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Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments

Hsuan-Cheng Wu, Xiantao Li

TL;DR

This work presents an exact unitary dilation that maps general linear Itô SDEs to stochastic Schrödinger equations on an enlarged Hilbert space, enabling native quantum simulation of both ensemble moments and trajectory realizations. By constructing a finite-dimensional tight-binding dilation with moment-matching properties, the authors prove a stochastic light-cone bound that controls ancilla resources and error propagation, and they develop two complementary algorithms: a Lindblad-based second-moment simulator and a trajectory-based weak-scheme simulator. The framework supports long-time simulation via segmentation with ancilla refresh and provides explicit constructions for the dilated Hamiltonian and coupling operators, with rigorous error analyses and numerical validations. This approach opens a pathway to quantum-accelerated stochastic simulation, filtering, and sampling for high-dimensional linear SDEs and related open-quantum-system dynamics, with potential extensions to non-Brownian noise and non-Markovian settings.

Abstract

We present a universal framework for simulating $N$-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous correspondence between the general linear SDE \[ dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j \] and a Stochastic Schrödinger Equation (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact: the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSE is {naturally implementable} on digital quantum processors, where the stochastic Wiener increments correspond directly to measurement outcomes of ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical simulations.

Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments

TL;DR

This work presents an exact unitary dilation that maps general linear Itô SDEs to stochastic Schrödinger equations on an enlarged Hilbert space, enabling native quantum simulation of both ensemble moments and trajectory realizations. By constructing a finite-dimensional tight-binding dilation with moment-matching properties, the authors prove a stochastic light-cone bound that controls ancilla resources and error propagation, and they develop two complementary algorithms: a Lindblad-based second-moment simulator and a trajectory-based weak-scheme simulator. The framework supports long-time simulation via segmentation with ancilla refresh and provides explicit constructions for the dilated Hamiltonian and coupling operators, with rigorous error analyses and numerical validations. This approach opens a pathway to quantum-accelerated stochastic simulation, filtering, and sampling for high-dimensional linear SDEs and related open-quantum-system dynamics, with potential extensions to non-Brownian noise and non-Markovian settings.

Abstract

We present a universal framework for simulating -dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous correspondence between the general linear SDE and a Stochastic Schrödinger Equation (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact: the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSE is {naturally implementable} on digital quantum processors, where the stochastic Wiener increments correspond directly to measurement outcomes of ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical simulations.
Paper Structure (33 sections, 11 theorems, 168 equations, 4 figures)

This paper contains 33 sections, 11 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setup: linear Itô SDEs
  3. Itô SDEs versus stochastic Schrödinger equations
  4. A dilation viewpoint: compiling linear SDEs into open quantum dynamics
  5. Our Contribution: Quantum Simulation via Moment-Matching Dilation
  6. Algorithm I: second-moment weak simulation.
  7. Algorithm II: pathwise trajectory simulation.
  8. Related works.
  9. Dilation of Stochastic Differential Equations
  10. Preliminaries: moment-matching dilation for ODEs
  11. The exact-mapping theorem for SDEs \ref{['eq:SDE0']}
  12. Finite-dimensional tight-binding dilations for SDE systems
  13. Error Analysis and Stochastic Light-Cone Property
  14. Implementation via Lindblad Simulation Algorithms
  15. The Dilated Master Equation
  16. ...and 18 more sections

Key Result

Theorem 1

Let $C_{\mathcal{L},T}$ denote the cost of simulating the dilated Lindblad dynamics for total time $T$ (including segmentation and refresh), and let $K_{\max}=\sup_t\|K(t)\|$ so that the number of segments satisfies $L=\Theta(TK_{\max})$. Let $\Lambda_T:=\tr(\Sigma_T)$. For segment $m$, define the s Then one can estimate $\mu=\tr(\Sigma_T O)$ to additive error $\varepsilon$ using total cost scalin

Figures (4)

  • Figure 1: Moment-matching dilation for \ref{['eq:example']} with $\sigma=1$. Both panels compare trajectories from \ref{['eq: example_weak']} with the projected dilated SSE \ref{['eq:dilated-SSE']}, driven by the same Brownian motion. Choosing an evaluation site closer to the origin improves the accuracy and extends the time horizon over which the projected dynamics remain reliable. In particular, the approximation deteriorates earlier for $p_*=0.4$ (\ref{['fig:p04']}) than for $p_*=0.1$ (\ref{['fig:p01']}).
  • Figure 2: Weak error $\mathrm{err}(\Delta t) = \bigl|\mathbb{E}[f(Y_{T,{\Delta t}}) - \mathbb{E}[f(X_T)]]\bigr|$ for the second-order scheme applied to \ref{['eq: example_weak']} with $f(\bm x)=\cos(x_1+x_2+x_3^2)$ at $T=1.0$. The sample size is $N_{\rm samp}=500000$. In the dilation on the geometric grid \ref{['eq: fj']}, we use M=500, and $h=2$.
  • Figure 3: Recovery of quadratic statistics for the SPDE \ref{['eq:example_Magnus_Linblad']}. Parameters are $\varepsilon =0.1$, $\beta=0.5$, $\sigma_1=0.5$, $\sigma_2=0.3$, and $T=1.0$. The evaluation site is $p_* = 5\times 10^{-6}$. The recovered second moment closely matches the reference solution over the simulated time interval.
  • Figure 4: Pointwise comparison of the second moment obtained from the second-moment equation and from the dilation-based recovery. Here $p_* = 5\times 10^{-6}$. The agreement improves as $p_*$ decreases, consistent with the localization effect of the projection.

Theorems & Definitions (15)

  • Theorem : Informal complexity: Algorithm I (Lindblad / second moments)
  • Theorem : Informal complexity: Algorithm II (trajectory generation)
  • Definition 1: Moment-matching dilation
  • Theorem 1: Deterministic moment-matching dilation
  • Proposition 1: Mean-Square Stability and Growth
  • Theorem 2: Exact recovery of the linear SDE
  • Theorem 3: Stochastic Light-Cone
  • Lemma 1: Lindblad equation for the dilated second moment
  • Proposition 2: Light-cone for the covariance
  • Theorem 4: Segment-wise Lindblad complexity
  • ...and 5 more