Deterministic Realization of Classical Dissipation on Quantum Computers

Preprint, 2026

Lattice Boltzmann (LB) on quantum devices must reconcile unitary gate evolution with the dissipative collision step. In the multiple-relaxation-time (MRT) class, we work in the common setting of modewise diagonal moment relaxation, δm_r' = λ_r δm_r with λ_r ∈ [−1, 1] (overrelaxation if λ_r < 0). Embedding that contraction in a unitary by block encoding or a linear combination of unitaries (LCU) typically yields subunitary success probability that decays multiplicatively across modes, sites, and time, a key bottleneck for quantum LB. For the dissipative MRT block alone, we give a block-encoding-free construction: a signed two-rail population encoding, then a completely positive trace-preserving (CPTP) map (per-rail amplitude damping with survival |λ_r| and, if λ_r < 0, a rail SWAP) so that, after the decode, the map agrees with classical MRT relaxation exactly (expectations of the rail number operators, common encoding–decode scale). Trace preservation gives success probability 1 for that substage. The main result is the dissipative MRT block; construction of the equilibrium moment vector m_eq = M f_eq (prescribed f_eq, host moment matrix M; notation as in Section ???), moment transforms, streaming, and boundaries are composed with it as in a standard host pipeline and lie outside the scope of the formal theorem. Hybrid and fully coherent encodings, adaptive scales, Carleman-based context, and a one-rail no-go in the same nonnegative population framework are in the main text. Audits of the open-channel map on a long LBM collide-stream simulation and on stencil-free inputs both match the target to machine precision.