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

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 $\tilde{O}(d^3/{\epsilon }^2)$ 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.

19 pages, 5 figures

References in corpus (32)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Quantum algorithm for solving linear systems of equations
• Planck 2018 results. X. Constraints on inflation
• A new approach to the evolution of cosmological perturbations on large scales
• Hamiltonian Simulation by Qubitization
• A general proof of the conservation of the curvature perturbation
• Optimal Hamiltonian Simulation by Quantum Signal Processing
• The inflationary prediction for primordial non-gaussianity
• Primordial black holes from single field models of inflation
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Primordial Black Holes and Slow-Roll Violation
• Primordial Black Hole production in Critical Higgs Inflation
• Correlation Functions in Stochastic Inflation
• Quantum diffusion during inflation and primordial black holes
• The exponential tail of inflationary fluctuations: consequences for primordial black holes
• Near-optimal ground state preparation
• Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production
• Black-box quantum state preparation without arithmetic
• A new algorithm for calculating the curvature perturbations in stochastic inflation
• Non-perturbative approach for curvature perturbations in stochastic-$\delta N$ formalism
• Power spectrum in stochastic inflation
• A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation
• Statistics of coarse-grained cosmological fields in stochastic inflation
• Statistical nature of infrared dynamics on de Sitter background
• Review on stochastic approach to inflation
• Hunting for quantum-classical crossover in condensed matter problems
• Quantum algorithms for approximate function loading
• Can all the infrared secular growth really be understood as increase of classical statistical variance?
• Clustering of primordial black holes from quantum diffusion during inflation
• Inflationary stochastic anomalies
• Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions
• Eigenvalue Estimation of Differential Operators

Cited by in corpus (3)

• Stochastic-tail of the curvature perturbation in hybrid inflation
• Eternal inflation near inflection points: a challenge to primordial black hole models
• Calculating the power spectrum in stochastic inflation by Monte Carlo simulation and least squares curve fitting