  We introduce the notion of combinatorial gauge symmetry -- a local
transformation that includes single spin rotations plus permutations of spins
(or swaps of their quantum states) -- that preserve the commutation and
anti-commutation relations among the spins. We show that Hamiltonians with
simple two-body interactions contain this symmetry if the coupling matrix is a
Hadamard matrix, with the combinatorial gauge symmetry being associated to the
automorphism of these matrices with respect to monomial transformations. Armed
with this symmetry, we address the physical problem of how to build quantum
spin liquids with physically accessible interactions. In addition to its
intrinsic physical significance, the problem is also tied to that of how to
build topological qubits.
