Physical braid implementation interface

Let $V$ be a nonzero finite-dimensional complex inner-product space and let $\rho :B_n\to U(V)$ be a fixed mathematical braid representation. A physical implementation record for a braid $b\in B_n$ consists of all of the following data and hypotheses:

1. a finite-dimensional physical Hilbert space ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$;
2. a $C^2$ path $H(s):{\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ of Hermitian endomorphisms, $0\le s\le 1$, and a norm-continuous constant-rank orthogonal spectral projection $P(s)$ of rank $m=\dim V$ that is a coherently followed band in the sense of Finite adiabatic path model, including its local-contour continuation, and is separated from its complement by a uniform positive full gap. The initial low-energy sector is $\mathrm{ran}P(0)$;
3. a unitary identification ${\iota }_0:V\to \mathrm{ran}P(0)$, an explicit preparation isometry $A_0:{\mathbb{C}}^m\to {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ with image exactly $\mathrm{ran}P(0)$, its declared preparation cost, and a stated slow-time schedule $T>0$;
4. a unitary final-band identification ${\iota }_1:V\to \mathrm{ran}P(1)$ and a separate identification of the band's transported action along the chosen exchange path, using ${\iota }_0$ and ${\iota }_1$, with $\rho (b)$ up to a stated scalar phase.

The first three clauses form an instance of the finite adiabatic path model Finite adiabatic path model on ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. Under those hypotheses, Finite JRS adiabatic projector bound bounds the deviation from ideal adiabatic band following. That estimate does not establish that a material realizes the fusion category, that the exchange path has the prescribed holonomy, or that the initial state can be prepared at no cost. Locality, temperature, quasiparticle separation, stray excitations, and tunnelling require their own physical error model.