a, b, We study the fully frustrated square-octagonal (a) and triangular (b) lattices with cylindrical boundary condition and width L up to 15 (L = 6 shown). FM couplers (Jij = −1.8) are indicated with blue lines in a; AFM couplers (Jij = 1 except on the boundary, where Jij = 1/2) are indicated with red lines in a and b. c, Owing to the combination of frustration and superposition, the AFM triangle has six classical ground states and under a small transverse field has six perturbative ground states. These states map to complex pseudospins as shown, indicated with black and white circles, respectively (real and imaginary axes are indicated by ‘Re’ and ‘Im’, respectively). Spin states of up, down and transverse-field-aligned superposition are denoted by ↑, ↓ and →, respectively, in c and by red, blue and white circles in d and e. d, e, Each plaquette in the triangular lattice has a pseudospin (indicated by arrows) defined by the spins of the plaquette. In a perturbative ground state (d), all spins in a given sublattice (indicated by numbers) take the same state, and consequently all pseudospins have the same orientation. The state shown in e has a shift in pseudospin orientation, which corresponds to an excitation: the two yellow triangles are in a classical ground state but have no energetic contribution from qubits aligning with the transverse field. The pseudospins seen in d and e are indicated in c.