Z2 Lattice Gauge Theory on Non-trivial Topology and Its Quantum Simulation

Abstract

Wegner duality is essential for Z2 lattice gauge theory, yet the duality on non-trivial topologies has remained implicit. We extend Wegner duality to arbitrary topology and dimension, obtaining a new class of Ising models, in which topology is encoded in non-local domain-wall patterns. Without the overhead of gauge constraints, simulating this model on an L*L torus requires only L*L qubits with two-body couplings, halving the conventional four-body coupled 2L*L qubits, enabling full experimental realization of Z2 lattice gauge theory on near-term devices.

Figures (5)

• Figure 1: Hodge decomposition of the string Hilbert space into contractible closed strings (red), harmonic fields (gold), and gauge transformations (blue), mathematically described by $\mathrm{im}(\partial)$, $\mathrm{ker}(\Delta)$, and $\mathrm{im}(\delta)$.
• Figure 2: Ground state of SI model at $g=0$: each sector (left/mid/right) has a unique ground state set by domain walls; left column with trivial topology reduces to the Ising model. Dual spin (red border) maps to a closed string in the corresponding $\mathbb{Z}_2$LGT sector.
• Figure 3: Two tetrahedrons sharing a link
• Figure 4: Quantum phase transition simulation of the model \ref{['eq:dual']}-\ref{['eq:dual-E']} on $L\times L$ tori. (a) Energy expectations. (b) Phase transition points $g_c$. (c) Gap $\Delta E$. (d) Wilson loops $\langle W_C\rangle$. 100,000 Trotter steps with step-size 0.01 controls $\langle x\rangle = \langle H_E\rangle$ and $\langle z\rangle = \langle H_M\rangle$ absolute errors below $10^{-3}$. Gradient is calculated by central differential method of step 0.01, with moving average filter window 0.01.
• Figure 5: A $2\times 2$ lattice with periodic boundary condition.