• dark matter simulations
• nonlinear quantum algorithms
• variational time evolution

The progress in our comprehension of dark matter structure and origin is challenged by the limitation imposed by classical computers to cosmological simulations.
Quantum computers have the potential of providing a quantum speed-up in same specific applications when compared to the corresponding classical computers.
In cosmology, this can be achieved for instance upon mapping the Vlasov-Poisson problem into its quantum counterpart defined by the Schroedinger-Poisson (SP) equation (Mocz et al. Physical Review D (Apr. 2018)).
In this work, we explore the challenges arising in the implementation of large scale cosmological simulations on quantum devices, through the use, e.g., of periodic boundary conditions, as already demonstrated by Mocz and Szasz The Astrophysical Journal (Mar. 2021).
In particular, we focus on the dynamics described by the SP equation, where a self-gravitating potential introduces nonlinearity in the problem.
The mapping of the nonlinearity onto a quantum device is solved using a classical-hybrid variational algorithm similar to the one proposed in Lubasch et al. Physical Review A(Jan. 2020).
The evolution of the wavefunction is carried out using a variational time evolution (VTE) approach, tailored for nonlinear self-consistent problems defined on a grid.
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As additional important contribution, we propose new quantum circuits for the evaluation of the matrix elements needed in the VTE algorithm, which improves the scaling of the depth from O(n^2) to O(n), allowing for possible near term implementations.

Finally, we estimated the upper and lower bound for the number of qubits needed for a simulation of a one- and three-dimensional (3D) systems with a resolution of $2048$ grid-points per dimension.
For a 3D cosmological relevant implementation, we find out that we could fully exploit circuit parallelization, trading single circuit measurements for more qubits (up to O(10^9)), significantly reducing therefore the total runtime.
An estimation of the number of non-Clifford gates needed in a fault-tolerant regime leads to a scaling of about O(n^2 K log_2(n) (where n is number of qubits used and K depends on the decomposition method chosen and the desired accuracy) for a one dimensional system.

We conclude that the solution of the quantum SP equation is a good candidate for future large scale dark matter simulations on quantum computers. While the quality of the results obtainable with noisy, near-term, quantum computers is difficult to predict since it depends crucially on the quality of the hardware and error mitigation schemes applicable, the impact of fault-tolerant quantum computers may be large and in close reach, as shown by our resource estimates.