Quantum-inspired method for solving the Vlasov-Poisson equations

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.

Figure 1

Tensor network diagrams depicting (a) the conversion of a vector into a MPS, and (b) solving the PDE using MPS. We first compute the time derivative, assumed to be a function of our state $f\left(t\right)$, within the MPS framework. This involves applying the desired operations [represented as matrix product operators (MPOs), depicted as 1-D chains of square tensors] to the state and summing all of the terms together. We then compute the state at the next time step. At this point, the bond dimension of $f(t+\mathrm{\Delta }t)$ is larger than its original value due to the various operations applied to the MPS, so the MPS needs to be recompressed. This is done by first canonicalizing the MPS using QR decompositions, and then performing the compression via SVD and retaining only the $D$ largest singular values at each bond (see the Appendixes for details). In these diagrams, $n$-dimensional tensors are represented by shapes with $n$ legs. Legs that are connected to each other represent tensor contractions along those dimensions. Tensors in canonical form are depicted using triangles and obey the property denoted in the yellow box, or that ${\sum }_{u,l}{A}_{l,r}\left(u\right)A_{l,r^{\prime }}^{*}\left(u\right)={\delta }_{r,r^{\prime }}$.

Figure 2

Landau damping without compression of the distribution functions. (a) Plot of energy density of electric field, electron distribution, and ion distribution over time for perturbation wave vector $k=0.75$. Results from gkeyll [5] are shown in dotted black as reference. (b) The corresponding ion and electron distribution functions at $t=60$.

Figure 3

Entanglement entropy for solution to 1D1V Landau damping at $t=60.0$. (top row) Different MPS orderings of a 2-D state. Arrows correspond to ordering of tensors from coarse to fine grid resolution. (middle row) Entanglement entropy of the ion and electron distributions for Landau damping with initial conditions $A=0.5$ and $k=0.75$ evolved to time $t=60$ using $2^L$ grid points per dimension. For the sequential orderings, the EE at the center bond corresponds to the EE if using a tensor train format. (bottom row) Root-mean-square (rms) error of the same distribution functions when compressed to bond dimension $D$ with respect to the original uncompressed representation for each grid resolution.

Figure 4

Compressed time evolution for nonlinear Landau damping with $k=0.75$ on a $2^6\times 2^8$ grid. (a)–(c) Ion and electron distributions (over one period in $x$) at $t=60$ obtained from compressed time evolution with $D=24$ and S3 ordering, with $D=16$ and S3 ordering, and with $D=16$ and IG ordering, respectively. (d) Electric-field energy density for different levels of compression for MPS with S3 ordering. (e) Relative error in the electric-field energy density of the compressed time evolution results with respect to the uncompressed case for different levels of compression. S3 ordering is used. Results for $D=24$ are within 10% of the uncompressed result for times less than $t=40$. Results for $D=64$ are exact with respect to the uncompressed result, meaning that each truncation error at each bond ${\epsilon }_j$ is less than ${10}^{-10}$, and thus no blue line is shown in this plot. (f) Electric-field energy density of the compressed time evolution for different MPS orderings at bond dimension $D=24$. Results for S1 and S2 ordering appear to be identical and are closely overlapping.

Figure 5

Buneman instability. (a)–(c) Distribution functions of the ions and electrons at normalized time $t=80$ obtained without compression, a bond dimension of $D=64$, and a bond dimension of $D=32$, respectively, using the S3 ordering. (d) Electric-field energy density for different levels of compression. The curve in dotted black is obtained using gkeyll for an external reference. (e) Electric-field energy density in the nonlinear regime for different MPS orderings (f), (g) Relative error in the electric field for different compression levels and different MPS orderings. Note that the interleaved orderings have larger error in the linear growth regime but perform better in the nonlinear regime. (h), (i) Rms error of the ion and electron distribution function after compression of the exact (uncompressed) results to $D=32$ at each time step, evaluated for different MPS orderings.

Figure 6

Approximate calculation of the nonlinear term for the Buneman instability. Ion and electron distribution functions are compressed to bond dimension $D=64$. (a) Traces of the electric-field energy in time with different levels of compression of the electric field ($D_F$). (b) Relative error in electric-field energy density with respect to the $D=64$ time evolution result but with no compression of the electric field. (c) Error in the electric field after compression to the specified $D_F$ for each time step.

Figure 7

Compressed time evolution for shock-wave formation on a ${512}^2$ grid. (a) Distribution of ions and electrons at a time of 1000 obtained using S3 ordering and bond dimension $D=64$. (b) Distributions at a time of 707 obtained using S3 ordering and bond dimension $D=32$. (c) Relative error in electric-field energy density and total-energy density obtained using the specified bond dimension and S3 ordering. (d) Rms error generated by compressing the uncompressed ion and electron distributions to a bond dimension of $D=32$ at each time step, evaluated for different MPS orderings. (e) Electric-field energy density computed using compressed time evolution with $D=32$ for different MPS orderings. (f) Electric-field energy density computed using compressed time evolution, with the inclusion of various noise-mitigation methods such as adding collisions and using a smaller time step, computed using MPS with $D=32$ and S3 ordering.

Figure 8

Electric-field energy densities the for shock-wave formation test case using the different compressed time evolution schemes detailed above. Results are for (a) S3 and (b) S1 orderings.

Figure 9

Ion and electron distributions at specified times for the Buneman instability with initial perturbation of wave vector $k=0.10$ and amplitude $A={10}^{-3}$. The different rows are results obtained with gkeyll [5], our code with no compression, and our code with compression to $D=64$ and $D=32$ at each time step (using S3 ordering).

Figure 10

Zeroth, first, and second moments of the distribution functions for the Buneman instability with initial perturbation of wave vector $k=0.10$ and amplitude $A={10}^{-3}$ at the specified times. Plots compare results obtained without compression, with compression to $D=64$ at each time step, and from gkeyll. MPS results are obtained using the S3 ordering. Plots on the top and bottom of each row correspond to the ion and electron distributions, respectively.