preparation of the state $\ket{U}$. Programming thus requires the ability to
create arbitrary superpositions of the twisted states $\ket{V_{g}}$. On the
other hand, MBQC is a computation protocol in which the programming is encoded
into a choice of measurement basis. The actual measurements in MBQC are local
and act on single qubits on a lattice, so we will not find this basis in
$\mathcal{H}_{S_{1}}$. However, as we will explain in section 7, the MBQC
measurements effectively projects onto basis elements in a code subspace666A
code subspace usually refers to a subspace of logical qubits. In this context,
the elements of the code subspace can be identified with coded operators via a
sort of state-operator correspondence.  and these _are_ represented in
$\mathcal{H}_{S_{1}}$. This basis is defined as follows. Consider fusing both
sides of equation (32) with the flux $U^{\dagger}$. Explicitly, if we define
$U$ as

$\displaystyle U$ $\displaystyle=c_{1}1+c_{X}X+c_{Z}Z+c_{ZX}ZX$ (33)

then

$\displaystyle U^{\dagger}$
$\displaystyle\equiv\sum_{g}\tilde{c}_{g}V_{g}=c^{*}_{1}1+c^{*}_{X}X+c^{*}_{Z}Z-c^{*}_{ZX}ZX,$
(34)

and we can define the state

$\displaystyle\ket{U^{\dagger}}$
$\displaystyle\equiv\sum_{g}\tilde{c}_{g}\ket{V_{g}}$ (35)

Now fusing both sides of (32) with $\ket{U^{\dagger}}$ gives777In (36), we
used the identity $\ket{U^{\dagger}}\boxtimes\ket{U}=\ket{1}$. This is a
property of the algebra of fluxes defined by (25), although it is intuitively
obvious from the matrix definition of $V_{g}$.

$\displaystyle\ket{1}=\sum_{g}c_{g}\ket{U^{\dagger}V_{g}}$ (36)

This equation implies that we can extract the computational output
$|c_{g}|^{2}$ by measuring $\ket{1}$ in a “rotated” flux basis
$\ket{U^{\dagger}V_{g}}$ . We can interpret measurements in the
$\ket{U^{\dagger}V_{g}}$ basis in terms of the path integral process obtained
by applying the adjoint map to turn the initial state $\ket{U}$ into a final
state: 888Notice that this diagram corresponds to applying the co product to
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} {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
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}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-26.88126pt}{-10.29784pt}{-25.60732pt}{-14.22638pt}{-25.60732pt}\pgfsys@curveto{-18.15492pt}{-25.60732pt}{-21.33957pt}{-26.88126pt}{-21.33957pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{21.33957pt}{-28.45276pt}\pgfsys@curveto{21.33957pt}{-26.88126pt}{18.15492pt}{-25.60732pt}{14.22638pt}{-25.60732pt}\pgfsys@curveto{10.29784pt}{-25.60732pt}{7.11319pt}{-26.88126pt}{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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G}\ket{g}\otimes\ket{g^{-1}}$ (37) and then taking the over lap with
$\bra{U}\otimes\bra{h}$. This gives $\braket{U^{\dagger}V_{g}}{1}$ provided we
observe that $\ket{(XZ)^{-1}}=-\ket{XZ}.$

(38)

In this “passive” version of the computation, the lab technician always
prepares the state $\ket{1}$ of trivial flux 999 we will later identify this
state with the cluster state, and the programming is implemented by the
rotation of the measurement basis

$\displaystyle\bra{V_{g}}\to\bra{U^{\dagger}V_{g}}$ (39)

While these two ways of implementing the computation differ from an
operational point of view, both methods work because the computational output
$|c_{g}|^{2}$ depend only on the relative rotation between the fluxes basis
$\\{\ket{V_{g}}\\}$ and $\\{\ket{U^{\dagger}V_{g}}\\}$. Naively, it may seem
that neither is more advantageous than the other, since measuring in the new
basis $\bra{U^{\dagger}V_{g}}$ maybe equally complicated as superposing
$\ket{V_{g}}$ to create the resource state $\ket{U}$. However, MBQC provides
an algorithm that effectively implements measurements of the code subspace in
the basis $\bra{U^{\dagger}V_{g}}$, without the need to perform arbitrary
superpositions. This drasticallly reduces the operational costs of programming
and makes MBQC a useful protocol.

To give an explicit description of MBQC, we will first map the TQFT states
$\mathcal{H}_{S^{1}}$ into entangled states on a lattice. In section 7, we
will show that measurements in the basis $\bra{U^{\dagger}V_{g}}$ can then be
implemented by a temporally ordered sequence of single qubit measurements in
which the choice of measurement angles at each step depends on the measurement
outcomes of the previous steps. Our main goal in the following sections is to
explain this MBQC measurement protocol from the point of view of extended
TQFT.

##### Commutative vs Non-commutative algebras

Finally, before moving on, we emphasize that even though the computational
output of MBQC is a classical variable valued in
$\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, it is important for the purposes of the
circuit simulation that the fluxes are valued in a nontrivial _projective_
representation, as indicated by the non-commutative algebra (25). An ordinary
representation of an abelian group like $\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$
is one dimensional, and the corresponding holonomies cannot produce a
nontrivial gate. Such representations would appear in ordinary
$\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$ Dijkgraaf Witten theory, which is a
_commutative_ algebra as supposed to the non commutative algebra of the
equivariant theory. Notice that in the theory with a commutative algebra of
fluxes, we can still form the superposition of flux eigenstates
$\ket{g}_{\text{abelian}}$ :

$\displaystyle\ket{U}_{\text{abelian}}=\sum_{g\in
G}c_{g}\ket{g}_{\text{abelian}}$ (40)

However, we cannot define the analogue of (35) and (36), which are necessary
ingredients for the programing of $U$ in the MBQC protocol. Alternatively, we
will show in the next section that MBQC fails to work in ordinary TQFT’s with
abelian algebras because it does not have the necessary entanglement edge
modes.

##  4 Edge modes and extended TQFT

As alluded to earlier, to give an explicit description of the circuit
simulation, we need to provide a TQFT definition of the Hilbert space for the
logical qubit, which is distinct from $\mathcal{H}_{S^{1}}$. Intuitively we
want to cut open the circle into an interval, and construct a suitable _edge
modes_ Hilbert space at the endpoints where the logical qubit lives. Moreover,
where as cobordisms of the closed TQFT carries a notion of Euclidean time that
flows from the initial to final boundary (from top to bottom), the circuit
model we are simulating requires a notion of circuit time that evolves the
qubit along a spatial slice from left to right. All of these ideas can be
incorporated into a coherent framework called _extended_ TQFT. In particular,
the sewing rules of the extended equivariant TQFT will link the algebra (25)
of fluxes to the algebra of representation matrices $V_{g}$ acting on the edge
modes. This defines a sort of state-operator correspondence that explains how
a circuit simulation can be encoded into the twisted states (22) of a closed
equivariant TQFT.

###  4.1 Category description of an open-closed TQFT

To appreciate the logic behind extended TQFT, it is useful formulate our
closed TQFT in the language of category theory. A category consists of a set
of objects and a set of relations between these objects called morphisms,
which compose in an associative fashion. In particular, the geometric category
$\mathcal{C}$ of closed, 1 dimensional manifolds has circles as the objects
and cobordisms between circles as the morphisms. The composition of these
morphisms correspond to the gluing of cobordisms. In this language, a 2D
closed TQFT is a map:

$\displaystyle Z_{\text{closed}}:\mathcal{C}\to\mathbf{Vect}_{\mathbb{C}}$
(41)

from $\mathcal{C}$ into the category $\mathbf{Vect}_{\mathbb{C}}$ of complex
vector spaces. The morphisms of $\mathbf{Vect}_{\mathbb{C}}$ are given by
linear maps with the usual composition rule. The fact that $Z_{\text{closed}}$
maps the gluing of cobordisms to composition of linear maps means that
$Z_{\text{closed}}$ is a functor from $\mathcal{C}$ to
$\mathbf{Vect}_{\mathbb{C}}$. The functor $Z_{\text{closed}}$ is therefore a
linear representation of 2D manifolds and describes their gluing and cutting
along co-dimension 1 manifolds.

A 2D extended TQFT generalizes this categorical description of the path
integral to allow for cutting and gluing along codimension 2 manifolds, which
in this case is just a point. It introduces two new ingredients, both of which
are relevant to MBQC. First, the extension of a closed TQFT
$Z_{\text{closed}}$ enlarges the source category $\mathcal{C}$ by including
labelled intervals  $a$$b$ as objects. We can view the labels $a,b$ as
abstract boundary conditions. The extended TQFT assigns a Hilbert space
$\mathcal{H}_{ab}$ to these intervals, which contains edge mode degrees of
freedom determined by $a,b$. The set of cobordisms is then enlarged to include
manifolds with corners, which have intervals as well as circles as boundaries.
In particular, there is a fusion of intervals

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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(42)

which defines the multiplication rule for states in the interval Hilbert space
$\mathcal{H}_{\text{open}}$:

$\displaystyle\mathcal{H}_{\text{open}}=\oplus_{a,b}\mathcal{H}_{ab}$ (43)

We refer to $\mathcal{H}_{\text{open}}$ as the open algebra. The open sector
of the TQFT is naturally suited to describe MBQC on a chain.

For ordinary TQFT (in the absence of symmetry), the open algebra generically
satisfies a different multiplication rule than closed algebra: in particular,
the open algebra can be non commutative, whereas the closed algebra is always
commutative.

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Even though they are generically not identical, the closed and open algebra
are related by the “zipper” cobordism

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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:\mathcal{H}_{S^{1}}\to\mathcal{H}_{aa}$
(45)

which cuts open the circle into an interval. The sewing relations of the
extended TQFT imply that this is an algebra homomorphism from the closed to
the open sector. Such a homomorphism must satisfy a set of open-closed sewing
constraints Moore:2006dw . The solutions to these constraints represent
different ways of extending the closed algebra into an _open-closed_ algebra
that unifies both sectors. In particular, for an ordinary TQFT, the open-
closed algebra is a _knowledgeable Frobenius algebra_

##### Edge modes and the boundary category

The second new ingredient introduced by the extended TQFT is the assignment of
a _boundary category_ $\mathcal{B}$101010Note that $\mathcal{B}$ is a target
category. i.e the extended TQFT includes a map $Z:\text{point}\to\mathcal{B}$
to codimension 2 surfaces which make up the endpoints of an interval. We will
choose $\mathcal{B}$ to be the category of boundary conditions, whose objects
are given by the abstract boundary labels $a,b,\cdots$. Morphisms between
boundary conditions $a$ and $b$ are boundary condition changing operators
$\hat{O}_{ab}$: such operators always form the structure of a vector space
(see Kapustin:2010ta  for a nice explanation of this fact). Concretely, in our
discussion of open TQFT, these boundary condition changing operators can be
identified with the elements of the interval Hilbert space $\mathcal{H}_{ab}$,
which we now view as a space of morphisms between $a$ and $b$:

$\displaystyle\mathcal{H}_{ab}=\text{Hom}(a,b)$ (46)

The fact that elements of $\mathcal{H}_{ab}$ are morphisms simply means that
they can be composed in an associative fashion. Their composition law is given
by the open fusion rule (42), which satisfies associativity as one of the
sewing relations.

The morphisms between objects at the endpoints of an interval represent a type
of “horizontal” evolution. This is the TQFT formulation of circuit time
evolution. Meanwhile, the identification of these horizontal morphisms with
quantum states suggests a type of quantum teleportation in which an entangled
state on an interval can be used to teleport a qubit between its endpoints.
Below, we will make these suggestive ideas precise in the context of extended
equivariant TQFT.

###  4.2 G-equivariant open-closed TQFT and matrix product states

We now consider the extension of G-equivariant TQFT. This again introduces
edge modes in an open sector in which intervals are the objects. These
intervals are marked with a based point at the left endpoint. As in the closed
sector, all cobordisms are equipped with a flat gauge bundle, which is
specified by indicating the holonomy between marked points on the initial and
final intervals

(47)

In addition, the open equivariant TQFT also includes a symmetry operation: .

$\displaystyle\vbox{\hbox{\includegraphics[scale={.1}]{stripsym.pdf}}}:\mathcal{H}_{ab}\to\mathcal{H}_{ab}$
(48)

The sewing relations for an open-closed equivariant TQFT were solved in
Moore:2006dw . In particular, the interval Hilbert space is given by

$\displaystyle\mathcal{H}_{ab}=\mathcal{V}^{*}_{a}\otimes\mathcal{V}_{b}=\text{Hom}(a,b)$
(49)

where $\mathcal{V}_{a}$ are projective representations of $G$. These
projective representations are the edge modes of the G-equivariant TQFT
Moore:2006dw . Notice that the structure of the Hilbert space (49) naturally
lends it self to the interpretation as the space of morphisms between
$\mathcal{V}_{a}$ and $\mathcal{V}_{b}$, since
$\mathcal{V}^{*}_{a}\otimes\mathcal{V}_{b}$ is isomorphic to the space of
linear maps between $\mathcal{V}_{a}$ and $\mathcal{V}_{b}$. This is made
explicit by choosing a basis $\ket{a,i}\in\mathcal{V}_{a}$,
$\ket{b,j}\in\mathcal{V}_{b}$ (we emphasize these states are distinct from
states in $\mathcal{H}_{ab}$). Then a basis for $\ket{\bar{i},j}$ for
$\mathcal{H}_{ab}$ is in one to one correspondence with operators from
$\mathcal{V}_{b}$ to $\mathcal{V}_{a}$:

$\displaystyle\ket{\bar{i},j}\equiv\ket{b,i}\otimes\ket{a,j}\leftrightarrow\ket{b,i}\otimes\bra{a,j}\quad
i=1,\cdots\dim\mathcal{V}_{a},\quad j=1,\cdots\dim\mathcal{V}_{b},$ (50)

and the open string fusion corresponds to the composition of these operators.
The open algebra is thus a matrix algebra:

$\displaystyle\quad\vbox{\hbox{ \leavevmode\hbox to43.08pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
21.53957pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-4.61287pt}{-4.61287pt}{-9.38945pt}{0.0pt}{-9.38945pt}\pgfsys@curveto{4.61287pt}{-9.38945pt}{7.11319pt}{-4.61287pt}{7.11319pt}{0.0pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@curveto{21.33957pt}{-12.45241pt}{7.11319pt}{-16.00035pt}{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{-12.45241pt}{-7.11319pt}{-16.00035pt}{-7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:\ket{i,\bar{j}}\otimes\ket{k,\bar{l}}\to\delta_{kj}\ket{i\bar{l}}$
(51)

##### Matrix product states

For applications to MBQC, we will be interested in G-equivariant TQFT’s that
describe the RG fixed point of an SPT phase. In this context, the interval
Hilbert space $\mathcal{H}_{ab}$ is a coarse grained description of a 1D
system with edge modes transforming in the same projective representation
111111 To simulate an arbitrary single qubit unitary actually requires a
maximally non commutative projective representation, in which the phases
$b(g,h)$ satisfy $b(g,h)\neq\b{(}h,g)$ when $g\neq h$ Else_2012  on both
boundaries. In this case, we set $a=b=\mathcal{V}$ and denote the interval
Hilbert space by

$\displaystyle\mathcal{H}_{\text{open}}=\mathcal{V}^{*}\otimes\mathcal{V}$
(52)

Each quantum state in $\mathcal{H}_{\text{open}}$ is then identified with a
linear map of $\mathcal{V}$ onto itself, which describes a quantum evolution
from the left endpoint to the right endpoint of the interval. In particular,
we will see that when $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, this horizontal
evolution can be interpreted as the standard teleportation of a qubit in
$\mathcal{V}$ from one endpoint to the other.

To see the relation to teleportation, it is useful to formulate the open TQFT
in terms of matrix product state (MPS) tensors. In this context MPS tensors
arise as intertwiners121212An intertwiner is a linear map between
representations that commute with the action of G between two projective
representations of $G$ with the same projective phase $b(g,h)$. One is given
by $\mathcal{H}_{\text{open}}=\mathcal{V}\otimes\mathcal{V}^{*}$, and the
other is the $b$ twisted regular representation of $G$ CHENG. The latter can
be defined abstractly via the action of $G$ on the twisted group algebra
$\mathbb{C}_{b}[G]$. As a vector space, this algebra is spanned by a basis
$\ket{g}$ labelled by group elements. Each element in $\mathbb{C}_{b}[G]$ is
thus a formal linear combination

$\displaystyle\ket{f}=\sum_{g\in G}f(g)\ket{g}$ (53)

where $f(g)$ is a complex valued wavefunction on the group. The action of an
element $h\in G$ in the regular representation is defined to be

$\displaystyle l(h)\ket{g}=b(h,g)\ket{hg}$ (54)

which corresponds to left multiplication by $\ket{h}$ in the $b$ twisted group
algebra.

The projective Peter Weyl theorem cheng2015character  implies that for every
nontrivial projective representations $\mathcal{V}$ of $G$, its representation
matrices $V_{g}^{ij}$ define an intertwiner131313This map is one to one if
there is a single projective irrep. associated to the phase $b(g,h)$. In
general the Peter Weyl theorem implies that
$\displaystyle\mathbb{C}_{b}[G]=\oplus_{R}\mathcal{V}_{R}\otimes\mathcal{V}_{R}^{*}$
(55) where $R$ labels the different irreps with the same phase. In this
general setting, the intertwining map between the two sides of (55) will
involve a direct sum of representation matrices.

$\displaystyle A$
$\displaystyle:\mathcal{V}\otimes\mathcal{V}^{*}\to\mathbb{C}[G]$
$\displaystyle A$ $\displaystyle=\sum_{i,j}\sum_{g\in
G}V_{g}^{ij}\ket{g}\bra{\bar{i}\,j}$ (56)

The interwiner $A$ can be represented graphically as a matrix product state
tensor

(57)

We will refer to $g$ as the “physical index” and $i,j$ as the edge mode index.
In the group basis $\ket{g}$, also called the _symmetry basis_ , this tensor
is given by $A_{g}^{ij}=V_{g}^{ij}$. However, to implement the MBQC protocol,
we will need to project $A$ onto a more general class of measurement basis
that is also labelled by group elements. For this reason, we will reserve the
symbol $A_{g}^{ij}$ to denote the components of $A$ in a general basis. In MPS
notation, the intertwining property of $A$ (with
$\mathcal{V}\otimes\mathcal{V}^{*}$ viewed as a representation of $G$ in the
left tensor factor) implies the transformation law

(58)

This follows directly from the representation property (25) of the matrices
$V_{g}$. This equation shows how the left-regular representation is equivalent
to the _edge modes symmetry_ acting on $\mathcal{V}$. Similarly, there is a
right-regular representation $r(h)$ which acts on the right by

$\displaystyle r(h)\ket{g}=b(g,h)\ket{gh^{-1}}$ (59)

with the associated transformation law ( change $V(g)$ to $V(g^{1})$

Usually, an MPS tensor is contracted with physical basis states $\ket{g}$ to
define a quantum state. For example, for each choice of $i,j$ we obtain a
state on $\mathcal{H}_{open}$ given by

$\displaystyle\ket{A_{ij}}$
$\displaystyle=\sum_{g}\braket{i}{A_{g}}{j}\ket{g}\in\mathcal{H}_{\text{open}}$
(61)

However, to relate our discussion of extended TQFT to teleportation, it is
more useful to view $A$ as a tensor that assigns a linear map (a morphism)
$A_{g}:\mathcal{V}\to\mathcal{V}$ to each quantum state $\ket{g}$. Such a
tensor can be viewed as a fluctuating quantum circuit on $\mathcal{V}$, which
we can write as:

$\displaystyle A=\sum_{g}A_{g}\ket{g},\qquad A_{g}:\mathcal{V}\to\mathcal{V}.$
(62)

In this formula, we read $A_{g}$ as an operator valued coefficient, so that a
measurement that projects $A$ on to $\ket{g}$ implements the linear map
$A_{g}$. In MPS notation, we express this linear map as:

$\displaystyle\quad\vbox{\hbox{ \leavevmode\hbox to43.08pt{\vbox
to32.56pt{\pgfpicture\makeatletter\hbox{\hskip
11.58092pt\lower-20.11684pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@lineto{19.91684pt}{11.38092pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@lineto{0.0pt}{-19.91684pt}\pgfsys@lineto{19.91684pt}{-19.91684pt}\pgfsys@lineto{19.91684pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-11.38092pt}{5.69046pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{5.69046pt}\pgfsys@lineto{31.2982pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{5.69046pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}\pgfsys@moveto{9.3894pt}{5.69055pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.6394pt}{2.2739pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$A$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{9.95863pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.90724pt}{-15.17082pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\footnotesize$g$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:\mathcal{V}\to\mathcal{V}.$
(63)

###  4.3 Teleportation by symmetry

We now give a circuit model description of the fluctuating operator $A$ which
shows that it describes the teleportation of edge mode qubits in
$\mathcal{V}$Wong:2022mnv . The conventional circuit diagram for teleportation
is shown in figure 6.

Figure 6: Circuit diagram for quantum teleportation.

To teleport a qubit $\ket{\text{in}}$ from Alice to Bob, they must share an
entangled resource state given by the bell pair
$\beta_{00}\equiv\frac{1}{\sqrt{2}}(\ket{11}+\ket{00})$. Alice then makes a
joint measurement on herself and $\ket{in}$ qubit in a bell basis

$\displaystyle\ket{\beta_{s_{1}s_{2}}}=X^{s_{2}}Z^{s_{1}}\ket{\beta_{00}}$
(64)

which teleports the qubit to Bob. Due to the randomness of Alice’s measurement
outcomes, The actual qubit received by Bob is rotated by a random _by product
operator_ $Z^{s_{1}}X^{s{2}}$. Thus, the teleportation protocol is captured by
a random circuit of the form (62), in which a measurement outcome
$(s_{1},s_{2})$ implements the unitary

$\displaystyle A_{s_{1}s_{2}}=\quad\vbox{\hbox{ \leavevmode\hbox
to43.08pt{\vbox to32.56pt{\pgfpicture\makeatletter\hbox{\hskip
11.58092pt\lower-20.11684pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@lineto{19.91684pt}{11.38092pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@lineto{0.0pt}{-19.91684pt}\pgfsys@lineto{19.91684pt}{-19.91684pt}\pgfsys@lineto{19.91684pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-11.38092pt}{5.69046pt}\pgfsys@lineto{0.0pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{5.69046pt}\pgfsys@lineto{31.2982pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{5.69046pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-8.5359pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}\pgfsys@moveto{9.3894pt}{5.69055pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.6394pt}{2.2739pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$A$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{9.95863pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.96863pt}{-15.22682pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\footnotesize$s_{1}s_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad=Z^{s_{1}}X^{s{2}}$
(65)

If we identify measurement outcomes $(s_{1},s_{2})$ in the bell basis with
group elements:

$\displaystyle g=(s_{1},s_{2})$ (66)

then the teleportation protocol can be identified with the fluctuating circuit
(62) defined by the intertwiner $A_{g}$. More precisely, this mapping works
because equation (64) shows that the bell basis forms an orbit under the
projective representation (30) of $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$.
This defines a group label $g=(s_{1},s_{2})$ for elements of this orbit with
respect to the reference state $\ket{\beta_{00}}$, which allows an
identification

$\displaystyle\ket{g}\to V(g)\ket{\beta_{00}}$ (67)

between the an abstract state $\ket{g}\in\mathbb{C}_{b}[G]$ and the bell
basis.

The symmetry that relates the measurement outcomes also allows Bob to mitigate
the randomness of the circuit: if Alice communicates to her measurement
outcomes to Bob via classical channels, he can apply an inverse operation that
undoes the effects of the byproduct, thereby retrieving the original qubit. In
this sense, teleportation can be also be viewed as implementing the identity
gate. This perspective is captured by the “zigzag” identity141414The
appearance of this 1 D TQFT is not a coincidence. There is another formulation
of 2D extended TQFT in which the interval itself is viewed as a “1-morphism”,
i.e. a cobordism between the two endpoints. In this language cobordisms
between intervals are 2-morphisms ( a morphism between morphism). In a sense
the extended TQFT provides rules that tie together TQFT in different
dimensions. In particular the zigzag identity is a sewing relation of the 1D
TQFT that is refered to as “Dualizability”  of a 1D TQFT, as illustrated in
figure 7.

Figure 7: The bell pair creation and bell measurement in the teleportation
protocol corresponds to the pairing and co-pairing cobordisms of a 1D TQFT.
The zigzag identity expresses the fact that these operations combine to form
the identity map.

##### The closed to open map

In section 3.3, we had claimed that holonomy $g$ that labels the states
$\ket{V_{g}}\in\mathcal{H}_{S^{1}}$ of the closed TQFT can be identified with
a quantum circuit $V_{g}$ that transports a virtual particle around the
circle. Our discussion above shows that this quantum circuit is a generalized
form of teleportation acting on edge modes that transform under a projective
representation of $G$.

In the language of extended TQFT, the identification between the twisted state
$\ket{V_{g}}$ and the teleportation of edge modes is given by the zipper
cobordism, which maps the states on the circle to those on the interval:

$\displaystyle\quad\vbox{\hbox{ \leavevmode\hbox to14.63pt{\vbox
to31.7pt{\pgfpicture\makeatletter\hbox{\hskip 7.31319pt\lower-28.65276pt\hbox
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}
\definecolor[named]{tikz@axis@bottom}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{0.75,0.75,0.75}\definecolor[named]{tikz@axis@top}{rgb}{.75,.75,.75}\definecolor{tikz@axis@middle}{rgb}{0.875,0.875,0.875}{}{{}}{}
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{}{}\pgfsys@moveto{-7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-22.22661pt}{-4.40256pt}{-18.62894pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{4.40256pt}{-18.62894pt}{7.11319pt}{-22.22661pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{-7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-22.22661pt}{-4.40256pt}{-18.62894pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{4.40256pt}{-18.62894pt}{7.11319pt}{-22.22661pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-14.22638pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.567}{0.0}{0.0}{0.2835}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44499pt}{-69.44499pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{-7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-22.22661pt}{-4.40256pt}{-18.62894pt}{0.0pt}{-14.22638pt}\pgfsys@curveto{4.40256pt}{-18.62894pt}{7.11319pt}{-22.22661pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
\definecolor[named]{tikz@axis@top}{rgb}{1,1,1}\definecolor{tikz@axis@middle}{rgb}{1,1,1}\definecolor[named]{tikz@axis@bottom}{rgb}{.75,.75,.75}\definecolor{tikz@axis@middle}{rgb}{0.875,0.875,0.875}{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{1.5715pt}{3.92854pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-3.92854pt}{2.84544pt}{-7.11319pt}{1.5715pt}{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-1.5715pt}{-3.92854pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{3.92854pt}{-2.84544pt}{7.11319pt}{-1.5715pt}{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{1.5715pt}{3.92854pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-3.92854pt}{2.84544pt}{-7.11319pt}{1.5715pt}{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-1.5715pt}{-3.92854pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{3.92854pt}{-2.84544pt}{7.11319pt}{-1.5715pt}{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.1134}{0.0}{0.0}{0.2835}{0.0pt}{0.0pt}\pgfsys@invoke{
}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44499pt}{-69.44499pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{1.5715pt}{3.92854pt}{2.84544pt}{0.0pt}{2.84544pt}\pgfsys@curveto{-3.92854pt}{2.84544pt}{-7.11319pt}{1.5715pt}{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-1.5715pt}{-3.92854pt}{-2.84544pt}{0.0pt}{-2.84544pt}\pgfsys@curveto{3.92854pt}{-2.84544pt}{7.11319pt}{-1.5715pt}{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:$
$\displaystyle\mathcal{H}_{S^{1}}\to\mathcal{H}_{\text{open}}$
$\displaystyle\ket{V_{g}}\to\ket{g}\equiv\sum_{ij}\sqrt{\dim
V}\,V_{g}^{ij}\ket{\bar{i}j}$ (68)

Applying the state-channel duality (50) to the basis states $\ket{\bar{i}j}$
then gives the identification of $\ket{V_{g}}$ with the quantum circuit
$V_{g}:\mathcal{V}\to\mathcal{V}$, which is the same linear map defined by the
MPS tensor in (63). This is the state-operator correspondence we alluded to at
the beginning of section 4.

We can check explicitly that (4.3) is an algebra homomorphism: using the fact
that the open algebra is the matrix algebra of the Pauli matrices, we apply
the representation property of these matrices 151515Explcitly we have
$\displaystyle\quad\vbox{\hbox{ \leavevmode\hbox to43.08pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip
21.53957pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-4.61287pt}{-4.61287pt}{-9.38945pt}{0.0pt}{-9.38945pt}\pgfsys@curveto{4.61287pt}{-9.38945pt}{7.11319pt}{-4.61287pt}{7.11319pt}{0.0pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@curveto{21.33957pt}{-12.45241pt}{7.11319pt}{-16.00035pt}{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{-12.45241pt}{-7.11319pt}{-16.00035pt}{-7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:\ket{g_{1}}\otimes\ket{g_{2}}$
$\displaystyle\to\sum_{ik}V^{ik}_{g_{1}}\ket{i}\bra{k}\sum_{lj}V^{lj}_{g_{2}}\ket{l}\bra{j}=\sum_{ikj}V^{ik}_{g_{1}}V^{kj}_{g_{2}}\ket{i}\bra{j}$
$\displaystyle=\sum_{ij}b(g_{1},g_{2})V^{ij}_{g_{1}g_{2}}\ket{i}\bra{j}=b(g_{1},g_{2})\ket{g_{1}g_{2}}$
(69) to obtain

$\displaystyle\mu\equiv\quad\vbox{\hbox{ \leavevmode\hbox to43.08pt{\vbox
to33.71pt{\pgfpicture\makeatletter\hbox{\hskip
21.53957pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-4.61287pt}{-4.61287pt}{-9.38945pt}{0.0pt}{-9.38945pt}\pgfsys@curveto{4.61287pt}{-9.38945pt}{7.11319pt}{-4.61287pt}{7.11319pt}{0.0pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@curveto{21.33957pt}{-12.45241pt}{7.11319pt}{-16.00035pt}{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{-12.45241pt}{-7.11319pt}{-16.00035pt}{-7.11319pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
;{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.8625pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\footnotesize$e$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad:\ket{g_{1}}\otimes\ket{g_{2}}=b(g_{1},g_{2})\ket{g_{1}g_{2}},$
(70)

which is the same as the closed multiplication (3.2) and the multiplication
rule of the twisted group algebra $\mathbb{C}_{b}[G]$. In this case, the map
(4.3) is also one to one, so the closed and open algebras are isomorphic. This
is a distinguishing feature of the equivariant TQFT, which sets it apart from
an ordinary TQFT where the closed algebra is always commutative while the open
algebra may not be.

##### Summary of extended TQFT and quantum circuits

For each 2D closed G-equivariant TQFT associated to a projective
representation $\mathcal{V}\in H^{2}(G,U(1))$, the extension of the TQFT
introduces an open algebra
$\mathcal{H}_{\text{open}}=\mathcal{V^{*}}\otimes\mathcal{V}$ corresponding to
the Hilbert spaces on an interval. Each state on
$\mathcal{V^{*}}\otimes\mathcal{V}$ is associated to operators $A_{g}$ which
can be interpreted as a quantum circuit acting an “edge mode” qubit in
$\mathcal{V}$. (In the group basis $A_{g}$ is just the projective
representation matrix $V_{g}$ associated to $\mathcal{V}$). When
$G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, this circuit corresponds to the
standard teleportation protocol. For general G, this defines a more general
notion of teleportation by symmetry.

In the extended TQFT language, the identification of holonomies around the
circle with a quantum circuit is expressed by the closed to open map (4.3).
This is an isomorphism which maps flux eigenstates on $\mathcal{H}_{S^{1}}$ to
quantum circuits $A_{g}$, which implement the parallel transport of TQFT edge
modes along an interval. Later we will identify this notion of parallel
transport with the entanglement holonomy defined in Czech:2018kvg

##  5 Towards a local description of MBQC

In the section 3, we formulated a global version of MBQC on a 1D circle using
the language of TQFT. To relate this to the conventional, local description of
MBQC in terms of single -qubit measurements, we need a procedure that maps the
states of the TQFT into states on a lattice of qubits. This discretization
procedure is really an encoding map that embedds the TQFT Hilbert space into a
code subspace on the lattice. This encoding is implemented by repeated
applications of the co product in the open algebra. This coproduct defines a
factorization map into an extended Hilbert space, which we will identify with
the Hilbert space of the lattice model. The factorization map introduces
_entanglement_ edge modes, which are virtual particles that corresponding to
the contracted edges of the MPS tensors $A_{g}$. These edge modes cannot be
accessed directly without cutting open a spatial slice. Nevertheless, their
entanglement holonomy can be extracted from the correlations among
measurements on a circular array of qubits. and constitutes the computation
output of MBQC in 1D. We will argue that this provides an operational meaning
to entanglement edge modes.

###  5.1 Factorization via the co-product

The open algebra $\mathcal{H}_{\text{open}}=\mathcal{V^{*}}\otimes\mathcal{V}$
is a Frobenius algebra, which has both a product and a co-product. The co-
product is given by

$\displaystyle\Delta:\mathcal{H}_{\text{open}}$
$\displaystyle\to\mathcal{H}_{\text{open}}\otimes\mathcal{H}_{\text{open}}$
$\displaystyle\ket{g}$
$\displaystyle\to\frac{1}{|G|}\sum_{g_{1},g_{2}}b^{-1}(g_{1},g_{2})\delta(g_{1}g_{2},g)\ket{g_{1}}\ket{g_{2}},$
(71)

As emphasized in Wong:2022eiu , $\Delta$ can be interpreted as a factorization
map in to an extended Hilbert space defined via the path integral evolution:

$\displaystyle\Delta=\quad\vbox{\hbox{ \leavevmode\hbox to43.08pt{\vbox
to34.14pt{\pgfpicture\makeatletter\hbox{\hskip
21.53957pt\lower-33.93854pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{-28.45276pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-23.76907pt}{-4.68369pt}{-18.77892pt}{0.0pt}{-18.77892pt}\pgfsys@curveto{4.68369pt}{-18.77892pt}{7.11319pt}{-23.76907pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{21.33957pt}{-28.45276pt}\pgfsys@curveto{21.33957pt}{-16.00035pt}{7.11319pt}{-12.45241pt}{7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-12.45241pt}{-21.33957pt}{-16.00035pt}{-21.33957pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
;{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.32813pt}{-30.60553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$e$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad$ (72)

This euclidean process splits a single interval into two intervals separated
by a “stretched” entangling surface, given by the small semicircular arc
labeled by $e$. This denotes an entanglement boundary condition which is
required to satisfy the shrinkability condition 161616This boundary condition
need not be unique, although it will be so in the examples we consider.

$\displaystyle\mu\circ\Delta=\quad\vbox{\hbox{ \leavevmode\hbox
to43.08pt{\vbox to57.31pt{\pgfpicture\makeatletter\hbox{\hskip
21.53957pt\lower-57.10551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}
{{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{-28.45276pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-23.76907pt}{-4.68369pt}{-18.77892pt}{0.0pt}{-18.77892pt}\pgfsys@curveto{4.68369pt}{-18.77892pt}{7.11319pt}{-23.76907pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{21.33957pt}{-28.45276pt}\pgfsys@curveto{21.33957pt}{-16.00035pt}{7.11319pt}{-12.45241pt}{7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{0.0pt}\pgfsys@curveto{-7.11319pt}{-12.45241pt}{-21.33957pt}{-16.00035pt}{-21.33957pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope ;
\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {}{{}}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{-28.45276pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@curveto{-7.11319pt}{-33.06563pt}{-4.61287pt}{-37.84221pt}{0.0pt}{-37.84221pt}\pgfsys@curveto{4.61287pt}{-37.84221pt}{7.11319pt}{-33.06563pt}{7.11319pt}{-28.45276pt}\pgfsys@lineto{21.33957pt}{-28.45276pt}\pgfsys@curveto{21.33957pt}{-40.90517pt}{7.11319pt}{-44.45311pt}{7.11319pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-7.11319pt}{-56.90552pt}\pgfsys@lineto{7.11319pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{-21.33957pt}{-28.45276pt}\pgfsys@curveto{-21.33957pt}{-40.90517pt}{-7.11319pt}{-44.45311pt}{-7.11319pt}{-56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
;{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.32813pt}{-30.60553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$e$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad=\quad\vbox{\hbox{
\leavevmode\hbox to14.63pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip 7.31319pt\lower-28.65276pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-7.11319pt}{0.0pt}\pgfsys@moveto{-7.11319pt}{0.0pt}\pgfsys@lineto{-7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{-28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad=\text{Identity
Map}.$ (73)

This condition ensures that the entangling surface is completely transparent,
and does not alter the correlation functions of the original, unfactorized
states. More generally, any hole surrounded by the boundary condition $e$ is
required to be shrinkable. 171717 In a 2D TQFT, the shrinkability constraint
can be captured by a single diagramatic relation Donnelly:2018ppr :
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implies all holes created by an entanglement boundary can be closed.. In
quantum information language, the shrinkability condition says that the
factorization map is an isometry which encodes the Hilbert space of one
interval into a code subspace inside the Hilbert space of two interval.

In general, the shrinkability constraint requires that the extended Hilbert
space contains a suitable set of _entanglement_ edge modes Donnelly:2018ppr .
These entanglement edge modes are manifest in representation basis for
$\mathcal{H}_{\text{open}}\otimes\mathcal{H}_{\text{open}}$

$\displaystyle\ket{V\,\bar{i}\,j}\otimes\ket{V\,\bar{k}\,,l},\qquad
i,j,k,l=1,\cdots\dim V,$ (75)

where the states labeled by $j$ and $k$ are the “extra” edge mode degrees of
freedom introduced by the entangling surface. Unlike the states at the
physical boundary labelled by $i$ and $l$, the entanglement edge modes are
always entangled under the embedding defined by $\Delta$. We can see this by
writing action of the co-product on the MPS wavefunctions $V_{il}(g)$

$\displaystyle\Delta:\sum_{g}V_{il}(g)\ket{g}\longrightarrow\sum_{g_{1},g_{2}}\sum_{j}b(g_{1},g_{2})^{-1}V_{ij}(g_{1})V_{jl}(g_{2})\ket{g_{1}}\ket{g_{2}},$
(76)

in which the entanglement edge mode index j is contracted. In the MPS
descriptions, these correspond to degrees of freedom on the contracted virtual
edges, which are not directly accessible by local measurements.

We can iterate this procedure to split the interval repeatedly:

$\displaystyle\mathcal{H}_{\text{open}}\overset{\Delta}{\longrightarrow}\mathcal{H}_{\text{open}}\otimes\mathcal{H}_{\text{open}}\overset{1\otimes\Delta}{\longrightarrow}\mathcal{H}_{\text{open}}\otimes\mathcal{H}_{\text{open}}\otimes\mathcal{H}_{\text{open}}\longrightarrow\cdots$
(78)

After $N$ applications of $\Delta$, the original Hilbert space
$\mathcal{H}_{\text{open}}$ is embedded into an extended Hilbert space
$\mathcal{H}_{\text{open}}^{\otimes N+1}$, which we associate with the Hilbert
on a lattice where each $\mathcal{H}_{\text{open}}$ factor describes a local
Hilbert space. This procedure encodes the TQFT states into entangled lattice
states that belong to the fixed point of a 1D SPT phase with symmetry $G$.

##### Cluster state on a chain

As an example of this encoding of TQFT states, consider applying the sequence
of embeddings (78) to the states $\ket{A_{ij}}$ defined in (61). This produces
entangled states on a 1D lattice

$\displaystyle\ket{A_{ij}}$
$\displaystyle\to\ket{A_{ij}}_{\text{lat}}\equiv\sum_{g_{1},\cdots
g_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}b(g_{N},g_{N-1})^{-1}b(g_{N-3},g_{N-4})^{-1}\cdots
b(g_{2},g_{1})^{-1}(A_{g_{N}g_{N-1}\cdots
g_{1}})_{ij}\ket{g_{N}}\otimes\ket{g_{2}}\cdots\ket{g_{1}}$
$\displaystyle=\sum_{g_{1},\cdots
g_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}b(g_{N},g_{N-1})^{-2}b(g_{N-3},g_{N-4})^{-2}\cdots
b(g_{2},g_{1})^{-2}(A_{g_{N}}A_{g_{N-1}}\cdots
A_{g_{1}})_{ij}\ket{g_{N}}\otimes\ket{g_{N-1}}\cdots\ket{g_{1}}$

When $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, the phases square to 1181818For
general G, these phases do not affect the implemenation of MBQC on these
resource states, so we just have

$\displaystyle\ket{A_{ij}}_{\text{lat}}=\sum_{g_{1},\cdots
g_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}(A_{g_{N}}A_{g_{N-1}}\cdots
A_{g_{1}})_{ij}\ket{g_{N}}\otimes\ket{g_{N-1}}\cdots\ket{g_{1}},\qquad
i,j=1,2.$ (80)

We will refer to these state as _cluster states_ on a chain191919These states
satisfy more general boundary conditions than the usual definition of the
cluster state, but we retain the same nomenclature for convenience..

##### Cluster state on a circle

In a similar fashion, we discretize TQFT states on a circle by first applying
the closed to open map on the twisted state
$\ket{V_{g}}\in\mathcal{H}_{S^{1}}$ followed by repeated iterations of
$\Delta$. The resulting cobordism is shown in figure 8, which describes the
embedding of $\ket{V_{g}}$ into the extended Hilbert space on the circle. The
image of $\ket{V_{g}}$ is the matrix product states

$\displaystyle\ket{V_{g}}\to\ket{V_{g}}_{\text{lat}}=\sum_{h_{1},\cdots
h_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}\operatorname{tr}(V_{g}A_{h_{N}}A_{h_{N-1}}\cdots
A_{h_{1}})\ket{h_{N}}\otimes\ket{h_{N-1}}\cdots\ket{h_{1}}$ (81)

Figure 8: A discretization of a quantum state $\ket{1}$ on the circle can be
obtained by first applying the closed to open map, and then repeatedly
applying the co-product.

We emphasize again that the lattice Hilbert space does not contain the
entanglement edge modes which live on the contracted virtual edges:
nevertheless these virtual degrees of freedom play an important role as the
logical qubit in MBQC.

###  5.2 The fully local description of the cluster state

When $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, the basis states $\ket{g}$ on
each lattice site can be identified with the entangled bell pairs
$\ket{\beta_{s_{1}s_{2}}}$ which appeared in the teleportation protocol in
figure 6. However, to implement MBQC, we need to obtain a fully local
description of the cluster state in which these bell pairs are replaced by un-
entangled product states. In terms of the teleportation protocol, this
corresponds to replacing bell measurements with single qubit measurements.
This is easily accomplished by observing that the bell pairs can be created
from a product state using a Cphase and Hadamard gate. In a basis
$\ket{s_{i}}$ of $X$ eigenstates satisfying

$\displaystyle X\ket{s_{i}}$ $\displaystyle=(-1)^{s_{i}}\ket{s_{i}},\qquad
s_{i}=0,1$ (82)

the Cphase gate is defined by while the Hadamard gate is defined by

$\displaystyle H=\frac{1}{\sqrt{2}}(X+Z)$ (84)

The bell pair preparation and bell measurements in the teleportation circuit
in figure 6 can then be replaced using the identities

(85)

The Hadamard gates will cancel, leaving behind a sequence of cphase gates that
defines the lattice MPS tensor:

(86)

To identify this circuit with the abstract MPS tensor (57), we need a
consistent assignment of group labels to the product state
$\ket{s_{1}}\otimes\ket{s_{2}}$. This can be achieved via a lattice version of
the projective representation $l(h)$ defined in (54) :

$\displaystyle\tilde{l}(00)$ $\displaystyle=1\otimes
1,\quad\tilde{l}(01)=X_{2}\otimes Z_{1},\quad\tilde{l}(10)=Z_{2}\otimes
1,\quad\tilde{l}(11)=Z_{2}X_{2}\otimes Z_{1}$ (87)

Given a reference state $\ket{0}\ket{0}$ on the lattice, we can use this
representation to assign group labels on the lattice:

$\displaystyle\ket{g}_{\text{lat}}\equiv\tilde{l}(g)\ket{0}\ket{0}$ (88)

This defines a lattice realization of the $b$ regular representation202020 We
could have defined $\tilde{l}$ by just imposing (54) on the lattice. But we
want to connect to the representation $\tilde{l}$ that is used in the SPT
literature. In order for the action of $\tilde{l}$ to match with the regular
representation, we need to define $\ket{g}_{\text{lat}}$ as in (88). : (54)

$\displaystyle\tilde{l}(h)\ket{g}_{\text{lat}}$
$\displaystyle\equiv\tilde{l}(h)\tilde{l}(g)\ket{0}\ket{0}$
$\displaystyle=b(h,g)\tilde{l}(hg)\ket{0}\ket{0}$
$\displaystyle=b(h,g)\ket{hg}_{\text{lat}}$ (89)

Since this representation is isomorphic to (54), we will henceforth drop the
tilde and the lat. subscripts. However it is important to remember that the
mapping of the abstract intertwiner $A_{g}$ to the lattice MPS tensor involves
a choice of reference state.

Expressing the MPS tensors in the cluster state (80) in terms of the quantum
circuit in (86) makes it manifest that the cluster state can be created from a
product state with a finite depth212121Here finite depth refers to the fact
that the number of layers in the unitary is independent of the system size
unitary. This can be achieved by simply re-ordering the sequence of
(commuting) cphase gates into two local Unitaries: The fact that such a finite
depth creation process exist is a general property of an SPT phase. Thus, we
can take a more general perspective in which the lattice serves as a blank
canvas on which we can engineer any G-equivariant TQFT, where $G$
characterizes the entanglement structure of states belonging to a code
subspace.

####  5.2.1 Hilbert space and flux sectors on the lattice

The discretization process described above maps the closed TQFT Hilbert space
$\mathcal{H}_{S^{1}}$ onto a Hilbert space $\mathcal{H}_{S^{1}}^{\text{lat}}$
on a periodic lattice of $N$ qubits. $\mathcal{H}_{S^{1}}^{\text{lat}}$ can be
organized into flux sectors in a manner that parallels the decomposition of
the closed TQFT Hilbert space (21):

$\displaystyle\mathcal{H}_{S^{1}}^{\text{lattice}}$
$\displaystyle=\oplus_{g}\mathcal{H}_{g}^{\text{lattice}},\qquad g\in G$
$\displaystyle\mathcal{H}_{g}^{\text{lattice}}$
$\displaystyle\equiv\text{span}\\{\ket{h_{1}}\otimes\ket{h_{2}}\cdots\ket{h_{N}};\,\prod_{i}h_{i}=g\\}$
(91)

On the lattice, the flux labels $g$ are measured by the charges that generate
the global symmetry of the SPT phase.

For example, when $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, these charges are
given by

$\displaystyle(Q_{1},Q_{2}),\qquad Q_{1}=\prod_{a=1}^{N-1}X_{2a-1},\qquad
Q_{2}=\prod_{a=1}^{2N}X_{2a}$ (92)

These charges are the image under $\Delta$ of the symmetry operation
$\alpha_{g}$ defined in (24), which forms the symmetry data of the closed
$\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$-equivariant TQFT. Since the $X$ basis
$\ket{h_{i}}$ diagonalizes this symmetry, it is referred to as the _symmetry_
basis in this context.

In its usual lattice formulation, the cluster state is viewed as the ground
state of the cluster Hamiltonian

$\displaystyle H=\sum_{a}Z_{a-1}X_{a}Z_{a+1}$ (93)

which consists of a sum of stabilizers $Z_{a-1}X_{a}Z_{a+1}$. The charges
$Q_{1}$ and $Q_{2}$ commute with $H$ and therefore define a symmetry of the
system. The sectors $\mathcal{H}_{g}^{\text{lattice}}$ do not mix under
operators that commute with the symmetry; it is in this sense that they form
“superselection” sectors. However, from the point of view of the lattice,
there are no superselection rules between flux sectors. Indeed we will see
that MBQC implements quantum computations by manipulating the definition of
these flux sectors.

##  6 Gauge symmetry from entanglement

In section 3, we motivated G-equivariant TQFT as the theory which arises upon
coupling a matter TQFT to a background gauge field. However, the abstract
formulation of this TQFT does not make reference to any local gauge
symmetries. This formalism circumvents the usual gauge redundant, local
degrees of freedom by working directly with the holonomy variables that label
the twisted states $\ket{V_{g}}$ . Formally, G-equivariant TQFT is just a
Turaev algebra that describes the fusion of fluxes.

On the other hand, once we have mapped the TQFT Hilbert space into a code
subspace on the lattice, we can define local gauge transformations on the
extended Hilbert space that leaves the twisted states
$\ket{V_{g}}_{\text{lat}}$ invariant. These are generated by stabilizers that
characterize the entanglement structure of $\ket{V_{g}}_{\text{lat}}$. In
addition to the Hilbert space definition, we will give an operational
interpretation of this gauge symmetry as transformations that preserve the
circuit simulation implemented by MBQC. From this definition, it will be
manifest that in MBQC, the entanglement holonomies defined in Czech:2018kvg
are exactly the holonomies of a G-equivariant TQFT.

###  6.1 Gauge symmetry and stabilzier transformations

In our extended TQFT formulation of MBQC a gauge symmetry arises because the
encoding map that embedds TQFT states into the extended Hilbert space is not
surjective. This is because the co-product $\Delta$ always maps onto a
subspace invariant under a combination of left/right edge mode symmetries. If
we denote the co product acting on the $ith$ block by

$\displaystyle\Delta^{i}:\mathcal{H}_{\text{open}}^{i}\to\mathcal{H}_{\text{open}}^{i}\otimes\mathcal{H}_{\text{open}}^{i+1}$
(94)

then this invariance is characterized by

$\displaystyle l(h)_{i+1}r(h_{i})\Delta^{i}=\Delta^{i}$ (95)

This implies that the discretized TQFT states satisfy the constraints

$\displaystyle
l(h)_{i+1}r(h)_{i}\ket{V_{g}}_{\text{lat}}=\ket{V_{g}}_{\text{lat}},\qquad
l(h)_{i+1}r(h)_{i}\ket{A_{IJ}}_{\text{lat}}=\ket{A_{IJ}}_{\text{lat}}$ (96)

in the bulk of the chain. These constraints can be viewed as gauge constraints
that generate _local_ gauge transformations in the extended Hilbert space.
They act on
$\mathcal{H}_{\text{open}}^{i}\otimes\mathcal{H}_{\text{open}}^{i+1}$
according to

$\displaystyle
l(h)_{i+1}r(h)_{i}\ket{g_{i}}\ket{g_{i+1}}=b(h,g_{i+1})b(g_{i},h^{-1})\ket{g_{i}h^{-1}}\ket{hg_{i+1}},$
(97)

Note that the phases in (97) are important because they make this local
representation of $G$ projective. To be explicit, lets consider
$G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$. Writing $l(h)$ and $r(h)$ in terms of
single qubit Pauli operators shows that (97) are the stabilizer constraints of
the cluster state.

$\displaystyle
Z_{a-1}X_{a}Z_{a+1}\ket{V_{g}}_{\text{lat}}=\ket{V_{g}}_{\text{lat}}$ (98)

The presence of the $X$ operator in the stabilizer is what makes these gauge
transformations projective, and distinguishes them from the usual gauge
transformations of lattice gauge theory. To highlight the difference with
lattice gauge theory, we will refer to the transformations (97) as MBQC gauge
symmetries.

###  6.2 Entanglement holonomy and deterministic MBQC

To understand the operational aspects of the MBQC gauge symmetry, it is useful
to begin by formulating deterministic MBQC in the MPS language
PhysRevA.76.052315  . In our extended TQFT formalism, deterministic MBQC
refers to a protocol for simulating a unitary gate $V_{g}$ using a twisted
state on the lattice:

$\displaystyle\ket{V_{g}}_{\text{lat}}=\sum_{h_{1},\cdots
h_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}\operatorname{tr}(V_{g}A_{h_{N}}A_{h_{N-1}}\cdots
A_{h_{1}})\ket{h_{N}}\otimes\ket{h_{N-1}}\cdots\ket{h_{1}}$ (99)

In deterministic MBQC, we measure each lattice site ( or pairs of lattice
sites in the case of the cluster state) in the symmetry basis $\ket{h_{i}}$,
moving in sequential order down the chain. As we explained below equation
(62), a measurement of a single block with outcome $h_{i}$ produces a
teleportation of the edge modes in $\mathcal{V}$ given by the linear map
$A_{h_{i}}$.

Figure 9: Each local measurement of the MPS describes the local parallel
transport of a qubit between the edge Hilbert spaces

According to Czech:2018kvg , $A_{h_{i}}$ should be viewed as a local parallel
transport of the edge modes. More precisely, we identify $h_{i}$ with a gauge
connection on a principal $G$ bundle over a circle, and $A_{h_{i}}$ describes
its action in the projective representation of $G$. Since each measurement
outcome is random, this local parallel transport is also random. Due to this
randomness, no information can be extracted from local measurements in any
subregion of the circle.

To extract information from measurements of the twisted states
$\ket{V_{g}}_{\text{lat}}$, we observe that they satisfy a global constraint
which fixes the total parallel transport around the circle to be $V_{g}$ \-
after all these are the fluxes they were meant to encode ! The constraint is
manifest in the MPS wavefunction

$\displaystyle\operatorname{tr}(V_{g}A_{h_{1}}A_{h_{2}}\cdots
A_{h_{N}})=(\text{phase})\operatorname{tr}(V_{g}A_{h_{1}\cdots h_{N}}),$ (100)

which vanishes unless

$\displaystyle g=\prod_{i}h_{i}.$ (101)

This implies that even though the local parallel transport $A_{h_{i}}$ is
random, the total parallel transport experienced by the logical qubit is:

$\displaystyle\prod_{i}A_{h_{i}}=\pm V_{g}$ (102)

Since there are no boundaries in the spatial geometry, this logical qubit has
to annihilate itself upon being teleported in a closed loop: for a circuit
model description of this process, we refer the reader to appendix A.1. Thus
the entanglement holonomy associated to measurements along the symmetry basis
coincides with the “physical” holonomy assigned by the equivariant TQFT.
Notice that the entanglement holonomy (102) is invariant under the action of
the stabilizer transformations (97). This is obvious in the MPS notation when
we apply the equivariant transformaiton laws (59) and (58), which pushes the
action of the gauge transformations (97) in to the virtual space, where they
correspond to acting with $V_{h}V^{\dagger}_{h}=1$ on a virtual edge. Since
the entanglement holonomy is exactly the MBQC circuit simulation, these are
also the transformations that preserve MBQC.

The constraint (101) has an important consequence for the processing of
information. Even though the individual measurement outcomes are completely
random, information is encoded nonlocally in the total set of outcomes around
the whole circle. For example, when $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$,
there are two classibal bits of information encoded in the quantity

$\displaystyle g=(\sum_{a\,\text{odd}}s_{a},\sum_{a\,\text{
even}}s_{a})\in\mathbb{Z}_{2}\times\mathbb{Z}_{2},$ (103)

where $(-1)^{s_{a}},a=1,\cdots 2N$ labels the X eigenvalues of the qubits.
This is a non local observable measured by the charges $Q_{i},i=1,2$ of
equation (92), and labels the different flux sectors on the circle. These
fluxes are computed directly from the measurement outcomes and constitutes the
computational output of deterministic MBQC. Following Wong:2022mnv  we will
refer to these fluxes as MBQC holonomies to distinguish them from the
entanglement holonomies (102).

Finally, note that the $X$ basis states corresponding to different measurement
outcomes $\\{s_{a}\\}_{a=1,\cdots 2N}$ with the same computational output
(103) are related by the gauge transformations (97). This is because the
stabilizers $K_{a}=Z_{a-1}X_{a}Z_{a+1}$ flip an even number of spins on either
the odd or even sublattice, so their effect on the measurement record is:

$\displaystyle K_{a}:\quad$ $\displaystyle s_{a-1}\to s_{a-1}+1$
$\displaystyle s_{a}\to s_{a}$ $\displaystyle s_{a+1}\to s_{a+1}+1$ (104)

Thus, “pure gauge” fluctuations capture the random, local degrees of freedom
that contain no information: they are the fluctuations that remain within a
single superselection sector denoted by $\mathcal{H}_{g}^{\text{lattice}}$ in
(7.2). On the other hand, the gauge invariant degrees of freedom carries
quantum information that is non-locally encoded in the measurement outcomes.
This provides an information theoretical interpretation of MBQC gauge
symmetry, which will generalize to the probabilistic MBQC discussed below.

##  7 Probabilistic MBQC and measurement adaptations

In the previous section, we gave an operational definition of the gauge
symmetry of deterministic MBQC. This involved measurements of a resource state
in an SPT phase, using the symmetry basis. Below we will give a gauge theory
formulation of the general MBQC protocol in which the qubits are measured in
an adaptive basis, which enables the simulation of a general unitary gate.

###  7.1 Programming by state preparation

Following the discussion in section 3.3, we begin by explaining how to produce
an arbitrary single qubit unitary $U=\sum_{g}c_{g}V_{g}$ using the same
measurement protocol defined for deterministic MBQC, but applied to a
modified, U -dependent resource state. This resource state arises via the
encoding of the TQFT state $\ket{U}=\sum_{g}c_{g}\ket{V_{g}}$ (introduced in
(32)) onto the lattice:

$\displaystyle\ket{U}_{\text{lat}}$
$\displaystyle\equiv\sum_{g}c_{g}\ket{V_{g}}_{\text{lat}}$ (105)
$\displaystyle=\sum_{h_{1},\cdots
h_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}\operatorname{tr}((\sum_{g}c_{g}V_{g})A_{h_{1}}A_{h_{2}}\cdots
A_{h_{N}})\ket{h_{1}}\otimes\ket{h_{2}}\cdots\ket{h_{N}}$
$\displaystyle=\sum_{h_{1},\cdots
h_{N}\in\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}}\operatorname{tr}(UA_{h_{1}}A_{h_{2}}\cdots
A_{h_{N}})\ket{h_{1}}\otimes\ket{h_{2}}\cdots\ket{h_{N}},$ (106)

Since $\ket{U}_{\text{lat}}$ does not belong to a single flux sector as
defined in (7.2) , single qubit measurements in the symmetry basis
$\ket{h_{i}}$ will no longer give a definite computational output (103).
Instead, each MBQC holonomy $g$ will occur with a probability distribution
given by

$\displaystyle\text{Prob}(g)$
$\displaystyle=|\bra{h_{1}}\otimes\bra{h_{2}}\cdots\braket{h_{N}}{U}|^{2}\text{
for states satsifying }\prod_{i}h_{i}=g$ $\displaystyle=|c_{g}|^{2}$ (107)

This defines the computational output, which is interpreted as simulation of

$\displaystyle U=\sum_{g}c_{g}V_{g}$ (108)

Notice that it is still true that the “pure gauge” fluctuations within each
flux sector do not carry any information. However, in addition to these
fluctuations, the state $\ket{U}_{\text{lat}}$ contains fluctuations that
mixes the different sectors, and these fluctuations have non trivial
information content that determines the unitary $U$.

###  7.2 Probabilistic MBQC and generalized MBQC fluxes

As discussed in section 3.3, rather than changing the resource state, MBQC
provides a novel way to extract the computational output $|c_{g}|^{2}$ by
measuring $\ket{1}_{\text{lat}}$ in a cleverly chosen product basis. We saw
previously that the closed TQFT analogue ((39))of this basis is obtained from
fusion of $\ket{V_{g}}$ with $\ket{U^{\dagger}}$, which gives the states
$\ket{U^{\dagger}V_{g}}$. Via our embedding map from the TQFT Hilbert space to
the lattice, we can define an analogue of this fusion between $\ket{U}_{lat}$
and $\ket{V_{g}}_{lat}$. In the MPS description, this corresponds to just
inserting an operator $U$ before closing the chain in the circle. This results
in the U-rotated twisted states:

$\displaystyle\ket{U^{\dagger}V_{g}}_{\text{lat}}\equiv\sum_{h_{1},\cdots
h_{N}}\operatorname{tr}(U^{\dagger}V_{g}A_{h_{1}}A_{h_{2}}\cdots
A_{h_{N}})\ket{h_{1}}\otimes\ket{h_{2}}\cdots\ket{h_{N}},$ (109)

These form a $|G|$ dimensional basis of a U-dependent code subspace
$\mathcal{H}_{\text{code}}(U)$ that contains the resource state
$\ket{1}_{\text{lat}}$.

Indeed we can check explicitly that

$\displaystyle\ket{1}_{\text{lat}}=\sum_{g}c_{g}(U)\ket{U^{\dagger}V_{g}}_{\text{lat}},$
(110)

which is the lattice analogue of (36). (Below we will suppress the dependence
of $c_{g}$ on $U$)

To understand how this equation relates to MBQC , we must consider the
following question : what single-qubit measurements of $\ket{1}_{\text{lat}}$
can be used to extract the probabilities $|c_{g}|^{2}$ for a general choice of
$U$ ? Notice that the states $\ket{U^{\dagger}V_{g}}_{\text{lat}}$ are as
entangled as the cluster state itself, so they do not make a good choice for a
simple measurement basis. Also, measurements in the symmetry basis
$\ket{h_{i}}$ will no longer work, since this will return the computation
output $(\sum_{a\,\text{odd}}s_{a},\sum_{a\,\text{ even}}s_{a})=(0,0)$ with
probablity 1.

The results of the seminal work RB01 provides an answer to this question by
defining a _temporally ordered_ measurement basis which depends on the choice
of circuit $U$ being simulated. Earlier in section 2, we defined this basis
for the case $G=\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$ via the observables
$O\big{(}(-)^{q_{i}}\alpha_{i}\big{)}$ (eq. (4) ) measured in the protocol.
Explicitly, this measurement basis consists of temporally ordered spin states:

$\displaystyle\ket{\vec{q},\vec{\mathbf{s}}}=\ket{q_{1},\mathbf{s}_{1}}\ket{q_{2},\mathbf{s}_{2}}\cdots\ket{q_{N},\mathbf{s}_{N}}$
(111)

where we used the bold font $\mathbf{s}_{i}=0,1$ to denote the spin pointing
along an angle $(-)^{q_{i}}\alpha_{i}$ in the XY plane, and $q_{i}$ are the
adaptive parameters defined in (5). The rotation of the measurement angles
enables the simulation of a general SU(2) gate, while the adaptations are
needed to mitigate the randomness of the outcomes.

##### Reference states and group labels

To understand the relation between the temporally ordered basis
$\ket{\vec{q},\vec{\mathbf{s}}}$ and the basis $\ket{U^{\dagger}V_{g}}$
generating the code subspace, it is useful to assign group labels to
$\ket{\vec{q},\vec{\mathbf{s}}}$. This was explained in Wong:2022mnv , but
since it is crucial in what follows, we will review the construction here.
First let us consider the temporally ordered measurements of a single block of
two qubits on a cluster state. Temporal order means that the measurement angle
of the second qubit is flipped when ever the first qubit is measured to be in
the state $\mathbf{s}_{1}=1$. For a general pair of outcomes
$(\mathbf{s}_{1},\mathbf{s}_{2})$, this choice of measurements induces a
transport of the edge mode given by Raussendorf:2002zji

$\displaystyle
A_{\mathbf{s}_{1},\mathbf{s}_{2}}(\alpha_{1},\alpha_{2})=B_{\mathbf{s}_{1},\mathbf{s}_{2}},U(\alpha_{1},\alpha_{2}),$
(112)

where $U$ captures the desired local circuit simulation

$\displaystyle U(\alpha_{1},\alpha_{2})\equiv
A_{00}(\alpha_{1},\alpha_{2})=\exp\left(-i\frac{\alpha_{2}}{2}X\right)\exp\left(-i\frac{\alpha_{1}}{2}Z\right)$
(113)

while the remaining operator is a random byproduct:

$\displaystyle\qquad
B_{\mathbf{s}_{1}\mathbf{s}_{2}}=X^{\mathbf{s}_{1}}Z^{\mathbf{s}_{2}}$ (114)

We can label the outcomes $(\mathbf{s}_{1},\mathbf{s}_{2})$ with group
elements because the lattice representation (88) of $l(g)$ permutes these
outcomes, as one can see from the commutation relations

$\displaystyle X_{i}\,O(\alpha_{i})$ $\displaystyle=O(-\alpha_{i})\,X_{i}$
$\displaystyle Z_{i}\,O(\alpha_{i})$ $\displaystyle=-O(\alpha_{i})\,Z_{i}$
(115)

and the explicit representation of $l(g)$ in (87). In particular, the
commutation relation of $O(\alpha_{i})$ with $X$ for non-zero $\alpha_{i}$
leads to the measurement adaptation of the second qubit. Thus, given a
reference state $\ket{0}_{\alpha_{1}}\ket{0}_{\alpha_{2}}$, we can then
define:

$\displaystyle\ket{\mathbf{h}}\equiv
l(\mathbf{h})\ket{0}_{\alpha_{1}}\ket{0}_{\alpha_{2}}$ (116)

This is the direct generalization of (88).

##### Circuit simulation in the temporally ordered basis

In terms of the basis (116), the local circuit simulation can be expressed in
MPS notation as [cite stephen]:

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