Constant-Factor Improvements in Quantum Algorithms for Linear Differential Equations

TL;DR

This paper establishes constant-factor bounds for a quantum differential equation solver, significantly reducing the estimated runtime costs and advancing the practical prospects of quantum algorithms for linear differential equations.

Contribution

It provides the first constant-factor bounds for the LCHS quantum solver, improving previous estimates by over two orders of magnitude and enhancing its practical applicability.

Findings

Achieved 100-200x reduction in runtime estimates for quantum linear differential equation solvers.

Developed tighter bounds and more efficient quantum compilation schemes.

Demonstrated potential for more practical quantum computing applications in differential equations.

Abstract

Finding the solution to linear ordinary differential equations of the form $\partial_t u(t) = -A(t)u(t)$ has been a promising theoretical avenue for \textit{asymptotic} quantum speedups. However, despite the improvements to existing quantum differential equation solvers over the years, little is known about \textit{constant factor} costs of such quantum algorithms. This makes it challenging to assess the prospects for using these algorithms in practice. In this work, we prove constant factor bounds for a promising new quantum differential equation solver, the linear combination of Hamiltonian simulation (LCHS) algorithm. Our bounds are formulated as the number of queries to a unitary $U_A$ that block encodes the generator $A$. In doing so, we make several algorithmic improvements such as tighter truncation and discretization bounds on the LCHS kernel integral, a more efficient quantum compilation scheme for the SELECT operator in LCHS, as well as use of a constant-factor bound for oblivious amplitude amplification, which may be of general interest. To the best of our knowledge, our new formulae improve over previous state of the art by at least two orders of magnitude, where the speedup can be far greater if state preparation has a significant cost. Accordingly, for any previous resource estimates of time-independent linear differential equations for the most general case whereby the dynamics are not \textit{fast-forwardable}, these findings provide a 100-200x reduction in runtime costs. This analysis contributes towards establishing more promising applications for quantum computing.