of applications of the methods and apparatuses disclosed herein include, but are not limited to, the graph coloring problem, the knapsack problem, the traveling salesman problem, the job scheduling problem, the transportation problem, the assignment problem, the network flow problem, the optimal design of a system, the optimal allocation of resources, or the optimal control of a system.

[0026] BRIEF DESCRIPTION OF THE DRAWINGS

[0027] To complete the description and to provide a better understanding of the disclosure, a set of drawings is provided. Said drawings form an integral part of the description and illustrate embodiments of the disclosure, which should not be interpreted as restricting the scope of the disclosure but just as examples of how the disclosure can be carried out. The drawings comprise the following figures:

[0028] Figure 1 shows a non-limiting example on how to embed a three-node graph coloring problem into a qudit representation.

[0029] Figure 2 shows a non-limiting example of a method, in accordance with some embodiments, for solving an integer optimization problem using the methods, devices, and systems disclosed herein.

[0030] Figure 3 shows a non-limiting example of a quantum-inspired optimization method.

[0031] Figure 4 shows a non-limiting example of a particular quantum-inspired optimization method, as some embodiments of the method described in Figure 3.

[0032] Figure 5 shows another non-limiting example of a particular quantum-inspired optimization method, as some embodiments of the method described in Figure 3. DETAILED DESCRIPTION

[0033] Figure 1 shows a non-limiting example on how to embed a three-node graph coloring problem into a qudit representation. In this example, the nodes of the graph are represented by the different qudits 101 , 102, 103. The edges of the graph are represented by the interactions between the qudits 104, 105, 106. The local state of each of the qudits is represented in the unit sphere as vectors 107, 108, 109, each of those associated with one of the qudits 104, 105, 106 respectively.

[0034] With this representation, the graph-coloring problem can be solved by finding the minimum of a cost function, which is defined as a function of the qudits. The cost function is written in terms of the values of the qudits. For a given set of nodes and edges forming a graph G, a non-limiting expression for a cost function is defined as

[0035] Where and 0 otherwise, and £ comprises all edges of the graph. When the graph coloring problem is solved, C reaches a minimum (in particular, for this specific cost function, C = 0 at the minimum).

[0036] In some embodiments, the cost function is written as a Hamiltonian operator acting on the qudits. In some embodiments, the cost function is written as a function of the angular momentum operator acting on the qudits. In some embodiments, the different colors are associated with the different eigenstates of one of the projections of the angular momentum operator L . In some embodiments, the cost function is a non-physical function of the parametrized qudits.

[0037] While graph coloring is an abstract problem, it serves as a model for various real-world technical problems. For instance, in frequency assignment for wireless communication systems, available frequencies (colors) must be assigned to transmitters (nodes) such that interfering transmitters (adjacent nodes) do not receive the same frequency, aiming to use the minimum number of frequencies. The qudits 101 , 102, 103 would represent transmitters, and their states 107, 108, 109 would represent the assigned frequency channel. The interactions 104, 105, 106 would represent potential interference between transmitters if they are assigned the same channel and are geographically close or share a coverage area. The objective of minimizing colors corresponds to minimizing the spectrum used.

[0038] Figure 2 shows a non-limiting example of a method, in accordance with some embodiments, for solving an integer optimization problem using the methods, devices, and systems disclosed herein.

[0039] The method includes loading the graph information into the system 201. The graph information includes the nodes and edges of the graph for which its associated solution to the graph coloring problem is to be found. Examples of these graphs include, but are not limited to, the graph shown in Figure 1. In some embodiments, the graph information includes any or all additional information relevant to the problem at hand.

[0040] The nature of this "graph information" and its constituent "nodes" and "edges" is determined by the specific technical problem being addressed.

[0041] For example, in a job scheduling problem within a manufacturing facility, the input data to be loaded 201 might comprise:

[0042] * A list of jobs to be processed, each with specific processing times on different machines.

[0043] * A list of available machines, each with its capacity and operational constraints.

[0044] * Precedence constraints (e.g., job A must complete before job B can start).

[0045] * Setup times required when a machine switches from one type of job to another.

[0046] This information is then translated into the "graph" structure for the optimization. Here, "nodes" could represent individual jobs or tasks (e.g., a specific job on a specific machine). The "integer variables" to be optimized (represented by qudits) could be the assignment of a job to a machine and a start time slot. The "local dimension" of a qudit associated with a job would then correspond to the number of possible valid (machine, time slot) assignments. The "edges" or, more generally, the interaction terms in the cost function, would encode the constraints: e.g., a high cost penalty if two jobs are assigned to the same machine at overlapping times, or if a precedence constraint is violated, or if total processing time on a machine exceeds its availability. The objective function to be minimized (part of the cost function) would typically be the makespan (total time to complete all jobs) or total tardiness.

[0047] As another non-limiting example, for optimal allocation of bandwidth in a communication network, the input data loaded 201 might comprise:

[0048] * The network topology: a set of communication nodes (routers, switches) and links connecting them, each link having a maximum bandwidth capacity.

[0049] * A list of requested data flows, each with a source node, a destination node, and a minimum required bandwidth and / or a desired quality of service (QoS) level.

[0050] * Potential paths for each data flow through the network.

[0051] In this scenario, the "integer variables" (represented by qudits) could be the discrete amount of bandwidth allocated to each data flow on each link of its chosen path, or the selection of a specific path from a set of possible paths. The "local dimension" of a qudit associated with a data flow could represent the number of discrete bandwidth allocation levels or the number of available paths. The cost function would aim to maximize overall network utility (e.g., sum of achieved QoS for all flows) or minimize congestion, subject to hard constraints (penalized by the cost function) such as, e.g., not exceeding link capacities and meeting minimum bandwidth requirements for critical flows. The "graph" here is directly the network topology, and interactions in the cost function would reflect shared link usage and capacity limits.

[0052] In some embodiments, the graph information is read from a file, a database, or another source of information. In some embodiments, the graph information is generated randomly, or using another method. In some embodiments, the graph information is received from user input means (e.g., the graph information is provided by a user), or from another source, e.g., from at least one device and / or another system(s) communicatively coupled therewith. In some embodiments, the graph information is received after the abstraction of a given system into its graph form. Examples of these abstractions include, but are not limited to, network graphs in electrical grids and / or a digital network, such as social networks.

[0053] In some embodiments, the graph information is preprocessed before being loaded into the system. Examples of preprocessing steps include, but are not limited to, filtering the graph, transforming the graph, normalizing the graph, splitting the graph, or merging the graph.

[0054] In some embodiments, the graph information is stored in at least one memory unit of the system. Examples of memory units include, but are not limited to, a hard drive, a solid-state drive, a random-access memory, or a cache memory. In some embodiments, the storage method comprises quantum hardware, including but not limited to quantum memories.

[0055] Once the graph is loaded into memory, the method includes setting a local dimension parameter 202, e.g., a local dimension of subsystems of the graph. In some preferred embodiments, the local dimension parameter is set to the number of colors that can be assigned to the nodes of the graph. In the context of the job scheduling example above, if a job can be assigned to one of 5 machines within 10 discrete time slots, the local dimension for the qudit representing that job could be up to 50 (or fewer if some assignments are invalid). In the bandwidth allocation example, if a flow can be allocated bandwidth in increments of 1 Mbps up to 10 Mbps, the local dimension could be 10. In some embodiments, the local dimension parameter is set to a fixed value, or to a variable value. In some embodiments, the local dimension parameter is set to a value that is a power of two, or to any other value. This allows for a more efficient representation of the integer variables of the integer optimization problem.

[0056] In some embodiments, the local dimension parameter is set by the user, or by any other source. In some embodiments, the local dimension parameter is set automatically by the system. In some embodiments, the local dimension parameter is set at least based on the graph information.

[0057] In some embodiments, all the nodes of the graph have the same local dimension parameter. In some embodiments, the nodes have different local dimensions. This is useful when we want to exclude some colors from a given region of the graph or, in the job scheduling example, if different jobs have different numbers of valid machine / time assignments, or in the bandwidth allocation example, if different flows have different maximum permissible bandwidths.

[0058] After defining the local dimension, the method includes setting an ansatz for the state 203. In some embodiments, the ansatz is chosen as a single classical state; that is, a single product state of each of the qudits. This allows for a more efficient representation of the state, by reducing the number of parameters of the state.

[0059] In some embodiments, the ansatz is chosen as a superposition of product states of the qudits. This allows for a better exploration of the configuration state of the system, by representing superpositions of different configurations.

[0060] In some embodiments, the ansatz is chosen as a tensor network written in terms of the qudits. This allows for a better capture of quantum effects, such as entanglement, during the optimization process. Examples of tensor networks include, but are not limited to, matrix product states, projected entangled pair states, or tree tensor networks.

[0061] In some embodiments, the ansatz is chosen as a variational state. This allows for a more flexible representation of the state, by allowing for the optimization of the parameters of the state. Examples of variational states include, but are not limited to, neural network states, quantum circuit states, or tensor networks. In some embodiments, the ansatz is chosen as a multiscale entanglement renormalization ansatz (MERA).

[0062] After defining the ansatz, the method includes setting the optimization algorithm 204, that will be used during the optimization process.

[0063] In some embodiments, the optimization algorithm is chosen as a classical algorithm. Examples of classical algorithms include, but are not limited to, gradient descent, ADAM, simulated annealing, genetic algorithms, reinforcement learning, or particle swarm optimization. This allows for a more efficient optimization process, by exploiting the capabilities and optimizations of classical hardware.

[0064] In some embodiments, the optimization algorithm is chosen as a quantum-inspired algorithm. Examples of quantum-inspired algorithms include, but are not limited to, quantum annealing, quantum-inspired annealing, quantum-inspired optimization, or quantum-inspired variational algorithms.

[0065] In some embodiments, quantum hardware is used to implement the optimization algorithm. Examples of quantum hardware include, but are not limited to, superconducting qudits, trapped ions, neutral atoms, or photonic quantum hardware. In some embodiments, the quantum hardware supports a qudit representation of the integer variables. Examples of these systems include, but are not limited to, multi-level quantum systems, atomic energy levels, photonic quantum modes (including discrete modulation of photonic modes). This allows for an efficient representation of the qudits comprising the integer variables. In some embodiments, the quantum hardware supports a superposition, and / or a tensor network representation of the state. This allows for an efficient representation of the state, by exploiting the capabilities of the quantum hardware.

[0066] In some embodiments, the optimization algorithm is chosen as a hybrid quantum-classical algorithm. Examples of hybrid quantum-classical algorithms include, but are not limited to, quantum-inspired optimization with classical post-processing, quantum-inspired optimization with classical pre-processing, or quantum-inspired optimization with classical optimization.

[0067] After selecting the optimization algorithm, the method includes setting the hyperparameters of the optimization process 205. In some embodiments, the hyperparameters are chosen as a fixed value. Examples of fixed hyperparameters include, but are not limited to, the learning rate of the optimization process, the number of iterations of the optimization process, the temperature of the optimization process, or the strength of the penalty term.

[0068] In some embodiments, these hyperparameters are chosen as a variable value. Examples of variable hyperparameters include, but are not limited to, the learning rate of the optimization process, the number of iterations of the optimization process, the temperature of the optimization process, or the strength of the penalty term. In some embodiments, the hyperparameters are optimized or tuned using a hyperparameter optimization process. Examples of hyperparameter optimization processes include, but are not limited to, grid search, random search, Bayesian optimization, reinforcement learning, or genetic algorithms.

[0069] In some embodiments, the hyperparameters are chosen by the user. In some embodiments, the hyperparameters are chosen automatically by the system. In some embodiments, the hyperparameters are chosen based on the graph information. In some embodiments, the hyperparametes are chosen at random from a probability distribution. Examples of these probability distributions include, but are not limited to, uniform or gaussian.

[0070] The method includes, after setting the hyperparameters, executing a Quantum-Inspired Optimization Method 206. This method finds the minimum of the cost function, by exploring the configuration space of the system. In preferred embodiments, the method 206 is performed using the graph information, the local dimensions, the state ansatz, the optimization algorithm, and the hyperparameters selected in the previous steps. Non-limiting examples of these methods are disclosed in further detail in the following sections.

[0071] After the optimization process is completed, the method includes returning, i.e. , outputting, the best classical solution found 207. In some embodiments, the best classical solution is obtained by projecting the qudit system into the local basis of the qudits. This provides a candidate solution for the integer optimization problem. Once this projection is completed, the method returns the best classical solution found. For the job scheduling example, the output 207 would be a concrete schedule specifying which machine processes which job and at what start time, leading to a minimized makespan. For the bandwidth allocation example, the output 207 would be a specific allocation of bandwidth in Mbps to each data flow on particular network links, resulting in improved network performance. These outputs are directly usable to control or configure the respective technical systems.

[0072] In some embodiments, methods, devices and systems according to the present disclosure also comprise providing one or more control commands for a target or for at least one controller monitoring and controlling the operation of the target (e.g., the machines, the network nodes, the network links, etc.), and / or the devices or system comprise at least one controller configured to provide control commands to the target. The one or more control commands are configured for adjusting operation of the target, either directly or through the at least one controller.

[0073] In some embodiments, the control commands are determined by the at least one controller based on the output 207 received, which may be provided to the at least one controller directly or a processed version thereof as processed, for example, by the device or system that solves the integer optimization problem. In some other embodiments, the control commands are determined by the device or system that solves the integer optimization problem and then provided to the at least one controller.

[0074] In some embodiments, some or all steps of the method can be repeated at least once, by changing the hyperparameters, the state ansatz, the optimization algorithm, or any other part of the method. This allows for a better exploration of the configuration space, and for a more efficient solution to the integer optimization problem.

[0075] In some embodiments, repetition of such steps is at least conducted after one or more control commands have been provided and new input data for the integer optimization problem may be considered. In this way, repeated optimization is possible as changes made to the operation of the target are considered for further optimization, if any is possible.

[0076] In some embodiments, the method is a computer-implemented method.

[0077] In some embodiments, the method is implemented on classical hardware. Examples of classical hardware include, but are not limited to, a computer, a server, a laptop, a tablet, or a smartphone.

[0078] In some embodiments, the method is implemented on a cloud computing platform. Examples of implementations include, but are not limited to, a virtual machine, a container, a serverless function, or a cloud service.

[0079] In some embodiments, the method is implemented on a hardware accelerator. Examples of hardware accelerators include, but are not limited to, a graphics processing unit, a tensor processing unit, a field-programmable gate array, or an application-specific integrated circuit.

[0080] In some embodiments, the method is implemented on a distributed system. Examples of distributed systems include, but are not limited to, a cluster of computers, a grid of computers, or a network of computers.

[0081] In some embodiments, the method is implemented on a decentralized system. Examples of decentralized systems include, but are not limited to, a peer-to-peer network, a blockchain network, or a distributed ledger.

[0082] In some embodiments, the method is implemented on a mobile device. Examples of mobile devices include, but are not limited to, a smartphone, a tablet, a laptop, or a wearable device.

[0083] In some embodiments, the method is implemented on an embedded system. Examples of embedded systems include, but are not limited to, a microcontroller, a microprocessor, a system-on-chip, or a field-programmable gate array.

[0084] In some embodiments, the method is implemented on quantum hardware. Examples of quantum hardware include, but are not limited to, superconducting qubits, trapped ions, or photonic quantum hardware. In some embodiments, the quantum hardware supports a qudit representation of the integer variables. Examples of these systems include, but are not limited to, multi-level quantum systems.

[0085] In some embodiments, the method is implemented on a quantum simulator. Examples of quantum simulators include, but are not limited to, a quantum circuit simulator, a quantum system simulator, or a quantum state simulator. In some embodiments, the quantum simulator comprises using quantum hardware for partial or total execution of the simulation.

[0086] In some embodiments, the method is implemented on hybrid quantum-classical hardware. Examples of hybrid quantum-classical hardware include, but are not limited to, quantum computers, quantum simulators, or quantum annealers.

[0087] In all previous embodiments, it is possible that different parts of the method are implemented on different hardware. This allows for a more efficient and scalable solution to the integer optimization problem.

[0088] Figure 3 shows a non-limiting example of a quantum-inspired optimization method 206 in accordance with some embodiments.

[0089] The method includes initializing control variables of the system 301 . Examples of these control variables include, but are not limited to, a current iteration n of the optimization process, a current state of the system, a current time t of the optimization process, current parameters of a cost function p, and / or current values of hyperparameters s.

[0090] In some embodiments, the control variables are initialized randomly. In some embodiments, the control variables are initialized based on the graph information, the local dimensions, the state ansatz, the optimization algorithm, and / or the hyperparameters resulting from one or more of the corresponding steps 201-205 described in relation to the embodiments of the method of Figure 2. In some embodiments, the control variables are initialized using a Reinforcement Learning method.

[0091] After initializing the control variables, the method includes evaluating the expression for the cost function 302.

[0092] In some embodiments, the cost function is evaluated as a function of the state. In some embodiments, this dependence is a continuous one. This allows for an efficient optimization process, by exploiting the capabilities of the optimization algorithm.

[0093] In some embodiments, the cost function is evaluated as a function of the angular momentum operator acting on the qudits. This allows for a more natural representation of the cost function in the embodiments where the integer variables are represented using spherical coordinates.

[0094] In some embodiments, the cost function is evaluated as a Hamiltonian operator acting on the qudits. This allows for a more efficient optimization process, by exploiting the capabilities of the optimization algorithm.

[0095] In some embodiments, the cost function is evaluated as a function of the hyperparameters. This allows for a more flexible optimization process, by allowing for the optimization of the hyperparameters, including, but not limited to, using computational graphs for accelerated performance.

[0096] In some embodiments, the cost function comprises at least a penalty term. One purpose of this term is to enforce the integer constraints of the integer optimization problem. In some embodiments, the penalty term is evaluated as a function of the angular momentum operator acting on the qudits. In some embodiments, the penalty term is evaluated as a Hamiltonian operator acting on the qudits.

[0097] In some embodiments, the cost function comprises at least a random potential. One purpose of this term is to promote the exploration of the solution space, by avoiding local minima. In some embodiments, the random potential is evaluated as a function of the angular momentum operator acting on the qudits. In some embodiments, the random potential is evaluated as a Hamiltonian operator acting on the qudits. In some embodiments, the random potential is a function of the time-step of the optimization process.

[0098] In some embodiments, the cost function is evaluated as a function of the control variables. This allows for a dynamic form of the cost function. In some embodiments, the cost function is evaluated as a function of the previous states of the system. This allows for a memory effect in the optimization process. In some embodiments, the cost function is a function of the current step n of the optimization process, and / or the current time t of the optimization process. This allows for a time-dependent form of the cost function. Examples of these time-dependent forms include, but are not limited to, a linear form, an exponential form, or a sinusoidal form.

[0099] In some embodiments, the cost function includes a transversal term. This term is used to promote the exploration of the solution space, by promoting entanglement between the qudits. In some embodiments, the transversal term is evaluated as a function of at least one projection of the angular momentum operator acting on the qudits. In some preferred embodiments, the projection used for the transversal term is not the same as the one used for the eigenstates of the colors.

[0100] After evaluating the cost function, the method includes executing 303 the defined optimization method at 204. This method finds the minimum of the cost function, by exploring the configuration space of the system. In some embodiments, the execution of the optimization method 303 is configured by the hyperparameters of step 205 of the embodiments of the method of Figure 2. These hyperparameters define a schedule s for the optimization process. Examples of these schedules include, but are not limited to, executing the optimization method for a fixed number of iterations, executing the optimization method until a convergence criterion is met, or executing the optimization method for a fixed amount of time.

[0101] In some embodiments, the schedule s is dependent on the control variables. This allows for a dynamic form of the optimization process. In some embodiments, the schedule s is dependent on the cost function. This allows for a more efficient optimization process, by adapting the schedule to the cost function. In some embodiments, the schedule s is dependent on the state of the system. This allows for a more flexible optimization process, by adapting the schedule to the state of the system.

[0102] In some embodiments, the schedule s is optimized using a hyperparameter optimization process. Examples of hyperparameter optimization processes include, but are not limited to, grid search, random search, Bayesian optimization, or genetic algorithms.

[0103] After executing the optimization method, the method includes evaluating whether the current state of the system needs to be rounded to a classical form, 304. This allows for having a candidate solution for the integer optimization problem after the optimization process 303 is completed.

[0104] In some embodiments, this rounding is carried out at least once during the optimization process. In some embodiments, this rounding is carried out at random stages of the optimization process. In some embodiments, this rounding is carried out at fixed times of the optimization process. In some embodiments, this rounding is carried out at the end of the optimization process. In some embodiments, this rounding is carried out at every time of the optimization process.

[0105] In some preferred embodiments, rounding means choosing the component of the current solution vector with the largest magnitude, setting that component to one and setting the remaining components to zero

[0106] When the current state of the system needs to be rounded, the method continues by rounding the state of the system 305. In some embodiments, the rounding is performed by projecting the qudit system into the local basis of the qudits. This provides a candidate solution for the integer optimization problem.

[0107] Either after rounding, or not rounding, the method includes evaluating the cost function on the current state 306. When the rounding did not take place, in some embodiments the value of the cost function is recovered directly from the optimization process 303.

[0108] After evaluating the cost function, the method includes evaluating whether the optimization process has converged enough 307. In some embodiments, the optimization process has converged enough when the cost function reaches a minimum value, or when a convergence criterion is met. In some embodiments, the convergence criterion is defined as a fixed value, or as a variable value. In some embodiments, the convergence criterion is defined as a function of the cost function, the state of the system, or the control variables. In some embodiments, multiple convergence criteria are defined, and the optimization process converges when any, a given selection, or all of them are met.

[0109] When the convergence criteria is met, the method includes rounding the state of the system to a classical one 308, identically to the step 305. In some embodiments, the rounding 305 has already taken place, and therefore step 308 is skipped.

[0110] When the convergence criteria are not met, the method includes evaluating whether one or more ending criteria set of the optimization process have been met 309. Examples of these ending criteria include, but are not limited to, a fixed number of iterations, a fixed amount of time, a fixed value of the cost function, or a minimum rate of change of the cost function. In some embodiments, the ending criteria are defined as a fixed value, or as a variable value. In some embodiments, the ending criteria are defined as a function of the cost function, the state of the system, or the control variables. In some embodiments, multiple ending criteria are defined, and the optimization process ends when any, a given selection, or all of them are met.

[0111] When the ending criteria are met, the method includes rounding the state of the system to a classical one 308, described above. In some embodiments, the rounding 305 has already taken place, and therefore step 308 is skipped.

[0112] When the ending criteria are not met, the method includes updating the control variables of the system 310. Examples of these updates include, but are not limited to, updating the current iteration n of the optimization process, updating the current state of the system, updating the current time t of the optimization process, updating the current parameters of the cost function p, and / or updating the current values of the hyperparameters s.

[0113] After updating the control variables, the method includes returning to step 302, and evaluating the expression for the cost function. This process is repeated until the optimization process converges, or until the ending criteria are met.

[0114] In some embodiments, several or all steps of the method are repeated at least once, by changing the hyperparameters, the state ansatz, the optimization algorithm, or any other part of the method. This allows for a better exploration of the configuration space, and for a more efficient solution to the integer optimization problem.

[0115] The method then returns the best classical solution found 311.

[0116] Figure 4 shows a non-limiting example of a particular quantum-inspired optimization method 206, as some embodiments of the method described in Figure 3.

[0117] The method includes initializing to zero the current time of the system t, which is used as the only control parameter of the cost function.

[0118] After initializing the current time, the method includes evaluating the expression for the cost function 401. In these embodiments, the cost function is defined as a function of the current time t of the system. In some embodiments, the cost function is defined as a continuous function of the current time t. Examples of these cost functions include, but are not limited to, the linear interpolation: c(e, t) = (i - t)Ez(e) + 1 EF(9 where E / e) is the initial energy of the system, EF(e) is the final energy of the system, and 9 codifies the qudit states. This cost function is designed to find the minimum of the energy of the system, by interpolating between the initial and final energy expressions of the system.

[0119] In some embodiments, the initial cost function E} and / or the final cost function EFis written in terms of the angular momentum operator acting on the qudits. Examples of these initial cost functions include, but are not limited to, the projections of the angular momentum operator on a given axis. In some embodiments, the axis used for the initial cost function is different from the one used for the eigenstates of the colors. As a non-limiting example, if the colors are associated with the eigenstates of the Lzoperator, the initial cost function can be associated with the eigenstates of the Lxoperator such as: where (L ) is the expectation value of the Lxoperator acting on the qudit i, and V is the set of vertices of the graph. Additional non-limiting examples for the initial cost function include, but are not limited to, a term that enforces an equal superposition of all qudit states.

[0120] In this particular, non-limiting example, an expression for the final energy reads where pLis the vector of probabilities of the qudit i, Jtjis the coupling matrix between the qudits i and j, and £ is the set of edges of the graph.

[0121] In some embodiments, the coupling matrix J^ is either 1 or 0, depending on whether the qudits i and j are connected by an edge. In some embodiments, the values of the coupling matrix Jtj are perturbed by a random potential. In some embodiments, the random potential is a function of the current time t of the system. In some embodiments, the random potential is a function of the hyperparameters of the problem. Examples of these perturbations include, but are not limited to, Jij = 1 + hij, where htj is a random variable with a given distribution. In some embodiments, the distribution of the random variable htj applies only to the existing edges of the graph (i.e. , htj = 0 if the original J^ = 0). This allows for the perturbation of only the existing edges of the graph, without introducing artificial connections. In some embodiments, the distribution of the random variable htj applies to all the edges of the graph. In some embodiments, the distribution of the random variable htj is given by a uniform distribution over the interval [0, h] where h is a hyperparameter of the problem. In some embodiments, h is itself a function of the current time t of the system.

[0122] In some embodiments, the cost function includes additional terms. Examples of these terms include, but are not limited to, the following modification of the cost function: where Ew(e) is a term that promotes a finite probability for all possible colors, even those that are initialized close to zero. As a non-limiting example, this term can be written as iv(0) = Y^ PT - log(Pi), lev where y is a hyperparameter of the problem. Positive values of y promote the presence of a finite probability for all colors, while negative values of y allow for the selection of a single color.

[0123] After evaluating the expression of the cost function, the method includes executing the optimization method 403. In these embodiments, the optimization method comprises executing s steps of a gradient descent algorithm whereas, in some other embodiments, other optimization methods are used. Examples of these methods include, but are not limited to, ADAM, simulated annealing, genetic algorithms, or particle swarm optimization.

[0124] After executing the optimization method, the method includes rounding the state of the system to a classical form 404. In this particular embodiment, the rounding is carried out at every step after the optimization process 403.

[0125] After rounding the state of the system, the method includes evaluating the cost function on the current state 405.

[0126] The method further evaluates at 406 whether the classical cost is a minimum. Using the non-limiting example used during the disclosure of Figure 1 as a non-limiting example, this involves evaluating whether the expression is zero. When the cost is zero, the method returns the best classical solution found 408.

[0127] When the cost is different from zero, the method evaluates at 407 whether the current time of the system is equal or larger to one. If so, this is used as a stop condition, and the method returns the best classical solution found 408. If not, the method continues by updating the current time of the system 409, and returns to step 402, evaluating the expression for the cost function at the new time.

[0128] The values of t = 0 and t = 1 as the initial and final times of the system are chosen as an example. However, other values can be used, depending on the problem at hand.

[0129] Figure 5 shows another non-limiting example of a particular quantum-inspired optimization method 206, as some embodiments of the method described in Figure 3.

[0130] In this case, the method has a fixed expression for the cost function. As a non-limiting example, this cost function is defined as where all terms have the same meaning as in the disclosure for Figure 4. The method executes the optimization method 502. In this particular embodiment, the optimization method comprises executing s steps of a gradient descent algorithm. However, similarly to the case of 403 and 303, this should not be understood as a limitation, and other optimization methods can be used. Examples of these methods include, but are not limited to, ADAM, simulated annealing, genetic algorithms, or particle swarm optimization.

[0131] After executing the optimization method, the method rounds the state of the system to a classical form 503. In this particular embodiment, the rounding is carried out at every step after the optimization process 502.

[0132] After rounding the state of the system, the method evaluates the cost function on the current state 504.

[0133] The method evaluates at 505 whether the classical cost is a minimum. Using the nonlimiting example used during the disclosure of Figure 1 as a non-limiting example, this involves evaluating whether the expression is zero. When the cost is zero, the method returns the best classical solution found 507. When the cost is different from zero, the method evaluates whether the classical cost is the same as the previous one 506. If so, the method returns the best classical solution found 507. If not, the method returns to step 502, and executes the optimization method again.

[0134] To further illustrate how the disclosed quantum-inspired optimization method solves concrete technical problems, the following non-limiting examples are provided, demonstrating the method's application from input ingestion to tangible technical output.

[0135] As a first example, A critical technical problem in modern manufacturing involves efficiently scheduling a set of N jobs (J1, ..., Jw) ontoaset of M machines (Ml t..., MM). The objective is typically to minimize the makespan, which is the total time required to complete all jobs, while adhering to one or more operational constraints. Successfully solving this problem enables improving throughput and / or optimizing resource utilization within the manufacturing facility.

[0136] In applying the disclosed method to this job scheduling problem, the initial step, loading graph information (Step 201), involves ingesting specific technical data pertinent to the manufacturing environment. This data includes, for each job Ji tits required processing time on each potential machine Mj, any release date before which the job cannot start, and its due date / );. Furthermore, data for each machine Mk, such as one, some or each of, e.g., its availability windows, operational capacity, and any setup times Sjtkincurred when switching from processing job Jj to job Ji on that machine, are also provided. Crucially, precedence constraints, often represented as a list of pairs ]j) indicating that job ]tmust be completed before job can commence, form part of this input.

[0137] Following data ingestion, the method proceeds to define the qudit representation and set the local dimension (Step 202). In a common embodiment for job scheduling, each job Jtis represented by a qudit. The "state" or "color" of this qudit then corresponds to a specific assignment for job ]tnamely its allocation to a particular machine Mkand a discrete start time slot Ts. Consequently, the local dimension d of the qudit associated with job ]tis determined by the number of such valid (machine, start_time_slot) combinations available for that job.

[0138] The core of the optimization lies in the definition and evaluation of a cost function C (as in Steps 302, 401), which is designed, preferably as meticulously as possible to yield a more accurate output, to represent the primary objective (e.g., makespan) and to penalize any violations of the manufacturing constraints. A non-limiting example of such a cost function could be formulated as:

[0139] C = a * Makespan + p * (Precedence_Violations) + y * £(Machine_Capacity_Violations) + 5 * (Simultaneous_Job_On_Machine_Violations), where a, p, y, and 5 are weighting hyperparameters. Within this function, the Makespan term is calculated based on the current assignment of jobs to machines and their scheduled times. The Precedence_Violations term introduces a penalty if a job ]tis scheduled to start before its prerequisite has finished. Similarly, Machine_Capacity_Violations penalizes scenarios where a machine is assigned more concurrent work than its capacity allows (if applicable), and Simultaneous_Job_On_Machine_Violations adds a penalty if two distinct jobs are assigned to the same machine during overlapping time intervals.

[0140] The quantum-inspired optimization method (Step 206) then systematically explores the configuration space of possible qudit states (job assignments) to find a configuration that minimizes this cost function C. Upon completion of the optimization process, the best classical solution found is returned as the output (Step 207). This output translates directly into a concrete and actionable manufacturing schedule, specifying for each job Ji tthe assigned machine Mkand its precise start time Ts. Accordingly, in some embodiments, one or more control commands are based on such output so that reconfiguration can be effected. In some cases, a predetermined model or an artificial intelligence processing may be used for establishing which control commands could cause the reconfiguration as known in the art, that fall within the scope of the present disclosure, but are not described in more detail in order not to obscure the teachings of the present disclosure. In this sense, in these cases, the predetermined model or artificial intelligence processing is configured for reconfiguring the machines to obtain the schedule. The technical effect of implementing such a schedule in the manufacturing facility is significant, leading to demonstrably reduced production times, improved utilization of machine resources, and better adherence to delivery deadlines, thereby solving a tangible and critical operational problem.

[0141] Another significant technical challenge addressed by the disclosed method is the optimal balancing of load in a power grid in some embodiments. This involves determining the ideal power output for a set of G generation units (which may include thermal, hydro, and renewable sources) to precisely meet the fluctuating electricity demand across the grid. The optimization must also minimize total generation costs, satisfy the operational constraints of each generator (i.e., generation unit), and respect the capacity limits of transmission lines to ensure overall grid stability.

[0142] The application begins by loading relevant graph information (Step 201). For power grid balancing, this input includes a detailed description of the grid topology, identifying all generation units, load centers (consumers), substations, and the network of transmission lines, along with their respective thermal limits or capacities. Specific data for each generator Gtis also crucial, encompassing its minimum and maximum operational output levels (Pmin,i> Pmax.i its generation cost function (often a quadratic cost curve), and its ramp-up and ramp-down rates, which dictate how quickly its output can change. Furthermore, a load forecast, providing the predicted power demand Dj at various load centers j for the target operational period, and transmission network parameters, necessary for power flow calculations (e.g., using DC power flow models or more complex AC models), are essential inputs.

[0143] In the subsequent step of setting the qudit representation and local dimension (Step 202), each generation unit Gtis typically represented by a qudit. The "state" or "color" of the qudit corresponding to generator Gi signifies a discrete power output level Ptchosen from within its permissible operating range [Pmtn,i>Pmax,i]- The local dimension dtfor this qudit is therefore the number of such distinct, discrete output levels considered for generator Gt.

[0144] The cost function C, evaluated during the optimization (e.g., Steps 302, 401), is formulated to minimize the total generation cost while rigorously enforcing physical and operational constraints. A representative cost function might take the form:

[0145] C = a * £(Cost_i(Pi)) + p * \ _Pi - .Dj\ + Y * £(Transmission_Line_Violations) + 6 * (Ramp_Rate_Violations), where a, p, y, and 5 are tunable weighting hyperparameters. The term £(Cost_i(Pi)) represents the sum of costs associated with generating power Ptfrom each unit i, according to their individual cost curves. The term - .Dj\ penalizes any mismatch between the total power generated and the total predicted demand Transmission_Line_Violations incurs a penalty if the calculated power flow on any transmission line, based on the proposed generation levels Ptand demands Dj , exceeds its established thermal limit. Finally, Ramp_Rate_Violations penalizes any scheduled change in a generator's output from its previous state that violates its specified ramp-up or ramp-down limits.

[0146] The quantum-inspired optimization process (Step 206) then searches for qudit configurations, i.e., sets of generator output levels, that minimize this comprehensive cost function C. The final output (Step 207) of this process is the set of determined power output levels {P1, P2< ■■■ PG } for all generation units. This resulting dispatch schedule, when implemented by the grid operator, achieves several technical effects: it leads to more cost-effective electricity generation, ensures a reliable power supply by accurately matching supply with demand, and critically, helps prevent grid overloads or potential blackouts, thus solving a vital infrastructure management problem. Accordingly, in some embodiments, one or more control commands are based on such output so that reconfiguration of the generators and / or power grid can be effected.

[0147] The disclosed method is also adept at addressing the technical problem of optimally allocating available bandwidth on network links among multiple competing data flows in some embodiments. The goal here is to maximize overall network performance, often measured by total throughput or achieved Quality of Service (QoS) for users, or to ensure fairness in resource distribution, all while strictly respecting the capacity constraints of individual network links.

[0148] The process commences with loading the graph information (Step 201), which for bandwidth allocation includes the network topology, defined as a set of nodes (e.g., routers, switches) and the links connecting them. Each link I is characterized by its maximum bandwidth capacity Capt. Information about the data flows themselves is also critical: a set of F data flows is considered, where each flow f is defined by its source and destination nodes. Optionally, multiple possible paths Pathfpthrough the network for each flow may be specified, along with a demand or utility function Uf (bf) that quantitatively describes the utility or QoS level achieved if flow f is allocated a bandwidth of bf.

[0149] Next, the qudit representation and local dimension are established (Step 202). In this context, each data flow f can be represented by a qudit. The "state" or "color" of this qudit then corresponds to a discrete level of bandwidth bf(e.g., measured in Mbps) allocated to that specific flow. The local dimension dffor the qudit associated with flow f is thus the number of distinct, discrete bandwidth levels being considered for allocation to that flow. In more complex scenarios where path selection is integrated into the optimization, the qudit state could represent a combined (path, bandwidthjevel) pair.

[0150] The cost function C, evaluated in steps such as 302 or 401 , is, in some cases, formulated to maximize the total utility derived from bandwidth allocation (or equivalently, minimize negative utility), subject to the hard constraints of link capacities. A non-limiting example of such a cost function is:

[0151] C = -a * Uf (bf) + P * Z_ Penalty(Load(, Capi).

[0152] Here, t / y (by)is the sum of utilities obtained for all flows, given their respectively allocated bandwidths bf. The term Loadi represents the total bandwidth allocated on a specific link I by all data flows that traverse it, calculated as Loadt= 7 {f using I} by. The 7 Penalty(Load(, Capi). term imposes a substantial penalty if the calculated Loadbon any link exceeds its Cap and is zero or a smaller congestion-related penalty otherwise. The parameters a and serve as weighting hyperparameters to balance the objectives of utility maximization and constraint adherence.

[0153] The quantum-inspired optimization method (Step 206) then seeks qudit configurations — that is, specific bandwidth allocations bffor each flow f — that minimize this cost function C (thereby maximizing the total utility). The output of the optimization process (Step 207) is the set of determined bandwidth allocations {b1;b2, ..., bF}. The technical effect of implementing this resulting bandwidth allocation scheme, for instance, in network routers or via a Software-Defined Networking (SDN) controller, is a direct improvement in overall network efficiency. It facilitates fair resource sharing among competing flows, enhances user experience by helping to meet QoS targets, and actively prevents network congestion, thereby solving a key technical challenge prevalent in modern communication systems. Accordingly, in some embodiments, one or more control commands are based on such output so that reconfiguration of network nodes and / or links can be effected.

[0154] An aspect of the disclosure relates to a method according to any one of the examples of Figures 3, 4 and 5. Another aspects of the disclosure relate to a computer program, a data processing apparatus or system, and a computer-readable medium configured to carry out a method according to any one of the examples of Figures 3, 4 and 5.

[0155] It is a purpose of this disclosure to provide improved quantum-inspired methods and apparatuses for integer optimization problems. The methods and apparatuses disclosed herein provide a more efficient and scalable solution to all these applications, compared to known integer optimization algorithms by enabling the formulation and resolution of complex technical constraints and objectives inherent in these real-world systems.

[0156] One non-limiting application of such methods and apparatuses is in solving the job scheduling problem. In this problem, a set of jobs must be assigned to a set of machines, such that the total time to complete all jobs is minimized. This problem is known to be NP-hard, and is of great importance in many industries. Particular embodiments of this problem include, but are not limited to, production and manufacturing timetable definition, project management, and / or resource allocation. As detailed above, the disclosed method ingests job parameters, machine characteristics, and constraints, maps jobs / assignments to qudit states, and optimizes a cost function representing makespan and penalties to output an actionable production schedule.

[0157] Another non-limiting application of such methods and apparatuses is in solving the transportation problem. In this problem, a set of goods must be transported from a set of suppliers to a set of customers, such that the total cost of transportation is minimized. This problem is known to be NP-hard, and is of great importance in logistics and supply chain management.

[0158] Another non-limiting application of such methods and apparatuses is in solving the optimal allocation of resources. In this problem, a set of resources must be allocated to a set of tasks, such that the total cost of allocation is minimized. This problem is known to be NP-hard. Particular embodiments of this problem include, but are not limited to, the optimal allocation of funds in a portfolio, the optimal allocation of energy in a power grid, the optimal allocation of bandwidth in a communication network, the optimal placement of registers in a microprocessor, the optimal allocation of memory in a computer system, the optimal allocation of air traffic control, and / or the optimal scheduling of sports tournaments. For instance, in allocating energy in a power grid, as exemplified earlier, the method takes generator capacities, costs, demand forecasts, and grid topology as inputs. Qudits represent generator output levels. The optimization minimizes a cost function reflecting generation costs and grid stability penalties, outputting a dispatch plan for generators. Similarly, for bandwidth allocation in a communication network, network topology, flow demands, and link capacities are input; qudits represent bandwidth assignments to flows; the optimization maximizes network utility subject to capacities, outputting an allocation plan.

[0159] Methods in accordance with embodiments of the present disclosure are, in some embodiments, entirely run by one or more processors and, thus, are computer-implemented methods. Such methods can be implemented, in some cases, by a data processing device or system comprising means for carrying out the methods. Also, such methods can be implemented, in some cases, in the form of a computer program product that comprises instructions which, when the program is executed by at least one computer, cause the at least one computer to carry out the method. The instructions or the computer program product may be comprised by a non- transitory computer-readable storage medium.

[0160] Alternatively, the instructions or the computer program product may be comprised by a data carrier signal carrying the instructions or the computer program product.

[0161] In this text, the term “includes”, “comprises” and derivations thereof (such as “including”, “comprising”, etc.) should not be understood in an excluding sense, that is, these terms should not be interpreted as excluding the possibility that what is described and defined may include further elements, steps, etc.

[0162] On the other hand, the disclosure is obviously not limited to the specific embodiment(s) described herein, but also encompasses any variations that may be considered by any person skilled in the art (for example, as regards the optimization method, state ansatz, application, or execution environment), within the general scope of the invention as defined in the claims.

[0163] This work has been partially funded by the European Union’s Horizon Europe research and innovation programme under grant agreement No. 101080086.

Claims

CLAIMS1 . A method for solving an integer optimization problem, the method comprising: a) defining a local representation of integer variables of the integer optimization problem as a set of qudits, where a local dimension of the qudits is set to a number of colors that can be assigned to each node of a graph, b) setting an state ansatz for the state of the qudits, c) setting a cost function as a function of the state, d) setting an optimization algorithm to minimize or maximize the cost function, and e) executing a quantum-inspired optimization method, wherein the quantum-inspired optimization method comprises: i. initializing control variables of the quantum-inspired optimization method, ii. evaluating the expression for the cost function, iii. executing the set optimization algorithm, iv. conditionally rounding the current state of the qudits to a classical form, v. evaluating the cost function on the current state of the qudits, vi. evaluating whether the optimization method has converged, and returning the solution found if the optimization method has converged, or vii. evaluating whether the predetermined ending criteria of the quantum- inspired optimization method have been met, and returning the classical solution found if the ending criteria have been met, or viii. updating the control variables of the quantum-inspired optimization method, and returning to step ii) until the quantum-inspired optimization method converges, or until the predetermined ending criteria are met.

2. The method of claim 1 , wherein the integer optimization problem is an instance of a graph coloring problem.

3. The method of claims 1 or 2, wherein the local representation of the qudits is written in terms of eigenstates of a component of the local angular momentum operator acting on the qudits.

4. The method of any of claims 1 to 3, wherein the state ansatz is set as a tensor network.

5. The method of any of claims 1 to 3, wherein the state ansatz is set as a variational state.

6. The method of any of claims 1 to 5, wherein the ending criteria are set before the executionof the quantum-inspired optimization method.

7. The method of any of claims 1 to 6, wherein the cost function contains at least a random potential term.

8. The method of any of claims 1 to 7, wherein the cost function is varied from an initial cost function to a different final cost function.

9. The method of any of claims 1 to 8, wherein the quantum-inspired optimization method is a gradient descent algorithm.

10. The method of any of claims 1 to 9, wherein the quantum-inspired optimization method is executed for a fixed number of iterations.

11. The method of any of claims 1 to 10, wherein the state of the qudits is rounded to a classical form at every step of the quantum-inspired optimization method.

12. The method of any of claims 1 to 11 , wherein all or any of steps i. to viii. of the quantum- inspired optimization method are repeated at least once, by changing the hyperparameters, the state ansatz and / or the optimization method.

13. The method of any of claims 1 to 12, wherein the integer optimization problem comprises a job scheduling problem within a manufacturing facility with input data at least comprising: a list of jobs to be processed by the manufacturing facility, a list of machines of the manufacturing facility, precedence constraints and setup times of machines of the manufacturing facility; wherein the cost function at least comprises minimizing a makespan or total tardiness of the jobs to be processed.

14. The method of any of claims 1 to 12, wherein the integer optimization problem comprises allocation of bandwidth in a communication network with input data at least comprising: a topology of the communication network; a list of requested data flows within the communication network; and a list of potential paths for each data flow through the communication network; wherein the cost function at least comprises maximizing overall utility of the communication network or minimizing congestion in the communication network.

15. The method of any of claims 1 to 12, wherein the integer optimization problem comprisesbalancing of load in a power grid with input data at least comprising: a topology of the power grid with at least a list of generation units and transmission lines of the power grid; operation constraints of each generation unit of the power grid; and capacity limits of each transmission line of the power grid; wherein the cost function at least comprises providing a power output for a set of the generation units that minimizes total generation costs and satisfies both operational constraints of each generation unit and capacity limits of transmission lines.

16. The method of any of claims 1 to 15, further comprising providing, based on an output of the integer optimization problem, one or more control commands to at least one controller or target for adjusting operation of one or more apparatuses associated with the integer optimization problem.

17. The method of claim 16, further comprising determining the one or more control commands based on the output of the integer optimization problem.

18. The method of any of claims 16 and 17, when depending upon claim 12, either directly or indirectly, wherein the repetition of steps of the quantum-inspired optimization method is conducted at least after providing the one or more control commands so that new inputs for the integer optimization problem are provided.

19. A computer program comprising instructions which, when the program is executed by a at least one computer, cause the at least one computer to carry out the methods of claims 1 to 18.

20. A data processing apparatus or system comprising means for carrying out the methods of claims 1 to 18.21 . A system comprising: at least one data processing apparatus or system according to claim 20, and one of: a manufacturing facility, a communication network, or a power grid.

22. A computer-readable medium comprising instructions which, when executed by at least one computer, cause the at least one computer to carry out the methods of claims 1 to 18.

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