The Pumped Langevin Dynamics Solver#
Stochastic differential equations#
Pumped Langevin dynamics, which is a modification of a typical Langevin dynamics (Khosravi et al 2022), can be described by the stochastic differential equations (SDE):
where \(p(t) = p_0 t/T\), with arbitrarily defined \(T=n_\mathrm{iter}\times dt\) to represent the evaluation time, and \(\lambda\) and \(\sigma\) are hyperparameters controlling the strength of the gradient and the diffusion terms, respectively. The preceding equation is very similar to the typical Langevin equation, but with addition of the term \(\big(-1+p-c_i^2\big)c_i\) to represent three physical processes: photon loss rate, the strength of the external pump field, and the nonlinear self-interaction induced by a nonlinear crystal (Khosravi et al 2022).
Similar to the Langevin solver, we have implemented a solver based on pumped Langevin dynamics.
We have developed this solver as a classical solver implemented solely on classical computers, that is, there is no optical simulation of pumped Langevin dynamics in our ccvm_simulators package.
Example script for solving a box-constrained quadratic programming problem using the Pumped Langevin solver#
The demo script, pumped_langevin_boxqp.py, in the examples folder simulates the Langevin dynamics for an example problem instance using the pumped Langevin solver. It can be executed from the project’s root directory using the following command in the terminal:
$ python ccvm/examples/pumped_langevin_boxqp.py
The script will print the solution, similar to the example output below.
Solution(
problem_size=20,
batch_size=1000,
iterations=15000,
...,
solve_time=5.422,
optimal_value=130.714,
best_value=120.533,
solution_performance={
'optimal': 0.004,
'one_percent': 0.394,
'two_percent': 0.908,
'three_percent': 0.991,
'four_percent': 1.0,
'five_percent': 1.0,
'ten_percent': 1.0
},
best_objective_value=130.653
)