Quantum Circuit for Binary Linear Equations Modulo 2
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing quantum algorithms for solving linear systems, such as VQLS and HHL, are limited to real and complex-valued matrices and have not been implemented to deal with binary-valued matrices, where coefficients and solution vectors are either 0s and 1s.
Innovation Solution
A quantum circuit design is developed that implements a matrix-vector product using modulo 2 arithmetic and a variational quantum algorithm with a brickwork layout ansatz to solve binary-valued linear equations. This design includes a quantum circuit with m+n qubits, where m and n are the dimensions of the binary coefficient matrix, and a parameterized component to optimize the solution using a cost function that applies penalties to computational basis states.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If conventional quantum algorithms (VQLS, HHL) are used to solve linear systems, then the solution can be obtained for real and complex-valued matrices, but these algorithms cannot handle binary-valued matrices where coefficients and solution vectors are 0s and 1s
Solution Approach 1:
The patent changes the parameter domain from real/complex numbers to binary values (0s and 1s) by implementing matrix-vector multiplication using modulo 2 arithmetic. This is achieved through quantum circuits that perform bitwise operations and XOR gates, fundamentally altering how the linear system is represented and solved, thereby enabling applicability to binary-valued matrices while maintaining solution accuracy.
Solution Approach 2:
The patent substitutes conventional quantum linear solver mechanisms (VQLS, HHL) with a novel quantum circuit architecture specifically designed for binary arithmetic. This involves replacing continuous-variable quantum operations with discrete quantum gate operations that implement modulo 2 addition and multiplication, creating a specialized solver that trades generality for binary-specific efficiency.
2Productivity
If a quantum circuit implements matrix-vector product with gates proportional to non-zero entries in coefficient matrix, then the computation efficiency is improved, but the circuit depth and gate count increase
Solution Approach 1:
The patent segments the matrix-vector multiplication process into modular quantum circuit components, where each non-zero entry in the coefficient matrix corresponds to a specific quantum gate or gate sequence. This segmentation allows the circuit to process only relevant matrix elements, improving computation efficiency by avoiding unnecessary operations on zero entries while maintaining a structured, manageable circuit architecture.
Solution Approach 2:
The patent implements partial action by designing the quantum circuit to perform operations only on the non-zero entries of the coefficient matrix. Rather than implementing a complete n×n matrix multiplication circuit, the design applies quantum gates selectively to positions where matrix elements are non-zero, thereby reducing the overall gate count and circuit depth proportionally to the sparsity of the matrix.
3Measurement precision
If a variational cost function is derived and optimized to produce a solution, then the accuracy of solving binary linear systems is improved, but the number of measurements and optimization iterations increases
Solution Approach 1:
The patent implements a variational feedback loop where the quantum circuit computes a cost function based on the current parameter settings, measures the output, and feeds this information back to a classical optimizer. The optimizer adjusts the parameters to minimize the cost function, which represents the error between the computed matrix-vector product and the target vector. This iterative feedback process continues until convergence, ensuring high solution accuracy for binary linear systems.
Solution Approach 2:
The patent employs preliminary action by using a brickwork layout ansatz with parameterized gates that are pre-configured to explore the solution space efficiently. This ansatz structure incorporates built-in correlations and entanglement patterns that guide the optimization process toward the correct solution more quickly, reducing the number of iterations needed compared to random parameter initialization.
Data Source
AI summary
A system and method is provided for solving systems of binary-valued linear equations using a quantum information processing (QIP) system. The present disclosure describes solving linear systems modulo 2 on a quantum computer. An exemplary method includes defining a quantum circuit implementing matrix-vector products with a number of gates proportional to the number of non-zero entries in the coefficient matrix and then deriving a variational cost function that may be optimized to produce a solution to the given system. Compared to other quantum linear solvers, the present disclosure may work on matrices of any size and rank.