Clifford group and stabilizer states from Chern-Simons theory

TL;DR

This paper constructs Clifford group generators and stabilizer states using Chern-Simons theory for specific Kac-Moody algebras, extending previous results in quantum algebra and topological quantum computation.

Contribution

It introduces a novel method to derive Clifford group elements and stabilizer states from Chern-Simons theory for certain algebraic structures, expanding the theoretical framework.

Findings

Constructed Clifford group generators from Chern-Simons theory for SU(2)1, U(N)N,N(K+N), and SU(N)1.

Extended previous algebraic results to new classes of Kac-Moody algebras.

Provided a topological quantum field theory perspective on stabilizer states.

Abstract

The construction of generators of the Clifford group and of stabilizer states from Chern-Simons theory is presented for the Kac-Moody algebras SU(2)1, U(N)N,N(K+N) with N = 2 and K = 1, and SU(N)1 extending results of Salton, et. al.

Equations

$$N_{ab}^c\text{ with a+b=cmod2}$$

$$S_{ab}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right),a,b=0,1$$

$$T_{ab}=\exp \left[2\pi i\left(h_a-\frac{c}{24}\right)\right]{\delta }_{ab}$$

$$h_a=\frac{C_2(a)}{3}$$

$$C_2(a)=j(j+1)=\frac{a}{4}(a+2)$$

$$T_{ab}=\exp \left[2\pi i\left(\frac{a(a+2)}{12}-\frac{1}{24}\right)\right]{\delta }_{ab}$$

$$c_{ADD}|a⟩|b⟩=|a⟩|a+b\mathrm{mod}2⟩$$

$$\frac{1}{\sqrt{2}}(\omega {)}_{ab}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

$$({\omega }^{-1/12}){\omega }^{a(a+2)/6}{T}_{ab}=\left(\begin{array}{cc}1& 0\\ 0& \omega \end{array}\right)=P_{ab}.$$

$${\mathcal{H}}_{T^2}^{{\otimes }_n}=(\text{mathbbm}{C}^2{)}^{{\otimes }_n}$$

$$U(N{)}_{K,N(N+K)}=\text{faktor}\left[SU(N{)}_K\times U(1{)}_{N(K+N)}\right]{\mathbb{Z}}_N$$

$$Q\equiv r\mathrm{mod}N$$

$$(R,Q)\simeq (\sigma (R),Q+N+K)$$

$$(R,Q)\simeq (R,Q+N(N+K))$$

$$0\le Q<N(N+K).$$

$$0\le l_N\le \dots \le l_1\le K.$$

$$S_{AB}=\sqrt{\frac{N}{N+K}}{S}_{ab}{e}^{-\text{nicefrac}2\pi i{Q}_A{Q}_{B}N(N+K)}$$

$$T_{AB}=\exp \left[2\pi i\left(h_A-\frac{c}{24}\right)\right]{\delta }_{AB}$$

$$c=\frac{N(NK+1)}{K+N}$$

$$h_A=\frac{\frac{1}{2}{C}_2(A)}{K+N}$$

$$h(R,Q)$$

$$=\frac{1}{2}\frac{C_2(R)}{(K+N)}+\frac{Q^2}{2N(N+K)}$$

$$=\frac{1}{2}\frac{C_2(R,Q)}{K+N}$$

$$T_{AB}=T_{ab}\exp \left[2\pi i\left(h_a+\frac{Q^2}{2N(N+K)}-\frac{c}{24}\right)\right]{\delta }_{AB}.$$

$$\text{tensor}N{}_{AB}^C$$

$$=\text{tensor}N{}_{ab}^c\sum _{Q_D}\exp \left[-2\pi i\frac{\left(Q_A+Q_B-Q_C\right)Q_D}{N(N+K)}\right]$$

$$=\text{tensor}N{}_{ab}^c{\delta }_{(Q_A+Q_B-Q_C)}$$

$$a+b=c\mathrm{mod}N,$$

$$\text{tensor}N{}_A{}_B^C=\text{tensor}N{}_a{}_b^c{\delta }_{Q_A+Q_B-Q_C},$$

$$\text{tensor}N{}_a{}_b^c,\text{with}a+b=c\mathrm{mod}N.$$

$$C_{ADD}|a⟩|b⟩=|a⟩|a+b\mathrm{mod}N⟩$$

$$S_{ab}=\frac{(-i{)}^{N(N-1)/2}}{\sqrt{N}(N+1{)}^{(N-1)/2}}\det M(a,b)$$

Full text

1 Introduction

We continue the study initiated by Salton, et. al. [1] of entanglement from topology in Chern-Simons theory. This is a topological quantum field theory in an arena where the Euclidean path integral provides a map between geometry and states. For these three-dimensional quantum field theories there is a mapping $Z$, the functional integral, which relates the 3-dimensional manifold $\mathcal{M}$ to a probability amplitude $Z(\mathcal{M})$. If $\mathcal{M}$ has a boundary, the boundary field configuration must be specified. That is, the path integral selects a state $\ket{\mathcal{M}}$ in a Hilbert space $\mathcal{H}_{\partial\mathcal{M}}$ associated to the boundary field configuration. If the boundary consists of several multiply connected components, $\mathcal{H}_{\partial\mathcal{M}}=\otimes^{n}_{i=1}\mathcal{H}_{\Sigma_{i}}$, where $\partial\mathcal{M}=\cup_{i=1}^{n}\Sigma_{i}$. This means that the different components $\Sigma_{i}$ are not coupled, so that the Euclidean path integral factorizes.

In this paper we focus on the preparation of stabilizer states constructed from Chern-Simons theory defined on the n-torus Hilbert space.

In section 2 we discuss the Kac-Moody algebra $SU(2)_{1}$, while in section 3 it is shown that the unitary Kac-Moody algebra [2] $U(N)_{k,N(N+k)}$ for $N=2$, $k=1$ shares the conclusions of section 2. Therefore Chern-Simons theory for both $SU(2)_{1}$, and $U(2)_{1,6}$ allow one to prepare arbitrary stabilizer states in $\mathcal{H}_{T^{2}}^{\otimes_{n}}$. The Clifford group is then constructed from Clifford gates applied to $\ket{0}^{\otimes_{n}}$. The Choi-Jamiolkowski isomorphism then allows the preparation of an arbitrary element of the Clifford group [3, 4].

In section 4 the analysis is extended to $SU(N)_{1}$, while in section 5 related issues are discussed.

2 $SU(2)_{1}$

The basis of the Kac-Moody algebra for $SU(2)_{1}$ is given in terms of Young tableau restricted to a single column, with basis $a=0,1$ where $a$ denotes the numbers of boxes of the tableau. We present the generators of the Clifford group for $SU(2)_{1}$ in terms of this basis, then the fusion matrix is

[TABLE]

The modular transformation matrices are [5], with standard normalization

[TABLE]

and [6]

[TABLE]

where $c=1$ is the central charge for $SU(2)_{1}$ and the conformal dimension

[TABLE]

with $C_{2}(a)$ the quadratic Casimir operator for the representation. In terms of angular momentum $j=a/2$, so that

[TABLE]

thus

[TABLE]

The generators of the Clifford group are the Hadamard gate, the phase gate, and the controlled addition gate $c_{ADD}$, which satisfies [3, 4]

[TABLE]

Equation (2.2) shows that $S_{ab}$ is in fact the Hadamard gate: It is convenient to define $\omega=\exp i\pi$. Then in (2.2)

[TABLE]

is the Hadamard gate. The phase gate $P_{ab}$ is

[TABLE]

Given (2.1) and (2.2) we inherit the construction of Figure $3$ of Salton et al [1] to construct the copy tensor, $c_{ADD}$, and a perfect tensor. Therefore for $SU(2)_{1}$ Chern-Simons theory one can prepare any stabilizer on the $n$-torus Hilbert space

[TABLE]

In section 5 we present this in a broader context.

3 $U(N)_{K,N(K+N)}$ for $N=2,K=1$

A

In order to understand the special case $U(2)_{1,6}$, we first present the representations of the general case, which is described in detail in section 2 of ref. [2], and which is summarized here. The essential feature is that

[TABLE]

which requires $K$ to be odd for consistency. Representations $(R,Q)$ of $SU(N)_{K}\times U(1)_{N(N+K)}$ must satisfy

[TABLE]

where $r$ is the number of boxes of the Young tableau associated to $R$. There is an equivalence relation

[TABLE]

where $\sigma$ is the simple current of $SU(N)_{K}$. Applying the simple current $N$ times, where $\sigma^{N}=1$, one obtains the equivalence

[TABLE]

so that $Q$ is restricted to the range

[TABLE]

The $U(N)$ representations $(R,Q)$ can be characterized by the extended Young tableau $\mathcal{R}$ with row lengths $\bar{l}_{i}\in\mathbb{Z}$, ($i=1$ to $N$). There is exactly one extended tableau $\mathcal{R}$ which satisfies

[TABLE]

Hence the primary fields of $U(N)_{K,N(K+N)}$, where $K$ is odd, are in one to one correspondence with the Young tableaux $\mathcal{R}$ with no more than $N$ rows and $K$ columns. The number of such tableaux is $\binom{N+K}{N}$.

The modular transformation matrix for the $U(N)_{K,N(K+N)}$ character is [2]

[TABLE]

where $S_{ab}$ is that of $SU(N)_{K}$. The subscripts $A$ or $a$ indicate whether one refers to $U(N)_{K,N(K+N)}$ or $SU(N)_{K}$. The modular transformation matrix for $\tau\to\tau+1$ is [2]

[TABLE]

where the central charge

[TABLE]

and

[TABLE]

where $h_{A}=h(R,Q)$, and

[TABLE]

and $Q=r\mod N$ from (3.2). Therefore

[TABLE]

The fusion matrix is

[TABLE]

where

[TABLE]

together with (3.2), describes the fusion matrix $\tensor{N}{{}_{ab}^{c}}$, as well as $Q$ charge conservation by virtue of (3.2).

B $N=2,K=1$; $U(2)_{1,6}$

Now specialize to $U(2)_{1,6}$, making use of the review of $U(N)_{K,N(K+N)}$ to discuss this case. The fusion matrix $\tensor{N}{{}_{ab}^{c}}$ satisfies $c=a+b\mod 2$, as does the fusion matrix of $SU(2)_{1}$, where $a,b,$ and $c$ are the number of boxes of a single column tableau. Then

[TABLE]

where now $Q_{A}=a\mod 2$ from (3.2), so that charge conservation is automatically satisfied. Restricting (3.7) to $N=2,K=1$, one can again inherit the construction of [1] to construct the $C_{ADD}$ gate. The Helmholtz gate and phase gate are essentially that of $SU(2)$, combined with $Q$ conservation.

4 $SU(N)_{1}$

Representations of $SU(N)_{1}$ are described by a single column tableau with $0\leq N-1$ boxes. The fusion tensor is

[TABLE]

Therefore, this case closely parallels that of $SU(2)_{1}$ in section 2. The $C_{ADD}$ gate will satisfy

[TABLE]

so that with $S_{ab}$, $\tensor{N}{{}_{a}{}_{b}^{c}}$, and the phase gate, one constructs a basis for the Clifford group.

The modular transformation matrix normalized as in [5] is

[TABLE]

where $a,b=0$ to $N-1$,

[TABLE]

with $i=1$ to $N$ and

[TABLE]

where

[TABLE]

and $r(a)=\sum_{i=1}^{N-1}l_{i}(a)$ is the total number of boxes in the reduced Young tableau corresponding to the representation $a$.

The modular transformation matrix [6]

[TABLE]

with the conformal dimension is

[TABLE]

with the quadratic Casimir operator

[TABLE]

with $r(a)$ as above, and

[TABLE]

where from (4.6)

[TABLE]

Note $C_{2}(R)$ is quadric in $a$, vanishing when $a=0$.

Given $\tensor{N}{{}_{a}{}_{b}^{c}}$ with $c=a+b\mod N$, the detailed expression for $S_{ab}$ is not explicitly needed for the construction of Figure 3 of [1]. From (4.7) to (4.11) one extracts an overall phase factor to obtain the phase gate

[TABLE]

This, together with $C_{ADD}$ and $S_{ab}$, generates the Clifford group [3, 4].

5 Related issues

In sections 2, 3 and 4 we generalized the results of Theorem 1 of [1] to $SU(2)_{1}$, $U(2)_{1,6}$ and $SU(N)_{1}$. The unifying feature which makes this possible is that the fusion tensors are all of the form

[TABLE]

This, together with the modular transformation matrices $T_{ab}$ and $S_{ab}$, allows one to construct the phase gate, and the $C_{ADD}$ gate, while $S_{ab}$ is (conjectured to be) an appropriate generalization of the Helmholtz gate. Thus, one can repeat the strategy of Figures 3 and 4, ff. of [1]. In particular, for $SU(N)_{1}$ the computation of Figure 4(a) gives $N^{2}$, that of Figure 4(b) yields $N^{4}$, etc., which means that the entanglement entropy of an arbitrary many torus system is

[TABLE]

As a consequence the $SU(N)_{1}$ fusion tensor is equivalent to $g=1$ GHZ state, independent of $N$ for the states that can be distilled between $A$, $B$, and $C$ for $\partial M=A\cup B\cup C$ for an arbitrary tripartition of the boundary torii. Similarly, one should expect analogous results for $Sp(N)_{1}$ Chern-Simons theory since the fusion matrix satisfies (5.1).

It is known that a universal topological computer based on $SU(2)_{K}$ requires $K\geq 3$ [7]. This is exemplified by the work of [8] which presents a detailed construction for $SU(2)_{3}$. Then level-rank duality shows that a universal topological quantum computer can be based on $SU(3)_{2}$ [9]. Level-rank duality then suggests that a universal topological quantum computer can be based on $SU(K)_{2}$, where $K\geq 3$.

Other applications of entanglement in Chern-Simons theory are discussed in refs. In particular, refs. [10, 11, 12, 13, 14] consider stabilizer states in $U(1)$ Chern-Simons theory.

In that context we follow [12], where upper-bounds are derived for $SU(2)_{K}$, given an $n$-component link $\mathcal{L}^{n}\subset S^{3}$, and two sublinks $\mathcal{L}_{A}^{m}$ and $\mathcal{L}_{\bar{A}}^{n-m}\subset S^{3}$ such that a separating surface $\Sigma_{A|\bar{A}}\subset S^{3}$ is a connected, compact, oriented two-dimensional surface without boundary, where: $(1)$ $\mathcal{L}_{A}^{m}$ is contained in the handlebody inside $\Sigma_{A|\bar{A}}$, $(2)$ $\mathcal{L}_{\bar{A}}^{n-m}$ is contained in the handlebody outside $\Sigma_{A|\bar{A}}$, and $(3)$ $\Sigma_{A|\bar{A}}$ does not intersect any of the components of $\mathcal{L}^{n}$. Reference [12] presents a trivial upper-bound on the entanglement entropy for $SU(2)_{K}$, i.e.

[TABLE]

and a tighter upper bound

[TABLE]

Specialize to $SU(2)_{1}$, where $S_{00}=1/\sqrt{2}$ and $S_{01}=-1/\sqrt{2}$, so that (5.3) becomes

[TABLE]

while (5.4) for $\min(g_{\Sigma})\geq 1$ becomes

[TABLE]

Further for $SU(N)_{1}$, recalling (5.2) we expect

[TABLE]

6 Acknowledgements

We thank Isaac Cohen, Alastair Grant-Stuart, and Andrew Rolph for assistance in the preparation of the paper.