A universal variational quantum eigensolver for non-Hermitian systems

Mathematical foundation

The critical aspect of enabling variational quantum algorithms to calculate the eigenvalues of a general matrix involves identifying a unitary transformation matrix that can distinctly expose the eigenvalues in a similar manner to the eigenvector matrix. In our devised variational quantum universal eigensolver (VQUE), we utilize the mathematical foundation of Schur’s decomposition theory as summarized below to ensure the existence of the solution:

An arbitrary square matrix \({\varvec{A}}\) can be transformed to a triangular matrix \({\varvec{T}}\) as: