Simulating closed- and open-system dynamics of Tavis--Cummings lattices on fault-tolerant quantum computers

Tavis--Cummings lattices couple cavities containing several emitters and host polaritonic many-body dynamics whose real-time correlations are costly to compute classically in two dimensions, especially when unequal emitter relaxation or dephasing rates break permutation symmetry. We analyze logical resources for fault-tolerant closed- and open-system simulation of an atom-resolved lattice with L cavities, N emitters per cavity, and photon cutoff nₘₐₓ, including target observable estimation. Building on ladder-operator block encoding, we construct circuits for photon hopping and local light--matter exchange and combine them to encode the lattice Hamiltonian. By updating source and destination photon occupations in place, the hopping circuit eliminates the auxiliary Llog₂(nₘₐₓ+1)-qubit photon register used by two-register implementations. Across three uniform-coefficient examples with $16$ to $256$ cavities, the controlled walk uses $4.75$--$8.11%$ fewer Toffoli gates than the paired two-register implementation under matched shared preparation, precision, control, and workspace conventions. For closed systems, we implement real-time evolution with generalized quantum signal processing and bound the cost of estimating retarded cavity spectral functions. For open systems with photon loss, emitter relaxation, and dephasing, we use a sampled first-order dilation and bound the runs and timesteps needed to estimate transient photon statistics such as g⁽²⁾(0). With state-preparation and measurement costs explicit, these estimates provide a reference for assessing fault-tolerant light--matter lattice simulation costs.