  In the current framework of Geometric Quantum Machine Learning, the canonical
method for constructing a variational ansatz that respects the symmetry of some
group action is by forcing the circuit to be equivariant, i.e., to commute with
the action of the group. This can, however, be an overzealous constraint that
greatly limits the expressivity of the circuit, especially in the case of
continuous symmetries. We propose an alternative paradigm for the
symmetry-informed construction of variational quantum circuits, based on
homogeneous spaces, relaxing the overly stringent requirement of equivariance.
We achieve this by introducing horizontal quantum gates, which only transform
the state with respect to the directions orthogonal to those of the symmetry.
We show that horizontal quantum gates are much more expressive than equivariant
gates, and thus can solve problems that equivariant circuits cannot. For
instance, a circuit comprised of horizontal gates can find the ground state of
an $\mathrm{SU}(2)$-symmetric model where the ground state spin sector is
unknown--a task where equivariant circuits fall short. Moreover, for a
particular subclass of horizontal gates based on symmetric spaces, we can
obtain efficient circuit decompositions for our gates through the KAK theorem.
Finally, we highlight a particular class of horizontal quantum gates that
behave similarly to general $\mathrm{SU}(4)$ gates, while achieving a quadratic
reduction in the number of parameters for a generic problem.
