On the order problem in construction of unitary operators for the Variational Quantum Eigensolver

Artur F. Izmaylov, Manuel Díaz-Tinoco, Robert A. Lang

One of the main challenges in the Variational Quantum Eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of N qubits is 4^N−1, which makes the choice of a polynomial subset of generators exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra - Lie group connection and corresponding closure relations that systematically eliminates the order problem.