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Because quantum simulation of molecular systems is expected to provide thestrongest advantage over classical computing methods for systems exhibitingstrong electron correlation, it is critical that the performance of VQEs beassessed for strongly correlated systems. For classical simulation, strongcorrelation often results in symmetry-breaking of the Hartree-Fock reference,leading to L\"owdin's well-known ``symmetry dilemma'' whereby accuracy in theenergy can be increased by breaking spin or spatial symmetries. Here, weexplore the impact of symmetry breaking on the performance of ADAPT-VQE usingtwo strongly correlated systems: (i) the ``fermionized" anisotropic Heisenbergmodel, where the anisotropy parameter controls the correlation in the system,and (ii) symmetrically-stretched linear \ce{H4}, where correlation increaseswith increasing H-H separation. In both of these cases, increasing the level ofcorrelation of the system leads to spontaneous symmetry breaking (parity and$\hat{S}^{2}$, respectively) of the mean-field solutions. We analyze the rolethat symmetry breaking in the reference states and orbital mappings of thefermionic Hamiltonians have on the compactness and performance of ADAPT-VQE. Weobserve that improving the energy of the reference states by breaking symmetryhas a deleterious effect on ADAPT-VQE by increasing the length of the ansatznecessary for energy convergence and exacerbating the problem of ``gradienttroughs".

Symmetry breaking slows convergence of the ADAPT Variational Quantum Eigensolver