A Geometric Theory of Fermion-to-Qubit Encodings
Exact fermion-to-qubit transformations are conventionally regarded as algorithmic tools that translate many-body Hamiltonians into qubit representations for quantum simulation. Here we show that they also define intrinsic geometric representations whose structure encodes physically meaningful information beyond spectral equivalence. We develop a geometric framework based on weighted hypergraphs and coupling-space representations constructed from the Bravyi–Kitaev (BK) and Xia–Bian–Kais (XBK) encodings. Applications to the Hubbard, spinless t–V, single-impurity Anderson, and Kitaev models demonstrate that these connectivity- and transport-based geometric descriptions consistently capture the structural evolution of encoded quantum Hamiltonians across distinct classes of many-body systems.
INTRODUCTION
Every exact representation of a physical system preserves its spectrum, but not every representation reveals the same physical structure. This work asks whether an exact fermion-to-qubit encoding can reveal physical structure beyond simply reproducing the spectrum of the original many-body Hamiltonian. Every Pauli string generated by a fermion-to-qubit transformation defines a multi-qubit interaction, and the complete encoded Hamiltonian therefore induces a weighted hypergraph whose structure evolves with the physical parameters of the underlying model. Rather than asking how the many-body wavefunction changes with interaction strength, we investigate how the geometry induced by the encoding reorganizes as the Hamiltonian is varied — using the Hubbard model, the paradigmatic model of strongly correlated electrons, as the primary setting.
METHODOLOGY
We combine two complementary representations of the encoded Hamiltonian: 1. BRAVYI–KITAEV (BK) HYPERGRAPH: Every encoded Hamiltonian is associated with a weighted hypergraph whose vertices are qubits and whose hyperedges are Pauli strings, weighted by their coupling coefficients. Mapping this hypergraph onto an equivalent weighted graph via clique expansion yields a combinatorial graph Laplacian, whose spectrum — and in particular its Fiedler eigenvalue (algebraic connectivity) — characterizes the global organization of the interaction network. 2. XIA–BIAN–KAIS (XBK) COUPLING-SPACE GEOMETRY: The same encoded Hamiltonian is separately mapped to an exactly equivalent diagonal Ising Hamiltonian, whose effective couplings define a probability measure over coupling space. The Wasserstein distance between coupling distributions at neighboring Hamiltonian parameters quantifies the rate of interaction-driven reorganization via optimal transport. For the Hubbard Hamiltonian in the BK representation, splitting the encoded operator into hopping and interaction sectors yields the dimensionless observable ρ(U) = λ2(Lhop)/λ2(Lint), comparing the algebraic connectivity of the kinetic and interaction hypergraphs.
RESULTS
The competition ratio ρ(U) follows an exact analytical dependence on interaction strength, ρ(U) = C/(αU), since the hopping-sector connectivity is a geometry-dependent constant while the interaction-sector connectivity scales linearly with U. This yields a characteristic interaction scale U* = C/α determined purely from the geometry of the encoding, without diagonalizing the many-body Hamiltonian. The Bravyi–Kitaev encoding separates into two distinct geometric universality classes — tapered and untapered — with the extrapolated tapered-encoding crossing scale (Uc/t = 8.87 ± 2.18) lying close to the interaction regime associated with the Mott crossover in the 2D Hubbard model. Extending the analysis to the full Laplacian spectrum reveals an exact internal partition: exactly one quarter of the interaction modes (Ntree = nq/4) belong to a tree-dominated sector inherited from the binary-tree architecture of the BK transformation, independent of lattice size. Independently, the XBK coupling-space Wasserstein distance develops a pronounced maximum in the same interaction regime where electronic double occupancy changes most rapidly. Applying the identical, unmodified construction to the 1D Hubbard, spinless t–V, single-impurity Anderson, and Kitaev models shows the same geometric reorganization signatures across metal-insulator crossovers, impurity (Kondo) screening, and topological phase transitions.
IMPLICATIONS
These results establish hypergraph geometry as a general framework for understanding fermion-to-qubit encodings: beyond enabling quantum simulation, exact encodings define geometric representations of many-body Hamiltonians whose structural organization — connectivity and coupling-space transport — carries physically meaningful information extractable without solving the many-body eigenvalue problem. The consistency of these signatures across the Hubbard, t–V, Anderson impurity, and Kitaev models suggests the framework captures general features of encoded quantum Hamiltonians rather than properties specific to one system, and may extend to other fermion-to-qubit transformations, quantum-chemical Hamiltonians, variational quantum algorithms, and quantum optimization problems.
REFERENCES
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