Finite anyon task and promise interface

Work in a fixed unitary braided fusion category $\mathcal{C}$ with the representative set $\mathrm{Irr}(\mathcal{C})$ of Multiplicity-aware fusion-tree basis. Fix ordered inputs $a_1,\dots ,a_n\in \mathrm{Irr}(\mathcal{C})$, $n\ge 1$, a parenthesized fusion tree, and a total charge $c\in \mathrm{Irr}(\mathcal{C})$ for which the fusion space is nonzero. Simple charges, multiplicity bases, and admissible fusion-tree vectors have the conventions of Multiplicity-aware fusion-tree basis. An encoded input subspace is specified by a fixed-charge isometry as in Fixed-charge encoding and code projector. Each task instance supplies either an initial unit vector in that subspace as an abstract quantum input, or a finite state-preparation procedure whose specified output is that vector and whose preparation operations and ancillas are recorded. The instance also gives a nonnegative rational error threshold and specifies the correctness relation and metric to which it applies.

A finite anyon decision task is a promise pair $(Y,N)$ of disjoint sets of binary strings with domain $D=Y\cup N$, together with a finite encoding of admissible fusion-tree input labels; a solver must answer yes on $Y$ and no on $N$, with no obligation outside $D$. A finite anyon search task is a domain $D$ and a finite-string witness relation, with a valid witness required only for encoded inputs in $D$. The input record states the number and order of strands, the finite alphabet of positive and inverse Artin generators available to the solver, any prescribed finite braid word, and the finite set of adjacent fusion-measurement locations. For distinct input charges, an adjacent exchange has the typed unitary F/R action from the current ordered-list fusion space to the space for the list with those charges exchanged; track that list after every generator. These colored maps and their Artin relations are supplied by Gauge covariance of fusion-space braid actions. If a single-space $B_n$ representation is required for the physical-interface record Physical braid implementation interface, use the finite orthogonal direct sum of fusion spaces over all distinct orderings of the fixed input multiset, with the same total $c$. Each generator maps each summand to its exchanged-list summand by the typed unitary; its inverse reverses that map. The colored Artin relations on each summand give the relations on the direct sum, hence a unitary $B_n$ action. Record the occupied summand and interpret measurements in its declared fusion-tree basis. Identical charges give the usual single fixed-space action. The physical-interface record specifies the mathematical action and separately states its physical realization hypotheses. The state-preparation field is an abstract task input and does not assert that a physical device prepares the state or realizes the braid. Input length and correctness-on-promises follow QA task and promise domain.