From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917158838075392 |
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| author | Li, Xiantao |
| author_facet | Li, Xiantao |
| contents | 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 $\dot x = L x$ with $L=-iH+K$ into a strictly unitary evolution on an enlarged Hilbert space: \[
( (l| \otimes I )
\mathcal T e^{-i \int ( I_A\otimes H +i F\otimes K) dt}
( |r) \otimes I )
= \mathcal T e^{\int L dt}, \] 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{Schrödingerization} [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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms Li, Xiantao Quantum Physics 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 $\dot x = L x$ with $L=-iH+K$ into a strictly unitary evolution on an enlarged Hilbert space: \[ ( (l| \otimes I ) \mathcal T e^{-i \int ( I_A\otimes H +i F\otimes K) dt} ( |r) \otimes I ) = \mathcal T e^{\int L dt}, \] 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{Schrödingerization} [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. |
| title | From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2507.10285 |