Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation

Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order $O(M^{-3/2})$, up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where $M$ denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple $(F_h, |r_h\rangle, \langle l_h|)$ can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from $O(\log M)$ to $O((\log M)^2)$, depending on the stage of the pipeline.