Financial combination optimization solution method and device based on quantum binary search

Abstract

The invention discloses a financial combination optimization solution method and device based on quantum binary search, and belongs to the technical field of quantum computing. According to the method, a financial combination problem proposed by a user is converted into a constraint equation, feasible solutions containing a special solution and a homogeneous solution base are obtained through solving, transition operators are constructed according to the homogeneous solution base, a series of transition operators act on an initial quantum state, quantum superposition states of all the feasible solutions are prepared on a quantum register, generation of infeasible solutions is avoided, and the quantum superposition states of all the feasible solutions are obtained. Complex penalty term design and parameter adjustment are avoided so as to maintain constraint conditions; according to the method, target function threshold judgment is performed in a feasible solution superposition state, a solution interval is gradually reduced in combination with a binary search strategy until an optimal solution of a financial combination problem proposed by a user is found, dependence on multiple quantum black boxes is reduced, the circuit structure is flexible, different hardware platforms are adapted, high efficiency can be kept when the feasible solution is sparse, and the method is suitable for large-scale popularization and application. And finally, on the premise of strictly satisfying the constraint, the search complexity is remarkably reduced, and the measurement success rate of the optimal solution is improved.

Description

Technical Field

[0001] This invention relates to the field of quantum computing technology, specifically to a method and apparatus for solving financial portfolio optimization problems based on quantum binary search. Background Technology

[0002] Constrained binary optimization problems are a key research area in computer science, with applications in fields such as artificial intelligence, logic, supply chain management, and theoretical computer science. The variables in these problems take values ​​of 0 or 1, and the solution must simultaneously satisfy a set of linear constraints. Finding the optimal solution to the objective function while satisfying these constraints is often NP-hard, as seen in problems like graph coloring, maximum cut, and the traveling salesman problem. Classical exact algorithms (such as branch and bound) exhibit exponentially increasing complexity as the problem size increases, far exceeding the computational resources of classical computers, making them inefficient for classical computers and unsuitable for large-scale scenarios.

[0003] Quantum computers have demonstrated enormous potential and advantages in solving integer factorization and searching unordered databases. For example, Shor's algorithm can solve integer factorization problems that are difficult for classical computers in polynomial time, and Grover's algorithm is one of the best unordered data search algorithms to date, showing a quadratic speedup. In 2019, Google performed random circuit sampling on its superconducting quantum chip and claimed to have achieved quantum supremacy. Thanks to the powerful parallelism and storage capabilities of quantum computing, quantum computers have a "quantum advantage" in solving these problems.

[0004] With the development of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) iteratively optimizes the quantum state distribution through parameterized quantum circuits to increase the probability of obtaining the optimal solution through measurement. However, the main drawback of QAOA is that the constraints are usually maintained by introducing penalty terms into the objective function or by using complex Hamiltonian encoding, making it difficult to guarantee that all solutions are strictly feasible in practice. Furthermore, IBM's Grover Adaptive Search (GAS) algorithm constructs oracle operators to identify and amplify solutions that satisfy the conditions, gradually approaching the optimal solution. However, this requires designing additional oracle operators for each constraint, and the complexity increases significantly with the number of constraints.

[0005] Focusing on the financial sector, constrained binary optimization problems are widely used in portfolio optimization, risk management, and pricing. These applications typically encounter extremely complex constraints, such as capital constraints and liquidity constraints, making the solution process even more difficult. Since feasible solutions constitute only a very small proportion of many practical problems—for example, in the 12-bit set covering problem, there are only 72 feasible solutions while the total solution space is as large as 4096—most existing algorithms inefficiently search within exponentially large solution spaces or rely on complex constraint construction.

[0006] Therefore, a new quantum method is needed that can strictly guarantee the feasibility of constraints and converge efficiently under the condition of sparse feasible solutions, so as to effectively solve NP-hard optimization problems and alleviate the complex constraints faced by quantum computing in the financial field. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides a method and apparatus for solving financial portfolio optimization based on quantum binary search. For complex constraints in financial portfolio optimization, the method performs objective function threshold judgment in the superposition of feasible solutions, and gradually narrows the solution interval using a binary search strategy until the optimal solution with complex constraints is found. This avoids complex penalty term design, reduces reliance on multiple oracles, and maintains high efficiency even when feasible solutions are sparse. Ultimately, under the premise of strictly satisfied constraints, the search complexity is significantly reduced, and the success rate of optimal solution measurement is improved.

[0008] To achieve the above-mentioned objectives, an embodiment provides a method for solving financial portfolio optimization based on quantum binary search, comprising the following steps: Step 1: Transform the financial portfolio problem proposed by the user into constraint equations, solve them to obtain feasible solutions containing particular solutions and homogeneous solution basis, construct transition operators based on homogeneous solution basis, apply a series of transition operators to the initial quantum state, and prepare the quantum superposition state of all feasible solutions on the quantum register; Step 2: Within the current search interval of the objective function value, calculate the midpoint of the current search interval as a threshold. By comparing the quantum black box, determine whether there is a feasible solution in the quantum superposition state with an objective function value better than the threshold. Update the search interval based on the determination result. Step 3: Iteratively update the search interval until it converges to a single objective function value, and approximate the optimal solution to the user's proposed financial portfolio problem by measuring the quantum register.

[0009] In one embodiment, when the financial portfolio problem proposed by the user is an unconstrained binary optimization problem, a quantum superposition state of all feasible solutions is prepared by applying a series of Hadamard gates on all qubits.

[0010] In one embodiment, the transition operator constructed in step 1 is used to switch the quantum register that satisfies... Two feasible solutions ground state and This enables the transformation from a single ground state to a uniform superposition state in a two-dimensional subspace.

[0011] In one embodiment, constructing the transition operator based on the homogeneous basis solution includes: Identify the support set of the homogeneous solution basis, where the support set represents the set of positions of the non-zero components in the homogeneous solution basis; Construct a preprocessing quantum circuit to encode any pair of quantum states on the support set that satisfy the difference relation into a standard form in which only the state of the first quantum bit is different and the rest of the quantum bits are in the specified computational ground state; Construct a core transformation quantum circuit that operates in the standard form and performs a superposition state generation operation controlled by the remaining qubits on the first qubit. Construct a post-processing quantum circuit, which is the inverse operation of the pre-processing quantum circuit; The transition operator is constructed by cascading the preprocessing circuit, the core transformation circuit, and the postprocessing circuit.

[0012] In one embodiment, the preprocessing quantum circuit includes: A series of CNOT gates are used to fan out the state of the first qubit in the support set to the other qubits in the support set; Based on the sign of each component of the homogeneous basis vector, for each qubit in the support set except the first qubit, if the sign of its corresponding homogeneous basis component is the same as the sign of the component corresponding to the first qubit, then a Pauli X gate is applied; if the signs are opposite, then no gate is applied.

[0013] In one embodiment, the core transformation quantum circuit includes a multi-control quantum gate for performing a multi-control Hadamard transformation on a target qubit conditioned on a control qubit and a single-bit phase rotation gate on the first qubit.

[0014] In one embodiment, when determining whether there is a feasible solution in the quantum superposition state with an objective function value better than the threshold, if the judgment result of the quantum black box is that there is, but the measurement probability of the corresponding quantum state is lower than a preset value, then the quantum state marked by the quantum black box with an objective function value better than the threshold is amplified in amplitude, and then the search interval is updated according to the judgment result.

[0015] In one embodiment, updating the search interval based on the judgment result includes: If a feasible solution exists where the objective function value is better than the threshold, then the lower bound of the search interval is updated to the threshold; otherwise, the upper bound of the search interval is updated to the threshold.

[0016] The present invention also provides a financial portfolio optimization solution device based on quantum binary search, including a memory and a processor. The memory is used to store a computer program, and the processor is used to implement the financial portfolio optimization solution method based on quantum binary search when the computer program is executed.

[0017] Compared with the prior art, the beneficial effects of the present invention include at least the following: This invention addresses the problem of complex constraints in financial portfolio optimization by providing a method and apparatus for solving financial portfolio optimization based on quantum binary search. By constructing transition operators, it achieves the transformation from a single ground state to a uniform superposition state in a two-dimensional subspace, avoiding the generation of infeasible solutions and maintaining the constraints without relying on penalty term parameter tuning. The objective function threshold is determined in the superposition state of feasible solutions, and the solution interval is gradually narrowed using a binary search strategy until the optimal solution with complex constraints is found. This avoids complex penalty term design, reduces reliance on multiple quantum black-box operators, and features a flexible circuit structure adaptable to different hardware platforms. It also maintains high efficiency even when feasible solutions are sparse. Ultimately, under the premise of strictly satisfied constraints, it significantly reduces search complexity and improves the success rate of optimal solution measurement, ultimately approximating the optimal solution to the user-proposed financial portfolio problem. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.

[0019] Figure 1 This is a flowchart illustrating the financial portfolio optimization solution method based on quantum binary search provided in an embodiment of the present invention.

[0020] Figure 2 This is a schematic diagram of the preparation process of the approximately uniform superposition state of the feasible solution provided in the embodiment; Figure 3 This is a schematic diagram of the overall process of quantum binary search provided in the embodiment. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of this invention clearer, the following description is provided in conjunction with the accompanying drawings and... The embodiments further illustrate the present invention in detail. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of protection of the invention.

[0022] To address the complex constraints in financial portfolio optimization, this paper proposes a quantum binary search-based solution method. Its core innovation lies in preparing a superposition of feasible solutions as the initial state, and combining this with a binary search mechanism and a comparative quantum black box (oracle) to efficiently find the optimal solution. This method rigorously guarantees constraint feasibility and achieves efficient convergence under conditions of sparse feasible solutions, thus alleviating the complex constraint problems faced by quantum computing in the financial field. Figure 1 As shown, it includes the following steps: S1. Transform the financial portfolio problem proposed by the user into a constraint equation, solve it to obtain a feasible solution containing a particular solution and a homogeneous solution basis, construct a transition operator based on the homogeneous solution basis, apply a series of transition operators to the initial quantum state, and prepare the quantum superposition state of all feasible solutions on the quantum register.

[0023] When the user-proposed financial portfolio problem is an unconstrained binary optimization problem, a series of Hadamard gates is used to directly prepare a uniform superposition state. For a constrained binary optimization problem, a constraint matrix is ​​constructed, and Gaussian elimination is used to obtain particular and homogeneous solutions. A superposition state of all feasible solutions is then prepared, ensuring that the evolution always remains within the feasible subspace. The details are as follows: 1. Solve the constraint equations: Solve the constraint equations Complete solution set: , in, For all solutions, For the first One solution. For index variables; Any feasible solution can be written as: , in, For a special solution, Let the dimension of the homogeneous solution space be . Let be the basis vector of the i-th homogeneous solution. These are the coefficients of the linear combination.

[0024] Therefore, all feasible solutions are exactly covered by a combination of particular solutions and homogeneous solution bases.

[0025] 2. Construct the transition operator: To realize the reversible mapping from one feasible solution to another in quantum circuits, this invention proposes the definition of a transition operator: For each homogeneous solution basis vector Construct the corresponding transition Hamiltonian In the implementation of the gate circuit, the homogeneous solution basis vectors are first determined. Support set and its size Only in The operation is applied to each relevant qubit. A series of CNOT gates fan out the value of the first qubit in the support set to the other qubits in the support set. Based on the sign of each component of the homogeneous solution basis vector, for each qubit in the support set other than the first qubit, if the sign of its corresponding homogeneous solution basis component is the same as the sign of the component corresponding to the first qubit, a Pauli X gate is applied; if the signs are opposite, no gate is applied. Thus, the encoding of the two corresponding feasible states on the support is uniformly reduced to a pair of standard ground states that differ only in the first and second digits (such as |01…1> and |11…1>).

[0026] Subsequently, with the tail Given a control qubit, a multi-controlled Hadamard rotation is applied to the target qubit, and this is achieved only within this two-dimensional subspace. The linear transformation is then performed, and finally the aforementioned CNOT gate and Pauli X gate are removed in reverse order, thus completing a transformation from the previous state while keeping other ground states unchanged. arrive Controlled transitions constitute transition operators.

[0027] The entire process involves only bit-nontrivial actions on the support set; the required number of two-bit gates and circuit depth vary with the size of the support set. Linear or near-linear growth makes it suitable for efficient implementation on superconducting quantum chips.

[0028] The constructed transition operator functions when the following conditions are met: At that time, the feasible solution state will be Evolves into another feasible solution state :

[0029] transition Hamiltonian by Hamiltonian simulation can achieve the following: , Sequential action The corresponding Hamiltonian simulation can expand the quantum space to the entire solution space. Furthermore, by introducing appropriate phase adjustments during the evolution, the quantum space can be expanded to include the entire solution space. Components apply additional Gate), denoted as transition operator : .

[0030] By repeatedly using the transition operator, the measurement probabilities of the evolving feasible solutions can be gradually brought to equilibrium. This increases the number of solutions in quantum space from 2... n Reduced to only containing 2 feasible terminations n / 2 .

[0031] like Figure 2The diagram illustrates the preparation and probabilistic evolution of the superposition of feasible solutions. A tree structure describes the expansion of the initial quantum state; different paths correspond to feasible solutions in the solution space, and their initial probability distributions differ significantly.

[0032] The diagram uses (a, b, c, d) to represent the ground state of the quantum evolution space. At the start of the first round, with 'a' as the starting point, measurement yields 'a' with a 100% probability. In evolution 1-1, 'a' evolves to 'b', with the probability split evenly, resulting in 50% probability of obtaining 'a' and 'b' respectively. In evolution 1-2, 'b' evolves to 'c', with the probability split evenly again, resulting in 50%, 25%, and 25% probability of obtaining 'a', 'b', and 'c' respectively. In evolution 1-3, 'c' evolves to 'd', with the probability split evenly, ultimately yielding 50%, 25%, 12.5%, and 12.5% ​​probabilities corresponding to 'a', 'b', 'c', and 'd' respectively.

[0033] The second and third rounds repeat this process. In each round, the probabilities of a and b are summed and evenly divided, the probabilities of b and c are summed and evenly divided, and the probabilities of c and d are summed and evenly divided. With the multiple iterations of the transition operator, the probabilities are gradually smoothed out among feasible solutions.

[0034]

[0035] Table 1 and Figure 2 The probability changes after the first to third iterations were recorded. It can be seen that the distribution of each ground state gradually approaches equilibrium, avoiding exponential decay of the minimum probability. This process demonstrates that through iterative correction, a uniform superposition state can be approximated from the biased initial state, providing a stable and reliable initial quantum state for subsequent quantum binary search.

[0036] S2. Within the current search interval of the objective function value, calculate the midpoint of the current search interval as a threshold. By comparing the quantum black box, determine whether there is a feasible solution in the quantum superposition state with an objective function value better than the threshold. Update the search interval based on the determination result.

[0037] In this embodiment, an initial search interval for the objective function is set. In each iteration, the midpoint is taken. By constructing an oracle, it is determined whether there exists an objective function value greater than a threshold in the superposition of feasible solutions. A feasible solution. If it exists, apply a Pauli X-gate to the auxiliary bit; when the result is 1, update the interval as follows. If it does not exist, then update the interval as follows: .

[0038] Furthermore, when determining whether there exists a target function value better than a threshold in a quantum superposition state... When considering feasible solutions, if the quantum black box's judgment indicates existence, but the measurement probability of the corresponding quantum state is lower than a preset value, then the objective function value marked by the quantum black box is better than the threshold. The quantum state is amplified to enhance its visibility in measurement, thereby improving the accuracy of judgment.

[0039] like Figure 3 The overall framework of the combinatorial optimization algorithm based on quantum binary search is shown below: Initialization: A uniform superposition state of feasible solutions is prepared by using transition operators and stored in a quantum register.

[0040] Objective function value encoding: The variables and objective function of the financial portfolio problem proposed by the user are encoded as quantum states, corresponding in a self-dictionary form.

[0041] Oracle module: Determine if the objective function value is greater than a threshold. The feasible solution is obtained, and the amplitude of the quantum state of the potential optimal solution is multiplied by -1 when satisfied, which is used as a marker for amplitude amplification.

[0042] Amplification: When a solution exists whose objective function value is greater than the threshold, its probability is amplified through Grover-type operations.

[0043] Binary search decision: Adjust the search interval based on the measurement results. If a better solution exists, increase the threshold; otherwise, decrease the threshold.

[0044] S3. By iteratively updating the search interval until the search interval converges to a single objective function value, the optimal solution to the financial portfolio problem proposed by the user is approximated by measuring the quantum register.

[0045] In the embodiments, such as Figure 3 As shown, after multiple rounds of binary search iterations, the search interval converges to a single objective function value, and the optimal solution to the user's proposed financial portfolio problem is approximated by measuring the quantum register.

[0046] The embodiment also provides a financial portfolio optimization solution device based on quantum binary search, including a memory and a processor. The memory is used to store a computer program, and the processor is used to implement the financial portfolio optimization solution method based on quantum binary search when the computer program is executed.

[0047] To verify the effectiveness of the proposed quantum binary search-based financial portfolio optimization solution method, it was compared with traditional methods and showed a significant complexity advantage: the number of search rounds is [missing information]. , The upper bound is the range of the objective function values. For example, if the upper bound of the objective function is 64, the optimal solution can be obtained in only 6 rounds of search. GAS is based on Grover's algorithm, with a complexity of O(n log n). ,in This method expands the search scope from... A solution, limited to This achieves a second-order acceleration. If n=12 and N=4096, there are only 72 feasible solutions. This method requires only about 6 rounds, while GAS expects about 7.5 iterations, resulting in a higher actual cost. As the scale increases, the relative advantage of this method becomes more obvious.

[0048] Strict feasibility: Through transition operators, the initial state is guaranteed to be an approximately uniform superposition of feasible solutions, avoiding the generation of infeasible solutions and not relying on penalty term parameter tuning. Compared with QAOA and GAS, this method naturally preserves the constraints.

[0049] Scalability and robustness: This invention is applicable to both constrained and unconstrained problems. The oracle can be implemented using phase estimation or a quantum dictionary, offering flexible circuit structure adaptability to different hardware platforms. For sparse feasible solutions, robustness can be further improved through probability smoothing and parallel preparation of multiple initial states.

[0050] In summary, this invention demonstrates significant advantages in terms of theoretical complexity, constraint handling, and hardware implementation.

[0051] The embodiments also included experimental verification in the following two application scenarios.

[0052] 1. The issue of assignment between the two departments Taking the bipartite assignment problem as an example, the input is the assignment constraints of a bipartite graph, which, in the form of inequality constraints, only contain the assignment variable. The kernel space basis of the assignment constraint is obtained through Gaussian elimination, and an approximately uniform superposition state is prepared over all perfect matches using transition operators. Subsequently, a binary search of the target cost is combined with a comparison with the quantum oracle, within the target value range... In this case, only The optimal assignment solution can be converged in just one iteration; at the same time, since no slack variables are introduced, the feasible solution is... The relatively high proportion of the middle part is beneficial to improving the convergence efficiency of quantum search.

[0053] 2. Site selection issues for non-capacity facilities Taking the non-capacity facility location problem as an example, the input consists of a combination of customer facility assignment constraints and inequality forms. Under the inequality constraint model, Gaussian elimination is performed on the constraint matrix to obtain a set of integer basis vectors describing "facility switching + customer reassignment". Based on this, a transition operator is constructed to extend any feasible solution into an approximately uniform superposition of all solutions that satisfy "each customer has one and only one open facility service".

[0054] Then, a binary search is performed on the total cost. In each round, a quantum oracle is used to determine whether there is a feasible solution with a cost not exceeding a threshold. Finally, in... The optimal facility location is found through iteration at each level; using only Under the inequality form, the proportion of feasible solutions is significantly higher than that under the equality form which introduces a large number of slack variables, thereby reducing amplitude dilution and improving search stability.

[0055] Therefore, for the problem of complex constraints in financial portfolio optimization, the threshold judgment of the objective function is performed in the superposition of feasible solutions. The solution interval is gradually narrowed by combining a binary search strategy until the optimal solution with complex constraints is found. This avoids the design of complex penalty terms, reduces the dependence on multiple oracles, and can maintain high efficiency when feasible solutions are sparse. Finally, under the premise of strict constraint satisfaction, the search complexity is significantly reduced and the success rate of optimal solution measurement is improved.

[0056] The specific embodiments described above illustrate the technical solution and beneficial effects of the present invention in detail. It should be understood that the above description is only the most preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for solving financial portfolio optimization based on quantum binary search, characterized in that, Includes the following steps: Step 1: Transform the financial portfolio problem proposed by the user into constraint equations, solve them to obtain feasible solutions containing particular solutions and homogeneous solution basis, construct transition operators based on homogeneous solution basis, apply a series of transition operators to the initial quantum state, and prepare the quantum superposition state of all feasible solutions on the quantum register; Step 2: Within the current search interval of the objective function value, calculate the midpoint of the current search interval as a threshold. By comparing the quantum black box, determine whether there is a feasible solution in the quantum superposition state with an objective function value better than the threshold. Update the search interval based on the determination result. Step 3: Iteratively update the search interval until it converges to a single objective function value, and approximate the optimal solution to the user's proposed financial portfolio problem by measuring the quantum register.

2. The financial portfolio optimization solution method based on quantum binary search according to claim 1, characterized in that, When the financial portfolio problem proposed by the user is an unconstrained binary optimization problem, a quantum superposition state of all feasible solutions is prepared by applying a series of Hadamard gates on all qubits.

3. The financial portfolio optimization solution method based on quantum binary search according to claim 1, characterized in that, The transition operator constructed in step 1 is used to switch the quantum register that satisfies... Two feasible solutions ground state and This enables the transformation from a single ground state to a uniform superposition state in a two-dimensional subspace.

4. The financial portfolio optimization solution method based on quantum binary search according to claim 3, characterized in that, The construction of transition operators based on homogeneous solution basis includes: Identify the support set of the homogeneous solution basis, where the support set represents the set of positions of the non-zero components in the homogeneous solution basis; Construct a preprocessing quantum circuit to encode any pair of quantum states on the support set that satisfy the difference relation into a standard form in which only the state of the first quantum bit is different and the rest of the quantum bits are in the specified computational ground state; Construct a core transformation quantum circuit that operates in the standard form and performs a superposition state generation operation controlled by the remaining qubits on the first qubit. Construct a post-processing quantum circuit, which is the inverse operation of the pre-processing quantum circuit; The transition operator is constructed by cascading the preprocessing circuit, the core transformation circuit, and the postprocessing circuit.

5. The financial portfolio optimization solution method based on quantum binary search according to claim 4, characterized in that, The preprocessing quantum circuit includes: A series of CNOT gates are used to fan out the state of the first qubit in the support set to the other qubits in the support set; Based on the sign of each component of the homogeneous basis vector, for each qubit in the support set except the first qubit, if the sign of its corresponding homogeneous basis component is the same as the sign of the component corresponding to the first qubit, then a Pauli X gate is applied; if the signs are opposite, then no gate is applied.

6. The financial portfolio optimization solution method based on quantum binary search according to claim 4, characterized in that, The core transformation quantum circuit includes a multi-control quantum gate for performing multi-control Hadamard rotation on the target qubit conditioned on the control qubit and a single-bit phase rotation gate on the first qubit.

7. The financial portfolio optimization solution method based on quantum binary search according to claim 1, characterized in that, When determining whether there is a feasible solution in the quantum superposition state with an objective function value better than the threshold, if the judgment result of the quantum black box is that it exists, but the measurement probability of the corresponding quantum state is lower than the preset value, then the quantum state marked by the quantum black box with an objective function value better than the threshold is amplified in amplitude, and the search interval is updated according to the judgment result.

8. The financial portfolio optimization solution method based on quantum binary search according to claim 7, characterized in that, The method of updating the search interval based on the judgment result includes: If a feasible solution exists where the objective function value is better than the threshold, then the lower bound of the search interval is updated to the threshold; otherwise, the upper bound of the search interval is updated to the threshold.

9. A financial portfolio optimization solution device based on quantum binary search, comprising a memory and a processor, wherein the memory is used to store a computer program, characterized in that, The processor is used to implement the financial portfolio optimization solution method based on quantum binary search as described in any one of claims 1 to 8 when executing the computer program.

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