Quantum Linear System Algorithm for General Matrices in System Identification

Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that [Formula: see text]. Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state [Formula: see text] corresponding to the solution of the linear system of equations in [Formula: see text] poly [Formula: see text] time for a general [Formula: see text] dimensional A, which is superior to existing quantum algorithms, where [Formula: see text] is the condition number, r is the rank of matrix A and [Formula: see text] is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.