From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms

• URL: http://arxiv.org/abs/2507.10285v1
• Date: Mon, 14 Jul 2025 13:51:38 GMT
• Title: From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms
• Authors: Xiantao Li,
• Abstract summary: We present a universal moment-fulfilling dilation that embeds any linear, non-Hermitian flow into a strictly unitary evolution.<n>We also unveil whole families of new dilations built from differential, integral, pseudo-differential, and difference generators.<n>As concrete demonstrations, we prove that a simple finite-difference dilation in a finite interval attains near-optimal oracle complexity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum speed-ups for dynamical simulation usually demand unitary time-evolution, whereas the large ODE/PDE systems encountered in realistic physical models are generically non-unitary. We present a universal moment-fulfilling dilation that embeds any linear, non-Hermitian flow $\dot x = A x$ with $A=-iH+K$ into a strictly unitary evolution on an enlarged Hilbert space: \[ ( (l| \otimes I ) \mathcal T e^{-i \int ( I_A\otimes H +i F\otimes K) dt} ( |r) \otimes I ) = \mathcal T e^{\int A dt}, \] provided the triple $( F, (l|, |r) )$ satisfies the compact moment identities $(l| F^{k}|r) =1$ for all $k\ge 0$ in the ancilla space. This algebraic criterion recovers both \emph{Schr\"odingerization} [Phys. Rev. Lett. 133, 230602 (2024)] and the linear-combination-of-Hamiltonians (LCHS) scheme [Phys. Rev. Lett. 131, 150603 (2023)], while also unveiling whole families of new dilations built from differential, integral, pseudo-differential, and difference generators. Each family comes with a continuous tuning parameter \emph{and} is closed under similarity transformations that leave the moments invariant, giving rise to an overwhelming landscape of design space that allows quantum dilations to be co-optimized for specific applications, algorithms, and hardware. As concrete demonstrations, we prove that a simple finite-difference dilation in a finite interval attains near-optimal oracle complexity. Numerical experiments on Maxwell viscoelastic wave propagation confirm the accuracy and robustness of the approach.

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