# Quantum cohomology determined with negative structure constants present

Ryan M. Shifler  Department of Mathematical Sciences, Henson Science Hall,
Salisbury University, Salisbury, MD 21801<EMAIL_ADDRESS>
###### Abstract.

Let $\mathrm{IG}:=\mathrm{IG}(2,2n+1)$ denote the odd symplectic Grassmannian
of lines which is a horospherical variety of Picard rank 1. The quantum
cohomology ring $\mathrm{QH}^{*}(\mathrm{IG})$ has negative structure
constants. For $n\geq 3$, we give a positivity condition that implies the
quantum cohomology ring $\mathrm{QH}^{*}(\mathrm{IG})$ is the only quantum
deformation of the cohomology ring $\mbox{H}^{*}(\mathrm{IG})$ up to the
scaling of the quantum parameter. This is a modification of a conjecture by
Fulton.

###### 2010 Mathematics Subject Classification:

Primary 14N35; Secondary 14N15, 14M15

##  1\. Introduction

Let $\mathrm{IG}:=\mathrm{IG}(2,2n+1)$ denote the odd symplectic Grassmannian
of lines which is a horospherical variety of Picard rank 1. This is the
parameterization of two dimensional subspaces of $\mathbb{C}^{2n+1}$ that are
isotropic with respect to a (necessarily) degenerate symmetric form. The
quantum cohomology ring $(\mathrm{QH}^{*}(\mathrm{IG}),\star)$ is a graded
algebra over $\mathbb{Z}[q]$ where $q$ is the quantum parameter and $\deg
q=2n$. The ring has a Schubert basis given by
$\\{\tau_{\lambda}:\lambda\in\Lambda\\}$ where

$\Lambda:=\\{(\lambda_{1},\lambda_{2}):2n-1\geq\lambda_{1}\geq\lambda_{2}\geq-1,\mbox{
}\lambda_{1}>n-2\Rightarrow\lambda_{1}>\lambda_{2},\mbox{ and
}\lambda_{2}=-1\Rightarrow\lambda_{1}=2n-1\\}.$

We define $|\lambda|=\lambda_{1}+\lambda_{2}$ for any $\lambda\in\Lambda$.
Then $\deg(\tau_{\lambda})=|\lambda|$. The ring multiplication is given by
$\tau_{\lambda}\star\tau_{\mu}=\sum_{\nu,d}c_{\lambda,\mu}^{\nu,d}q^{d}\tau_{\nu}$
where $c_{\lambda,\mu}^{\nu,d}$ is the Gromov-Witten invariant of
$\tau_{\lambda}$, $\tau_{\mu}$, and the Poicaré dual of $\tau_{\nu}$. Unlike
the homogeneous $G/P$ case, the Gromov-Witten invariants may be negative. For
example, in $\mathrm{IG}(2,5)$, we have

$\tau_{(3,-1)}\star\tau_{(3,-1)}=\tau_{(3,1)}-q\mbox{ and
}\tau_{(2,1)}\star\tau_{(3,-1)}=-\tau_{(3,2)}+q\tau_{1}.$

The quantum Pieri rule has only non-negative coefficients and is stated in
Proposition 2.3. See [Shifler:thesis, pech:quantum, pasquier:horo,
mihalcea.shifler:qhodd, LMS, curvIG, mihai:odd, GPPS] for more details on
$\mathrm{IG}$.

######  Definition 1.1.

For any given collection of constants
$\\{a_{\mu}\in\mathbb{Q}:\mu\in\Lambda\\}$, a quantum deformation with the
corresponding basis $\\{\sigma_{\lambda}:\lambda\in\Lambda\\}$ is defined as a
solution to the following system:

$\displaystyle\tau_{\lambda}=\sigma_{\lambda}+\sum_{j\geq
1}\left(\sum_{|\mu|+2nj=|\lambda|}a_{\mu}q^{j}\sigma_{\mu}\right),\lambda\in\Lambda.$

######  Remark 1.2.

It is always possible to re-scale the quantum parameter $q$ by a positive
factor $\alpha>0$, or equivalently, multiply each Gromov-Witten invariant
$c_{\lambda,\mu}^{\nu,d}$ by $\alpha^{-d}$. We only consider the $\alpha=1$
case in this manuscript.

To contextualize the significance of quantum deformations we review the
following conjecture by Fulton for Grassmannians and its extension to a more
general case by Buch in [buch.Pos.Grass, Conjecture 1].

######  Conjecture 1.

Let $X=G/P$ be any flag variety of simply laced Lie type. Then the Schubert
basis of $\mathrm{QH}^{*}(X)$ is the only homogeneous $\mathbb{Q}[q]$-basis
that deforms the Schubert basis of $\mbox{H}^{*}(X,\mathbb{Q})$ and multiplies
with non-negative structure constants.

This conjecture is shown to hold for any Grassmannian and a few other examples
in [buch.Pos.Grass]. Li and Li proved the result for symplectic Grassmannians
$\mathrm{IG}(2,2n)$ with $n\geq 3$ in [PosIG]. The condition that the root
system of $G$ be simply laced is necessary since the conjecture fails to hold
for the Lagrangian Grassmannian $\mathrm{IG}(2,4)$ as shown in
[buch.Pos.Grass, Example 6]. However, this conjecture is not applicable to
$\mathrm{IG}(2,2n+1)$ since negative coefficients appear in quantum products
for any $n$. We are able to modify the positivity condition on Fulton’s
conjecture to arrive at a uniqueness result for quantum deformations.

######  Definition 1.3.

For $\mathrm{IG}(2,2n+1)$ we will use (**) to denote the condition that the
coefficients of the quantum multiplication of $\sigma_{(1,1)}$ and any
$\sigma_{\mu}$ in the basis $\\{\sigma_{\lambda}:\lambda\in\Lambda\\}$ are
polynomials in $q$ with non-negative coefficients.

We are ready to state the main result.

######  Theorem 1.4.

Let $n\geq 3$. Suppose that $\\{\sigma_{\lambda}:\lambda\in\Lambda\\}$ is a
quantum deformation of the Schubert basis
$\\{\tau_{\lambda}:\lambda\in\Lambda\\}$ of $\mathrm{QH}^{*}(\mathrm{IG})$
such that Condition (**) holds. Then $\tau_{\lambda}=\sigma_{\lambda}$ for all
$\lambda\in\Lambda$.

######  Remark 1.5.

The methods used in this manuscript are motivated by those of Li and Li in
[PosIG]. In particular, multiplication by $\tau_{(1,1)}$ is critical to prove
the result.

In Section 2 we prove the main result for the $|\lambda|<2n$ case, state the
quantum Pieri rule, and give identities for later in the paper; in Section 3
we prove the main result for the $|\lambda|=2n$ case; and in Section 4 we
prove the main result for the $|\lambda|>2n$ case. Theorem 1.4 follows from
Propositions 2.1, 3.1, and 4.2.

##  2\. Preliminaries

We begin the section with a proposition that reduces the number of possible
quantum deformations that we need to check. This is accomplished by using the
grading. The proposition also states our main result for the $|\lambda|<2n$
case.

######  Proposition 2.1.

We have the following results.

  1. (1)

We have that
$\tau_{\lambda}=\sigma_{\lambda}+\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}.$

  2. (2)

If $|\lambda|<2n$ then $\tau_{\lambda}=\sigma_{\lambda}$.

###### Proof.

The first part follows since $|\lambda|\leq(2n-1)+(2n-2)=4n-3<4n=2\deg q$ for
any $\lambda\in\Lambda$. The second part follows immediately from the grading.
∎

######  Remark 2.2.

A key part of our proof strategy is to utilize Part (2) of Proposition 2.1.
The result is often used without reference in the paper.

Next we state the quantum Pieri rule for $\mathrm{IG}(2,2n+1)$.

######  Proposition 2.3.

[pech:quantum, Theorem 1] The quantum Pieri rule.

$\tau_{1}\star\tau_{a,b}=\left\\{\begin{array}[]{ll}\tau_{a+1,b}+\tau_{a,b+1}&\quad\mbox{if
}a+b\neq 2n-3\mbox{ and }a\neq 2n-1,\\\
\tau_{a,b+1}+2\tau_{a+1,b}+\tau_{a+2,b-1}&\quad\mbox{if }a+b=2n-3,\\\
\tau_{2n-1,b+1}+q\tau_{b}&\quad\mbox{if }a=2n-1\mbox{ and }0\leq b\leq
2n-3,\\\ \tau_{2n-1}&\quad a=2n-1\mbox{ and }b=-1,\\\
q(\tau_{2n-1,-1}+\tau_{2n-2})&\quad a=2n-1\mbox{ and
}b=2n-2.\par\end{array}\right.$
$\tau_{1,1}\star\tau_{a,b}=\left\\{\begin{array}[]{ll}\tau_{a+1,b+1}&\quad\mbox{if
}a+b\neq 2n-4,2n-3\mbox{ and }a\neq 2n-1,\\\
\tau_{a+1,b+1}+\tau_{a+2,b}&\quad\mbox{if }a+b=2n-4\mbox{ or }2n-3,\\\
q\tau_{b+1}&\quad\mbox{if }a=2n-1\mbox{ and }b\neq 2n-3,\\\
q(\tau_{2n-1,-1}+\tau_{2n-2})&\quad a=2n-1\mbox{ and
}b=2n-3.\end{array}\right.$

The quantum Pieri rule yields identities that are useful to clarify the proofs
in this paper. In particular, multiplication by
$\displaystyle\Pi_{i=1}^{t}\tau_{(1,1)}$ is a significant part of our strategy
to prove Theorem 1.4.

######  Lemma 2.4.

We have the following identities.

  1. (1)

Let $t\leq n-2$. Then

$\sigma_{(t,t)}=\tau_{(t,t)}=\Pi_{i=1}^{t}\tau_{(1,1)}=\Pi_{i=1}^{t}\sigma_{(1,1)}.$

  2. (2)

Let $|\lambda|\geq 2n$, $t:=2n-\lambda_{1}$, and $\lambda_{2}+t\neq 2n-2$.
Then

$\tau_{(t,t)}\star\tau_{\lambda}=q\tau_{(\lambda_{2}+t)}.$

  3. (3)

We have that

$\displaystyle\Pi_{i=1}^{n-1}\tau_{(1,1)}=\tau_{(n-1,n-1)}+\tau_{(n,n-2)}.$

  4. (4)

Let $\lambda=(n+1,n-1)$. Then

$\displaystyle\left(\Pi_{i=1}^{n-1}\tau_{(1,1)}\right)\star\tau_{\lambda}=q\tau_{(2n-1,-1)}+q\tau_{(2n-2)}.$

  5. (5)

If $2t+|\mu|\leq 2n-3$ then

$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\mu}=\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)\star\tau_{\mu}=\tau_{(\mu_{1}+t,\mu_{2}+t)}=\sigma_{(\mu_{1}+t,\mu_{2}+t)}.$

  6. (6)

If $2t+|\mu|=2n-2$ or $2n-1$ then

$\displaystyle\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)\star\tau_{\mu}=\tau_{(\mu_{1}+t,\mu_{2}+t)}+\tau_{(\mu_{1}+t+1,\mu_{2}+t-1)}.$

###### Proof.

Part (1) is clear since $2t\leq 2n-4$. For Part (2),
$\tau_{(1,1)}\star\tau_{(t-1,t-1)}\star\tau_{\lambda}=\tau_{(1,1)}\star\tau_{(2n-1,\lambda_{2}+t-1)}=q\tau_{(\lambda_{2}+t)}.$
For Part (3),
$\tau_{(1,1)}\star\Pi_{i=1}^{n-2}\tau_{(1,1)}=\tau_{(1,1)}\star\tau_{(n-2,n-2)}=\tau_{(n-1,n-1)}+\tau_{(n,n-2)}.$
For Part (4),
$\displaystyle\tau_{(1,1)}\star\left(\Pi_{i=1}^{n-2}\tau_{(1,1)}\right)\star\tau_{\lambda}=\tau_{(1,1)}\star\tau_{(2n-1,2n-3)}=q\tau_{(2n-1,-1)}+q\tau_{(2n-2)}.$
Part (5) is clear. For Part (6), we have
$\tau_{(1,1)}\star\left(\Pi_{i=1}^{t-1}\tau_{(1,1)}\right)\star\tau_{\mu}=\tau_{(1,1)}\star\tau_{(\mu_{1}+t-1,\mu_{2}+t-1)}=\tau_{(\mu_{1}+t,\mu_{2}+t)}+\tau_{(\mu_{1}+t+1,\mu_{2}+t-1)}.$
This completes the proof. ∎

##  3\. The $|\lambda|=2n$ case

In this section we will assume that $|\lambda|=2n$. By Proposition 2.1 it must
be the case that $\tau_{\lambda}=\sigma_{\lambda}+aq.$ We show $a\leq 0$ in
two parts. Lemma 3.2 considers the $\lambda_{1}\geq n+2$ case and Lemma 3.3
consider the $\lambda=(n+1,n-1)$ case. The strategy for both lemmas is to
multiply both sides of $\sigma_{\lambda}=\tau_{\lambda}-aq$ by
$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)$ and rewrite the right side of the
equation as a sum of basis elements in
$\\{\sigma_{\lambda}:\lambda\in\Lambda\\}.$ We show $a\geq 0$ in Lemma 3.4 as
a straight forward application of the quantum Pieri rule. The main proposition
of this section is stated next.

######  Proposition 3.1.

Let $|\lambda|=2n$. If $\tau_{\lambda}=\sigma_{\lambda}+aq$ and Condition (**)
holds then $\tau_{\lambda}=\sigma_{\lambda}$.

###### Proof.

This is an immediate consequence of Lemmas 3.2, 3.3, and 3.4. ∎

We state and prove the next lemma.

######  Lemma 3.2.

Let $|\lambda|=2n$ and $\lambda_{1}\geq n+2$. If
$\tau_{\lambda}=\sigma_{\lambda}+aq$ and Condition (**) holds then $a\leq 0$.

###### Proof.

Let $t:=2n-\lambda_{1}\leq n-2$. Note that $t+\lambda_{1}=2n$. Then we have
the following by multiplying $\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)$ to
both sides of $\sigma_{\lambda}=\tau_{\lambda}-aq$ and using Part (1) of Lemma
2.4.

$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\lambda}=\tau_{(t,t)}\star\tau_{\lambda}-a\sigma_{(t,t)}q.$

By Part (2) of Lemma 2.4 we have
$\tau_{(t,t)}\star\tau_{\lambda}=q\tau_{(\lambda_{2}+t)}=q\sigma_{(\lambda_{2}+t)}.$
So,

$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\lambda}=q\sigma_{(\lambda_{2}+t)}-a\sigma_{(t,t)}q.$

It follows from Condition (**) that $a\leq 0$. ∎

We will now prove $a\leq 0$ for the $\lambda=(n+1,n-1)$ case.

######  Lemma 3.3.

Let $\lambda=(n+1,n-1)$. If $\tau_{\lambda}=\sigma_{\lambda}+aq$ and Condition
(**) holds then $a\leq 0$.

###### Proof.

Recall from Part (3) of Lemma 2.4 that
$\Pi_{i=1}^{n-1}\tau_{(1,1)}=\tau_{(n-1,n-1)}+\tau_{(n,n-2)}$ and from Part
(4) of Lemma 2.4 we have that
$\displaystyle\left(\Pi_{i=1}^{n-1}\tau_{(1,1)}\right)\star\tau_{\lambda}=q\tau_{(2n-1,-1)}+q\tau_{(2n-2)}.$
Multiplying by $\Pi_{i=1}^{n-1}\sigma_{(1,1)}$ to both sides of
$\sigma_{\lambda}=\tau_{\lambda}-aq$ and substituting in the identities yields

$\displaystyle\left(\Pi_{i=1}^{n-1}\sigma_{(1,1)}\right)\star\sigma_{\lambda}$
$\displaystyle=$
$\displaystyle\left(\Pi_{i=1}^{n-1}\tau_{(1,1)}\right)\star\tau_{\lambda}-a\left(\Pi_{i=1}^{n-1}\tau_{(1,1)}\right)q$
$\displaystyle=$ $\displaystyle
q\tau_{(2n-1,-1)}+q\tau_{(2n-2)}-aq\left(\tau_{(n-1,n-1)}+\tau_{(n,n-2)}\right)$
$\displaystyle=$ $\displaystyle
q\sigma_{(2n-1,-1)}+q\sigma_{(2n-2)}-aq\left(\sigma_{(n-1,n-1)}+\sigma_{(n,n-2)}\right).$

It follows from Condition (**) that $a\leq 0$. ∎

We conclude the section by showing that $a\geq 0$ in the next lemma.

######  Lemma 3.4.

Let $|\lambda|=2n$. If $\tau_{\lambda}=\sigma_{\lambda}+aq$ and Condition (**)
holds then $a\geq 0$.

###### Proof.

Let $\lambda^{j}=(n+1+j,n-1-j)$ for all $j=0,1,2,...,n-2$. Assume that
$\tau_{\lambda^{j}}=\sigma_{\lambda^{j}}+a_{j}q.$ Then for all $0\leq j\leq
n-2$ it follows from the quantum Pieri rule that
$\tau_{(1,1)}\star\tau_{(n+j,n-2-j)}=\tau_{\lambda^{j}}.$ Since
$\tau_{(n+j,n-2-j)}=\sigma_{(n+j,n-2-j)}$ by Part (2) of Lemma 2.1, we have
that

$\displaystyle\sigma_{(1,1)}\star\sigma_{(n+j,n-2-j)}=\tau_{(1,1)}\star\tau_{(n+j,n-2-j)}=\tau_{\lambda^{j}}=\sigma_{\lambda^{j}}+a_{j}q.$

It follows from Condition (**) that $a_{j}\geq 0$ for all $j=0,\cdots,n-2$. ∎

##  4\. The $|\lambda|>2n$ case

In this section we will assume that $|\lambda|>2n$. By Proposition 2.1 it must
be the case that
$\tau_{\lambda}=\sigma_{\lambda}+\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}.$
We show $a_{\mu}\leq 0$ in Lemma 4.1. The strategy for this lemma is to
multiply both sides of
$\sigma_{\lambda}=\tau_{\lambda}-\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}$
by $\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)$ and rewrite the right side of
the equation as a sum of basis elements in
$\\{\sigma_{\lambda}:\lambda\in\Lambda\\}.$ Then we will state and prove the
main result of this section as Proposition 4.2.

######  Lemma 4.1.

Let $|\lambda|>2n$. If
$\tau_{\lambda}=\sigma_{\lambda}+\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}$
and Condition (**) holds then $a_{\mu}\leq 0$.

###### Proof.

If $|\lambda|>2n$ then $\lambda_{1}\geq n+2$. Let $t:=2n-\lambda_{1}\leq n-2$.
We will multiply $\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)$ to both sides of
$\sigma_{\lambda}=\tau_{\lambda}-\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}.$
By Part (2) of Lemma 2.4 we have that
$\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)\star\tau_{\lambda}=q\tau_{(\lambda_{2}+t)}.$
Since $\lambda_{2}+t<\lambda_{1}+t=2n$, we have that

$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\lambda}=q\sigma_{(\lambda_{2}+t)}-\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\left(\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}\right).$

Next observe that $2t+|\mu|=2t+|\lambda|-2n=2n-\lambda_{1}+\lambda_{2}\leq
2n-1.$ So, one of the following must occur:

  * •

If $2t+|\mu|\leq 2n-3$ then by Part (5) of Lemma 2.4 we have

$\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\mu}=\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)\star\tau_{\mu}=\tau_{(\mu_{1}+t,\mu_{2}+t)}=\sigma_{(\mu_{1}+t,\mu_{2}+t)}.$

  * •

If $2t+|\mu|=2n-1$ or $2n-2$ then by Part (6) of Lemma 2.4 we have

$\displaystyle\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\mu}$
$\displaystyle=$
$\displaystyle\left(\Pi_{i=1}^{t}\tau_{(1,1)}\right)\star\tau_{\mu}=\tau_{(\mu_{1}+t,\mu_{2}+t)}+\tau_{(\mu_{1}+t+1,\mu_{2}+t-1)}$
$\displaystyle=$
$\displaystyle\sigma_{(\mu_{1}+t,\mu_{2}+t)}+\sigma_{(\mu_{1}+t+1,\mu_{2}+t-1)}.$

After substituting, one of the following must occur:

  * •

The first possible outcome is

$\displaystyle\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\lambda}=q\sigma_{(\lambda_{2}+t)}-\left(\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{(\mu_{1}+t,\mu_{2}+t)}\right).$

  * •

The second possible outcome is

$\displaystyle\left(\Pi_{i=1}^{t}\sigma_{(1,1)}\right)\star\sigma_{\lambda}=q\sigma_{(\lambda_{2}+t)}-\left(\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\left(\sigma_{(\mu_{1}+t,\mu_{2}+t)}+\sigma_{(\mu_{1}+t+1,\mu_{2}+t-1)}\right)\right).$

It follows from Condition (**) that $a_{\mu}\leq 0$. ∎

######  Proposition 4.2.

Let $|\lambda|>2n$. If
$\tau_{\lambda}=\sigma_{\lambda}+\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}$
and Condition (**) holds then $a_{\mu}=0$.

###### Proof.

We proceed by way of induction. Suppose $\tau_{\lambda}=\sigma_{\lambda}$ for
all $|\lambda|\leq s$ where $s\geq 2n$. Consider $|\lambda|=s+1.$ Since
$|\lambda|\geq 2n+1$ for $|\lambda|=s+1$, and by an application of the quantum
Pieri rule, we have that
$\tau_{(1,1)}\star\tau_{(\lambda_{1}-1,\lambda_{2}-1)}=\tau_{\lambda}.$
Observe that
$\tau_{(\lambda_{1}-1,\lambda_{2}-1)}=\sigma_{(\lambda_{1}-1,\lambda_{2}-1)}$
by the inductive hypothesis. Then

$\sigma_{(1,1)}\star\sigma_{(\lambda_{1}-1,\lambda_{2}-1)}=\tau_{(1,1)}\star\tau_{(\lambda_{1}-1,\lambda_{2}-1)}=\tau_{\lambda}=\sigma_{\lambda}+\sum_{|\mu|+2n=|\lambda|}a_{\mu}q\sigma_{\mu}.$

So, $a_{\mu}\geq 0$ by Condition (**). By Lemma 4.1, we have $a_{\mu}\leq 0$.
Therefore, $a_{\mu}=0$. The result follows. ∎

## References


# Fold-Transversal Clifford Gates for Quantum Codes

Nikolas P. Breuckmann  Department of Computer Science, University College
London, WC1E 6BT London, United Kingdom<EMAIL_ADDRESS>http://nikobreu.website Simon Burton  Institute of Physics, Jagiellonian
University, Łojasiewicza 11, 30-348 Kraków, Poland<EMAIL_ADDRESS>http://arrowtheory.com

###### Abstract

We generalize the concept of folding from surface codes to CSS codes by
considering certain dualities within them. In particular, this gives a general
method to implement logical operations in suitable LDPC quantum codes using
transversal gates and qubit permutations only.

To demonstrate our approach, we specifically consider a $[[30,8,3]]$
hyperbolic quantum code called Bring’s code. Further, we show that by
restricting the logical subspace of Bring’s code to four qubits, we can obtain
the _full_ Clifford group on that subspace.

##  1 Introduction

We show how symmetries of CSS quantum codes can be utilized to implement
encoded Clifford gates, using only transversal gates and qubit permutations.
Our scheme is not only trivially fault-tolerant, but it also incurs no time-
overhead and does not require any additional ancilla qubits. It can be
understood as a generalization of the concept of folding surface codes due to
Moussa [1], see Figure 1.

Folding of topological quantum codes is part of the folklore relating surface
codes and color codes: folded surface codes give color codes, or
alternatively, unfolded color codes give surface codes [2].

i Moussa’s original construction on a surface code. The $ZX$-duality $\tau$ is
reflecting along a diagonal.

ii The Steane code is self-dual, and so the identity permutation serves as a
$ZX$-duality.

iii On a $\\{5,5\\}$-tiled hyperbolic surface, a $ZX$-duality is shown
reflecting along a geodesic.

Figure 1: How $ZX$-dualities define folds of quantum codes. Qubits are shown
as grey circles. The green/blue faces correspond to the $Z$\- and $X$-checks.
The $ZX$-duality $\tau$ maps between the two types of checks. The codes (i)
and (iii) are built from the homology of a tiling (cellulation) shown in
black.

We generalize the concept of folding to arbitrary CSS codes by introducing
_fold-transversal gates_. While our approach works for any sufficiently
symmetric CSS code, we believe it is particularly suited to implement logical
gates in low-density parity-check (LDPC) quantum codes. These codes have
recently attracted a lot of attention, see [3] for a recent review. This is
partially due to the fact that constant overhead fault-tolerant quantum
computation can be realized with these codes [4, 5]. Recent developments have
shown that it is possible to construct high-performance LDPC quantum codes
from classical codes [6, 7, 8]. In particular, [8] gave a construction that
was conjectured to have optimal asymptotic parameter scaling, which was
subsequently proven in [9].

While the construction of LDPC quantum codes has made big advances, it is
still unclear how to best manipulate the encoded information. The constant
overhead in [4] was achieved by implementing gates via ancillary states.
However, in order to keep the qubit overhead constant in their proof, the
implementation of gates had to be serialized, making the scheme less
practical.

For specific families of LDPC quantum codes, a few techniques to fault-
tolerantly implement logical gates have been introduced. For hyperbolic
surface codes, Breuckmann et al. considered code deformations in order to
perform $\operatorname{CNOT}$-gates [10]. Further, Lavasani–Barkeshli
discussed the implementation of other Clifford gates on hyperbolic surface
codes by using ancillary states [11]. Krishna–Poulin [12] considered
generalizations of code deformation techniques of the surface code to
hypergraph product codes in order to implement Clifford gates. On the other
hand, Burton–Browne [13] showed that it is not possible to obtain transversal
logical gates outside of the Clifford group using certain hypergraph product
codes.

More recently, Cohen et al. generalized lattice surgery between two logical
operators of an arbitrary LDPC quantum code [14]. The surgery is facilitated
by an ancillary surface code that grows with the code distance. Hence, for
codes with large distance, it is necessary to serialize the quantum circuit to
avoid a large overhead, as in [4].

In comparison, the logical gates constructed in our work do not incur any
qubit overhead, as they are realized within the quantum code directly,
avoiding the need for ancilla qubits. We also avoid any time overhead, as
logical gates are implemented by a transversal circuit of single- and two-
qubit gates, which can be applied in parallel. This also means that our
implementation is compatible with any decoding scheme, as it can be executed
between rounds of error correction.

We explicitly construct fault transversal gates for a certain $[[30,8,3]]$
hyperbolic surface code that we call Bring’s code. We show that we can
generate a subgroup of the Clifford group $\mathcal{C}_{8}$ using transversal
circuits. By adding a single logical entangling gate, which can be obtained
using other techniques, we generate the full Clifford group $\mathcal{C}_{8}$.
Alternatively, when we restrict the logical subspace to four qubits, we can
generate all of $\mathcal{C}_{4}$, using transversal circuits only.

In this work we do not consider any constraints on the locality of qubit
interactions. In particular, we make use of qubit permutations, which in
principle, can map any qubit to any other qubit. Thus, in order to take full
advantage of our construction, a potential hardware architecture ideally
supports non-local interactions between qubits. Fortunately, there have been
several advances in experimental setups that can realize such unconstrained
connectivity. Linear optical quantum computation, for example, is naturally
flexible in terms of connectivity [15, 16, 17]. The Moussa folding
construction is used by the fusion-based photonic scheme proposed in [18]. In
a similar vain, modular architectures, in which modules are interlinked by a
photonic interface, allow for non-local connectivity [19, 20, 21]. Other
approaches to quantum computation, such as mobile qubits [22] or qubits
coupled to a common cavity mode [23, 24], even allow for fast any-to-any
connectivity between qubits. In particular, [25] proposes an architecture
based on Rydberg atoms, or alternatively, ion-traps, in which it is possible
to have fast, long-range interactions between hundreds of physical qubits.

Bring’s code is small enough that this scheme could be implemented on NISQ
hardware. Together with the fold-transversal gates it provides a simple set-up
in which fault-tolerant Clifford gates could be demonstrated.

In Section 2 we introduce the theory of _fold-transversal_ gates. When
discussing stabilizer codes, we largely employ the language of homology which
is introduced in Section 2.1. In Section 2.2 we give the definition of a _$ZX$
-duality_ of a homological code, and show how the symmetries of the code
interact with the $ZX$-dualities. We then apply this to a class of highly
symmetric two-dimensional hyperbolic surface codes in Section 2.3. These codes
all have $ZX$-dualities, which are described in Theorem 5. Armed with this
theory of $ZX$-dualities, in Section 2.4 we characterize when these dualities
give rise to _fold-transversal_ Clifford gates on the logical subspace of the
code. In Section 3 we discuss the example of _Bring’s code_ and work out in
detail the fold-transversal gates and their action on the logical subspace.

##  2 Folding along a $ZX$-duality

###  2.1 CSS quantum codes

A _stabilizer code_ is defined by a subgroup $\mathcal{S}$ of the Pauli group
$\mathcal{P}_{n}$ which is abelian and does not contain $-I$. We call the
elements of some distinguished set of generators the _(stabilizer) checks_ of
the stabilizer code. A stabilizer code is a _Calderbank-Shor-Steane (CSS)
code_ if there exists a generating set of $\mathcal{S}$ such that each
generator acts as either Pauli-$X$ or Pauli-$Z$ on all qubits in its support.
Hence, we can define a CSS code $C$ in terms of two binary matrices
$H_{X}\in\mathbb{F}_{2}^{r_{X}\times n}$ and
$H_{Z}\in\mathbb{F}_{2}^{r_{Z}\times n}$ where each row corresponds to the
support indication vector of a Pauli-$X$ or Pauli-$Z$ generator, respectively.
Note we do not require $H_{X}$ or $H_{Z}$ to have full rank. As $X$-type and
$Z$-type Pauli operators commute if and only if the overlap of their supports
contains an even number of qubits we must have

$\displaystyle H_{X}H_{Z}^{\top}=0.$ (1)

In the language of homology, a CSS code $C$ is equivalent to a chain complex
over the field $\mathbb{F}_{2}=\mathbb{Z}/2$:

$C=\\{\mathbb{F}_{2}^{r_{Z}}\xrightarrow{H_{Z}^{\top}}\mathbb{F}_{2}^{n}\xrightarrow{H_{X}}\mathbb{F}_{2}^{r_{X}}\\}.$

We can think of $H_{Z}^{\top}$ and $H_{X}$ as _boundary operators_
$\partial_{2}$ and $\partial_{1}$, respectively:

${C_{2}}$${C_{1}}$${C_{0}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$

where $C_{2}=\mathbb{F}_{2}^{r_{Z}}$, $C_{1}=\mathbb{F}_{2}^{n}$ and
$C_{0}=\mathbb{F}_{2}^{r_{X}}$. The elements of $C_{i}$ are called _$i$
-chains_. Note that the space of $i$-chains comes with a natural basis due to
the construction of $C_{i}$ as a free vector space on a basis. The fact that
“the boundary of a boundary is zero”, i.e. $\partial_{1}\,\partial_{2}=0$, is
guaranteed by Equation 1. The group generated by the $Z$-type stabilizer
operators $\mathcal{S}_{Z}$ corresponds to boundaries
$B_{1}=\operatorname{im}\partial_{2}$ and the $Z$-type Pauli-operators
normalizing the $X$-type stabilizer operators $N(\mathcal{S}_{X})_{Z}$
corresponds to cycles $Z_{1}=\ker\partial_{1}$. Similarly, $\mathcal{S}_{X}$
and $N(\mathcal{S}_{Z})_{X}$ correspond to coboundaries
$B^{1}=\operatorname{im}\delta_{0}$ and cocycles $Z^{1}=\ker\delta_{1}$,
respectively, where $\delta_{0}=\partial_{1}^{\top}=H_{X}^{\top}$ and
$\delta_{1}=\partial_{2}^{\top}=H_{Z}$. The $Z$-type logical operators of the
code are found as cycles modulo boundaries, which is the (first) _homology_ of
the code, $\mathcal{H}_{1}:=Z_{1}/B_{2}.$ Dually, the $X$-type logical
operators are cocycles modulo coboundaries. This is the (first) _cohomology_
of the code, $\mathcal{H}^{1}:=Z^{1}/B^{1}.$ A (co)cycle that is not trivial
modulo a (co)boundary is called an _essential (co)cycle_.

###  2.2 $ZX$-Dualities

Given a CSS Code $C=(H_{Z},H_{X})$, the _dual_ CSS code is

$C^{\top}=\\{\mathbb{F}_{2}^{r_{X}}\xrightarrow{H_{X}^{\top}}\mathbb{F}_{2}^{n}\xrightarrow{H_{Z}}\mathbb{F}_{2}^{r_{Z}}\\}.$

A _self-dual_ CSS code $C$ has $C=C^{\top}.$

An _isomorphism_ of CSS codes $\sigma:C\to C^{\prime}$ is a triple of
permutation matrices $(\sigma_{Z},\sigma_{n},\sigma_{X})$ where

$\displaystyle\sigma_{Z}$
$\displaystyle:\mathbb{F}_{2}^{r_{Z}}\to\mathbb{F}_{2}^{r^{\prime}_{Z}},$
$\displaystyle\sigma_{n}$
$\displaystyle:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}^{n^{\prime}},$
$\displaystyle\sigma_{X}$
$\displaystyle:\mathbb{F}_{2}^{r_{X}}\to\mathbb{F}_{2}^{r^{\prime}_{X}},$

such that the following diagram commutes:

${\mathbb{F}_{2}^{r_{Z}}}$${\mathbb{F}_{2}^{n}}$${\mathbb{F}_{2}^{r_{X}}}$${\mathbb{F}_{2}^{r^{\prime}_{Z}}}$${\mathbb{F}_{2}^{n^{\prime}}}$${\mathbb{F}_{2}^{r^{\prime}_{X}}}$$\scriptstyle{\sigma_{Z}}$$\scriptstyle{H^{\top}_{Z}}$$\scriptstyle{\sigma_{n}}$$\scriptstyle{H_{X}}$$\scriptstyle{\sigma_{X}}$$\scriptstyle{H^{\prime\top}_{Z}}$$\scriptstyle{H^{\prime}_{X}}$

The _automorphism group_ of a CSS code $C$ is the group of self-isomorphisms
$\mathrm{Aut}(C)=\\{\sigma:C\to C\\}.$

We are also interested in weaker notions of isomorphism between codes that
allow for swapping the $X$\- and $Z$-checks. Given a CSS code
$C=(H_{Z},H_{X})$ we define the _underlying classical_ code $C/\sim_{ZX}$ to
be given by the parity check matrix $H$ as:

$C/\sim_{ZX}:=\Bigl{\\{}\mathbb{F}_{2}^{n}\xrightarrow{H:=\left(\begin{smallmatrix}H_{X}\\\
H_{Z}\end{smallmatrix}\right)}\mathbb{F}_{2}^{r}\Bigr{\\}}.$

where $r:=r_{X}+r_{Z}.$ This classical code remembers all the checks of the
quantum code, but it has forgotten whether each check is an $X$\- or $Z$-type
check.

An _automorphism_ of a classical code with parity check matrix
$H:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}^{r}$ is a pair of permutation matrices
$(\sigma_{n},\sigma_{r})$ where

$\displaystyle\sigma_{n}$
$\displaystyle:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}^{n},$
$\displaystyle\sigma_{r}$
$\displaystyle:\mathbb{F}_{2}^{r}\to\mathbb{F}_{2}^{r},$

such that the following diagram commutes:

${\mathbb{F}_{2}^{n}}$${\mathbb{F}_{2}^{r}}$${\mathbb{F}_{2}^{n}}$${\mathbb{F}_{2}^{r}}$$\scriptstyle{\sigma_{n}}$$\scriptstyle{H}$$\scriptstyle{\sigma_{r}}$$\scriptstyle{H}$

Given a CSS code $C=(H_{Z},H_{X})$ we write the automorphisms of the
underlying classical code $C/\sim_{ZX}$ as

$\mathcal{G}:=\mathrm{Aut}(C/\sim_{ZX}).$

The subgroup of $\mathcal{G}$ that fixes the $X$\- and $Z$-sectors corresponds
precisely to $\mathrm{Aut}(C)$ defined above. In more detail, these elements
of $\mathcal{G}$ are pairs $(\sigma_{n},\sigma_{r})$ where $\sigma_{r}$ has
block form

$\sigma_{r}=\left(\begin{array}[]{cc}\sigma_{X}&0\\\
0&\sigma_{Z}\end{array}\right)$

and then $(\sigma_{Z},\sigma_{n},\sigma_{X})$ is the corresponding
automorphism of $C$.

The elements $\tau$ of $\mathcal{G}$ that swap the $X$\- and $Z$-sectors are
called _$ZX$ -dualities_. These are pairs $\tau=(\tau_{n},\tau_{r})$ such that
$\tau_{r}$ has the block form:

$\tau_{r}=\left(\begin{array}[]{cc}0&\tau_{Z}\\\
\tau_{X}&0\end{array}\right).$

In this case, $\tau=(\tau_{Z},\tau_{n},\tau_{X})$ is an isomorphism $\tau:C\to
C^{\top},$ and we call such a code _self- $ZX$-dual_. The set of all
$ZX$-dualities of $C$ we denote by $\mathcal{D}_{ZX}$.

######  Lemma 1.

Given a self-$ZX$-dual CSS code $C$, the set of $ZX$-dualities for $C$ is a
coset of $\mathrm{Aut}(C)$ in $\mathcal{G}$:

$\mathcal{D}_{ZX}=\tau\mathrm{Aut}(C),$

where $\tau$ is any choice of a $ZX$-duality for $C$.

###### Proof.

It is not hard to see that from a given $ZX$-duality $\tau:C\to C^{\top}$ and
an isomorphism $\sigma:C\to C$ we get another $ZX$-duality from the
composition $\tau\sigma$. Conversely, given another $ZX$-duality
$\tau^{\prime}=(\tau_{n}^{\prime},\tau_{r}^{\prime})$ we find that

$\displaystyle\tau_{r}^{-1}\tau^{\prime}_{r}$
$\displaystyle=\left(\begin{array}[]{cc}0&\tau_{Z}\\\
\tau_{X}&0\end{array}\right)^{-1}\left(\begin{array}[]{cc}0&\tau^{\prime}_{Z}\\\
\tau^{\prime}_{X}&0\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\tau_{X}^{-1}\tau^{\prime}_{X}&0\\\
0&\tau_{Z}^{-1}\tau^{\prime}_{Z}\end{array}\right)\in\mathrm{Aut}(C).$

Therefore we have shown the identity $\mathcal{D}_{ZX}=\tau\mathrm{Aut}(C).$ ∎

######  Corollary 2.

For a self-$ZX$-dual code $C$, the number of $ZX$-dualities equals the size of
the automorphism group of $C$:

$|\mathcal{D}_{ZX}|=|\mathrm{Aut}(C)|.$

Given CSS codes $C=(H_{Z},H_{X})$ and
$C^{\prime}=(H^{\prime}_{Z},H^{\prime}_{X})$ we can form the _direct sum_ CSS
code:

$C\oplus C^{\prime}:=(H_{Z}\oplus H^{\prime}_{Z},H_{X}\oplus H^{\prime}_{X}).$

######  Lemma 3.

Given a CSS code $C$ with $ZX$-duality $\tau:C\to C^{\top}$ then $C\oplus C$
has a $ZX$-duality without any fixed points.

###### Proof.

We define the $ZX$-duality given by $\tau\oplus\tau$ followed by the
permutation that swaps the two summands:

$C\oplus C\xrightarrow{\tau\oplus\tau}C^{\top}\oplus
C^{\top}\xrightarrow{\mathrm{swap}}C^{\top}\oplus C^{\top}=(C\oplus
C)^{\top}.$

∎

In what follows we will often abuse notation and write $\tau(i)$ for the
action of $\tau_{n}$ on the $i$-th basis vector of $\mathbb{F}_{2}^{n}$.

###  2.3 Hyperbolic surface codes

In this section we review the construction of hyperbolic triangle groups ([26]
Section 2.4), and how they give rise to two dimensional hyperbolic surface
codes [27, 28]. We then find the $ZX$-dualities in this context.

The Poincaré disc model of the hyperbolic plane is written as
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}.$ The group of orientation preserving
automorphisms of $\mathbb{D}$ is given by the Möbius transforms,

$\displaystyle\mathrm{Aut}(\mathbb{D}^{+})$
$\displaystyle=\Bigl{\\{}z\mapsto\frac{az+b}{cz+d}\ \Bigl{|}\
a,b,c,d\in\mathbb{R},$ $\displaystyle\ \ \ \ \ ad-bc=1\Bigr{\\}}$
$\displaystyle\cong\mathrm{PSL}(2,\mathbb{R}).$

By also including the reflection given by complex conjugation
$z\mapsto\bar{z}$, we generate the full automorphism group
$\mathrm{Aut}(\mathbb{D}).$

The Coxeter reflection group for the $\\{5,5\\}$-tiling of $\mathbb{D}$ has
presentation

$\displaystyle R_{5,5}:=\langle a,b,c\ |\
a^{2},b^{2},c^{2},(ab)^{5},(ac)^{2},(bc)^{5}\rangle.$

The generators $a,b,c$ here correspond to reflections that _destabilize_ a
vertex, edge, or face respectively. We can summarize this presentation using
the Coxeter diagram:

See Figure 2ii. The rotations $ab,ac,bc$ correspond to the rotation around a
face, edge or vertex respectively. The subgroup generated by these rotations
we write as

$R^{+}_{5,5}=\langle ab,ac,bc\rangle.$

This subgroup has index 2 in $R_{5,5}.$ Next we choose a normal subgroup
$\Gamma$ of $R^{+}_{5,5}$ with finite index, that acts freely on $\mathbb{D}.$
The surface we are interested in is the compact Riemann surface

$S=\mathbb{D}/\Gamma.$

This surface is tiled by triangles, which we can identify with the elements of
the group $\mathcal{G}^{+}:=R^{+}_{5,5}/\Gamma$ once we choose a fiducial tile
to act upon. Up to conjugation, the stabilizer subgroup of a face, edge or
vertex is generated by the rotations $ab,ac,bc\in\mathcal{G}^{+}$,
respectively. Therefore, the chain complex of our quantum code is obtained
from the free vector spaces generated by cosets of these:

$\displaystyle C=\Bigl{\\{}\ \mathbb{F}_{2}^{r_{Z}}$
$\displaystyle:=\mathbb{F}_{2}[\mathcal{G}^{+}/\langle ab\rangle]$
$\displaystyle\xrightarrow{H_{Z}^{\top}}\mathbb{F}_{2}^{n}$
$\displaystyle:=\mathbb{F}_{2}[\mathcal{G}^{+}/\langle ac\rangle]$
$\displaystyle\xrightarrow{H_{X}}\mathbb{F}_{2}^{r_{X}}$
$\displaystyle:=\mathbb{F}_{2}[\mathcal{G}^{+}/\langle bc\rangle]\
\Bigr{\\}}.$ (2)

Proceeding with the theory developed in Section 2.2 we now seek the
automorphism group $\mathcal{G}$ of the underlying classical code of $C$. We
can do this using another Coxeter reflection group $R_{5,4}$ for the
$\\{5,4\\}$-tiling of $\mathbb{D}$. This has presentation

$R_{5,4}:=\langle d,e,f\ |\
d^{2},e^{2},f^{2},(de)^{5},(df)^{2},(ef)^{4}\rangle.$

The generators $d,e,f$ here correspond to reflections that destabilize a
vertex, edge, or face respectively. The rotations $de,df,ef$ correspond to the
rotation around a face, edge or vertex respectively. We can summarize this
presentation using the Coxeter diagram:

See Figure 2i.

######  Lemma 4.

$R_{5,5}$ has an outer automorphism that swaps the generators $a,c$ while
fixing $b$. Moreover, $R_{5,5}$ is a subgroup of $R_{5,4}$ in which the outer
automorphism is conjugation by $f$.

###### Proof.

The outer automorphism is clear from the presentation of $R_{5,5}$. There is a
group monomorphism $R_{5,5}\rightarrowtail R_{5,4}$ given by sending the
generators

$a\mapsto e,\ b\mapsto d,\ c\mapsto f^{-1}ef,$

and conjugation of these generators by $f$ has the required effect. ∎

######  Theorem 5.

For the hyperbolic code $C$ defined in Section 2.3 we have the following:

$\displaystyle\mathrm{Aut}(C)$ $\displaystyle=R_{5,5}/\Gamma,$
$\displaystyle\mathrm{Aut}(C/\sim_{ZX})$ $\displaystyle=R_{5,4}/\Gamma.$

Furthermore, $C$ is self-$ZX$-dual as exhibited by the $ZX$-duality $\tau=f\in
R_{5,4}$.

###### Proof.

(Sketch.) By Lemma 4 we can think of $R_{5,5}$ as a subgroup of $R_{5,4}$,
where the latter has an additional generator $f$. The reflection $f$ divides
each of the fundamental triangles of $R_{5,5}$ with internal angles $\pi/2$,
$\pi/5$, $\pi/5$ into two triangles with internal angles $\pi/2$, $\pi/4$,
$\pi/5$, cf. Figures 2i and 2ii.

By studying the cosets which give rise to $C$ and $C/\sim_{ZX}$ in the
universal cover, and noting that $\Gamma$ is normal and acts fixed-point free,
our arguments regarding the cosets are preserved under the covering map
induced by $\Gamma$ onto the finite surface. ∎

Using this result and Lemma 1 we obtain all the $ZX$-dualities of our
hyperbolic code $C$ as

$\mathcal{D}_{ZX}=fR_{5,5}/\Gamma.$

A similar result holds for other symmetric Coxeter reflection groups $R_{l,l}$
for $l\geq 3$. There is always a subgroup inclusion $R_{l,l}\rightarrowtail
R_{l,4}$ coming from the outer automorphism of $R_{l,l}.$ In the case of
$R_{3,3}$ this group is $\operatorname{S}_{4}$ the (finite) symmetry group of
the tetrahedron, and $R_{3,4}$ is the symmetry group of the cube. The
inclusion $R_{3,3}\rightarrowtail R_{3,4}$ corresponds to the inclusion of a
tetrahedron inside a cube. When $l=4$, we get the group $R_{4,4}$ which is the
(infinite) symmetry group of the square two-dimensional lattice. The inclusion
$R_{4,4}\rightarrowtail R_{4,4}$ exhibits an inclusion of this lattice into a
$1/\sqrt{2}$ scaled and rotated copy of the same lattice.

i The $\\{5,4\\}$-tiling of the Poincaré hyperbolic disc model. Vertices,
edges and faces are marked with red, blue and green dots respectively. This
tiling has symmetry group the Coxeter reflection group $R_{5,4}$.

ii The $\\{5,5\\}$-tiling of the Poincaré hyperbolic disc model. Vertices,
edges and faces are marked with red, blue and green dots respectively. This
tiling has symmetry group the Coxeter reflection group $R_{5,5}$.

iii Bring’s curve as a quotient of the $\\{5,5\\}$-tiled hyperbolic disc.
Faces are identified according to the number scheme. Each edge is associated
with a qubit, shown with a grey circle, and faces/vertices define
$Z$/$X$-check operators.

Figure 2:  Hyperbolic codes are defined using the homology of compact
hyperbolic surfaces. We construct these tiled surfaces as quotients of the
$\\{5,5\\}$-tiling (ii), for example Bring’s code (iii). We find
$ZX$-dualities of the hyperbolic codes by forgetting the distinction between
vertices and faces in (ii), which yields the $\\{5,4\\}$-tiling (i).

###  2.4 Fold-Transversal Clifford Gates

Some of the $ZX$-dualities in $\mathcal{D}_{ZX}$ can be used to implement
logical gates using circuits of low, constant depth.

Given a $ZX$-duality $\tau$, a unitary operator on the $n$ qubits is called
_fold-transversal_ when it is a tensor product of single qubit unitaries, and
two-qubit unitaries with support on the orbits $\\{(i,\tau(i))\\}_{i=1,...,n}$
of $\tau.$

A unitary operator on the $n$ qubits of a CSS code $C$ is called an _encoded
logical gate_ when it commutes with the codespace projector.

####  2.4.1 Hadamard-type fold-transversal gates

The simplest such gate is of what we call _Hadamard-type_ , meaning it
exchanges logical $X$-operators with logical $Z$-operators. Given a
$ZX$-duality $\tau\in\mathcal{D}_{ZX}$ we define this as the following fold-
transversal operator on the $n$ physical bits:

$\displaystyle H_{\tau}=\bigotimes_{\begin{subarray}{c}i=1,..,n\\\
i<\tau(i)\end{subarray}}\operatorname{SWAP}_{i,\tau(i)}\
\bigotimes_{i=1}^{n}H_{i}$

######  Theorem 6.

Given a $ZX$-duality $\tau$ for a CSS code $C$ then $H_{\tau}$ is an encoded
logical gate.

###### Proof.

The operator $H_{\tau}$ acts on elements of the Pauli group $\mathcal{P}_{n}$
via conjugation. Using the identities $H_{i}X_{i}H_{i}^{\dagger}=Z_{i}$ and
$H_{i}Z_{i}H_{i}^{\dagger}=X_{i}$, as well as the fact that $\tau$ maps
between $X$\- and $Z$-checks, we see that $H_{\tau}$ leaves the stabilizer
group $\mathcal{S}$ invariant. Thus, the normalizer of the stabilizer group
$N(\mathcal{S})$ must also be invariant. In other words, $H_{\tau}$ leaves the
code space invariant and maps logical operators onto logical operators. ∎

Note that $H_{\tau}$ is generally not simply a logical Hadamard, exchanging
pairs of logical Pauli-operators $\bar{X}_{i}$ with $\bar{Z}_{i}$, as $\tau$
does not have to be compatible with the choice of logical basis. More
severely, in general, we do not have $H_{\tau}^{2}=I$. This is only the case
if $\tau^{2}=\operatorname{id}$.

When we construct $H_{\tau}$ for the surface code, using the $ZX$-duality
$\tau$ shown in Figure 1i, we obtain the Hadamard-gate described by Moussa
[1].

####  2.4.2 Phase-type fold-transversal gates

Given a $ZX$-duality $\tau\in\mathcal{D}_{ZX}$ we define the following fold-
transversal operator on the $n$ physical qubits:

$\displaystyle S_{\tau}=\bigotimes_{\begin{subarray}{c}i=1,...,n\\\
i=\tau(i)\end{subarray}}S_{i}^{(\dagger)}\
\bigotimes_{\begin{subarray}{c}i=1,...,n\\\
i<\tau(i)\end{subarray}}\operatorname{CZ}_{i,\tau(i)}$

for some choice of $S_{i}$ and $S_{i}^{\dagger}$ on the fixed points
$i=\tau(i).$ We call this a _phase-type_ gate.

######  Theorem 7.

Let $\tau\in\mathcal{D}_{ZX}$ be a $ZX$-duality for a CSS code $C$, which is
self-inverse ($\tau^{2}=\operatorname{id}$) and whose set of stabilized qubits
is even, i.e. $|\\{i=1,...,n\mid\tau(i)=i\\}|\equiv 0\mod 2$. Further, suppose
that each $X$-check $X^{\otimes s}$ with support $s\subset\\{1,\dotsc,n\\}$
has (a) even overlap with the set of invariant qubits
$\\{i=1,...,n|\tau(i)=i\\}$ and (b) contains an even number of two element
orbits $\\{i,\tau(i)\\}$, where $i\neq\tau(i)$. Then there exists a phase-type
gate $S_{\tau}$ that is an encoded logical gate for $C$.

###### Proof.

Due to condition (a) there exists a partition of the set of invariant qubits
of $\tau$ into $A$ and $B$ such that each $X$-check is half supported on $A$
and half supported on $B$. We define the following unitary operator on the $n$
physical bits:

$\displaystyle S_{\tau}=\bigotimes_{i\in A}S_{i}\ \bigotimes_{j\in
B}S_{j}^{\dagger}\ \bigotimes_{\begin{subarray}{c}i=1,...,n\\\
i<\tau(i)\end{subarray}}\operatorname{CZ}_{i,\tau(i)}$

Let us verify that $S_{\tau}$ leaves the stabilizer group $\mathcal{S}$
invariant. We do this by showing that $S_{\tau}$ sends check operators to
other elements of $\mathcal{S}$. By virtue of $S_{\tau}$ being Clifford, this
then extends to a permutation of $\mathcal{S}$. As $S_{\tau}$ is diagonal in
the computational basis, it commutes with all $Z$-checks. Consider an
$X$-check $X^{\otimes s}$ with support $s\subset\\{1,\dotsc,n\\}$. First, let
us assume that $s$ does not contain qubits invariant under $\tau$, i.e.
$s\cap(A\cup B)=\emptyset$. Note that for single-qubit Pauli-$X$ operators we
have that

$\operatorname{CZ}_{i,j}X_{i}\operatorname{CZ}_{i,j}^{\dagger}=X_{i}Z_{j}.$

Further, due to condition (b) we have that $|s\cap\tau(s)|$ is even, so that
we can commute the Pauli-$X$ and Pauli-$Z$ operators. Hence, the operator
$X^{\otimes s}$ is mapped by $S_{\tau}$ onto the operator $X^{\otimes
s}Z^{\otimes\tau(s)}$. By virtue of $\tau$ mapping $X$-checks onto $Z$-checks,
we know that $Z^{\otimes\tau(s)}\in\mathcal{S}$. Now, let us assume that
$s\cap(A\cup B)\neq\emptyset$. For Pauli-$X$ operators we have that
$S_{i}X_{i}S_{i}^{\dagger}=iX_{i}Z_{i}$ and
$S_{i}^{\dagger}X_{i}S_{i}=-iX_{i}Z_{i}$. The check $X^{\otimes s}$ is mapped
onto

$i^{|s\cap A|}(-i)^{|s\cap B|}X^{\otimes s}Z^{\otimes\tau(s)}.$

Each phase gate $S_{i}$ ($S_{i}^{\dagger}$) introduces an unwanted factor of
$i$ ($-i$). However, by assumption on $\tau$, we are guaranteed that $|s\cap
A|=|s\cap B|$, so that these factors cancel out. ∎

As for the Hadamard-type gates, we have shown that the phase-type gate
$S_{\tau}$ leaves the code space invariant and maps logical operators onto
logical operators. We stress again that the action of $S_{\tau}$ will
generally be different from applying a logical $\bar{S}_{i}$ on each logical
qubit. However, we do have $S_{\tau}^{2}=I$ by construction.

When we construct $S_{\tau}$ for the surface code, using the $ZX$-duality
$\tau$ shown in Figure 1i, we obtain the $S$-gate described by Moussa [1].
Note there is another $ZX$-duality that rotates the surface code by 90
degrees. However, this rotation fixes a single physical qubit and so we cannot
construct a phase-type encoded logical gate from this $ZX$-duality.

The next result shows we can get a phase-type encoded logical gate by
_stacking_ two copies of a self-$ZX$-dual code.

######  Corollary 8.

Let $\tau\in\mathcal{D}_{ZX}$ be a $ZX$-duality for a CSS code $C$ which is
self-inverse, then $C\oplus C$ has phase-type encoded logical gate:

$\displaystyle
S_{\tau}=\bigotimes_{i=1,...,n}\operatorname{CZ}_{i,n+\tau(i)}.$

###### Proof.

We use the $ZX$-duality defined in Lemma 3. This $ZX$-duality has no fixed
points and is also self-inverse so the result follows from the previous
theorem. ∎

###  2.5 The Symplectic Group

Instead of dealing with unitary Clifford operators directly, we consider a
well-known relation to the _symplectic group_
$\operatorname{Sp}_{2n}(\mathbb{F}_{2})$.

By definition, we have that the Pauli group $\mathcal{P}_{n}$ is a normal
subgroup of the Clifford group $\mathcal{C}_{n}$. The quotient group
$\mathcal{C}_{n}/\mathcal{P}_{n}$ is isomorphic to the symplectic group
$\operatorname{Sp}_{2n}(\mathbb{F}_{2})$ [29]. In other words, the following
short sequence is exact:

${\mathcal{P}_{n}}$${\mathcal{C}_{n}}$${\operatorname{Sp}_{2n}(\mathbb{F}_{2})}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

where $\iota$ is the canonical embedding map and $\pi$ the quotient map. This
implies that any element of the Clifford group $U\in\mathcal{C}_{n}$ can be
specified by an element of the symplectic group
$\pi(U)\in\operatorname{Sp}_{2n}(\mathbb{F}_{2})$ and a representative of the
Pauli group $P\in\mathcal{P}_{n}$. This fact is well-known and used e.g. for
sampling random elements of the Clifford group [30]. Note that we can
trivially implement logical Pauli operators $\bar{P}\in\mathcal{P}_{k}$
transversally.

From now on, we will use the notation $\tilde{U}:=\pi(U)$ for any operator
$U\in\mathcal{C}_{n}$.

##  3 Bring’s code

We demonstrate our construction on a hyperbolic surface code, called Bring’s
code, which encodes 8 logical qubits into 30 physical qubits. Hyperbolic codes
are good candidates for constructing interesting fold-transversal gates, as
they are finite-rate and have large symmetry groups [27]. One reason for
choosing Bring’s code in particular is that its symmetries can be visualized
as a 3D polyhedron (see Figure 3). Also, it is small enough to analyze the
group generated by the fold-transversal gates with a computer algebra system,
such as GAP [31]. Larger hyperbolic codes already encode too many logical
qubits to do so.

We briefly discuss some further examples, such as hypergraph product codes,
balanced product codes and block codes, in Appendix B.

###  3.1 Construction

Using the notation from Section 2.3 for Bring’s code we define $\Gamma$ equal
to the normal closure in $R^{+}_{5,5}$ of the group generated by $(abcb)^{3}.$
This single generator is enough to define the quotient of the hyperbolic disc
$\mathbb{D}$, see Figure 2iii. Somewhat surprisingly, this quotient has an
immersion into three dimensional space known as the _great dodecahedron_. This
is a Kepler-Poinsot polyhedron derived from the dodecahedron, see Figure 3. It
has pentagonal faces, of which five meet at any vertex. By identifying the
edges of the great dodecahedron with qubits and associating vertices/faces
with $X$-/$Z$-checks, we obtain a quantum code with parameters $[[30,8,3]]$.

Figure 3:  The great dodecahedron representation of Bring’s code. Faces are
pentagons which intersect. A single face is highlighted yellow. Each edge is
associated with a qubit and vertices/faces define $X$/$Z$-check operators.
This figure was made with Stella4D [32].

From a topological perspective, the great dodecahedron is a tiling
(cellulation) of a genus-4 surface, $S=\mathbb{D}/\Gamma.$ This tiling
consists of 12 vertices, 30 edges and 12 faces. We refer to the derived code
as _Bring’s code_ , as the underlying surface is known as _Bring’s curve_ ,
named after Erland Samuel Bring, who studied the algebraic equation that
defines the surface as a complex curve. Bring’s code was first defined in [27]
and further analyzed in [10] and [33].

From Theorem 5, the group of symmetries of Bring’s code is
$\mathrm{Aut}(C)=R_{5,5}/\Gamma$ which in this case is found to be the
permutation group $\operatorname{S}_{5}$. In terms of the presentation of
$R_{5,5}$ we have,

$\displaystyle\mathrm{Aut}(C)$ $\displaystyle=R_{5,5}/\Gamma$
$\displaystyle=\langle a,b,c\mid a^{2},b^{2},c^{2},$ $\displaystyle\ \ \ \ \
(ab)^{5},(ac)^{2},(bc)^{5},(abcb)^{3}\rangle$
$\displaystyle=\operatorname{S}_{5}.$

We also make the definitions $r:=ab$ and $s:=bc$ which are rotations around a
face and vertex, respectively. Also from Theorem 5, we have the larger
symmetry group which includes symmetries that exchange vertices and faces,

$\displaystyle\mathrm{Aut}(C/\sim_{ZX})$ $\displaystyle=R_{5,4}/\Gamma$
$\displaystyle=\operatorname{S}_{5}\times\operatorname{C}_{2}.$

###  3.2 Logical bases

In the homological representation of CSS codes, the logical $Z$-operators
correspond to a basis of the homology group $\mathcal{H}_{1}$. Essential
cycles on Bring’s surface can be associated to the 20 faces of the
icosahedron, which forms the convex hull of the great dodecahedron, see Figure
3. First we note that a chain consisting of three edges of a triangle is a
closed loop and hence it is clearly a cycle, i.e. an element of
$Z_{1}=\ker\partial_{1}$. Direct computation shows that it is also not a
boundary, i.e. an element of $B_{1}=\operatorname{im}\partial_{2}$. In fact,
the set of all 20 chains associated to the triangles span the first homology
group $\mathcal{H}_{1}$. However, as $k=\dim\mathcal{H}_{1}=8$, this spanning
set can not be independent. Picking a suitable subset of these triangles gives
rise to a basis.

Figure 4: The basis $\mathfrak{B}_{\mathcal{H}_{1}}$ of the first homology
group $\mathcal{H}_{1}$. Each element is a cycle of length three shown with
thick lines of one color. Equivalently, this is a choice of logical
$Z$-operators $\bar{Z}_{1},\dotsc,\bar{Z}_{8}$. Each triangular face of this
planar embedding (of which there are 20, including the outer face) corresponds
to a face of the convex hull of the great dodecahedron, which is the
icosahedron.

A particularly nice basis is shown in Figure 4. The basis elements correspond
to eight faces of the icosahedron. These faces are related by the action of
the subgroup
$\operatorname{C}_{2}\times\operatorname{C}_{2}\times\operatorname{C}_{2}\subset\operatorname{S}_{5}$.

Figure 5: The convex hull of the great dodecahedron is the icosahedron. We can
place a cube inside the icosahedron such that the eight vertices of the cube
lie on the mid-points of eight faces of the icosahedron. The symmetries of the
cube are also symmetries of the icosahedron. The orientation of the cube
inside the icosahedron corresponds to a choice of basis in $\mathcal{H}_{1}$
where each triangle touched by a vertex of the cube is a basis vector, cf.
Figure 4.

This choice of basis has a geometric representation that can be seen in Figure
5. We can place the icosahedron inside a cube, such that eight faces of the
icosahedron can be associated with the eight vertices of the cube. The group
$\operatorname{C}_{2}\times\operatorname{C}_{2}\times\operatorname{C}_{2}$ is
a subgroup of the symmetry group of the cube which has a regular action on its
vertices. It is generated by reflections corresponding to the three planes
intersecting the cube in the middle. This placement of the icosahedron inside
the cube is not unique and different placements correspond to different
choices of a basis.

The logical $X$-operators correspond to a basis of the cohomology group
$\mathcal{H}^{1}$. Similarly as for the homology basis, we can associate the
essential cocycles with triangles of the icosahedron: for a given triangle
there are three edges (not belonging to the triangle) which are adjacent to
exactly one of the triangle’s vertices and are not neighboring any of the
triangle’s edges. The chain formed by these edges is an essential cocycle. By
associating eight triangles with the vertices of a cube we obtain a basis of
$\mathcal{H}^{1}$. Unfortunately, choosing the bases of $\mathcal{H}_{1}$ and
$\mathcal{H}^{1}$ in this fashion can not give rise to a symplectic basis of
$\mathcal{H}_{1}\oplus\mathcal{H}^{1}$.

In order to represent the logical gates that we are going to construct in
terms of symplectic matrices, we need to fix ordered bases of
$\mathcal{H}_{1}$ and $\mathcal{H}^{1}$. We fix the basis
$\mathfrak{B}_{\mathcal{H}_{1}}$ of $\mathcal{H}_{1}$ as shown in Figure 4
with the order $\\{1,7,9\\}$, $\\{2,12,23\\}$, $\\{3,5,19\\}$,
$\\{10,16,18\\}$, $\\{21,26,30\\}$, $\\{8,25,28\\}$, $\\{11,17,29\\}$,
$\\{14,20,27\\}$. The ordered basis elements of
$\mathfrak{B}_{\mathcal{H}^{1}}$, given in terms of supports on the edges (cf.
Figure 4), are $\\{1,6,11\\}$, $\\{2,8,22\\}$, $\\{4,5,25\\}$,
$\\{10,13,17\\}$, $\\{24,28,30\\}$, $\\{12,19,26\\}$, $\\{9,16,27\\}$,
$\\{15,20,29\\}$.

The bases $\mathfrak{B}_{\mathcal{H}_{1}}$ and
$\mathfrak{B}_{\mathcal{H}^{1}}$ give rise to a basis
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ of the symplectic vector
space $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ in the obvious way. The
symplectic product between the elements of
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ is given by the matrix
$\Phi$ with $\Phi_{i,j}=\langle b^{i},b_{j}\rangle$ with
$b^{i}\in\mathfrak{B}_{\mathcal{H}^{1}}$ and
$b_{j}\in\mathfrak{B}_{\mathcal{H}_{1}}$.

$\displaystyle\Phi=\left(\begin{array}[]{rrrrrrrr}1&0&0&0&0&0&1&0\\\
0&1&0&0&0&1&0&0\\\ 0&0&1&0&0&1&0&0\\\ 0&0&0&1&0&0&1&0\\\ 0&0&0&0&1&1&0&0\\\
0&1&1&0&1&0&0&0\\\ 1&0&0&1&0&0&0&1\\\ 0&0&0&0&0&0&1&1\end{array}\right)$

As $\Phi$ is not diagonal,
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ is not a symplectic
basis.

We can fix this by defining alternative bases as follows: Let
$b^{{}^{\prime}}_{i}=b_{i}$ for $i\in\\{1,\dotsc,5\\}$,
$b^{{}^{\prime}}_{6}=b_{2}+b_{3}+b_{5}+b_{6}$,
$b^{{}^{\prime}}_{7}=b_{1}+b_{4}+b_{7}$ and
$b^{{}^{\prime}}_{8}=b_{1}+b_{4}+b_{7}+b_{8}$, defining a new basis
$\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}_{1}}$ of the first homology group.
Further, let $b^{{}^{\prime}i}=b^{i}$ for $i\in\\{1,\dotsc,5\\}$,
$b^{{}^{\prime}6}=b^{2}+b^{3}+b^{5}+b^{6}$, $b^{{}^{\prime}7}=b^{8}$ and
$b^{{}^{\prime}8}=b^{1}+b^{4}+b^{7}$, defining a new basis
$\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}^{1}}$ of the first cohomology group.
The resulting basis
$\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ of the
combined symplectic space $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ is a
symplectic basis.

###  3.3 Permutation gates

Before we discuss the fold-transversal gate constrictions, we note that some
non-trivial gates can be implemented simply by permuting qubits [34].

By definition, the automorphisms of Bring’s code directly correspond to the
symmetries of Bring’s surface. The subgroup of symmetries of Bring’s surface,
which maps vertices to vertices and faces to faces, is $\operatorname{S}_{5}$
and acts transitively on the 30 edges. This action is also faithful, so that
we can write the generators $a$, $b$ and $c$ of $\operatorname{S}_{5}$ in
terms of cycles of $\operatorname{S}_{30}$, i.e. cycles permuting the edge
labels:

$\displaystyle
a=(2,3)(4,5)(6,8)(7,9)(10,13)(11,14)(12,15)(16,19)(18,20)(21,26)(22,27)(23,28)(24,29)$
$\displaystyle
b=(1,2)(3,6)(4,7)(5,10)(9,16)(11,17)(13,21)(14,22)(15,23)(18,24)(19,26)(20,30)(25,28)$
$\displaystyle
c=(2,4)(3,5)(6,11)(7,12)(8,14)(9,15)(10,18)(13,20)(17,25)(21,27)(22,26)(23,29)(24,28)$

This action of $\operatorname{S}_{5}$ on the edges extends to a linear
representation $\rho$ on the space of 1-chains.

Since clearly $Z_{1}$ and $B_{1}$ are invariant subspaces under any
permutation matrix in the image of $\rho$, there exists a representation
$\sigma_{\mathcal{H}_{1}}:\operatorname{S}_{5}\to\operatorname{GL}(H_{1})$ on
the first homology group. Similarly, $Z^{1}$ and $B^{1}$ are invariant
subspaces as well, so that we obtain a representation
$\sigma_{\mathcal{H}^{1}}:\operatorname{S}_{5}\to\operatorname{GL}(H^{1})$ on
the first cohomology group. See Section A.1 for the matrix representations.

The group of logical gates induced by permutations of the edges is generated
by

$\displaystyle\tilde{\sigma}_{x}=\left(\begin{array}[]{c|c}\tilde{\sigma}_{\mathcal{H}_{1}}(x)&0\\\
\hline\cr 0&\tilde{\sigma}_{\mathcal{H}^{1}}(x)\end{array}\right)$

where $x\in\\{a,b,c\\}$.

As the representation is faithful, we have that
$\langle\tilde{\sigma}_{a},\tilde{\sigma}_{b},\tilde{\sigma}_{c}\rangle\cong\operatorname{S}_{5}<\operatorname{Sp}_{16}(\mathbb{F}_{2})$.
This can be verified using a computer algebra system, such as GAP [31].

###  3.4 $ZX$-Dualities

The group of automorphisms of Bring’s code is
$\mathrm{Aut}(C)=\operatorname{S}_{5}$ which has order $5!=120$. Furthermore,
Bring’s code is self-ZX-dual, by Theorem 5, with self-ZX-duality

$\tau_{0}:=f\in R_{5,4}.$

By Corollary 2 we know that there are 120 $ZX$-dualities exchanging $X$\- and
$Z$-checks of Bring’s code. Among these, the number of self-inverse dualities
($\tau^{2}=\operatorname{id}$) is 20. For the implementation of the Hadamard-
and phase-type gates in Section 3.5 we will consider the $ZX$-duality
$\tau_{0}$.

Figure 6 shows the orbits of edges under the action of $\tau_{0}$. As
$\tau_{0}$ is self-inverse each edge is either mapped onto a partner
($\tau_{0}(e_{1})=e_{2}$ and $\tau_{0}(e_{2})=e_{1}$), or it is fixed
($\tau_{0}(e)=e$). In the figure, the pairs are given the same color, while
the fixed edges are drawn in black. The same information is shown in Figure 7
in the planar layout.

We have picked the ordered bases $\mathfrak{B}_{\mathcal{H}_{1}}$ and
$\mathfrak{B}_{\mathcal{H}^{1}}$ in Section 3.2 such that $\tau_{0}$ maps
between the pairs of basis elements, i.e. $\tau_{0}(b_{i})=b^{i}$ for
$b_{i}\in\mathfrak{B}_{\mathcal{H}_{1}}$ and $\tau_{0}(b^{i})=b_{i}$ for
$b^{i}\in\mathfrak{B}_{\mathcal{H}^{1}}$.

Figure 6: The $ZX$-duality $\tau_{0}$ on Bring’s code. Qubits are associated
with edges. Qubits colored in black are left invariant under the $ZX$-duality.
Qubits not left invariant come in pairs, as $\tau_{0}^{2}=\operatorname{id}$,
and given the same color. Figure 7: The orbits of $\tau_{0}$ shown in the
planar layout.

###  3.5 Fold-transversal gates

We will now use the $ZX$-duality $\tau_{0}$ from the previous section to
construct fold-transversal Clifford gates for Bring’s code. This is
straightforward, as $\tau_{0}$ and the bases of
$\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ were carefully chosen.

####  3.5.1 Hadamard-type gate

We have picked the bases $\mathfrak{B}_{\mathcal{H}_{1}}$ and
$\mathfrak{B}_{\mathcal{H}^{1}}$ such that $\tau_{0}(b_{i})=b^{i}$ and
$\tau_{0}(b^{i})=b_{i}$ for each $b_{i}\in\mathfrak{B}_{\mathcal{H}_{1}}$ and
$b^{i}\in\mathfrak{B}_{\mathcal{H}^{1}}$. Therefore, the representation of
$\tilde{H}_{\tau_{0}}$ in the basis of
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ takes the simple form

$\displaystyle\tilde{H}_{\tau_{0}}=\left(\begin{array}[]{c|c}0&I_{8}\\\
\hline\cr I_{8}&0\end{array}\right).$

Changing to the symplectic basis
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}^{{}^{\prime}}$ yields

$\displaystyle{\tilde{H}}_{\tau_{0}}^{{}^{\prime}}=\left(\begin{array}[]{c|c}\makebox[17.98058pt]{\huge$0$}&\leavevmode\hbox{\set@color$\begin{matrix}I_{6}&&\\\
&0&1\\\ &1&1\end{matrix}$}\\\
\hline\cr\leavevmode\hbox{\set@color$\begin{matrix}I_{6}&&\\\ &1&1\\\
&1&0\end{matrix}$}&\makebox[17.98058pt]{\huge$0$}\end{array}\right).$

####  3.5.2 Phase-type gate

First we note that $\tau_{0}$ fulfills all requirements listed in Section
2.4.2: it is self-dual, the number of stabilized qubits is six, there are
either none or two stabilized qubits in the support of each $X$-check and no
$X$-check contains an orbit of cardinality two. All of this can be verified by
inspecting Figure 7.

The representation of $\tilde{S}_{\tau_{0}}$ in the basis
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$ takes the simple form

$\displaystyle\tilde{S}_{\tau_{0}}=\left(\begin{array}[]{c|c}I_{8}&I_{8}\\\
\hline\cr I_{8}&0\end{array}\right).$

Changing to the symplectic basis
$\mathfrak{B}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}^{{}^{\prime}}$ yields

$\displaystyle{\tilde{S}}_{\tau_{0}}^{{}^{\prime}}=\left(\begin{array}[]{c|c}\makebox[17.98058pt]{\large$I_{8}$}&\makebox[17.98058pt]{\large$0$}\\\
\hline\cr\leavevmode\hbox{\set@color$\begin{matrix}I_{6}&&\\\ &1&1\\\
&1&0\end{matrix}$}&\makebox[17.98058pt]{\large$I_{8}$}\end{array}\right).$

###  3.6 Achievable gates

Let $G_{\tau_{0}}$ be the group generated by the permutation gates and fold-
transversal gates, i.e. it is generated by the set
$\\{\tilde{\sigma}_{a},\tilde{\sigma}_{b},\tilde{\sigma}_{c},\tilde{H}_{\tau_{0}},\tilde{S}_{\tau_{0}}\\}$.
We observe, using GAP [31], that the group $G_{\tau_{0}}$ is isomorphic to
$\operatorname{Sp}_{8}(\mathbb{F}_{2})\times\operatorname{C}_{2}$. We can also
verify that $G_{\tau_{0}}$ does in fact not depend on the choice of
$ZX$-duality $\tau_{0}$. Further, a single $ZX$-duality generates the same
group as taking all possible Hadamard-type and phase-type gates of Bring’s
code.

####  3.6.1 Full Clifford group

Adding the symplectic equivalent of a single $\operatorname{CNOT}$\- or
$\operatorname{CZ}$-gate between any of the logical qubits to the set
$\\{\tilde{\sigma}_{a},\tilde{\sigma}_{b},\tilde{\sigma}_{c},\tilde{H}_{\tau_{0}},\tilde{S}_{\tau_{0}}\\}$
elevates it to a generating set of $\operatorname{Sp_{16}(\mathbb{F}_{2})}$.
Hence, we can obtain the full Clifford group $\mathcal{C}_{8}$.

This can be done in several ways: In [10] the authors show how to use Dehn
twists to implement logical $\operatorname{CNOT}$-gates. Their technique
requires either an increase of the qubit degree111Meaning the cumulative total
number of other qubits a single qubit has to be connected with. The
instantaneous qubit degree is constant throughout the procedure. up to
$O(\log(n))$ or, alternatively, an increase in overhead due to ancilla qubits.
The temporal overhead of this technique is $O(d^{2})=O(\log(n)^{2})$.
Alternatively, the authors of [14] suggest a generalization of lattice surgery
facilitated by an ancillary surface code. This results in an overhead in
ancilla qubits scaling as $O(d^{2})$.

####  3.6.2 Fold-transversal Clifford group

Implementing entangling gates can certainly be done using lattice surgery and
code deformation techniques. However, it would be much more appealing to
achieve all logical Clifford gates fold-transversally. It turns out that this
can indeed be done: Let
$\tilde{\sigma}_{r}=\tilde{\sigma}_{a}\tilde{\sigma}_{b}$ and
$\tilde{\sigma}_{s}=\tilde{\sigma}_{b}\tilde{\sigma}_{c}$.

Let $\tilde{\sigma}_{r}=\tilde{\sigma}_{a}\tilde{\sigma}_{b}$ and
$\tilde{\sigma}_{s}=\tilde{\sigma}_{b}\tilde{\sigma}_{c}$. Under the action of
$G_{\tau_{0}}$ the symplectic space $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$
decomposes into two invariant subspaces $V$ and $W$. Both, $V$ and $W$ have
dimension eight. The subspace $V$ is symplectic and gives rise to a set of
Pauli-operators acting on four qubits. The subspace $W$, on the other hand, is
maximally isotropic ($W^{\perp}=W$), in other words, it is a Lagrangian
subspace. A basis of both sub-spaces can be found in Section A.2. The group
$G_{\tau_{0}}$ acts faithfully as $\operatorname{Sp}_{8}(\mathbb{F}_{2})$ on
$V$. Therefore, we obtain the full Clifford group $\mathcal{C}_{4}$ when we
restrict the logical subspace to the one induced by $V$.

##  4 Conclusion

We have introduced a method to implement Clifford gates in CSS quantum codes
using symmetries that we call $ZX$-dualities. A $ZX$-duality defines a fold-
transversal operator, which is a quantum circuit with gates supported on the
orbits of $\tau$. In particular, we defined Hadamard-type gates $H_{\tau}$ and
phase-type gates $S_{\tau}$, derived from suitable $ZX$-dualities $\tau$. For
the surface code with $\tau$ being a literal fold (see Figure 1i), this is
equivalent to the construction of Moussa [1]. Further, we explicitly discussed
Bring’s code as an example, showing that it is possible to generate the
Clifford group $\mathcal{C}_{4}$ by restricting the logical subspace.

It would be interesting to understand the set of fold-transversal gates for a
family of LDPC quantum codes, such as hyperbolic codes or those discussed in
Appendix B. While one can verify the construction using a computer algebra
system, as we have done here, this has its limitations. This is because the
groups generated by the fold-transversal and permutation gates become very
large, even for a moderate number of encoded qubits $k$.

Finally, we expect our results to generalize to other code families that
support a notion of Fourier duality: non-CSS qubit stabilizer codes, qudit
stabilizer codes, and subsystem codes.

## Acknowledgements

N.P.B. acknowledges support through the EPSRC Prosperity Partnership in
Quantum Software for Simulation and Modelling (Grant No. EP/S005021/1). S.B.
acknowledges financial support by the Foundation for Polish Science through
TEAM-NET project (contract no. POIR.04.04.00-00-17C1/18-00).

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##  Appendix A Matrices

Here we give further details on associated matrices coming from the
construction of Bring’s code in Section 3.

###  A.1 Qubit permutation

In the basis $\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}_{1}}$ we have

$\displaystyle\tilde{\sigma}_{\mathcal{H}_{1}}(a)=\left(\begin{array}[]{rrrrrrrr}1&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&0\\\ 0&1&1&0&0&0&0&1\\\ 0&0&0&1&0&1&1&1\\\ 0&0&0&0&1&0&0&0\\\
1&1&0&0&0&0&1&0\\\ 1&0&0&0&1&0&1&0\\\
0&1&0&0&1&1&1&1\end{array}\right)\qquad\tilde{\sigma}_{\mathcal{H}_{1}}(b)=\left(\begin{array}[]{rrrrrrrr}0&1&1&0&0&0&0&1\\\
1&0&0&0&0&1&1&1\\\ 1&0&0&1&0&1&0&0\\\ 0&0&1&0&1&0&0&1\\\ 0&0&0&1&0&1&1&1\\\
0&1&1&0&1&0&0&0\\\ 1&1&0&1&1&0&1&0\\\ 1&0&0&1&0&0&1&1\end{array}\right)$
$\displaystyle\tilde{\sigma}_{\mathcal{H}_{1}}(c)=\left(\begin{array}[]{rrrrrrrr}1&0&0&0&0&1&1&1\\\
0&0&1&0&0&1&1&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&1&0&0&1&0&0&1\\\
0&1&0&1&0&0&1&0\\\ 0&0&1&1&0&0&1&0\\\ 0&1&1&0&0&1&1&1\end{array}\right)$

and in the basis $\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}^{1}}$ we have
$\tilde{\sigma}_{\mathcal{H}^{1}}(x)=\tilde{\sigma}_{\mathcal{H}_{1}}(x)^{\top}$,
for $x\in\\{a,b,c\\}$.

Note that the matrices operate on _row vectors_ from the _right_.

###  A.2 $G_{\tau_{0}}$-Invariant subspaces of
$\mathcal{H}_{1}\oplus\mathcal{H}^{1}$

A symplectic basis of the symplectic space
$V\subset\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ in the basis
$\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$:

$\displaystyle\left(\begin{array}[]{rrrrrrrr|rrrrrrrr}0&1&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 1&1&1&1&1&0&1&0&0&0&0&0&0&0&0&0\\\
1&0&0&1&0&1&1&1&0&0&0&0&0&0&0&0\\\ \hline\cr
0&0&0&0&0&0&0&0&0&1&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&1&0&0&1&0&1&1&0\\\
0&0&0&0&0&0&0&0&0&1&1&0&1&1&1&1\end{array}\right)$

A basis of the Lagrangian space $W\subset\mathcal{H}_{1}\oplus\mathcal{H}^{1}$
in the basis
$\mathfrak{B}^{{}^{\prime}}_{\mathcal{H}_{1}\oplus\mathcal{H}^{1}}$:

$\displaystyle\left(\begin{array}[]{rrrrrrrr|rrrrrrrr}1&0&0&0&1&1&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&1&0&0&0&0&0&0&0&0\\\ 0&0&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&1&1&0&0&0&0&0&0&0&0\\\ \hline\cr
0&0&0&0&0&0&0&0&1&0&0&0&1&1&1&1\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1\\\
0&0&0&0&0&0&0&0&0&0&1&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&1&0&0&0&0&1&0\end{array}\right)$

In the basis of V above, the action of the generators of
$G_{\tau_{0}}\cong\operatorname{Sp}_{8}(\mathbb{F}_{2})$ restricted to $V$ is
given by the following matrices:

$\displaystyle\tilde{\sigma}_{r}|_{V}=\left(\begin{array}[]{rrrr|rrrr}1&0&0&1&0&0&0&0\\\
1&0&1&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 1&1&0&0&0&0&0&0\\\ \hline\cr
0&0&0&0&1&0&0&1\\\ 0&0&0&0&1&0&1&1\\\ 0&0&0&0&0&1&0&0\\\
0&0&0&0&1&1&0&0\end{array}\right)\qquad\tilde{\sigma}_{s}|_{V}=\left(\begin{array}[]{rrrr|rrrr}0&0&0&1&0&0&0&0\\\
0&0&1&1&0&0&0&0\\\ 1&0&0&1&0&0&0&0\\\ 0&1&0&1&0&0&0&0\\\ \hline\cr
0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&0\\\ 0&0&0&0&1&1&0&0\\\
0&0&0&0&0&1&0&1\end{array}\right)$
$\displaystyle\tilde{H}_{\tau_{0}}|_{V}=\left(\begin{array}[]{rrrr|rrrr}0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&1\\\ \hline\cr
1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&1&1&0&0&0&0\\\
0&0&1&0&0&0&0&0\end{array}\right)\qquad\tilde{S}_{\tau_{0}}|_{V}=\left(\begin{array}[]{rrrr|rrrr}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ \hline\cr
1&0&0&0&1&0&0&0\\\ 0&1&0&0&0&1&0&0\\\ 0&0&1&1&0&0&1&0\\\
0&0&1&0&0&0&0&1\end{array}\right)$

Note that matrices operate on _row vectors_ from the _right_.

##  Appendix B Further Examples

Here we discus some further examples of fold-transversal gates and the groups
they generate. We do not go through all the details, as we did for Bring’s
code. However, the derivation of these results follows along the same steps.

###  B.1 Hyperbolic Surface Codes

We list some more examples of $\\{5,5\\}$ hyperbolic surface codes obtained
from the construction in Section 2.3, the first of which is Bring’s code. The
table shows the parameters $[[n,k,d]]$ for the code $C$, and the order of the
automorphism group $\mathrm{Aut}(C)$ which is equal to the number of
$ZX$-dualities.

Code parameters | $|\mathcal{D}_{ZX}|=|\mathrm{Aut}(C)|$
---|---
$[[30,8,3]]$ | 120
$[[40,10,4]]$ | 160
$[[80,18,5]]$ | 320
$[[150,32,6]]$ | 600
$[[180,38,4]]$ | 720
$[[330,68,6]]$ | 1320
$[[480,95,?]]$ | 1920

###  B.2 Hypergraph Product Code

Consider the $[6,2,4]$ code defined by the parity check matrix

$\displaystyle\left(\begin{array}[]{rrrr}1&1&0&0\\\ 0&1&0&0\\\ 1&0&1&0\\\
0&0&1&0\\\ 1&0&0&1\\\ 0&0&0&1\end{array}\right)$

acting on row vectors. The hypergraph product of this code with its dual gives
a $[[52,4,4]]$ quantum code with checks of weights 3, 4 and 5. The
automorphism group of the code, as defined in Section 2.2, is
$(\operatorname{S}_{3}\times\operatorname{S}_{3})\rtimes\operatorname{C}_{2}$.
This can be seen from the fact that $\operatorname{S}_{3}$ is the symmetry
group of the input code and we have the exchange of $X$\- and $Z$-checks,
giving the semi-direct product with $\operatorname{C}_{2}$. There are four
$ZX$-dualities and all of them fulfill the requirements of the theorems in
Section 2.4. Using a suitable choice of basis for
$\mathcal{H}_{1}\oplus\mathcal{H}^{1}$, we can construct eight Hadamard- and
phase-type gates. Together with the permutation gates they generate the group
$(\operatorname{A}_{6}\times\operatorname{A}_{6})\rtimes\operatorname{D}_{4}$.
This is an index-45696 subgroup of $\operatorname{Sp}_{8}(\mathbb{F}_{2})$.

As for Bring’s code, we can obtain the full group symplectic group
$\operatorname{Sp}_{8}(\mathbb{F}_{2})$ by adding a single entangling gate,
e.g. by using surgery with an ancillary surface code [14]. Alternatively, the
largest symplectic subgroup of
$(\operatorname{A}_{6}\times\operatorname{A}_{6})\rtimes\operatorname{D}_{4}$
is $\operatorname{Sp}_{4}(\mathbb{F}_{2})$, so we can obtain the full Clifford
group on two logical qubits when restricting the code space accordingly.

###  B.3 Balanced Product Code

Balanced product quantum codes are a generalization of the hypergraph product
codes, allowing for the construction of quantum codes that can achieve higher
encoding rates and larger distances [8]. The balanced product between two
classical codes, that share a common symmetry, can be understood as the
hypergraph product between the two, with the symmetry subsequently factored
out. We refer to [3, 8] for more details. Symmetry groups play a prominent
role in the construction of balanced product codes, so that they are amenable
to our scheme.

For symmetric classical codes, let us consider the extended binary Golay code
$\mathcal{G}_{24}$ with parameters $[24,12,8]$ and its dual
$\mathcal{G}_{24}^{\top}$. It turns out that, coincidentally, the parity
checks of the Golay code can also be derived from the great dodecahedron
(Figure 3), see [35] for details. The automorphism group of this parity check
matrix of $\mathcal{G}_{24}$ is thus the same as that of Bring’s code
$\operatorname{S}_{5}\times\operatorname{C}_{2}$ (see Section 3.1). The
alternating group $\operatorname{A}_{5}$ is an (up to conjugation) unique
subgroup. The balanced product code
$\mathcal{G}_{24}\otimes_{\operatorname{A}_{5}}\mathcal{G}_{24}^{\top}$ is a
$[[20,4,3]]$ code with check weights 4, 5 and 7. It has 10 $ZX$-dualities,
giving rise to Hadamard-type gates. However, only one of the $ZX$-dualities
satisfies the requirements of Theorem 7, giving rise to a single phase-type
gate. The group of permissible qubit permutations is
$\operatorname{C}_{2}\times\operatorname{S}_{4}$ and it has a faithful
representation on $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$. The permutation
gates, the Hadamard-type gates and the phase-type gate generate the group
$\operatorname{S}_{3}\times\operatorname{S}_{4}$, which is an index-329011200
subgroup of $\operatorname{Sp}_{8}(\mathbb{F}_{4})$. Adding a single
entangling gate generates the full symplectic group
$\operatorname{Sp}_{8}(\mathbb{F}_{2})$.

Alternatively, consider the smaller subgroup
$H=\operatorname{C}_{2}\times(\operatorname{C}_{5}\rtimes\operatorname{C}_{4})$.
The balanced product code $\mathcal{G}_{24}\otimes_{H}\mathcal{G}_{24}^{\top}$
is a $[[30,6,3]]$ code. Similarly, the permutation gates, the Hadamard-type
gates and the phase-type gate generate an index-135491300609753088000 subgroup
of $\operatorname{Sp}_{12}(\mathbb{F}_{4})$. Adding a single entangling gate
generates the full symplectic group $\operatorname{Sp}_{12}(\mathbb{F}_{2})$.

A $[[60,12,5]]$ code can be obtained by taking the balanced product over the
even smaller subgroup $\operatorname{C}_{3}\rtimes\operatorname{C}_{4}$.
Unfortunately, the space $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ is
24-dimensional and thus too large for our computational methods.

###  B.4 Block Code

The following example shows that the fold-transversal gates do not always
suffice to generate large groups. Consider the $[[16,4,4]]$ code, obtained by
concatenating the $[[4,2,2]]$ code with itself. It is a weakly self-dual CSS
code with classical code defined by the check matrix

$\displaystyle\left(\begin{array}[]{rrrrrr}1&1&1&0&0&0\\\ 1&1&0&1&0&0\\\
1&1&0&0&1&0\\\ 1&1&0&0&0&1\\\ 1&0&1&0&0&0\\\ 1&0&0&1&0&0\\\ 1&0&0&0&1&0\\\
1&0&0&0&0&1\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
0&1&1&0&0&0\\\ 0&1&0&1&0&0\\\ 0&1&0&0&1&0\\\ 0&1&0&0&0&1\end{array}\right)$

A natural basis of $H_{1}$ and $H^{1}$ is

$\displaystyle\left(\begin{array}[]{rrrrrrrrrrrrrrrr}1&1&0&0&1&1&0&0&0&0&0&0&0&0&0&0\\\
1&1&0&0&0&0&0&0&0&0&0&0&1&1&0&0\\\ 0&1&1&0&0&1&1&0&0&0&0&0&0&0&0&0\\\
0&1&1&0&0&0&0&0&0&0&0&0&0&1&1&0\end{array}\right)$

There exist 20 $ZX$-dualities, giving rise to 8 distinct Hadamard-type gates
on $\mathcal{H}_{1}\oplus\mathcal{H}^{1}$. Only one of the $ZX$-dualities
satisfies the requirements of Theorem 7, giving rise to a phase-type gate. The
group of permissible qubit permutations is
$\operatorname{C}_{2}\times\operatorname{S}_{4}$. However, unlike in the
previous examples, the induced representation on
$\mathcal{H}_{1}\oplus\mathcal{H}^{1}$ is not faithful, as its image is
isomorphic to the dihedral group $\operatorname{D}_{6}$. The permutation
gates, the Hadamard-type gates and the phase-type gate generate the group
$\operatorname{C}_{2}\times\operatorname{S}_{3}\times\operatorname{S}_{3}$.
Unlike in the previous examples, note even adding all
$\tilde{\operatorname{CZ}}_{i,j}$-gates does not give rise to the full
symplectic group $\operatorname{Sp}_{8}(\mathbb{F}_{2})$, but to the
orthogonal group $\operatorname{O}^{+}_{8}(\mathbb{F}_{2})$, which is an
index-272 subgroup.


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