Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

💡 Research Summary

This paper introduces a groundbreaking advancement in quantum algorithms for solving systems of linear ordinary differential equations (ODEs). The authors address the general problem of simulating dynamics governed by du(t)/dt = -A(t)u(t) + b(t), where A(t) can be a non-Hermitian matrix, making the evolution non-unitary.

The core innovation is a profound generalization of the recently proposed Linear Combination of Hamiltonian Simulation (LCHS) method. The original LCHS approach decomposed A(t) into its Hermitian part L(t) (assumed positive semi-definite) and anti-Hermitian part H(t), expressing the evolution operator as an integral: T exp(-∫A) = ∫