Deterministic Realization of Classical Dissipation on Quantum Computers

Lattice Boltzmann (LB) on quantum devices must reconcile unitary gate evolution with the dissipative \emph{collision} step. In the multiple-relaxation-time (MRT) class, we work in the common setting of \emph{modewise diagonal} moment relaxation, $\delta m_r^{\mathrm{\prime }}={\lambda }_r\delta m_r$ with ${\lambda }_r\in [-1,1]$ (overrelaxation if ${\lambda }_r<0$). 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. \emph{For the dissipative MRT block alone} we give a \emph{block-encoding-free} construction: a signed \emph{two-rail} population encoding, then a completely positive trace-preserving (CPTP) map (per-rail amplitude damping with survival $\mathrm{\mid }{\lambda }_r\mathrm{\mid }$ and, if ${\lambda }_r<0$, 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~$m^{\mathrm{e}\mathrm{q}}=M{f}^{\mathrm{e}\mathrm{q}}$ (prescribed~$f^{\mathrm{e}\mathrm{q}}$, host moment matrix~$M$; notation as in Section~\ref{subsec:generic-mrt}), 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.