Method and system for solving a problem

Abstract

The invention relates to a computer-implemented method for solving a problem, the problem being a decision problem or optimization problem, in which method at least one subproblem of the problem is identified. An occupancy is determined for at least one variable of the problem by solving the at least one subproblem using a quantum computer. A solution to the problem is generated using the determined occupancy of the at least one variable of the problem.

Description

[0001] Description

[0002] title

[0003] Method and system for solving a problem

[0004] The invention relates to a method and a system for solving a decision problem or optimization problem. In particular, the invention relates to methods and systems for solving decision problems or optimization problems using quantum computers, quantum annealers, or quantum simulators.

[0005] State of the art

[0006] The satisfiability problem (SAT) in propositional logic is a well-known and well-studied NP-complete problem. SAT solvers such as the Davis-Putnam-Logemann-Loveland (DPLL) algorithm use the backtracking technique, in which the values ​​of free Boolean variables are determined using Boolean Constraint Propagation (BCP). Conflicts are resolved by reversing branching steps. Conflict-Driven Clause Learning (CDCL) extends DPLL by learning from conflicts and applying backtracking non-chronologically, enabling effective applications in areas such as hardware or software design.

[0007] The satisfiability problem in propositional logic is an example of a more general logic problem or optimization problem that has numerous practical applications. Ideally, a program for solving a logic problem or optimization problem generates all valid or optimal solutions. For more complex or larger logic problems, the solution process can be accelerated using heuristics. By limiting the runtime, even suboptimal solutions can be used, which may, however, be sufficient in practice.

[0008] Furthermore, it is known to use quantum computers to solve logic or optimization problems. For example, DE 10 2022 202698 A1 describes a method for operating a quantum register, in which an objective function is optimized.

[0009] Disclosure of the invention

[0010] The invention provides a method and a system for solving a problem having the features of the independent claims.

[0011] Preferred embodiments are the subject of the respective subclaims.

[0012] According to a first aspect, the invention accordingly relates to a computer-implemented method for solving a problem, wherein the problem is a decision problem or optimization problem. At least one subproblem of the problem is determined. For at least one variable of the problem, an assignment is determined by solving the at least one subproblem using a quantum computing device. A solution to the problem is generated using the determined assignment of the at least one variable of the problem.

[0013] According to a second aspect, the invention relates to a system for solving a problem, wherein the problem is a decision problem or optimization problem. A computing device determines at least one subproblem of the problem. A quantum computing device determines an assignment for at least one variable of the problem by solving the at least one subproblem. The computing device generates a solution to the problem using the determined assignment of the at least one variable of the problem.

[0014] Advantages of the invention

[0015] The invention provides a hybrid approach in which at least one subproblem of a problem to be solved is first determined, which can then be solved using a quantum computing device. Using the variable assignment determined by the quantum computing device, the actual solution to the problem can be determined more quickly.

[0016] The quantum computing device can thus be used to determine a heuristic assignment of variables, which can accelerate the solution process. Thus, better and / or valid solutions can be calculated in less time.

[0017] The underlying physical idea is that solutions can be found more quickly using quantum mechanical algorithms. For example, tunneling processes through energy barriers in quantum annealing can potentially optimize the subproblem faster or more optimally than using classical computers. The resulting solutions can then be used as heuristics.

[0018] A quantum computing device can be understood as a computing device that operates on the basis of quantum mechanical states. In particular, the quantum computing device can perform calculations based on superposition states and / or the entanglement of states. The quantum computing device can, in particular, be a quantum computer, quantum annealer, or quantum simulator.

[0019] The calculation of subproblems by the quantum computing device to determine a variable assignment is particularly advantageous in current so-called NISQ (noisy intermediate-scale quantum) systems. These are currently available implementations of quantum computers that have a relatively small number of physical qubits, meaning that relevant problems cannot yet be completely solved, for example, due to a lack of qubits for effective quantum error correction. However, the NISQ systems can be powerful enough to solve subproblems. The determined partial solutions can then be considered, for example, by a classical logic solver when solving the complete problem.

[0020] The problem is a decision problem or an optimization problem. A decision problem is a problem in which only "yes" (1) and "no" (0) are possible outcomes. In a decision problem, the decision is made whether an input is accepted (output 1) or rejected (output 0).

[0021] In an optimization problem, the best possible solution is found by considering a target criterion. An optimization method that solves an optimization problem sometimes only produces local optima, but this is often sufficient in practice.

[0022] According to a further embodiment of the method for solving the problem, the determined assignment of the at least one variable of the problem is an initial assignment. In the event of a conflict, a further assignment of the at least one variable of the problem is determined when generating the solution to the problem. The further assignment can in turn be carried out using the quantum computing device, for example by adding additional constraints. Even if the quantum computing device initially generates an incorrect assignment, the solution to the problem can still be generated more quickly under certain circumstances. According to a further embodiment of the method for solving the problem, the generation of the solution to the problem is carried out using the solution for the at least one sub-problem using a classical computing device (i.e. a classical computer).The final solution, taking into account the variable assignment, can therefore be achieved by a classical logic solver.

[0023] According to a further embodiment of the method for solving the problem, determining the at least one subproblem of the problem comprises decomposing the problem into subproblems depending on a specification of the quantum computing device. In particular, subproblems can be determined in this way that enable the use of the quantum computing device in the first place. In other words, the problem is decomposed into smaller subproblems such that the quantum computing device can solve at least one of these subproblems based on its specification.

[0024] According to a further embodiment of the method, the subproblems can be specified by the algorithm, for example by the decision subroutines of a DPLL or CDCL algorithm.

[0025] According to a further embodiment of the method for solving the problem, a plurality of subproblems of the problem are determined. A subproblem of the plurality of subproblems is solved using the quantum computing device if a number of variables of the subproblem is smaller than a predetermined value. The value depends in particular on a specification of the quantum computing device. In particular, the value can depend on the number of physical qubits of the quantum computing device. A qubit mapping can be taken into account, which specifies how logical qubits are mapped to physical qubits. The variables of the subproblem can be expressed using logical qubits. For implementation, the logical qubits are converted into physical qubits, whereby the conversion can also depend, for example, on any quantum error correction that may be used.The number of possible variables is then limited by the number of physical qubits.

[0026] According to a further embodiment of the method for solving the problem, a plurality of subproblems of the problem are determined. One subproblem of the plurality of subproblems is solved using a classical computer if a number of variables of the subproblem is greater than a predetermined value. The term "classical" here refers to an implementation not based on the principles of quantum mechanics. In this case, the number of variables may be too large to solve the subproblem using the quantum computing device. In this case, a classical solution can be generated, in particular using classical heuristics.

[0027] For example, graph problems can be solved using minimum cuts or maximum cuts as heuristics. These heuristics can incorporate domain knowledge. Examples are so-called look-back heuristics or scoring tables. Using these tables, variables can be selected that have appeared particularly frequently in logical conflicts, as this can accelerate the learning of new constraints. With look-ahead heuristics, variables are assigned values ​​in such a way that they generate many subsequent implications, which then lead to the assignment of as many other variables as possible.

[0028] According to a further embodiment of the method for solving the problem, it can be provided that the subproblem is further decomposed or the decomposition is already carried out in such a way that the subproblem always has a number of variables that is smaller than the predetermined value. This makes it possible for all subproblems to be solved using the quantum computing device. According to a further embodiment of the method for solving the problem, the problem is a SAT problem, wherein the problem is solved by a SAT solution algorithm. The subproblem of the problem comprises the assignment of at least one variable from a plurality of unassigned variables in a decision step of the SAT solution algorithm. In one embodiment, the SAT solution algorithm is the DPLL algorithm or a variant thereof. The DPLL algorithm checks whether a predetermined set of clauses is satisfiable, i.e.whether the variables in the clauses have an assignment that satisfies the clauses. The DPLL algorithm uses unit propagation and proceeds recursively by splitting the problem into subproblems (splitting rule). In a decision step, a free variable (a literal) is selected and a value is assigned to this variable, i.e., the variable is assigned a value. It is then checked whether the clauses can be satisfied. In the event of a conflict, the opposite truth value is chosen, and the recursive check is performed again. The assignment of this variable with a truth value (1 or 0) is performed using the quantum computing device.

[0029] According to another embodiment of the method for solving the problem, the SAT solution algorithm is the CDCL algorithm or a variant thereof, which, in further development of the DPLL algorithm, includes clause learning and non-chronological backtracking. In the decision step, the quantum computing device again selects the free variable and assigns it.

[0030] In another embodiment of the method for solving the problem, the problem is the satisfiability modulo theories (SMT) problem.

[0031] According to a further embodiment of the method for solving the problem, the at least one subproblem of the problem is determined by restricting or truncating the problem. The number of free variables of the at least one subproblem is smaller than a number of free variables of the problem. For example, an optimization problem with constraints is to be solved with a predetermined number of free variables. During truncation, a subset of free variables is selected, and all expressions containing a non-selected free variable are set to zero. The reduced subproblem for the selected free variables is then solved.

[0032] According to a further embodiment of the method for solving the problem, determining the at least one subproblem of the problem comprises decomposing the problem into a plurality of disjoint subproblems. The problems thus do not overlap. This can avoid potential conflicts that may arise, for example, if the solutions to overlapping subproblems cannot be reconciled.

[0033] According to a further embodiment of the method for solving the problem, the subproblem of the problem comprises a nondeterministic decision routine of the problem. The method can thus be applied to symbolic methods or logic solvers that have a subroutine (i.e., a subproblem) that nondeterministically assigns an initial value to Boolean variables. Examples are the decision steps or subroutines of the DPLL algorithm or CDCL algorithm. In particular, the method is applicable to SAT solvers or to decision logic modulo other theories (satisfiability modulo theories, SMT). Another area of ​​application concerns answer set solvers.

[0034] According to a further embodiment of the method for solving the problem, a model of at least a part of a computer program or a hardware device is received in a logical representation. The problem is generated based on the model. The problem is a satisfiability problem for verifying the computer program or the hardware device. By means of the verification, it can be determined whether the computer program or the hardware device conforms to the specification. In particular, verification of safety-critical parts of the hardware device can be performed. When verifying a computer program, for example, it can be determined whether a variable assignment exists that generates an error.

[0035] The solution to the problem can have numerous other practical applications, such as material simulation, shift planning, logistics, person recognition (such as pose recognition), network flow optimization, process planning, system design, production planning, or the like.

[0036] Further advantages, features and details of the invention will become apparent from the following description, in which various embodiments are described in detail with reference to the drawings.

[0037] Short description of the drawings

[0038] They show:

[0039] Figure 1 is a schematic block diagram of a system for solving a problem according to an embodiment of the invention; and

[0040] Figure 2 is a flowchart of a method for solving a problem according to an embodiment of the invention.

[0041] The numbering of procedural steps is for clarity and generally does not imply a specific chronological order. In particular, several procedural steps can be performed simultaneously.

[0042] Description of the embodiments

[0043] Figure 1 shows a schematic block diagram of a system 1 for solving a problem. The problem is a decision problem or optimization problem. The decision problem can, in particular, be a SAT problem. The system 1 comprises a computing device 2, such as a processor, microprocessor, microcontroller, integrated circuit, a field-programmable gate array (FPGA), or the like. The computing device 2 can thus, in particular, be a conventional computer.

[0044] The system 1 further comprises a quantum computing device 3. This can be a quantum computer, a quantum annealer, or a quantum simulator. The quantum computing device 3 can, in particular, be a NISQ system, for example, with fewer than 1000 qubits, which is sensitive to ambient noise and quantum decoherence effects.

[0045] The system 1 further comprises a storage device 4, such as a hard disk, a memory card, or the like. Instructions that can be executed on the computing device 2 are stored in the storage device 4. Furthermore, specifications of the problem or an algorithm for solving the problem can be stored on the storage device 4. Furthermore, measurement results of the quantum computing device 3 can be stored on the storage device 4.

[0046] A computing device 2 determines at least one subproblem of the problem. For example, the problem is broken down into a plurality of disjoint subproblems. The quantum computing device 3 solves a subproblem and thereby generates an assignment for at least one variable of the problem. The variable can, in particular, be a Boolean variable. The number of assigned variables can be smaller than the number of variables in the original problem; thus, the variables do not need to be fully assigned.

[0047] The quantum computing device 3 can generate the solution, for example, using the Quantum Approximate Optimization Algorithm (QOAO), which is a well-known heuristic method for solving binary optimization problems. A quantum annealer, as a possible implementation of the quantum computing device 3, can be specialized for solving specific problems. In particular, the problem can be solved using adiabatic quantum computations. For this purpose, a first Hamiltonian function Ho can be specified, and a system is prepared in a known ground state of the first Hamiltonian function Ho. Adiabatic time evolution then continuously transitions to a second Hamiltonian function Hi, so that the system is ideally in the ground state of the second Hamiltonian function Hi at the end of the process.

[0048] The second Hamiltonian function Hi is chosen such that a minimum corresponds to the minimum of the function to be minimized. The minimum of the function can thus be determined through measurements. In practice, a large number of measurements can be performed to reduce the influence of situations in which the system is not in the desired ground state at the end of the evolution.

[0049] A quantum simulator as a possible implementation of the quantum computing device 3 comprises a quantum system that can be controlled and thereby imitates another quantum system.

[0050] The variables used and assigned by quantum computing device 3 correspond to the variables of the original problem. Computing device 2 uses the calculated assignment of the at least one variable to generate a solution to the problem.

[0051] For example, the original problem can be truncated to create one or more subproblems. The number of free variables is reduced, meaning that the subproblem has fewer free variables than the original problem.

[0052] For example, the original problem has four free variables, x_1, x_2, x_3, x_4, each of which can take the possible values ​​0 or 1. The problem is an optimization problem, meaning that the objective is to find an assignment of the variables that minimizes a given function under constraints. The function to be minimized can be given, for example, by: x_1 + x_2 - x_3 - (x_4 + 1) 2 where the constraints are given by: x_1 + x_2 < 1 , x_3 - x_4 = 1 , x_2 - x_3 = 0.

[0053] Depending on the specifications of the quantum computing device 3, a number of variables of the subproblem are determined.

[0054] For example, if quantum computing device 3 has only two qubits, two variables are selected. For example, variables x_1 and x_2 can be selected heuristically. These variables correspond to logical qubits and can be implemented by the physical qubits of quantum computing device 3.

[0055] For example, the mapping of variables to physical qubits can be done using the Quantum Approximate Optimization Algorithm (QAOA). This algorithm minimizes a function with a desired approximation ratio.

[0056] Quantum computing device 3 solves the subproblem obtained by truncating the problem, i.e., by setting all expressions containing the unselected variables to zero. Quantum computing device 3 thus finds a solution to the problem x 1 + x 2 with the constraint: x_1 + x_2 < 1.

[0057] Alternatively, the subproblem can be formulated without optimization, and the quantum computing device 3 only solves the necessary constraints. The generated assignment of the variables x_1 = 0 and x_2 = 0 is used in a decision step to solve the overall problem.

[0058] In this case, this is the minimal or optimal solution to the subproblem. Since the decision step only involves a heuristic choice of variable assignments, it is also possible that the problem is further solved based on non-optimal or even incorrect solutions to the subproblem (e.g., x_1 = 1 and x_2 = 1). In this case, a conflict arises due to the assignment, and a different assignment is chosen accordingly.

[0059] According to one embodiment, the remaining problem can also be solved using the quantum computing device 3, in the example with the variables x_3 and x_4.

[0060] According to one embodiment, the remaining problem is solved using classical algorithms, such as classical variable assignment heuristics in a classical logic solver. The solution to the original problem is thus performed by the (classical) computing device 2 based on the solution to the subproblem.

[0061] The numerical values ​​mentioned are to be understood as examples only. In particular, the quantum computing device 3 can have a larger number of physical qubits, and the problem will generally also have a larger number of free variables. To solve the original problem, repeated decomposition into subproblems may also be necessary or advantageous. If, for example, further variable assignment is required, at least one subproblem can be determined again, and this can again be solved using the quantum computing device 3.

[0062] Furthermore, it may happen that the solution generated by quantum computing device 3 leads to a conflict, i.e., the assignment leads to an inconsistency, so that the problem cannot be solved. Some algorithms can then add additional constraints so that the incompatibility of assignments of certain variables can be learned from the conflict. In this case, it can be provided that the subproblem is solved again with the additional constraints using quantum computing device 3. Thus, in the event of a conflict when generating the solution to the problem, another assignment of at least one variable of the problem is determined.

[0063] The problem can be decomposed into subproblems depending on a specification of the quantum computing device 3. For example, the number of physical qubits of the quantum computing device 3 can be taken into account. This number can limit the number of free variables that can be processed by the quantum computing device 3.

[0064] For example, each free variable can be expressed by exactly one physical qubit. In this case, the subproblems can be determined such that the number of free variables in the subproblems is smaller than the number of physical qubits of the quantum computing device 3. In this case, the problem is thus broken down in such a way that the technical characteristics of the quantum computing device 3 are taken into account, thus making the use of the quantum computing device 3 possible in the first place. The use can be maximized by allowing each subproblem to be solved by the quantum computing device 3. In further embodiments, it can also be provided that a subproblem is only solved using a quantum computing device 3 if the number of variables in the subproblem is smaller than a predetermined value, for example, smaller than the number of physical qubits of the quantum computing device 3.Subproblems for which this is not the case, and for which the number of variables of the subproblem reaches or exceeds a given value, can be solved using a classical algorithm.

[0065] In a further embodiment of the method for solving the problem, the problem is a SAT problem. The SAT problem can be solved, for example, using a DPLL algorithm, a CDCL algorithm, or the like. Such algorithms include so-called decision steps, in which a variable is selected from the as yet unassigned free variables and assigned a logical value. This step is typically non-deterministic and is conventionally performed using heuristics. The system 1 can be configured to perform the assignment of the free variables using the quantum computing device 3. For this purpose, for example, a subproblem can be determined, which results from reducing the problem to certain free variables.

[0066] System 1 can be configured to verify a computer program and a hardware device. During verification, the computer program or the hardware device (or at least a part thereof) is logically represented by a model. Verification corresponds to solving a SAT problem of the model. This SAT problem is solved using quantum computing device 3.

[0067] Figure 2 shows a flowchart of a method for solving a problem. The method can be implemented, in particular, using the system described above. Conversely, the system described above can be configured to implement the method described below. The problem is a decision problem or optimization problem.

[0068] In a method step S1, a specification of a problem is obtained or determined, for example in the form of an algorithm for solving the problem.

[0069] In a process step S2, at least one subproblem of the problem is identified. The subproblems may already be specified (e.g., by the algorithm used) or may be generated. For example, the problem can be truncated to generate subproblems. The subproblems can be disjoint, meaning that subproblems do not have any common free variables. Each free variable of the original problem is thus contained in at most one subproblem.

[0070] In a method step S3, one of the subproblems is selected, and it is checked whether the subproblem can be solved by a quantum computing device 3. This check can be performed taking into account a specification of the quantum computing device 3, for example, taking into account the number of physical qubits, possible error correction, a fidelity of the quantum computing device 3, etc. For example, it can be required that the number of free variables of the subproblem be smaller than a predetermined value.

[0071] In the positive case, the quantum computing device 3 solves the subproblem and thereby generates an assignment of at least one variable of the problem, S4.

[0072] In the negative case, the subproblem can be solved by a classical algorithm, for example, using a classical heuristic, whereby an assignment of at least one variable of the problem can also be determined, S5. In further embodiments, it can also be provided that the problem is already broken down into subproblems in step S2 in such a way that all subproblems can be solved by the quantum computing device 3. In a method step S6, it is checked whether further subproblems exist, which are then in turn solved in the manner described.

[0073] If no further subproblems exist, a solution to the problem is generated using the determined assignment of at least one variable of the problem, S7. The determined assignment can be an initial assignment, which is used to further solve the problem (e.g., instead of a classical heuristic).

[0074] The solution can be generated entirely by a classical computer. However, it can also be provided that the solution is generated using the quantum computing device 3. If, for example, a further decomposition of the problem into subproblems is necessary, steps S3 to S6 can be performed again.

Claims

Claims 1 . A computer-implemented method for solving a problem, wherein the problem is a decision problem or optimization problem, comprising the steps: Determining (S2) at least one sub-problem of the problem; Determining (S4) an assignment for at least one variable of the problem by solving the at least one sub-problem using a quantum computing device (3); and Generating (S7) a solution to the problem using the determined assignment of at least one variable of the problem.

2. The method according to claim 1, wherein the determined assignment of the at least one variable of the problem is an initial assignment, and wherein in the event of a conflict when generating the solution to the problem, a further assignment of the at least one variable of the problem is determined.

3. The method according to claim 1 or 2, wherein generating the solution to the problem using the solution to the at least one sub-problem is performed using a classical computing device.

4. Method according to one of the preceding claims, wherein determining the at least one sub-problem of the problem comprises decomposing the problem into sub-problems depending on a specification of the quantum computer (3) or quantum annealer (3).

5. Method according to one of the preceding claims, wherein a plurality of sub-problems of the problem are determined, and wherein a sub-problem of the plurality of sub-problems is solved using the quantum computing device (3) if a number of variables of the sub-problem is smaller than a predetermined value.

6. Method according to one of the preceding claims, wherein a plurality of subproblems of the problem are determined, and wherein a subproblem of the plurality of subproblems is solved using a classical computing device if a number of variables of the subproblem is greater than a predetermined value.

7. The method according to any one of the preceding claims, wherein the problem is a satisfiability (SAT) problem, wherein the problem is solved by a SAT solution algorithm, and wherein the subproblem of the problem comprises assigning at least one variable from a plurality of unassigned variables in a decision step of the SAT solution algorithm.

8. Method according to one of the preceding claims, wherein the at least one subproblem of the problem is determined by restricting or truncating the problem such that a number of free variables of the at least one subproblem is smaller than a number of free variables of the problem.

9. The method according to any one of the preceding claims, wherein determining the at least one subproblem of the problem comprises decomposing the problem into a plurality of disjoint subproblems.

10. Method according to one of the preceding claims, wherein the subproblem of the problem comprises a non-deterministic decision routine of the problem. 11 . Method according to one of the preceding claims, further comprising the steps: Receiving a model of at least a portion of a computer program or hardware device in a logical representation; and Generating the problem based on the model, where the problem is a satisfiability problem for verifying the computer program or the Hardware device is 12. Method according to one of the preceding claims, wherein the quantum computing device (3) is a quantum computer, quantum annealer or quantum simulator.

13. A system (1) for solving a problem, wherein the problem is a decision problem or optimization problem, comprising: a computing device (2) configured to determine at least one subproblem of the problem; and a quantum computing device (3) configured to determine an assignment for at least one variable of the problem by solving the at least one subproblem; wherein the computing device (2) is further configured to generate a solution to the problem using the determined assignment of the at least one variable of the problem.

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