[Paper Review] Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling

Pedro C. S. Costa, Philipp Schleich+2 more|arXiv (Cornell University)|Dec 15, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper improves quantum algorithms for nonlinear differential equations by introducing higher-order time integration via truncated Taylor series, rescaling to enhance success probability, and tighter error bounds for Carleman linearisation. The approach achieves near-linear scaling in time and logarithmic dependence on error, enabling efficient quantum solution of nonlinear PDEs like reaction-diffusion equations when combined with high-order spatial discretisation.

ABSTRACT

The solution of large systems of nonlinear differential equations is needed for many applications in science and engineering. In this study, we present three main improvements to existing quantum algorithms based on the Carleman linearisation technique. First, by using a high-precision technique for the solution of the linearised differential equations, we achieve logarithmic dependence of the complexity on the error and near-linear dependence on time. Second, we demonstrate that a rescaling technique can considerably reduce the cost, which would otherwise be exponential in the Carleman order for a system of ODEs, preventing a quantum speedup for PDEs. Third, we provide improved, tighter bounds on the error of Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We show that providing a stability criterion independent of the discretisation can conflict with the use of the rescaling due to the difference between the max-norm and 2-norm. An efficient solution may still be provided if the number of discretisation points is limited, as is possible when using higher-order discretisations.

Motivation & Objective

• To overcome exponential complexity bottlenecks in prior quantum Carleman linearisation algorithms for nonlinear ODEs and PDEs.
• To reduce the number of discretisation points required for stable and accurate solution by employing higher-order finite difference schemes.
• To resolve the conflict between 2-norm rescaling (for high success probability) and max-norm stability criteria (for PDEs) in quantum PDE solvers.
• To provide tighter, analytically derived bounds on the truncation error of Carleman linearisation for improved accuracy control.
• To enable sublinear scaling in the number of discretisation points by ensuring the success probability of amplitude amplification remains high.

Proposed method

• Applying a truncated Taylor series method to solve the linearised ODE system, achieving near-linear dependence on evolution time T.
• Introducing a rescaling technique that transforms the original dynamics to boost the amplitude of the solution component in the Carleman vector, improving success probability.
• Using higher-order finite difference stencils for spatial discretisation of PDEs, reducing the number of grid points needed and improving overall complexity.
• Deriving tighter analytical bounds on the error introduced by truncating the Carleman linearisation process, improving accuracy control.
• Constructing a block-encoding of the Carleman matrix using a linear combination of unitaries, with complexity proportional to the number of basis states in the encoding.
• Analyzing the interplay between stability in the max-norm (PDE requirement), 2-norm rescaling (for quantum efficiency), and ODE solver stability to identify conditions under which the algorithm remains efficient.

Experimental results

Research questions

• RQ1 Can rescaling the original dynamics in Carleman linearisation improve the success probability of amplitude amplification, enabling sublinear scaling in the number of discretisation points?
• RQ2 Does using a truncated Taylor series for time evolution lead to near-linear scaling in the total evolution time T, improving over prior methods?
• RQ3 Can higher-order spatial discretisation reduce the number of grid points required for stability and accuracy, thereby improving quantum algorithm complexity?
• RQ4 What is the precise relationship between the max-norm stability criterion of PDEs and the 2-norm rescaling required for efficient quantum state preparation?
• RQ5 How tight are the error bounds for Carleman linearisation truncation, and can they be analytically derived to ensure accuracy in quantum algorithms?

Key findings

• The truncated Taylor series method for solving the linearised ODE system results in near-linear scaling with respect to the total evolution time T, significantly improving over prior methods.
• Rescaling the original dynamics enables a high success probability for amplitude amplification, which is essential for achieving sublinear scaling in the number of discretisation points.
• Tighter analytical bounds on the Carleman linearisation truncation error are derived, providing improved accuracy control and reducing overestimation of error.
• Higher-order finite difference schemes reduce the number of required grid points, which is critical for maintaining efficiency when combined with rescaling and 2-norm-based amplitude amplification.
• The conflict between max-norm stability (required for PDEs) and 2-norm rescaling (required for quantum efficiency) is identified as a key challenge, but the algorithm remains efficient if the 2-norm stability condition is satisfied.
• The overall quantum algorithm achieves logarithmic dependence on the error and near-linear dependence on time, enabling a practical quantum speedup for nonlinear PDEs under specific conditions.