Quantum simulation for the 1D heat equation with a central Gaussian barrier potential via Schrödingerization

Abstract

We present a quantum simulation of the 1D heat equation with a central Gaussian barrier using the Schrödingerization method. The non-unitary nature of the heat equation is subverted through a warped phase transformation. This transforms the heat equation into a unitary operator that can be evolved in a quantum circuit. Quantum simulations were performed using Qiskit with n_x = 6 position registers and n_p = 2, 4, 6 momentum registers. The accuracy of the quantum simulation was evaluated by getting the l² error relative to the classical matrix exponentiation method. Results show that increasing the momentum registers n_p and reducing the timestep Δt both reduce the simulation error, correctly reproducing the M-shaped solution induced by the Gaussian barrier.