Mapping of Fermionic Lattice Models for Ising Solvers
We present an end-to-end, symmetry-aware pipeline that converts interacting fermionic and quantum-spin models into annealer-ready QUBOs while preserving low-energy physics. The workflow combines Bravyi–Kitaev encoding, exact Z2 symmetry tapering, Xia–Bian–Kais (XBK) diagonalization to a Z-only form, and k→2 local quadratization, with ground energies recovered via a Dinkelbach fixed-point over the resulting Ising objective. We validate the approach across a complexity ladder spanning a frustrated 2D Ising model run on a D-Wave Advantage QPU, finite-temperature 1D Ising checks, a genuinely quantum XXZ spin target, and interacting t–V fermions in 1D and 2D, establishing a practical pathway for mapping quantum matter to current annealers.
INTRODUCTION
Interacting quantum lattices exhibit superconductivity, quantum magnetism, correlated metals, and topological order, but predictive simulation is hard: Hilbert spaces grow exponentially and quantum statistics impose long-range constraints. Classical solvers each cover only part of this space — exact diagonalization is definitive but small-scale, quantum Monte Carlo suffers from the sign problem in frustrated or fermionic regimes, and tensor networks degrade with entanglement and dimensionality. Quantum annealers offer a complementary route: thousands of qubits with persistent couplings that directly minimize Ising/QUBO cost functions, provided the physical Hamiltonian — typically noncommuting and often multi-body — can be translated into a diagonal, two-local Ising form without losing essential physics. Jordan–Wigner and Bravyi–Kitaev encodings produce a Pauli Hamiltonian suitable for gate-based hardware, but this Pauli form is not annealer-ready because X/Y terms are non-diagonal; the Xia–Bian–Kais (XBK) method resolves this by embedding the problem in a replicated register and rewriting all X/Y structure as Z-only couplings.
METHODOLOGY
We assemble a deterministic, five-stage, symmetry-aware pipeline: 1. BRAVYI–KITAEV (BK) ENCODING: Map the fermionic or spin Hamiltonian to an exact Pauli-operator form with short typical operator strings, using space-filling (Hilbert-curve) lattice orderings in higher dimensions to keep geometric neighbors adjacent in index space. 2. EXACT Z2 SYMMETRY TAPERING: Identify commuting Pauli symmetries via the nullspace of the symplectic parity-check matrix, and project the Hamiltonian onto a fixed symmetry sector, removing qubits without approximation — performed before diagonalization, since symmetries are hidden once operators are made diagonal. 3. XIA–BIAN–KAIS (XBK) EMBEDDING: Replicate each qubit and rewrite every X and Y operator as a Z-bilinear across a reference copy and its replicas, controlled by sector-dependent signs, yielding a purely diagonal (Z-only) Ising Hamiltonian. 4. QUADRATIZATION: Reduce any remaining k-local (k>2) Ising terms to two-local form using auxiliary binary variables and Rosenberg/Choi-style penalty gadgets. 5. CLASSICAL/QUBO SOLVING: Recover ground-state energies via a Dinkelbach fixed-point iteration over the discrete Rayleigh quotient H'p(s)/Cp(s), which converges monotonically and is solved sector-by-sector.
RESULTS
Across a validation ladder of increasing physical complexity: (i) a frustrated 2D Ising model (20×20) run on a D-Wave Advantage QPU reproduces the known ferromagnet–stripe transition near frustration ratio R ≈ 0.5, confirmed by magnetization, Binder cumulant, and structure-factor observables; (ii) finite-temperature checks on the 1D ferromagnetic Ising chain recover standard finite-size scaling trends, validating the thermal-sampling module; (iii) a genuinely quantum spin-½ XXZ chain (1×4) matches exact diagonalization across the Néel transition at V/t = 2; and (iv) interacting spinless t–V fermions in 1D (rings L = 2–8) show ED-level ground-state energies with the expected kink near V/t ≈ 2, while a 2D 2×2 cluster tracks ED slopes up to a small, uniform offset attributable to finite XBK replication. A replication-factor study on a 4-qubit system quantifies the accuracy–overhead trade-off: error falls by roughly an order of magnitude from replication r=2 to r=4, with diminishing returns beyond r ≈ Nq, while wall-clock cost grows steeply (41.5s → 1044.8s from r=2 to r=6). The pipeline is also shown to be portable beyond lattices, correctly resolving the equilibrium C–C and C–H bond geometry of benzene as a molecular test case.
IMPLICATIONS
The primary contribution is a unified, end-to-end, symmetry-aware mapping pipeline — BK → tapering → XBK → quadratization → QUBO — validated across classical Ising models, quantum spin chains, fermionic lattice Hamiltonians, and a molecular electronic-structure example, establishing formal annealer compatibility across a broad class of Hamiltonians. The auxiliary-variable and replication overhead is not fundamental: established constructions such as Ishikawa-type reductions and structured minor-embeddings can substantially reduce it in practice. Because the resulting QUBO encodes all Hamiltonian parameters simultaneously, parameter sweeps require no circuit recompilation or ansatz retraining, positioning annealing-based mapping as a practical, complementary route to gate-based quantum simulation as hardware connectivity and qubit counts continue to improve.
REFERENCES
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