Simulating the Lindblad Master Equation in Multi-Qubit Phase Space with Deep Learning

Abstract

The cost of simulating an open quantum many-body system is set by the density matrix, which grows exponentially with the number of particles. Phasespace representations replace it with a function, the Q-function, which encodes the state of N spin-1/2 particles as a scalar field on N spheres. The Lindblad master equation then becomes a partial differential equation for that function, on a domain growing linearly with N. This thesis presents a solution that employs a neural network trained on no data. The initial condition is built into the ansatz, and the training signal is the residual of the equation itself, supplemented only by an exact equation of motion for the Pauli moments and a penalty on unphysical harmonic content. On the transverse-field Ising chain under on-site amplitude damping, the reconstructed states reach worst-time fidelities of 0.9994, 0.9970, and 0.9787 at two, three, and four sites against exact integration. The moment equation of motion is decisive: without it, the same field reaches only 0.9801 at three sites. Enforcing it requires phase-space integrals, evaluated here on a quadrature grid whose cost grows exponentially with N. A Monte-Carlo estimator removes that cost but plateaus at 0.9830, limited by a variance intrinsic to the correlators being estimated