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

Xiantao Li

TL;DR

The paper introduces a universal moment-matching dilation that embeds any linear non-Hermitian flow into unitary evolution on an enlarged space, enabling Encode–Evolve–Evaluate implementations that connect and generalize Schrödingerization and LCHS. It develops multiple dilation families, including SBP-based compact-domain dilations and Bargmann–Fock ancillas, each with tunable parameters, and proves near-optimal quantum complexity for the simple finite-difference dilation, complemented by Maxwell viscoelastic numerical benchmarks. The framework provides a flexible, hardware-aware design space for quantum simulation of dissipative and growth-prone dynamics, and it suggests avenues for extending to nonlinear problems via Carleman embeddings. Overall, the work furnishes both exact dilation criteria and practical, segmented quantum algorithms with error-control mechanisms suitable for diverse quantum hardware platforms.

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 $\dot x = L x$ 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.

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

TL;DR

The paper introduces a universal moment-matching dilation that embeds any linear non-Hermitian flow into unitary evolution on an enlarged space, enabling Encode–Evolve–Evaluate implementations that connect and generalize Schrödingerization and LCHS. It develops multiple dilation families, including SBP-based compact-domain dilations and Bargmann–Fock ancillas, each with tunable parameters, and proves near-optimal quantum complexity for the simple finite-difference dilation, complemented by Maxwell viscoelastic numerical benchmarks. The framework provides a flexible, hardware-aware design space for quantum simulation of dissipative and growth-prone dynamics, and it suggests avenues for extending to nonlinear problems via Carleman embeddings. Overall, the work furnishes both exact dilation criteria and practical, segmented quantum algorithms with error-control mechanisms suitable for diverse quantum hardware platforms.

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 with into a strictly unitary evolution on an enlarged Hilbert space: provided the triple satisfies the compact moment identities for all 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.

Paper Structure

This paper contains 20 sections, 18 theorems, 171 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem setup and mathematical framework
  3. A dilation using a function space on a compact domain
  4. Numerical discretizations using SBP
  5. Error Estimation using a Light-cone Property
  6. Geometrically refined grids with an improved light-cone property
  7. Segmentation-wise simulation and oblivious amplitude amplification
  8. Numerical Illustration: 2-D Maxwell Viscoelasticity model
  9. Dilation with a Bargmann--Fock Ancilla
  10. Further Discussions
  11. Implementation by LCU
  12. ODEs with transient or persistent growth
  13. Summary and outlook
  14. Lightcone property of the finite difference operator
  15. Finite–Propagation Property of the Semi-Discrete Scheme
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\dot{\bm x}(t)=(-iH(t)+K(t))\bm x(t)$ with $H(t)=H(t)^\dagger$ and $K(t)=K(t)^\dagger$ (allowing gain). Using the differential generator $F=p\partial_p+\tfrac12$ and a moment-matching ancilla triple $(F,\ket{r},\bra{l})$, we construct an exact unitary dilation $\widetilde{H}(t)=I_A\!\otimes H(t

Figures (7)

  • Figure 1: Snapshots of the semi-discrete solution for $\bm v' = -\theta F_h \bm v$ with $v_j(0) \propto p_j^{1/\theta - 1/2}$.
  • Figure 2: Circuit diagram for the segmented simulation. The algorithm proceeds in time steps of duration $\tau$. In each segment, the joint evolution $U_k = \mathcal{T} \exp(-i \int \widetilde{H}ds )$ is applied. The system state evolves from $|x(t)\rangle$ to $|x(t+\tau)\rangle$ (conditioned on success), while the ancilla is restored to the reference state $|r_h\rangle$ via Oblivious Amplitude Amplification (OAA) before the next segment begins.
  • Figure 3: Snapshot of the pressure field $p=-K(\epsilon-\gamma)$ in a $2\times2$ periodic domain.
  • Figure 4: Pressure time series at $(x,y)=(1/4,1/4)$ for the Maxwell viscoelastic wave. Colored curves show the dilated dynamics for varying readout points $p_\ast$. Smaller $p_\ast$ tracks the reference for longer times, while larger $p_\ast$ leads to earlier deviations, illustrating the boundary light-cone effect.
  • Figure 5: Edge-density dynamics for the $\mathcal{P}T$-SSH dimer: simulation using CV dilation (dots) vs. exact results using matrix exponential (solid).
  • ...and 2 more figures

Theorems & Definitions (34)

  • Theorem 1: Informal main result
  • Theorem 2: Moment-fulfilling and Exact Dilation
  • proof
  • Corollary 1
  • Example 1: Schrödingerization via warped–phase transformation PhysRevLett.133.230602
  • Example 2: Linear combination Hamiltonian simulations (LCHS) ALL23
  • Example 3: Integral–kernel dilation
  • Example 4: Dilation via a pseudodifferential generator
  • Example 5: Dilations using difference operators
  • Lemma 1
  • ...and 24 more