Quantum algorithm for linear matrix equations

We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in control theory and physics. Our approach constructs the solution matrix X/s in a block-encoding, where s>0 is a rescaling factor needed for normalization. This allows one to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. We discuss potential applications of our approach and comment on open problems.