Variational quantum algorithm for low-dimensional systems in the Pauli basis

In the last decade, variational quantum algorithms implemented on modern quantum computers have successfully solved practical problems of optimization, quantum chemistry, and machine learning. We propose new variational quantum algorithm based on a Monte Carlo scheme that uses a random selection of the generators for a unitary transformation, and also uses optimization of the objective functional employing the annealing or Metropolis-Hastings algorithm. The states of the quantum system in the form of a density operator and its model Hamiltonian are represented by expansions in the Pauli basis. In the algorithm, the state of the system is changed by means of a random choice of the Pauli generator that determines the unitary transformation of the state. The efficiency of the annealing algorithm directly depends on the equiprobable choice of the transition from one state to the second, so the work uses a compromise version of the uniform distribution of operators on the SU (2 ⁿ ) group - the direct product of the SU (2) group, where n is the number of qubits in the system. The random choice of a single-qubit operator (consistent with the Haar measure on SU (2)) is implemented in Hopf coordinates on the group manifold (the three-sphere). The results of testing the algorithm show that it can be effective for low-dimensional systems.

variational quantum algorithm, annealing algorithm, unitary transformation, Pauli basis, Hamiltonian expansion, uniform distribution of a random variable on a three-dimensional sphere, Hopf coordinates