  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.
