We present a simple criterion for solvability of lattice spin systems on the basis of graph theory and simplicial homology. The lattice systems satisfy algebras with graphical representations. It is shown that the null spaces of adjacency matrices of the graphs provide conserved quantities of the systems. Furthermore, when the graphs belong to a class of simplicial complexes, the Hamiltonians are found to be mapped to bilinear forms of Majorana fermions, from which the full spectra of the systems are obtained. In the latter situation, we find a relation between conserved quantities and the first homology group of the graph, and the relation enables us to interpret the conserved quantities as flux excitations of the systems. The validity of our theory is confirmed in several known solvable spin systems including the one-dimensional (1D) transverse-field Ising chain, the 2D Kitaev honeycomb model, and the 3D diamond lattice model. We also present new solvable models on a 1D trijunction, 2D and 3D fractal lattices, and the 3D cubic lattice.
The applications of free fermions in statistical physics and quantum computation have long been well established. In particular, the Jordan–Wigner (JW) transformation can map certain spin-
1/2 chains to free fermionic systems, revealing their complete solvability. On the other hand, free fermionic systems are also classically simulable and therefore provide useful benchmarks for experimental implementations of quantum computers. Recently, Free Fermions in Disguise (FFD) have been introduced: these are new spin-
1/2 chain models that can be mapped to free fermions, but not through the standard Jordan–Wigner transformation. In this thesis, we address the problem of constructing quantum circuits from Free Fermions in Disguise. This is challenging because not all circuits built from these models remain free fermionic, in contrast to the JW-diagonalizable case. We present a systematic approach to demonstrate that some circuits previously proposed in the literature are indeed free fermionic. We then study the circuit dynamics of certain observables expressible in terms of fermionic operators for small system sizes, performing an exact-diagonalization check comparing the spin and fermionic evolutions. These results raise new questions about the classical simulability of free fermions in disguise, which appears to be a more subtle issue than in standard JW-diagonalizable models.