  The simulation of strongly correlated many-electron systems is one of the
most promising applications for near-term quantum devices. Here we develop and
apply a family of quantum simulation algorithms for electronic structure, based
on the integration (or contraction) of the Schr\"odinger equation onto the
two-electron space. We showcase the family of algorithms by resolving the
energy splittings of the ortho-, meta-, and para-isomers of benzyne ${\textrm
C_6} {\textrm H_4}$ including the strongly correlated biradical para-benzyne.
The quantum solution of the anti-Hermitian part of the contracted Schr\"odinger
equation (qACSE) provides a scalable approach with low circuit depth and few
variational parameters that has its foundations in two-electron reduced density
matrix (2-RDM) theory. Error in the 2-RDM is mitigated through projection of
the 2-RDM measured by tomography onto the approximate set of physically
realistic 2-RDMs defined by the 2-positive $N$-representability conditions. The
relative energies exhibit single-digit millihartree errors, and the computed
natural-orbital occupations capture significant differences in the electron
correlation of the isomers including the biradical nature of the {\em
para}-isomer.
