Quantum algorithm for the Vlasov simulation of the large-scale structure formation with massive neutrinos

TL;DR

This paper introduces a quantum algorithm to simulate the large-scale structure formation of the universe with massive neutrinos, significantly reducing computational resources and enabling practical extraction of neutrino density perturbations.

Contribution

The paper presents the first quantum algorithm for LSS simulation that efficiently encodes neutrino phase space distribution and extracts the power spectrum with guaranteed accuracy.

Findings

Reduces space complexity to polylogarithmic in grid points

Uses quantum amplitude estimation for power spectrum extraction

Provides a practical quantum approach for cosmological simulations

Abstract

Investigating the cosmological implication of the fact that neutrino has finite mass is of importance for fundamental physics. In particular, massive neutrino affects the formation of the large-scale structure (LSS) of the universe, and, conversely, observations of the LSS can give constraints on the neutrino mass. Numerical simulations of the LSS formation including massive neutrino along with conventional cold dark matter is thus an important task. For this, calculating the neutrino distribution in the phase space by solving the Vlasov equation is a suitable approach, but it requires solving the PDE in the $(6+1)$-dimensional space and is thus computationally demanding: Configuring $n_\mathrm{gr}$ grid points in each coordinate and $n_t$ time grid points leads to $O(n_\mathrm{gr}^6)$ memory space and $O(n_tn_\mathrm{gr}^6)$ queries to the coefficients in the discretized PDE. We propose a quantum algorithm for this task. Linearizing the Vlasov equation by neglecting the relatively weak self-gravity of the neutrino, we perform the Hamiltonian simulation to produce quantum states that encode the phase space distribution of neutrino. We also propose a way to extract the power spectrum of the neutrino density perturbations as classical data from the quantum state by quantum amplitude estimation with accuracy $\epsilon$ and query complexity of order $\widetilde{O}((n_\mathrm{gr} + n_t)/\epsilon)$. Our method also reduces the space complexity to $O(\mathrm{polylog}(n_\mathrm{gr}/\epsilon))$ in terms of the qubit number, while using quantum random access memories with $O(n_\mathrm{gr}^3)$ entries. As far as we know, this is the first quantum algorithm for the LSS simulation that outputs the quantity of practical interest with guaranteed accuracy.