Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians

Abstract

Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von Neumann equation rendering it in a suitable form for quantum simulations. Departing from the conventional method of expanding the density matrix in the Liouville space formed by matrices unit we express the von Neumann equation in terms of Pauli strings. This provides several advantages related to the quantum tomography of the density matrix and the formulation of the unitary gates that generate the time evolution. The use of Pauli strings facilitates the quantum tomography of the density matrix whose elements are purely real. As for any other basis of Hermitian matrices, this eliminates the need to calculate the phase of the complex entries of the density matrix. This approach also enables to express the evolution operator as a sequence of commuting Hamiltonian gates of Pauli strings that can readily be synthetized using Clifford gates. Additionally, the fact that these gates commute with each other along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. The algorithm is demonstrated for three Hamiltonians using the IBM noisy quantum circuit simulator.