Quantum operation control layout for quantum computing

Abstract

According to one embodiment, a method for determining a quantum operation control layout for quantum computation on a quantum system is provided. The quantum computation is performed on components of the quantum system arranged according to a mesh. The vertices of the mesh represent possible locations of the components of the quantum system. Each cell of the mesh indicates a quantum interaction between the components of the quantum system arranged in that cell is possible during the quantum computation. The method includes providing a dataset including data representing hyperedges of a hypergraph. The method includes determining a set of generalized cycles. The generalized cycles include hyperedges of the hypergraph or hyperedges of an expanded hypergraph. The expanded hypergraph includes at least the hyperedges of the hypergraph and additional hyperedges. A maximum length of a generalized cycle in the set of generalized cycles is not greater than a maximum number of vertices of a cell of the mesh. The method includes determining a mesh mapping that maps data representing the hyperedges of the hypergraph or the expanded hypergraph to vertices of the mesh. Each generalized cycle of a constrained subset of the set of generalized cycles consists of a hyperedge that maps to a cell of the mesh. The method includes generating a quantum operation control layout. The quantum operation control layout includes data indicating layout vertices of the mesh. Each layout vertex corresponds to a hyperedge mapped according to the mesh mapping. The layout includes data indicating a set of layout vertices, each of which consists of layout vertices in cells of the mesh that correspond to a generalized cycle in a constraint subset of the generalized cycle.

Description

[Technical field]

[0001]

[0003] Embodiments described herein relate to methods of determining a quantum operation control layout for a quantum computation on a quantum system, the quantum operation control layout itself, computer program products including the quantum operation control layout, and methods of performing a quantum computation on a quantum system using the quantum operation control layout. Further embodiments are directed to systems for determining a quantum operation control layout for a quantum computation on a quantum system and / or for performing a quantum operation on a quantum system using the quantum operation control layout, in particular systems configured to perform the methods described herein. [Background technology]

[0002] Computing devices based on classical information processing, i.e., computing devices that do not utilize quantum mechanical effects, began as hardwired computers capable of performing only specific operations. The transition to fully programmable computers revolutionized the field and inaugurated the information age. Currently, quantum computing devices, i.e., computing devices that utilize quantum mechanical effects to solve computational problems in addition to using classical information processing, are probably still at a stage comparable to hardwired computers in that they can only tackle computational problems for which they are specifically designed.

[0003] EP 3113084 A1 describes a method and apparatus for solving computational problems using a quantum system. The quantum computational method / apparatus accepts computational problems, in particular NP-hard or NP-complete computational problems, such as the (classical) Ising spin model with N spins and all-to-all pairwise interactions. The quantum method / apparatus encodes the computational problem into a simplex problem Hamiltonian of the quantum system with adjustable parameters. For example, for a (classical) Ising spin model with N spins and all-to-all pairwise interactions between the N spins, each term in the simplex problem Hamiltonian can be considered to correspond to one of the pairwise interactions. Thus, there are N(N-1) / 2 simplex terms of the problem Hamiltonian acting on a corresponding number of quantum bits (qubits) of the quantum system, and a similar number of adjustable parameters. The qubits of the quantum system represent the parity of the spins of the Ising spin model, with the state |0> indicating an antiparallel arrangement of the corresponding spins of the Ising spin model and the state |1> indicating a parallel arrangement.

[0004] Furthermore, in EP 3113084 A1 a short-range Hamiltonian is provided to compensate for the increased number of degrees of freedom of the quantum system compared to the Ising spin model. The short-range Hamiltonian is the sum of at least N(N-1) / 2-N constraint Hamiltonians, where each constraint Hamiltonian acts with constraint strength C on up to four qubits forming a plaquette of a square lattice containing the qubits of the quantum system. The constraint Hamiltonian ensures consistency with the Ising spin model in that it enforces the existence of an even number (0, 2, etc.) of states |0> in the subset of qubits corresponding to spins with antiparallel alignment in a closed loop, beyond the spins in the Ising spin model.

[0005] The ground state of the final Hamiltonian, which is the sum of the problem Hamiltonian and the short-range Hamiltonian, or at least a thermal state close to the ground state, contains information about the solution of the computational problem encoded in the parameters of the problem Hamiltonian. Measuring the quantum system in such a state reveals information about the solution of the computational problem. The ground state of the final Hamiltonian, or a thermal state close to the ground state, can be reached by quantum annealing, i.e., an adiabatic sweep from the ground state of the initial Hamiltonian to the ground state of the final Hamiltonian, as described in EP 3113084. Alternatively, the ground state can be reached by counter-diabatic driving using a Hamiltonian with an additional anti-adiabatic part, as described in PCT / EP2019 / 066916. Both adiabatic and anti-adiabatic quantum computing can be considered as analog quantum computing. A digital version of quantum computing using quantum gates is described in PCT / EP2019 / 052528. A state approximating the ground state of the final Hamiltonian can be reached by a sequence of unitary operators acting on the initial state. The unitary operators are propagators of the initial Hamiltonian, the problem Hamiltonian and the short-distance Hamiltonian, and the sequence of unitary operators, and their parameters can be optimized using classical feedback algorithms. The unitary operators can be implemented by highly parallelizable application of quantum gates acting on the local or nearest neighbors of the qubits in a square lattice.

[0006] Because the (classical) computational problem is encoded in the parameters of the problem Hamiltonian, these methods and apparatus provide a fully programmable quantum computing architecture, as opposed to a hardwired quantum computing device. Quantum computing architectures are also scalable. However, scaling can be resource intensive. For example, as the number of spins N in the (classical) Ising spin model grows, the size of the quantum system (number of qubits) grows quadratically with N. Furthermore, EP 3113084 A1 describes that the method / apparatus can be applied to the Ising spin model with three-body interactions and can be implemented in a three-dimensional lattice of qubits in a quantum system, and states that the method / apparatus can be generalized to rZ-body interactions. Quantum operations in a quantum system of qubits arranged on a three-dimensional lattice may be possible, but may be difficult to perform. Furthermore, rZ-body interactions lead to implementation in higher dimensional lattices, following the teachings of EP 3113084, but this is impractical due to the limited number of spatial dimensions in our world.

[0007] Thus, there is a need for improvement. Summary of the Invention

[0008] According to one embodiment, a method for determining a quantum operation control layout for a quantum computation on a quantum system is provided. The quantum computation is performed on components of the quantum system arranged according to a mesh. The vertices of the mesh represent possible locations of the components of the quantum system. Each cell of the mesh indicates that a quantum interaction between the components of the quantum system arranged in that cell is possible during the quantum computation. The method includes providing a dataset including data representing hyperedges of a hypergraph. The method includes determining a set of generalized cycles. The generalized cycles include hyperedges of the hypergraph or hyperedges of an expanded hypergraph. The expanded hypergraph includes at least the hyperedges of the hypergraph and additional hyperedges. Wherein a maximum length of a generalized cycle of the set of generalized cycles is not greater than a maximum number of vertices of a cell of the mesh. The method includes determining a mesh mapping that maps the data representing the hypergraph or hyperedges of the expanded hypergraph to vertices of the mesh. Wherein each generalized cycle of a constrained subset of the set of generalized cycles consists of a hyperedge that is mapped to a cell of the mesh. The method includes generating a quantum operation control layout. The quantum operation control layout includes data indicative of layout vertices of the mesh. Each layout vertex corresponds to a hyperedge mapped according to the mesh mapping. The quantum computation control layout includes data indicating a layout vertex set. Each layout vertex set consists of layout vertices in cells of the mesh that correspond to a generalized cycle of a constraint subset of the generalized cycle.

[0009] According to another embodiment, a quantum operation control layout for quantum computation on a quantum system or a computer program product including the quantum operation control layout is provided. The quantum computation is performed on components of the quantum system arranged according to a mesh. The vertices of the mesh represent possible locations of the components of the quantum system. Each cell of the mesh indicates a quantum interaction between the components of the quantum system arranged in that cell is possible during the quantum computation. The quantum operation control layout includes data indicative of layout vertices of the mesh and data indicative of a layout vertex set. Each layout vertex set is composed of layout vertices in a cell of the mesh. The layout vertices may correspond to hyperedges of a hypergraph or an expanded hypergraph that are mapped to the layout vertices according to the mesh mapping, and the layout vertices of each layout vertex set correspond to hyperedges that form a generalized cycle of the hypergraph or the expanded hypergraph.

[0010] According to a further embodiment, there is provided a method of performing a quantum computation on a quantum system, where the quantum computation is performed on components of the quantum system arranged according to a mesh. The vertices of the mesh represent possible locations of the components of the quantum system. Each cell of the mesh indicates that a quantum interaction between the components of the quantum system arranged in that cell is possible during the quantum computation. The method includes providing a quantum computation control layout. The quantum computation control layout includes data indicative of layout vertices of the mesh and data indicative of a layout vertex set. Each layout vertex set is made up of layout vertices in cells of the mesh. The layout vertices may correspond to hyperedges of a hypergraph mapped to the layout vertices according to a mesh mapping, the layout vertices of each layout vertex set corresponding to hyperedges that form a generalized cycle of the hypergraph. The method includes providing the components of the quantum system in a spatial arrangement such that for each layout vertex of the mesh there is a component, and for each layout vertex set quantum interactions are possible between the components corresponding to the layout vertices of the layout vertex set. The method includes applying a local field to the components corresponding to the layout vertex for each layout vertex associated with a non-zero weight. The method includes performing, for each layout vertex set, a quantum interaction between the components corresponding to the layout vertices of the layout vertex set. The method includes measuring some or all of the components of the quantum system.

[0011] The embodiments are also directed to methods of operating the systems described herein, and to the use of the systems to perform methods according to the embodiments described herein.

[0012] Further advantages, features, aspects and details which can be combined with the embodiments described herein are evident from the dependent claims, the description and the drawings. [Brief description of the drawings]

[0013] A full and enabling disclosure to one of ordinary skill in the art is set forth more particularly in the remainder of the specification, including references to the accompanying drawings. [Figure 1] 1 illustrates a mesh that can be used in embodiments described herein. [Diagram 2] 1 illustrates a mesh that can be used in embodiments described herein. [Diagram 3] 1 illustrates a hypergraph that can be used in the embodiments described herein. [Figure 4] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Diagram 5] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 6] 1 illustrates a hypergraph that can be used in the embodiments described herein. [Figure 7] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 8] 1 illustrates an expanded hypergraph that can be used in embodiments described herein. [Figure 9] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 10] 1 illustrates a hypergraph that can be used in the embodiments described herein. [Figure 11] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 12] 1 illustrates a hypergraph that can be used in the embodiments described herein. [Figure 13]1 illustrates a regular generalized cycle of a hypergraph that can be used in embodiments described herein. [Figure 14] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 15] 1 illustrates a hypergraph with side conditions that can be used in embodiments described herein. [Figure 16] 1 illustrates an irregular generalized cycle of a hypergraph with side conditions that can be used in embodiments described herein. [Figure 17] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 18] 1 illustrates an irregular generalized cycle of a hypergraph with side conditions that can be used in embodiments described herein. [Figure 19] 1 illustrates a quantum operation control layout for quantum computing according to, resulting from, or for use with embodiments described herein. [Figure 20] 1 illustrates a system for solving a computational problem, a system for determining a quantum operation control layout for a quantum computation, and a system for performing a quantum computation, according to embodiments described herein. [Figure 21] 1 illustrates a system for solving a computational problem, a system for determining a quantum operation control layout for a quantum computation, and a system for performing a quantum computation, according to embodiments described herein. [Figure 22] 1 illustrates a method for determining a quantum operation control layout for quantum computing according to embodiments described herein. [Diagram 23] 1 illustrates a method for performing quantum computing according to embodiments described herein. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0014] Reference will now be made in detail to various exemplary embodiments, one or more examples of which are illustrated in the figures. Each example is provided by way of explanation and not meant to be limiting. For example, features illustrated or described as part of one embodiment can be used on or in conjunction with other embodiments to yield still further embodiments. This disclosure is intended to include such modifications and variations.

[0015] In the description of the drawings, the same reference numbers refer to the same or similar components. Generally, only the differences with respect to individual embodiments are described. The structures shown in the drawings are not necessarily drawn to scale and may include details that are depicted in an exaggerated manner to facilitate a better understanding of the embodiments.

[0016] Some embodiments described herein relate to methods for determining a quantum operation control layout for quantum computing in a quantum system.

[0017] Information about quantum systems

[0018] A quantum system has components, specifically, physical qubits. The components of the quantum system may be arranged according to a mesh. The mesh may be a two-dimensional mesh. The mesh may be embedded in a two-dimensional manifold, such as a plane. The vertices of the mesh represent possible sites for the components of the quantum system. Each cell of the mesh indicates a possible quantum interaction between the components of the quantum system arranged in that cell during a quantum computation. The cells may be two-dimensional. The cells may have a boundary that, when mapped onto the manifold, can be visualized as a closed polyline (a closed polygonal chain) between the vertices of the cell. Here, a polyline is a series of connected line segments, where the last line segment in the series connects back to the first line segment in the series. The vertices and some of the boundary may be shared between adjacent cells. The quantum computation control layout may indicate a set of layout vertices that indicate which layout vertices are occupied by the components of the quantum system during a quantum computation, and which interactions take place between each set of layout vertices during the quantum computation. The quantum computation control layout may control the quantum computation.

[0019] For example, FIG. 1 shows a mesh 100 of 5×5 vertices and 4×4 cells. Each cell contains four vertices. The mesh in FIG. 1 is a quadrilateral mesh. FIG. 1 can also be considered as a portion of an infinite mesh of that form, i.e. a mesh of practically unlimited size. Cell 110 has vertices 111, 112, 113 and 114. The cell boundary 115 is represented here as a polyline in the form of a square connecting vertices 111, 112, 113 and 114. Cell 110 represents components of a quantum system when they are placed at vertices 111, 112, 113 and 114 and interact with each other during quantum computing. The physics of the quantum system may determine, for example, whether any three components placed in that cell, or even all four components placed in that cell, can interact with each other as in a digital, gate-based style of quantum computing, or whether only all four components can interact with each other as in some forms of analog quantum computing. The number of vertices (v) of the cells (c) of the mesh may be determined by the number of vertices (v) of the cells (c) of the mesh.c ) is the number of vertices contained in the cell. The maximum number of vertices in a cell of a mesh (v max ) is the maximum number of vertices in a mesh cell (v max =maxv c For example, in the mesh shown in Figure 1, each cell has four vertices, so the maximum number of vertices a cell in the mesh can have is also four.

[0020] A mesh generally indicates by its cells which components of a quantum system can interact during a quantum computation when placed at the vertices of the mesh. The vertices of a mesh need not be associated with one fixed location in space. For example, the mesh of FIG. 2 is the same as the mesh of FIG. 1, but when viewed as a geometric object, the geometric object of FIG. 2 is sheared compared to the geometric object of FIG. 1. If a mesh indicates concrete spatial locations that components of a quantum system can occupy, it is referred to herein as a geometric mesh. A geometric mesh can be considered as a rigid body, although potentially extensible. Thus, for a geometric mesh, the vertices cannot be "moved" by mapping them to different spatial locations by different mappings (embeddings), but are fixed by one concrete mapping.

[0021] Which components of a quantum system can interact or are designed to interact during a quantum computation depends on the physical implementation. For example, if qubits are arranged in a square lattice, as shown as black dots in Figures 1 and 2, in one physical implementation, any plaquette of four qubits within a square of the lattice may be able to interact (shown as a star-like shape in the upper right corner of Figure 1). Meanwhile, in another physical implementation, any four qubits within a parallelogram may be able to interact by nearest neighbor interactions in the square lattice (shown as black lines and boxes in the upper right corner of Figure 2, where the black lines represent CNOT gates and the black boxes represent local fields). This makes no difference to the mesh and its vertices and cells, nor to the quantum computation control layout, which shows the layout vertices and layout vertex sets involved in the quantum computation, and can be applied to both physical implementations.

[0022] The mesh provides physical information about the quantum system, in particular information about which components can interact with which other components during quantum computation. Since quantum interactions in many physical implementations are only feasible between components that are close to each other (short-range interactions), in this case the mesh, and in particular its cells, can be considered to reflect the close or short distances of the quantum system. A method for determining a quantum operation control layout for quantum computation on a quantum system explicitly or implicitly takes into account physical information about the mesh and the quantum system contained therein. For explicit consideration, data about the structure and / or size of the mesh can be passed as input to the method. Alternatively, the method can implicitly assume a particular structure and / or size of the mesh.

[0023] input

[0024] Many computational problems of interest, in particular NP-hard optimization problems, but also NP-complete decision problems, can be mapped onto the Ising spin model, which is itself an NP-complete decision form. Specifically, such problems can be mapped onto the problem of finding the ground state energy of the classical Hamiltonian function shown in the upper equation of Equation 1 below, or the corresponding quantum Hamiltonian operator shown in the lower equation of Equation 1 below.

number

[0025] Many of the aforementioned computational problems map more naturally, i.e., as the number of spins decreases, to a spin model that includes not only pairing interactions but also k-body interactions, where k is greater than 2. That is, the computational problem at hand can be rephrased (mapped) to the problem of finding the ground-state energy of the spin-model Hamiltonian shown in Equation 2 below.

number

[0026] The number of terms in the spin model Hamiltonian grows quadratically in N when all-to-all interactions, as well as single- and two-body interactions, are present, and grows exponentially in N when all-to-all interactions, as well as k-body interactions for all k from 1 to N, are present. However, in many computational problems, the number K of terms in the spin model Hamiltonian grows much slower with N. For a given N, the number K may in particular be smaller than: Vector h, matrix J and tensors R, T etc. may be sparse.

number

[0027] A spin model Hamiltonian with k-body interactions can be associated with a hypergraph with N nodes and k hyperedges. The hypergraph may be simple, i.e., it may not have multiple hyperedges containing the same node. A hyperedge is a non-empty set with elements from a set of N nodes. A hypergraph can be viewed as a pair (X,E), where X is a set of nodes (node ​​set) and E is a set of hyperedges (hyperedge set). In correspondence with the spin model, a hyperedge denotes a k-body interaction between spins. Here, k is the cardinality of the hyperedge, i.e., the number of elements in a non-empty subset of N nodes. A hypergraph is a graph if it is 2-uniform, i.e., if it contains only hyperedges that are edges (meaning that all hyperedges have cardinality 2). A hypergraph is connectable, i.e., starting from any node, it is possible to reach all other nodes by following the hyperedges. If the hypergraph is not connected, the method can be applied to all connected components individually. An unconnected hypergraph can be associated with a separable spin model Hamiltonian.

[0028] As an example, FIG. 3 shows a hypergraph 200 with five nodes labeled with index 1...5, the last of which is exemplarily shown in FIG. 3 with reference number 214. The nodes correspond to five spins. FIG. 3 shows five hyperedges: two hyperedges {1,2}, {4,5} (shown with black lines connecting the respective nodes) that are edges, and three hyperedges {1,2,3}, {3,4,5}, {1,2,4,5} with boxes drawn around them. For illustrative purposes, hyperedge {1,2,3} is shown with reference number 222 and hyperedge {1,2,4,5} is shown with reference number 212. The hypergraph shown in FIG. 3 can be considered to be associated with a spin model Hamiltonian shown in the following equation 4:

number

[0029] Data representing a hypergraph or hyperedges of a hypergraph can be provided as input in a method for determining a quantum computation control layout for quantum computation on a quantum system. The data represents a hypergraph if the hypergraph can be derived from the data, and the data represents a hyperedge if the hyperedge can be derived from the data. The data can be in any standard form that identifies a hyperedge set or the entire hypergraph, or in any form. A hypergraph may be derivable from a hyperedge set, especially if all nodes are included in some hyperedge, especially if the hypergraph is connected. For example, in the example of FIG. 3 above, each of the graphical depiction, the description as a set, and the description by the spin model allow the hypergraph to be derived and qualify as data representing the hypergraph. Herein, it is sufficient to provide a hyperedge set, since the hyperedges together include all nodes of the hypergraph, and the depiction / description allows the nodes in the hyperedges to be identified, and the entire hypergraph can be derived from the data representing the hyperedges of the hypergraph.

[0030] For hyperedges of hypergraphs, hypergraphs, and data representing hyperedges or hypergraphs, there are natural connections to spin models with k-body interactions, but the computational problem does not have to be encoded in the hypergraph by first mapping the computational problem to the spin model.

[0031] Generalized cycle determination

[0032] A set of generalized cycles can be determined from the hyperedges of a hypergraph. A generalized cycle can be a regular generalized cycle or an irregular generalized cycle. Irregular generalized cycles are discussed below. A regular generalized cycle means a set of hyperedges such that each node included as an element in a hyperedge of the set of hyperedges appears even times as an element in all hyperedges of the set of connected hyperedges. For example, in the example hypergraph shown in Figure 3, the set of hyperedges {{1,2,4,5},{1,2},{4,5}} forms a regular generalized cycle because nodes 1, 2, 4 and 5 appearing in one of the three hyperedges of the set appear twice in all three hyperedges. Also, the set of hyperedges {{4,5},{1,2},{3,4,5},{1,2,3}} forms a regular generalized cycle. A regular generalized cycle is a cycle of edges in the graph, i.e., the edges of the graph are 2-uniform, i.e., the edges of the graph are 1-uniform ...

[0033] The length of a regular generalized cycle is the number of hyperedges of the regular generalized cycle, i.e. the cardinality of the set of hyperedges forming the regular generalized cycle. The maximum length of a regular generalized cycle of a set of regular generalized cycles is the maximum of the lengths of these regular generalized cycles. The maximum length of a regular generalized cycle of a set of regular generalized cycles determined in this manner may not be greater than the maximum number of vertices of a cell of the mesh. The set of regular generalized cycles determined in this manner may include all regular generalized cycles of the hypergraph whose length is less than or equal to the maximum number of vertices of a cell of the mesh. Thus, the determination of the set of regular generalized cycles takes into account the properties of the mesh. The mesh itself reflects the physical properties of the quantum system on which the quantum computation is performed.

[0034] Without wishing to be bound by any particular theory, the following is offered as an illustration of the importance of regular generalized cycles to quantum computations performed in quantum systems. In terms of a spin model that can be associated with a hypergraph, a regular generalized cycle corresponds to a particular interaction between particular spins. To illustrate, first consider a 2-uniform hypergraph with node set {1,2,3} and edge set {{1,2},{2,3},{1,3}} (weights omitted). This 2-uniform hypergraph is a graph and has one regular generalized cycle of length 3, i.e., {{1,2},{2,3},{1,3}}, which is the cycle of the graph. The spin model associated with spins labeled 1, 2, and 3 has the following Hamiltonian (weights omitted):

number

[0035] To see this, consider that original spins 1 and 2 are in configuration 01 (i.e., physical qubit 1, representing the interaction between spins 1 and 2, or the corresponding hyperedge in the hypergraph, is in state |0>), then original spins 2 and 3 can only enter configurations 10 or 11, since these are the only ones the spin direction of the third qubit can still be chosen. In the first case, physical qubit 2, representing the interaction between spins 2 and 3, or the corresponding hyperedge in the hypergraph, is in state |0>. The state of physical qubit 3, representing the interaction between spins 1 and 3, has already been determined, since the spin configurations of spins 1 and 3 are fixed to 00 by a cyclic property, and the state of physical qubit 3 is |1>. In the second case, physical qubit 2 is in state |1>, and the spin configurations of spins 1 and 3 are fixed to 01 by a cyclic property, and the state of physical qubit 3 has already been determined, and the state of physical qubit 3 is |0>. In all other spin configurations, the number of physical qubits in state |0> is also even.

[0036] Next, consider a hypergraph with node set {1,2,3} and edge set {{1},{2,3},{1,2,3}} (weights omitted). The hypergraph has one regular generalized cycle of length 3: {{1},{2,3},{1,2,3}}. The spin model associated with spins labeled 1, 2, and 3 has the following Hamiltonian (weights omitted):

number

[0037] To see this, consider that original spin 1 is in spin configuration 0. Then physical qubit 1, which represents the “interaction” (field) of spin 1 or the corresponding hyperedge in the hypergraph, will be in state |0>. Original spins 2 and 3 can be in spin configurations 01, 10 or 00, 11. The first two configurations correspond to state |0> of physical qubit 2, which represents the interaction between spins 2 and 3 or the corresponding hyperedge in the hypergraph. The last two configurations correspond to state |1> of physical qubit 2. The state of physical qubit 3, which represents the interaction between spins 1, 2 and 3, has also already been determined. In the first case, the spin configurations of spins 1, 2 and 3 are fixed to 001 or 010, so physical qubit 3 is in state |1>. This is because there is an even number of spin downs / zeros in the spin configurations of the three spins. In the second case, the spin configurations of spins 1, 2 and 3 are 000 or 011, so physical qubit 3 is in state |0>. In all other spin configurations, the number of physical qubits in state |0> is also even.

[0038] That is, if physical qubits are associated with hyperedges of a hypergraph that form a regular generalized cycle, then only an even number (0, 2, etc.) of physical qubits can be in state |0>. Regular generalized cycles thus impose constraints on the states of the components of a quantum system if consistency with the spin model is to be maintained, and therefore consistency with the hypergraph encoding the computational problem is to be maintained. Regular generalized cycles whose maximum length is less than or equal to the maximum number of vertices in a cell of the mesh impose constraints that are designed to be enforced in quantum computation.

[0039] Determining Mesh Mapping

[0040] If we map the hyperedges of the hypergraph, each of which is a k-body interaction of the spin model associated with the hypergraph, onto the components of the quantum system, in particular physical qubits, and map the weights / interaction strengths onto local fields acting on the components of the quantum system, then the degrees of freedom of the quantum system are greater than the degrees of freedom of the spin model. The constraints imposed by the regular generalized cycles of the hypergraph can reduce the degrees of freedom of the quantum system by one. The consistency between the state of the quantum system and the state of the spin model associated with the hypergraph is provided by a set of regular generalized cycles, which impose constraints on the quantum system that reduce the degrees of freedom of the quantum system to those of the spin model associated with the hypergraph. The term constraining a subset of regular generalized cycles is used for the set of regular generalized cycles that provide this consistency. Here, "subset" is used because this set is a subset of all regular generalized cycles of the hypergraph, in particular a subset of the set of all regular generalized cycles whose maximum length does not exceed the maximum number of vertices of a cell of the mesh. However, this does not imply that we must construct a set of all regular generalized cycles whose maximum length does not exceed the maximum number of vertices in a cell of the mesh, nor does it imply that the constraint subset is a proper subset of it.

[0041] For example, if the degree of the hypergraph is N, i.e. the cardinality of the node set is N, the associated spin model typically has N spins / degrees of freedom, and the size of the hypergraph is K, i.e. the cardinality of the hyperedge set is K, and the Hamiltonian has K terms representing k-body interactions between spins, then quantum systems often have K degrees of freedom, since their components represent K hyperedges / k-body interactions. Consistency may then be provided by at least KN constraints imposed by the regular generalized cycles of the hypergraph. If the hypergraph has only hyperedges with even cardinality, i.e. k is even for all k-body interactions of the associated spin models, then the ground state of the Hamiltonian of the spin model is degenerate, since spin-up and spin-down can be exchanged (there is no energy gap between spin-up / spin-down). Quantum systems do not exhibit this global symmetry, so in this case another regular generalized cycle and imposed constraints are added, totalling at least K-N+1. Thus, in this example, the constraint subset includes at least KN or at least K-N+1 regular generalized cycles. The constraint subset could include more regular generalized cycles, but these additional regular generalized cycles are not independent of the first KN or K-N+1 regular generalized cycles and therefore do not further reduce the degrees of freedom. Such extra constraints can be used to lower the energy of the constraint Hamiltonian imposed on the quantum system. Thus, the energy scale separation between the energy of the constraint Hamiltonian and the energy of the local fields that form the problem Hamiltonian of the quantum system is reduced.

[0042] The method can include determining a mesh mapping that maps hyperedges of the hypergraph or data representing the hyperedges to vertices of a mesh. The vertices of the mesh to which the hyperedges are mapped are called layout vertices. The layout vertices correspond to components of a quantum system used in quantum computation. In the mesh mapping, each regular generalized cycle of the constraint subset is composed of hyperedges that are mapped to cells of the mesh. Mapping hyperedges to cells of the mesh means mapping these hyperedges to vertices of only that cell. In this way, the cells of the mesh can physically enforce the constraints imposed by the regular generalized cycles, since quantum interactions between components placed at the vertices of that cell can interact during quantum computation. Thus, the mesh mapping takes into account the physical properties of the quantum system represented by the mesh.

[0043] For example, for meshes as shown in Figures 1 and 2, each cell has four vertices, so the maximum number of vertices for a cell in a mesh is four. In the example in Figure 3, there are three regular generalized cycles of hypergraphs of length at most four, namely, {{1,2,4,5},{1,2},{4,5}, {{1,2,4,5}, {1,2,3}, {3,4,5}}, and {{2,4}, {1,5}, {1,2,3}, {3,4,5}}. Mesh mapping can map the five hyperedges in the hyperedge set {{1,2}, {4,5}, {1,2,3}, {3,4,5}, {1,2,4,5}} to five different vertices of the mesh as shown in Figure 4, where "12" is used as the label of the node to which the hyperedge {1,2} is mapped, and so on. Mesh mapping is injective. In the mesh mapping, the regular generalized cycles {{1,2,4,5},{4,5},{1,2}} and {{1,2},{4,5},{1,2,3},{3,4,5}} form a constrained subset of the set of regular generalized cycles whose maximum length does not exceed 4. Since the hypergraph has N=5 nodes and K=5 hyperedges, and there is global symmetry despite the odd-cardinality hyperedges (the expected energy remains the same whether all spin-ups are exchanged for spin-downs or vice versa), K-N+1=1 regular generalized cycles are sufficient. However, in this example, two regular generalized cycles allow all hyperedges to be mapped and included in the regular generalized cycle. As can be seen in Figure 4, the hyperedges of any regular generalized cycle from the constrained subset are mapped within the respective cells of the rectangular mesh of Figure 1 or Figure 2, and each regular generalized cycle from the constrained subset is mapped to a different cell of the mesh.

[0044] The mesh mapping is constructed by mapping the hyperedges of a first regular generalized cycle onto the vertices of a first cell of the mesh. If there exists a second regular generalized cycle that has at least one hyperedge in common with the first regular generalized cycle, the hyperedges of the second regular generalized cycle may be mapped onto the vertices of neighboring cells of the mesh. Neighboring cells of the first cell are cells that have at least one vertex in common with the first cell. At least one hyperedge that the second regular generalized cycle has in common with the first regular generalized cycle is mapped onto the corresponding at least one vertex of the mesh that is common to the first cell and its neighboring cells. This process can be repeated. The construction of the mesh mapping is completed when all the regular generalized cycles that form the constraint subset have been mapped onto the mesh. In this way, the mesh mapping is constructed such that the condition that the hyperedges of the regular generalized cycles are mapped onto vertices that belong to the same cell of the mesh is satisfied.

[0045] Quantum Computation Control Layout

[0046] The method can include generating a quantum operation control layout. The quantum operation control layout can control quantum operations of a quantum computation, for example, when loaded into a quantum processing unit (QPU) of a quantum processing system. Thus, the quantum operation control layout can be a control program for the quantum computation. The quantum operation control layout includes data indicative of layout vertices of a mesh. Each layout vertex corresponds to a hyperedge mapped according to the mesh mapping. For example, with respect to the example of FIG. 4, five vertices of the mesh labeled 1245, 45, 12, 123, and 345 are layout vertices included in the quantum operation control layout 300. Each of the five layout vertices corresponds to one of the five hyperedges {1,2,4,5}, {4,5}, {1,2}, {1,2,3}, {3,4,5} of the hypergraph of FIG. 3. The bottom left layout vertex is illustratively indicated by reference numeral 312, and the bottom right layout vertex is illustratively indicated by reference numeral 322.

[0047] In FIG. 4, the labels of the layout vertices, i.e., 12, 45, 123, 345, and 1245, provide information about the mesh mapping, i.e., about the hyperedges mapped to the layout vertices. However, the labeling may be arbitrary. If the data indicating the layout vertices of the mesh lacks information about the mesh mapping, more specifically, if there is no information in the quantum arithmetic control layout that allows identifying which hyperedges of the hypergraph have been mapped to which layout vertices, the quantum arithmetic control layout is called intransparent. The quantum computation can continue to be performed, controlled by the quantum arithmetic control layout. However, without a connection to the hypergraph, the person performing the quantum computation does not know which problem is actually being solved. When returning the result of a quantum computation to another person who knows the relationship to the hypergraph, if that person knows which computation problem has been mapped to the hypergraph, that person can calculate the ground state energy of the associated spin model or calculate the solution to the computation problem encoded in the hypergraph. If a quantum operation control layout contains information that can identify which hyperedges in the hypergraph are mapped to which layout vertices (e.g., by providing information about the mesh mapping), then the quantum operation control layout is called transparent.

[0048] The quantum computation control layout can include weights for the hyperedges, where the weight of each hyperedge is associated with the layout vertex to which the corresponding hyperedge is mapped. For example, a weighted hyperedge is the pair ({1, 2}, J 12 ), where the first element of the pair represents the hyperedge and the second element of the pair represents the weight. This weighted hyperedge is a set of layout vertices 12 and weights J 12 Pairs including (12,J 12 ) In the case of weighted hypergraphs, the edge set may contain pairs of elements such as

[0049] Additionally, the quantum arithmetic control layout includes data indicative of layout vertex sets. Each layout vertex set is comprised of layout vertices in a cell of the mesh. The layout vertices of the layout vertex set correspond to hyperedges of regular generalized cycles of the constraint subset of the regular generalized cycles. In an opaque quantum arithmetic control layout, this correspondence would not be apparent. Providing this correspondence would allow the quantum arithmetic control layout to be transparent if information about the mesh mapping could be derived therefrom. A mesh mapping that maps the hyperedges to the layout vertices of the mesh could be displayed to simultaneously map the regular generalized cycles of the constraint subset to the layout vertex set. The layout vertices and the layout vertex set form a second hypergraph whose nodes are layout vertices and whose layout vertex set are hyperedges. Because the layout vertices are part of the mesh, and specifically, are part of the cells that indicate which quantum interactions are possible and have adjacency relationships, the layout vertices provide additional information related to physical properties of the quantum system that is not conveyed by the second hypergraph itself.

[0050] In FIG. 4, the quantum computation control layout 300 includes two layout vertex sets 310 and 320, i.e., the layout vertex sets represented by the triangles and squares between corresponding layout vertices, i.e., the layout vertex sets {1245, 45, 12} and {45, 12, 123, 345}. This information is completed in the example of FIG. 4 by information about the mesh, which in this example is a rectangle. FIG. 4 is a graphical representation of a quantum computation control layout. The quantum computation control layout can be provided by a suitable data structure in a machine-readable format. The quantum computation control layout can, for example, be stored in a memory, displayed on a display device, and / or transmitted over a network and can, for example, serve as an input to a quantum computation controller for controlling the quantum computation.

[0051] quantum computing

[0052] In a further embodiment, a method for performing a quantum computation on a quantum system is provided. The quantum computation is performed on components of the quantum system, in particular physical qubits. A quantum computation control layout may be provided as described herein, for example a quantum computation control layout obtained by the method for determining a quantum computation control layout as described herein. The quantum computation control layout may be loaded, for example, into a controller of the quantum computation. The components of the quantum system are spatially arranged such that a component is present at every layout vertex of the mesh, and quantum interactions are possible between components corresponding to the layout vertices in each set of layout vertices.

[0053] Quantum computation can be controlled by a quantum arithmetic control layout. The quantum system is governed by an initial Hamiltonian H 0 For each layout vertex associated with a nonzero weight, a local field with a strength corresponding to the nonzero weight is applied to the components corresponding to that layout vertex. For each layout vertex set, quantum interactions between the components corresponding to the layout vertices of that layout vertex set are performed. The cardinality c of a layout vertex set implies c-body interactions between the components corresponding to the layout vertices of that layout vertex set. The strength of quantum interactions (two-body interactions or more) between two or more components may be greater than the strength of any of the local fields (one-body interactions), with the possible exception of local fields that impose constraints (see, for example, the special element mechanism below). The state of the quantum system may evolve to a final state during the quantum computation. The final state may be a ground state or a final Hamiltonian H f Alternatively, the final Hamiltonian H f It is possible that the ground state of and such a state have high fidelity.

[0054] Let M be a mesh, L(M) be the quantum computing control layout, and V MLet be a set of layout vertices of the quantum arithmetic control layout, and VS be a set of layout vertex sets of the quantum arithmetic control layout. The initial Hamiltonian is shown in the following formula 7.

number

[0055] Final Hamiltonian H f is the problem Hamiltonian H P and the short-distance Hamiltonian H S The problem Hamiltonian may be the sum of

number

number

[0056] The evolution of the quantum system during quantum computation can be done by adiabatic sweeps (quantum annealing) as described in EP 3113084, by antiadiabatic driving using a Hamiltonian with an additional antiadiabatic part along the way as described in PCT / EP2019 / 066916, or by gate-based highly parallelizable quantum computation as described in PCT / EP2019 / 052528, all of which are incorporated herein by reference.

[0057] The method may include measuring some or all of the components of the quantum system. The measurement of at least some of the components of the quantum system provides a readout of the final state of the quantum system. The readout may include information about the ground state energy of the spin model and, ultimately, information about the computational problem encoded in the hypergraph associated with the spin model. For computational problems of class NP, the trial solution thus generated may be checked in polynomial time by classical computation to determine whether it is a solution to the computational problem. If not, the quantum computation may be repeated until the trial solution is a solution, or a finite number of iterations may be performed and then the best trial solution may be selected as an approximate solution.

[0058] Information about the spin model can be included in the quantum system redundantly. It is sufficient to measure a read-out group of components, selected such that the corresponding mesh layout vertices are associated with hyperedges in the hypergraph via mesh mapping. This can be used to create paths with distinct endpoints across all nodes in the hypergraph. There are no repeated nodes or repeated hyperedges in the paths. Various such read-out groups can be exploited. For example, in FIG. 4, the components associated with layout vertices 12, 123, 345, and 45 form a first read-out group, because the corresponding hyperedges create paths with distinct endpoints across all nodes 1,...,5, as shown in Eq. (10).

number

number

[0059] If the quantum computing control layout is transparent, one readout group of components can be measured. Alternatively, multiple readout groups of components or all of the components can be measured, the latter meaning measuring all readout groups. The additional information can be used, for example, for consistency checks or classical error correction. In the case of an opaque quantum computing control layout, all components can be measured and a result returned, so it is not possible to know which components form a readout group.

[0060] Regular generalized cycles containing special elements and auxiliary components

[0061] Consider the regular generalized cycle {{1,2,4,5},{4,5},{1,2}} of the example of Fig. 3. It is mapped under the mesh mapping to the layout vertex set {1245,45,12}, shown as a triangle with reference numeral 310 in Fig. 4. This layout vertex set leads to three-body interactions between corresponding components of a quantum system in quantum computing. At the same time, the regular generalized cycle {{4,5},{1,2},{1,2,3},{3,4,5}} leads to the layout vertex set {45,12,123,345}, shown as a square with reference numeral 320 in Fig. 4. The layout vertex set 320 indicates four-body interactions between corresponding components of a quantum system in quantum computing.

[0062] However, there may be quantum systems in which both kinds of k-body interactions, particularly three-body interactions, are difficult or impossible to implement directly. For example, if the components are arranged in a square lattice, it is practical to have local fields and interactions between all four components of the square, but not directly between three of the four components. Thus, a three-body interaction can be realized by placing an auxiliary component, particularly an auxiliary physical qubit, in an empty corner of a square in the square lattice, fixing the state of the auxiliary component via a local field, and implementing a four-body interaction between the three components and the auxiliary component. The local field can be set stronger than the interaction strength of the four-body interaction to constrain the auxiliary component to remain in a fixed state. The effective interaction is the interaction of the three components in the other three corners of the square of the square lattice. This mechanism may be particularly useful for some analog forms of quantum computing. In digital, gate-based forms of quantum computing interactions between any subset of the components of a cell can typically be implemented by one or more gates and / or local operations, and an auxiliary physical qubit to effectively realize interactions with fewer than all components of the cell is typically not implemented.

[0063] A method for determining a quantum operation control layout for quantum computation can also take into account such physical properties of quantum systems. A regular generalized cycle can include one or more special elements, denoted by XX. The special elements XX are considered here as hyperedges. For example, the regular generalized cycle of length 3 {{1,2,4,5},{4,5},{1,2}} in the example of FIG. 3 is extended by the special element XX to form a regular generalized cycle of length 4 {{1,2,4,5},{4,5},{1,2},XX}. A regular generalized cycle can be extended by one or more occurrences of the special element XX, and its length increases by the number of occurrences of the special element XX. Formally, a regular generalized cycle can be considered as a multiset, but only the special element XX may appear repeatedly. There, the special element XX is considered not to interfere with the property that any node appearing in a hyperedge appears even times in all hyperedges of the connected regular generalized cycles (RGC property). Alternatively, a special element XX can be considered to satisfy the RGC property itself. A special element XX can be considered as a loop {X,X} added to the hypergraph of some node X (this is why the notation XX was chosen). The hyperedges of this modified hypergraph are then considered as multisets, and the hypergraph may not be simple, since many occurrences of the loop {X,X} may be required. However, no mathematical considerations are necessary for the special element XX, as there is no ambiguity about how it is treated by the method.

[0064] A regular generalized cycle containing one or more occurrences of a special element XX can have a length equal to the maximum number of vertices of a cell of the mesh, or at least equal to the number of vertices of some cells of the mesh. Each occurrence of a special element XX in a regular generalized cycle will create a corresponding number of special layout vertices xx in the quantum arithmetic control layout, meaning that all layout vertices corresponding to the elements of the regular generalized cycle and the special layout vertices belong to the same cell of the mesh. For example, a regular generalized cycle {{1,2,4,5},{4,5},{1,2},XX} is mapped to the layout vertex set {1245,45,12,xx}, shown in FIG. 5 as a square with reference sign 390. The special element XX is treated here as a hyperedge, so the mesh mapping maps it to a layout vertex xx, shown with reference sign 392. When constructing a mesh mapping by placing images of the hyperedges of a regular generalized cycle on the vertices of cells of the mesh, all special elements XX or corresponding special layout vertices xx coincide with each other. This is justified because during quantum computation, all components of a fixed quantum state corresponding to a special layout vertex xx are fixed to the same quantum state, e.g., by a large local field. The special layout vertex is identifiable even in a non-transparent quantum computation-controlled layout, e.g., by special labeling or very large weights.

[0065] Using components of a quantum system that remain in a fixed state during quantum computation, the method uses special elements XX and special layout vertices xx to indirectly allow some k-body interactions to be implemented when only some k+l-body interactions are feasible for a particular quantum system. If k-body interactions can be implemented directly, using the special element mechanism may not be preferable because it requires additional components such as physical qubits, occupying space where components that normally participate in quantum computation can be placed. Furthermore, placing additional components in a fixed state may require large local fields that may adversely affect the quantum computation (energy scale separation between normal local fields, strength of constraint Hamiltonian, and even larger local fields to fix the quantum state). The physical capabilities of the quantum system as to which interactions can be implemented directly can be explicitly considered in the method of determining the quantum operation control layout of the quantum computation, for example in the form of data provided. Based on this information, it can be selected whether to include regular generalized cycles with special elements in the determination of regular generalized cycles. Alternatively, the method can be designed to implicitly satisfy the physical functions of a particular quantum system, since regular generalized cycles may always be allowed to include special elements or may not always be allowed to include special elements.

[0066] Expanded hypergraphs and ancilla constituents

[0067] Some hypergraphs may not have a sufficient number of regular generalized cycles whose length is less than or equal to the maximum number of vertices in a cell of the mesh. For example, consider the hypergraph {{1,2},{2,3},{3,4},{4,5},{5,6},{1,6}}, denoted by reference numeral 201 in FIG. 6. The hypergraph 201 is associated with a spin model with pairwise interactions in the Hamiltonian. If six spins are arranged on a ring such that spins 1 and 6 are nearest neighbors, then all pairwise interactions in this example will be nearest neighbor interactions.

[0068] If the mesh is hexagonal, such as mesh 101 in FIG. 7, each cell contains six vertices, and the maximum number of vertices for a cell in a mesh is six. There is one regular generalized cycle of length six, i.e., the set of hyperedges {{1,2},{2,3},{3,4},{4,5},{5,6},{1,6}}. Weights associated with the hyperedges are omitted here. Since there are N=6 nodes and K=6 hyperedges, and the cardinality of all hyperedges is even (i.e., 2), K-N+1=1 constraints are required, and one constraint is imposed by this regular generalized cycle, forming a constraint subset. Since there is only this one regular generalized cycle in the hypergraph, the constraint subset is now equal to the set of all regular generalized cycles in the hypergraph 201. 7 shows a quantum operation control layout 301 having layout vertices 12, 23, 34, 45, 56, and 16, and a layout vertex set {12, 23, 34, 45, 56, 16}, in which the bottom layout vertex is exemplarily marked with reference numeral 332 and the layout vertex set is denoted with reference numeral 330.

[0069] If the mesh is quadrilateral, each cell contains four vertices, and the maximum number of vertices for a cell in a mesh is 4. The hypergraph does not contain any regular generalized cycles of length less than or equal to 4. Nevertheless, one constraint imposed by regular generalized cycles is necessary to provide consistency between the spin model associated with the hypergraph and the quantum system whose physical properties (interaction functions) are reflected by the mesh.

[0070] To generate a quantum computationally controlled layout for the example mesh of squares, the hypergraph can be expanded by adding additional hyperedges between the nodes of the hypergraph, where the additional hyperedges are associated with a weight of zero. The hypergraph is transformed into an expanded hypergraph, where the order of the hypergraph remains the same but the size increases. Regular generalized cycles are determined in the expanded hypergraph whose required length is at most four, i.e., no more than the maximum number of vertices in a cell of the mesh.

[0071] Under mesh mapping, the additional hyperedges are treated like any other hyperedge and are mapped to layout vertices. The layout vertices to which the additional hyperedges are mapped are associated with weight zero. In quantum computing, the components of the quantum system that correspond to such layout vertices are called associated components, specifically associated qubits, and the local fields do not act on them or are formulated in a different way such that a local field of strength zero corresponds to weight zero. The constraint Hamiltonian acts on the components that correspond to the layout vertices in the layout vertex set. This is regardless of whether these layout vertices come from additional hyperedges of the expanded hypergraph that are not included in the hypergraph by the mesh mapping, or from hyperedges of the hypergraph.

[0072] Adding an additional hyperedge increases the size K of the hypergraph to size K+1 of the enlarged hypergraph. If there are not enough regular generalized cycles of the required length in the hypergraph to impose a constraint, adding one additional hyperedge, thereby generating one regular generalized cycle of the required length that imposes one constraint, would provide no benefit in providing consistency between the spin model and the quantum system. This is because the enlarged hypergraph of size K+1 requires one more regular generalized cycle than the hypergraph of size K, so there is still an insufficient number of regular generalized cycles of the required length. Adding c additional hyperedges to provide an enlarged hypergraph with at least c+1 additional regular generalized cycles of the required length could move toward providing consistency between the spin model and the quantum system.

[0073] For example, the hypergraph of Figure 6 may be expanded by two additional hyperedges {1,4} and {3,6}, denoted by reference numerals 247 and 245, respectively, to form the expanded hypergraph {{1,2},{2,3},{3,4},{4,5},{5,6},{1,6},{1,4},{3,6}}, denoted by reference numeral 202 in Figure 8. Three regular generalized cycles of length 4 or less can be determined: {{1,2},{2,3},{3,4},{1,4}}, {{3,4},{4,5},{5,6},{3,6}}, and {{1,4},{3,4},{3,6},{1,6}}, which form the constraint subset. Here, (K+2)-N+1=(6+2)-6+1=3 regular generalized cycles and a corresponding number of imposed constraints are required and distributed. The quantum arithmetic control layout 302 has layout vertices {12, 23, 34, 45, 56, 16, 14, 16}, of which the layout vertex with label 45 corresponding to the hyperedge 232 is exemplarily indicated with reference numeral 332. The quantum arithmetic control layout includes vertex sets {12, 23, 34, 14}, {34, 45, 56, 36} and {14, 34, 36, 16}, which are shown as squares in FIG. 9. The layout vertices 14 and 36, which correspond to the added hyperedges 247 and 245, are indicated with reference numerals 347 and 345. When the quantum arithmetic control layout drives a quantum computation in a quantum system, the associated components of the quantum system corresponding to the layout vertices 14 and 36 are not acted upon by the local field. Comparing Figures 7 and 9, although the quantum computation control layouts 301 and 302 in these two figures are different, they are derived from the same hypergraph for different meshes, i.e. the hexagonal mesh 101 shown in Figure 7 and, for example, the square mesh 100 shown in Figure 1. Since the shape of the mesh reflects the physical properties of the quantum system, this example shows once again that the method takes into account, and therefore the output depends on, the physical properties of the quantum system reflected by the shape of the mesh.

[0074] Further, expanding the hypergraph with one or more additional hyperedges to form an expanded hypergraph helps to form a mesh mapping that is consistent with the requirement that for every layout vertex set that corresponds to a regular generalized cycle, the layout vertices of each layout vertex set are contained within one cell of the mesh. For example, FIG. 10 shows a 2-uniform hypergraph, and FIG. 11 shows a quantum arithmetic-controlled layout that includes layout vertex 46. Layout vertex 46 is a vertex of the mesh to which the mesh mapping maps an additional hyperedge {4,6} of the expanded hypergraph. There are 17 layout vertices corresponding to the 17 hyperedges of the expanded hypergraph, and there are 8 nodes in the hypergraph and the expanded hypergraph. The layout vertex set of the quantum arithmetic-controlled layout provides 10 constraints that are sufficient for K-N+1=17-8+1=10.

[0075] Side conditions and irregular generalized cycles

[0076] In a further example, a 2-uniform hypergraph 203 is provided as shown in Fig. 12, and the mesh considered in the method is a quadrilateral mesh. In the hypergraph shown in Fig. 12, there are five regular generalized cycles of length less than or equal to 4, where 4 is the maximum number of vertices in a cell of a quadrilateral mesh. The five regular generalized cycles of length less than or equal to 4 are {{1,2}, {2,6}, {1,6}}, {{3,4}, {4,5}, {3,5}}, {{1,4}, {4,5}, {5,6}, {1,6}}, {{1,4}, {3,4}, {2,3}, {1,2}} and {{2,3}, {3,5}, {5,6}, {2,6}}, which are shown in Fig. 13. The right-most regular generalized cycle shown in Fig. 13 is exemplarily given the reference sign 240. Since K-N+1=9-6+1=4, four of these five regular generalized cycles are sufficient to form the constraint subset. Figure 14 shows a quantum computation control layout with layout vertices {12,23,34,45,56,16,26,35,14} corresponding to the hyperedges under the mesh mapping, layout vertex sets {23,12,14,34} and {13,16,56,45} shown as squares, and layout vertex sets {34,45,35} and {12,26,16} shown as triangles. Here, the layout vertex sets correspond to the four constraint subsets of the five regular generalized cycles. Layout vertex set 340 corresponds to regular generalized cycle 240 of Figure 13.

[0077] The spin model associated with a hypergraph may include one or more side conditions. For example, the Hamiltonian of the spin model, shown in Eq. (12), for some i, j, k, l, σ z (i) =1 or -1, σ z (i) σ z (j) =1 or -1, σ z (i) σ z (j) σ z (k) =1 or -1, σz (i) σ z (j) σ z (k) σ z (l) It may be subject to side conditions of the form =1 or -1.

number

[0078] In the quantum system in which quantum computation is performed, each component corresponds to a layout vertex of the mesh, which in turn corresponds to a hyperedge of the hypergraph. A hyperedge with a side condition (+1 or -1) means that the corresponding component of the quantum system does not need to participate in the quantum computation, and can absorb and remove its influence. z (a) σ z (b) σ z (c) The first effect of =±1 is an energy contribution of +1 or -1 times the weight of the hyperedge to the ground state energy of the final Hamiltonian. This is the sum of the σ z (a) σ z (b) σ z (c) This is because ... corresponds to the local operator of the problem Hamiltonian shown in the following equation (13) and arises from the side condition shown in the following equation (14).

number

number

[0079] The second effect concerns regular generalized cycles that contain one or more hyperedges with side conditions. For example, in the side-conditioned hypergraph of Figure 15, the regular generalized cycle {{2,3},{1,2},{1,4},{3,4}} has a final Hamiltonian H f This is finally related to the constraint Hamiltonian shown in Equation 15 below.

number

number

number

number

[0080] Thus, the regular generalized cycle {{2,3},{1,2},{1,4},{3,4}} of length 4 can be reduced to an irregular generalized cycle {{1,2},{1,4},{3,4}} of length 3 associated with the pair ({{2,3}},-1). The first element of the pair is a set containing the hyperedges with side conditions that complete the irregular generalized cycle to a regular generalized cycle. In this example, this is the hyperedge {2,3} with side condition "-1". Every node that appears in one hyperedge of the irregular generalized cycle and that appears an odd number of times in all hyperedges of the irregular generalized cycle is balanced by a constrained, restrained hyperedge. The second element of the pair is the product of the side conditions of the hyperedges of the set that forms the first element of the pair. The second element is therefore either +1 or -1, in this example -1. In general, an irregular generalized cycle of length r is a set {h 1 ,···,h r} and {h 1 ,···,h r}∪{h c1 ,···,h cS} is a regular generalized cycle of length r+s, and c 1 ,···,c S is the hyperedge {h c1 ,···,h cS}, which is the irregular generalized cycle property (IGC property).

number

[0081] In the mesh mapping, only hyperedges that are not affected by side conditions are mapped to layout vertices. The layout vertex set of layout vertices that correspond to hyperedges without side conditions of an irregular generalized cycle is associated with a value of +1 or -1 depending on the value of the following equation (20) even in the opaque quantum arithmetic control layout:

number

[0082] For example, continuing with the example hypergraph of FIG. 15, the irregular generalized cycle {{1,2},{1,4},{3,4}} is shown in FIG. 16 at 250. The corresponding quantum computation control layout, including the corresponding layout vertex set 350, is shown in FIG. 17. In FIG. 17, "23|-1" is used as a shorthand notation for the pair ({{2,3}},-1) associated with the irregular generalized cycle. In quantum computation, the constraints imposed by the layout vertex set are enforced by a constraint Hamiltonian. The constraint Hamiltonian acts with strength C on the layout vertex set shown by the upper right triangle, the lower right square, and the lower left triangle, and with strength -C on the upper left triangle.

[0083] The length of an irregular generalized cycle is r even if the length of a regular generalized cycle completed by s hyperedges with side conditions is r+s. When a set of generalized cycles is determined in this manner, the generalized cycles may be regular and / or irregular generalized cycles. The maximum number of vertices v in a cell of a mesh is max Determining a generalized cycle of length not exceeding v contains s hyperedges with side conditions. max+s The method may include determining an irregular generalized cycle of

[0084] In the example hypergraph of Fig. 14, there are two regular generalized cycles of length 5, each containing one hyperedge {2,3} with constraint "-1". These two regular generalized cycles are shown in Fig. 18, the one shown in the lower right corner is exemplarily provided with reference number 260. Thus, there are two irregular generalized cycles of length 4, which are the same for the v of the quadrilateral mesh. max = 4, i.e. {{3,5},{5,6},{1,6},{1,2}} and {{3,4},{4,5},{5,6},{2,6}}, both associated with pair ({{2,3}},-1). Since the hyperedge {2,3} with side condition "-1" is not mapped to a layout vertex and is not represented by a component of the quantum system during quantum computation, there are K-N+1 = 8-6 + 1 = 3 generalized cycles required to impose the three constraints. Thus, the two irregular generalized cycles, together with the regular generalized cycle of length 4 {{1,4},{4,5},{5,6},{1,6}}, form the constraint subset. FIG. 19 illustrates a quantum computation control layout with corresponding layout vertices and layout vertex sets. Layout vertex set 360 is the layout vertex set corresponding to irregular generalized cycle 260 of FIG. 18. In quantum computing, the constraints imposed by the layout vertex set are applied by a constraint Hamiltonian that operates with the strength −C of the layout vertex set shown in the upper left and lower right squares, and the strength C of the lower left square.

[0085] The quantum operation control layouts shown in Figures 17 and 19 provide different quantum computation control, but the readout of the final state of the quantum system may contain the solution to the same computation problem encoded in the hypergraph shown in Figure 15. The quantum operation control layout of Figure 19 may be selected, for example, when three-body interactions between components of a quantum system arranged in cells of a square mesh are difficult to implement in the particular quantum system in which the quantum computation is performed, but four-body interactions are easier to implement. The method of determining the quantum operation control layout may take into account such characteristics of the quantum system. For example, max , i.e., by giving preference to generalized cycles (both regular and irregular) whose length is equal to the maximum number of vertices in a cell of the mesh, and then by using the special element mechanism described here to reduce their length to v max We need to consider generalized cycles of shorter length by increasing the length to v. Finally, the length of all generalized cycles so determined is v max become.

[0086] The mechanism of using special elements and auxiliary components and / or the mechanism of using additional hyperedges (enlarged hypergraphs) and auxiliary components can also be used in connection with irregular generalized cycles. An irregular generalized cycle may include one or more additional hyperedges and / or may include one or more special elements. As described above, the method is described for regular generalized cycles, but the method can include irregular generalized cycles as well. Thus, whenever both types can be used, the term "generalized cycle" including both types can replace the term "regular generalized cycle" in the above disclosure.

[0087] Determination of quantum operation control layout and system for quantum computing

[0088] Exemplary systems and apparatus for determining a quantum operation control layout for quantum computing, and / or for performing quantum computing on a quantum system, and / or for solving (classical) computational problems using quantum computing are described with respect to Figures 20 and 21.

[0089] FIG. 20 illustrates a system 400 for solving a computational problem 412 using quantum computation performed by a quantum computing system 500. In the embodiment illustrated in FIG. 20, the computational problem 412 is stored in a first classical computing system 410. A classical computing system may refer to a computing system that operates on bits or other classical units of information. A classical computing system may include a central processing unit (CPU) for processing information represented by bits and / or a memory for storing information represented by bits. A classical computing system may include one or more conventional computers, such as personal computers (PCs), and / or a network of conventional computers. The first classical computing system 410 transmits (401) the computational problem 412 to a second classical computing system 420. It should be understood that transmitting, receiving, encoding, decoding, storing, loading, and other conventional tasks are performed on or with data representing the computational problem, hypergraph, quantum arithmetic control layout, etc., or on or with data from which these entities are derived. For simplicity, the description omits reference to such data and speaks of "submitting a computation problem," "encoding a hypergraph," "storing quantum computation control layouts," and so on.

[0090] The second classical computing system 420 encodes the computation problem 412 into a hypergraph (422). The hypergraph is associated with a corresponding spin model so that a solution to find the ground state energy of the spin model can be translated back into a solution of the computation problem. In FIG. 20, the hypergraph 203 is illustratively shown as a hypergraph generated by the second classical computing system 420. The second computing system 420 transmits the hypergraph to a system 650 (402) to determine a quantum operation control layout for a quantum computation in a quantum system. The system 650 may be a third classical computing system, such as a conventional computer or a network of conventional computers, a computer cluster or a network of computer clusters, or a cloud computing environment. The system 650 may be configured to perform a method of determining a quantum operation control layout for a quantum computation described herein. In FIG. 20, the system 650 performs a computer-implemented method 600, and in the depicted example, the system 650 is used to determine a quantum operation control layout 303 from the hypergraph 203.

[0091] The computer-implemented method 600 is shown generally in FIG. 22 and includes providing 610 hyperedges of a hypergraph, such as hypergraph 203. This may include receiving the hypergraph via a network component and / or loading the hypergraph from a memory. The method 600 includes determining 620, by a processor of the system 650, a set of generalized cycles from the hypergraph or the expanded hypergraph, while taking into account the characteristics of the mesh, in particular the maximum number of vertices of a cell of the mesh, as described herein. The method 600 includes determining 630, by a processor of the system 650, a mesh mapping that maps hyperedges of the hypergraph (or the expanded hypergraph) to vertices of the mesh, such that each generalized cycle of a constrained subset of the set of generalized cycles consists of hyperedges that are mapped to cells of the mesh, as described herein. The determination of the generalized cycles 620 may precede the determination of the mesh mapping 630, although the determination of the generalized cycles 620 and the determination of the mesh mapping 630 may be intertwined and are therefore shown next to each other in FIG. 22. Method 600 includes generating 640 a quantum operation control layout, such as quantum operation control layout 303, where the generating is performed by a processor of system 650. The quantum operation control layout includes layout vertices of a mesh and layout vertex sets, where each layout vertex set is made up of layout vertices in cells of the mesh that correspond to generalized cycles of a constrained subset of the generalized cycles determined by determining 620. As noted above, a "quantum operation control layout including layout vertices" or the like should be understood as a quantum operation control layout including data representing the layout vertices or the like, although for simplicity references to the data and suitable data structures are omitted.

[0092] The quantum operation control layout determined by the system 650 can be stored in a memory of the system 650. The quantum operation control layout, such as the quantum operation control layout 303 shown in FIG. 20, is transmitted (403) to the second classical computing system 420. The second classical computing system 420 transmits (404) the quantum operation control layout to the quantum computing system 500 in a transparent or opaque form. Thus, in FIG. 20, the quantum computing system is shown receiving the quantum operation control layout 303-1 at the input section 510. Here, the quantum operation control layout 303-1 can be the quantum operation control layout 303 or an opaque form thereof.

[0093] Quantum operation control layout 303-1 can control quantum computation on quantum computing system 500. Quantum processing unit (QPU) 520 loads quantum operation control layout 303-1 from input section 510 (501) and controls local operations on the physical qubits of quantum system 530 and interactions between the physical qubits specified by quantum operation control layout 303-1 (502). In FIG. 20, the qubits are arranged in a lattice, here a square lattice, with qubits participating in the quantum computation shown as black dots, such as qubits 532 and 534. Other parts of the lattice are shown as circles, such as point 536. These other parts may be empty or occupied by qubits that do not participate in the quantum computation. When QPU 520 evolves quantum system 530 from an initial state to a final state under the control of quantum operation control layout 303-1, quantum system 530 or some of its qubits are measured (503) by measurement unit 540. Such measurements are also referred to as readouts.

[0094] The quantum computing system 500 may perform a method 700 of performing a quantum computation on a quantum system 530, as shown diagrammatically in FIG. 23. The quantum computation is performed on the qubits of the quantum system 530. The method 700 includes providing 710 a quantum computation control layout, which may include loading the quantum computation control layout for execution of control instructions contained therein by the QPU 520. The method 700 includes providing 720 the qubits of the quantum system in a spatial arrangement, e.g., a two-dimensional lattice, such that there is a qubit for every layout vertex of the mesh, and for each set of layout vertices, quantum interactions are possible between components corresponding to the layout vertices of that set of layout vertices. Providing 720 the qubits in the spatial arrangement may include or consist of addressing appropriate qubits of a set of qubits fixedly arranged at spatial locations, e.g., in a two-dimensional lattice. The method 700 includes applying 730 a local field (local computation) to the qubits corresponding to that layout vertex, for each layout vertex associated with a non-zero weight. Method 700 includes, for each layout vertex set, performing 740 quantum interactions between qubits corresponding to layout vertices of the layout vertex set. Applying 730 the local field and performing 740 the quantum interactions may be performed by QPU 520 in a particular manner and order depending on the type of driving the quantum system from an initial state to a final state (adiabatic driving, antiadiabatic driving, gate-based quantum interaction, etc.). Method 700 includes measuring 750 some or all of the components of the quantum system using measurement unit 540. The result of measurement 750 is the result of a quantum computation.

[0095] The measurement result of the quantum computation is sent to the second classical computation system 420 (405). The second classical computation system 420 includes a verification unit 424 that receives the measurement result and checks whether the measurement result contains a solution to the problem of finding the ground state energy of the spin model associated with the hypergraph 203 (spin model problem) (406). If yes, the verification unit 424 calculates a solution to the computation problem 412 from the solution of the spin model problem and sends the solution of the computation problem 412 to the first classical computation system 410 (408). The first classical computation system receives the solution to the computation problem. The solution to the computation problem is shown in FIG. 20 with reference numeral 414. If the measurement result does not contain a solution to the spin model problem, the second classical computation system 420 instructs the quantum processing system 500 to repeat the quantum computation (407). The quantum computation may be repeated until a solution to the spin model problem, and ultimately to the computational problem 412, is found, or may be repeated a predetermined finite number of times, and if no solution is found in the predetermined finite number of iterations, the best approximate solution is sent to the first classical computation system.

[0096] In the embodiment shown in FIG. 20, including the embodiment of the system 400 for determining a solution to a computational problem, the embodiment of the system 650 for determining a quantum operation control layout for a quantum computation on a quantum system, and the embodiment of the quantum computing system 500 for performing a quantum computation on a quantum system under the control of a quantum operation control layout, different services can be identified. For example, from the perspective of the first classical computing system 410, the second classical computing system 420 provides a service of returning a solution 414 to a computational problem 412. The second classical computing system 420 can be seen as providing an interface through which the solution of the computational problem 412 can be sought, the (classical) information flow across the interface being indicated by references 401, 408. From the perspective of the second classical computing system 420, the system 650 provides a service of returning a quantum operation control layout for a quantum computation on a quantum system, given a hypergraph encoding the computational problem and a mesh containing information about the quantum. Also, the quantum computing system 500 provides a service of returning the results (measurements / readouts) of the quantum computation on the quantum system, given a quantum operation control layout for controlling the quantum computation, to the second classical computing system 420. The second classical computing system 420 can query the quantum computing system 500 to return a description of the mesh, and can pass that description of the mesh along with the description of the hypergraph to the system 650, unless a particular mesh is assumed by default by the system 650. The system 650 can be seen to provide an interface by which the quantum computation control layout can be queried, and the (classical) information flow across the interface can include information indicated by references 402 and 403 (hypergraph and quantum computation control layout). It can also include information about the mesh passed to the system 650. The quantum computing system 500 can be seen to provide an interface by which the results of the quantum computation can be queried, and the (classical) information flow across the interface can include information indicated by references 404, 405, and 407 (quantum computation control layout, measurement results, iteration commands).It may also include information about the mesh that is passed by quantum computing system 500 to classical computing system 420 upon request.

[0097] An entity requesting a service does not need to have knowledge of how the service is performed. For example, details of how the quantum computation is performed do not need to be known to the second classical computing system 420. Conversely, an entity performing a service does not need to know more information than is passed to its interface. For example, the quantum computing system 500 does not need to know the computation problem 412 or the hypergraph 203, only the (possibly opaque) quantum computation control layout 303-1 that drives the quantum computation. The service may be a remote service provided by the best available infrastructure in different parts of the world. The first classical computing system 410, the second classical computing system 420, the system for determining quantum computation control layout for quantum computation on a quantum system 650, and the quantum computing system 500 may be separate systems connected by a network for the exchange of classical information between them. The separate systems may be under the control of different parties. Alternatively, any subset of these systems, including a subset of all the systems, may be integrated into a common system. The exchange of information through the interfaces of the systems integrated into the common system then becomes an internal process within the common system. The common system may be under the control of one party, while the remaining systems, if any, external to the common system may be under the control of another party or parties.

[0098] FIG. 21 shows a system 450 for solving a computation problem 412 using quantum computation performed by a quantum computing system 550. The quantum computing system 550 is a common system that integrates the functionality of the first and second classical computing systems 410, 420 and the functionality of the quantum computing system 500 of FIG. 20. The information flow indicated by reference numbers 405 to 408 is internal within the quantum computing system 550. The QPU 520 is shown diagrammatically to use a different type of quantum computation on the quantum system 530, whose qubits may have different physical properties and addressability compared to those of FIG. 20. For example, there is a qubit at location 536 of the lattice participating in the quantum computation, while location 534 is empty or at least unused. In the example shown in FIG. 21, external services are still required from the system 650. The system 650 is shown to directly return the quantum operation control layout 303 determined from the hypergraph 203 to the input unit 510.

[0099] The methods and systems described herein provide several advantages. For example, the methods and systems can handle any k-body interactions in a spin model with N spins, with k ranging from 1 to N, corresponding to a hypergraph where the cardinality of the hyperedges can assume values ​​of k from 1 to N. Many computational problems are mapped to such spin models with k-body interactions with less overhead, particularly with a reduced number of spins and interactions between spins, as opposed to spin models that only allow two-body or possibly three-body interactions. Since quantum computing ultimately represents the k-body interactions / hyperedges using building blocks such as physical qubits, smaller quantum systems with fewer building blocks may be required to solve the computational problem. Furthermore, while quantum computation control layouts for controlling quantum computations can be determined for a two-dimensional mesh, the spin model includes k-body interactions, and the hypergraph can include hyperedges of cardinality k for any high k from 1 to N. This means that quantum computations can be performed for building blocks of quantum systems arranged in two spatial dimensions, e.g., a plane, as opposed to schemes that require the spatial dimension to be k. Since our world has only three spatial dimensions, this is infeasible for k greater than two, and impossible for k greater than three. Moreover, the number of physical qubits required for a quantum computation scales simply with K, the number of terms in the Hamiltonian of the hypergraph order or associated spin model, since this translates into the number of layout vertices under the mesh mapping. 2In contrast to inflexible schemes that scale on the order of magnitude or more, this allows for a significant reduction in the number of physical qubits required for quantum computation. Furthermore, the present methods and systems take into account the physical properties of the quantum system, in particular those expressed by the shape and possibly the size of the mesh (e.g., square mesh vs. hexagonal mesh), and thus allow for optimizing quantum operation control layouts for quantum computations on different quantum systems. Thus, for a given size of quantum system on which quantum computation is performed, the methods and systems described herein can make the solution of larger / more complex classical computation problems computable. Conversely, for a given size / complexity of a classical computation problem, the methods and systems described herein may require a smaller size of the quantum system used for quantum computation.

[0100] It should be understood that the numbers of nodes, hyperedges, layout vertices, layout vertex sets, and quantum system components shown in the figures and described herein are used for purposes of illustration and explanation, and that the actual numbers may deviate therefrom, and in particular may be much greater.

[0101] According to a further embodiment, there is provided a method for determining a quantum operation control layout for quantum computing on a quantum system, the method may be performed on a system for determining a quantum operation layout, the system for determining a quantum operation layout may be a classical computing system and may include a processor and a memory.

[0102] A quantum system has components. The components of a quantum system may be physical qubits, i.e., physical two-level quantum objects. The components of a quantum system may be physical qudits, i.e., physical d-level quantum objects, of which only two levels may be used. The components may be or include the following: These include superconducting qubits, e.g. transmon or flux qubits; trapped ions, e.g. alkaline earth or alkaline earth positively charged ions such as Ca40+; ultracold atoms, e.g. ultracold neutral alkali atoms, which can be trapped in optical lattices or large spacing lattices from laser fields; quantum dots, e.g. fabricated from GaAs / AlGaAs heterostructures, where the state of the physical qubit is encoded in the spin state; impurities in solid crystals, such as NV centers, which are point defects in diamond crystals, or other impurities, such as color centers related to chromium impurities, rare earth ions in solid crystals, or defect centers in silicon carbide; optical qubits, where the polarization of the photon can form two quantum levels of the qubit (e.g. vertical polarization representing spin up and horizontal polarization representing spin down). In photonic systems, the interaction between photons is performed via a polarizing beam splitter, and depending on the measurement result of the auxiliary entangled photon, additional auxiliary entangled photons and classical post-selection can be performed.

[0103] A quantum system may have Q components. The number Q of components of a quantum system may be 9, 16, 25, 50, 100, 1000, or 10000 or more. The number Q of components may be less than or equal to 20000, 10000, 1000, 500, 200, or 100. The number Q of components may be within a range given a lower and upper limit by any of the lower and upper limits listed above, so long as a range is formed.

[0104] The components of a quantum system can be located on a two-dimensional surface (manifold). The two-dimensional surface can be planar or can include curvature. The components can be located on a two-dimensional lattice, such as a square, rectangular, triangular or hexagonal lattice. The components of a quantum system can be located in three-dimensional space (on a three-dimensional surface). The components can be located on a three-dimensional lattice, such as a cubic, rectangular or other honeycomb lattice. The surface or lattice need not be physical, but merely describes the spatial location of the components.

[0105] Quantum computations are performed on components of a quantum system arranged according to a mesh. A mesh has vertices and cells. A cell contains vertices. A vertex may be an element of a different cell. A cell may be represented as a set of vertices, but may have additional structure. A mesh includes or is a pair of a vertex set and a cell set, where the elements of the vertex set are the vertices of the mesh, and the elements of the cell set are the cells of the mesh. A mesh may have at least the structure of a hypergraph. Two cells may be adjacent if they have at least one vertex in common. The size of a mesh is the number of cells of the mesh. The vertices of the mesh represent possible locations of components of a quantum system. A quantum system is arranged according to a mesh when components are arranged such that their spatial positions correspond to the vertices of the mesh. Each cell of the mesh indicates that quantum interactions are possible between components of a quantum system arranged in that cell during quantum computation. Components are arranged in a cell when their spatial positions correspond to the vertices of that cell. A mesh does not have to be physical, but can simply describe properties of a quantum system, such as which components can interact with which other components during quantum computation.

[0106] A mesh may have additional structure. The cells of a mesh may have boundaries. The boundary of a cell may be a non-empty subset of the cell, i.e. a subset of the set of vertices that form the cell, but may have additional structure, such as the structure of a tuple of a set of tuples. Vertices that are elements of the boundary are said to be on the boundary of the cell. Vertices of a cell that are not included in the boundary of the cell are said to be in the interior of the cell. The boundary of a cell may be a proper subset of the cell, i.e. there need not be any vertices in the interior of the cell. A mesh may not contain any vertices that are in the interior of any cell of the mesh. If a cell has a boundary, two cells may be adjacent if they have at least one vertex in common that is an element of the boundaries of the two cells.

[0107] The mesh can be embedded in an m-dimensional manifold. m can be specifically 2 or 3, e.g. 2. The 2-dimensional manifold may be specifically a plane. An embedding of a mesh is a mapping of the vertices of the mesh to the points of an m-dimensional manifold. A mesh is embeddable if there is at least one such embedding. One limiting factor for such an embedding to exist is that the mesh describes properties of the quantum system, specifically properties that components located at the vertices of the cells must be able to interact during the quantum computation. Since a mesh is a description of a quantum system whose components are located in a 2-dimensional manifold, such as a plane, or in 3-dimensional space, an embedding usually does exist. The embedding can be known in the method that determines the quantum computation control layout, or by the system that performs the method, but is not required. The embedding may be known only to the system that performs the quantum computation, or to the party that owns the system. The mesh may be 2-dimensional. The cells of the mesh can be embedded in 2 dimensions, i.e. in a 2-dimensional manifold, such as a plane. The mesh may be 3-dimensional. Embeddable meshes are called spatial meshes. An embedded mesh, i.e., a mesh that has a particular embedding in an m-dimensional manifold, is a geometric mesh (allowing for scaling of the manifold and therefore the geometric mesh). Figures 1 and 2 show a spatial mesh and two possible embeddings that turn it into two different geometric meshes.

[0108] For a 2D mesh, the boundary of a cell may be a tuple of vertices. The tuples forming the boundary of a cell may be shown to be geodesic segments (such as line segments in a plane) between images of adjacent elements (vertices) of the tuple under a potential embedding of the mesh into a 2D manifold. The first and last elements of the tuple are considered to be adjacent and represent the image of the boundary of that cell in the 2D manifold, enclosing the area of ​​the cell. For a 3D mesh, the boundary of a cell may be a set of tuples of vertices. The set of tuples forming the boundary of a cell may be shown to be line segments between images of adjacent elements (vertices) of any tuple under a potential embedding of the mesh into a 3D Euclidean space. The first and last elements of the tuple are considered to be adjacent and represent the faces of that cell, and all the faces together represent the image of the boundary of the cell in 3D space, enclosing the volume of the cell. Depending on the properties of the quantum system, the embedding may possess or be required to possess some or all of the following properties for some or all of the cells of the mesh: The properties are: the images of a cell's boundary, i.e., the geodesic lines / line segments connecting the images of the vertices of a cell in a manifold, must not self-intersect; the images of the vertices of a cell that are in the interior of the cell (if they exist) must be in the area of ​​the cell in a 2D manifold or in the volume of the cell in 3D space; the images of the boundaries of different cells must not intersect, i.e., the geodesic lines / line segments connecting the images of the vertices of different cells must not intersect in the manifold (but they may coincide); the area or volume of a cell defined by the images of its boundary must be convex; the images of the boundaries of a mesh's cells form a cover of the 2D manifold or of the 3D space (or of a finite part of it); the images of the boundaries of a mesh's cells form a tessellation of the 2D manifold or of the 3D space (or of a finite part of it).

[0109] A mesh may be regular, e.g., quadrilateral or hexagonal or cuboid. In a regular mesh, each cell contains the same number of vertices. The number of vertices of a cell is called the vertex count of the cell. All cells of a mesh may have the same number of vertices. All cells of a mesh may have 3, 4, 6 or 8 vertices. The maximum number of vertices of a cell of a mesh is the maximum number of vertex counts of cells of a mesh. The maximum number of vertices of a cell of a mesh may be 3, 4, 6 or 8. A mesh may be of finite or unlimited size. A mesh may have M vertices, such that M≦Q.

[0110] The method can include providing information about the quantum system. The information about the quantum system can be explicitly passed as an input to the method or can be implicitly assumed by the method. The information about the quantum system can be any information element, including information about a mesh that itself contains information about the quantum system, or any information element described herein in connection with the quantum system. The method can include providing information about quantum computation to be performed on the quantum system, such as information about a type of quantum computation (analog quantum computation or digital, gate-based quantum computation), a subtype of quantum computation (e.g., adiabatic or inverse adiabatic quantum computation as subtypes of analog quantum computation), or other information such as a set of gates available for digital, gate-based quantum computation, etc.

[0111] Specifically, mesh data may be provided, which may represent a mesh or may represent at least one information element regarding a mesh. Mesh data represents a mesh if information about all vertices and cells and their neighboring relationships can be obtained from the mesh data. The information element may be any of the following: a dimension of the (spatial) mesh or its cells; the number of vertices; the number of cells; the number of vertices in a cell; an arrangement of the cells of the spatial mesh or a geometric mesh; a particular regularity (quad, hexagon, etc.); a maximum number of vertices of a cell of the mesh. The method may include assuming default values ​​for information elements that are not represented by the mesh data of the dataset, i.e. cannot be derived from the mesh data. For example, if only a maximum number of vertices of a cell of the mesh with a value of 4 was represented by the mesh data, the method may assume a two-dimensional quadrilateral mesh of practically unlimited size.

[0112] The method may include providing a mesh as input, whether the mesh is derived from mesh data, from mesh data and at least partially from default values, or only from default values, or may alternatively implicitly assume a particular form of mesh, such as a two-dimensional quadrilateral or hexagonal mesh of substantially unlimited size.

[0113] The method includes providing a dataset including data representing hyperedges of a hypergraph. The mesh data may be provided as part of the dataset or separately. Providing the dataset may include receiving the dataset over a network from a remote classical computing system and / or loading the dataset from a memory. The dataset, or at least the data representing the hyperedges, may be provided to a processor. The method may include deriving a hypergraph from the data representing the hyperedges. The dataset may include data representing nodes of the hypergraph. The method may include deriving a hypergraph from the dataset or from the data representing hyperedges of the hypergraph, respectively. The data representing the hyperedges, nodes, hypergraph, etc. allow the hyperedges, nodes, hypergraph, etc. to be derived from the data. The method may include converting at least one of the dataset, the data representing the hyperedges, the data representing the nodes, and the hypergraph into a standard data format. The derivation and / or conversion may be performed by a processor.

[0114] The number of nodes in a hypergraph, i.e., the degree of the hypergraph, may be N. The number of nodes in a hypergraph, N, may be 5, 10, 20, 40, 50, 100, or 200 or more. The number of hyperedges in a hypergraph, i.e., the size of the hypergraph or the cardinality of the hyperedge set, may be K. Hyperedges with side conditions are not counted and their effect is absorbed into an irregular generalized cycle. The number of hyperedges in a hypergraph, K, may be 9, 16, 25, 50, 100, 1000, or 10000 or more. The number of hyperedges in a hypergraph, K, may be less than or equal to 20000, 10000, 1000, 500, 200, or 100. The number of hyperedges in a hypergraph, K, may be within a range given by any of the lower and upper limits above, as long as a range is formed thereby.

[0115] The cardinality of at least one hyperedge of a hypergraph may be odd. The cardinality of at least one hyperedge of a hypergraph may be 1, and / or the cardinality of at least one hyperedge of a hypergraph may be at least 3. The cardinality of at least one hyperedge of a hypergraph can be at least 4. The maximum cardinality of the hyperedges of a hypergraph, i.e., the maximum value of the cardinality of the hyperedges of a hypergraph, also called the rank (rank(H)) of the hypergraph, may be 2, 3, 4, 5, 6 or more, and may be in the range from 2 to N or in the range from 3 to N. The number K of hyperedges of a hypergraph may be less than the following number 21, or may be less than the following number 22.

Number

Number

[0116] The hyperedges of a hypergraph can be associated with weights. The weights and / or the association between the weights and the hyperedges may be included in a data set, or may be respectively included in the data representing the hyperedges of the hypergraph. The hypergraph may be a weighted hypergraph. The weights can be integers, numbers from a finite set of integers, numbers from a finite set of real numbers, or real numbers. The absolute value of the weights can be bounded below by a constant w min and can be bounded above by a constant w maxThe upper limit can be set thereby. The hypergraph is at least one of simple, non-separable, connected, k-uniform, l-bound, and d-regular. Here, k is an integer in the range from 2 to N, and l is an integer in the range from 2 to N - 1. When l = rank(H) < N, the hypergraph is l-bound. When the degree of all nodes of the hypergraph is d, the hypergraph is d-regular. The degree of a node is the number of hyperedges containing the node. When no node with degree zero is included, the hypergraph is not separated.

[0117] The hyperedges of the hypergraph can be associated with side conditions. Some hyperedges of the hypergraph can be associated with their respective side conditions. The side conditions and / or the association between the side conditions and the hyperedges may be included in the dataset or may be respectively included in the data representing the hyperedges of the hypergraph. The side conditions may be indicated by a sign (plus sign or minus sign) or a coefficient +1 or -1.

[0118] The method includes determining a set of generalized cycles. Determining the set of generalized cycles may be performed by a processor of a system for determining a quantum arithmetic control layout. A generalized cycle is a set of hyperedges. A cardinality of a generalized cycle is called the length of the generalized cycle. A maximum length of a generalized cycle of a set of generalized cycles is the maximum value of the lengths of the generalized cycles of the set of generalized cycles. A generalized cycle may include a hyperedge of a hypergraph or may include a hyperedge of an expanded hypergraph. That is, an element of a generalized cycle is a hyperedge of a hypergraph or an expanded hypergraph. A set of generalized cycles includes or consists of regular generalized cycles. A set of generalized cycles may include or consist of irregular generalized cycles. A set of generalized cycles may include regular and irregular generalized cycles. A generalized cycle may include special elements as described herein. Special elements are considered as hyperedges and count the length of a generalized cycle. A non-zero weight may be assigned to the special elements. The absolute value of the nonzero weight of a particular element may be greater than the absolute value of any weight associated with a hyperedge in the hypergraph. That is, w max It may be larger than that.

[0119] The expanded hypergraph includes at least the hyperedges of the hypergraph and additional hyperedges. The additional hyperedges can include at least one node of the hypergraph, and the nodes of the expanded hypergraph are the same as the nodes of the hypergraph. The expanded hypergraph can include two or more additional hyperedges. A weight of zero is assigned to the additional hyperedges of the expanded hypergraph, i.e., the hyperedges not included in the hypergraph. The number of nodes of the expanded hypergraph, i.e., the degree of the expanded hypergraph, can be N. The number of hyperedges of the expanded hypergraph, i.e., the size of the expanded hypergraph, can be as shown in the following number 23. The following relationship shown in number 24 may hold.

Number

Number

[0120] A maximum length of a generalized cycle of the set of generalized cycles is not greater than a maximum number of vertices of a cell of the mesh. The method may include deriving the maximum vertex number from a vertex number of a cell of the mesh. Alternatively, the maximum number of vertices may be implicitly assumed in the method. A length of a regular generalized cycle of the set of generalized cycles may range from 3 to a maximum number of vertices of a cell of the mesh. A length of an irregular generalized cycle of the set of generalized cycles may range from 2 to a maximum number of vertices of a cell of the mesh. A length of a generalized cycle of the set of generalized cycles may range from 3 to a maximum number of vertices of a cell of the mesh. A length of a generalized cycle of the set of generalized cycles may be equal to a maximum number of vertices of a cell of the mesh.

[0121] The method includes determining a mesh mapping that maps hyperedges of the hypergraph or the expanded hypergraph to vertices of a mesh, or that maps data representing hyperedges of the hypergraph or the expanded hypergraph to vertices of the mesh. The determination of the mesh mapping may be performed by a processor of the system for determining a quantum computing control layout. Hyperedges with side conditions may not be mapped to vertices of the mesh. In the mesh mapping, each generalized cycle of a constrained subset of the set of generalized cycles consists of a hyperedge that is mapped to a cell of the mesh. A hyperedge is mapped to a cell of the mesh if the hyperedge is mapped to a vertex that belongs to the cell. The constrained subset may be one of a proper subset and an improper subset of the set of generalized cycles. In determining the mesh mapping, the mesh or at least some information elements about the mesh may be explicitly considered as parameters of the mesh mapping. Alternatively, some form of mesh may be implicitly assumed in the mesh mapping.

[0122] The cardinality of the constraint subset may be at least KN. The cardinality of the constraint subset may be at least [K]-N. The cardinality of the constraint subset may be at least K-N+1, or at least [K]-N+1. The cardinality of the constraint subset may be equal to KN or K-N+1 if the hyperedges of the expanded hypergraph are mapped under a mesh mapping. Alternatively, the cardinality of the constraint subset may be equal to [K]-N or [K]-N+1 if the hyperedges of the expanded hypergraph are mapped under a mesh mapping. The constraint subset may include at least one of a regular generalized cycle and an irregular generalized cycle. The constraint subset may include or consist of at least one regular generalized cycle. The constraint subset may include or consist of at least one irregular generalized cycle. The union of the generalized cycles of the constraint subset of the generalized cycles may include all hyperedges of the hypergraph or the expanded hypergraph.

[0123] The mesh mapping is constructed by mapping hyperedges of a first generalized cycle of the set of generalized cycles to vertices of cells of the mesh and mapping hyperedges of a second generalized cycle of the set of generalized cycles to vertices of adjacent cells of the mesh, where the first generalized cycle and the second generalized cycle have at least one hyperedge in common, and the at least one hyperedge is mapped to at least one corresponding vertex of the mesh. This process of mapping hyperedges of generalized cycles of the set of generalized cycles can be repeated until the mapped generalized cycles form a constraint subset. Mapping the hyperedges of the first generalized cycle, the second generalized cycle, etc. can include placing an image of the first generalized cycle, the second generalized cycle, etc. on the mesh. The image is an image under the mesh mapping being constructed.

[0124] The method includes generating a quantum operation control layout. Generating the quantum operation control layout may be performed by a processor of a system for determining a quantum operation control layout. The quantum operation control layout may be viewed as a control program that, when executed by a quantum processing unit, causes the quantum processing unit to perform the quantum operations encoded therein. The quantum operation control layout includes data indicative of or representative of layout vertices of a mesh. Each layout vertex corresponds to a hyperedge that is mapped according to a mesh mapping. The quantum operation control layout includes data indicative of or representative of a layout vertex set. Each layout vertex set is made up of layout vertices in a cell of the mesh, where the layout vertices correspond to elements of a generalized cycle of the constraint subset. The correspondence is provided by the mesh mapping. The mesh mapping is injective and invertible on its image.

[0125] Depending on the information about the quantum system available to the method for determining a quantum operation control layout, the content and format of the data indicating or representing the layout vertices may vary, as may the content and format of the data indicating or representing the layout vertex sets. For example, if the method obtains only mesh data, the method may return the mesh and an indication of which vertices of the mesh are layout vertices and which sets of these vertices form the layout vertex set. If the method obtains information about the quantum computation to be performed, such as information that the quantum computation is performed as a digital gate-based quantum computation having a particular set of gates, the method may return a list of gates to be performed, where the list specifies between which components the gates are performed and in what order. This list will be a quantum operation control layout of a particular content and format that takes into account knowledge about the quantum system and the quantum computation, but is a special form of data indicating or representing the layout vertices / layout vertex sets, as it is data from which the layout vertices and layout vertex sets can be derived. The method may include providing information about the type of quantum computation to be performed. In this method, generating the quantum operation control layout may include adapting the format and / or content of the quantum operation control layout depending on the information about the type of quantum computation to be performed.

[0126] The quantum arithmetic control layout may include data associating layout vertices with weights of hyperedges of the hypergraph or expanded hypergraph that are mapped to the layout vertices by the mesh mapping. The quantum arithmetic control layout may include data associating layout vertex sets with constraint weights. The constraint weights of a layout vertex set may have the same absolute value or the same value for all layout vertex sets. The constraint weights of layout vertex sets corresponding to regular generalized cycles may be the same. The absolute values ​​of the constraint weights of layout vertex sets corresponding to irregular generalized cycles may be the same. The values ​​of the constraint weights of some layout vertex sets corresponding to irregular generalized cycles may be the opposite of the values ​​of the constraint weights of some other layout vertex sets corresponding to irregular generalized cycles. If the sum of the values ​​is zero, then a value is the opposite of another value. The absolute value of the constraint weight is greater than the absolute value of any weight of a hyperedge of the hypergraph, i.e. w max The absolute value of the constraint weights of a set of layout vertices may be less than the absolute value of a special weight associated with a layout vertex corresponding to a special element of a generalized cycle.

[0127] The quantum arithmetic control layout may be an opaque quantum arithmetic control layout. The quantum arithmetic control layout may be a transparent quantum arithmetic control layout. The transparent quantum arithmetic control layout may include data representing a mesh mapping. Additionally or alternatively, the transparent quantum arithmetic control layout may include data representing a generalized cycle of a constrained subset of the generalized cycle.

[0128] The method may include outputting the quantum operation control layout. Outputting the quantum operation control layout may include storing the quantum operation control layout in a memory and / or transmitting the quantum operation control layout over a network.

[0129] According to further embodiments, a quantum operation control layout is provided. According to further embodiments, a computer program product is provided that includes the quantum operation control layout. The quantum operation control layout may be a quantum operation control layout obtained by a method for determining a control layout for quantum computing on a quantum system according to embodiments described herein. The quantum operation control layout may be a direct product of a method for determining a quantum operation control layout according to embodiments described herein. The computer program product may include a quantum operation control layout embodied in a machine-readable form, for example in the form of symbols or bits. The computer program product may be an internal or external storage, such as a memory of a classical computing system, a hard drive, SSD, flash memory on a card or stick, CD, DVD, etc. The quantum operation control layout may be embodied as a downloadable symbol stream or bit stream and may be transmitted over a communication channel, such as a network, to a remote location away from the system for determining the quantum operation control layout.

[0130] The quantum computing control layout may be configured to control quantum computation on a quantum system. When loaded into a memory of a quantum computing system and / or processed by a quantum computing unit, the quantum computing control layout causes the quantum computing system to perform a quantum computation, in particular a quantum computation according to a method for performing a quantum computation on a quantum system according to embodiments described herein. The quantum computation is performed on components of the quantum system arranged according to a mesh. The vertices of the mesh represent possible locations of the components of the quantum system. Each cell of the mesh indicates that a quantum interaction between the components of the quantum system arranged in that cell is possible during the quantum computation. The quantum system and its components, the mesh and its vertices, and the cells may have any of the properties described herein.

[0131] The quantum arithmetic control layout includes data representing layout vertices of a mesh and data representing a layout vertex set. Each layout vertex set is composed of layout vertices in a cell of the mesh. The layout vertices may correspond to hyperedges of a hypergraph or an expanded hypergraph that are mapped to the layout vertices according to the mesh mapping. The layout vertices of each layout vertex set correspond to hyperedges that form a generalized cycle of the hypergraph or the expanded hypergraph. The quantum arithmetic control layout may include data representing weights associated with the layout vertices. The weights associated with the layout vertices may correspond to weights of hyperedges of the hypergraph or an expanded hypergraph that are mapped to the layout vertices by the mesh mapping. The quantum arithmetic control layout may include data representing constraint weights associated with the layout vertex set. All layout vertices may be included in a union of the layout vertex set. The quantum arithmetic control layout may include a mesh, data representing the mesh, or data representing at least one information element related to the mesh. The layout vertices and the layout vertex set may be identified in relation to the mesh. The meshes, information elements about the meshes, layout vertices, layout vertex sets, hyperedges, hypergraphs, expanded hypergraphs, generalized cycles, mesh mappings, weights, constraint weights, and quantum operation control layouts may have any of the properties described herein.

[0132] According to a further embodiment, a method of performing a quantum computation on a quantum system is provided. The method may be performed by a quantum computing system. The quantum computing system may include a memory, a quantum processing unit (QPU), and a measurement unit. The quantum computation is performed on components of the quantum system. The method includes providing a quantum computation control layout as described herein. Providing the quantum computation control layout may include receiving the quantum computation control layout over a network and / or loading the quantum computation control layout from a memory of the quantum computing system. The method of performing a quantum computation may include determining a quantum computation control layout for a quantum computation as described herein. The method of performing a quantum computation includes providing components of a quantum system in a spatial arrangement such that for each layout vertex of a mesh there is a component, and for each set of layout vertices quantum interactions are possible between components corresponding to layout vertices of the set of layout vertices. The quantum system and its components, including any arrangement thereof, as well as the mesh, its vertices, and cells may have any of the properties described herein.

[0133] A method of performing a quantum computation in a quantum system includes, for each layout vertex associated with a non-zero weight, applying a local field to the components corresponding to the layout vertex. The method includes, for each set of layout vertices, performing a quantum interaction between the components corresponding to the layout vertices of the layout vertex set. The application of the local field and the performance of the quantum interaction may be driven by a quantum processing unit of the quantum processing system. The method includes measuring some or all of the components of the quantum system. The measurement may be performed by a measurement unit of the quantum processing system. The method of performing a quantum computation may further include any of the features described herein.

[0134] According to a further embodiment, a method for solving a computational problem is provided. The computational problem may be a classical computational problem, e.g., an NP-hard or NP-complete computational problem, as described herein. The method may include encoding the computational problem into a hypergraph. The hypergraph may be associated with a spin model in that nodes of the hypergraph correspond to spins of the spin model and hyperedges correspond to interactions between spins of the spin model, and finding a ground state of the spin model may be equivalent to finding a solution to the computational problem. The method may further include obtaining or determining / generating a quantum operation control layout based on the hypergraph. The method for solving the computational problem may include a method for determining a quantum operation control layout as described herein. The method may include performing a quantum computation controlled by the quantum operation control layout. The method for solving the computational problem may include a method for performing a quantum computation as described herein. The method for solving the computational problem may include obtaining a measurement result of the quantum computation as a trial solution and determining whether the trial solution is a solution to the computational problem. If the trial solution is not a solution to the computational problem, the method may include repeating the execution of the quantum computation until a solution is found, or repeating the execution of the quantum computation a finite number of times and selecting the best trial solution as an approximate solution to the computational problem. The methods for solving computational problems may be performed by classical and quantum computing systems described herein.

[0135] According to further embodiments, a system for determining a quantum operation control layout is provided. The system may be a classical computing system and may include a processing unit / processor and a memory. The system for determining a quantum operation control layout may be configured to perform a method for determining a quantum operation control layout according to embodiments described herein. The components of the system may be configured to perform individual features of the method as described herein. Furthermore, a system for performing quantum computing is provided. The system for performing quantum computing may be a quantum processing system and may include a quantum processing unit, a measurement unit, and any other components described herein. The system and its components may be configured to perform a method for performing quantum computing or individual features of the method according to embodiments described herein. The system for performing quantum computing may be configured to perform a quantum computation under the control of a quantum operation control layout described herein when the quantum operation control layout is loaded into the memory of the system and / or processed by the quantum processing unit. An embodiment is directed to a quantum operation control layout according to embodiments described herein, which, when executed as a control program by a system performing quantum computing, causes the system to perform a method for performing quantum computing described herein. Furthermore, a system for solving a computation problem is provided. The system may include at least one classical computing system for encoding the computational problem into a hypergraph, determining a quantum computation control layout, and determining whether a quantum computation measurement result contains a solution to the computational problem, and may include a quantum computing system for performing a quantum computation on the quantum system controlled by the quantum computation control layout. The system for solving a computational problem and its components may be configured to perform methods and individual features of methods for solving a computational problem as described herein.Further embodiments are directed to the use of a system for determining a quantum operation control layout to perform a method for determining a quantum operation control layout according to embodiments described herein, the use of a system for performing quantum computing on a quantum system to perform a method for performing quantum computing described herein, and the use of a system for solving a computational problem to perform a method for solving a computational problem described herein.

[0136] While the above relates to embodiments, other and further embodiments may be devised without departing from the scope, which is determined by the claims.

Claims

1. A method for determining a quantum operation control layout (300, 301, 302, 303) for a quantum computation on a quantum system (530), the quantum computation being performed on components (532, 534) of the quantum system arranged according to a mesh (100, 101), vertices (111, 112, 113, 114) of the mesh representing possible locations of the components of the quantum system, each cell (110, 120) of the mesh indicating a quantum interaction between the components of the quantum system arranged in that cell being possible during the quantum computation, the method comprising: providing a dataset including data representing hyperedges (212, 222, 232) of a hypergraph (200, 201, 203); determining a set of generalized cycles (240, 250, 260), the generalized cycles including hyperedges of the hypergraph or including hyperedges of an expanded hypergraph (202), the expanded hypergraph including at least the hyperedges of the hypergraph and additional hyperedges (245, 247), the maximum length of a generalized cycle in the set of generalized cycles being no greater than the maximum number of vertices of the cells of the mesh; determining a mesh mapping that maps data representing the hyperedges of the hypergraph or the expanded hypergraph to the vertices of the mesh, wherein each generalized cycle of a constraint subset of the set of generalized cycles consists of a hyperedge that is mapped to a cell of the mesh; generating the quantum computation controlled layout including data indicative of layout vertices (312, 322, 345, 347, 392) of the mesh, each layout vertex corresponding to a hyperedge mapped according to the mesh mapping, and including data indicative of layout vertex sets (310, 320, 330, 340, 350, 360), each layout vertex set consisting of layout vertices in cells of the mesh that correspond to generalized cycles of the constraint subset of generalized cycles; A method for determining a quantum operation control layout, comprising:

2. the mesh is two-dimensional; The method of claim 1.

3. a length of the generalized cycles of the set of generalized cycles ranges from 3 to the maximum number of vertices of the cells of the mesh or is equal to the maximum number of vertices of the cells of the mesh; The method according to claim 1 or 2.

4. the number of nodes in the hypergraph is N, the number of hyperedges in the hypergraph is K, and the cardinality of the constraint subset is at least K−N; A method according to any one of claims 1 to 3.

5. the number of nodes in the hypergraph is N, the number of hyperedges in the hypergraph is K, and K is less than N(N-1) / 2; 5. The method according to any one of claims 1 to 4.

6. the hyperedges of the hypergraph are associated with weights, and the quantum computation control layout includes data associating layout vertices with the weights of the hyperedges of the hypergraph or the expanded hypergraph that are mapped to the layout vertices by the mesh mapping, and additional hyperedges of the expanded hypergraph that are not included in the hypergraph are assigned a weight of zero.

6. The method according to any one of claims 1 to 5.

7. the quantum arithmetic control layout is a transparent quantum arithmetic control layout including at least one of data representing the mesh mapping and data representing the generalized cycles of the constrained subset of generalized cycles; 7. The method according to any one of claims 1 to 6.

8. a union of generalized cycles of the constraint subsets of generalized cycles includes all hyperedges of the hypergraph or the expanded hypergraph, and / or the generalized cycles of the constraint subsets of generalized cycles connect all hyperedges of the hypergraph or the expanded hypergraph.

8. The method according to any one of claims 1 to 7.

9. the cardinality of at least one hyperedge of the hypergraph is odd and / or the cardinality of at least one hyperedge of the hypergraph is at least 3; 9. The method according to any one of claims 1 to 8.

10. the constraint subset includes at least one of a regular generalized cycle (240) and an irregular generalized cycle (250, 260); 10. The method according to any one of claims 1 to 9.

11. the mesh mapping is constructed by mapping the hyperedges of a first generalized cycle of the set of generalized cycles to vertices of cells of the mesh and mapping the hyperedges of a second generalized cycle of the set of generalized cycles to vertices of adjacent cells of the mesh; the first generalized cycle and the second generalized cycle have at least one hyperedge in common, the at least one hyperedge being mapped to a corresponding at least one vertex of the mesh, and repeating this process of mapping hyperedges of generalized cycles of the set of generalized cycles until the mapped generalized cycles form the constraint subset.

11. The method according to any one of claims 1 to 10.

12. 12. The method of claim 1 , comprising at least one of the following: (a) a) a step of: (a) providing mesh data representing or including at least one information element relating to said mesh, and deriving said mesh from said mesh data, where default values ​​are used in the case where information relating to said mesh is missing. (b) providing information regarding the type of quantum computation to be performed; Generating the quantum operation control layout includes adapting a form and / or content of the quantum operation control layout to be executed in response to the information regarding the type of the quantum operation.

13. A computer program product that causes a computer to execute the quantum operation control layout determination method according to claim 1.

14. A method of performing a quantum computation in a quantum system (530), the quantum computation being performed in components (532, 534) of the quantum system, the method comprising: Providing a quantum computing control layout (300, 301, 302) according to claim 13; providing components of the quantum system in a spatial arrangement such that for each layout vertex of the mesh there is a component, and for each set of layout vertices quantum interactions are possible between components corresponding to layout vertices of that set of layout vertices; for each layout vertex associated with a non-zero weight, applying a local field to the component corresponding to that layout vertex; for each layout vertex set, performing quantum interactions between components corresponding to said layout vertices of that layout vertex set; measuring some or all of the components of the quantum system; A method for performing quantum computing, comprising:

Images