Quantum calculation method for solving combinatorial optimization problem

Abstract

There is provided a quantum computing method for obtaining an optimal solution to a problem having a plurality of discrete variables, where the problem is represented by a cost function, the method comprising:-generating a graph structure from the cost function,-dividing the graph structure into at least two separate sub-graph structures, where each sub-graph structure comprises a subset of the plurality of variables,-dividing the sub-graph structures into at least two separate sub-graph structures, mapping each subgraph structure to a local cost function represented as a local cost Hamiltonian, determining, for each local cost Hamiltonian, all eigenstates corresponding to an energy below a predetermined cutoff energy using a quantum processing device, wherein each variable in the subset of the plurality of variables is represented by a qubit of the quantum processing device,-recombining the determined eigenstate, and-approximating a ground state from the recombined eigenstate, where the ground state represents the optimal solution.

Description

Technical Field

[0001] The present invention relates to a quantum computing method for obtaining an optimal solution to a combinatorial optimization problem and a corresponding quantum computing system. Background Art

[0002] Combinatorial optimization problems arise in many areas of industry and logistics. Typical examples include maximum satisfiability (MaxSAT) problems such as the traveling salesman problem, the job shop scheduling problem, or the employee scheduling problem. Each of these problems can be expressed as a search for a bit string (i.e., a sequence of zeros and ones) such that the cost function of the corresponding problem is minimized. The number of possible bit strings grows exponentially with the length of the bit string, which determines the system size. Therefore, for moderate system sizes, finding an exact solution becomes infeasible. However, quantum information processing platforms can provide the option of faster algorithms than classical computers.

[0003] Currently, there are two main methods for solving combinatorial optimization problems.

[0004] The first approach is called quantum annealing or adiabatic quantum computing. In quantum annealing, the cost function of the problem is formulated as a quantum mechanical Hamiltonian, whose ground state corresponds to the solution to the problem. The ground state is prepared by using cooling or adiabatic dynamics. Currently, there are devices with thousands of quantum bits (qubits) that implement quantum annealing. However, it is not clear whether the qubits in these devices work properly, that is, whether they really use quantum properties.

[0005] The second method is the quantum approximate optimization algorithm (QAOA), which aims to generate the ground state of the Hamiltonian operator through a sequence of quantum gates in a digital quantum computer. In this process, a state is prepared in the form of a quantum mechanical superposition of all possible bit strings, and a parameterized gate sequence is used to filter the best candidate solution of this solution from the quantum mechanical superposition. The measurement of the energy expectation value of the prepared state provides information about the proximity of this prepared state to the sought ground state. This information is used to optimize the parameters of the gate sequence to maximize the approximation.

[0006] The disadvantage of the two known methods, quantum annealing and QAOA, is that the quantum processor requires as many qubits as there are bits in the bit string. For example, in the case of a bit string sequence consisting of a hundred zeros and ones, a hundred qubits are required to find the optimal solution. In practice, relevant applications in industry are based on thousands of binary different variables, so that the corresponding bit strings consist of thousands of bits. Therefore, as long as the quantum information processing platform includes far fewer than a few thousand normally working qubits, the requirement for the number of qubits for the known quantum methods is a big limitation.

[0007] At present, quantum computers only include about a hundred qubits or less. In addition, the gates (i.e., the mechanisms for manipulating qubits) are not perfect, and quantum information systems are susceptible to noise. Error correction is an ongoing research topic in the development of quantum hardware, which will still take some time, and it is not yet clear whether a fully error-corrected quantum computer of a practically relevant size will ever appear. The implementation of a single error-correcting qubit is already very complex and requires multiple qubits. Therefore, error-correcting quantum computers include an even smaller number of error-correcting qubits. Summary of the invention

[0008] It is therefore an object of the present invention to provide a quantum method that is able to find solutions to combinatorial optimization problems for practical applications on quantum computers whose capacity is strongly limited relative to the problem size.

[0009] Solution according to the invention The solution according to the invention is a quantum computing method as specified in independent claim 1. Preferred and / or advantageous embodiments are specified in the dependent claims. Furthermore, a corresponding system for carrying out the method according to the invention is specified in claim 13.

[0010] Therefore, a quantum computing method for obtaining an optimal solution to a problem with multiple discrete variables is provided. The problem is represented by a cost function. The method comprises: - Generate a graph structure from this cost function, - dividing the graph structure into at least two separate sub-graph structures, wherein each sub-graph structure includes a subset of the plurality of discrete variables, - Map each subgraph structure to a local cost function represented as a local cost Hamiltonian, - for each local cost Hamiltonian, determining, using a quantum processing device, all eigenstates corresponding to energies below a predetermined cutoff energy, wherein each variable in the subset of the plurality of discrete variables is represented by a qubit of the quantum processing device, - the eigenstates determined by the recombination, and - approximating a ground state from the recombined eigenstates, wherein the ground state represents the optimal solution.

[0011] The step of approximating the ground state can be performed multiple times.

[0012] A cost function representing a problem with multiple discrete variables assigns a cost value to each combination of the multiple discrete variables. For typical problem sizes encountered in industry, i.e., thousands or more discrete variables, the possible combinations of the variables are so numerous that a brute force calculation to determine all possible cost values ​​is infeasible even on the most advanced supercomputing systems.

[0013] However, it is found that the cost function can generally be represented by a graph structure. Generating this graph structure representation has the following advantages: clustering methods can be applied to the graph in order to partition the graph structure and thus the original problem size into smaller disjoint subgraph structures. Each subgraph structure comprising a subset of multiple discrete variables can be solved independently. Therefore, the method according to the present invention is adapted to overcome the limitations of current and short-term quantum processing hardware. Although the original problem with a large number of discrete variables is too large for current and short-term quantum processing hardware, after partitioning into smaller subsets, each of the subsets can be solved independently on a quantum processing device.

[0014] However, it is found that typical combinatorial optimization problems are inherently frustrating. Due to the frustrating nature of the graph structure, the subgraphs cannot be considered to be completely decoupled from each other. Therefore, the optimal solution to the original problem cannot be found in a set that includes the best solution of each of the subgraph structures.

[0015] The inventors first discovered that the frustration problem of the cost function can be solved not only by determining the best solution for each subgraph structure (i.e., the ground state of each local cost Hamiltonian), but also by determining all eigenstates corresponding to energies below a predetermined cutoff energy. It has been recognized that all eigenstates corresponding to energies below a predetermined cutoff energy can be determined using QAOA without further modification of its quantum operations including quantum gates and measurements, the only modification being in the classical evaluation cost function. Therefore, it is extremely convenient to implement the method according to the present invention in current quantum information processing platforms.

[0016] By reorganizing the eigenstates of all subgraphs, the original combinatorial optimization problem with multiple discrete variables can be simplified to a problem with the determined eigenstates. Therefore, the number of discrete variables of the original problem is effectively reduced to the number of determined eigenstates. By determining the basis state of the system in the reorganized eigenstate basis, the optimal solution of the combinatorial optimization problem can be approximated.

[0017] In one embodiment, the at least two subgraph structures are interconnected. The coupling of a subgraph structure to an adjacent subgraph structure is quantified by a coupling strength. The predetermined cutoff energy of each subgraph structure is determined by summing the coupling strengths of each of the couplings of the subgraph structures.

[0018] It is found that this coupling strength provides a useful energy scale to determine which eigenstate of the local cost Hamiltonian of the subgraph structure should be taken in. This determination of the cut-off energy has the advantage that the method according to the invention can be applied to any type of combinatorial optimization problem as long as the coupling strength between each subgraph structure is known and / or can be determined.

[0019] In one embodiment, the determined eigenstates are reorganized by generating a representation of the cost function in a reduced Hilbert space spanned by the determined eigenstates.

[0020] The Hilbert space can be viewed as a vector space with a modified scalar product. In the simplified Hilbert space, the basis is formed by the determined eigenstates and is therefore finite. The determined eigenstates can be considered as specific combinations of multiple discrete variables of the subgraph structure.

[0021] This recombination of the determined eigenstates has the advantage that the number of discrete variables of the original problem is effectively reduced to the number of determined eigenstates. By approximating the ground state in a reduced Hilbert space, an optimal solution is obtained.

[0022] In one embodiment, after the determined eigenstates are reorganized into the simplified Hilbert space, a simplified graph structure is generated from the representation of the cost function in the simplified Hilbert space. The simplified graph structure can be divided into at least two disjoint subgraph structures and processed according to the above method steps.

[0023] Preferably, the at least two separate subgraph structures are determined by using a heuristic clustering method from a group of methods, the group of methods including at least the Louvain method.

[0024] In one embodiment, the local cost Hamiltonian for each subgraph structure includes a penalty term, wherein the penalty term is adapted to add a predetermined penalty value to the energy expectation value of each quantum state having an energy below the cutoff energy, such that the quantum processing device is adapted to first determine all eigenstates corresponding to energies below the cutoff energy.

[0025] The penalty term has the structure of a projection operator, where the summation is over all quantum states with energies below the cutoff energy, and where each summand is scaled by a penalty value that depends on the corresponding quantum state. This structure has the advantage that the penalty term does not require any modification of the actual quantum gates used in the QAOA. Instead, the penalty term simply adds an energy value to the determined energies corresponding to the quantum states, thereby preferentially determining quantum states with energies below the cutoff energy. This effect arises from the way expectation values, in particular energy expectation values, are calculated in quantum mechanics.

[0026] Preferably, the quantum processing device is adapted to perform a further QAOA to approximate a ground state of the cost function in a reduced Hilbert space.

[0027] Similar to determining the eigenstates of the local cost Hamiltonian, the quantum processing device can also be used to determine the cost Hamiltonian in the simplified or renormalized Hilbert space. The QAOA for determining the ground state in the simplified Hilbert space differs from the first QAOA in that gates and circuits are configured based on the reorganized eigenstates.

[0028] In one embodiment, the cost function representing the problem may be represented by a set of models including at least an Ising spin glass model.

[0029] The Ising spin glass model is characterized by pairwise interactions between bodies, so-called two-body interactions. In this case, the bodies are given by vertices of a graph structure to which the cost function is mapped. Each vertex is assigned a set of two states, e.g., |0> and |1> or |down> and |up>, and the coupling value defines the interaction / coupling strength between the two vertices. In other words, each vertex of the Ising spin glass model represents a discrete variable of the combinatorial optimization problem.

[0030] In one embodiment, each of the at least two subgraph structures is constrained such that a cardinality of a subset of the plurality of discrete variables of the subgraph structure is less than or equal to a number of qubits of the quantum processing device.

[0031] This constraint has the advantage that each subgraph structure is small enough to be solved using QAOA on a quantum processing device.

[0032] In one embodiment, each of the at least two subgraph structures is constrained such that the cardinality of the subset of the plurality of discrete variables of the subgraph structure is less than or equal to the number of error correction qubits of the quantum processing device.

[0033] In one embodiment, the graph structure includes a plurality of vertices, wherein the plurality of vertices are connected by a plurality of connection points. Each vertex represents a discrete variable, and each connection point represents an interaction between at least two discrete variables.

[0034] This graph structure can not only represent two-body interactions between vertices, but also connections in the sense of three-body or more interactions between three or more vertices.

[0035] A quantum computing system is also provided, comprising at least one quantum processing device, the quantum processing device comprising a plurality of quantum bits, and the quantum processing device being adapted to perform a quantum computing method according to the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Some embodiments of the present invention are explained below with reference to the accompanying drawings. As shown below: Figure 1A schematic diagram of method steps for obtaining an optimal solution to a combinatorial optimization problem according to the method of the present invention; Figure 2 For an illustration of the average qubit reduction for an embodiment of the method according to the invention, two iteration steps were used for system sizes ranging from 12 to 32 discrete variables. DETAILED DESCRIPTION

[0037] In the following, the technical basis of the method according to the present invention is emphasized by briefly introducing the key concepts of quantum computing.

[0038] In classical computing, the basic unit of information is the bit, which takes on a value of either 0 or 1. In quantum computing, the basic unit of information is encoded in a quantum bit (or qubit for short). Currently, the implementation of qubits in the real world is a highly researched topic. The key goals are qubits that are noise-resistant, tunable, and easy to mass-produce. A notable version of the qubit is implemented in a superconducting state. Other technologies are based on trapped ions or neutral atoms.

[0039] Similar to a bit, a qubit also includes a first state and a second state, however, these two states of a qubit are quantum states. Therefore, a qubit can also be in any superposition state established by these two states. The superposition state is established by the sum of the two states, where each state is scaled by a probability amplitude as a complex number. The probability amplitude is normalized, which means that the sum of their squared values ​​is 1. It should be noted that, in contrast to a classical bit from which information can be simply read out, the complete information encoded in the superposition state of a qubit cannot be retrieved by only one measurement. More precisely, the qubit is repeatedly prepared and measured multiple times in the same state. Then, the probability amplitude of the first state can be approximated, for example, by dividing the number of observations of the first state by the total number of measurements. Similarly, the probability of observing the second state can be approximated.

[0040] It can be shown that, as long as it is not measured, a qubit can evolve continuously in time on the so-called Bloch sphere, thereby being in any of the infinite number of possible states defined by the Bloch sphere. Continuous variables are inherently accounted for and can be exploited in quantum information processing platforms.

[0041] When two or more qubits are combined into a composite system, more complex quantum states can be achieved. The dimension of the composite system is the product of the dimensions of each subsystem. n The dimension of a composite system of qubits varies with Scales. In other words, the number of continuous variables that can be processed in parallel according to quantum properties scales exponentially in the qubit count. For up to 500 qubits, the number of dimensions is already greater than the number of atoms in the visible universe. This fact forms the basis for the dramatic increase in computational speed that quantum computers can achieve over any state-of-the-art classical computer, known as quantum advantage.

[0042] The quantum state of the qubit system introduced above evolves continuously without measurement, making it possible to process an exponential number of probability amplitudes in parallel. In classical computing, bits are manipulated via logic gates. In quantum computing, a similar principle is introduced through so-called quantum gates. Since the time evolution operator in quantum mechanics is unitary, the gates that manipulate qubits must also meet this requirement, that is, the evolution of the quantum state must be reversible.

[0043] In quantum computing, qubit gates can be represented by any 2 x 2 unitary matrix, in particular a Pauli matrix. For example, a NOT gate for a single qubit can be implemented by a Pauli X matrix. For multi-qubit gates, i.e., gates that act on multiple qubits, a notable example is the controlled NOT gate, also commonly referred to as a CNOT. Every other multi-qubit gate can be decomposed into a CNOT and a single-qubit gate, so that a finite number of single-qubit gates are actually required to implement the universal set of gates as known in classical computing.

[0044] In quantum computing, qubits and the gates used to manipulate them are combined into complete quantum circuits. In short, quantum circuits follow a similar scheme to their classical counterparts: - preparing the initial input state, for example, by cooling the system to an appropriate temperature; - Apply a sequence of quantum gates; -Measure the state of some or all of the qubits.

[0045] Then, a bridge is built between combinatorial optimization problems that occur every day in industry and quantum computers. Below, it is explained how to formulate combinatorial optimization problems in a form that can be solved on quantum computers. Combinatorial optimization may be a very good candidate for the application of quantum computers because quantum computers have the ability to explore the Hilbert space (i.e., the solution space that is transformed into the quantum world) using the superposition of basis states. In other words, quantum computers are able to process many possible candidate solutions in parallel.

[0046] Combinatorial optimization problems are characterized by cost functions that map a countable set of solutions to real values, usually expressed as cost values. The goal of the optimization process is to find a solution from the set of solutions such that the cost function returns the lowest cost value. In the following, we will focus on solutions whose solutions are of length nIn order to solve combinatorial optimization problems using quantum computers, the classical cost function needs to be converted into a Hamiltonian. A Hamiltonian is an operator that assigns energy values ​​to quantum states of a system. The quantum state of a Hamiltonian with the lowest energy is called the ground state. Therefore, by converting the classical cost function into a Hamiltonian, the combinatorial optimization problem is transformed into the problem of finding the ground state of the corresponding Hamiltonian.

[0047] The problem of approximating the ground state of the Hamiltonian can be solved using a variational quantum solver. One possible algorithm for variational optimization is QAOA, which approximates the ground state of the system.

[0048] The ground state of a system is characterized by the minimum energy value of the system assigned to it by the Hamiltonian operator. Variational optimization is based on the Ritz-Rayleigh variational principle: an arbitrary wave function is selected on the basis of the exact eigenstates of the Hamiltonian operator. It can be seen that the energy expectation value of the selected wave function is greater than or equal to the ground state energy. Therefore, when the selected wave function is parameterized according to one or more parameters, the parameters can be optimized so that the energy expectation value of the wave function is minimized. Therefore, the wave function characterized by the parameters with the minimum energy expectation value can be regarded as the approximated ground state of the Hamiltonian. Therefore, the energy value corresponding to the Hamiltonian acting on the approximated ground state is regarded as the ground state energy. Depending on the parameterization of the wave function, the approximated ground state may even correspond to the exact ground state.

[0049] In QAOA, the parameterization of the wave function is based on two parameter vectors. On the computational basis implemented by quantum circuits, the expectation value of the prepared wave function is approximated by its repeated measurements. Then, the optimal parameters are determined by iteratively calling the classical optimizer, making QAOA actually a hybrid algorithm.

[0050] As mentioned above, the cost function of a combinatorial optimization problem can be represented by a Hamiltonian. The Hamiltonian is usually in the form of a sum over all active qubits, where the sum is over every product between the Pauli Z-gates acting on the qubits.

[0051] Therefore, the Hamiltonian may include an identity term, which only produces a global phase and can therefore be ignored. The Hamiltonian may also include a linear term, which also does not carry important information. The first relevant term in the Hamiltonian is a quadratic term. This quadratic term defines the sum over all pairs of two Z gates scaled by a coupling value, which depends on the Z gates involved. Therefore, this coupling value encodes a two-body interaction between the two qubits. A Hamiltonian containing only quadratic terms is called an Ising spin glass model. It is known that, in general, finding a configuration that produces the ground state of the Ising spin glass model is a non-deterministic polynomial (NP) problem. Many highly relevant optimization problems in industry can be mapped to this Hamiltonian, including the traveling salesman problem and the maximum cut problem. Then, the Ising spin glass Hamiltonian can be mapped to a graph, or represented by a graph, so that the tools of graph theory can be used to take advantage of the present invention.

[0052] The Hamiltonian may also include cubic or higher order terms, including coupling values ​​representing three-body or multi-body interactions.Such Hamiltonians may also be partitioned into clusters and thus may be solved using the present invention.

[0053] A graph includes a plurality of vertices (also referred to as nodes) and connection points between the vertices. In the case of a quadratic Hamiltonian, these connection points are simplified to edges between the vertices. In the following, embodiments of the present invention are considered with quadratic terms of the Hamiltonian. Nevertheless, the concepts discussed below are similarly applicable to Hamiltonians including cubic terms and higher-order terms.

[0054] In the case of a quadratic Hamiltonian, each edge comprises a pair of vertices, also called endpoints, which are connected via corresponding edges. In addition, a weight is attached to each edge of the graph. The weight is given by the coupling value between the two corresponding endpoints. Each term of the cost Hamiltonian is converted into an undirected edge, the weight of which corresponds to the coupling value. In addition, vertices are assigned as state attributes to represent the state of discrete variables. Thus, the original combinatorial optimization problem is completely mapped onto the graph, which can then be used to determine the energy assigned to its quantum state. In the case where the cost Hamiltonian includes an additional linear term, this term can be taken into account by assigning an additional local constant defined by a scaling factor of the linear term to each vertex.

[0055] Optimization problems have been mapped onto graphs with the goal of partitioning the graph structure into subsystems. Subsystems are subsequently also referred to as subgraph structures or communities of the graph. In general, graphs are intended to have sets of vertices that are grouped together and act relatively independently of other vertices in the graph. This property is especially found in graphs representing social networks and can be exploited by partitioning the graph into disjoint subsets, called communities. While vertices within a subset are densely connected, communities have only a small number of connection points between each other. The union of all communities yields the entire graph.

[0056] In order to partition the graph structure into subsystems, algorithms that partition the graph into communities can be used. One particularly useful algorithm is the Louvain algorithm based on modularity optimization. Modularity is a metric that quantifies how well the graph is partitioned into different communities. Note that community sizes can be unconstrained, so minimizing the inter-community edges is not enough. Modularity is normalized between 0 and 1, where a value of 0 indicates that the modularity is no better than a random assignment of communities, and values ​​close to 1 indicate strong community structure.

[0057] The Louvain method is a heuristic method for extracting community structures of large graphs in short computation time. In contrast, exact modularity optimization is computationally difficult. Other methods, especially heuristic methods, can be similarly applied to partition graph structures into disjoint communities.

[0058] Figure 1 A schematic overview of the method steps of an embodiment of the method according to the invention using a quadratic cost Hamiltonian is shown.

[0059] First, the The combinatorial optimization problem represented by is mapped to the cost Hamiltonian as described above where χ represents the solution space and C represents the cost function. In this example, the cost Hamiltonian has a quadratic form, thus representing the Ising spin glass model. After mapping the cost Hamiltonian onto the graph structure, the graph is divided into communities using the method described above. Each community is assigned a ,……, After clustering the graph, the original cost Hamiltonian can be expressed as the sum of each of the local cost Hamiltonians and the inter-community terms.

[0060] Instead, for each subsystem with a local cost Hamiltonian, QAOA is used to find all quantum states with energies below a predetermined cutoff energy. These quantum states include the ground state and some excited states of each subsystem. In this process, the eigenstates of the corresponding subsystem are selected as the computational basis states. k By adding a penalty term to all quantum states of the energy of the excited state to modify the local cost Hamiltonian, we can effectively find the k Excited state. Since the quantum circuit generated by QAOA does not depend on which eigenstate is the lowest energy state, the modified local cost Hamiltonian including the penalty term can be implemented using exactly the same gate sequence as the original local cost Hamiltonian. The penalty term only modifies the classical energy evaluation of the measured computational basis state.

[0061] Since we not only focus on kexcited state, and pays attention to all lower quantum states, so this iterative process provides some robustness in the case that QAOA cannot immediately find the lowest possible quantum state. The penalty term is adjusted at each iteration, depending on which excited state is to be found.

[0062] For example, when trying to find the four lowest energy states of a subsystem, where in the second iteration, the third quantum state is found by QAOA instead of the second quantum state, then the local cost Hamiltonian applied to the third iteration includes penalty terms for the ground state and the third quantum state, but not the second quantum state. Therefore, in the third iteration of QAOA, the missing second quantum state can be found. This iterative process continues until a certain threshold is reached, that is, the energy assigned to the quantum state exceeds the cutoff energy.

[0063] The penalty term is assigned its magnitude by a penalty value. On the one hand, the penalty value must be large enough so that the same quantum state is not found twice, on the other hand, choosing a too large penalty value changes the energy landscape and has a negative impact on the optimization process. It was found that the penalty value is optimal when it is determined from the difference between the energy assigned to the quantum state to be penalized and the ground state energy of the community plus the cutoff energy.

[0064] The cutoff energy for each cluster depends not only on the community itself, but is found to be most useful when based on inter-community edges. Therefore, the inter-community term of the original cost Hamiltonian can be used to find an upper bound condition when determining the useful range of this cutoff energy.

[0065] It is found that the cutoff energy of the chosen community is given by the sum of the coupling strengths of each inter-community edge connected to that particular community. Therefore, a configuration of the qubit, i.e., a quantum state inside that particular community, which has an energy that exceeds the sum of the ground state energy and the cutoff energy of that particular community, cannot be part of the ground state of the whole system. This stems from the fact that the communities are isolated from each other except for the inter-community edges. The contribution of these inter-community edges at most increases the energy term given by the cutoff energy. Therefore, in the state found In all combinations of , the ground state |GS〉 exists within the cutoff energy range of each community.

[0066] Then, the determined quantum state is reorganized.

[0067] According to an embodiment of the present invention, the determined quantum states are reorganized so that QAOA can be applied to the reorganized problem to find the ground state of the cost Hamiltonian. To this end, the determined quantum state of each subgraph structure is assigned to a new qubit state. The number of qubits required varies with each community. The quantum state determined within The number of qubits grows logarithmically. The newly assigned qubit states span the simplified Hilbert space. In other words, the quantum states of the determined subgraph structure form the basis states of the simplified Hilbert space.

[0068] The cost Hamiltonian expressed in the basis of the reduced Hilbert space (subsequently also referred to as the renormalized cost Hamiltonian) consists of two terms. The first term of the cost Hamiltonian is a summation over different communities, where the summand is a diagonal matrix, where the diagonal elements are given by the energy levels determined in the corresponding community. Thus, each entry assigns the corresponding energy of the determined quantum state found in the previous optimization process to the reduced qubit state. The second term is a summation over all pairs of communities, where the summand is a diagonal matrix, where the diagonal entries assign inter-community coupling weights to each reduced pair of qubit states.

[0069] Based on the renormalized cost Hamiltonian, the gates for QAOA can be generated. The diagonal matrices are swapped so that the gates corresponding to the summands can be applied separately in the gate sequence of the QAOA algorithm.

[0070] Instead of directly solving the renormalized cost Hamiltonian, it is also possible to consider repeating the previous method steps for the renormalized cost Hamiltonian. In other words, the renormalized cost Hamiltonian can be mapped onto a graph structure. Subsequently, the graph structure can be divided into a number of disjoint subgraph structures, each of which is assigned a local cost Hamiltonian. For each subgraph structure, the local cost Hamiltonian derived from the renormalized cost Hamiltonian is solved, that is, all eigenstates with energies below a predetermined cutoff energy are determined. The new cutoff energy is again given by the sum of all inter-community couplings. The quantum states of each determined community are recombined, and a further renormalized Hamiltonian is determined in a further simplified Hilbert space. This process can be repeated until the ground state of the cost Hamiltonian is found. The iterative process is particularly advantageous for graphs with a strong hierarchical community structure.

[0071] Therefore, the method according to the present invention can be summarized as follows: 1. Given a combinatorial optimization problem instance, choose the number of iterations; 2. Convert the cost function of the combinatorial optimization problem into a graph; 3. Use clustering algorithm to divide the graph into subgraphs; 4. Determine the cutoff energy for each subgraph; 5. Solve the local cost Hamiltonian of each subgraph structure to determine the quantum states with energies below the cutoff energy; 6. Express the determined quantum state in a simplified basis; 7. Calculate the renormalized Hamiltonian on a simplified basis; 8. If the current iteration is not the last one, go to step 2; 9. Solve the entire system, that is, determine the ground state of the renormalized Hamiltonian corresponding to the optimal solution to the combinatorial optimization problem.

[0072] The inventors show that the method according to the present invention can determine the ground state of the cost Hamiltonian of different graph structures.

[0073] Figure 2 Shown is a graph depicting the average qubit reduction when using the method according to the invention with two iterative steps.

[0074] In these simulations, the number of discrete variables ranged from 12 to 32, resulting in 12 to 32 vertices in each graph. For 12 and 16 vertices, the results for each vertex are the average of 1000 data samples. For higher vertex counts, the average for each vertex consists of the average of 100 data samples. The bars pointing in the y direction represent the standard deviation.

[0075] It was found that in the case of 3-regular graphs, the reduction in qubits ranged from about 39% to 49%. In the graph structure, reductions ranging from approximately 53% up to 60% were detected.

[0076] We also tested reducing the cutoff energy by 50% to reduce the number of determined quantum states carried from each community. For the reduced cutoff energy version, the reduction in the number of qubits ranged from about 56% to 68% for 3-regular graphs and The chart shows the increase from approximately 59% to 66%.

[0077] These results indicate that the hardware requirements of quantum information processing platforms are significantly reduced. They also show a clear trend that the hardware requirements will be reduced even further as the system size increases. Note that 3-regular graphs and Graphs typically have weak community structures. Therefore, for graph structures that can be naturally partitioned into communities, the potential for even greater qubit reduction is expected.

Claims

1. A quantum computing method for obtaining an optimal solution to a problem with multiple discrete variables, wherein the problem is represented by a cost function, the method comprising: - generating a graph structure from said cost function, - dividing the graph structure into at least two separate sub-graph structures, wherein each sub-graph structure includes a subset of the plurality of variables, - Map each subgraph structure to a local cost function represented as a local cost Hamiltonian, - for each local cost Hamiltonian, determining, using a quantum processing device, all eigenstates corresponding to energies below a predetermined cutoff energy, wherein each variable in the subset of the plurality of variables is represented by a qubit of the quantum processing device, - the eigenstates determined by the recombination, and - approximating a ground state from the reorganized eigenstates, wherein the ground state represents the optimal solution.

2. The quantum computing method according to claim 1, wherein the at least two sub-graph structures are interconnected, wherein the coupling of one sub-graph structure to an adjacent sub-graph structure is quantified by a coupling strength, and wherein the predetermined cutoff energy of each sub-graph structure is determined by summing the coupling strengths of each of the couplings of the sub-graph structures.

3. A quantum computing method according to any one of the preceding claims, wherein the quantum processing device is adapted to execute a first quantum approximate optimization algorithm for determining the eigenstates of each local cost Hamiltonian.

4. A quantum computing method according to any one of the preceding claims, wherein the determined eigenstates are reorganized by generating a representation of the cost function in a reduced Hilbert space spanned by the determined eigenstates.

5. The quantum computing method according to the preceding claim, further generating a simplified graph structure from the representation of the cost function in the simplified Hilbert space, and recursively repeating the method steps according to claim 1 for the simplified graph structure.

6. The quantum computing method according to any one of the preceding claims, wherein the at least two separate subgraph structures are determined by using a heuristic clustering method from a group of methods, the group of methods including at least the Louvain method.

7. A quantum computing method according to any one of the preceding claims, wherein the local cost Hamiltonian for each subgraph structure includes a penalty term, wherein the penalty term is adapted to add a predetermined penalty constant to the energy value of each quantum state having an energy below the cutoff energy, so that the quantum processing device is adapted to first determine all eigenstates corresponding to energies below the cutoff energy.

8. The quantum computing method according to any one of the preceding claims 4 to 7, wherein the quantum processing device is adapted to execute another quantum approximate optimization algorithm for approximating the basis state of the cost function in the simplified Hilbert space.

9. A quantum computing method according to any one of the preceding claims, wherein the cost function representing the problem can be represented by a set of models, the set of models including at least an Ising spin glass model.

10. The quantum computing method according to any one of the preceding claims, wherein each of the at least two subgraph structures is constrained so that the cardinality of the subset of the plurality of variables of the subgraph structure is less than or equal to the number of quantum bits of the quantum processing device.

11. The quantum computing method according to the preceding claim, wherein each of the at least two subgraph structures is constrained so that the cardinality of a subset of the plurality of variables of the subgraph structure is less than or equal to the number of error correction qubits of the quantum processing device.

12. A quantum computing method according to any one of the preceding claims, wherein the graph structure comprises a plurality of vertices, wherein the plurality of vertices are connected by a plurality of connection points, wherein each vertex represents a discrete variable, and each connection point represents an interaction between at least two discrete variables.

13. A quantum computing system comprising at least one quantum processing device, the quantum processing device comprising a plurality of quantum bits, the quantum computing system being adapted to perform the quantum computing method according to any one of the preceding claims.

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