发表机构
Chiba University（千叶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究KLM构造的扭曲Burau辫子表示的量子门实现，确定其谱与分解，构造正交基使生成元由相位反射和CNOT门组成，给出逼近误差界，证明非Clifford与纠缠性质。
AI 中文摘要
我们研究了由Katz-Long-Moody (KLM)构造从字符扭曲Burau输入得到的辫子表示，该构造对应于KZ型方程的乘法中间卷积。对于$n\ge2$，设$q=e^{2\pi ia}$，$\tau=e^{2\pi ic}$，其中$a,c>0$且$nc+(n+1)a<1$。已知的形式和符号结果给出了$\lambda=e^{2\pi il}$，$0<l<1-nc-(n+1)a$时的正定不变形式。对于固定输入，酉实现唯一到同时酉共轭。我们解决两个问题。首先，我们确定$M=n(n+1)$维输出的谱和不变直和项。生成元谱恢复$(q,\tau)$。输出分解为两个平凡和两个约化Burau直和项，以及一个与Long约化构造同构的$n(n-1)$维直和项，在$n=3,4$时一般不可约。若$a$或$c$为无理数，则每个生成元在任意有限维Weyl-Heisenberg框架中都是非Clifford的；若$1,a,c$在$\mathbb{Q}$上线性无关，则每个生成元对任意非平凡二分都是纠缠的。完整辫子像不是投影稠密的。对于一般正定酉输入，在非例外参数下完整辫子输出的不可约性迫使自由群输入算子交换。其次，我们构造一个标准正交基，使每个生成元有一个大小为$2(n+1)$的块，$n-2$个大小为二的块，其余为单位。每个生成元是$2n$个秩一相位反射的乘积，并且在固定二进制编码中，是一个由单量子比特酉门和CNOT门组成的$O(M)$长度字，无需辅助量子比特，包括方向制备及其逆。我们给出有限字母逼近、条件和误差界，以及与量子Shannon参考编译器的匹配资源比较。门界在编码量子比特数上指数增长，并排除经典预处理。
英文摘要
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 $n\ge2$, 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 $\mathbb{Q}$, 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.
