Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization
Abstract
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) d Xt = (A(t) Xt + F(t) Xt⊗ s) d t + B(t) d Wt for an ℝN-valued random process Xt, aiming at quantum speed-up with respect to the dimension N. We introduce auxiliary processes Yjt that approximate Xt⊗ j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of Xt by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order J in CL, and we can estimate the target expectation with a polylog(N) number of queries to unitaries encoding coefficient matrices A, F, and B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as J increases.
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BibTeX
@misc{miyamoto2026quantum,
title = {{Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization}},
author = {Koichi Miyamoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27416},
eprint = {2610.27416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27416},
}
RIS
TY - UNPB
TI - Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization
AU - Koichi Miyamoto
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27416
UR - https://arxiv.org/abs/2610.27416
AB - Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) d X_t = (A(t) X_t + F(t) X_t^⊗ s) d t + B(t) d W_t for an ℝ^N-valued random process X_t, aiming at quantum speed-up with respect to the dimension N. We introduce auxiliary processes Y^j_t that approximate X_t^⊗ j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of X_t by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order J in CL, and we can estimate the target expectation with a polylog(N) number of queries to unitaries encoding coefficient matrices A, F, and B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as J increases.
ER -