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We consider the question of how correlated the system hardness is betweenclassical algorithms of electronic structure theory in ground state estimationand quantum algorithms. To define the system hardness for classical algorithmswe employ empirical criterion based on the deviation of electronic energiesproduced by coupled cluster and configuration interaction methods from theexact ones along multiple bonds dissociation in a set of molecular systems. Forquantum algorithms, we have selected the Variational Quantum Eigensolver (VQE)and Quantum Phase Estimation (QPE) methods. As characteristics of the systemhardness for quantum methods, we analyzed circuit depths for the statepreparation, the number of quantum measurements needed for the energyexpectation value, and various cost characteristics for the Hamiltonianencodings via Trotter approximation and linear combination of unitaries (LCU).Our results show that the quantum resource requirements are mostly unaffectedby classical hardness, with the only exception being the state preparationpart, which contributes to both VQE and QPE algorithm costs. However, there areclear indications that constructing the initial state with a significantoverlap with the true ground state (>10%) is easier than obtaining the statewith an energy expectation value within chemical precision. These resultssupport optimism regarding the identification of a molecular system where aquantum algorithm excels over its classical counterpart, as quantum methods canmaintain efficiency in classically challenging systems.

Probing Quantum Efficiency: Exploring System Hardness in Electronic Ground State Energy Estimation