Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation

• URL: http://arxiv.org/abs/2509.16777v2
• Date: Fri, 03 Oct 2025 08:39:39 GMT
• Title: Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation
• Authors: Seonggeun Park,
• Abstract summary: We present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions.<n>On the analytical side, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation.<n>We prove that the resulting scheme achieves a global error bound of order $O(M-3/2)$, up to exponentially small boundary effects.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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