Quantum gates from the middle convolution of twisted Burau representations

We study braid representations obtained from character-twisted Burau inputs by the Katz-Long-Moody (KLM) construction, which corresponds to multiplicative middle convolution for KZ-type equations. For n 2, let q=e 2πia, τ=e 2πic, where a,c>0 and nc+(n+1)a<1. Known form and signature results give a positive-definite invariant form for λ=e 2πil, 0<l<1-nc-(n+1)a. For fixed input, unitary realizations are unique up to simultaneous unitary conjugation. We address two questions. First, we determine spectra and invariant summands of the M=n(n+1) -dimensional output. The generator spectrum recovers (q,τ). The output decomposes into two trivial and two reduced Burau summands, and an n(n-1) -dimensional summand isomorphic to Long's reduced construction, generically irreducible for n=3,4. If a or c is irrational, every generator is non-Clifford in every finite-dimensional Weyl-Heisenberg frame; if 1,a,c are linearly independent over Q, every generator is entangling for every nontrivial bipartition. The full braid image is not projectively dense. For general positive-definite unitary inputs, irreducibility of the full braid output at a nonexceptional parameter forces the free-group input operators to commute. Second, we construct one orthonormal basis giving each generator one block of size 2(n+1), n-2 blocks of size two, and identity elsewhere. Each generator is a product of 2n rank-one phase reflections and, in a fixed binary encoding, an O(M) -length word in one-qubit unitaries and CNOTs, without auxiliary qubits and including direction preparation and its inverse. We give finite-alphabet approximation, conditioning and error bounds, and a matched-resource comparison with a Quantum Shannon reference compiler. The gate bound is exponential in the encoded qubit count and excludes classical preprocessing.