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. Assuming block-encoding access to the matrices, our approach constructs the solution matrix X/x also in a block-encoding, where x > 0 is a rescaling factor needed for normalization. This allows us to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. The query and gate complexities of the quantum circuit that implements this block-encoding are almost linear in a condition number that depends on A and B, and depend logarithmically in the dimension and inverse error. We show how our quantum circuits can solve BQP-complete problems efficiently, discuss potential applications and extensions of our approach, its connection to Riccati equation, and comment on open problems.
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