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.

Tensorized Pauli decomposition algorithm

“…They are essential in quantum simulation too as they are used in the description of Hamiltonians of many physical systems that can be mapped onto spin models in quantum many body physics [5], as well as describing electronic or molecular Hamiltonians in quantum chemistry [6]. For ground state preparation of materials [7], chemistry problems [8], generic Hamiltonians [9], and time-dependent Hamiltonians [10], the respective Hamiltonian's Pauli decomposition is crucial for all simulation-based classical and quantum algorithms; the number of Pauli terms directly dictates the complexity of the simulation task. In principle, any Hamiltonian-as it can be expressed as a linear combination of tensor products of Pauli matrices-can be simulated using a quantum simulator [11], leading also to many quantum annealing protocols [12] that rely on the prior decomposition of the evolutionary Hamiltonian into Pauli strings.…”