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arXiv:2303.16490 · Computers & Fluids

Quantum algorithm for collisionless Boltzmann simulation of self-gravitating systems

Soichiro Yamazaki, Fumio Uchida, Kotaro Fujisawa, Koichi Miyamoto, Naoki Yoshida

Published Dec 19, 2024• 1 citation
arXivHTMLPDF

Abstract

The collisionless Boltzmann equation (CBE) is a fundamental equation that governs the dynamics of a broad range of astrophysical systems from space plasma to star clusters and galaxies. It is computationally expensive to integrate the CBE directly in a multi-dimensional phase space, and thus the applications to realistic astrophysical problems have been limited so far. Recently, Todorova & Steijl (2020) proposed an efficient quantum algorithm to solve the CBE with significantly reduced computational complexity. We extend the algorithm to perform quantum simulations of self-gravitating systems, incorporating the method to calculate gravity with the major Fourier modes of the density distribution extracted from the solution-encoding quantum state. Our method improves the dependency of time and space complexities on N v , the number of grid points in each velocity coordinate, compared to the classical simulation methods. We then conduct some numerical demonstrations of our method. We first run a 1+1 dimensional test calculation of free streaming motion on 64 × 64 grids using 13 simulated qubits and validate our method. We then perform simulations of Jeans collapse, and compare the result with analytic and linear theory calculations. It will thus allow us to perform large-scale CBE simulations on future quantum computers. • We propose a quantum algorithm for solving Boltzmann equation. • Reservoir method is improved to be able to simulate variable force field systems. • Quantum Fourier transform and tomography enables efficient information extraction. • Test calculations assuming noiseless 13 qubits have been conducted.

DOI
10.1016/j.compfluid.2024.106527

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The collisionless Boltzmann equation (CBE) is a fundamental equation that governs the dynamics of a broad range of astrophysical systems from space plasma to star clusters and galaxies. It is computationally expensive to integrate the CBE directly in a multi-dimensional phase space, and thus the applications to realistic astrophysical problems have been limited so far. Recently, Todorova & Steijl (2020) proposed an efficient quantum algorithm to solve the CBE with significantly reduced computational complexity. We extend the algorithm to perform quantum simulations of self-gravitating systems, incorporating the method to calculate gravity with the major Fourier modes of the density distribution extracted from the solution-encoding quantum state. Our method improves the dependency of time and space complexities on N v , the number of grid points in each velocity coordinate, compared to the classical simulation methods. We then conduct some numerical demonstrations of our method. We first run a 1+1 dimensional test calculation of free streaming motion on 64 × 64 grids using 13 simulated qubits and validate our method. We then perform simulations of Jeans collapse, and compare the result with analytic and linear theory calculations. It will thus allow us to perform large-scale CBE simulations on future quantum computers. • We propose a quantum algorithm for solving Boltzmann equation. • Reservoir method is improved to be able to simulate variable force field systems. • Quantum Fourier transform and tomography enables efficient information extraction. • Test calculations assuming noiseless 13 qubits have been conducted.
A novel quantum algorithm for solving the Boltzmann-Maxwell equations of the 6D collisionless plasma is proposed. The equation describes the kinetic behavior of plasma particles in electromagnetic fields and is known for the classical first-principles equations in various domains, from space to laboratory plasmas. We have constructed a quantum algorithm for a future fault-tolerant large-scale quantum computer to accelerate its costly computation. This algorithm consists mainly of two routines: the Boltzmann solver and the Maxwell solver. Quantum algorithms undertake these dual procedures, while classical algorithms facilitate their interplay. Each solver has a similar structure consisting of three steps: Encoding, Propagation, and Integration. We conducted a preliminary implementation of the quantum algorithm and performed a parallel validation against a comparable classical approach. IBM Qiskit was used to implement all quantum circuits.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.
The simulation of gas flows for reentry problems can require to resolve a larger range of spatial scales in order to accurately model the physics of the flow field. In hypersonic flows, the modeling additionally requires to cover a large range in the velocity subspace of the phase space. Scale resolving methods like the Direct Simulation Monte Carlo (DSMC) method have been developed but can become computationally very expensive and unfeasible. Therefore, approximation methods allowing coarser grids are used which, however, may not be sufficiently accurate. To enable more efficient simulation methods, first approaches of a quantum lattice-gas model have been developed but these model do not scale efficiently with the grid resolution. Building on these quantum algorithms, promising modeling approaches using a quantum lattice Boltzmann method (QLBM) with amplitude encoding have shown to efficiently deal with very large grids and fine discretizations. Fine resolution of position and velocity space in the context of lattice Boltzmann methods (LBM) translate to many grid cells and a large set of velocity vectors in the LBM velocity discretization stencil, which can be encoded to scale efficiently in a qubit representation using the QLBM. For the investigation of collisionless flows a method based on Todorova et al. has been adopted and extended (cf. Fig. 1) to build upon our previous publication. The streaming and boundary interaction are modeled as two independent modular blocks, where the streaming block is adapted to be performed independently of the boundary. Additionally, our quantum algorithm includes the summation of the velocity distribution functions to calculate the mass density distribution directly in the quantum simulation. In contrast to a summation by classical post processing, the quantum summation block enables further development to extract reduced data from the quantum computer. This is a key point to not having to extract the entire high dimensional quantum state vector. In this study, a hypersonic collisionless flow around a flat plate with specular reflection is simulated using a quantum lattice Boltzmann method and compared with results using the established DSMC method for modeling rarefied flows. The quantum simulation is performed using a quantum algorithm simulator. Further approaches on how to extend the specular reflection model to a diffusive reflecting model are sketched out.
A fully quantum algorithm for solving the one-dimensional linear advection-diffusion equation using the Lattice Boltzmann method as a numerical procedure is presented in this work. We start by presenting a state of the art of the current usage of quantum algorithms for solving ordinary and partial differential equations. We then describe two algorithms for the one-dimensional Lattice Boltzmann method with two degrees of freedom. The first one is an existing hybrid quantum-classical algorithm with measurements at each time step, and the second one is our improved version, viz. a fully quantum algorithm where only one measurement is needed at the end of the algorithm. The fully quantum algorithm is first executed on a quantum simulator and then compared with a classical approach. Subsequently, the fully quantum algorithm is run on a quantum system with 133 qubits to investigate the effect of noise and the depth of the circuit on the output state. We find fluctuations in the final result due to the decoherence noise of the qubits.
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 ngr grid points in each coordinate and nt time grid points leads to O(ngr6) memory space and O(ntngr6) 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 ε and query complexity of order Õ[(ngr+nt)/ε] . Our method also reduces the space complexity to O[polylog(ngr/ε)] in terms of the qubit number, while using quantum random access memories with O(ngr3) 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. Published by the American Physical Society 2024
The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as O(Nvt)O({N}_{v}t)O(Nv​t) for a velocity grid with Nv{N}_{v}Nv​ grid points and simulation time ttt, our quantum algorithm scales as $O[\text{polylog}({N}_{v})t/\ensuremath{\delta}]$ where $\ensuremath{\delta}$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\ensuremath{\delta}$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.