  We propose VQE circuit fabrics with advantageous properties for the
simulation of strongly correlated ground and excited states of molecules and
materials under the Jordan-Wigner mapping that can be implemented linearly
locally and preserve all relevant quantum numbers: the number of spin up
($\alpha$) and down ($\beta$) electrons and the total spin squared. We
demonstrate that our entangler circuits are expressive already at low depth and
parameter count, appear to become universal, and may be trainable without
having to cross regions of vanishing gradient, when the number of parameters
becomes sufficiently large and when these parameters are suitably initialized.
One particularly appealing construction achieves this with just orbital
rotations and pair exchange gates. We derive optimal four-term parameter shift
rules for and provide explicit decompositions of our quantum number preserving
gates and perform numerical demonstrations on highly correlated molecules on up
to 20 qubits.
