# The Quantum Internet: an Efficient Stabilizer states Distribution Scheme

Seid Koudia3  3Department of Physics “Ettore Pancini”
University of Naples Federico II
Naples, 80125 Italy
<EMAIL_ADDRESS>
###### Abstract

Quantum networks constitute a major part of quantum technologies. They will
boost distributed quantum computing drastically by providing a scalable
modular architecture of quantum chips, or by establishing an infrastructure
for measurement based quantum computing. Moreover, they will provide the
backbone of the future quantum internet, allowing for high margins of
security. Interestingly, the advantages that the quantum networks would
provide for communications, rely on entanglement distribution, which suffers
from high latency in protocols based on Bell pair distribution and bipartite
entanglement swapping. Moreover, the designed algorithms for multipartite
entanglement routing suffer from intractability issues making them unsolvable
exactly in polynomial time. In this paper, we investigate a new approach for
graph states distribution in quantum networks relying inherently on local
quantum coding –LQC– isometries and on multipartite states transfer.
Additionally, single-shot bounds for stabilizer states distribution are
provided. Analogously to network coding, these bounds are shown to be
achievable if appropriate isometries/stabilizer codes in relay nodes are
chosen, which induces a lower latency entanglement distribution. As a matter
of fact, the advantages of the protocol for different figures of merit of the
network are provided.

###### Index Terms:

Quantum communication, Quantum switch, Environment-assisted communication.

##  I Introduction

Quantum networks [1, 2] lie at the heart of many future quantum technologies,
and are the basis of the quantum internet [3, 4, 5]. They rely on genuinely
quantum effects, allowing them to overcome classical approaches, or allowing
them to achieve unprecedented tasks [6, 7, 8, 9]. Indeed, quantum networks
promise a better security of information exchange among parties through
different quantum key distribution protocols, as well as other tasks suck as
quantum secret sharing, quantum leader election, distributed quantum computing
and quantum sensing [10].

Quantum multipartite entanglement is the backbone of most of the applications.
In particular, for many party-protocols, multipartite states are targeted, in
order to fulfill the promises of the functionalities of the underlying quantum
network [11]. One of the most important classes that allow for the achievement
of the advantages in many applications, as well as providing a fluid
connectivity in the network are graph states. These states are famously known
in the design of quantum error correcting codes under the name of stabilizer
states [12]. This is an important class of entangled states, not only for its
use to connect different quantum nodes for the establishment of a modular
architecture for distributed quantum computing, but it also allows for new
types of secure quantum computing either in the cloud, i.e., as blind quantum
computing, or in a distributed way, i.e, as in measurement based quantum
computing.

Nevertheless, the main obstacle that faces quantum networks –in the first
place– is the distribution of these resourceful multipartite entangled states
[13, 8]. In the one hand, Different protocols in the literature have been
studying the problem of entanglement distribution in quantum networks from
different perspectives. Yet, many of the known protocols rely on the
intermediate distribution of instances of Bell pair states, which, after
different local operations on repeater nodes, establishes links between the
clients and a central node which teleports the target graph state [14, 15]. A
typical repeater node in today’s quantum networks is capable of a single
effective functionality: forwarding -swapping- bipartite entanglement. But
there is no intrinsic reason why we must assume this is the only function ever
permitted to nodes and, in application-level overlay networks and multihop
networks, for example, allowing nodes to have a wider variety of functions
makes sense. Clearly, this nature of quantum repeaters poses a scalability
issue of the proposed protocols.

Moreover, the designed entanglement routing protocols, so far, deal mainly
with point-to-point entanglement establishment [16, 17, 18, 19]. Indeed,
future concrete application of quantum networks, should deal with distributing
multipartite entangled states, rather than solely the end-to-end Bell pair
distribution. Many efforts have been put forward in this direction, although
they suffer from major problems. In contrast to the end-to-end entanglement
routing which might be solved with algorithms that are a variety of Dijkstra’s
algorithm -which is able to find an optimal route in polynomial time in the
number of nodes in the network-, multipartite entanglement distribution
protocols suffer from the fact that they rely on NP-hard problems such as the
minimum spanning tree or the Steiner tree. As a matter of fact, these can be
approximated by algorithms that run in a polynomial time but they are never
exact.

In the other hand, harnessing the full capacities of communication networks
should be taken into account when studying entanglement distribution. Namely,
constrained quantum networks should be considered for entanglement
distribution, when the use of different point-to-point channels building the
network is limited, or when the time slots allowed for the entanglement
distribution required by the clients are fixed. These assumptions are more
practical for many reasons. Mainly, this is because the quantum hardware, such
as quantum memories, are expensive to be taken unlimited [20]. Additionally,
it is more practical to consider relatively lower time-slots of entanglement
distribution to increase the communication data rate and decrease the latency,
or to make different protocols more robust to adversarial attacks on security
schemes relying on multipartite entanglement.

###  I-A Previous Work

Quantum entanglement routing has been the focus of many works in the past few
years. A design of an optimal routing scheme has been presented in [21], with
the end-to-end entanglement rate was set as the optimality figure of merit. In
[22], the authors have presented a remote entanglement distribution scheme for
linear repeater chains, where an optimal route maximizing the end-to-end
entanglement is determined alongside with the optimal sequence of the
entanglement swaps. Distributed entanglement routing algorithms, with latency
taken into account has been the purpose of [23]. The problem of optimizing
end-to-end entanglement in a many source-destination scenario has been studied
in details as a multicommodity flow problem in [24]. In [25], a decentralized
adaptive routing scheme has been developed, in which the imperfection of
quantum memories is taken into account. A multipath routing approach for
multiple end-to-end entanglement has been thoroughly investigated in [26].

In [27], the authors handled the case where quantum repeaters are allowed to
perform quantum encoding, where it has been shown that the later increases
drastically the end-to-end entanglement rates with respect to usual protocols.
This has been recently experimentally investigated in NV centers in [28]. The
effect of this type of intermediate encoding on end-to-end key rate generation
in QKD has been studied in [29].

For Multipartite entanglement generation and distribution in quantum networks
relying on a central node connected by EPR pairs to the remote clients has
been the focus of [30]. In the same spririt, multipartite entanglement
distribution relying on finding an optimal spanning tree of the remote client
nodes has been investigated in [14]. Graph states distribution relying on
finding the Steiner tree spanning the remote nodes, and the use of local
graphical operations on remote nodes has been elaborated in [31]. Similar
Graph states distribution protocols with lower EPR resource consumption have
been given in [32, 33]. Protocols relying on an inherentely multipartite
approach for multipartite entanglement distribution have been designed in
[34]. More related to our paper, with a different approach, is [35] where
repeater nodes in the quantum network are allowed to perform quantum coding.

###  I-B Outline and Contribution

In the present paper, we consider quantum intermediate relay nodes where the
contents of outgoing states are causal functions of the contents of received
states, referring to this mapping as Local Quantum Coding. First, we show that
Local Quantum Coding –LQC– in intermediate nodes, can be used to distribute
the particular class of graph states in a distributed way among remote client
nodes. This feature will be proved to be important for the following reasons:

  * •

It is useful to beat the scalability issues of the previous protocols.

  * •

Providing a distributed architecture for multipartite entanglement
distirbution

Second, we show that the scheme of using LQC, is nothing but a particular
network coding strategy [36, 37, 38]. This allows to achieve the single-shot
capacity of the appropriate quantum network topologies, which normal
entanglement routing fails to fulfill. To this aim:

  * •

Different bounds for stabilizer states distribution capabilities of the
underlying network would be obtained in terms of min-cuts capacities
corresponding to different bipartitions of the remote client nodes requiring
the graph state.

By considering the quantum network as a tensor network with a particular
instance of tensors being specific coding isometries, the advantages of the
advantages of our scheme are given in terms of the following figure of merits
and applications:

  * •

We show that by allowing the intermediate nodes in the quantum network to use
LQC, an exponential latency reduction in the entanglement distribution time is
achieved.

  * •

We show that the use of LQC reduces drastically the memory qubits overhead of
the entanglement distribution cost.

  * •

We discuss how by allowing intermediate use of quantum error correcting codes,
distributed storage can be achieved in quantum networks.

The paper is structured as follows. In Section. II some preliminaries are be
given. In particular, graph states and their entanglement structure will be
highlighted. Additionally, the concept of tensor networks along with the
notion of min-cut will be presented. In Section. III, our model of the
distribution quantum network will be elaborated. An equivalence between our
model and a network of stabilizer states contracted by Bell pairs is given in
Section. IV. In Section. V our results comprising the different bounds on the
consumed resources to single-shot distribute a target graph state will be
established. Section. VI illustrates the obtained results and bounds for some
specific network topologies. In particular, the protocol will be benchmarked
to Bell pair based protocols with the time complexity of the graph states
distribution and the number of memory qubits taken as figure of merits. We
also study the application of choosing the LQC schemes to be valid stabilizer
quantum error correcting schemes in order to achieve distributed quantum
storage within quantum networks. We finish the paper with conclusions and
future work in Section. VII.

##  II Preliminaries

In this section, we provide the necessary preliminaries and tools allowing for
the understanding of the derived results.

###  II-A Graph states

Graph states admit a simple description in terms of mathematical graphs [12].
A graph is a pair $(G,V)$ of a finite set $V=\\{1.\dots,N\\}$ and a set
$E\subset[V]^{2}$. The elements of $V$ are called the vertices of the graph
and the elements of $E$ are called its edges. Additionaly, we denote by
$N_{a}$ the set of neighboring vertices of the vertex $a$.

The graph state $\ket{G}$ that corresponds to the graph $G$ is the pure state
given as

$\ket{G}=\prod_{\\{a,b\\}\in E}U_{ab}\ket{+}^{V}$ (1)

where

$\ket{+}^{V}=\otimes_{a\in V}\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})^{a}$ (2)

Similarly, $U_{ab}$ is the controlled $Z$ gate between the vertices $a$ and
$b$.

It is known that any graph state is equivalent to a stabilizer state under a
certain class of local operations, the so-called local Clifford operations
[12], which map the set of stabilizer states onto itself. Accordingly, graph
states can be defined uniquely as the common eigenvector with eigenvalues
equal to one in $\mathbb{C}^{V}$ to the set of independent commuting operators

$K_{a}=\sigma_{x}^{a}\sigma_{z}^{N_{a}}$ (3)

for all $a\in V$.

###  II-B Entanglement structure of graph states

Generally, the many interesting entanglement structure properties of any
multipartite state are captured by the reduced state after tracing out parts
of the system. Specifically, for a given pure multipartite state $\ket{\Psi}$.
For a specific bipartition of the state, namely $\\{A,B\\}$, the reduced state
on $A$

$\rho^{A}_{\Psi}=\mathrm{Tr}_{B}(\rho)$ (4)

where $\rho=\ket{\Psi}\bra{\Psi}_{AB}$. The entanglement rank of the state
$\ket{\Psi}$ in the bipartition $\\{A,B\\}$, quantifies the amount of Bell
pairs needed to create such bipartition, and is given as [12]:

$r_{A}(\ket{\Psi})=\log_{2}[rank(\rho^{A}_{\Psi})]$ (5)

###  II-C The min-cut

For a network $(G,V,S,T)$ a cut $(\tilde{S},\tilde{T})$ is a partition
$V=\tilde{S}\cup\tilde{T}$ such that $\tilde{S}\subset S$ and
$\tilde{T}\subset T$. The min-cut $\mathrm{MC}$ associated with the network is
defined to be the minimum over all possible cuts of the product [39, 36]

$\prod_{(u,v)\in E,u\in\tilde{S},v\in\tilde{T}}d_{(u,v)}$ (6)

Namely,

$\text{MC}=\min_{(\tilde{S},\tilde{T})}\prod_{(u,v)\in
E,u\in\tilde{S},v\in\tilde{T}}d_{(u,v)}$ (7)

It is much more convenient in the remaining of the paper, to work with the
min-cut with respect to $\log_{2}$ of the dimensions, hence the min-cut would
be given by

$\text{MC}=\min_{(\tilde{S},\tilde{T})}\sum_{(u,v)\in
E,u\in\tilde{S},v\in\tilde{T}}\log_{2}(d_{(u,v)})$ (8)

Working with this definition of the min-cut would allow us to work later with
the entanglement ranks instead of the dimensions of the states distributed.

##  III The Model

Figure 1: Scheme for distributing multipartite entanglement states in a
quantum network. The network is composed by twelve clients. each served by a
relay node. To the right is a magnification of a relay node, which exploits an
LQC $S=\sum_{i=1}^{d}U^{(i)}\otimes|i\rangle\langle i|$ to distribute multi-
partite entanglement to remote nodes.

In our model of quantum networks we are embedding all relay nodes with the
possibility of establishing an LQC strategy to generate locally graph states,
instead of just doing swapping operations and joint bipartite measurements.
Formally, each node $a\in V$ is able to generate a graph state $\ket{G_{a}}$:

$\ket{G_{a}}=\sum_{i=1}^{d}U^{(i)}_{a}\ket{0}^{n_{a}}$ (9)

following a coherent control strategy according to the isometry:

$S_{a}=\sum_{i=1}^{d}U^{(i)}_{a}\otimes|i\rangle\langle i|_{c}$ (10)

where $d$ is the dimension of the control degree of freedom. Indeed, after
measurement of the control degree of freedom in the Fourier basis, the state
in Eq. 9, as well as local unitary equivalent to it, are post-selected.

Each party of the obtained graph state $\ket{G_{a}}$, locally generated at
node $a$, would be distributed to the neighboring relay nodes $\\{N_{a}\\}$.
Each part serves as a control degree of freedom, in its turn, of an
appropriate LQC process $\sum_{i}U^{i}_{b}\otimes|i\rangle\langle i|$, in the
corresponding neighboring node $b$. This is illustrated in Fig. 1. A more
specific scheme relying on indefinite causal ordering to generate entanglement
has been established and analyzed in details in [7].

In the next section, we show that by considering a network whose nodes
$\\{a\\}_{a\in V}$ are endowed with coding isometries $\\{S_{a}\\}_{a\in V}$,
indeed, a graph state can be distributed appropriately among the sink nodes,
by studying the equivalence to a dual model of the network.

##  IV Distributed Graph States Generation

In this section, we show formally, how the endowment of quantum networks with
chosen LQC in its relay nodes, allows for the distributed generation of graph
states.

######  Lemma 1.

Let a quantum network whose relay nodes are endowed with controlled-
isometries. The isometries $\\{S_{x}\\}_{x\in v}$ are assumed to be able to
generate locally graph states in an appropriate way, upon a control degree of
freedom received from the immediate previous nodes $x\in N_{y}$ according to

$S_{y}=\sum_{i=1}^{2^{r}}U^{(i)}_{y}\otimes|i\rangle\langle i|_{x}$ (11)

with $r$ being the number of –incoming– control qubits. This model of the
quantum network is equivalent to a network whose vertices $\\{x\\}$ are
endowed with a graph state $\\{\ket{G_{x}}\\}$. Each state of these is
contracted with the neighbooring state according to the edges $\\{e\\}$ which
represent EPR pairs, as

$\bra{e_{xy}}\ket{G_{x}}\otimes\ket{\bar{G}_{y}}$ (12)

where $\ket{\bar{G}_{y}}$ is the augmented graph state given by

$\ket{\bar{G}_{y}}=\ket{0}^{\bar{a}_{y}}\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}+\ket{1}^{\bar{a}_{y}}\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$
(13)

for all $a_{y}\in V_{y}$.

###### Proof.

In what follows we give a sketch of the proof. In general any two graph states
$\ket{G_{x}}$ and $\ket{G_{y}}$ have the following schmidt decomposition on
single qubit bipartitions [12]:

$\displaystyle\ket{G_{x}}$
$\displaystyle=\ket{0}^{a_{x}}\ket{G_{x}/a_{x}}+\ket{1}^{a_{x}}\sigma_{Z}^{N_{a_{x}}}\ket{G_{x}/a_{x}}$
$\displaystyle\ket{G_{y}}$
$\displaystyle=\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}+\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$
(14)

with $\braket{G_{i}/a_{i}}{\sigma_{Z}^{N_{a_{i}}}}{G_{i}/a_{i}}=0$ if the
underlying graph $G_{i}$ is connected. Similarly, an augmentation graph state
$\ket{\bar{G}_{y}}$ of the state $\ket{G_{y}}$ on node $a_{y}$ has the the
decomposition

$\ket{\bar{G}_{y}}=\ket{0}^{\bar{a}_{y}}\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}+\ket{1}^{\bar{a}_{y}}\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$
(15)

On the one hand, by contracting the two states with a Bell pair, the resulting
state would be given by

$\ket{G_{x}/a_{x}}\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}+\sigma_{Z}^{N_{a_{x}}}\ket{G_{x}/a_{x}}\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$
(16)

On the other hand, we let one qubit from $\ket{G_{x}}$ to be sent to node $y$
to control a LQC strategy $S_{y}$ given as

$S_{y}=U_{y}^{0}\otimes|0\rangle\langle
0|^{x}+U_{y}^{1}\otimes|1\rangle\langle 1|^{x}$ (17)

where

$\displaystyle U_{y}^{0}\ket{0}^{\otimes n_{y}}$
$\displaystyle=\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}$ $\displaystyle
U_{y}^{1}\ket{0}^{\otimes n_{y}}$
$\displaystyle=\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$ (18)

generating the graph state $\ket{G_{y}}$ after measuring the distributed qubit
from $x$ in the coherent basis. Indeed, the overall state obtained before
measuring the control is given by

$\ket{G_{x}/a_{x}}\ket{0}^{a_{y}}\ket{G_{y}/a_{y}}+\sigma_{Z}^{N_{a_{x}}}\ket{G_{x}/a_{x}}\ket{1}^{a_{y}}\sigma_{Z}^{N_{a_{y}}}\ket{G_{y}/a_{y}}$
(19)

The procedure extends easily to $r$ incoming qubits, in the same spirit,
leading to the equivalence of the two approaches. ∎

To get a feel of this lemma lets suppose that the relay nodes $x,y$ choose
appropriate LQC strategies enabling the generation of $GHZ$ states
$\ket{GHZ}_{x}$ and $\ket{GHZ}_{y}$ respectively. By sending a single qubit
from $\ket{GHZ}_{x}$ in order to control the LQC in $y$, the overall state is
given up to a local unitary by

$\frac{1}{\sqrt{2}}(\ket{00}_{x}\ket{000}_{y}+\ket{11}_{x}\ket{111}_{y})$ (20)

it is easy to notice that this state is equivalent to the state
$\bra{EPR}_{xy}\ket{GHZ}_{x}\ket{GHZ}_{y}$.

######  Theorem 1.

The state distributed between the clients in the quantum network where the
relay nodes use appropriate LQC strategies of single qubit unitaries is given
by

$\ket{\Psi}=\otimes_{x}^{|V|}\bra{G_{x}}\otimes_{e}^{|E|}\ket{e}$ (21)

This overall state is a graph state.

###### Proof.

The state in 21 results directly from a direct application of Lemma. 1 to many
contractions between the relay nodes leading to the overall state $\ket{\Psi}$
between the clients. Therefore, we only give the proof for the statement about
the fact that the over state is a graph state.

First we should note that any graph state is Local Clifford equivalent to a
stabilizer state. Hence, in what comes we will limit ourselves to stabilizer
states. We should point out that any stabilizer state is characterized by the
set of generators of its stabilizer group. These are elements of the local
Pauli group that mutually commute, forming an abelian group. Let
$\ket{\Phi}_{AB}$ be a stabilizer state whose stabilizer group is $S$.
Similarity, Let $\ket{\chi}_{A}$ be a stabilizer state with corresponding
stabilizer group $T$. The contraction of the two states
$\ket{\Psi}_{B}=\braket{\chi}{\Phi}_{B}$ can be written in density operator
notation as

$\displaystyle|\Psi\rangle\langle\Psi|_{B}$
$\displaystyle=\bra{\chi}_{A}|\Phi\rangle\langle\Phi|_{AB}\ket{\chi}_{A}$ (22)

By using properties of the trace operator and of pure density matrices, we can
write the density operator $|\Psi\rangle\langle\Psi|_{B}$ in a trace form as:

$|\Psi\rangle\langle\Psi|_{B}=\mathrm{Tr}(|\chi\rangle\langle\chi|_{A}|\Phi\rangle\langle\Phi|_{AB})$
(23)

To simlify the trace formula, we can use the fact that the density matrices of
the stabilizer states $\ket{\chi}_{A}$ and $\ket{\Phi}_{AB}$ might be written
respectively as:

$\displaystyle|\chi\rangle\langle\chi|$
$\displaystyle=\frac{1}{|S|}\sum_{s_{AB}\in S}{s_{AB}}$
$\displaystyle|\Phi\rangle\langle\Phi|$
$\displaystyle=\frac{1}{|T|}\sum_{t_{A}\in T}{t_{A}}$ (24)

Therefore, we have

$|\Psi\rangle\langle\Psi|_{B}=\sum_{s_{AB}}\sum_{t_{A}}\frac{1}{|S||T|}\mathrm{Tr}(s_{AB}t_{A})$
(25)

The trace in the last equality is non-vanishing if and only if

$s_{AB}=t_{A}\otimes u_{B}$ (26)

This is due to the fact that stabilizer group elements are a product of
traceless Pauli matrices, which make them traceless in their turn. This means
that the sum is carried only over the elements $s_{AB}$ of the form
$t_{A}\otimes u_{B}$. It is worthnoting that both $t_{A}$ and $u_{B}$ are both
Local Pauli operators with support on $A$ and $B$ respectively. By this, we
conclude that:

$|\Psi\rangle\langle\Psi|_{B}=\frac{|K|\cdot
2^{|A|}}{|S||T|}\sum_{u_{B}}u_{B}$ (27)

The factor $2^{|A|}$ is due to the fact that
$\mathrm{Tr}(\mathbb{I}^{A})=2^{|A|}$, where $\mathbb{I}^{A}$ is the identity
on the subspace $A$. Furthermore, because all the state $\ket{G_{x}}$ or
$\ket{e}$ are graph states, they should have full rank stabilizer groups,
therefore the full cardinally of the stabilizers $S$ and $T$ should be
respectively given by:

$\displaystyle|S|$ $\displaystyle=2^{|A|+|B|}$ $\displaystyle|T|$
$\displaystyle=2^{|A|}$ (28)

By plugging this into Eq. 27 we obtain:

$\displaystyle|\Psi\rangle\langle\Psi|_{B}$ $\displaystyle=\frac{|K|\cdot
2^{|A|}}{2^{2|A|+|B|}}\sum_{u_{B}}u_{B}$
$\displaystyle=\frac{|K|}{2^{|A|+|B|}}\sum_{u_{B}}u_{B}$ (29)

Now we can notice that in order for $|\Psi\rangle\langle\Psi|_{B}$ to be a
graph state, the elements $\\{u_{B}\\}$ should form a group and satisfy

$\displaystyle|\\{u_{B}\\}|=2^{B}$
$\displaystyle[u_{b},u^{\prime}_{b}]=0\quad\forall
u_{B},u^{\prime}_{B}\in\\{u_{B}\\}$ (30)

The first condition is to have a full rank stabilizer group, and the second
guarantees that the group is indeed abelian. The first condition requires that
$|K|$ should satisfy:

$|K|=2^{|A|}$ (31)

In order to complete the proof, we should whow that $\\{u_{B}\\}$ form an
abelian group of local Pauli operators, making the state $\ket{\Psi}_{B}$ a
stabilizer state.
As a matter of fact, we note that since $\\{t_{A}\otimes u_{B}\\}$ are
elements of an abelian group, they should pairwise commute. In order to prove
that the operators $\\{t_{A}\otimes u_{B}\\}$ form a group we note that for
any two elements $t^{1}_{A}\otimes u^{1}_{B}$ and $t^{2}_{A}\otimes
u^{2}_{B}$, their product is of the form $t^{1}_{A}t^{2}_{A}\otimes
u^{1}_{B}u^{2}_{B}$, hence of the form $t_{A}\otimes u_{B}$. This proves the
closure of the set. It is easy to note that any operator in this set is its
own inverse, hence the set of operators $\\{t_{A}\otimes u_{B}\\}$ is an
abelian group. In turn, since $\\{t_{A}\\}$ form an abelian group, similarly,
$\\{t_{A}\otimes u_{B}\\}$ form abelian subgroup of the stabilizer $S$,
$\\{u_{B}\\}$ should form an abelian group as well. Consequently, the abelian
group $\\{u_{B}\\}$ should describe a stabilizer state. By considering
$\ket{\Phi}_{AB}$ in 21 to be the stabilizer state given by
$\otimes_{e}^{|E|}\ket{e}$ and $\ket{\chi}_{A}$ to be the stabilizer state
given by $\otimes_{x}^{|V|}\ket{x}$, and by applying the previous discussion,
we conclude that the state $\ket{\Psi}$ shared between the clients is indeed a
graph state. ∎

It is worth-mentioning that the proof of Theorem. 1, is device-independent, as
it is solely relying on the group theoretical structure of stabilizer states.
Thus, the proof holds independently of the technological platform implementing
the LQC generating entanglement.

##  V Single-Shot network Capacities

In this section, we establish entanglement distribution bounds in quantum
networks using network coding methods. we quantify the different resources
needed for the Establishment of target graph states with known entanglement
ranks.

###  V-A Entanglement Distribution Bounds

######  Theorem 2.

A graph state with a list of bipartite entanglement ranks $\\{r_{AB}\\}$ can
be distributed in a quantum network by a single use of the latter if and only
if:

$MC_{\\{A,B\\}}\geq
r_{AB}=\text{rank}_{\mathbb{F}_{2}}(\Gamma_{AB})=\log_{2}[\text{rank}(\rho^{G}_{A})]$
(32)

for every bipartition $\\{A,B\\}$ of the target graph state.

###### Proof.

We are gonna prove this theorem by contradiction. A bipartition of the state
of the client nodes into $\\{A,B\\}$ would induce a bipartition of the network
into a cut $(\tilde{S},\tilde{T})$ with the client nodes $A,B$ satisfying
$A\subset\tilde{S}$ and $B\subset\tilde{T}$ respectively. We can regard $A$
and $B$ as source and sink nodes respectively and we can collect them in a
single node as an effective network without affecting the communication
capacities between the bipartitions. It is known that the entanglement that
can be established between the source and sink nodes in a tensor network
should satisfy [39]

$Q(\\{A,B\\})\leq MC_{\\{A,B\\}}$ (33)

For bi-partite pure entangled states the entanglement that can be extracted
$E_{D}(A,B)$ (entanglement distillation) and the entanglement needed to create
the state $E_{F}$ (the entanglement of formation) are equal and they satisfy

$\displaystyle E_{D}(A,B)$ $\displaystyle=E_{F}(A,B)$ $\displaystyle=S(A)$
$\displaystyle=rank(\mathrm{Tr}_{B}(\rho))$ (34)

Accordingly, for graph states and by using Eq. 5, we have that

$\displaystyle Q(\\{A,B\\})$
$\displaystyle=\text{rank}_{\mathbb{F}_{2}}(\Gamma_{AB})$ $\displaystyle\leq
MC_{\\{A,B\\}}$ (35)

∎

Although it is obvious, the fact that by distributing EPR pairs between a
central node and clients on a star topology, any graph state can be
distributed among the clients, can be derived from the previous theorem in
network information theoretical settings.

######  Corollary 1.

The star network topology can distribute any graph state between the leafs of
the network from the central node by a single use of the network.

###### Proof.

On the one hand, we note that the entanglement rank $r_{AB}$ for any
bipartition $\\{A,B\\}$ of a graph state is bounded from above as:

$r_{AB}\leq\min\\{|A|,|B|\\}$ (36)

On the other hand, the min-cut of any bipartition of the clients in the star
network topology satisfies:

$MC_{\\{A,B\\}}=\min\\{|A|,|B|\\}$ (37)

Therefore, we have that

$r_{AB}\leq MC_{\\{A,B\\}}$ (38)

and hence, by Theorem. 2 the star topology of the network allows for the
distribution of any graph state. ∎

Similarly, we can use the results of Theorem. 2 to state the following
corollary for GHZ states distribution in arbitrary quantum networks.

######  Corollary 2.

Any topology of a connected quantum network distributes , by a single shot, a
GHZ state between the clients

###### Proof.

The proof is straightforward from Theorem. 2 by noting that the entanglement
ranks $r_{AB}$ on any bipartition $\\{A,B\\}$ of the GHZ state satisfy

$r_{AB}=1$ (39)

On the other hand, the min-cut of any bipartition of the clients in any
connected network satisfies:

$MC_{\\{A,B\\}}\geq 1$ (40)

Therefore, we have that

$r_{AB}\leq MC_{\\{A,B\\}}$ (41)

and hence, by Theorem. 2, any connected topology of the quantum network can
distribute a GHZ state between the clients by single use of the network. ∎

###  V-B Achieveability of the ultimate bounds for entanglement distribution

Figure 2: The time complexity of the distribution of a target stabilizer state
in different depths of the networks. Left: The time complexity of a network of
connectivity $n=3$. Right: The time complexity of a network of connectivity
$n=4$.

In order to better harness the capabilities of a given quantum network for
entanglement distribution, the design of appropriate LQC strategies in relay
nodes is needed. This is equivalent to finding the optimal coding strategies
in relay nodes in network coding in order to achieve the capacity of a given
network. As a matter of fact, our network architecture relying on LQC for
entanglement distribution, might be regarded as a quantum network coding
counterpart of the classical network coding paradigm.

As a matter of fact, suppose we endow all the relay nodes in the quantum
network with a Linear local quantum coding strategy (LLQC) given by:

$\displaystyle S_{x}$ $\displaystyle=U^{0}_{x}\otimes|0\rangle\langle
0|+U^{1}_{x}\otimes|1\rangle\langle 1|$ $\displaystyle=I^{\otimes
n}\otimes|0\rangle\langle 0|+X^{\otimes n}\otimes|1\rangle\langle 1|$ (42)

This strategy is equivalent to a quantum repetition code, generating GHZ
states in each node. It is easy to notice that this strategy is identical to
the isometry in Eq.17 up to a local unitary given by $I\otimes H^{\otimes
n-1}$. Indeed this strategy can only generate entanglement ranks no higher
than $r_{AB}=1$, making it not a suitable candidate for achieving the network
capacity bound in general. To illustrate this, we consider different sub-
network structures appearing in the network of Fig. LABEL:Fig:09. For the
chain subnetwork (blue nodes), the isometry in Eq. 42 performed in the blue
dashed node

$S_{0}=I_{0}\otimes|0\rangle\langle 0|_{1}+X_{0}\otimes|1\rangle\langle
1|_{1}$ (43)

leads to the same performance of an entanglement swapping. Formally, the
effect of the LQC 43 in the dashed node is equivalent, by Lemma. 1, to

$\bra{EPR}_{0}\ket{EPR}_{01}\ket{EPR}_{02}=\ket{EPR}_{12}$ (44)

which is exactly entanglement swapping performed at node $0$. Differently, the
performance of the same isometry, in the dashed nodes of sub-networks (orange
nodes) and (end nodes), is not equivalent to entanglement swapping, and leads
to better entanglement distribution rates as will be discussed in the next
section. This suggests that our multiparty protocol provides an advantage over
the Bell pair based approach when the entanglement distribution network has
high connectivity.

##  VI Advantages

In this section we illustrate the advantages of the use of intermediate
quantum coding in quantum networks accordign to different figures of merit.

###  VI-A Exponential Latency and Memory Overhead Reduction

Figure 3: The number of memory qubits required for the distribution of a
target graph states within networks of differents depths. Left: The number of
memory qubits required in a network of depth three. Right: The number of
memory qubits required in a network of depth four.

To illustrate these advantages we consider regular networks with different
degrees of connectivity. For instance, we will consider a tree network where
each node has the same number of children nodes. We will consider also that
the end relay nodes are connected to the same number of clients. In such a
network the number of relay nodes is given by $N\sim O(n^{p-1})$, where $p$ is
the depth of the network –the smallest distance from the central node and the
clients–. Accordingly, the complexities of the single shot protocol with the
isometry 42 in each relay node, and the Bell pair based, are given
respectively by

$\displaystyle\sim O(p)$ $\displaystyle\sim O(n^{p})$ (45)

Clearly, we have an exponential time complexity reduction in the depth of the
network for the distribution of GHZ like states. The results are illustrated
in the plots of Fig. 2. We can easily see that the more the deeper the quantum
network is the more clear the advantage of using a multipartite distribution
strategy is.

The similar investigation can be conducted for the advantage over the
reduction of memory qubits in the network for GHZ. The figure of merit that we
use in this case is the maximal number of memory qubits/per node used in the
network to achieve the task of distributing a target graph state between the
clients. For the LQC approach, the maximum number of memory qubits needed in
each relay node is equal to the number of controlled qubits $n^{a}_{c}$
required to generate locally the entangled state with an appropriate LQC
strategy. For the case of the previous regular network, this is maximized by
the number of connectivity $n+1$ of the node according to Lemma. 1. Thus, the
maximum number of memory qubits $n_{m}$ satisfies

$\max_{a\in V/C}(n_{m})=n^{a^{*}}_{c}\leq n+1$ (46)

Differently, in the Bell pair-based approach, the highest possible number of
memory qubits $n_{m}$ in each node is of the order of the number of clients in
the network

$\max_{a\in V/C}(n_{m})\sim n^{p}$ (47)

The results are illustrated in Fig. 3. We can notice how the entanglement
distribution based on multipartite protocols is beneficial for networks with
growing depth and connectivity.

###  VI-B Noise resilience

As a noise model, we assign to every entangled state distributed in the
quantum network in Fig. LABEL:Fig:09 a probability of failure $p$. Namely, the
noise model for the entanglement distribution is given by

$\mathcal{N}(\rho)=(1-p)\rho+pe$ (48)

where $e$ is the error flag which will abort the distribution protocol.
Accordingly, the error propagation in the network for the distribution of
$n$-EPR pairs would be given by

$\mathcal{N}^{\otimes n}(\rho^{\otimes n})=\sum_{k=0}^{n}{n\choose
k}(1-p)^{n-k}p^{k}\rho^{\otimes n-k}\otimes e^{\otimes k}$ (49)

Clearely, the probability that the protocol does not abort is when $k=0$ which
is given by

$p_{sucess}=(1-p)^{n}$ (50)

As we can easily notice, the less qubit channels are used to establish the
target state between the clients the less noisy is the protocol.

For the case of the tree network of depth $p=2$, our protocol outperforms the
protocol relying on central node and EPR pairs distribution. The performance
of the two protocols is illustrated in Fig. 4. It is clear that for different
target graph states we have

$p_{sucess}^{LQC}>p_{sucess}^{EPR}$ (51)

showing that our protocol is more robust to noise than other protocols.

Figure 4: A plot comparing the probability of success of the distribution of
different graph states in the framework of the paper with the EPR based
approach.

###  VI-C Distributed Quantum Storage

A distributed storage system is an infrastructure that can split data across
multiple physical servers, and often across more than one data center. It
typically takes the form of a cluster of storage units. The storage system
includes features that serves as a mechanism for data synchronization and
coordination between cluster storage nodes that are geographically dispersed.
Moreover, it has the intelligence to detect and respond to failures and cyber
attacks. The objective is to achieve very low latency by storing data
physically near the location it will be used.

Fundamentally, the transition from classical distributed storage to quantum
distributed storage of quantum data, cannot be smooth due to the no-cloning
theorem. Although, our distributed stabilizer states generation discussed in
Sec. IV, using intermediate quantum encoding, can be harnessed to enhance
quantum storage of quantum data significantly.

Consider the network in Fig. 5, where the bulk nodes (orange) denoted by
$\\{b_{i}\\}$ want to store their single qubit data $\\{q_{b_{i}}\\}$ into the
neighbooring cluster nodes (red) denoted as $\\{c_{i}j\\}$. The best that each
node can do to store its qubit, independentely of the other bulk nodes, is to
encode it in a three qubit code given by a GHZ-like state

$\ket{GHZ}^{b_{i}}=\frac{1}{\sqrt{2}}\big{(}\ket{000}^{c_{i1}c_{i2}c_{i3}}+\ket{111}^{c_{i1}c_{i2}c_{i3}}\big{)}$
(52)

It can easily be noticed that by encoding in such codes, single qubit errors
that might occur during the storage cannot be corrected by no means, making
the encoding vulnerable to noise and adversarial attacks in the transmission.
Differently, if the bulk nodes (orange) collaborates and look for local codes
that harness the full capacity of the network as discussed in Sec. V, they
might achieve better performance. to do so, let the bulk nodes choose to
encode their single qubits in a five qubit code each. The five qubit code,
denoted as $[[5,1,3]]$, can be described by a cyclic five qubits graph state,
which have maximal entanglement ranks among five qubit graph states.
Equivalently, it might be described by the stabilizer generators given by[40,
41]:

$\displaystyle XZZXI,IXZZX,XIXZZ,ZXIXZ$ (53)

This can encode at most a single logical qubit, and can correct single qubit
errors and up to two erasure errors.

######  Corollary 3.

By contracting the three $[[5,1,3]]$ codes with Bell pairs corresponding to
edges according to Eq. 21, a nine qubit code $[[9,3,3]]$ is established
between remote cluster nodes. This nine qubit code has as generators

$\displaystyle XZZXIXZZX,XIXXZZXZZ,IXZZXIIXZ$ $\displaystyle
ZXIIXZZXI,ZYYXZZIII,YXXYXXIII$ (54)

It can be simply checked that these nine qubit Pauli operators are commuting.
Moreover, it is shown in Appendix. A that they are independent of each other.
Therefore, they provide a stabilizer code storing three logical qubits
-encoded locally in the three bulk nodes-, in nine physical qubits-shared
between the remote cluster nodes-. Furthermore, it is shown in details in the
Appendix that the established nine qubit code can correct for single qubit
errors and up to two erasure errors. As a matter of fact, the remotely
established code between remote cluster nodes, and storing the three qubits
from the bulk nodes is denoted by $[[9,3,3]]$. Not surprisingly, the obtained
code satisfies, indeed, the quantum singleton bound given by[40, 41]:

$k\leq n-2(d-1)$ (55)

We can generalize the previous example of quantum storage to regular networks
with a fixed number of neighbors. As a matter of fact we have the following
proposition:

Figure 5: A network where the bulk nodes (orange) store a single qubit quantum
data and wants to store it in its neighbooring cluster nodes (red)

######  Proposition 1.

Let a quantum storage network where the boundary $B$, containing the quantum
registers, and a bulk containing $m$ nodes each of which encodes $k$ logical
qubits in a stabilizer code $[[l,k,d]]$ of $l$ physical qubits and distance
$d$. Then, by contracting the different codes, a stabilizer code
$[[|B|,mk,D]]$, encoding the $mk$ qubits, can be established between the
quantum registers $|B|$, where its distance $D$ satisfies:

$D\leq\frac{|B|+ml-2m(d-1)-2km}{2}+1$ (56)

###### Proof.

Suppose we have a network with bulk nodes having $k$ qubits encoded in a
stabilizer quantum error correcting code $[[l,k,d]]$, where $l$ is the number
of neighboring nodes. The code-space of such code is given by

$|\psi\rangle\langle\psi|_{i}=\frac{1}{2^{l-k}}\sum_{b_{i}}u_{b_{i}}$ (57)

where $\\{u_{b_{i}}\\}$ are the elements of the corresponding stabilizer group
of node $i$. Suppose also that we have $m$ nodes in the bulk of the network
and we have the storing registers sitting at the boundary $B$ of the network.
According to Theorem. 1 the state on the boundary would be given by the
projector

$\displaystyle|\Psi\rangle\langle\Psi|_{B}$
$\displaystyle=\mathrm{Tr}(\otimes_{i=1}^{m}|\psi\rangle\langle\psi|_{i}|e\rangle\langle
e|^{\otimes(lm+|B|)})$
$\displaystyle=\frac{1}{2^{(n-k)m+lm+|B|}}\sum_{s_{AB}}\sum_{t_{A}}\mathrm{Tr}(s_{AB}t_{A})$
$\displaystyle=\frac{1}{2^{(|B|-km)}}\sum_{u_{B}}u_{B}$ (58)

The last two equalities follow from the derivation of Theorem. 1. with $km$ is
the number of logical qubits stored into the $|B|$ quantum registers.
Additionally, according to the quantum singleton bound, the best distance $D$,
i.e., minimum Pauli error weight that cannot be corrected, that this
stabilizer code can have should satisfy:

$km\leq|B|-2(D-1)$ (59)

therefore:

$D\leq\frac{|B|-km}{2}+1$ (60)

Differently, the single codes sitting in the bulk satisfy

$km\leq ml-2m(d-1)$ (61)

by adding the two inequalities 59 and 61 we get:

$2km\leq|B|-2(D-1)+ml-2m(d-1)$ (62)

and hence, $D$ should satisfy:

$D\leq\frac{|B|+ml-2m(d-1)-2km}{2}+1$ (63)

∎

Proposition. 1 can be regarded as a straightforward result of known entropic
inequalities. To better understand this, let each of the $m$ nodes containing
$K$ EPR pairs where half of these pairs are encoded into a $l$-qubits quantum
code with distance $d$ and the other half is kept as a purifying system
$R_{i}$. We can split the encoded $l$ qubits part into three partitions
$A_{i}F_{i}C_{i}$ where $\dim(A_{i})=\dim(F_{i})=2^{d-1}$ and
$\dim(C_{i})=2^{l-2(d-1)}$. The global system is thus given by
$RAFC=R_{i}^{\otimes_{i=1}^{m}}A_{i}^{\otimes_{i=1}^{m}}F_{i}^{\otimes_{i=1}^{m}}C_{i}^{\otimes_{i=1}^{m}}$.
In the one hand, it is known that as long as the quantum code has distance
$d$, any partition of $d-1$ qubits of the codespace is independent of the
encoded state [40, 41]. Formally, we should have

$\displaystyle S(R_{i}A_{i})=S(R_{i})+S(A_{i})$ $\displaystyle
S(R_{i}F_{i})=S(R_{i})+S(F_{i})$ $\displaystyle
S(R)=S(AFC)=\sum_{i=1}^{m}S(R_{i})=mk$

Accordignly, we can have

$\displaystyle S(R)=S(FC)-S(A)=\sum_{i=1}^{m}S(R_{i})=mk$ $\displaystyle
S(R)=S(AC)-S(F)=\sum_{i=1}^{m}S(R_{i})=mk$ (65)

Applying the subbaditivity property of the entropy to the last two
inequalities we get

$\displaystyle mk=S(R)\leq S(C)+S(F)-S(A)$ $\displaystyle mk=S(R)\leq
S(C)+S(A)-S(F)$ (66)

Therefore

$\displaystyle mk$ $\displaystyle\leq S(C)$
$\displaystyle=\sum_{i=1}^{m}S(C_{i})$ $\displaystyle\leq m(l-2(d-1))$ (67)

In the pther hand, we note that the contraction of the codes maps the
entanglement in the bulk into the boundary $B$ as $RAFC\rightarrow RB$ with

$S(R)=\min\\{mk,|B|\\}$ (68)

Provided that $|B|>km$ and that the state on the boundary is indeed a quantum
code according to Theorem. 1, and in the same spirit as before we have:

$\displaystyle S(R)=mk$ $\displaystyle\leq S(C_{B})$
$\displaystyle\leq\dim(C_{B})$ (69)

where $\dim(C_{B})=|B|-2(D-1)$.

We can easily verify that the example of the Network in Figure. 5, with
$|B|=9$, $l=5$, $m=3$, $d=3$ and $k=1$ satisfy Proposition 1 in the sense that
we have the singleton bound $D=3\leq 4$, which is the smallest distance code
for $|B|=9$ physical quantum registers and encoding $k=3$ logical qubits.

Clearly, the collective efforts of the bulk nodes -by harnessing the full
capacity of the network- to establish a common code for distributed storage is
advantageous than encoding separately in a centralized way. This gives more
resilience to errors that might occur during the storage time or during the
transmission process, in the meanwhile, it provides more resilience to losses,
or to dishonest participation, when the stored information is to be retrieved.
Therefore, this shows the importance of choosing appropriate quantum network
encoding for providing secure quantum cryptographical schemes for
distributively storing quantum data. Indeed, this would not be possible if the
network relies solely on bipartite entanglement distribution and bipartite
entanglement swapping.

##  VII Conclusions and Future Work

A new approach for entanglement distribution in quantum networks has been
presented. The approach relies on linear quantum coding strategies in relay
nodes, which is an inherently multiparty strategy. We have provided the
usefulness of this approach in terms of single shot capacities of the quantum
network, making it analogous to the classical network coding paradigm.
Moreover, a performance analysis has been elaborated for particular ordered
network topologies. We have shown that for some networks, the time complexity
that the LQC approach achieves, is exponentially reduced –in the depth
parameter of the network– with respect to the time complexity that the usual
approach relying on EPR pairs distribution with a central node gives.
Similarly, a polynomial advantage –in the connectivity parameter- on the
maximal number of memory qubits needed throughout the network is observed.
These figure of merits, illustrates the advantage of the LQC paradigm in
practical communication networks with high depth and high connectivity.
Additionally, a resilience to noise has been illustrated for some specific
model, where the probability of success of distributing some target graph
states, using LQC, exceeds with significant amount its counterpart in the EPR
based approach. As an application we have shown how the proposed protocol when
uses quantum error correcting codes as LQC can be useful for distributed
quantum storage. Indeed, all the advantages of LQC were assuming that the
distribution network is given. Although, in real case scenarios, the
distribution network should, of course, be determined.

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##  Appendix A Proof of Corollary. 3

$H=\setcounter{MaxMatrixCols}{18}\begin{bmatrix}[ccccccccc|ccccccccc]1&0&0&1&0&1&0&0&1&0&1&1&0&0&0&1&1&0\\\
1&0&1&1&0&0&1&0&0&0&0&0&0&1&1&0&1&1\\\ 0&1&0&0&1&0&0&1&0&0&0&1&1&0&0&0&0&1\\\
0&1&0&0&1&0&0&1&0&1&0&0&0&0&1&1&0&0\\\ 0&1&1&1&0&0&0&0&0&1&1&1&0&1&1&0&0&0\\\
1&1&1&1&1&1&0&0&0&1&0&0&1&0&0&0&0&0\\\ \end{bmatrix}$ (70)

The proof of Corollary. 3 requires explicit computation of the contractions of
the three five qubit codes $[[5,1,3]]$ with EPR pairs. Without any loss of
generality, we let the EPR pairs used for contraction to be given by

$\ket{e}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ (71)

This is indeed a graph state with stabilizer group given explicitly by the
elements:

$II,\quad XZ,\quad ZX,\quad-YY$ (72)

Since $3$ EPR pairs are needed for contraction withing the bulk nodes, the
overall state for contraction is $\ket{e}^{\otimes 3}$ with stabilizer group
with elements given by the tensor product of the individual elements of the
stabilizer group in 72. The cardinality of this group is indeed given by
$4^{3}=64$ elements. We don’t count all the possibilities of combinations of
the elements, but we give an example that illustrates the procedure. If we
allow the nodes to be contracted by the stabilizer elements:

$ZX,\quad XZ,\quad XZ$ (73)

on each of the edges of the bulk, then the global stabilizer element of the
contraction is given by

$\underbrace{XZ}\underbrace{ZX}\underbrace{XZ}$ (74)

where the under-braces refer to operators that should act on different bulk
nodes. Moreover, the elements of stabilizer group of the code $[[5,1,3]]$ are
given by:

$\displaystyle ZXI\underbrace{XZ},\quad ZZX\underbrace{IX},\quad
XZZ\underbrace{XI},\quad IXZ\underbrace{ZX}$ $\displaystyle-
IYX\underbrace{XY},\quad-YYZ\underbrace{IZ},\quad-ZIZ\underbrace{YY},\quad-
YIY\underbrace{XX}$ $\displaystyle-ZYY\underbrace{ZI},\quad-
XYI\underbrace{YX},\quad-XXY\underbrace{IY},\quad-IZY\underbrace{YZ}$
$\displaystyle-YZI\underbrace{ZY},\quad-YXX\underbrace{YI},\quad
XIX\underbrace{ZZ},\quad III\underbrace{II}$ (75)

The underbraces refer to the parts of the codes to be contracted with the
neighboring bulk nodes. There are two neighboring bulk nodes to each bulk
node, therefore, two physical qubits have to be contracted. If we contract 74
with the stabilizer group of the three individual codes A the following nine
qubits residual element remains:

$ZXIIXZZXI$ (76)

In fact, we have $64$ nine qubit residual elements after full contraction.
They indeed form a group as we have shown in the proof of Theorem. 1. In order
to prove that this residual group encodes three qubits, we have to show that
it contains six independent generators. Clearely, proving that the six
elements

$\displaystyle XZZXIXZZX,XIXXZZXZZ,IXZZXIIXZ$ $\displaystyle
ZXIIXZZXI,ZYYXZZIII,YXXYXXIII$ (77)

belonging to the residual group, are independent, would be sufficient. To do
so, we switch to the symplectic representation of the stabilizer elements [41,
40]. Accordignly, the above elements might be represented by the matrix $H$
given by 70. It is clear that the last two rows of the matrix $H$ cannot be
written as any possible linear combination over $GF(2)$ of the other four
rows. In the meanwhile the last two rows are independent from each other. From
the last three entries of the $Z$ part of the matrix $H$ we can notice that
the first four rows are linearly independent. Therefore, the rows of the
matrix $H$ are linearly independent of each other. Hence, the corresponding
stabilizer elements in A are indeed generators of the residual stabilizer
group.

Now we should prove that the nine qubit code generated by the stabilizer
elements in A has distance $3$. To do this, we should show that any $9$-qubit
Pauli operator of weight two should anticommute with at least one of the
generators A. Let $g_{l}$ be a generator of the residual nine qubit stabilizer
group, and let $O_{B}$ be a two qubit Pauli acting on the nine qubits. Form
the proof of Theorem. 1 the following is true

$g_{l}=\mathrm{Tr}_{A}(s_{BA}t_{A})$ (78)

for some $s_{BA}$, an element of the stabilizer group of the three five qubit
codes and $t_{A}$ an element of the stabilizer group corresponding to the
contracting edges. Accordingly, we have

$\displaystyle O_{B}g_{l}$ $\displaystyle=\mathrm{Tr}_{A}(O_{B}\otimes
I_{A})\mathrm{Tr}_{A}(s_{BA}t_{A})=\mathrm{Tr}_{A}(O_{B}\otimes t_{A}s_{BA})$
(79)

Since $\\{s_{BA}\\}$ is a stabilizer group correcting for at most two Pauli
errors, then there is at least one element $s_{BA}$ that satisfies

$\\{O_{B}\otimes I_{A},s_{BA}\\}=0$ (80)

Hence, there is at least one generator $g_{l}$ of the residual stabilizer
group satisfying

$\\{O_{B},g_{l}\\}=0$ (81)

Therefore, the nine qubit code can correct for any single qubit error. The
same argument applies to show that the distance of the code is indeed $D=3$.


# Pre-burst neutrinos of gamma-ray bursters accompanied by high-energy photons

Jie Zhu  Bo-Qiang Ma  School of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China  Center for
High Energy Physics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing, China

###### Abstract

Previous researches on high-energy neutrino events from gamma-ray bursters
(GRBs) suggest a neutrino speed variation $v(E)=c(1\pm
E/E^{\nu}_{\mathrm{LV}})$ with ${E}^{\nu}_{\rm LV}=(6.4\pm 1.5)\times
10^{17}~{}{\rm GeV}$, together with an intrinsic time difference
${\Delta{t}_{\rm in}=(-2.8\pm 0.7)\times 10^{2}~{}{\rm s}}$, which means that
high-energy neutrinos come out about 300 s earlier than low-energy photons in
the source reference system. Considering the possibility that pre-bursts of
neutrinos may be accompanied by high-energy photons, in this work we search
for high-energy photon events with earlier emission time from 100 to 1000 s
before low-energy photons at source by analyzing Fermi Gamma-ray Space
Telescope (FGST) data. We perform the searching of photon events with energies
larger than 100 MeV, and find 14 events from 48 GRBs with known redshifts.
Combining these events with a $1.07~{}\rm{TeV}$ photon event observed by the
Major Atmospheric Gamma Imaging Cherenkov telescopes (MAGIC), we suggest a
pre-burst stage with a long duration period of several minutes of high energy
neutrino emissions accompanied by high energy photons at the GRB source.

###### keywords:

pre-burst, gamma ray burst, light speed variation, neutrino speed variation,
Lorentz invariance violation

††journal: Phys. Lett. B 820 (2021) 136546,
https://doi.org/10.1016/j.physletb.2021.136546

##  1 Basic formulas for speed variation analysis

Lorentz invariance is a basic assumption in Einstein’s relativity. However, it
is speculated from quantum gravity that the Lorentz invariance might be broken
at the Planck scale ($E_{\mathrm{Pl}}\simeq 1.22\times
10^{19}~{}\mathrm{GeV}$). Amelino-Camelia et al. [1, 2] first suggest testing
Lorentz violation by comparing the arrival times between high energy and low
energy particles from gamma-ray bursters (GRBs). For a particle propagating in
the quantum space-time with energy $E\ll E_{\rm Pl}$ , the Lorentz violation
(LV) modified dispersion relation can be written in a general form as the
leading term in Taylor series

$E^{2}\simeq p^{2}c^{2}+m^{2}c^{4}-s_{n}E^{2}\left(\frac{E}{E_{{\rm
LV},n}}\right)^{n},$ (1)

where $E$, $p$, $m$ represent the energy, momentum and mass of the particle
respectively, $c$ is the light speed in vacuum, $s_{n}=\pm 1$ indicates
whether high-energy particles travel faster ($s_{n}=-1$) or slower
($s_{n}=+1$) than low-energy particles, $n=1$ or $n=2$ as usually assumed
which shows the leading order of the influence from LV, and $E_{\rm{LV},n}$
represents the nth-order Lorentz violation scale. Since photons and ultra-high
energy neutrinos are both ultra-relativistic particles, it is reasonable to
set $m=0$ in the discussion. Using the relation $v=\partial E/\partial p$, we
can get the modified velocity relation with energy $E$

$v(E)=c\left[1-s_{n}\frac{n+1}{2}\left(\frac{E}{E_{{\rm
LV},n}}\right)^{n}\right].$ (2)

With the expansion of the universe taken into consideration [3], the time
difference from LV between two particles from the same source with energy of
$E_{\rm{h}}$ and $E_{\rm{l}}$ is

$\Delta t_{\rm LV}=s_{n}\frac{1+n}{2H_{0}}\frac{E_{\rm h}^{n}-E_{\rm
l}^{n}}{E_{{\rm LV},n}}\int^{z}_{0}\frac{(1+z^{\prime})^{n}{\rm
d}z^{\prime}}{\sqrt{\Omega_{\rm m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$ (3)

where $z$ is the redshift of the source of the two particles, $H_{0}$,
$\Omega_{\rm{m}}$ and $\Omega_{\rm{\Lambda}}$ are universe constants. Here we
adopt the present day Hubble constant [4] $H_{0}=67.3\pm
1.2~{}\rm{km~{}s}^{-1}\rm{Mpc}^{-1}$, the pressureless matter density [4]
$\Omega_{\rm{m}}=\mathrm{0.315^{+0.016}_{-0.017}}$ and the dark energy density
[4] $\Omega_{\Lambda}=\mathrm{0.685^{+0.017}_{-0.016}}$. In this work, we
focus on the circumstances of $n=1$, so Eq. (3) can be expressed as

$\Delta t_{\rm LV}=s(1+z)\frac{K}{E_{\rm LV}},$ (4)

where $s=\pm 1$ is the sign factor and

$K=\frac{E_{\rm h}-E_{\rm
l}}{H_{0}}\frac{1}{1+z}\int^{z}_{0}\frac{(1+z^{\prime}){\rm
d}z^{\prime}}{\sqrt{\Omega_{\rm m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$ (5)

is the LV factor. In our discussion, $E_{\rm{h}}$ ($\sim 1~{}\rm{GeV}$ or
higher) is much more higher than $E_{\rm{l}}$ ($\sim 100~{}\rm{keV}$), so it
is reasonable to take $E_{\rm{l}}$ as 0.

The observed arrival time difference $\Delta t_{\rm obs}$ between two
particles detected on the Earth is actually caused by two factors: the LV time
correction $\Delta t_{\rm LV}$ in the propagation and the intrinsic time
difference $\Delta t_{\rm in}$ at the source [5, 6, 7]. $\Delta t_{\rm in}$ is
only related to the intrinsic mechanism of the GRB source. Hence we have

$\Delta t_{\rm obs}=t_{\rm h}-t_{\rm l}=\Delta t_{\rm LV}+(1+z)\Delta t_{\rm
in}.$ (6)

where $t_{\rm h}$ and $t_{\rm l}$ represent the arrival times of high-energy
and low-energy particles. Considering Eq. (5), we rewrite Eq. (6) as

$\frac{\Delta t_{\rm obs}}{1+z}=\Delta t_{\rm in}+s\frac{K}{E_{\rm LV}}.$ (7)

According to Eq. (7), there would be a linear relation between $\Delta t_{\rm
obs}/(1+z)$ and $K$ if the energy dependence speed variation does exist, and
particles emitted at a same intrinsic time in the source reference frame would
fall on a straight line in the $K$-$\Delta t_{\rm obs}/(1+z)$ plot. The slope
of the line represents $s/E_{\rm{LV}}$ and the intercept represents the
intrinsic time difference $\Delta t_{\rm in}$ between high-energy particles
and low-energy photons.

##  2 Physics picture from previous researches

__

As sources with high-energy particle emissions, GRBs are natural laboratories
for high energy physics, and the particles from GRBs have the potential to
reveal novel features of the universe. Because the energies of the particles
from GRBs can be much more higher than energies of the particles from
laboratories on the Earth, we can use them to test the predictions from
theoretical physics, e.g., the Lorentz violation. There are many researches on
the tests of LV with high-energy particles from GRBs, and they focus on
photons and neutrinos.

An attempt to test LV from high-energy photons was in Ref. [8]. In this work,
it suggests a light speed variation of $v(E)=c(1-E/E^{\gamma}_{\mathrm{LV}})$
with $E^{\gamma}_{\mathrm{LV}}=3.6\times 10^{17}~{}\mathrm{GeV}$ and an
intrinsic time difference $\Delta t_{\rm{in}}=(-10.7\pm 1.5)~{}\rm{s}$, which
means high-energy photons come out earlier than low-energy ones at the GRB
source and the speed of high-energy photons is slower than low-energy ones. In
this physics picture, it suggests a pre-burst stage that high-energy photons
come out about 10 seconds earlier than low-energy photons at the GRB source,
and because of light speed variation, high-energy photons travel slower than
low-energy photons, and the light speed difference and the long cosmological
distances lead to an expectation that one usually observes low-energy photons
earlier than high-energy photons, as is indeed the case in the earlier
observations of GRB data. This conclusion was soon supported by a remarkable
high energy event of GRB160509A in Ref. [9]. More recently, a direct evidence
for the pre-burst stage is shown in Ref. [10] with the observation of several
pre-burst events which are earlier detected than the prompt low energy
photons.

Unlike photons, neutrinos are able to escape from dense astrophysical
environments. Therefore ultra-high energy cosmic neutrinos can be observed on
the Earth and provide a powerful tool to explore LV physics [11, 12, 13, 14].
Analysis on PeV and TeV GRB neutrinos [15] suggests a neutrino speed variation
of $v(E)=c(1-sE/E^{\nu}_{\mathrm{LV}})$ with $E^{\nu}_{\mathrm{LV}}=6.5\times
10^{17}~{}\mathrm{GeV}$ and $s=\pm 1$ and an intrinsic time difference $\Delta
t_{\rm{in}}=(1.7\pm 3.6)\times 10^{3}~{}\rm{s}$. This suggestion shows that
the speed of neutrinos can be both slower and faster than the light speed
constant $c$, and the LV scale is not the same as the LV scale of photons but
of the same order of magnitude. In the case of PeV GRB neutrinos, the time
delay caused by LV can be really large, e.g., the largest time difference in
Ref. [15] is 75 days, and it is much more larger than $10^{3}~{}\rm{s}$ as the
order of magnitude of $\Delta t_{\rm{in}}$ determined by the data of PeV GRB
neutrinos, so considering the uncertainty of the energy and the constants of
the universe we can not expect an accurate value of $\Delta t_{\rm{in}}$. In a
later research on near-TeV GRB neutrinos from IceCube [16], it is suggested
that $E^{\nu}_{\mathrm{LV}}=(6.4\pm 1.5)\times 10^{17}~{}\rm{GeV}$ and $\Delta
t_{\rm in}=(-2.8\pm 0.7)\times 10^{2}~{}{\rm s}$, which is more accurate and
shows that high-energy neutrinos come out about $300~{}\rm{s}$ earlier from
the GRB source than low-energy photons in the source reference frame.

Researches on high-energy photons and neutrinos reveal the following physics
picture of GRBs. In the reference frame of the source GRB, about $300~{}\rm s$
before the low-energy photon burst, a pre-burst stage of high-energy neutrinos
firstly happens. The ultra-high energies of neutrinos lead to big time lags or
leads because of velocity variation, so we can observe ultra-high-energy
neutrino events even 2 or 3 months later or before observing the GRB lights.
Then about $10~{}\rm s$ before the low-energy photon burst, a pre-bust stage
of high-energy photons happens. Different from neutrino events, high energies
of the photon events only lead to time lags, so most of these events can be
observed after the low-energy photon burst and only few of them can be
observed before the low-energy photon burst. At the pre-burst stage of
neutrinos, high-energy neutrinos can interact with the matters of the GRB
source at the edge, so these neutrino events can be accompanied by some high-
energy photon events in the source reference frame. So to find the high-energy
photon events at the pre-burst stage of neutrinos will be an indirect support
for the physics picture of GRB neutrinos. However considering the expansion of
the universe and the light speed variation, we can not simply search for
photon events just 300 s before a GRB. According to Eq. (6), time difference
at the source will be magnified by a factor $(1+z)$, so we need to search for
high-energy photon events with a larger time range.

##  3 Data acquisition and analysis

We search for high energy photon events from the Fermi Gamma-ray Space
Telescope (FGST) data. FGST consists of the Fermi Large Area Telescope (LAT)
[17] and the Gamma-Ray Burst Monitor (GBM) [18]. LAT aims to collect high-
energy events while GBM aims to collect low-energy events. The GBM data can be
downloaded from the Fermi website [19] while the LAT data need to be retrieved
and downloaded from this website [20]. Considering that redshift plays an
important role in our analysis, we focus on GRBs which are not only detected
by FGST but also have redshifts recorded. In this work we search 48 GRBs from
GRB 080916C to GRB 210204A. The trigger time $t_{\rm{trig}}$ of GBM is usually
assumed as the onset time of GRBs, so in the following discussion we take
$t_{\rm{trig}}$ as zero point of time. Detailed discussion on the criteria of
arrival time of low-energy photon events $t_{\rm l}$ is shown in Ref. [21],
however in this work the absolute value of arrival time of high-energy photon
events $t_{\rm h}$ is much more larger than $t_{\rm l}$, so we simply take
$t_{\rm l}=t_{\rm{trig}}=0$. For data of high-energy photon events from LAT,
we need to specify an exact search range. We select LAT data with
reconstructed energies in the $100~{}\rm{MeV}$-$100~{}\rm{GeV}$ range as Ref.
[22] did, and the lower limit is chosen to reject events with poorly
reconstructed directions and energies. Since the time difference at the source
will be magnified by the factor $(1+z)$, we set a larger time range to ensure
that photons emit from -1000 s to -100 s in the source reference frame are
selected. For the searching angle radius and background analysis, we still
choose the same strategy as Ref. [22] did. We use a fixed-radius circular
region of interest set at $12^{\circ}$ to ensure that 99% events are selected,
and then we estimate the probability of each $\gamma$-ray being associated
with the GRB by using the Fermi ScienceTool [23] _gtsrcprob_ , and then we
choose events that have probabilities $>0.8$ of being associated with the
GRBs. The data are listed in Table 3. We notice that the Major Atmospheric
Gamma Imaging Cherenkov telescopes (MAGIC) [24] found an ultra-high-energy
photon events with energy 1.07 TeV and arrival time 73.6 s [25, 26], and the
same analysis on the MAGIC event below suggests that it may come from the pre-
burst stage of neutrinos, so we also list it in the table.

[t] The data of observed pre-burst events GRB $z$ $t_{\rm h}$ (s) $E_{\rm
obs}$ (GeV) $E_{\rm source}$ (GeV) $\frac{\Delta t_{\rm obs}}{1+z}(\rm s)$
$K(\times 10^{17}~{}\rm{s}\cdot\rm{GeV})$ $p$ 090510A 0.903 -1543.07 0.317
0.603 -810.86 0.764 0.82 090902B 1.822 -513.126 1.17 3.30 -181.83 3.77 0.84
091003A 0.8969 -1156.97 0.327 0.620 -609.93 0.785 0.87 141028A 2.33 -1230.13
0.109 0.362 -369.41 0.372 0.80 170405A 3.51 -1888.34 0.247 1.11 -418.70 0.886
0.85 -1004.15 0.332 0.794 -420.15 0.979 0.85 181010A 1.39 -830.676 0.272 0.649
-347.56 0.801 0.88 -780.3 1.28 3.06 -326.49 3.78 0.93 -697.326 0.333 0.795
-291.77 0.981 0.88 191011A 1.722 -1048.13 0.614 1.67 -385.06 1.95 0.88
-398.586 0.301 0.624 -192.55 0.792 0.88 201021C 1.07 -320.235 0.321 0.665
-154.70 0.845 0.86 -294.255 0.129 0.267 -142.15 0.339 0.83 201221D 1.046
-952.789 0.446 0.912 -465.68 1.16 0.83 190114C by MAGIC 0.4245 73.6
$1.07\times 10^{3}$ $1.52\times 10^{3}$ 51.667 $1.577\times 10^{3}$

  * 1.

Data of high-energy photon events from FGST and MAGIC. $z$ is the redshift of
the GRB, $t_{\rm h}$ is the arrival time of the event when $t_{\rm trig}$ is
set to zero, $E_{\rm obs}$ is the energy of the the event observed by the
telescope while $E_{\rm source}$ is its energy in the source GRB, and $p$ is
the probability that the event is associated with the GRB calculated by the
Fermi ScienceTool  _gtsrcprob_. $\Delta t_{\rm obs}/(1+z)$ and $K$ are parts
of Eq. (7), and because of $t_{\rm l}=t_{\rm trig}=0$, here $\Delta t_{\rm
obs}=t_{\rm h}-t_{\rm l}=t_{\rm h}$. Data of $t_{\rm h}$ and $E_{\rm obs}$ is
from LAT, and the references for the redshifts of the GRBs are
[27](090510A),[28](090902B),[29](091003A),[30](141028A),[31](170405A),[32](181010A),[33](191011A),
[34](201021C),[35](201221D),[36, 37](190114C), and the data of the MAGIC event
are from Ref. [26].

(a)

(b)

Figure 1: The $\Delta t_{\rm{obs}}/(1+z)$ versus $K$ plot for photon data in
Table 3 and in Refs. [8, 9] (denoted as Work by Xu & Ma), and for neutrino
data in Ref. [16]. In the upper panel (a) we draw events in Table 3 and in
Refs. [8, 9] and in the lower panel (b) we add neutrino events. In both
figures the black line is the mainline of Ref. [8] as reference, and the edges
of the red (dark) area share the same slope of the black line, and the
intercepts of the two edges are -140 s and -820 s. In the lower panel the deep
blue area is the maximum envelope of the fitting result in Ref. [16], and the
light blue area keeps the same slope of the neutrino events [16] but
intercepts changed to -140 s and -820 s.

Combining with data of high-energy photon and neutrino events from Refs. [8,
9, 16], we draw these events in the $K$-$\Delta t_{\rm obs}/(1+z)$ plot shown
in Fig. 1. In the upper panel Fig. 1a we only include photon events from Refs.
[8, 9], and in the lower panel Fig. 1b we also add near-TeV neutrino events
from Ref. [16]. Drawing a straight line through a point with slop
$1/E^{\gamma}_{{\rm{LV}}}$ from Refs. [8, 9], we can get $\Delta t_{\rm in}$
from the intercept which represents the time difference between the event and
low-energy photon bursts in the source reference frame. In Fig. 1a, we draw a
red area with the same slope from Refs. [8, 9] and intercepts from -140 s to
-820 s, we can see that all of the events fall inside the area, including the
1.07 TeV photon event observed by MAGIC. This figure suggests that the events
from FGST with negative arrival time are related with the 1.07 TeV event from
MAGIC with positive arrival time, and they all come from the same time frame
before the low-energy photon bursts of GRBs. In Fig. 1b, we add near-TeV
neutrino events from Ref. [16] as blue points. The redshifts of these neutrino
GRB sources are unknown, and in Refs. [15, 16] an approximate estimation of
redshifts with errors is adopted, so here we draw these events with error
bars. The deep blue area is the maximum envelope of the fitting result in Ref.
[16] and the intercepts are from -210 s to -350 s. In Ref. [16] the fitting
result comes from the data of 12 events while other 6 events are abandoned, as
shown is Fig. 1b that 12 points fall on the deep blue area while the rest are
not. Combing the results of both photons and neutrinos, we suggest that these
neutrinos and photons come from the same time frame of GRBs, and inspired by
this we enlarge the intercepts of neutrinos to the red ones, and the result is
shown as the light blue area. We see clearly that 5 more points of neutrinos
are included in the area, and if the rest one point are considered as valid
data, the lower limit of the intercept can be more lower. The consistence of
the intercepts of both neutrinos and photons suggests that they come from the
same time frame of GRBs, in other words the photon events analysed in this
work indirectly support the conclusion of pre-burst neutrino emissions in Ref.
[16].

##  4 Conclusion

In this work we search 48 GRBs with redshifts to find pre-burst high energy
photon events from FGST data, and analysis on these observed events from FGST
suggests that they are related with a 1.07 TeV photon event from MAGIC and a
number of near-TeV GRB neutrinos from IceCube. The combination of them reveals
a picture that there is a pre-burst stage with a long duration period of
several minutes of neutrino emissions accompanied with high-energy photons,
within a time interval of about 140 to 820 seconds before the low-energy
photon burst of a GRB in the source reference frame. This picture provides a
comprehensive understanding of the observed high energy photon events and
neutrino events, and strongly supports the photon and neutrino speed
variations suggested from phenomenological analyses. Of course more detections
of high energy photon and neutrino events are still necessary to check the
picture revealed in this work.

Acknowledgements: This work is supported by National Natural Science
Foundation of China (Grant No. 12075003).

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