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

Quantum algorithms for scientific computing and their applications have been actively studied. In this paper, we propose a quantum algorithm for estimating the first eigenvalue of a differential operator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">L</a:mi></a:math> on <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msup><c:mi mathvariant="double-struck">R</c:mi><c:mi>d</c:mi></c:msup></c:math> and its application to cosmic inflation theory. A common approach for this eigenvalue problem involves applying the finite-difference discretization to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi mathvariant="script">L</e:mi></e:math> and computing the eigenvalues of the resulting matrix, but this method suffers from the curse of dimensionality, namely, the exponential complexity with respect to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>d</g:mi></g:math>. Our first contribution is the development of a 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mover accent="true"><h:mi>O</h:mi><h:mo>̃</h:mo></h:mover><h:mrow><h:mo>(</h:mo><h:msup><h:mi>d</h:mi><h:mn>3</h:mn></h:msup><h:mo>/</h:mo><h:msup><h:mi>ε</h:mi><h:mn>2</h:mn></h:msup><h:mo>)</h:mo></h:mrow></h:mrow></h:math> with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>d</j:mi></j:math> and estimation accuracy <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>ε</k:mi></k:math>, which is polynomial in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>d</l:mi></l:math> and shows 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.