  Density matrix electronic structure theory is used in many quantum chemistry
methods to "alleviate" the computational cost that arises from directly using
wave functions. Although density matrix based methods are computationally more
efficient than wave functions based methods, yet significant computational
effort is involved. Since the Schr\"odinger equation needs to be solved as an
eigenvalue problem, the time-to-solution scales cubically with the system size,
and is solved as many times in order to reach charge or field self-consistency.
We hereby propose and study a method to compute the density matrix by using a
quadratic unconstrained binary optimization (QUBO) solver. This method could be
useful to solve the problem with quantum computers, and more specifically,
quantum annealers. The method hereby proposed is based on a direct construction
of the density matrix using a QUBO eigensolver. We explore the main parameters
of the algorithm focusing on precision and efficiency. We show that, while
direct construction of the density matrix using a QUBO formulation is possible,
the efficiency and precision have room for improvement. Moreover, calculations
performing Quantum Annealing with the D-Wave's new Advantage quantum processing
units is compared with classical Simulated annealing, further highlighting some
problems of the proposed method. We also show some alternative methods that
could lead to a better performance of the density matrix construction.
