The Tridiagonal Propagator

The Bose-Hubbard Hamiltonian — bosonic particles hopping between lattice sites with on-site interactions — is the standard model for ultracold atoms in optical lattices and coupled nonlinear optical cavities. Simulating its quantum dynamics requires exponentiating the Hamiltonian, which for multi-mode systems involves matrices of dimension D that grows exponentially with the number of modes and the occupation number cutoff.

Tridiagonalization reduces the Hamiltonian matrix to a form with nonzero entries only on the main diagonal and the two adjacent diagonals. For a general D×D matrix, tridiagonalization is a standard numerical technique (Lanczos, Householder). But the result here is stronger: the Bose-Hubbard Hamiltonian can be rendered exactly tridiagonal using tools from number theory — the specific arrangement of Fock states that achieves tridiagonality is not arbitrary but is dictated by number-theoretic properties of the mode occupation numbers.

The tridiagonal form enables symplectic split-operator propagation: the time evolution operator e^{-iHt} is factored into a product of simple factors, each involving only the tridiagonal matrix. The split-operator scheme preserves the symplectic structure (unitarity) of quantum mechanics exactly, regardless of the time step — no accumulation of unitarity-violating errors. The tridiagonal structure makes each factor efficient to apply, reducing the cost from O(D²) to O(D) per time step.

The through-claim: the quantum dynamics of bosonic multimode systems has a hidden structure — exact tridiagonality — that makes simulation dramatically cheaper. The structure comes from number theory (the organization of Fock space), not from physics approximations. The speedup is exact, not approximate, and the symplectic propagation preserves unitarity exactly. The connection between number theory and quantum simulation is through the combinatorial structure of the Hilbert space basis.