Quantum

Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization

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Briefings

A new quantum algorithm using Carleman linearization solves high-dimensional nonlinear stochastic differential equations by approximating nonlinear terms with auxiliary processes. The method achieves exponential error decay with truncation order and requires polylogarithmic queries for unitaries encoding system coefficients, offering a potential speedup over classical approaches for specific SDEs.

Key points

• The algorithm uses Carleman linearization to approximate nonlinear terms in a high-dimensional SDE by introducing auxiliary processes.
• Quantum ODE solvers are applied to the linearized system to estimate expectations of linear functions of the original process.

Context

This work addresses a long-standing challenge in quantum computing: constructing algorithms for high-dimensional nonlinear stochastic differential equations. The method provides a pathway to quantum speedup by leveraging the linearity of the linearized system derived from Carleman linearization.

Limitations

The study is limited to SDEs where nonlinear and noise terms are sufficiently small relative to the linear term. The exponential decay of the CL error is observed numerically for a specific Langevin dynamics example rather than proven universally. The polylogarithmic query complexity applies only under the stated smallness conditions.

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