Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation

TL;DR

This paper introduces a new quantum algorithm for efficiently estimating the first eigenvalue of differential operators, with applications to cosmic inflation, overcoming the curse of dimensionality present in classical methods.

Contribution

The paper develops a quantum algorithm based on singular value transformation that improves eigenvalue estimation for high-dimensional differential operators.

Findings

Query complexity scales as d^3/psilon^2, polynomial in dimension d.

Numerical results suggest the method effectively estimates eigenvalues with well-overlapping trial functions.

Potential application to decay rate estimation in stochastic inflation models.

Abstract

Quantum algorithms for scientific computing and their applications have been studied actively. In this paper, we propose a quantum algorithm for estimating the first eigenvalue of a differential operator $\mathcal{L}$ on $\mathbb{R}^d$ and its application to cosmic inflation theory. A common approach for this eigenvalue problem involves applying the finite-difference discretization to $\mathcal{L}$ and computing the eigenvalues of the resulting matrix, but this method suffers from the curse of dimensionality, namely the exponential complexity with respect to $d$. Our first contribution is the development of a new quantum algorithm for this task, leveraging recent quantum singular value transformation-based methods. Given a trial function that overlaps well with the eigenfunction, our method runs with query complexity scaling as $\widetilde{O}(d^3/\epsilon^2)$ with $d$ and estimation accuracy $\epsilon$, which is polynomial in $d$ and shows an improvement over existing quantum algorithms. Then, we consider the application of our method to a problem in a theoretical framework for cosmic inflation known as stochastic inflation, specifically calculating the eigenvalue of the adjoint Fokker--Planck operator, which is related to the decay rate of the tail of the probability distribution for the primordial density perturbation. We numerically see that in some cases, simple trial functions overlap well with the first eigenfunction, indicating our method is promising for this problem.