2 The Burau representation

Let ${\mathrm {Br}}_n$ be the braid group on n strands. That is,

$$ \begin{align*}{\mathrm{Br}}_n=\left\langle \sigma_1, \ldots, \sigma_{n-1} | \; \begin{array}{ll} \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} , \; &i=1,\ldots, n-2 \\ \sigma_i\sigma_j = \sigma_j\sigma_i, &|i-j|>2. \end{array} \right\rangle.\end{align*} $$

In this section, we review the Burau representation of ${\mathrm {Br}}_n$ following the expositionFootnote ² of Squier [Reference Squier58]. The Burau representation of ${\mathrm {Br}}_n$ on the vector space $B_{n,s}$ spanned by $ \{E_1, \ldots , E_{n-1} \}$ is defined by

(2.1) $$ \begin{align} \sigma_i(E_j) = \begin{cases} E_i + E_{i-1} & \text{ if } j=i-1, \\ -s^{-2} E_i & \text{ if } j=i, \\ s^{-2} E_i + E_{i+1} & \text{ if } j=i+1, \\ E_j & \text{ otherwise}, \end{cases} \end{align} $$

where s is a complex number.

Consider a new basis of $B_{n,s}$ given by $ \{f_1, \ldots , f_{n-1} \},$ where

$$\begin{align*}f_j=E_1+(1+s^2)E_2 + (1+s^2+s^4)E_3 + \cdots + (1+s^2+\cdots + s^{2j-2}) E_j. \end{align*}$$

In this basis, $ \sigma _j (f_i) = f_i $ if $i\neq j-1,j$ . In the ordered basis $\{f_{j-1},f_j \}$ , the action of this generator is given by

(2.2) $$ \begin{align} \sigma_i = X_i D_B X_i^{-1} \quad \quad X_i = \begin{pmatrix} s^{i+2}-s^i & s^{2i+2}-1 \\ s^{i}-s^{i+2} & s^{2i}-s^{2} \end{pmatrix} \quad \quad D_B= \begin{pmatrix} -s^{-2} & 0 \\ 0 & 1 \end{pmatrix} . \end{align} $$

Squier found a non-degenerate Hermitian sesquilinear pairing on the Burau representation [Reference Squier58]. In the basis $\{E_1, \ldots , E_{n-1} \}$ it is

$$\begin{align*}(E_i, E_j) = \begin{cases} s+s^{-1} & \text{ if } j=i, \\ -s^{i-j} & \text{ if } j=i \pm 1, \\ 0 & \text{ otherwise. } \end{cases} \end{align*}$$

The basis $\{{f}_1, \ldots , {f}_{n-1} \} $ is orthogonal with respect to this form with

(2.3) $$ \begin{align} ({f}_i, {f}_j) = \delta_{i,j} \cdot s^{-2i+1} \frac{(1-s^{2i})(1-s^{2i+2})}{(1-s^2)^2} . \end{align} $$

We now come to the foundational result of Squier.

3 Unrolled $\mathfrak {sl}_2$

In this section, we recall the algebra ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ and a category of modules over this algebra. Let ${\mathbb {C}}$ be the complex numbers and ${\ddot {\mathbb {C}}}=({\mathbb {C}}\setminus {\mathbb {Z}})\cup 2{\mathbb {Z}}.$ Let $q=e^{\frac {\pi \sqrt {-1}}{2}}$ be a $4^{th}$ -root of unity. We use the notation $q^x=e^{\frac {\pi \sqrt {-1} x}{2}}$ . For $n\in {\mathbb {N}}$ , we also set

$$ \begin{align*}{\left\{x\right\}}{\kern-1.2pt}={\kern-1.2pt}q^x{\kern-1.2pt}-{\kern-1.2pt}q^{-x},\!\!\quad{\left[x\right]}{\kern-1.2pt}={\kern-1.2pt}\frac{{\left\{x\right\}}}{{\left\{1\right\}}},\!\!\quad{\left\{n\right\}}!={\left\{n\right\}}{\left\{n{\kern-1.2pt}-{\kern-1.2pt}1\right\}}\cdots{\left\{1\right\}},{\quad\!\!\text{and}\!\!\quad}{\left[n\right]}!{\kern-1.2pt}={\kern-1.2pt}{\left[n\right]}{\left[n{\kern-1.2pt}-{\kern-1.2pt}1\right]}\cdots{\left[1\right]}.\end{align*} $$

Note that for $q=e^{\frac {\pi \sqrt {-1}}{2}}$ , we have the identities

(3.1) $$ \begin{align} [\alpha+2] = -[\alpha], \qquad [2-\alpha] = [\alpha]. \end{align} $$

3.1 The Drinfel’d–Jimbo quantum group

Let ${U_q{\mathfrak {sl}(2)}}$ be the ${\mathbb {C}}$ -algebra given by generators $E, F, K, K^{-1}$ and relations:

(3.2) $$ \begin{align} KK^{-1}&=K^{-1}K=1, & KEK^{-1}&=q^2E, & KFK^{-1}&=q^{-2}F, & [E,F]&=\frac{K-K^{-1}}{q-q^{-1}}. \end{align} $$

The algebra ${U_q{\mathfrak {sl}(2)}}$ is a Hopf algebra where the coproduct, count, and antipode are defined by

(3.3) $$ \begin{align} \Delta(E)&= 1\otimes E + E\otimes K, &\varepsilon(E)&= 0, &S(E)&=-EK^{-1}, \end{align} $$

(3.4) $$ \begin{align} \Delta(F)&=K^{-1} \otimes F + F\otimes 1, &\varepsilon(F)&=0,& S(F)&=-KF, \end{align} $$

(3.5) $$ \begin{align} \Delta(K)&=K\otimes K, &\varepsilon(K)&=1, & S(K)&=K^{-1}. \end{align} $$

Let ${\overline U_q{\mathfrak {sl}(2)}}$ be the algebra ${U_q{\mathfrak {sl}(2)}}$ modulo the relations $E^2=F^2=0$ .

3.2 A modified version of ${U_q{\mathfrak {sl}(2)}}$

Let ${U_q^{H}{\mathfrak {sl}(2)}}$ be the ${\mathbb {C}}$ -algebra given by generators $E, F, K, K^{-1}, H$ and relations in (3.2) along with the relations:

$$ \begin{align*} HK&=KH, & [H,E]&=2E, & [H,F]&=-2F. \end{align*} $$

The algebra ${U_q^{H}{\mathfrak {sl}(2)}}$ is a Hopf algebra where the coproduct, counit, and antipode are defined in (3.3)–(3.5) and by

$$ \begin{align*} \Delta(H)&=H\otimes 1 + 1 \otimes H, & \varepsilon(H)&=0, &S(H)&=-H. \end{align*} $$

Define the unrolled quantum group ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ to be the Hopf algebra ${U_q^{H}{\mathfrak {sl}(2)}}$ modulo the relations $E^2=F^2=0$ .

Let V be a finite-dimensional ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ -module. An eigenvalue $\lambda \in {\mathbb {C}}$ of the operator $H:V\rightarrow V$ is called a weight of V and the associated eigenspace is called a weight space. A vector v in the $\lambda $ -eigenspace of H is a weight vector of weight $\lambda $ , i.e., $Hv=\lambda v$ . We call V a weight module if V splits as a direct sum of weight spaces and ${q}^H=K$ as operators on V, i.e., $Kv=q^\lambda v$ for any vector v of weight $\lambda $ . Let $\mathcal {C}$ be the category of finite-dimensional weight ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ -modules.

Since ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ is a Hopf algebra, $\mathcal {C}$ is a tensor category, where the unit $\mathbb {I}$ is the one-dimensional trivial module ${\mathbb {C}}$ . Moreover, $\mathcal {C}$ is ${\mathbb {C}}$ -linear: hom-sets are ${\mathbb {C}}$ -modules, the composition and tensor product of morphisms are ${\mathbb {C}}$ -bilinear, and $\operatorname {End}_{\mathcal {C}}(\mathbb {I})={\mathbb {C}}\mathrm {Id}_{\mathbb {I}}$ . When it is clear, we denote the unit $\mathbb {I}$ by ${\mathbb {C}}$ . We say a module V is simple if it has no proper submodules. For a module V and a morphism $f\in \operatorname {End}_{\mathcal {C}}(V)$ , we write ${\left \langle {f}\right \rangle }_V=\lambda \in {\mathbb {C}}$ if $f-\lambda \mathrm {Id}_V$ is nilpotent. If V is simple, then Schur’s lemma implies that $\operatorname {End}_{\mathcal {C}}(V)={\mathbb {C}}\mathrm {Id}_V$ . Thus, for $f\in \operatorname {End}_{\mathcal {C}}(V)$ , we have $f={\left \langle {f}\right \rangle }_V \mathrm {Id}_V$ .

We will now recall the fact that the category $\mathcal {C}$ is a ribbon category. Let V and W be objects of $\mathcal {C}$ . Let $\{v_i\}$ be a basis of V and $\{v_i^*\}$ be a dual basis of $V^*=\mathrm {Hom}_{\mathbb {C}}(V,{\mathbb {C}})$ . Then,

$$ \begin{align*} \mathrm{coev}_V \colon {\mathbb{C}} \rightarrow V\otimes V^{*}, \text{ given by } 1 \mapsto \sum v_i\otimes v_i^*, \mathrm{ev}_V \colon & V^*\otimes V\rightarrow {\mathbb{C}}, \text{ given by } f\otimes w \mapsto f(w) \end{align*} $$

are duality morphisms of $\mathcal {C}$ . In [Reference Ohtsuki48], Ohtsuki truncates the usual formula of the h-adic quantum ${\mathfrak {sl}(2)} R$ -matrix to define an operator on $V\otimes W$ by

(3.6) $$ \begin{align} R={q}^{H\otimes H/2} \sum_{n=0}^{1} \frac{\{1\}^{2n}}{\{n\}!}{q}^{n(n-1)/2} E^n\otimes F^n, \end{align} $$

where $q^{H\otimes H/2}$ is the operator given by

$$ \begin{align*}q^{H\otimes H/2}(v\otimes v') =q^{\lambda \lambda'/2}v\otimes v'\end{align*} $$

for weight vectors v and $v'$ of weights of $\lambda $ and $\lambda '$ . The R-matrix is not an element in ${\overline U_q^{H}{\mathfrak {sl}(2)}}\otimes {\overline U_q^{H}{\mathfrak {sl}(2)}}$ . However, the action of R on the tensor product of two objects of $\mathcal {C}$ is a well-defined linear map. Moreover, R gives rise to a braiding $c_{V,W} \colon V\otimes W \rightarrow W \otimes V$ on $\mathcal {C}$ defined by $v\otimes w \mapsto \tau (R(v\otimes w)),$ where $\tau $ is the permutation $x\otimes y\mapsto y\otimes x$ . This braiding follows from the invertibility of the R-matrix. An explicit inverse (see [Reference Brown, Dimofte, Garaoufalidis and Geer7, Section 2.1.2] and [Reference Ohtsuki48]) is given by

(3.7) $$ \begin{align} R^{-1}= (\sum_{n=0}^{1} (-1)^n \frac{\{1\}^{2n}}{\{n\}!}{q}^{-n(n-1)/2} E^n\otimes F^n) {q}^{-H\otimes H/2}. \end{align} $$

Let $\theta $ be the operator given by

(3.8) $$ \begin{align} \theta=K^{}\sum_{n=0}^{1} \frac{\{1\}^{2n}}{\{n\}!}{q}^{n(n-1)/2} S(F^n){q}^{-H^2/2}E^n, \end{align} $$

where $q^{-H^2/2}$ is an operator defined on a weight vector $v_\lambda $ by $q^{-H^2/2}.v_\lambda = q^{-\lambda ^2/2}v_\lambda .$ Ohtsuki shows that the family of maps $\theta _V:V\rightarrow V$ in $\mathcal {C}$ defined by $v\mapsto \theta ^{-1}v$ is a twist (see [Reference Murakami45, Reference Ohtsuki48]).

Now the ribbon structure on $\mathcal {C}$ yields right duality morphisms

(3.9) $$ \begin{align} \stackrel{\longleftarrow}{\operatorname{ev}}_{V}=\mathrm{ev}_{V}c_{V,V^*}(\theta_V\otimes\mathrm{Id}_{V^*})\text{ and }\stackrel{\longleftarrow}{\operatorname{coev}}_V =(\mathrm{Id}_{V^*}\otimes\theta_V)c_{V,V^*}\mathrm{coev}_V \end{align} $$

which are compatible with the left duality morphisms $\{\mathrm {coev}_V\}_V$ and $\{\mathrm {ev}_V\}_V$ . These duality morphisms are given explicitly by

$$ \begin{align*} \stackrel{\longleftarrow}{\operatorname{coev}}_{V} \colon & {\mathbb{C}} \rightarrow V^*\otimes V, \text{ where } 1 \mapsto \sum v_i^* \otimes K^{}v_i, \\ \stackrel{\longleftarrow}{\operatorname{ev}}_{V} \colon & V\otimes V^*\rightarrow {\mathbb{C}}, \text{ where } v\otimes f \mapsto f(K^{-1}v). \end{align*} $$

The quantum dimension $\operatorname {qdim}(V)$ of an object V in $\mathcal {C}$ is defined by

$$\begin{align*}\operatorname{qdim}(V)= {\left\langle{\stackrel{\longleftarrow}{\operatorname{ev}}_V\circ \mathrm{coev}_V}\right\rangle}_{\mathbb{I}}=\sum v_i^*(K^{-1}v_i) . \end{align*}$$

For $g\in {\mathbb {C}}/2{\mathbb {Z}}$ , define $\mathcal {C}_{g}$ as the full subcategory of weight modules whose weights are all in the class g (mod $2{\mathbb {Z}}$ ). Then, $\mathcal {C}=\{\mathcal {C}_g\}_{g\in {\mathbb {C}}/2{\mathbb {Z}}}$ is a ${\mathbb {C}}/2{\mathbb {Z}}$ -graded category (where ${\mathbb {C}}/2{\mathbb {Z}}$ is an additive group). Let $V\in \mathcal {C}_g$ and $V'\in \mathcal {C}_{g'}$ . Then, the weights of $V\otimes V'$ are congruent to $g+g' \mod 2{\mathbb {Z}}$ , and so the tensor product is in $\mathcal {C}_{g+g'}$ . Also, if $g\neq g'$ then $\mathrm {Hom}_{\mathcal {C}}(V, V')=0$ since morphisms in $\mathcal {C}$ preserve weights. Finally, if $f\in V^*=\mathrm {Hom}_{\mathbb {C}}(V,{\mathbb {C}})$ , then by definition the action of H on f is given by $(Hf)(v)=f(S(H)v)=-f(Hv)$ and so $V^{*}\in \mathcal {C}_{-g}$ . We call the part of the category where $g=0,1$ , the singular part of the category, and refer to the objects in this part of the category as singular objects. The category $\mathcal {C}_g$ is non-semisimple if g is singular, otherwise g is called generic and $\mathcal {C}_g$ is semisimple.

We now consider the following class of finite dimensional highest weight modules. For each $\alpha \in {\mathbb {C}}$ , we let $V_\alpha $ be the two-dimensional highest weight ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ -module of highest weight $\alpha + 1$ . The module $V_\alpha $ has a basis $\{v_0,v_{1}\}$ whose action is given by

(3.10) $$ \begin{align} H.v_i=(\alpha + 1-2i) v_i,\quad E.v_i= \frac{{\left\{i\right\}}{\left\{i-\alpha\right\}}}{{\left\{1\right\}}^2} v_{i-1} ,\quad F.v_i=v_{i+1}. \end{align} $$

For all $\alpha \in {\mathbb {C}}$ , the quantum dimension of $V_\alpha $ is zero:

$$ \begin{align*}\operatorname{qdim}(V_\alpha)= \sum_{i=0}^{1} v_i^*(K^{-1}v_i)= \sum_{i=0}^{1} q^{-(\alpha + 1-2i)} = q^{-\alpha - 1}\frac{1-q^{4}}{1-q^{2}}=0.\end{align*} $$

For $a\in {\mathbb {Z}}$ , let ${\mathbb {C}}^H_{2a}$ be the one dimensional module in $\mathcal {C}_{\bar 0}$ where both E and F act by zero and H acts by $2a$ . Every simple module of $\mathcal {C}$ is isomorphic to exactly one of the modules in the list.

• • ${\mathbb {C}}^H_{2a}$ , for $a\in {\mathbb {Z}}$ .

• • $V_\alpha $ for $\alpha \in ({\mathbb {C}}\setminus {\mathbb {Z}})\cup 2 {\mathbb {Z}}$ .

Let $P_0$ be the projective and indecomposable module with highest weight $2$ , defined in Proposition 6.2 of [Reference Costantino, Geer and Patureau-Mirand10, Proposition 6.2]. Moreover, any indecomposable projective weight module has a highest weight, and such a module $P\in \mathcal {C}_{\overline 0}\cup \mathcal {C}_{\overline 1}$ with the highest weight $2(k+2)-2$ is isomorphic to $P_0\otimes {\mathbb {C}}^H_{2k}$ .

The simple module ${\mathbb {C}}^H_{0}$ has an indecomposable projective cover $P_0$ of dimension $4$ . A detailed description of this module can be found in [Reference Costantino, Geer and Patureau-Mirand10, Proposition 6.1]. A summary can be found in Figure 1. The vectors $w^Y$ , for $Y \in \{R,H,S,L\}$ have weights $2,0,0,$ - $2,$ respectively (under the action of H). It is easy to check that $\operatorname {End}(P_0) \cong {\mathbb {C}}[x]/(x^2),$ where

(3.11) $$ \begin{align} x \colon P_0 \rightarrow P_0 \quad \quad \quad w^H \mapsto w^S. \end{align} $$

Figure 1:

The weight space structure of the module $P_0$ .

A key ingredient in the construction of so-called non-semisimple TQFTs from the representations of ${\overline U_q^{H}{\mathfrak {sl}(2)}}$ is the notion of a modified trace [Reference Geer, Kujawa and Patureau-Mirand22]. Taking the modified trace of the identity gives a notion of modified dimension, which can be viewed as renormalizing the representations whose usual quantum dimension is zero. The modified dimension of $V_{\alpha }$ is given by

$$\begin{align*}{\mathsf d}(V_{\alpha}) = {\mathsf d}(\alpha) := -\frac{2 \sin \left(\frac{\pi \alpha }{2}\right)}{\sin (\pi \alpha )} = -\frac{1}{[\alpha+1]}, \end{align*}$$

where the last identity holds because q is a fourth root of unity.

If $\alpha , \beta $ , and $\alpha +\beta $ are generic, then it is well known (see, e.g., [Reference Costantino, Geer and Patureau-Mirand10, Theorem 5.2]) that

(3.12) $$ \begin{align} V_{\alpha} \otimes V_{\beta} \cong V_{\alpha+\beta+1} \oplus V_{\alpha+\beta-1} . \end{align} $$

In the next few sections, we will often use a graphical calculus describing this category of representations described in [Reference Costantino, Geer and Patureau-Mirand9]. All diagrams are read from bottom to top.

Our main focus in the rest of the article is the study of representations of the braid group ${\mathrm {Br}}_n$ on various morphism spaces in the category of representations of the unrolled quantum group.

4 Unitary representations of the braid group: generic part of the category

In this section, we will study representations coming from the generic part of the category. Let $\alpha $ be irrational so that $V_{\alpha }$ is a projective simple. For integers $k,n$ with $n\geq 1$ let $\mathcal {H}_{n,k,\alpha }:=\mathrm {Hom}(V_{n \alpha +k}, V_{\alpha }^{\otimes n})$ . Note that this is non-zero if and only if ${k=n-1,n-3,\ldots , 1-n}$ . We choose specific morphisms

(4.1)

coming from the conventions of [Reference Costantino, Geer and Patureau-Mirand9, Reference Costantino and Murakami11]. The vector spaces $\mathcal {H}_{n,k,\alpha }$ can be equipped with a non-degenerate Hermitian pairing, which will be utilized throughout this article. For details on this construction, see [Reference Geer, Lauda, Patureau-Mirand and Sussan23].

It is useful to have the notion of an $\mathcal {H}_{n,k,\alpha }$ path. This is a path $\textbf {p}=( \alpha = x_0, x_1,\ldots , x_{n-2}, n\alpha +k)$ in Pascal’s triangle (4.2) from $\alpha $ to $n \alpha +k$ by taking southeast or southwest steps.

(4.2)

We will also label such paths by direction sequences $\textbf {s}$ of L’s and R’s, which indicates at each step whether the path is going to the left or right. These sequences will have $n-1$ entries. For example, there is the path $(\alpha ,2\alpha -1,3\alpha )$ given in (4.3). The corresponding direction sequence is $(L,R)$ .

(4.3)

Proof The bijection maps a path $\textbf {p}=(\alpha ,x_1,\ldots , x_{n-2}, n\alpha +k)$ to the homomorphism encoded by the tree

(4.4)

We will usually abbreviate the label $V_{\beta }$ of an edge of a tree in (4.4) simply by $\beta $ .

Lemma 4.2 gives a basis for $\mathcal {H}_{n,k,\alpha }$ indexed by fusion diagrams representing homomorphisms $V_{n\alpha +k} \rightarrow V_{\alpha }^{\otimes n}$ constructed from the generators in (4.1). Different ways of combining the generators (4.1) with identity morphisms to construct homomorphism $V_{n\alpha +k} \rightarrow V_{\alpha }^{\otimes n}$ give rise to different bases for $\mathcal {H}_{n,k,\alpha }$ .

Lemma 4.3 We have the following change of basis formulas for the space of morphisms:

(4.5)

(4.6)

(4.7)

(4.8)

The space $\mathcal {H}_{n,k,\alpha }$ has a non-degenerate Hermitian sesquilinear pairing [Reference Geer, Lauda, Patureau-Mirand and Sussan23].

Proposition 4.4 The non-degenerate Hermitian pairing on $\mathcal {H}_{n,k,\alpha }$ is orthogonal with respect to the basis $\textbf {p}=(\alpha ,x_1,\ldots , x_{n-2}, x_{n-1}=n\alpha +k)$ . On these basis vectors, the pairing is

(4.9) $$ \begin{align} \langle \textbf{p} | \textbf{p} \rangle = \prod_{j=0}^{n-2} \frac{[x_j+\alpha][(x_j+\alpha)^{\frac{x_{j+1}-x_j-\alpha+1}{2}}]}{[(x_j+1)^{\frac{x_j+\alpha+1-x_{j+1}}{2}}] [(\alpha+1)^{\frac{x_j+\alpha+1-x_{j+1}}{2}}]} {\mathsf d}(V_{n \alpha+k}). \end{align} $$

Proof This is a lengthy but straightforward calculation using the definitions of [Reference Geer, Lauda, Patureau-Mirand and Sussan23] where it is proved that there is an anti-linear involution $\dagger $ on the morphisms of the category of Hermitian modules for the unrolled quantum group. For two morphisms $f,g$ with the same source and target, the pairing is defined to be the modified trace of $f^{\dagger } g$ .

The proposition follows from the straightforward calculations:

(4.10)

Lemma 4.5 [Reference Costantino, Geer and Patureau-Mirand9, Equation Ng]

There is an action of the braid group ${\mathrm {Br}}_2$ on the one-dimensional morphism spaces $\mathrm {Hom}(V_{\alpha +\beta -1},V_{\alpha } \otimes V_{\beta })$ and $\mathrm {Hom}(V_{\alpha +\beta +1},V_{\alpha } \otimes V_{\beta })$ given by

These formulas could be written uniformly as

(4.11)

We continue with the following easy lemma.

Lemma 4.6 Consider paths

$$ \begin{align*} \textbf{p} &=(\alpha,x_1,\ldots,x_{i-1},x_i,x_{i+1},\ldots, x_{n-2}, n\alpha+k) \\ \textbf{p}' &=(\alpha,x_1,\ldots,x_{i-1},x_i',x_{i+1},\ldots, x_{n-2}, n\alpha+k). \end{align*} $$

If the morphisms

(4.12)

span $\mathrm {Hom}(V_{x_{i+1}},V_{x_{i-1}}\otimes V_{\alpha }\otimes V_{\alpha })$ , then the braiding operator $\sigma _{i+1}$ preserves the subspace spanned by $\textbf {p}$ and $\textbf {p}'$ .

We now restrict ourselves to the special case $\mathcal {H}_{n,n-3,\alpha }$ . Consider the basis of direction sequences

(4.13) $$ \begin{align} e_i=(R, \ldots, R, L, R, \ldots, R), \end{align} $$

where the L is in position i. One easily calculates that that in this basis $\sigma _1(e_j)=q^{\frac {(\alpha +1)^2}{2}} e_j$ , except $\sigma _1(e_1)=q^{\frac {(\alpha -1)^2}{2}} e_1$ . Furthermore, $\sigma _i(e_j)=q^{\frac {(\alpha +1)^2}{2}} e_j$ if $i\geq 2$ and ${j \neq i-1,i}$ . Finally, in the ordered basis $\{e_{i-1},e_i\}$ , the action of this generator is given by

(4.14) $$ \begin{align} \sigma_i = A_i^{-1} \sigma A_i, \end{align} $$

where

(4.15) $$ \begin{align} \sigma= \begin{pmatrix} q^{\frac{(\alpha-1)^2}{2}} & 0 \\ 0 & q^{\frac{(\alpha+1)^2}{2}} \end{pmatrix} \ , \quad A_i = \begin{pmatrix} \operatorname{\mathsf{d}}(2\alpha-1) [-\alpha-1] & \operatorname{\mathsf{d}}(2\alpha-1) [-\alpha-1] \\ \operatorname{\mathsf{d}}(2\alpha+1)[(i+1) \alpha + i-3] & \operatorname{\mathsf{d}}(2\alpha+1)[(i-1) \alpha + i-3] \end{pmatrix}\! .\nonumber\\ \end{align} $$

In the basis $\{e_1,\ldots ,e_{n-1} \}$ , the Jucys–Murphy elements (see (4.25) later on for a graphical definition of these elements) are given by

(4.16) $$ \begin{align} J_j = \mathrm{Diag} \hspace{.05in} q^{j(\alpha+1)^2}\left\{\underbrace{q^{(-2\alpha-2)}, q^{(-2\alpha-2)}, \ldots, q^{(-2\alpha-2)}}_{j-1}, q^{(-2j-2)\alpha-(2j-2)}, \underbrace{1, \ldots, 1}_{n-1-j} \right\} .\nonumber\\ \end{align} $$

Proof Define an isomorphism of vector spaces $\phi \colon B_{n,s} \rightarrow \mathcal {H}_{n,n-3,\alpha } $ , where

(4.17) $$ \begin{align} \phi(f_k){\kern-1pt}={\kern-1pt}(-1)^{k+1} \frac {((iq^{\alpha})^{k+1}-(iq^{\alpha})^{-k-1}) ((iq^{\alpha})^{k}-(iq^{\alpha})^{-k})} {((iq^{\alpha})^{2}-(iq^{\alpha})^{-2}) ((iq^{\alpha})^{}-(iq^{\alpha})^{-1})} \left( \frac {[2\alpha+2] \sqrt{[\alpha+1][n\alpha+n-2]}} {[2\alpha][3\alpha+1]\cdots[n\alpha+n-2]} \right) e_k \nonumber\\ \end{align} $$

for $k=1,\ldots ,n-1$ . The second factor in the parenthesis is just needed to normalize the isomorphism of representations so that it becomes an isomorphism of unitary representations.

Consider the matrix $A_j$ in (4.15). Up to a scalar, it is equal to

(4.18) $$ \begin{align} \widehat{A}_j = \begin{pmatrix} [\alpha+1] & [\alpha+1] \\ [(j+1)\alpha+j-3] & [(j-1)\alpha+j-3] \end{pmatrix} . \end{align} $$

Then, one calculates

(4.19) $$ \begin{align} (\widehat{A}_j)^{-1} = \begin{pmatrix} [(j-1)\alpha+j-3] & -[\alpha+1] \\ - [(j+1)\alpha+j-3] & [\alpha+1] \end{pmatrix} . \end{align} $$

Letting $s=iq^{\alpha }$ , one obtains that

(4.20) $$ \begin{align} (\widehat{A}_j)^{-1} = c \begin{pmatrix} s^{2}-s^{2j} & s^{j}-s^{j+2} \\ 1-s^{2j+2} & s^{j+2}-s^{j} \end{pmatrix} \end{align} $$

for some scalar c. Thus, up to a scalar, $\sigma _j$ acts on the basis elements $\{e_{j-1},e_j \}$ by

(4.21) $$ \begin{align} (\widehat{A}_j)^{-1} \begin{pmatrix} -s^{-2} & 0 \\ 0 & 1 \end{pmatrix} \widehat{A}_j . \end{align} $$

Next, notice that (2.2) could be written as

(4.22) $$ \begin{align} \sigma_j= X_j \Gamma_j D_b \Gamma_j^{-1} X_j^{-1}, \quad \quad \Gamma_j = (-1)^{j+1} \begin{pmatrix} \frac{s^2-s^{-2}}{s^j-s^{-j}} & 0 \\ 0 & \frac{s^2-s^{-2}}{(s^{j-1}-s^{-(j-1)})(s^{j}-s^{-j})(s^{j+1}-s^{-(j+1)})} \end{pmatrix} . \end{align} $$

Clearly, the diagonal matrices $D_b$ and $ \begin {pmatrix} -s^{-2} & 0 \\ 0 & 1 \end {pmatrix}$ are the same. Letting $\phi _{j-1,j}$ be the diagonal matrix containing entries $j-1$ and j of the map $\phi $ , it is straightforward to show that $\phi _{j-1,j} X_j \Gamma _j=(\widehat {A}_j)^{-1} $ up to a scalar, so $\phi $ is a projective isomorphism of representations.

Note that the pairing on $\mathcal {H}_{n,n-3,\alpha }$ coming from Proposition 4.4 is

$$\begin{align*}\langle e_k | e_k \rangle = \frac {[(k+1) \alpha +k-1] \prod_{j=1}^{k-1} [(j+1) \alpha +j-1]^2 \prod_{j=k+1}^{n-1} [(j+1) \alpha + j-3]^2} {[k\alpha+k][\alpha+1]} {\mathsf d}(V_{n\alpha+n-3}). \end{align*}$$

Compare this to Squier’s form on the Burau representation from (2.3). After substituting $s=iq^{\alpha }$ , it is straightforward to show that $\phi $ is an isometry.

Proposition 4.8 There is an isomorphism of projective representations of the braid group

$$\begin{align*}\Lambda^k \mathcal{H}_{n,n-3,\alpha} \cong \mathcal{H}_{n,n-1-2k,\alpha} .\end{align*}$$

Proof Consider the linear isomorphism $\phi \colon \Lambda ^k \mathcal {H}_{n,n-3,\alpha } \rightarrow \mathcal {H}_{n,n-1-2k,\alpha } $ mapping $e_{l_1} \wedge \cdots \wedge e_{l_k}$ to a sequence $\textbf {s}$ with L’s in positions $l_1,\ldots ,l_k$ , with $l_1 < \cdots < l_k$ .

First, consider the braid group action on $\mathcal {H}_{n,n-1-2k,\alpha }$ . For the action of $\sigma _i$ , the relevant part of the tree is the part containing the $(i-1)$ st and ith leaves. For such a tree, let $\gamma $ be the number of R’s minus the number of L’s in the corresponding direction sequence in the first $i-2$ entries.

If a direction sequence $\textbf {s}$ has L’s in entries $i-1$ and i, then the relevant part of the corresponding tree is

In this case, $\sigma _i$ acts on this basis element by the scalar $q^{\frac {(\alpha -1)^2}{2}}$ .

If a direction sequence $\textbf {s}$ has R’s in entries $i-1$ and i, then the relevant part of the corresponding tree is

In this case, $\sigma _i$ acts on this basis element by the scalar $q^{\frac {(\alpha +1)^2}{2}}$ .

Now consider the case where we have two basis elements whose direction sequences are the same in all entries except the $(i-1)$ st and ith entries. In the first of the two basis elements, the sequence has L’s and R’s in entries $i-1$ and $i,$ respectively. The other basis element has R and L in entries $i-1,i$ . The corresponding trees are

(4.23)

One calculates using Lemma 4.3 that $\sigma _i$ acts by the matrix

$$\begin{align*}A_{i,\gamma}^{-1} \sigma A_{i,\gamma}, \end{align*}$$

where

(4.24) $$ \begin{align} A_{i,\gamma}= \begin{pmatrix} {\mathsf d}(2\alpha-1) [-\alpha-1] & {\mathsf d}(2\alpha-1) [-\alpha-1] \\ {\mathsf d}(2\alpha+1) [(i+1)\alpha+\gamma-1] & {\mathsf d}(2\alpha+1) [(i-1)\alpha+\gamma-1] \end{pmatrix}, \end{align} $$

where recall that $\sigma $ is given in (4.15).

Next, we compute the action of $\sigma _i$ on basis elements of $\Lambda ^k \mathcal {H}_{n,n-3,\alpha }$ .

The basis element in the exterior power corresponding to a direction sequence with R’s in entries $i-1,i$ is a wedge $e_{s_1} \wedge \cdots \wedge e_{s_k}$ , where none of the subscripts are $i-1$ or i. In this case, using (4.14), the action of $\sigma _i$ is given by the scalar $q^{\frac {k(\alpha +1)^2}{2}}$ .

The basis element in the exterior power corresponding to a direction sequence with L’s in entries $i-1,i$ is a wedge $e_{s_1} \wedge \cdots \wedge e_{s_k}$ , where the subscripts $i-1$ and i appear in the expression. Then, $\sigma _i$ acts by the scalar $q^{\frac {(\alpha +1)^2}{2}}$ on $k-2$ of the factors. Recall that on the basis $\{e_{i-1}, e_i \}$ that $\sigma _i$ acts by the matrix $A_i^{-1} \sigma A_i$ . A straightforward computation then yields that

$$\begin{align*}\sigma_i(e_{s_1} \wedge \cdots \wedge e_{s_k}) = q^{\frac{(\alpha-1)^2}{2} + \frac{(k-1)(\alpha+1)^2}{2} } e_{s_1} \wedge \cdots \wedge e_{s_k}. \end{align*}$$

Finally, we consider the two basis elements $w_1, w_2$ in (4.23). These elements correspond to wedges $e_{s_1} \wedge \cdots \wedge e_{i-1} \wedge e_{s_r} \wedge \cdots \wedge e_{s_k}$ (with $s_r \neq i$ ) and $e_{s_1} \wedge \cdots \wedge e_{s_c} \wedge e_i \wedge \cdots \wedge e_{s_k}$ (with $s_c \neq i-1$ ) respectively. Then, $\sigma _i$ acts by the scalar $q^{\frac {(\alpha +1)^2}{2}}$ on $k-1$ of the factors and by the matrix $A_i^{-1} \sigma A_i$ on the parts of the wedges $e_{i-1} \wedge e_{s_r}$ and $e_{s_c} \wedge e_i$ .

Assume for the moment that $A_i^{-1} \sigma A_i = A_{i,\gamma }^{-1} \sigma A_{i,\gamma } $ . Then, comparing the actions of $\sigma _i$ on $ \Lambda ^k \mathcal {H}_{n,n-3,\alpha } $ and $\mathcal {H}_{n,n-1-2k,\alpha }$ , we see that they are the same up to a factor of $q^{\frac {(k-1)(\alpha +1)^2}{2}}$ . Thus, the actions are the same projectively.

Finally, we need to check $A_i^{-1} \sigma A_i = A_{i,\gamma }^{-1} \sigma A_{i,\gamma } $ . The top rows of the matrices $A_i$ and $A_{i,\gamma }$ are the same, so we just need to analyze the bottom row. The bottom left entries are $[(i+1)\alpha +i-3]$ and $[(i+1)\alpha +\gamma -1],$ respectively. We claim that $\gamma -1$ and $i-3$ differ by an even number. Let $a_L$ be the number of L’s in the first $i-2$ entries and let $a_R$ be the number of R’s in the first $i-2$ entries. Then, $a_L+a_R=i-2$ and $-a_L+a_R=\gamma $ . Then, the claim easily follows. By (3.1), the bottom left entries are the same up to $\pm 1$ . Similarly, the bottom right entries are the same up to $\pm 1$ . Then, it follows easily that $A_i^{-1} \sigma A_i = A_{i,\gamma }^{-1} \sigma A_{i,\gamma } $ .

Proof Let $\mathcal {H}=\mathcal {H}_{n, k, \alpha } $ . The fact that the representation is (possibly indefinite) unitary follows from [Reference Geer, Lauda, Patureau-Mirand and Sussan23, Proposition 5.8]. Throughout this proof, we will use basis elements $\textbf {p}=(x_0=\alpha ,x_1,\ldots , x_{n-2}, x_{n-1}=n\alpha +k)$ from Lemma 4.2.

Consider the elements $J_i$ , for $i=1,\ldots ,n-1$ in the braid group ${\mathrm {Br}}_n$ :

(4.25)

These braid elements act diagonally on the basis spanned by trees (4.4). This is clear for $J_1$ . For $J_i$ with $i>1$ , this follows from the fact that strands slide past trivalent vertices as in (4.26).

(4.26)

Let G denote the closure of the image of the representation in the unitary group $\mathrm {U}(\mathcal {H})$ , so $G \subset \mathrm {U}(\mathcal {H})$ . G is a Lie group and let $\mathfrak {g}$ be the corresponding real Lie algebra.

Recall that $\mathcal {H}$ has an orthogonal basis indexed by a set of paths via the bijection given in Lemma 4.2.

Let $K \subset \mathrm {U}(\mathcal {H})$ be the maximal torus subgroup of diagonal matrices. Note that the corresponding Cartan subalgebra is a maximal compact Cartan subalgebra. Using (4.11), one computes

$$ \begin{align*}J_i \cdot \textbf{p}=q^{\frac{x_i^2-x_{i-1}^2-\alpha^2+1}{2}}.\end{align*} $$

For $k=n-3$ , the vector space is $(n-1)$ -dimensional. Recall the ordered basis $\{e_1, \ldots , e_{n-1} \}$ from earlier, where

$$\begin{align*}e_i=(\alpha, 2\alpha+1, 3\alpha+2, \ldots, i\alpha+(i-1), (i+1)\alpha+(i-2), (i+2)\alpha+(i-1), \ldots, n\alpha+(n-3)) . \end{align*}$$

In this basis, recall the matrix for $J_j$ (for $j=1,\ldots , n-1$ ) from (4.16). These $J_i$ generate a dense subgroup of K. Let $\mathfrak {h}=\mathrm {Lie}(K) \subset \mathfrak {g}$ . Note that $\mathfrak {g} \subset \mathfrak {u}(\mathcal {H})$ has multiplicity-free root spaces.

Any $\mathfrak {g}$ -submodule of $\mathcal {H}$ is a weight module. If $\mathcal {H}$ is an irreducible $\mathfrak {g}$ -module, then $\mathfrak {g}=\mathfrak {u}(\mathcal {H})$ .

Consider the graph of paths where the vertices are paths and two vertices are connected if the corresponding paths differ by one labeled edge in their corresponding trees. If paths $\textbf {p}$ and $\textbf {p}'$ are connected by an edge in the graph, then $\textbf {p}' \in \mathfrak {g} \cdot \textbf {p}$ using operators $\sigma _i$ and $J_i$ for some i. Since the graph of paths is connected, we obtain $\mathfrak {g}=\mathfrak {u}(\mathcal {H})$ .

When $\mathrm {U}(\mathcal {H})$ is a compact Lie group, it is well known that the exponential map is surjective. This is no longer true in the non-compact case. For example, it fails for $\mathrm {SL}(2,{\mathbb {R}}) \cong \mathrm {SU}(1,1)$ . There are suitable variations of this property that suit our purposes. The exponential map is surjective for $\mathrm {PSU}(p,p)$ [Reference Djokovic and Thǎńg15, Main Theorem]. See also [Reference Djokovic and Hofmann13, Theorem 4.7] for a survey. In that theorem, there is a surjectivity result for a certain quotient of $\mathrm {SU}(p,q)$ with $p \neq q$ .

The image of the exponential map is actually dense for the adjoint group of $\mathfrak {su}(p,q)$ with $p \neq q$ [Reference Djokovic and Thǎńg16, Theorem C]. By [Reference Djokovic and Thǎńg14, Theorem 4.5], the image of the exponential map is actually dense in $\mathrm {SU}(p,q)$ for $p \neq q$ . See also [Reference Neeb47, Section IV] for similar results.

Thus, using the commutative diagram (4.27), $G=\mathrm {U}(\mathcal {H})$ and, in particular, ${\mathrm {Br}}_n$ generates a dense subgroup of $\mathrm {PSU}(\mathcal {H})$ .

(4.27)

5 Perspectives on quantum computation

In this section, we consider the reduced Burau representation for the braid group on three strands and restrict ourselves to values of $\alpha $ that make the Hermitian pairing positive definite. Thus, the image of the braid group in this section lies in $\mathrm {PSU}(2)$ . This then serves as a model for a single qubit quantum computer.

Consider the two-dimensional vector space $\mathcal {H}_{3,\alpha }$ spanned by $ \{v_1, v_2 \},$ where

(5.1)

This is an orthogonal basis with pairing from Proposition 4.4:

(5.2) $$ \begin{align} n_1^2 &:=\langle v_1 | v_1 \rangle = \frac{[2\alpha]}{[\alpha+1]}=2\sin(\frac{\pi\alpha}2) ,\nonumber\\ n_2^2 &:=\langle v_2 | v_2 \rangle = \frac{-[2\alpha] [3\alpha+1]}{[\alpha+1]^2}=2\sin(\frac{\pi\alpha}2)(1-2\cos(\pi\alpha)) . \end{align} $$

Define a normalizing matrix to go from the ${v_1}, {v_2}$ basis to an orthonormal basis by

$$\begin{align*}N= \begin{pmatrix} \frac{1}{n_1} & 0 \\ 0 & \frac{1}{n_2} \end{pmatrix} . \end{align*}$$

Consider the following basis of $\mathcal {H}_{3,\alpha }$ :

By Lemma 4.3,

$$\begin{align*}v_1=- {\mathsf d}(2\alpha+1) [\alpha+1] v_1' - {\mathsf d}(2\alpha-1) [\alpha+1] v_2' \end{align*}$$

$$\begin{align*}v_2={\mathsf d}(2\alpha+1) [3\alpha+3] v_1' - {\mathsf d}(2\alpha-1) [\alpha+1] v_2' \end{align*}$$

and so the change of basis matrix between the basis $\{v_1, v_2 \}$ and $ \{v_1', v_2' \}$ is given by

$$\begin{align*}A= \begin{pmatrix} - {\mathsf d}(2\alpha+1) [\alpha+1] & {\mathsf d}(2\alpha+1) [3\alpha+3] \\ - {\mathsf d}(2\alpha-1) [\alpha+1] & - {\mathsf d}(2\alpha-1) [\alpha+1] \end{pmatrix} \;\; = \;\; \frac{1}{[2\alpha]} \begin{pmatrix} - [\alpha+1] & [3\alpha+3] \\ [\alpha+1] & [\alpha+1] \end{pmatrix} . \end{align*}$$

Note that $\operatorname {\mathsf {d}}(2\alpha +1) = -1/[2\alpha +2] = 1/[2\alpha ]$ and $\operatorname {\mathsf {d}}(2\alpha -1)=-1/[2\alpha ]$ .

From Section 4, there is an action of the braid group ${\mathrm {Br}}_3$ , where $\sigma _1$ crosses the two top left strands and $\sigma _2$ crosses the two top right strands. Just as in the previous section, after a lengthy calculation, using Lemma 4.5 and a change of basis from $\{{v_1},{v_2} \} $ to $\{{v_1'},{v_2'} \} $ one gets the matrices for $\sigma _1$ and $\sigma _2 $ in the basis $\{{v_1}, {v_2}\}$ to be

In the Fibonacci category, the half-twist braiding operator $\sigma _1 \sigma _2 \sigma _1$ gives rise to the Hadamard gate in $\mathrm {PSU}(2)$ . We record here a formula for this braid in our category of interest. Note that when $\alpha =\frac {1}{2}$ , the half-twist is the Hadamard operator up to a scalar. However, we exclude such an $\alpha $ from this section since it is rational.

Lemma 5.3 The braid group generators of ${\mathrm {Br}}_3$ associated to the parameter $\alpha $ satisfy

(5.3) $$ \begin{align} \sigma_1\sigma_2\sigma_1 &= \frac{(-1)^{3/4} e^{\frac{1}{4} i \pi \alpha (3 \alpha +2)}}{1-e^{i \pi \alpha }} \left( \begin{array}{cc} 1 & -1+2\cos(\pi \alpha) \\ -1 & -1 \\ \end{array} \right) ,\end{align} $$

(5.4) $$ \begin{align} (\sigma_1\sigma_2\sigma_1)^2 &= i e^{\frac{3}{2} i \pi \alpha ^2}\left( \begin{array}{cc} 1& 0 \\ 0 & 1 \\ \end{array} \right). \end{align} $$

Thus, $\sigma _1 \sigma _2 \sigma _1$ is an involution in $\mathrm {PSU}(2)$ .

In the orthonormal basis, these operators have the form:

(5.5) $$ \begin{align} N^{-1} \sigma_1\sigma_2\sigma_1 N= \frac{(-1)^{3/4} e^{\frac{1}{4} i \pi \alpha (3 \alpha +2)}}{1-e^{i \pi \alpha }} \left( \begin{array}{cc} 1 & \varsigma \\ \varsigma & -1 \\ \end{array} \right), \qquad \varsigma= -\frac{n_2}{n_1}, \end{align} $$

with $n_1$ and $n_2$ defined in (5.2), so that $\varsigma ^2=1-2\cos (\pi \alpha )$ and (5.4) becomes

(5.6) $$ \begin{align} N^{-1} (\sigma_1\sigma_2\sigma_1)^2 N =\left( \begin{array}{cc} i e^{\frac{3}{2} i \pi \alpha ^2} & 0 \\ 0 & i e^{\frac{3}{2} i \pi \alpha ^2} \\ \end{array} \right) . \end{align} $$

Next, we consider the following bi-invariant metric on $\mathrm {PSU}(2)$ :

(5.7) $$ \begin{align} d(x,y)= \sqrt{1-\frac{|\mathrm{Tr}(x^\dagger y)|}{2}} . \end{align} $$

Earlier, it was shown that $\sigma _1, \sigma _2$ generate a dense subgroup of $\mathrm {PSU}(2)$ . Then, it is easy to see that the operators $\sigma _1 $ and $N^{-1} \sigma _1 \sigma _2 \sigma _1 N$ from (5.5) also generate a dense subgroup.

The Solovay–Kitaev theorem provides an algorithm to approximate up to an error $\varepsilon $ a given element $M \in \mathrm {PSU}(2)$ by elements in our generating set $\sigma _1, N^{-1} \sigma _1 \sigma _2 \sigma _1 N$ (and their inverses), using a word of order $\mathrm {Log}^c(\frac {1}{\varepsilon }),$ where $c \geq 1$ . It is well known that c cannot be less than $1$ . We believe that with our generating set, we could obtain the optimal bound of $c=1$ .

In Figure 2, we give numerical evidence illustrating the efficiency of approximating elements of $\mathrm {PSU}(2)$ using braiding operators on the basis from (5.1).

Figure 2:

For each fixed value of $\alpha $ , we perform a brute-force search over all weaves of length up to 24 to find the unitary transformation that best approximates a given target gate, measured using the operator norm. The plot shows the minimal error $\varepsilon (U(\textbf {n}), T)$ achieved for unitaries $U(\textbf {n})$ constructed from weaves as in equation (5.8) with target gate T, across the sampled values of $\alpha $ .

We consider a brute force search over the best possible approximation achievable using a fixed number N of braid generators. The quality of the approximation is measured using the operator norm where the distance between operators U and V is given by $\varepsilon (U,V)=\|U-V\|,$ where $\|O\|$ is the square root of highest eigenvalue of $O^{\dagger }O$ . In this example, we take $N=24$ and consider approximations of the $\mathrm {PSU}(2)$ matrices $iX$ , $iZ$ , and T (or $\pi /8$ phase). We perform such a brute force search for varying values of $\alpha $ and plot the minimal error achieved with the fixed $N=24$ braid generators.

We follow [Reference Hormozi, Zikos, Bonesteel and Simon31] and restrict our search to a subclass of all possible braids known as weaves. Weaves are braids in which a single mobile quasiparticle interlaces between fixed and immutable quasiparticles. It was argued in [Reference Simon, Bonesteel, Freedman, Petrovic and Hormozi57] that this restricted class of braids could still be used for universal quantum computation and may have advantages in physical implementations. Hence, we consider approximations of a target unitary by unitaries $ U(\textbf {n} )$ obtained from weaves taking the form

(5.8) $$ \begin{align} U(\textbf{n} ):=\sigma_1^{n_m}\sigma_2^{n_{m-1}} \dots \sigma_1^{n_3}\sigma_2^{n_2} \sigma_1^{n_1}, \end{align} $$

for $\textbf {n}=(n_m,n_{m-1},\dots , n_2,n_1)$ , where we can assume that all the $n_i$ are even integers and that $n_i \neq 0$ for $1< i <m$ . The exponents being even translates into the mobile quasiparticle wrapping entirely around one of the neighboring quasiparticles and returning to the center position.

As a point of reference, we have also included the best approximation obtainable from braiding on a similarly defined fusion basis in the context of Fibonacci anyons associated with $\mathrm {SU}(2)$ at a fifth root of unity. This is a popular model for universal TQC. For all of the gates considered, the braiding defined on the fusion basis (5.1) can achieve an order of magnitude improvement over the best approximation obtained in the Fibonacci model for some value of $\alpha $ .

The Fibonacci result is depicted as a horizontal line in the graphs representing the best achievable error by weaves with 24 generators in the Fibonacci model. In Figure 2a, there are only a few values of $\alpha $ which improve upon the Fibonacci model’s best approximation. However, in Figure 2b, there is a much larger range of values of $\alpha $ that produce improvements over the Fibonacci model in approximating the $iZ$ gate. Figure 2c shows that most values of $\alpha $ give a much better approximation of the $\pi /8$ phase gate than the Fibonacci model. Finally, in Figure 2d, we graph all the values of $\alpha $ , which offer improvements over the Fibonacci model. The values around $\alpha =0.6$ offer improvements over the Fibonacci model for all gates considered.

Remark 5.4 Finding an optimal set of topological generators is a challenging question that has connections to many different areas of mathematics. There are even many different notions of optimal. One particular perspective is the notion of efficiently universal mentioned earlier, meaning that it saturates the lower bound of the exponent in the Kitaev–Solovay Theorem. In the work of Chuang, Harrow, and Recht [Reference Harrow, Recht and Chuang29], it was shown that a finite topologically generating set saturates this bound if their corresponding mixing operator on $L^2(\mathrm {SU}(2))$ is close in some sense to the projection operator onto constant functions. Using the work of Lubotzky, Phillips, and Sarnak [Reference Lubotzky, Phillips and Sarnak40, Reference Lubotzky, Phillips and Sarnak41], they deduce that the so-called V-gates (5.9)

(5.9) $$ \begin{align} V_x= \frac{1}{\sqrt{5}} \begin{pmatrix} 1+2i & 0 \\ 0 & 1-2i \end{pmatrix}, \quad \quad V_y= \frac{1}{\sqrt{5}} \begin{pmatrix} 1 & 2 \\ -2 & 1 \end{pmatrix}, \quad \quad V_z= \frac{1}{\sqrt{5}} \begin{pmatrix} 1 & 2i \\ 2i & 1 \end{pmatrix} \end{align} $$

are efficiently universal. For explicit algorithms, see [Reference Blass, Bocharov and Gurevich5, Reference Bocharov, Gurevich and Svore6, Reference Ross50]. These algorithms use solutions of certain grid problems and diophantine equations.

Before explicit algorithms were constructed for these V-gates, analogous questions for the more standard Clifford $+T$ -gates

(5.10) $$ \begin{align} H= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad \quad T= \begin{pmatrix} 1 & 0 \\ 0 & e^{\pi i / 4} \end{pmatrix} , \end{align} $$

were studied in [Reference Giles and Selinger27, Reference Kliuchnikov, Maslov and Mosca34–Reference Kliuchnikov, Maslov and Mosca36, Reference Ross and Selinger51, Reference Selinger56].

For the Fibonacci category, it was shown in [Reference Kliuchnikov, Bocharov and Svore33] that the generating gate set coming from the standard $\sigma _1, \sigma _2$ (or an equivalent set) is efficiently universal, in the sense that in order to approximate an arbitrary element of $\mathrm {PSU}(2)$ up to an error $\varepsilon $ , one needs a braid whose length is order $\mathrm {Log}(\frac {1}{\varepsilon })$ . This is the fastest possible speed-up that could be achieved from the Solovay–Kitaev theorem. Their proof is number-theoretic.

Based on experimental evidence, we conjecture that our operators $\sigma _1, \sigma _1^{-1}, N^{-1} \sigma _1 \sigma _2 \sigma _1 N, N^{-1} \sigma _1^{-1} \sigma _2^{-1} \sigma _1^{-1} N$ are efficiently universal. It is well known that an optimal generator of $\mathrm {U}(1)$ is $e^{2 \pi i \phi }$ , where $\phi $ is the golden ratio (up to an integer). See [Reference Graham and van Lint28, Reference Stier59] for a more in-depth analysis. This then tells us which $\alpha =4\phi $ modulo $1$ gives the best approximations of diagonal matrices, which in turn may improve the length of our quantum circuit even more.

There are also notions of golden and super golden gate sets due to Sarnak [Reference Sarnak53] and developed in [Reference Evra and Parzanchevski17, Reference Parzanchevski and Sarnak49]. By definition, these gates cover the Lie group in an optimal way and there are efficient algorithms to write an arbitrary element in the Lie group in terms of the generating gate set. For a precise definition, see [Reference Evra and Parzanchevski17, Definition 2.8]. Proving that a gate set is golden or super golden uses some deep theorems in number theory. The so-called icosahedral super golden gates enjoy certain advantages over other sets [Reference Blackman and Stier3]. It would be interesting to prove that the gates we consider here are golden. They cannot be super golden since, by definition, the elements of the generating gate set must be of finite order, and the elements $\sigma _1, \sigma _2$ that we consider certainly do not possess this property.

6 Unitary representations of the braid group: singular part of the category

In this section, we will consider the singular part of the category from Section 3.2. Representations of the braid group coming from morphism spaces here, lie in discrete subgroups of the corresponding unitary groups and thus have no value for quantum computation.

The simple module $V_0$ in the singular part of the category is projective. Recall that in Figure 1, a basis for the module $P_0$ which is the projective cover of ${\mathbb {C}}^H_0$ is given. There is a unique isomorphism up to scalar $\phi \colon V_0 \otimes V_0 \cong P_0$ , where

$$\begin{align*}\phi(v_0 \otimes v_0) = w^R, \quad \phi(v_1 \otimes v_1) = w^L, \quad \phi(v_0 \otimes v_1) = w^H, \quad \phi(v_1 \otimes v_0) = w^S+q w^H. \end{align*}$$

There is an isomorphism $V_0^{\otimes 3} \cong V_0 \otimes P_0 \cong ({\mathbb {C}}_2 \oplus {\mathbb {C}}_0^{\oplus 2} \oplus {\mathbb {C}}_{-2}) \otimes V_0$ .

So, we have a potential model for a qubit

$$\begin{align*}\mathrm{Hom}(V_0^{}, V_0^{\otimes 3}) = \operatorname{Span} \{ {\pi_1}, {\pi_2} \}, \end{align*}$$

where $\pi _1$ and $\pi _2$ are defined as follows:

$$\begin{align*}\pi_1^{}(v_0)= v_1 \otimes v_0 \otimes v_0 + q^{-1} v_0 \otimes v_1 \otimes v_0, \quad \quad \pi_1^{}(v_1)= v_1 \otimes v_0 \otimes v_1 + q^{-1} v_0 \otimes v_1 \otimes v_1 , \end{align*}$$

$$\begin{align*}\pi_2^{}(v_0)= v_0 \otimes v_1 \otimes v_0 + q^{-1} v_0 \otimes v_0 \otimes v_1, \quad \quad \pi_2^{}(v_1)= v_1 \otimes v_1 \otimes v_0 + q^{-1} v_1 \otimes v_0 \otimes v_1. \end{align*}$$

We depict these morphisms by

where the vertices labeled $1$ and $2$ denote the induced maps $P_0 \otimes V_0 \rightarrow V_0$ coming from the maps $\pi _1$ and $\pi _2$ under the identification $\phi \colon V_0 \otimes V_0 \cong P_0$ .

In this basis, the matrices of the braid group generators $\sigma _1$ and $\sigma _2$ are given by

(6.1) $$ \begin{align} {\sigma_1}= q^{\frac{1}{2}} \begin{pmatrix} 1 & -q \\ 0 & 1 \end{pmatrix} , \quad \quad \sigma_2= q^{\frac{1}{2}} \begin{pmatrix} 1 & 0 \\ -q & 1 \end{pmatrix}. \end{align} $$

Then, one could compute their Hermitian adjoints in $\mathrm {Hom}(V_0^{\otimes 3}, V_0^{}).$

$$ \begin{align*} {\pi_1^{\dagger}}(v_0 \otimes v_0 \otimes v_0) = 0 & & {\pi_2^{\dagger}}(v_0 \otimes v_0 \otimes v_0) = 0 \\ {\pi_1^{\dagger}}(v_1 \otimes v_0 \otimes v_0) = q v_0 & &{\pi_2^{\dagger}}(v_1 \otimes v_0 \otimes v_0) = 0 \\ {\pi_1^{\dagger}}(v_0 \otimes v_1 \otimes v_0) = v_0 & &{\pi_2^{\dagger}}(v_0 \otimes v_1 \otimes v_0) = qv_0 \\ {\pi_1^{\dagger}}(v_0 \otimes v_0 \otimes v_1) = 0 & & {\pi_2^{\dagger}}(v_0 \otimes v_0 \otimes v_1) = v_0 \\ {\pi_1^{\dagger}}(v_1 \otimes v_1 \otimes v_0) = 0 & & {\pi_2^{\dagger}}(v_1 \otimes v_1 \otimes v_0) = q v_1 \\ {\pi_1^{\dagger}}(v_1 \otimes v_0 \otimes v_1) = q v_1 & & {\pi_2^{\dagger}}(v_1 \otimes v_0 \otimes v_1) = v_1 \\ {\pi_1^{\dagger}}(v_0 \otimes v_1 \otimes v_1) = v_1 & & {\pi_2^{\dagger}}(v_0 \otimes v_1 \otimes v_1) = 0 \\ {\pi_1^{\dagger}}(v_1 \otimes v_1 \otimes v_1) = 0 & & {\pi_2^{\dagger}}(v_1 \otimes v_1 \otimes v_1) = 0. \end{align*} $$

It is straightforward to check

$$\begin{align*}\pi_i^{\dagger} \pi_j^{} = \begin{cases} \mathrm{Id} &\text{ if } i \neq j \\ 0 &\text{ otherwise }. \end{cases} \end{align*}$$

Thus, the vector space $ \mathrm {Hom}(V_0^{}, V_0^{\otimes 3}) $ has a non-degenerate Hermitian pairing

(6.2) $$ \begin{align} B = {\mathsf d}(V_{0}) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. \end{align} $$

The dual space is

$$\begin{align*}\mathrm{Hom}(V_0^{\otimes 3}, V_0^{}) = \operatorname{Span} \{ \pi_1^{\dagger}, \pi_2^{\dagger} \}. \end{align*}$$

These basis vectors are graphically depicted as follows:

Recall the endomorphism $x \colon P_0 \rightarrow P_0$ (3.11), which becomes identified with an endomorphism $x \colon V_0 \otimes V_0 \rightarrow V_0 \otimes V_0$ .

Lemma 6.2 We have equalities of morphisms

$$\begin{align*}(x \otimes \mathrm{Id}) \circ \pi_2^{} = \pi_1^{} \quad \quad (x \otimes \mathrm{Id}) \circ \pi_1^{} =0. \end{align*}$$

It is straightforward to check that the endomorphism x of $V_0 \otimes V_0$ satisfies the property: $x^{\dagger }=x$ .

Lemma 6.3 We have equalities of morphisms

$$\begin{align*}\pi_2^{\dagger} \circ (x \otimes \mathrm{Id}) = \pi_1^{\dagger} \quad \quad \pi_1^{\dagger} \circ (x \otimes \mathrm{Id}) = 0. \end{align*}$$

One could calculate the matrices for $\sigma _1^{\dagger }, \sigma _2^{\dagger }$ with respect to the indefinite pairing (6.2):

$$\begin{align*}\sigma_1^{\dagger} = q^{-\frac{1}{2}} \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix} , \quad \quad \sigma_2^{\dagger} = q^{-\frac{1}{2}} \begin{pmatrix} 1 & 0 \\ q & 1 \end{pmatrix}. \end{align*}$$

More generally, consider the vector space

$$\begin{align*}\mathcal{H}_{n,0} = \mathrm{Hom}(V_0, V_0^{\otimes 2n+1}). \end{align*}$$

Proof The fact that there is a (possibly indefinite) unitary representation of the braid group follows from [Reference Geer, Lauda, Patureau-Mirand and Sussan23, Proposition 5.8]. The dimension is determined by a straightforward character calculation.

Part of a basis of the space is given by trees in (6.3).

(6.3)

At each vertex, where two edges labeled $V_0$ meet to produce an edge labeled $P_0$ , we assume this vertex is the unique isomorphism (up to scalar) $\phi \colon V_0 \otimes V_0 \rightarrow P_0$ . At each vertex, where edges labeled $P_0$ and $V_0$ meet and output $V_0$ , there are two linearly independent morphisms. Thus, this subspace has dimension $2^n$ . The signature of this subspace is calculated easily using the computation for $\mathcal {H}_{1,0}$ .