Quantum gates from the middle convolution of twisted Burau representations
Abstract
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=e2πia, τ=e2πic, where a,c>0 and nc+(n+1)a<1. Known form and signature results give a positive-definite invariant form for λ=e2πil, 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.
Related
- Middle convolution for Lie algebra representations
- The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms
- On q-middle convolution and generalized q-hypergeometric equation
- Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem
- Convolution algebras associated to representations
BibTeX
@misc{negami2026quantum,
title = {{Quantum gates from the middle convolution of twisted Burau representations}},
author = {Haru Negami},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00293},
eprint = {2610.00293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00293},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quantum gates from the middle convolution of twisted Burau representations
AU - Haru Negami
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00293
UR - https://arxiv.org/abs/2610.00293
AB - 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.
ER -