Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 3 / c / Solution

Ignoring the common normalization $1/(2\sqrt{2})$, direct multiplication givesThese four states span the positive eigenspace of $Z\otimes X\otimes Z$. The parent term annihilating that local MPS support is therefore the complementary projector

Translated terms have stabilizers $K_j=Z_{j-1}{X}_j{Z}_{j+1}$. Two such Pauli strings either do not overlap nontrivially or have two X--Z anticommutations, so all $K_j$ commute. Their positive eigenspaces have a common state, making the parent Hamiltonian frustration free. Each alternating global X symmetry flips the two Z factors of every $K_j$ or neither, and therefore commutes with every local term.

Solved by gpt-5.6-sol high.