Quantum algorithm for linear matrix equations

• URL: http://arxiv.org/abs/2508.02822v1
• Date: Mon, 04 Aug 2025 18:45:05 GMT
• Title: Quantum algorithm for linear matrix equations
• Authors: Rolando D. Somma, Guang Hao Low, Dominic W. Berry, Ryan Babbush,
• Abstract summary: We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C.<n>We show how our quantum circuits can solve BQP-complete problems efficiently.
• Score: 0.025206105035672277
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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. Rather than encoding the solution in a quantum state in a fashion analogous to prior quantum linear algebra solvers, our approach constructs the solution matrix X in a block-encoding, rescaled by some factor. 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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