We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group $B_3$, specialised at $t=e^{iω}$ and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries $A$, $B$, the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling $p_{\mathrm{fixed}}$, defining the fixed-order ceiling $p_{\mathrm{fixed}}^*$ and the witness gaps $Δ_{\rm sw}(ω)=p_{\mathrm{switch}}(ω)-p_{\mathrm{fixed}}^*$ and $Δ_{\rm test}(ω)=p_{\mathrm{test}}(ω)-p_{\mathrm{fixed}}^*$. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast $Δ_{\rm int}(ω):=Δ_{\rm test}(ω)-Δ_{\rm sw}(ω)=p_{\rm test}(ω)-p_{\rm switch}(ω)$. Across the Squier positivity region, $Δ_{\rm int}(ω)$ takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (non-Abelian) order control, while $Δ_{\rm sw}(ω)>0$ certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control that reproduces the characteristic interference pattern expected from a \emph{Gedankenexperiment} in anyonic statistics.

翻译：我们提出一种基于辫群$B_3$约化Burau表示（在$t=e^{iω}$处特化并通过Squier埃尔米特形式幺正化）构建的频率可调二维非阿贝尔操作顺序控制方案。该开关耦合两个非对易量子比特酉算符$A$、$B$后，可获得单次Helstrom成功概率的闭式表达式与固定顺序上限$p_{\mathrm{fixed}}$，由此定义固定顺序最优值$p_{\mathrm{fixed}}^*$及见证间隙$Δ_{\rm sw}(ω)=p_{\mathrm{switch}}(ω)-p_{\mathrm{fixed}}^*$和$Δ_{\rm test}(ω)=p_{\mathrm{test}}(ω)-p_{\mathrm{fixed}}^*$。非阿贝尔混合器可增强或抑制基础开关优势，我们通过干涉对比度$Δ_{\rm int}(ω):=Δ_{\rm test}(ω)-Δ_{\rm sw}(ω)=p_{\rm test}(ω)-p_{\rm switch}(ω)$量化该效应。在Squier正性区域内，$Δ_{\rm int}(ω)$同时呈现正值（相长干涉）与负值（相消干涉），这是矩阵值（非阿贝尔）顺序控制的标志性特征，而$Δ_{\rm sw}(ω)>0$则认证了代数因果不可分离性。数值模拟同时确认了增强与抑制两种状态，由此建立了能复现任意子统计\emph{思想实验}特征干涉图样的最小$B_3$辫群控制方案。