$Y_{1}Y_{2}$, $X_{1}Y_{2}$, and $Y_{1}X_{2}$ (i.e. effective $X$ and $Y$ Pauli
operators of the color code at $0\mod 5$ rounds) commute with $V(Z_{1}Z_{2})$
measurements. Therefore, once we infer these plaquettes, they survive the
$V(Z_{1}Z_{2})$ round and are inferred again during the next period of the
protocol. It may appear strange that the measurements of such detectors are
inferred in an “out-of-time” way, i.e. as an example, for $P_{r}(X_{1}Y_{2})$,
$P_{r}(X_{1})$ is inferred at $t=3$ while $P_{r}(Y_{2})$ is inferred at $t=4$,
from which the values are combined to infer $P_{r}(X_{1}Y_{2})$ at $t=4$.
However, this is not problematic because outside of the $V(Z_{1}Z_{2})$
rounds, the two layers are completely decoupled and the measurements are not
“causally related” so long as they are performed between the same two
consecutive $V(Z_{1}Z_{2})$ measurement rounds. The shaded region of each
detector in Fig. 12 denotes the temporal support in each layer which is “out-
of-time” for many detectors (including $P_{r}(X_{1}Y_{2})$).

As we see, the structure of detectors in the DA color code is substantially
more complex than in the honeycomb and CSS honeycomb codes. This is further
exemplified by constructing detectors for sequences where we sandwich an
automorphism with a padding sequence; in particular, we illustrate this for
$\hat{M}(\pi)\circ\hat{M}(\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})})\circ\hat{M}(\pi)$
(see Fig. 13) and
$\hat{M}(\pi)\circ\hat{M}(\varphi_{(\texttt{gb})})\circ\hat{M}(\pi)$ (see Fig.
14). Notice that some detectors start before or end after the steps shown in
the tables. One can check that these detectors start and end in the preceding
or subsequent padding sequence, respectively, and the start or end of such a
detector coincides with that of detectors of the padding sequence.

There is a further subtlety concerning error correction in these codes.
Specifically, we purposefully choose a basis of detectors such that no
detectors involve the outcomes of $V(Z_{1}Z_{2})$ measurements, whenever the
measurement rounds preceding and following the $V(Z_{1}Z_{2})$ round are the
same. This is because, as we show below, the error correction will work up to
correcting measurement errors or circuit-level errors equivalent to
measurement errors at these $V(Z_{1}Z_{2})$ rounds. However, for the exact
reason that the preceding and following measurement rounds are the same, we
can argue that measurement outcomes of $V(Z_{1}Z_{2})$ measurements are not
required to update the logical operators and stabilizers. Since these
measurement outcomes are not used either for logical update, stabilizer
update, or detector construction, we can safely discard them. Therefore, even
if error correction works up to correcting these errors, it can be declared
successful, as our logical information will not have been corrupted.

Let us discuss this subtlety in more detail now. An example of an error that
is equivalent to a $V(Z_{1}Z_{2})$ measurement outcome error for the
$\varphi_{(\texttt{gb})}$ automorphism is
$\varepsilon_{4}^{\ell}\varepsilon_{6}^{\ell}$ acting on the same red(blue)
edge in layer $\ell=1$ ($\ell=2$). In fact, such undetectable errors tell us
that the two same rounds of measurements (in our case, it is
${{E_{r}}}(X_{1}),E_{b}(X_{2})$ repeated both at step 4 and step 6) are
indistinguishable by our code from an error detection perspective if they are
separated by a single $V(Z_{1}Z_{2})$ round. To be more accurate, it is not
entirely true that such an error is completely undetectable. The time and the
green plaquette where such an error has occurred can still be found if we use
a $P_{g}(Z_{1}Z_{2})$-type detector involving measurements from round $5$, but
better spatial resolution is not possible.

One can understand the effect of an error
$\varepsilon_{4}^{\ell}\varepsilon_{6}^{\ell}$ as a measurement error during
the $V(Z_{1}Z_{2})$ round because if we consider the action of such an error
on a logical state, we find that it can be replaced by changing the sign of
the intervening $V(Z_{1}Z_{2})$ measurement outcome. Explicitly, consider a
red edge on the lattice denoted by $\langle i,j\rangle$. Then labelling
measurement outcomes by $m_{t}^{i}$ where $t$ denotes the time stamp and $i$
denotes the site in case of a $V(Z_{1}Z_{2})$ check, we can write the action
of the error $\varepsilon_{4,i}^{1}\varepsilon_{6,i}^{1}$ on some logical
state $\ket{L}$ of the code at time step 4 as

$\displaystyle\ket{\varepsilon_{4,i}^{1}\varepsilon_{6,i}^{1}}$
$\displaystyle=\varepsilon^{1}_{6,i}\left(1+m_{6}X_{1,i}X_{1,j}\right)\left(1+m_{5}^{i}Z_{1,i}Z_{2,i}\right)\left(1+m_{5}^{j}Z_{1,j}Z_{2,j}\right)\varepsilon^{1}_{4,i}\left(1+m_{4}X_{1,i}X_{1,j}\right)\ket{L}$
$\displaystyle=X_{1,i}\left(1+m_{6}X_{1,i}X_{1,j}\right)\left(1+m_{5}^{i}Z_{1,i}Z_{2,i}\right)\left(1+m_{5}^{j}Z_{1,j}Z_{2,j}\right)X_{1,i}\left(1+m_{4}X_{1,i}X_{1,j}\right)\ket{L}$
$\displaystyle=\left(1+m_{6}X_{1,i}X_{1,j}\right)\left(1+m_{5}^{i}X_{1,i}Z_{1,i}X_{1,i}Z_{2,i}\right)\left(1+m_{5}^{j}Z_{1,j}Z_{2,j}\right)\left(1+m_{4}X_{1,i}X_{1,j}\right)\ket{L}$
$\displaystyle=\left(1+m_{6}X_{1,i}X_{1,j}\right)\left(1-m_{5}^{i}Z_{1,i}Z_{2,i}\right)\left(1+m_{5}^{j}Z_{1,j}Z_{2,j}\right)\left(1+m_{4}X_{1,i}X_{1,j}\right)\ket{L}.$
(20)

As we see, the effect of the error is to flip the sign of $m_{5}^{i}$. More
generally, such a correlated error whose support is right before and after
$V(Z_{1}Z_{2})$ round will flip the sign of a $V(Z_{1}Z_{2})$ measurement. In
a spacetime picture, such an error creates a pair of anyons and annihilates
them after two time steps.

Figure 13: Detectors in two layers corresponding to the padded
$\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ automorphism. The
detectors that extend past the sequence are started or completed within
another adjacent padding sequence. The notation is the same as in Fig. 12. The
detectors for the padded $\varphi_{(\texttt{xr})(\texttt{yg})(\texttt{zb})}$
can be obtained by swapping the colors and layers accordingly to the
measurement sequence. Additionally, instead of using padded
$\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ and
$\varphi_{(\texttt{xr})(\texttt{yg})(\texttt{zb})}$ as two of the generators,
one can use padded $\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ and the
padding itself repeated odd number of times.

We circumvent this issue by discarding the outcomes of $V(Z_{1}Z_{2})$
measurements whenever they are made. Although these rounds are necessary
between the end of a previous and the next automorphism sequence, measurement
outcomes during these rounds can be safely discarded as they are also not used
to perform logical string updates, stabilizer updates, and error detection.

At the same time, when a $V(Z_{1}Z_{2})$ measurement occurs via the same
scenario as round 9 in the padded $\varphi_{(\texttt{gb})}$ sequence (see Fig.
14), in which the preceding and following measurements are different, there
are no longer undetectable error chains and this subtlety no longer appears.

Next, let us discuss the detectors for the padded protocols for
$\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ and
$\varphi_{(\texttt{gb})}$ shown in Figs. 13 and 14, respectively. The sequence
for the padded $\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ looks
identical to the honeycomb code in the second layer and performs three
measurements (instead of four) in the first layer. Therefore, one might expect
the detectors to look similar to those of the padding sequence itself (Fig.
12). The difference is that the detectors of type $R^{(2)}$ are temporarily
discontinued (which does not turn out to negatively impact error
detectability) and a short detector $G^{(4)}$ is added. For the padded
$\varphi_{(\texttt{gb})}$ sequence, the protocol is different; some of the
detectors change shape while others are added/removed. The most important
distinction is that the outcomes of $V(Z_{1}Z_{2})$ measurements at round 9 of
this sequence can be collected and used for error correction, as previously
discussed. This allows us to introduce new detectors $G^{(3)}$ and $B^{(3)}$
that involve the $V(Z_{1}Z_{2})$ round of measurements.

Next, we consider every possible distinct single-qubit error, which
corresponds to an examination of all distinct $\varepsilon^{\ell}_{t}$ and
$\varepsilon^{\ell}_{t}\varepsilon^{\ell}_{t+1}$ errors. First consider a
$\mathbb{F}_{2}$-valued vector space of errors $V_{\varepsilon}$. Single qubit
errors such as $\varepsilon^{\ell}_{t}$ furnish a basis for $V_{\varepsilon}$,
and will be denoted ${\hat{\varepsilon}}^{\ell}_{t}$. Further define an
$\mathbb{F}_{2}$-valued vector space of detectors labeled by $V_{D}$, whose
basis vectors are isomorphic to the basis detectors $\hat{D}^{i}$ (we added a
hat to emphasize that we are dealing with the basis detectors). For each
single-qubit error, we identify the set of detectors that have support on the
respective timelike edge and light up if this error has occurred. We express
this set as a vector $s({\hat{\varepsilon}}^{\ell}_{t})\in V_{D}$. The inner
product $\langle s({\hat{\varepsilon}}^{\ell}_{t}),\hat{D}^{i}\rangle=1$ if
error $\varepsilon^{\ell}_{t}$ lights up detector $D^{i}$ and $0$ otherwise.

There exists a natural map $H:V_{\varepsilon}\to V_{D}$ defining a parity
check matrix of a classical code. Its action is

$s({\hat{\varepsilon}}^{\ell}_{t})=H{\hat{\varepsilon}}^{\ell}_{t}.$ (21)

For each error of the type $\varepsilon^{\ell}_{t}$ occurring after a round
corresponding to the decoupled toric codes, we find that at least two
detectors of different colors ‘light up’, and moreover, the intersection of
the detectors give the time and the edge where the error has occurred (the
exception are errors of the $\varepsilon^{\ell}_{4}\varepsilon^{\ell}_{6}$
type, which, as we have already discussed, form an undetectable and
inconsequential error. If an error occurred after a $V(Z_{1}Z_{2})$ round, at
least three detectors of three different colors light up, revealing the
‘vertical’ edge where the error has occurred. For example, an error
$\varepsilon_{1}^{1}$ in the sequence implementing $\pi$ lights up detectors
$G_{1}^{(2)}$ and $B_{1}^{(1)}$. An error $\varepsilon_{5}^{1}$ lights up
detectors $R_{2}^{(1)}$, $R_{2}^{(2)}$, $G_{2}^{(2)}$, $G_{2}^{(3)}$ and
$B_{2}^{(2)}$.

Figure 14: Detectors in two layers for the corresponding to the padded
$\varphi_{(\texttt{gb})}$ automorphism. The detectors that extend past the
sequence are started or completed within another adjacent padding sequence.
The notation is the same as in Fig. 12.

In fact, we can show that the detectors shown in Figs. 12, 13 and 14
constitute a basis for detecting all possible single-qubit Pauli errors. In
particular, all of these detectors are independent because each of them
includes at least one type of measurement that is not used for measuring
plaquettes in any other detector. Moreover, from analyzing the rows of matrix
$H$, we find that each single-qubit error of the type $\varepsilon^{\ell}_{t}$
can be detected and corrected (up to measurement outcome errors on the
$V(Z_{1}Z_{2})$ rounds) and moreover all possible pairwise products
$\varepsilon^{\ell}_{t}\varepsilon^{\ell}_{t+1}$ can also be detected and
corrected.

###  III.6 Fault tolerance and threshold in the dynamic automorphism color
code

The decoding graph of a DA color code is constructed by placing detectors at
the vertices and connecting pairs of detectors that can light up
simultaneously in response to the single-qubit Pauli errors (either
$\varepsilon^{\ell}_{n}$ or $\varepsilon^{\ell}_{n}\varepsilon^{\ell}_{n+1}$).
The decoding graph has a complicated and irregular structure, but the DA color
code for any automorphism can be constructed from a few automorphism
generators plus padding, which gives a finite number of kinds of error
detection events. In fact, when showing that any single-qubit Pauli error can
be corrected, we implicitly considered each type of edge that will occur on
the decoding graph (explicit calculation is bulky and is omitted for brevity,
but is straightforward).

The question of a proper definition of the distance in dynamic codes is still
being discussed in the literature. Here, we define the distance of a DA color
code as the minimum between the weight of the smallest logical operator (where
the minimum is taken over all time slices) and the weight of the smallest
consequential but undetectable spacetime error chain. As a reminder, in
(2+1)D, we define the logical membrane for each logical operator
$\mathcal{L}^{(i)}$ to be
$\mathcal{L}_{0}^{(i)}\cup\mathcal{L}_{1}^{(i)}\cup...\cup\mathcal{L}_{T}^{(i)}$,
and the difference between $\mathcal{L}_{t}^{(i)}$ and
$\mathcal{L}_{t+1}^{(i)}$ is an element of the ISG at round $t$ that is found
according to the update rule for the respective logical operator. We then
argue that the distance $d$ of our code is $O(L)$ where $L$ is the linear size
of the code, because the only undetectable objects that anticommute with
logical membranes of the DA color code are the ones that form a homologically
nontrivial loop around the torus in their spatial support.

Finally, it is likely that there exists a lower bound on threshold of these
codes from a mapping to a random-bond Ising model on a regular three-
dimensional graph with connectivity $z$ which is the maximum connectivity of
the decoding graph for the DA color code (which for all our examples is upper
bounded by 4). The phase transition of the random-bond Ising model will lower
bound the threshold of the maximum likelihood decoder [47]. It would also be
interesting to find an efficient decoder based on projection [48] or
restriction [49], or alternative [39, 50, 51, 52] decoders for the color code.
When it comes to practical error correction for the DA color code, it should
be possible to perform an exhaustive numerical search over the space of
measurement sequences implementing a given automorphism and padding sequences
in order to find a sequence maximizing the threshold.

##  IV TQFT perspective on dynamic automorphism codes

In this section, we will study DA codes from the perspective of TQFTs, which
allows us to study the universal topological properties of such codes. We will
restrict the discussion to parent Pauli topological codes in two spatial
dimensions on prime-dimensional $p$-qudits. In particular, this applies to
qubits where $p=2$. Such Pauli topological codes have been shown to be
equivalent up to a finite depth unitary circuit to a finite stack of
$\mathbb{Z}_{p}$ toric codes [39, 53, 54] under an extra assumption of
translation invariance. For example, the color code defined on prime-
dimensional qudits is equivalent to two copies of the $\mathbb{Z}_{p}$ toric
code [40]. A full analysis of more general Abelian TQFTs in two dimensions
will be given in an upcoming work [55].

In Subsec. IV.1 we review the basic concepts that are needed for defining
condensation sequences from a parent model that is equivalent to several
copies of a prime-dimensional toric code. Next, in Subsec. IV.2, we will
introduce the concept of reversible condensations between two child theories
obtained from the same parent theory. Reversible condensations are the
building blocks for constructing sequences that implement automorphisms of a
child theory. In Subsec. IV.3, we explain how to find shared basis of
deconfined anyons between two child theories that is conserved upon a
reversible condensation between these theories. Given a sequence of reversible
condensations, a shared basis is changed between each two consecutive pairs.
Tracking this update allows us to determine the automorphism implemented by
the sequence, which is explained in Subsec. IV.4.

In Subsec. IV.5, we define the concept of the condensation graph, which charts
the space of possible reversible condensations, and show how to assign
isomorphisms to its edges and automorphisms to its cycles. As an example, we
explore these condensation graphs for the cases when the parent theory is a
single color code (i.e. the graph for the “Floquet color code” [14, 13]) and
when the parent theory is two copies of a color code (i.e. the graph for the
DA color code). Lastly, in Subsec. IV.6 we explain a technique that allows,
starting from a condensation sequence implementing an arbitrary automorphism,
to design sequences for other desired automorphisms.

###  IV.1 TQFT review of Abelian anyons

We will assume that the parent theory is equivalent to the toric code of some
Abelian group $G=\prod_{p}\mathbb{Z}_{p}^{N_{p}}$, where the product runs over
all primes $p$ and $N_{p}\in\mathbb{Z}_{\geq 0}$ denotes the number of copies
of that prime-dimensional qudit toric code. The analysis in this section
factors into each prime $p$ separately. Therefore, without loss of generality,
we can further simplify and analyze the case $G=\mathbb{Z}_{p}^{N_{p}}$ for
some prime $p$.

An anyon in the parent theory can be specified by a vector $c\in
M=\mathbb{F}_{p}^{2N_{p}}$101010In this section, we use non-calligraphic
symbols ($M,A,\ldots)$ for the vector spaces to distinguish them from
equivalent modular tensor category notation $(\mathcal{M},\mathcal{A},\ldots)$
used elsewhere in the paper. Moreover, fusion is replaced by the addition of
vectors. where the first $N_{p}$ unit vectors correspond to electric charges e
for each copy of the $\mathbb{Z}_{p}$ the toric code, and the last $N_{p}$
unit vectors are the respective magnetic charges m. Note that for clarity, we
use the regular and not typewriter font for anyons in the vector notation.

The self-statistics of an anyon is given by the $\mathbb{F}_{p}$-valued
quadratic form

$\displaystyle q(c)$ $\displaystyle=c^{T}\mu c,$ $\displaystyle\mu$
$\displaystyle=\begin{pmatrix}0_{N_{p}\times N_{p}}&\mathbbm{1}_{N_{p}\times
N_{p}}\\\ 0_{N_{p}\times N_{p}}&0_{N_{p}\times N_{p}}\end{pmatrix}$ (22)

and the mutual statistics of two anyons $c_{1}$ and $c_{2}$ can be computed by
the inner product

$\displaystyle\langle c_{1},c_{2}\rangle$
$\displaystyle=q(c_{1}+c_{2})-q(c_{1})-q(c_{2})=c_{1}^{T}\lambda c_{2},$
$\displaystyle\lambda=\begin{pmatrix}0_{N_{p}\times
N_{p}}&\mathbbm{1}_{N_{p}\times N_{p}}\\\ \mathbbm{1}_{N_{p}\times
N_{p}}&0_{N_{p}\times N_{p}}\end{pmatrix}.$ (23)

We remark that this is a special case of a more general statement that an
Abelian modular tensor category can be fully characterized by an Abelian group
along with a $U(1)$-valued quadratic form (see for example Ref. [56]).

Recall that a choice of condensation is a set of bosons that all braid
trivially with each other [44, 43]. In this formalism, we can specify such a
condensation by a subspace $A\subset M$ which satisfies $q(a_{1})=0$ and
$\langle a_{1},a_{2}\rangle=0$ for all $a_{1},a_{2}\in A$. Let $|A|=p^{N_{A}}$
for some $N_{A}<N_{p}$. Given such a condensation, consider the orthogonal
subspace $A^{\perp}=\\{b\in M|\ \forall a\in A,\langle a,b\rangle=0\\}$.
Physically, this corresponds to anyons in $M$ that braid trivially with all
anyons in $A$. By definition, $A$ is a subspace of $A^{\perp}$ so we may
quotient out $A$ to obtain $C=A^{\perp}/A$ which labels the deconfined anyons
in the child theory111111This will ultimately be associated to logical
operators of the child code. . These anyons correspond to equivalence classes
in the parent theory $M$. Note that $C\cong\mathbb{F}_{p}^{2N_{C}}$ where
$N_{C}=N_{p}-N_{A}$. Thus an anyon in the child theory can labeled by a vector
in $A^{\perp}$, up to an equivalence of addition by vectors in $A$.

It is useful to define a representative of an anyon $c\in C$ within $M$, which
we denote $\rho(c)$ (we will often abbreviate this as a representative of
$c$). The representative is not unique and depends on the choice of the linear
map $\rho:C\rightarrow M$.

To formalize the notion of a representative, we first define the quotient map
$\kappa:A^{\perp}\rightarrow A^{\perp}/A=C$. Next, we define a section to be
any linear map $s:C\rightarrow A^{\perp}$ such that $\kappa\circ
s=\text{id}_{C}$. The section, along with the inclusion map
$\iota:A^{\perp}\rightarrow M$, can then be used to defined the representative
map as $\rho=\iota\circ s$.

To conclude, we can define a representative $\rho(c)$ for any $c\in C$ and
choice of map $\rho$. Note that since the elements of $\rho(C)$ represent
every coset of $A$ in $A^{\perp}$, we have $\rho(C)+A=A^{\perp}$.

###  IV.2 Reversible condensations

Recently, Ref. [12] formalized the notion of a reversible pair of stabilizer
groups. Let us develop an analogous notion of a reversible pair of
condensations. First, we provide an informal definition. Take the parent
theory $M$ and consider condensing $A_{1}$ in one region of spacetime and
$A_{2}$ in the other region (see Fig. 15(a)). This results in child theories
$C_{1}$ and $C_{2}$ in the two regions separated by an interface. We say that
$A_{1}$ and $A_{2}$ are a reversible pair of condensations (with respect to
the parent $M$) if the resulting interface is transparent on both sides, by
which we mean any anyon from $C_{1}$ can go through and become some anyon in
$C_{2}$ and vice versa121212Invertibility is a sufficient but not necessary
condition. For example, the domain wall could host multiple channels for a
single anyon, making it non-invertible.

The above definition sets some constraints on what pairs of condensations can
be allowed for constructing meaningful sequences. For example, it requires
that $A_{1}$ and $A_{2}$ are of equal size, since $C_{1}$ and $C_{2}$ have to
be isomorphic. Henceforth, we will make the assumption that
$|A_{1}|=|A_{2}|=p^{N_{A}}$ for some $N_{A}$.

Let us now make a formal definition in the context of Abelian anyons. $A_{1}$
and $A_{2}$ are a reversible pair of condensations if there exists a shared
and complete set of representatives for both $C_{1}$ and $C_{2}$ in $M$. That
is, there exist linear maps $\rho_{1}$ and $\rho_{2}$ such that
$\rho_{1}(C_{1})=\rho_{2}(C_{2})$, $\rho_{1}(C_{1})+A_{1}=A_{1}^{\perp}$, and
$\rho_{2}(C_{2})+A_{2}=A_{2}^{\perp}$. (see Fig. 15(b)).

Figure 15:  Two definitions of a reversible pair of condensations $A_{1}$ and
$A_{2}$ in $M$. (a) Informal definition: condensing $A_{1}$ in on region of
spacetime and $A_{2}$ in the other region of a parent theory $M$ results in
child theories $C_{1}$ and $C_{2}$ separated by a transparent interface. (b)
Formal definition in the context of Abelian anyons: there exists a shared and
complete set of representatives (black squares) for both $C_{1}$ and $C_{2}$
in $M$.

###  IV.3 Transitions between child theories

Consider a reversible pair of condensations, $A_{1}$ and $A_{2}$, and their
corresponding child theories $C_{1}$ and $C_{2}$. For every anyon in $C_{1}$
there will exist a representative that will braid trivially with condensation
$A_{2}$ and thus will become an anyon in child theory $C_{2}$. We call finding
such a representative an “update” because starting with an arbitrary
representative (in $M$) of a given anyon in $C_{1}$, we can “update” it by
fusing with some anyon in $A_{1}$ such that it becomes an element of
$A_{2}^{\perp}$. This allows us to consider it (project it to) an anyon of
$C_{2}$. Reversibility guarantees that these anyons map onto complete $C_{2}$.

Let us now provide an explicit calculation for how to update representatives
of anyons when going from one child theory to another by a reversible
condensation. We note that this update procedure directly translates into
updating logical operators in DA codes by considering logical strings instead
of deconfined anyons and multiplying by measured operators instead of fusing
with condensed anyons.

Assume that we have a condensation $A_{1}$ and a map $\rho_{1}$ which realizes
representatives $\rho_{1}(C_{1})$. A priori, we are not guaranteed that
$\rho_{1}(C_{1})\subset A_{2}^{\perp}$. Thus, we would like to construct an
alternative map $\tilde{\rho}_{1}(C_{1})$ such that
$\tilde{\rho}_{1}(C_{1})\subset A_{2}^{\perp}$, and is moreover a complete set
of representatives for $C_{2}$. This construction will allow us to determine
an update map between condensation rounds, which we will explain later.

We proceed as follows. Let us define $N_{Q}$ to be the number of generators of
$Q=A_{1}\cap A_{2}$ and $N_{\overline{Q}}=N_{A}-N_{Q}$, where $N_{A}$ is the
number of generators in $A_{1}$ (and $A_{2}$). First, we pick an arbitrary
basis for $Q$:

$\displaystyle Q=\text{span}\\{\hat{a}_{Q}^{(i)}\\}$ (24)

for $i=1,\ldots,N_{Q}$. Next, we may pick a basis for the bosons in $A_{1}$
and $A_{2}$ as

$\displaystyle A_{1}$ $\displaystyle=\text{span}\\{\hat{a}^{(i)}_{1}\\},$
$\displaystyle A_{2}$ $\displaystyle=\text{span}\\{\hat{a}^{(i)}_{2}\\}$ (25)

where $i=1,\ldots,N_{A}$, with the constraint

$\displaystyle\hat{a}^{(i+N_{\overline{Q}})}_{1}=\hat{a}^{(i+N_{\overline{Q}})}_{2}=\hat{a}_{Q}^{(i)}$
(26)

for $i=1,\ldots,N_{Q}$, corresponding to a shared basis of bosons in $Q$.

Let us define the matrix $B$ which has components

$\displaystyle B^{ij}=\langle\hat{a}^{(i)}_{2},\hat{a}^{(j)}_{1}\rangle$ (27)

As a matrix, we find that it must take the form

$\displaystyle B=\begin{pmatrix}\beta_{N_{\overline{Q}}\times
N_{\overline{Q}}}&0_{N_{\overline{Q}}\times N_{Q}}\\\ 0_{N_{Q}\times
N_{\overline{Q}}}&0_{N_{Q}\times N_{Q}}\end{pmatrix}$ (28)

for some matrix $\beta$. The zero matrices follow from the fact that $\langle
A,Q\rangle=\langle B,Q\rangle=\langle Q,Q\rangle=0$.

Let $L\in\rho_{1}(C_{1})$ be a given representative. For each representative,
we would like to find a different representative $\widetilde{L}$ that also
lives in $A_{2}^{\perp}$ and, moreover, $\text{span}\\{\widetilde{L}\\}$ forms
a complete set of representatives for $C_{2}$. For each $L$, finding an
appropriate $\widetilde{L}$ can be achieved by adding appropriate elements of
$A_{1}$ to the original representative. That is, assume

$\displaystyle\widetilde{L}=L+\sum_{i=1}^{N_{A}}\hat{a}^{(i)}_{1}W^{i}$ (29)

for some $\mathbb{F}_{p}$-valued vector $W^{i}$. Since we want
$\widetilde{L}\in A^{\perp}_{2}$, this implies that

$\displaystyle\langle\hat{a}^{(k)}_{2},\widetilde{L}\rangle=\langle\hat{a}^{(k)}_{2},L\rangle+\sum_{i=1}^{N_{A}}B^{ki}W^{i}=0$
(30)

for $k=1,\ldots,N_{A}$. However, note that for
$k=N_{\overline{Q}}+1,\ldots,N_{A}$, the equation is trivially satisfied since
$\hat{a}^{(k)}_{2}\in Q$, so we actually only need to solve for $W^{ij}_{t}$
for $k=1,\ldots,N_{\overline{Q}}$. Thus, writing
$W=\begin{pmatrix}\omega_{N_{\overline{Q}}}\\\ 0_{N_{Q}}\\\ \end{pmatrix}$ for
some vector $\omega$, we instead can solve

$\displaystyle\langle\hat{a}^{(k)}_{2},L\rangle+\sum_{i=1}^{N_{\overline{Q}}}\beta^{ki}\omega^{i}=0,$
(31)

for $k=1,\ldots,N_{\overline{Q}}$. Assuming $\beta^{ki}$ is invertible over
$\mathbb{F}_{p}$, the solution is

$\displaystyle\omega^{i}=-\sum_{k=1}^{N_{\overline{Q}}}(\beta^{-1})^{ik}\langle\hat{a}^{(k)}_{2},L\rangle.$
(32)

In conclusion, we have found a new set of representatives which generate
$\tilde{\rho}_{1}(C_{1})$ given by

$\displaystyle\widetilde{L}=L-\sum_{i=1}^{N_{\overline{Q}}}\hat{a}_{1}^{(i)}(\beta^{-1})^{ik}\langle\hat{a}^{(k)}_{2},L\rangle.$
(33)

The desired representative map $\widetilde{\rho}_{1}$ is then defined via its
action $\kappa_{1}(\widetilde{L})\mapsto\widetilde{L}$. Note that the above
update Eq. (33) is linear in $L$. Thus, we may use it to define the update map
$\nu_{1}:\rho_{1}(C_{1})\rightarrow\widetilde{\rho}_{1}(C_{1})$ which acts as
$L\mapsto\widetilde{L}$. We will sometimes refer to Eq. 33 as an “update rule”
or an “update formula”.

We have found a complete set of representatives that is shared between $C_{1}$
and $C_{2}$. Namely, we have a map $\widetilde{\rho}_{1}$ such that
$\widetilde{\rho}_{1}(C_{1})+A_{1}=A_{1}^{\perp}$ and
$\widetilde{\rho}_{1}(C_{1})+A_{2}=A_{2}^{\perp}$. This can be used to define
a canonical isomorphism between $C_{1}$ and $C_{2}$. Namely, we may define
$\lambda_{21}:C_{1}\rightarrow C_{2}$ where
$\lambda_{21}=\kappa_{2}\circ\widetilde{\rho}_{1}$.

We remark again that the update formula described here translates naturally to
the update of logical operators in the corresponding DA code. Namely, the
logical operators can be modified to commute with the next round of
measurements by multiplying them with the measurements from the current round,
or equivalently, by multiplying them by string operators of the condensed
anyons.

###  IV.4 Computing automorphisms for a condensation sequence

Assume that we have a sequence of condensations $A_{t}$ for $t=0,\ldots,T$
with $A_{0}=A_{T}$, where each $A_{t}$ and $A_{t+1}$ are a pair of reversible
condensations for $t=0,\ldots,T-1$. We may now ask what automorphism is
applied between the initial and final steps when we realize the same child
theory the second time, i.e. we arrive at $C_{T}=C_{0}$.

One possible way is to work directly with anyons in the parent code (see Fig.
16). Start with $C_{0}$ and pick any representative $\rho_{0}(C_{0})$. We then
may use the update formula in Eq. (33) to update them to the next round.

Each condensation $A_{t}$ will participate in two reversible pairs: one with
$A_{t-1}$ and one with $A_{t+1}$. Thus we will need to represent it using two
different bases during the calculation. Hence, define
$A_{t}=\text{span}\\{\hat{\texttt{a}}^{(i)}_{t,{\pm}}\\}$ to be the basis for
participating in the reversible condensation with $A_{t\pm 1}$. Thus given
$L_{0}=\rho_{0}(c)$ for $c\in C$ we may solve for $L_{T}$ recursively via
$L_{t+1}=\nu_{t}(L_{t})$ where

$\displaystyle\nu_{t}(L_{t})=L_{t}-\sum_{i,k=1}^{N_{\overline{Q}_{t+1,t}}}\hat{a}_{t,+}^{(i)}(\beta_{t+1,t}^{-1})^{ik}\langle\hat{a}^{(k)}_{t+1,-},L_{t}\rangle$
(34)

where $N_{\overline{Q}_{t+1,t}}$ is the rank of the matrix
$B_{t+1,t}^{ij}=\langle\hat{\texttt{a}}^{(i)}_{t+1,-},\hat{\texttt{a}}^{(j)}_{t,+}\rangle$
and $\beta_{t+1,t}$ is the restriction of the matrix to the invertible part.

The automorphism $\varphi$ in $C_{0}$ therefore acts as

$\displaystyle\varphi=\kappa_{T}\circ(\nu_{T-1}\circ\cdots\circ\nu_{0})\circ\rho_{0}$
(35)

Alternatively, suppose all the canonical isomorphisms $\lambda$ have been
computed in advance, then one can readily compose them to obtain the
automorphism

$\displaystyle\varphi=\lambda_{T,T-1}\circ\cdots\circ\lambda_{21}\circ\lambda_{10}$
(36)

${\cdots=}$${\rho_{1}(C_{1})}$${\widetilde{\rho}_{1}(C_{1})}$${=}$${\rho_{2}(C_{2})}$${\widetilde{\rho}_{2}(C_{2})}$${=\cdots}$${\cdots}$${C_{1}}$${C_{2}}$${\cdots}$$\scriptstyle{\nu_{1}}$$\scriptstyle{\kappa_{1}}$$\scriptstyle{\kappa_{1}}$$\scriptstyle{\nu_{2}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\lambda_{10}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\tilde{\rho}_{1}}$$\scriptstyle{\lambda_{21}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\tilde{\rho}_{2}}$$\scriptstyle{\lambda_{32}}$
Figure 16: Summary of the representative updates, which can be used to compute
automorphisms. Here, $\rho_{t}:C_{t}\rightarrow M$ maps the anyons of the
child theory to their representatives in parent theory $M$. Anyons in
$\rho_{t}(C_{t})$ are updated via $\nu_{t}$ is defined via the update formula
Eq. (33). This gives a complete set of shared representatives between $C_{t}$
and $C_{t+1}$, and moreover allows us to define of an isomorphism
$\lambda_{t+1,t}$ between $C_{t}$ and $C_{t+1}$.

###  IV.5 Condensation graph

The emergence of automorphisms as a consequence of a sequence of reversible
condensations motivates us to define the condensation graph. We define a
vertex for each condensation and connect a pair of vertices by an edge if the
corresponding pair of condensations is reversible. We can furthermore assign
an isomorphism to each edge given by $\lambda_{21}:C_{1}\rightarrow C_{2}$ and
its inverse $\lambda_{12}=\lambda_{21}^{-1}:C_{2}\rightarrow C_{1}$. Fixing a
base point, every cycle can be assigned an automorphism given by multiplying
all the isomorphisms along the cycle.

####  IV.5.1 Example: condensing a parent color code

Ref. [14] introduced the example where the parent tat is a single copy of the
color code CC is turned into a single toric code theory
$\text{TC}(\texttt{c}\texttt{\textsigma})$ by condensing a single boson c. Let
us study the corresponding condensation graph as our first example.

To do this, we first introduce a second, rearranged, magic square,
complementary to the Mermin square in Eq. (1). Recall that the color code
contains nine non-trivial bosons and six fermions. The relation between the
fermions and bosons can be summarized by the following “fermion magic square”:

${\large\raisebox{0.2pt}{
\begin{tabular}[]{c|c|c|c|}\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$\overline{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{f}_{\texttt{r}}}}}$\hfil\lx@intercol
&\lx@intercol\hfil$\overline{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{f}_{\texttt{g}}}}}$\hfil\lx@intercol
&\lx@intercol\hfil$\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{f}_{\texttt{b}}}}}$\hfil\lx@intercol\\\
\cline{2-4}\cr${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{f}_{\texttt{r}}}}$&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{{ry}}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{bx}}}&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>\cline{2-4}\cr${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{f}_{\texttt{g}}}}$&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{bz}}}&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{{gy}}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{{rx}}}\\\
\cline{2-4}\cr${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{f}_{\texttt{b}}}}$&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{{gx}}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{{rz}}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{by}}}\\\
\cline{2-4}\cr\end{tabular}}\triangleq\raisebox{0.2pt}{
\begin{tabular}[]{c|c|c|c|}\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil{fe}\hfil\lx@intercol
&\lx@intercol\hfil{1f}\hfil\lx@intercol
&\lx@intercol\hfil{fm}\hfil\lx@intercol\\\
\cline{2-4}\cr{mf}&{{em}}&{{m1}}&{{ee}}\\\
\cline{2-4}\cr{f1}&{{1e}}&{{ff}}&{{1m}}\\\
\cline{2-4}\cr{ef}&{{mm}}&{{e1}}&{{me}}\\\ \cline{2-4}\cr\end{tabular}}}$ (37)

Here, the squares on the left and on the right correspond to the anyon labels
for the color code and two copies of the toric code, respectively. The above
magic squares satisfies the following properties:

  1. 1.

The product of three bosons on individual rows or columns is the corresponding
fermion denoted on the side of the respective row or column.

  2. 2.

Each boson is a product of two of the fermions: one from the row label and one
from the column label.

  3. 3.

The mutual statistics of two bosons that share the same row or column is $-1$.
Otherwise, the mutual statistics is trivial.

The last point actually follows naturally from the viewpoint of the 3-fermion
(3F) theory as follows. All fermions in a 3F theory have pairwise mutual
statistics $-1$. Thus, if two bosons only differ by a column or a row, then
using the property 2, they must differ by a single fermion, which means that
they will braid non-trivially. On the other hand, if they differ by both a row
and a column, then both the fermion in 3F and $\overline{\text{3F}}$ differ,
which means they braid trivially.

Let us show that the condensation graph is exactly the “dual” of this fermion
magic square. There are nine possible condensations corresponding to each
boson in the magic square. Thus, we can construct the condensation graph as
follows. First, we assume the magic square is placed on periodic boundary
conditions. Take the vertices to be the bosons that are to be condensed
(plaquettes of the fermion magic square) and connect the vertices with an edge
if the plaquettes in the magic square share an edge (which are exactly the
bosons that have non-trivial mutual statistics).

Figure 17: The condensation graph for the color code corresponds to the torus
grid graph $T_{3,3}$. a) A contractible path of condensations gives rise to a
trivial automorphism. b) A non-contractible path of condensations (such as the
Hastings-Haah sequence as shown) gives rise to an $e\rightarrow m$
automorphism. c) The CSS honeycomb sequence wraps around both cycles of the
torus, resulting in a trivial automorphism.

The resulting graph is called the torus grid graph $T_{3,3}$ because it can be
depicted as a $3\times 3$ grid with periodic boundary conditions as shown in
Fig. 17. We can also alternatively denote the vertices with coordinates
$(i,j)\in\mathbb{Z}_{3}^{2}$ where in the 3F notation, $i=-1,0,1$ correspond
to the fermion
$\texttt{f}_{\texttt{r}},\texttt{f}_{\texttt{g}},\texttt{f}_{\texttt{b}}$, and
$j=-1,0,1$ correspond to
$\overline{\texttt{f}}_{\texttt{r}},\overline{\texttt{f}}_{\texttt{g}},\overline{\texttt{f}}_{\texttt{b}}$.
In color code notation, the color label is given by mapping
$i+j\in\\{-1,0,1\\}$ to $\\{\texttt{b},\texttt{g},\texttt{r}\\}$ and the Pauli
flavor is given by mapping $i-j\in\\{-1,0,1\\}$ to
$\\{\texttt{x},\texttt{y},\texttt{z}\\}$.

We can now study what automorphisms are assigned to the the different cycles
of this graph. In order to aid in the calculation of the automorphisms, it is
helpful to make a choice for the representative anyons. Since the child anyons
correspond a single toric code, we will make the following choice to identify
anyons that braid trivially with the condensation c and anyons as the the
toric code

$\displaystyle\kappa(\texttt{c}\texttt{\textsigma}^{\prime})$
$\displaystyle=\texttt{e}\text{ for }\sigma^{\prime}\neq\sigma$
$\displaystyle\kappa(\texttt{c}^{\prime}\texttt{\textsigma})$
$\displaystyle=\texttt{m}\text{ for }c^{\prime}\neq c$ (38)

With this choice, note that every reversible pair of condensations is of the
form c and $\texttt{c}^{\prime}\texttt{\textsigma}^{\prime}$ for
$c^{\prime}\neq c$ and $\sigma^{\prime}\neq\sigma$ (this is the “tree code
condition” of Ref. [13]). This implies that
$\texttt{c}\texttt{\textsigma}^{\prime}$ maps to e in the child for c but maps
to m in the child for $\texttt{c}^{\prime}\texttt{\textsigma}^{\prime}$.
Similarly, for $\texttt{c}^{\prime}\texttt{\textsigma}$. In conclusion, with
this choice we find that every isomorphism on this graph swaps e and m of the
child toric code!

The above isomorphisms can be used to conveniently compute all automorphisms
on this condensation graph. First, consider an arbitrary contractible loop on
this torus (Fig. 17a). Such a loop is always composed of an even number of
edges, and therefore, the automorphism is trivial.

On the other hand, let us consider the Hastings-Haah measurement sequence,
which corresponds to the cycle
$\texttt{rx}\rightarrow\texttt{gy}\rightarrow\texttt{bz}\rightarrow\texttt{rx}$.
Notice that this path is a non-contractible cycle around the torus (Fig. 17a).
Indeed, it is known that automorphism associated with this path is the $e-m$
automorphism [1], which agrees with the fact that the sequence has an odd
length.

The above two facts allow us to derive an intriguing property of this torus.
Any contractible cycle on the torus will also result in a trivial
automorphism. While any non-contractible loop around the $(1,0)$ or $(0,1)$
cycles of the torus will result in the $\texttt{e}-\texttt{m}$ automorphism!
More generally, any path $(n,m)$ implements the $\texttt{e}-\texttt{m}$
automorphism if $n+m$ is odd and the trivial automorphism if $n+m$ is even. A
notable example is the sequence corresponding to the CSS honeycomb code [13,
14, 15] wraps around the $(1,1)$ cycle of the torus (Fig. 17c). Thus, it
implements a trivial automorphism. To conclude, we can colloquially say that
each hole of the torus contains such an $\texttt{e}-\texttt{m}$ automorphism.

One might be concerned that choosing a particular choice of basis of $e$ and
$m$ in the child will affect the final automorphism calculated. To illustrate
that this is not the case, suppose we make a basis transformation by applying
an $\texttt{e}-\texttt{m}$ automorphism to a vertex. This will modify the
isomorphism for all edges entering that vertex to be the trivial isomorphism.
However, note that this does not change the property of the product of
isomorphisms around any closed loop, since any path entering that vertex must
also exit it, thus the number of $\texttt{e}-\texttt{m}$ isomorphisms along
the loop always changes by an even number of times.

Readers familiar with lattice gauge theory might find the above argument very
familiar. Indeed, one can abstractly view this graph as having a $C$-valued
bundle on every vertex. Isomorphisms connecting different vertices can be
thought of as “connections”, which are not “gauge invariant”. However, the
“integration of a connection around a closed loop” is gauge-invariant,
corresponding to an automorphism. Choosing a basis for $C$ at each vertex can
be thought of as “gauge fixing” while changing a basis can be thought of as a
“gauge transformation”.

####  IV.5.2 Products of condensation graphs

To build up to describing the condensation graph for the DA color code, we
first consider a subclass of condensations $\texttt{c}\texttt{\textsigma}_{1}$
and $\texttt{c}^{\prime}\texttt{\textsigma}_{2}^{\prime}$ in two parent color
codes (compared to the DA color code, we are only missing the condensation to
$\widetilde{\text{CC}}$, which we will add back shortly). The corresponding
condensation graph can be simply described by taking a product of the two
graphs.

More generally, suppose we take a product of two parent theories
$M=M_{1}\boxtimes M_{2}$ and only consider a subset of condensations of the
form $A=\langle A_{1}\boxtimes I_{2},I_{1}\boxtimes A_{2}\rangle$. Reversible
condensations to $A$ are of the form $A=\langle A_{1}^{\prime}\boxtimes
I_{2},I_{1}\boxtimes A_{2}\rangle$ or $A=\langle A_{1}\boxtimes
I_{2},I_{1}\boxtimes A_{2}^{\prime}\rangle$. Therefore, if the condensation
graphs are $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$, the condensation graph of
the product is exactly the (Cartesian) product
$\mathcal{G}_{1}\Box\mathcal{G}_{2}$.

Note that within the product graph, new cycles of the form

${\langle A_{1}\boxtimes I_{2},I_{1}\boxtimes A_{2}\rangle}$${\langle
A_{1}\boxtimes I_{2},I_{1}\boxtimes A_{2}^{\prime}\rangle}$${\langle
A_{1}^{\prime}\boxtimes I_{2},I_{1}\boxtimes A_{2}\rangle}$${\langle
A_{1}^{\prime}\boxtimes I_{2},I_{1}\boxtimes
A_{2}\rangle}$$\scriptstyle{\text{id}\boxtimes\lambda_{2}}$$\scriptstyle{\lambda_{1}\boxtimes\text{id}}$$\scriptstyle{\lambda_{1}^{-1}\boxtimes\text{id}}$$\scriptstyle{\text{id}\boxtimes\lambda_{2}^{-1}}$
(39)

are introduced. However, we see that the corresponding isomorphism is trivial.
Thus, the only non-trivial isomorphisms of the graph are those that are
already present in $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$.

####  IV.5.3 Condensation graph of the DA color code

First, let us discuss the condensation graph where the parent code is two
copies of the color code $\text{CC}_{1}\times\text{CC}_{2}$ and condensations
of the form
$A=\text{span}\\{\texttt{c}_{1}\texttt{\textsigma}_{1},\texttt{c}_{2}\texttt{\textsigma}_{2}\\}$
which leads to the child
$C=\text{TC}(\texttt{c}_{1}\texttt{\textsigma}_{1})\times\text{TC}(\texttt{c}_{2}\texttt{\textsigma}_{2})$.
Because the condensations are products, we conclude that the corresponding
condensation graph is $T_{3,3,3,3}=T_{3,3}\Box T_{3,3}$.

The above graph can be visualized on a 4-torus. One way to describe such a
graph is to label the vertices by $(i,j,k,l)\in\mathbb{Z}_{3}^{4}$. Using the
same basis for anyons as in the single color code, the edges and the
corresponding isomorphisms are then

$\displaystyle(i,j,k,l)$ $\displaystyle\rightarrow(i+1,j,k,l)$
$\displaystyle(\texttt{xr})(\texttt{yg})(\texttt{zb})$ (40)
$\displaystyle(i,j,k,l)$ $\displaystyle\rightarrow(i,j+1,k,l)$
$\displaystyle(\texttt{xr})(\texttt{yg})(\texttt{zb})$ (41)
$\displaystyle(i,j,k,l)$ $\displaystyle\rightarrow(i,j,k+1,l)$
$\displaystyle(\texttt{zr})(\texttt{yg})(\texttt{xb})$ (42)
$\displaystyle(i,j,k,l)$ $\displaystyle\rightarrow(i,j,k,l+1)$
$\displaystyle(\texttt{zr})(\texttt{yg})(\texttt{xb})$ (43)

for $i,j,k,l\in\mathbb{Z}_{3}$ and we remind that
$(\texttt{xr})(\texttt{yg})(\texttt{zb})$ and
$(\texttt{zr})(\texttt{yg})(\texttt{xb})$ in the color code convention
corresponds to $\texttt{e}_{\texttt{1}}\leftrightarrow\texttt{m}_{\texttt{1}}$
and $\texttt{e}_{\texttt{2}}\leftrightarrow\texttt{m}_{\texttt{2}}$ in the
toric code convention, respectively.

Now, we will add the vertex corresponding to condensing
$A_{\widetilde{\text{CC}}}=\langle\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}},\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}\rangle$
(which corresponds to measuring $V(Z_{1}Z_{2})$ in the code). The other
vertices in $T_{3,3,3,3}$ that share an edge with $A_{\widetilde{\text{CC}}}$
are of the form
$\langle\texttt{c}_{1}\texttt{\textsigma}_{1},\texttt{c}_{2}\texttt{\textsigma}_{2}\rangle$
where $c_{1}\neq c_{2}\in\\{r,g,b\\}$ and $\sigma_{1},\sigma_{2}\in\\{x,y\\}$.
Thus $A_{\widetilde{\text{CC}}}$ connects to 24 other vertices.

Isomorphism $(\lambda)$ | Condensation $(A)$ | Isomorphism $(\lambda)$ | Condensation $(A)$
---|---|---|---
$1$ |  $\texttt{rx}_{\texttt{1}}$ | $\texttt{ry}_{\texttt{1}}$ | $(\texttt{xy})$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{ry}_{\texttt{1}}$
$\texttt{bx}_{\texttt{2}}$ | $\texttt{by}_{\texttt{2}}$ | $\texttt{by}_{\texttt{2}}$ | $\texttt{bx}_{\texttt{2}}$
$(\texttt{rg})$ | $\texttt{gx}_{\texttt{1}}$ | $\texttt{gy}_{\texttt{1}}$ | $(\texttt{xy})(\texttt{rg})$ |  $\texttt{gx}_{\texttt{1}}$ | $\texttt{gy}_{\texttt{1}}$
$\texttt{bx}_{\texttt{2}}$ | $\texttt{by}_{\texttt{2}}$ | $\texttt{by}_{\texttt{2}}$ | $\texttt{bx}_{\texttt{2}}$
$(\texttt{gb})$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{ry}_{\texttt{1}}$ | $(\texttt{xy})(\texttt{gb})$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{ry}_{\texttt{1}}$
$\texttt{gx}_{\texttt{2}}$ | $\texttt{gy}_{\texttt{2}}$ | $\texttt{gy}_{\texttt{2}}$ | $\texttt{gx}_{\texttt{2}}$
$(\texttt{rb})$ | $\texttt{bx}_{\texttt{1}}$ | $\texttt{by}_{\texttt{1}}$ | $(\texttt{xy})(\texttt{rb})$ | $\texttt{bx}_{\texttt{1}}$ | $\texttt{by}_{\texttt{1}}$
$\texttt{rx}_{\texttt{2}}$ | $\texttt{ry}_{\texttt{2}}$ | $\texttt{ry}_{\texttt{2}}$ | $\texttt{rx}_{\texttt{2}}$
$(\texttt{rgb})$ | $\texttt{gx}_{\texttt{1}}$ | $\texttt{gy}_{\texttt{1}}$ | $(\texttt{xy})(\texttt{rgb})$ | $\texttt{gx}_{\texttt{1}}$ | $\texttt{gy}_{\texttt{1}}$
$\texttt{rx}_{\texttt{2}}$ | $\texttt{ry}_{\texttt{2}}$ | $\texttt{ry}_{\texttt{2}}$ | $\texttt{rx}_{\texttt{2}}$
$(\texttt{rbg})$ | $\texttt{bx}_{\texttt{1}}$ | $\texttt{by}_{\texttt{1}}$ | $(\texttt{xy})(\texttt{rbg})$ | $\texttt{bx}_{\texttt{1}}$ | $\texttt{by}_{\texttt{1}}$
$\texttt{gx}_{\texttt{2}}$ | $\texttt{gy}_{\texttt{2}}$ | $\texttt{gy}_{\texttt{2}}$ | $\texttt{gx}_{\texttt{2}}$
Table 5: Isomorphisms
$\lambda:\widetilde{\text{CC}}\rightarrow\text{TC}(\texttt{c}_{1}\texttt{\textsigma}_{1})\boxtimes\text{TC}(\texttt{c}_{2}\texttt{\textsigma}_{2}).$
We have chosen a representative for the anyons according to Eq. (3) and for
the toric codes as Eq. (38).

The isomorphisms from $A_{\widetilde{\text{CC}}}$ are calculated using the
update formula Eq. (33) and are summarized in Table 5. Because the table
contains all the generators of all 72 automorphisms, this shows that we can
get all of them. Recall that we have chosen a representative for the anyons of
the effective color code according to Eq. (3) and for the toric codes as Eq.
(38).

With this, let us derive the automorphisms shown in Table 1. Consider the
first sequence

$t=$0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{gy}_{\texttt{1}}$ | $\texttt{bz}_{\texttt{1}}$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$
$\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$ | $\texttt{bx}_{\texttt{2}}$ | $-$ | $\texttt{gz}_{\texttt{2}}$ | $\texttt{bx}_{\texttt{2}}$ | $\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$

We have $\lambda_{10}=\lambda_{54}^{-1}=\text{id}$. From rounds $1-4$ we
accrue $\varphi_{(\texttt{xr})(\texttt{yg})(\texttt{zb})}$ three times and
$\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ two times. Thus, combining
all isomorphisms, we find that this sequence gives
$\varphi_{(\texttt{xr})(\texttt{yg})(\texttt{zb})}$. The second sequence gives
$\varphi_{(\texttt{zr})(\texttt{yg})(\texttt{xb})}$ through an identical
calculation. Lastly, the third sequence

$t=$0 | 1 | 2 | 3 | 4
---|---|---|---|---
$\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{bz}_{\texttt{1}}$ | $\texttt{rx}_{\texttt{1}}$ | $\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$
$\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$ | $\texttt{bx}_{\texttt{2}}$ | $\texttt{ry}_{\texttt{2}}$ | $\texttt{gx}_{\texttt{2}}$ | $\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$

has $\lambda_{10}=\text{id}$,
$\lambda_{21}=\lambda_{32}=\varphi_{(\texttt{xz})(\texttt{rb})}$,
$\lambda_{43}=\varphi_{(\texttt{gb})}$. Thus, this sequence gives the
automorphism $\varphi_{(\texttt{gb})}$. We remark that this argument shows
that we would also have gotten the same automorphism had we replaced the
second condensation with $\texttt{bz}_{\texttt{1}}$ and
$\texttt{rz}_{\texttt{2}}$, for example.

###  IV.6 A method to construct desired automorphisms

Let us finally provide a method to construct a sequence of condensations that
produces the desired automorphism on a child anyon theory. Let us first show
that we can construct a desired isomorphism from a condensation sequence.
Assume we have a reversible sequence of condensations $A_{0},\ldots,A_{T}$,
where, unlike the previous section, we do not need to assume $A_{0}=A_{T}$.
Since each consecutive pair of condensations is reversible, we can associate
an isomorphism $\lambda$ from the child theory $C_{0}=A_{0}^{\perp}/A_{0}$ to
the child theory $C_{T}=A_{T}^{\perp}/A_{T}$.

Now, let us assume that there exists an automorphism $\varphi$ of the parent
such that $\varphi(A_{0})=A_{0}$ and $\varphi(A_{T})=A_{T}$. Note that this
induces well-defined automorphisms on the child models $C_{0}$ and
$C_{T}$131313To see that an automorphism that preserves a condensation $A$
defines an automorphism on the child theory $C$, first note that $\varphi$
restricts to the action on $A^{\perp}$ since if $b\in A^{\perp}$, the $\langle
b,a\rangle=0$ for all $a\in A$, Applying $\varphi$ we have that
$\langle\varphi(b),\varphi(a)\rangle=0$. But since $\varphi(a)\in A$ we
conclude that $\varphi(b)\in A^{\perp}$ also. Next, we may quotient
$A^{\perp}$ by $A$ to get $C=A^{\perp}/A$. Again, since $\varphi(A)=A$ this
guarantees that $\varphi(b)$ is in the same coset as $\varphi(ba)$.. Let us
further assume that this induced automorphism is a trivial automorphism
$\psi_{0}=\text{id}_{C_{0}}$ on $C_{0}$ but a non-trivial automorphism
$\psi_{T}$ on $C_{T}$.

We can apply this automorphism of the parent theory $\varphi$ to the entire
sequence. This produces a new condensation sequence
$\varphi(A_{0}),\varphi(A_{1}),\ldots,\varphi(A_{T-1}),\varphi(A_{T})$. This
sequence can be equivalently written as
$A_{0},\varphi(A_{1}),\ldots,\varphi(A_{T-1}),A_{T}$. As a consequence of this
modification, an isomorphism from $C_{0}$ to $C_{T}$ of the original sequence
is now changed to
$\psi_{T}\circ\lambda\circ\psi_{0}^{-1}=\psi_{T}\circ\lambda$.

Thus, consider the sequence
$A_{T},A_{T-1},\ldots,A_{1},A_{0},\varphi(A_{1}),\ldots,\varphi(A_{T-1}),A_{T}$
where we apply the original sequence in reverse followed by the new sequence.
This induces an automorphism $\psi\circ\lambda\circ\lambda^{-1}=\psi$ on
$C_{T}$.

To give an example, let us show how we can use this method to obtain an
automorphism $\varphi_{(\texttt{xy})}$, which for the color code is equivalent
to implementing a transversal $S$ gate. The parent is equivalent to two copies
of the color code, and consider the condensation sequence

$\displaystyle A_{0}$
$\displaystyle=\\{1,\texttt{gz}_{\texttt{1}},\texttt{rz}_{\texttt{2}},\texttt{bz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}\\},$
$\displaystyle A_{1}$
$\displaystyle=\\{1,\texttt{rx}_{\texttt{1}},\texttt{gx}_{\texttt{2}},\texttt{rx}_{\texttt{1}}\texttt{gx}_{\texttt{2}}\\},$
$\displaystyle A_{2}$
$\displaystyle=\\{1,\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}},\texttt{gz}_{\texttt{1}}\texttt{gz}_{\texttt{2}},\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}\\}.$
(44)

We choose the automorphism of the parent theory
$\varphi={\varphi_{(\texttt{xy})}}_{1}\boxtimes\text{id}_{2}$. That is, we
only permute x and y type anyons in the first copy. This leaves $A_{0}$ and
$A_{2}$ invariant. First consider the child at round zero
$C_{0}=\text{TC}(\texttt{gz}_{\texttt{1}})\times\text{TC}(\texttt{rz}_{\texttt{2}})$.
The equivalence classes of the anyons in the first toric code copy are

$\displaystyle[\texttt{e}_{1}]$
$\displaystyle=\\{\texttt{rz}_{\texttt{1}},\texttt{bz}_{\texttt{1}},\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}},\texttt{bz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}\\}$
$\displaystyle[\texttt{m}_{1}]$
$\displaystyle=\\{\texttt{gx}_{\texttt{1}},\texttt{gy}_{\texttt{1}},\texttt{gx}_{\texttt{1}}\texttt{rz}_{\texttt{2}},\texttt{gy}_{\texttt{1}}\texttt{rz}_{\texttt{2}}\\}$
(45)

which are invariant under the action of $\varphi$. Thus
$\psi_{0}=\text{id}_{C_{0}}$ as desired. On the other hand,
$C_{2}=\widetilde{\text{CC}}$, and $\varphi$ acts as
$\psi_{2}=\widetilde{\varphi}_{(\texttt{xy})}$.

This allows us to design a cycle of condensations that starts and ends with an
effective color code and use the trick above to implement a desired
automorphism. Specifically, based ont he sequence above, let us choose the
cyclic sequence

$\displaystyle A_{0}^{\prime}$ $\displaystyle=A_{2},$ $\displaystyle
A_{1}^{\prime}$ $\displaystyle=A_{1},$ $\displaystyle A_{2}^{\prime}$
$\displaystyle=A_{0},$ $\displaystyle A_{3}$
$\displaystyle=\varphi(A_{1})=\\{1,\texttt{ry}_{\texttt{1}},\texttt{gx}_{\texttt{2}},\texttt{ry}_{\texttt{1}}\texttt{gx}_{\texttt{2}}\\}$
$\displaystyle A_{4}^{\prime}$ $\displaystyle=A_{2}$ (46)

performs the automorphism $\widetilde{\varphi}_{(\texttt{xy})}$ on the child
$\widetilde{\text{CC}}$. In terms of measurements on stabilizer codes, this is
exactly sequence 19 in Appendix A.

Later in Sec.VI, we will also use this trick to design measurement sequences
that perform the $(\texttt{ry})$ automorphism of the 3D color code (see table
12) as well as the automorphism that is equivalent to implementing a
transversal $T$ gate [24, 57] (see Table 16).

##  V Full Clifford group with dynamic automorphism color code on a triangle

In Sec. II we showed how to implement all automorphisms of the color code via
a sequence of measurements. On a torus, the action of the color code
automorphisms on the logical operators furnishes a subgroup of the real
Clifford group on four logical qubits. Here, we show that the full (complex)
Clifford group can be achieved on two logical qubits by constructing a DA
color code on a triangle with Pauli boundaries. By extension, the full
Clifford group on $N$ logical qubits can be realized on $N$ layers of a
triangular DA color code.

To achieve automorphisms that realize complex Clifford gates, one needs to
consider a color code on a manifold with a boundary. For example, a color code
on a triangle with Pauli boundaries (first introduced in [25]) encodes a
single logical qubit and achieves automorphisms corresponding to all possible
permutations of logical strings. In terms of gates, these automorphisms form a
basis for all single-qubit Clifford gates. This color code can be considered
as a child code of a parent bilayer color code with specific bulk and boundary
condensations. Further taking two layers of the (child) color code on a Pauli
triangle and augmenting the single qubit gates with an additional automorphism
that realizes the iSWAP gate, we form a basis for the Clifford group on two
logical qubits.

We start by showing in Subsec. V.1 how a sequence of reversible condensations
for a theory without a boundary plus a fixed boundary condition at one rounds
gives rise to a boundary condensation sequence at all other rounds. Such
boundary condensation sequences can be translated into boundary measurement
sequences upon which the corresponding logical strings can terminate without
violating a boundary stabilizer. The length of the logical operators for such
a code at each round is proportional to the geometric distance between the
boundaries of the same kind. In Subsec. V.2 we work out (bulk and boundary)
condensation sequences that produce the automorphisms corresponding to a full
Clifford group on two logical qubits living on two (bilayer) Pauli triangles.
These condensation sequences can be mapped to an explicit set of measurements
on a triangular patch, which allows us to construct a DA color code with Pauli
boundaries in Subsec. V.3. Lastly, we address the question of error correction
of proposed protocols in Subsec. V.4; in particular, we discuss the basis of
bulk and boundary detectors of the DA color code.

Although this section addresses the DA color code that realizes color code
ISGs on a Pauli triangle only, the case of the color triangle [45] is
discussed in Appendix C.

###  V.1 Condensation sequences in the presence of boundaries

Let us explain how we extend the concept of reversible condensations to child
theories with boundaries. Roughly speaking, the boundaries are associated with
condensing certain representatives of anyons in the parent theory and
therefore must also be updated in accordance. This translates into a series of
updated measurement sequences near the boundaries which allow the associated
logicals (and their updates) to terminate at each round. Without loss of
generality, we will explain the case of a single gapped boundary, but the
construction readily generalizes to multiple gapped boundaries, as is the case
for the color code on a triangle.

Consider a child theory $C$. A gapped boundary of $C$ can be constructed by
first choosing a condensation $B\subset C$. $B$ has the further property that
it is Lagrangian in $C$, namely $B=B^{\perp}$ where orthogonality is based on
the inner product defined on $C$ (see Sec. IV.1). Condensing such a Lagrangian
subspace guarantees that there are no remaining deconfined anyons after the
condensation (that is, the resulting theory is trivial). Thus, given $C$
defined on an infinite plane, a gapped boundary of $C$ can be constructed by
condensing $B$ on, for example, the right half-plane. This realizes $C$ on the
left half of the plane, with condensation to the vacuum on the right half.

Next, recall that $C$ itself is a child theory which resulted by starting with
a parent $M$ and condensing $A$. We would now like to construct $C$ with a
boundary starting from $M$. To do so, we find a representative for every anyon
in $B$, which forms a subspace $\rho(B)\subset M$. Therefore, starting from
$M$ on an infinite plane, we can condense $A$ on the left half of the plane
and condense $A+\rho(B)$ on the right half. Note that $A+\rho(B)$ is
Lagrangian in $M$141414When $M$ is an Abelian modular category and $A$ is a
condensible subset of anyons we have $|A^{\perp}||A|=|M|$, see Ref. [58]. We
also have $|A^{\perp}/A|=|A^{\perp}|/|A|$. An algebra $B$ is Lagrangian in a
modular category $C$ iff $|C|=|B|^{2}$. We wish to show $A+\rho(B)$ is
Lagrangian in $M$. Note that $A\cap\rho(B)=0$ since $B$ is a non-trivial set
of anyons in $C$ and $C=A^{\perp}/A$. We also have that $|B|=|\rho(B)|$. Hence
$|A+\rho(B)|^{2}=|A|^{2}|\rho(B)|^{2}=|A|^{2}|B|^{2}=|A|^{2}|C|=|A|^{2}\left|{A^{\perp}}/{A}\right|=|A||A^{\perp}|=|M|$.
This implies that $A+\rho(B)$ is Lagrangian in $M$.. This matches the
intuition that condensing both $A$ and $\rho(B)$ must result in the trivial
theory. After the above condensation, the anyons in $B$ condense on this
boundary to the vacuum.

We can now analyze the case when we have a pair of reversible condensations in
the bulk, $A_{1}$ and $A_{2}$. Given a Lagrangian subspace
$B_{1}=B_{1}^{\perp}\subset C_{1}$, there exists $\rho_{1}$ such that
$\rho_{1}(B_{1})\subset A_{2}^{\perp}$. Moreover,
$B_{2}=\kappa_{2}(\rho_{1}(B_{1}))$ is Lagrangian in $C_{2}$ since
$|B_{2}|^{2}=|B_{1}|^{2}=|C_{2}|$. Let us define
$A_{\partial,12}=\rho_{1}(B_{1})$. We would like to show that the following
condensations constitute a reversible pair:

  1. 1.

Condensing $A_{1}$ on the left and $A_{1}+A_{\partial,12}$ on the right.

  2. 2.

Condensing $A_{2}$ on the left and $A_{2}+A_{\partial,12}$ on the right.

To see this, we can reinterpret the construction of the gapped boundary as
follows. Since $A_{\partial,12}$ appears in both condensations and the
condensations commute, let us first condense only $A_{\partial,12}$ on the
right half-plane. This results in $M$ on the left plane and
$C_{\partial,12}=A_{\partial,12}^{\perp}/A_{\partial,12}$ on the right plane.
Let us now consider each half plane separately. On the left half-plane, the
remaining condensation pair $A_{1}$ and $A_{2}$ is reversible by assumption.
On the right plane, the condensation pair $\kappa_{\partial,12}(A_{1})$ and
$\kappa_{\partial,12}(A_{2})$ are also reversible with respect to
$C_{\partial,12}$, because there are no deconfined anyons left on either side.
This shows our claim.

Let us discuss the implications of reversible condensations in the presence of
a gapped boundary. Since $A_{\partial,12}$ are only shared among a pair of
reversible condensations $A_{1}$ and $A_{2}$, this means that condensations
must also be updated for each pair of condensation rounds that follow. In
fact, since $A_{\partial,12}=\rho_{1}(B_{1})$, where $B_{1}\subset C_{1}$,
this means that the boundary condensations must be updated in the exact same
way as the representatives of child theory anyons are updated! Explicitly,
given bulk condensations $A_{0},A_{1},\ldots A_{T}=A_{0}$ (i.e. on the left
half-plane), and a Lagrangian boundary condensation $B_{0}\subset C_{0}$, the
condensations of the right half plane are fixed to be

$\displaystyle A_{0}$ $\displaystyle+\rho_{0}(B_{0});$ (47) $\displaystyle
A_{1}$ $\displaystyle+A_{\partial,01};$ $\displaystyle A_{\partial,01}$
$\displaystyle=\rho_{0}(B_{0})$ (48) $\displaystyle A_{2}$
$\displaystyle+A_{\partial,12};$ $\displaystyle A_{\partial,12}$
$\displaystyle=\nu_{1}(A_{\partial,01})$ (49) $\displaystyle\vdots$
$\displaystyle\vdots$ (50) $\displaystyle A_{T}$
$\displaystyle+A_{\partial,T-1\ T};$ $\displaystyle A_{\partial,T-1\ T}$
$\displaystyle=\nu_{T-1}(A_{\partial,T-2\ T-1})$ (51)

where $\nu$ updates the anyons according to the update formula, see Eq.(33).

Going to dynamic automorphism codes, we recall that a measurement sequence can
be translated from a condensation sequence by considering a Hamiltonian
realization of the parent and each of the child theories and by translating
condensations into measurements (see for example Eq. (52) for the DA color
code). Note that in doing so, we have the freedom of redefining the basis of
anyons of $A_{t}+A_{\partial,t-1t}$. In particular, we may choose a basis that
corresponds to low-weight measurements or another that has advantageous error-
correction properties.

The boundary update condition for condensations translates into the following
condition on measurements: the boundary measurements have to be such that, if
a logical string terminates (condenses) at a boundary at a given round, this
logical string, updated according to the bulk update rule, has to condense at
the same boundary in the subsequent round. Intuitively, one might think about
this the following way: within the child code, each boundary can be put in a
one-to-one correspondence with the set of logical strings that can terminate
at this boundary. For each logical string, there exists a representative that
is also a logical string of the next round. The condition demands that this
representative terminates on the same boundary in the next round; in the
presence of several boundaries, it means that the shape and position of the
logical operators do not change (however, they might shift by an edge upon
being updated). In turn, that means that 2+1D logical membranes remain
“continuous” in spacetime and do not rotate, which could cause uncorrectable
errors.

###  V.2 Condensation protocols on single and double triangles with Pauli
boundaries

####  V.2.1 Automorphisms on Pauli triangles

Figure 18: Unfolding the Pauli triangle color code into a pair of toric codes
in (a) double toric code and (b) color code notation. Here and in subsequent
drawings, the solid boundary is of $Z$ type, the dashed one is of $X$ and the
wavy one is the $Y$ boundary. here is an $\texttt{e}-\texttt{m}$ domain wall
along the line where the toric codes join, and the transformations of anyons
crossing this domain wall are shown on both sides of the arrows. Each Pauli
boundary corresponds to condensing a Lagrangian algebra object generated by a
corresponding Pauli column in the color code magic square. The bosons
condensed at each boundary are shown in gray. (c) The logical operator strings
of the Pauli triangle color code and our convention for the logical Pauli
operators. We display the trivalent junction as an rx and a rz logical strings
that join to form ry before meeting the $Y$ boundary; similar logic applies to
the other colors.  Figure 19: A sequence of anyon condensations from the
parent bilayer color code that implements $\varphi_{(\texttt{rbg})}$
automorphism in a DA color code on a triangle. Specifically, the condensations
in the bulk correspond to the $\varphi_{(\texttt{rbg})}$ protocol from Table
6. At $t=0$, we start in an effective color code with $\mathtt{rz_{1}rz_{2}}$
and $\mathtt{gz_{1}gz_{2}}$ condensations in the bulk and appropriate
condensations at the boundary that give Pauli boundaries of the effective
color code. Between rounds 1 and 3, the top row shows the triangle picture
with boundary condensations worked out as described in Sec. V.1, such that
each transition preserves the logical information. Rounds 1 and 3 correspond
to a toric code with an $\texttt{e}-\texttt{m}$ domain wall at each round. The
bottom row at these rounds shows the usual unfolding picture for these rounds.
The other generating sequence for a single triangle, corresponding to the
$\varphi_{(\texttt{rb})}$ automorphism, as well as the double triangle
sequence can be worked out analogously and the result is shown in Table 7.

First, let us consider a color code on a triangle with Pauli boundaries as
shown in Fig. 18. There are three such boundaries labeled by $\sigma=X,Y,Z$
where the $\sigma$ boundary is obtained by condensing the Lagrangian subgroup
$\\{1,\texttt{r}\texttt{\textsigma},\texttt{g}\texttt{\textsigma},\texttt{b}\texttt{\textsigma}\\}$
(i.e. all the bosons in the $\sigma$ column of the magic square) at the
boundary. In the presence of these boundaries, the automorphisms must leave
the Lagrangian subgroup for each boundary invariant. Therefore, the only
boundary-preserving automorphisms are the color-permuting automorphisms, for
which there are six of them in total.

Starting from two layers with the parent color code theory on a triangle as
described above, let us show how to construct a single child color code on a
triangle with Pauli boundaries. For now, assume that we work on an infinite
plane, and consider bulk condensations acting everywhere on the plane and
boundary condensations acting on disjoint thirds of the plane. The
partitioning of the plane is chosen so that a triangular patch is formed from
the set of sites that involves bulk measurements but no boundary measurements.
Thus, an effective (child) color code can be obtained by condensing
$\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$ and
$\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$ (in which case
$\texttt{gz}_{\texttt{1}}\texttt{gz}_{\texttt{2}}$ is also condensed)
everywhere in space and then forming effective $X,Y$, and $Z$ boundaries by
condensing appropriate bosons of the effective color code (shown in Eq. (3))
at the respective boundaries. Such a condensation pattern is shown at $t=0$ in
Fig. 19.

The distinct measurement sequences giving rise to color-permuting
automorphisms in the bulk are shown in the first two rows of Table 6. The
first two sequences in Table 6 correspond to $\varphi_{(\texttt{rb})}$, which
swaps green and blue anyon labels, and the $\mathbb{Z}_{3}$ automorphism
$\varphi_{(\texttt{rbg})}$ that cyclically permutes the colors. In fact, any
of the 6 automorphisms can be obtained from these generating measurement
sequences. Alternatively, we can obtain sequences realizing other
automorphisms by simply using the sequences in Table 6 and appropriately
permuting the colors in the measurement sequences.

Gate/Aut | $t$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6
$H\sim\varphi_{(\texttt{rb})}$ | $V(Z_{1}Z_{2})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{1})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(Z_{1})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{1})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(Z_{1})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{1})$ | $V(Z_{1}Z_{2})$
${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{2})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(Z_{2})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(X_{2})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(Z_{2})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{2})$
$SH\sim\varphi_{(\texttt{rbg})}$ | $V(Z_{1}Z_{2})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{1})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(Z_{1})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{1})$ | $V(Z_{1}Z_{2})$ |  |
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(X_{2})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(Z_{2})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{2})$ |  |
iSWAP | $V(Z_{1}Z_{2})$ | ${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{1})$ | $V(Z_{1}Z_{3})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{1})$ | $V(Z_{1}Z_{2})$ |  |
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{2})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(X_{2})$ |  |
$V(Z_{3}Z_{4})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{3})$ | $V(Z_{2}Z_{4})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{E_{g}}}(X_{3})$ | $V(Z_{1}Z_{2})$ |  |
${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{E_{r}}}}}(X_{4})$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{E_{b}}}(X_{4})$ |  |
Table 6: The first two rows show the bulk measurement sequences for the two
generators of the $S_{3}$ subgroup of color permutations of the color code
automorphisms. These sequences are used in a Pauli triangle dynamic color
code, which generates the Clifford group on one qubit. To obtain the full
Clifford group, we augment it with the two-qubit logical iSWAP gate, shown in
the last row. We discuss the boundary measurements in subsection V.3 and how
the two-qubit gate is identified is explained in subsection V.2.3.

The sequences for automorphisms shown in Table 6 were chosen such that the
corresponding logical gates on a triangle have a nice form. We also restrain
ourselves to CSS protocols in the bulk (i.e. only $X$ and $Z$ measurements)
because the codes obtained from them are easier to analyze. Moreover, as we
demonstrate later on, the chosen protocols for a single Pauli triangle DA
color code are error-correcting both in the bulk and on the boundaries.

The automorphisms shown in the first two rows of Table 6 allow us to realize
all single-qubit Clifford gates. To complete the generators for a multi-qubit
Clifford group, we need to achieve a two-qubit entangling gate. The last row
of the table shows a sequence that yields an automorphism corresponding to an
iSWAP gate. Together with the single qubit generators, this furnishes a
Clifford group on two qubits. For the iSWAP gate, we require two triangle DA
color codes, each of which contributes one logical qubit. This sequence will
be discussed in greater detail in Subsec. V.3.

####  V.2.2 Condensation sequences for the dynamic automorphism color code on
a triangle

The sequences of condensations producing generators for color permutations can
be worked out according to Sec. V.1. We start with the effective color code on
a Pauli triangle having performed
$\texttt{rz}_{\texttt{1}}\texttt{rz}_{\texttt{2}}$ and
$\texttt{bz}_{\texttt{1}}\texttt{bz}_{\texttt{2}}$ bulk condensations
(corresponding to measuring $V(Z_{1}Z_{2})$), as well as boundary
condensations that produce effective $X,$ $Y$ and $Z$ boundaries, as shown in
Fig. 19 at $t=0$. Next, we choose the sequences of bulk condensations
corresponding to the measurements shown in Table 6, and work out the boundary
condensations in all rounds from the requirement that a representative of each
logical operator has to be passed to each next round and has to condense on
the same boundary each time. As a remark, we keep the notations for the $X$,
$Y$ and $Z$ boundaries static for simplicity. The true boundary labels for the
measurement rounds when the bulk theory is that of the toric code are “rough”
and “smooth” and are permuted each round.

$t$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$H\sim\varphi_{(\texttt{rb})}$ | Bulk | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$
Bdry | $X$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>|  | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}}}}$
$\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}}}}$ | $\sim{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}}}}$
$Y$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}}}}$
$\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}}}}$ | $\sim{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}}}}$
$Z$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>|  | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}}}}$
$\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}}}}$ | $\sim{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}}}}$
$SH\sim\varphi_{(\texttt{rbg})}$ | Bulk | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  |  |  |
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ |  |  |  |
Bdry | $X$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}}}}$ |  |  |  |
$\widetilde{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| $\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>|  | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}}}}$ |  |  |  |
$Y$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{by}}}}$ |  |  |  |
$\widetilde{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| $\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gy}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gy}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{1}}}}}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{rx}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{ry}}}}$ |  |  |  |
$Z$ | $\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}}}}$ | $\sim{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{gz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ |  | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | $\sim\widetilde{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}}}}$ |  |  |  |
$\widetilde{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| $\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>|  | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gz}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\times}{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\texttt{bz}_{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | $\sim\widetilde{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}}}}$ |  |  |  |
iSWAP | Bulk | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{3}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{1}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}$ |  |  |  |  |
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{2}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{4}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}$ |  |  |  |  |
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{3}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{4}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{3}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{1}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{3}}}}}$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\texttt{gx}_{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{3}}}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rz}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{4}}}}}$ |  |  |  |  |
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{3}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{4}}}}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{rx}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\texttt{4}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{2}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{4}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bx}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{4}}}}}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{3}}}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{bz}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\texttt{4}}}}}$ |  |  |  |  |