# Quantum information spreading and scrambling in a distributed quantum
network: A Hasse/Lamport diagrammatic approach

Kiran Adhikari  Institute for Communications Engineering
Technical University of Munich
MQV
<EMAIL_ADDRESS>Christian Deppe  Institute for Communications
Engineering
Technical University of Munich
6G-life & MQV
<EMAIL_ADDRESS>
###### Abstract

Large-scale quantum networks, known as quantum internet, hold great promises
for advanced distributed quantum computing and long-distance quantum
communication. It is essential to have a proper theoretical analysis of the
quantum network and explore new applications and protocols that justify
building such an extensive network. We propose a novel diagrammatic way of
visualizing information flow dynamics within the quantum network, which
preserves the causal relationship between different events at different nodes.
This facilitates synchronization among network nodes, studies the error
propagation, and allows for tracking valuable quantum resources. Additionally,
We propose a quantum information scrambling protocol, where a specific node
scrambles secret quantum information across the entire network. This protocol
ensures that a malicious party would need access to a significant subset of
the network to retrieve the information.

###### Index Terms:

Hasse diagram, Lamport diagram, quantum information scrambling, quantum secret
sharing, synchronization

##  I Introduction

Recently, there has been a growing interest from both experimental and
theoretical perspectives in building large-scale quantum networks called
quantum internet [1, 2, 3, 4, 5]. Quantum internet offers exciting
opportunities for advanced quantum information processing, distributed quantum
computing, quantum metrology, and long-distance quantum communication [6, 7,
8, 9, 10, 11, 12, 13, 14]. Because of technological advancement [15, 16, 17,
18, 19], it becomes crucial to explore these networks from the theoretical
perspective as well [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. By
studying information flows within these networks, we can effectively track the
information dynamics and error propagation and ensure seamless synchronization
among network nodes. At the same time, finding new protocols and applications
of such distributed quantum networks justifies building a quantum internet in
the first place. Therefore, the theoretical analysis of the quantum network
design and the finding novel applications of such networks are of fundamental
importance in large-scale quantum networks.

Quantum information scrambling has been studied extensively in the context of
quantum many-body systems [32, 33, 34]. However, we are unaware of any
connections between quantum information scrambling and distributed quantum
networks. In this context, we propose a novel protocol called quantum
information scrambling protocol, where the information from one node gets
scrambled across the entire network such that the external malicious party
cannot reconstruct it from having control of only a fraction of the nodes in
the network. In Section II, we propose a novel way of diagrammatically showing
the flow of quantum information in the network. In Section III, we present an
application of a quantum network called quantum information scrambling
protocol, which utilizes the diagrammatic representation of Section II.

###  I-A Definition and conventions

A quantum network has two fundamental elements: nodes processing quantum
information and edges representing quantum channels. These nodes can be any
entity able to perform local operations (LO) as dictated by the principles of
quantum mechanics, for example, a client, a quantum repeater, a universal
quantum computer, or an extremely powerful quantum data center (QDC) with
multiple universal quantum computers. Conversely, a quantum channel is an edge
between two arbitrary nodes, allowing the exchange of a quantum system between
them. Examples include an optical fiber, a superconducting microwave
transmission line, or an optical free-space link.

We can represent the structure of a quantum network by a graph $G=(V,E)$, as
shown in Figure 1, with a set $V$ of vertices and a set $E$ of directed edges.
Vertices $V=\\{P_{1},P_{2},....,P_{N}\\}$ indicates $N$ nodes with each vertex
$P_{i}$ representing a classical or quantum information processing node. A
directed edge $e=v\xrightarrow{}v^{\prime}$ represents a quantum channel
$\mathcal{N}^{v\xrightarrow{}v^{\prime}}$. A set of cuts $\partial V$ among
the edge, at certain time $t$, can divide the quantum network into $L$
different non overlapping sub networks
$V_{1}^{t}=\\{P_{1},...,P_{i}\\},...,V_{i}^{t}=\\{P_{j+1},...P_{k}\\},....,V_{L}^{t}=\\{P_{l},....P_{N}\\}$
where each $V_{i}^{t}$ is a subset of the graph at time $t$. Often, we will
remove superscript $t$ for convenience. Furthermore, a quantum network $G$ is
denoted by $G_{\phi}$ if every edge $e\in G_{\phi}$ represents a maximally
entangled state. If the entanglement in a $G_{\phi}$ network is a free
resource, we denote it by $G_{\phi}^{\infty}$.

Mathematically, a function of an arbitrary node $X$ can be represented by a
local operation where, after getting quantum state $\rho$ as an input, return
a quantum state $\sigma_{k}$ with probability $p_{k}$ as an output, where
$\sigma_{k}:=M_{k}^{X}\rho(M_{k}^{X})^{\dagger}/p_{K}$ with Kraus operators
$M_{k}^{X}$ satisfying $\sum_{k}(M_{k}^{X})^{\dagger}M_{k}^{X}=1^{X}$ and
$p_{k}=\text{Tr}[(M_{k}^{X})^{\dagger}M_{k}^{X}\rho]$. Suppose there are $N$
nodes in a quantum network, and we label the $i$th node by $P_{i}$. A quantum
state in such a network lives in a Hilbert space:
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes.....\mathcal{H}_{N}$
where $\mathcal{H}_{i}\simeq\mathcal{C}^{d_{i}}$, $d_{i}<\infty$, corresponds
to Hilbert space of node $P_{i}$. Furthermore, a quantum channel
$\mathcal{N}^{X\xrightarrow{}Y}$ from a node $X$ to a node $Y$ is a
completely-positive trace-preserving (CPTP) map where,
$\mathcal{N}^{X\xrightarrow{}Y}(\rho):=\text{Tr}_{E^{\prime}}[U^{XE\xrightarrow{}YE^{\prime}}(\rho\otimes\ket{0}\bra{0})(U^{XE\xrightarrow{}YE^{\prime}})^{\dagger}]$
where $U^{XE\xrightarrow{}YE^{\prime}}$ is a unitary operator on Hilbert space
$\mathcal{H}^{X}\otimes\mathcal{H}^{E}$ to
$\mathcal{H}^{Y}\otimes\mathcal{H}^{E^{\prime}}$ and a state $\ket{0}^{E}$ of
auxiliary system $E$.

We will also need the concept of system and subsystem sizes to use results
such as the decoupling theorem [35] of quantum information theory. The size of
the node $P_{i}$ is denoted by $|P_{i}|$, and it is related to the dimension
by $d_{P_{i}}=2^{|P_{i}|}$. If in a arbitrary protocol, each node $P_{i}$
dedicates $n_{P_{i}}$ qubits, then the size of that node would be
$|P_{i}|=n_{P_{i}}$ and the dimension would be
$d_{P_{i}}=2^{|P_{i}|}=2^{n_{P_{i}}}$. Then, the size of subnetwork $V_{i}$ is
$|V_{i}|=\sum_{P_{j}\in V_{i}}|P_{j}|$, and it’s dimension is
$d_{V_{i}}=2^{|V_{i}|}$. The size of the entire network $G=(V,E)$ with $P$
different subnetworks would be
$|V|=\sum_{i=1}^{P}|V_{i}|=\sum_{i=1}^{N}|P_{i}|$ with network’s dimension
being $d=2^{|V|}$. For example, if there are $n$ qubits in total participating
in some arbitrary protocol in a quantum network, then the network size would
be just $n$, dimension $2^{n}$. One can also generalize it to the case of
qudits.

One primary goal of introducing the diagrammatic approach is to track various
resources spent in the quantum network while running some arbitrary protocol.
In a resource theoretic language [36], $[c\to c]$ would denote the
communication resource of a noiseless classical bit channel, $[q\to q]$ a
noiseless qubit channel, $[cc]$ a shared, non-local bit of shared randomness
and $[qq]$ a shared, noiseless EPR pair. Furthermore, other than having two-
party quantum resources such as EPR pair, it is possible to have multi-party
quantum resources in a network. For this, we introduce the notation
$((x_{1},y_{1}),(x_{2},y_{2}),..(x_{i},y_{i})..(x_{R},y_{R}))$ where $R$
indicates the number of species of resources and $x_{i}$ indicates the amount
of $y_{i}$ species. Of course, the definition of species depends upon the
problem in hand. For example, $((3,GHZ),(4,W))$ would imply that we have 3 GHZ
states and 4 W states shared among three nodes. $R=2$ as we have two different
species of quantum resources, $GHZ$ and $W$ entangled states if we define
species as three party entanglement in non equivalent way. We can also easily
include information about which nodes are sharing the given resource by adding
one more parameter in the notation $(x_{2},y_{2},P_{i}P_{j}....P_{k})$. Here,
$P_{i}P_{j}....P_{k}$ denotes the nodes where the resource $y_{i}$ is shared
upon. One primary goal of introducing the diagrammatic approach is to track
various resources spent in the quantum network while running some arbitrary
protocol. In a resource theoretic language, $[c\to c]$ would denote the
communication resource of a noiseless classical bit channel, $[q\to q]$ a
noiseless qubit channel, $[cc]$ a shared, non-local bit of shared randomness
and $[qq]$ a shared, noiseless EPR pair. Furthermore, other than having two-
party quantum resources such as EPR pair, it is possible to have multi-party
quantum resources in a network. For this, we introduce the notation
$((x_{1},y_{1}),(x_{2},y_{2}),..(x_{i},y_{i})..(x_{R},y_{R}))$ where $R$
indicates the number of species of resources and $x_{i}$ indicates the amount
of $y_{i}$ species. Of course, the definition of species depends upon the
problem at hand. For example, $((3,GHZ),(4,W))$ would imply that we have 3 GHZ
states and 4 W states shared among three nodes. $R=2$ as we have two different
species of quantum resources, $GHZ$ and $W$ entangled states if we define
species as three-party entanglement in non equivalent way. We can also easily
include information about which nodes share the given resource by adding one
more parameter in the notation $(x_{2},y_{2},P_{i}P_{j}....P_{k})$. Here,
$P_{i}P_{j}....P_{k}$ denotes the nodes where the resource $y_{i}$ is shared
upon.

Figure 1: Quantum network with subnetworks: A network is represented by a
graph $G=(V,E)$ where vertices represent quantum information processing nodes
and edges represent quantum channels. A subnetwork
$V_{A}=\\{P_{0},P_{1},P_{2},P_{3},P_{4},P_{5}\\}$ is shown with a red shade
while subnetwork $V_{B}=\\{P_{6},P_{7},P_{8},P_{9},P_{10}\\}$ is represented
with green shade.

#### Quantum Information Scrambling

The phenomenon of scrambling of quantum information has recently emerged as a
fundamental concept across multiple disciplines, including many-body physics,
quantum chaos, complexity theory, and black holes [32, 33, 37, 38, 39]. Here,
we present a concise overview of information scrambling while more extensive
insights can be explored in the literature [32, 40].

We can roughly understand quantum information scrambling as a phenomenon of
thermalization of quantum information and is thus crucial in exploring how
quantum information spreads in quantum systems. To grasp the essence of it, it
is easier to express it using the framework of quantum circuits. Suppose we
have a unitary $U_{AB}:A\otimes B\xrightarrow{}C\otimes D$, and external
system $R^{\prime}$ purifying $A$. The unitary is of scrambling nature if
$R^{\prime}$ and $C$ are almost independent for all or most $C$ of size
smaller than some parameter $l$. How one selects a subset $C$ from the
composite system $CD$ becomes arbitrary as long as it remains below $l$.
Mathematically, this implies that arbitrary subsystem $C$ of size less than
$l$ approximately decouples from $R^{\prime}$:
$\rho_{R^{\prime}C}\stackrel{{\scriptstyle\epsilon}}{{\approx}}\rho_{R^{\prime}}\otimes\rho_{C}$.
Hence, the outcome of any measurement on $C$ is statistically independent of
any measurement on $R^{\prime}$ and can provide no information about
$R^{\prime}$. The parameter $l$ depends upon the purity of $B$. It is possible
to change the purity of $B$ by entangling $B$ with some external system
$B^{\prime}$. The entire setup is depicted in Figure 2.

Figure 2: The quantum information of $A$ interacts with system $B$ via a
scrambling unitary $U_{AB}$. System $B$ is purified by external system
$B^{\prime}$ while $A$ is purified by reference system $R$. Thus, the total
initial state is $\ket{RA}\ket{BB^{\prime}}$ while the final state after
scrambling unitary $U_{AB}$ is $\ket{\psi}_{RB^{\prime}CD}=U_{AB}\otimes
I_{RB^{\prime}}\ket{RA}\ket{BB^{\prime}}$.

.

##  II Diagrammatic representation of Quantum information spreading

In this section, we propose a novel way of diagrammatically representing the
information flow within a quantum network motivated by the formalism of Hasse
diagram [41] and Lamport diagrams [42].

Causality in a single node is trivial, as one event is the cause for the next.
But, it quickly gets complicated in cooperating nodes that exchange
information to solve a common task. In the realm of quantum mechanics,
preserving causal relationships can be even trickier due to the presence of
entanglement. Nonetheless, despite this intricacy, a causal framework should
persist due to the no-signaling theorem [36], which states the impossibility
of sending information just using entanglement alone. Furthermore, if we have
sufficient non-local quantum resources like entanglement, all non-local
quantum computations can be shifted to local quantum computations and
classical communication. With this, most multi-party quantum protocols can be
phrased as a synchronization problem among different nodes.

Considering the large scale of quantum networks, which may comprise numerous
nodes with many qubits each, creating a quantum circuit diagram for the entire
network is impractical. Instead, this diagrammatic approach focuses on
maintaining a casual relationship across events in different nodes, thus
providing a robust tracking mechanism for information dynamics and error
propagation within the network. This approach also facilitates the seamless
synchronization of different nodes within the network while tracking resource
expenses, thereby enabling the successful implementation of quantum protocols.

###  II-A Space-time diagram

The concept of the Hasse/Lamport diagram has found prior applications within
the realm of classical computing (see [43, 44, 45, 46]). Furthermore, it has
been employed in physics, particularly in specific quantum gravity models (see
[47, 41]). In this work, we use the Hasse/Lamport diagrammatic approach to
visually depict quantum protocols.

The evolution of distributed execution is given by a Hasse space-time diagram
where each node $P_{i}$ has a corresponding horizontal line that tracks the
progress of that node. The binary relationship arrow indicates the casual
relationship from one event to another. A solid dot can mark a local event. A
line at the end of the arrow can indicate the message received event. Messages
can be sent via quantum or classical channels. Because of the Hasse-like
structure, the diagram has interesting properties. Suppose $x$, $y$, and $z$
indicate three events, and if there is an arrow connecting between two events,
we indicate it by a binary relation $\prec$ with properties:

  1. 1.

Acyclic: x $\prec$ y and y $\prec$ x $\Rightarrow$ x = y

  2. 2.

Transitive: y $\prec$ y and y $\prec$ z $\Rightarrow$ x $\prec$ z

The transitive property guarantees that we don’t have a loop in the diagram,
which otherwise would imply a future affecting the past and violate causality.
Space-time diagram 3 shows an example of a space-time diagram with four nodes.
In the following sections, we will only use the arrow in the message send
event for convenience and drop the line at the message received event.

Figure 3: Example of a space-time diagram with four nodes. The dots represents
local event while arrows represents message send event.

###  II-B Local event

Defining an event in a quantum network can be trickier because of non-local
resources such as entanglement and the requirement to track both quantum and
classical states. In this section, we propose a novel definition of a local
event based on the notion of density matrix and its change. The quantum state
of a node $P_{i}$ can be mathematically described by a local density matrix
$\rho_{i}$ obtained by tracing out the rest of the network,
$\rho_{i}=\Tr_{\\{V/P_{i}\\}}(\rho)$, where $\rho$ is the quantum state of the
entire network. This characterizes both the classical and quantum state of the
node $P_{i}$. We say a local event occurred at node $i$ whenever this local
(reduced) density matrix $\rho_{i}$ changes. The change in the density matrix
can be computed using $L_{1}$ norm which for any operator $M$ is defined as
$||M||_{1}=\text{Tr}\sqrt{M^{\dagger}M}$. With this, we propose a following
definition of a local event.

######  Definition 1 (Local event)

A local event is recorded at node $P_{i}$ at time $t^{\prime}$ if
$||\rho_{i}(t^{\prime})-\rho_{i}(t^{\prime\prime})||_{1}\geq\epsilon$, where
$t^{\prime\prime}$ is the time of the immediate previous event at the same
node $P_{i}$. The threshold parameter $\epsilon$ depends upon the protocol and
its requirements.

This definition is also physically motivated because for
$||\rho_{i}(t^{\prime})-\rho_{i}(t^{\prime\prime})||_{1}<\epsilon$,
$\text{Tr}(\Pi(\rho_{i}(t^{\prime})-\rho_{i}(t^{\prime\prime})))<\epsilon$ for
any projection operator $\Pi$. This implies that the probability outcome of
any experiment between two density matrices $\rho_{i}(t^{\prime})$ and
$\rho_{i}(t^{\prime\prime})$ differs by at most $\epsilon$ when
$||\rho_{i}(t^{\prime})-\rho_{i}(t^{\prime\prime})||_{1}<\epsilon$. Therefore,
if the density matrix doesn’t change up to a certain threshold, operationally,
it is unnecessary to call it an event as no new information can be obtained
from it.

This way of defining a local event is also ideal from the perspective of
causality. Suppose two nodes $P_{i}$ and $P_{j}$ share a bipartite system and
are even allowed to be entangled. No signaling theorem or, in general, no
communication theorem [36] of quantum mechanics says that nothing $P_{i}$
chooses to do using local computations on his side will have any effect on the
local density matrix of $P_{j}$. With this definition of a local event, we can
then guarantee that the event at one node cannot affect other nodes unless the
information is transferred via a quantum or classical channel. This serves as
a foundation for maintaining causality in the space-time diagram. A future of
an event $x$ denotes all events casually affected by $x$, and the past of an
event $x$ are all the events that could have affected $x$. In the space-time
diagram 4, a dot represents a local event at that node.

Figure 4: Local event denoted by a dot

###  II-C Quantum and Classical Communication

Based on the definition of a local event, for one node to casually affect the
other node, it must use a communication channel. The communication in a
quantum network between two nodes can be via quantum or classical channel
[48].

In the space-time diagram, quantum processes are typically represented with a
solid line, while classical processes are typically represented with a dotted
line. Furthermore, usually, non-local computations are not allowed. In the
space-time diagram depicted in Figure 5, we represent a message transfer
through a classical channel using a dotted slant arrow while a message
transfer through a quantum channel using a solid slant arrow. The labels
indicate the unit message, which could, for example, be bit or qubit. This
allows one to track the resource consumption and generation in the network.

Figure 5: Space-time diagram with the following sequences: local event at node
S represented by a solid dot, classical communication from S to R via dotted
arrow, local event at S, quantum communication from S to R, shared quantum
resources between R and S, shared quantum resources between R and S, local
event at R, classical communication from R to S following resource
consumption, local event at S, and then quantum message from S to R after the
resource consumption. The labels $a$, $b$, $c$, $d$, $e$, and $f$ indicates
the amount of unit resources. For example, $a$ and $e$ could indicates number
of bits, while $b$ and $f$ qubits. Similarly, $c$ indicates amount of
maximally entangled qubits between $R$ and $S$ and $d$ indicates the amount of
shared classical resources.

###  II-D Shared resources

A fundamental way a quantum network differs from a classical one is by
introducing new resources, such as quantum entanglement, which cannot be
achieved via classical resources alone. We represent classical and quantum
resources using dotted and solid curves rather than arrows. The shared
classical resource between two nodes, R and S, is denoted by a dotted curved
shape ending with dots at each node as shown in Figure 5. In contrast, a solid
curve denotes shared quantum resources. The label in the diagram denotes the
amount of shared resources. For example, it could be the amount of classical
common randomness or entanglement. Because a maximally entangled pair is a
fundamentally important resource in a quantum network, we represent it by a
solid angle shape rather than a curve, terminating with dots at each node.
Similarly, we can also represent shared common randomness by a similar shape
but a dotted one.

###  II-E Resource generation

Using space-time diagrams, we will now show how to generate resources between
two nodes. Because it is a shared resource, local processes cannot generate it
alone. Therefore, it requires a communication channel, albeit quantum or
classical. Furthermore, quantum resources cannot be generated via classical
channel communication alone.

As an example, let us consider two ways entanglement can be generated between
nodes R and S. The first approach entails R preparing a maximally entangled
pair and transmitting one of the qubits from the pair to node S through a
quantum channel, as shown in the space-time diagram 6(a). In the second
approach, a third node Q generates a maximally entangled pair and sends one
qubit from the pair to node R and another to S, which results in the
establishment of a shared quantum resource between nodes R and S. This
approach is depicted in space-time diagram 6(b).

(a)

(b)

(c)

(d)

Figure 6: Resource generation and consumption. (a) and (b) shows resource
generation among R and S. (c) and (d) shows two fundamental protocol,
superdense coding and quantum teleportation respectively, that can be
implemented by consuming these resources.

###  II-F Resource consumption

The exciting thing about quantum resources is that, by consuming them, one can
do nontrivial tasks that would have otherwise been impossible. Therefore, it
is necessary to distinguish between usual communication and communication
after resource consumption. In a space-time diagram, we can represent a
message sent after consuming quantum or classical resources using a classical
channel by a dotted double-line arrow. We can represent the usage of the
quantum channel after consuming these resources by a solid double-line arrow.
as shown in the space-time diagram 5. The double line here doesn’t imply that
we are sending two units of messages. Instead, the label in the space-time
diagram will tell how many units of messages are transferred. For example, we
will show two fundamental protocols in a space-time diagram, which consumes
quantum entanglement to achieve specific nontrivial tasks.

####  II-F1 Superdense coding

Super dense coding protocol uses pre-shared entangled qubits to encode two
classical bits of information by transmission of a single qubit:

$[qq]+[q\rightarrow q]\geq 2[c\rightarrow c]$

In the space-time diagram 6(c), R and S share a pair of EPR pairs. S performs
an encoding on her half of the EPR pair and then sends it to R via a quantum
channel denoted by a double-line solid arrow. Note that even though S sends a
single qubit, it is represented by a double-line arrow in a space-time
diagram.

####  II-F2 Teleportation

Quantum teleportation protocol uses pre-shared entanglement to transfer a
quantum state from one node to another via classical communication only

$[qq]+2[c\to c]\geq[q\to q].$

The protocol is shown in the space-time diagram 6(d) where dotted double line
arrow is to indicate usage of classical channel after consuming quantum
resources. In the space-time diagram 6(d), the label $2$ indicates sending two
classical bits of messages.

###  II-G Multiparty case

(a)

(b)

(c)

Figure 7:  Entanglement swapping protocol used for building quantum repeater
(a) Node R and Node Q are maximally entangled, while Node Q and Node S are
maximally entangled. The goal is to get Node R and Node S maximally entangled.
Node Q performs a local computation and sends classical information to Node S.
This results in Node R and Node S being maximally entangled. (b) Space-time
diagram showing entanglement swapping protocol among 5 nodes in three time
steps. (c) Entanglement swapping protocol illustrated in a conventional
literature style

The true potential of these space-time diagrams arises for the multiparty
case. For two-party cases or a network with relatively few nodes, quantum
circuits could suffice and might even offer an advantage. However, for the
multiparty scenario, with a large number of nodes and a large number of
processes, the quantum circuit description would be highly impractical.
Nonetheless, space-time diagrams can offer a means to extract numerous
essential features of such complicated network protocols.

In the space-time diagram 7(a), we’ve illustrated a crucial three-party
protocol used in quantum repeater, called entanglement swapping, involving
three nodes Q, R, and S. At the end of the protocol, one can observe that R
and S are entangled by checking the dots on the end of a curved line. Moving
to the space-time diagram depicted in Figure 7(b), we’ve expanded the scenario
to involve more nodes and presented in Figure 7(c), a commonly used
illustration in the literature to depict the same concept.

One can have multiparty classical and quantum resources for the multiparty
case, which we denote by a repeated curved $\mathcal{W}$ shape with a hinge
touching each participating node. Again, the shape will be dotted for
classical resources and solid for quantum. We can use the following notation
to label the resource
$((x_{1},y_{1}),(x_{2},y_{2}),..(x_{i},y_{i})..(x_{N},y_{N}))$, where $N$
indicates the number of species of resources and $x_{i}$ indicates the amount
of $y_{i}$ species. We can also use this notation
$(x_{2},y_{2},P_{i}P_{j}....P_{k})$ to label the resources to indicate which
nodes are participating. However, this information can also be obtained from
the space-time diagram itself.

For example, in the space-time diagram for a three-party case, it is possible
to have two different families of shared quantum resources, such as GHZ or W
entangled states. Suppose we label it as $((3,GHZ),(4,W))$, then we have 3 GHZ
states and 4 W states shared among three nodes. Here, we refer to GHZ and W
states as two different families as they are not equivalent under Local
Operation and Classical Communication (LOCC); however, how one categorizes
different types of resources depends on their definition. The three-party
controlled teleportation, a variation of teleportation protocol, is shown in
the space-time diagram 9 to demonstrate this multiparty resources and
consumption concept.

(a)

(b)

Figure 8:  (a) LU protocol where local nodes perform local operation but no
communication among nodes is allowed. (b) LOCC protocol where nodes can
perform local operation but only classical communication is allowed among
nodes. In the subsequent round, another node may perform an operation
depending on measurement from other nodes. The label $\ket{\psi}$ and
$\ket{\phi}$ denote initial and final multiparty quantum state indicated by
curved line touching multiple nodes. Figure 9: Space-time diagram showing
third party controlled teleportation from node Q to node S using GHZ states
and only classical communication. The resource inequality is:
$[qqq]_{QRS}+2[c\to c]_{Q\to R}+2[c\to c]_{R\to S}\geq[q\to q]_{Q\to S}$ where
$[qqq]_{QRS}$ indicates the GHZ state shared between Q, R and S.

There are two important protocols called local unitary equivalence (LU) and
equivalence under local operations and classical communication (LOCC)
equivalence, which doesn’t require the use of a quantum channel. Two state
vectors are considered equivalently entangled under LU if they differ only by
a local unitary basis.

$\ket{\psi}\equiv_{\text{LU}}\ket{\phi}\xrightarrow{}\ket{\psi}=(U_{1}\otimes....\otimes
U_{N})\ket{\phi}$ (1)

Figure 8(a) shows the space-time diagram for such a protocol. Similarly, we
say that two states are considered LOCC-equivalent if they can be transformed
into each other through a protocol that only involves the usage of a classical
channel as shown in space-time diagram 8(b).

###  II-H Noise and types of operations

One crucial advantage of a space-time diagram is that it helps to analyze the
error propagation in a quantum network. We can denote a noisy local operation
and communication with a cross, as shown in Figure 10.

So far, we have assumed that all local operations are equally powerful.
However, it is unrealistic, especially for distributed quantum networks, where
some nodes might be more powerful. One can introduce different types of local
operations based on their protocol. One way of classifying local operations
would be based on the complexity the node can handle. For example, one can
have type I and type II operations, where type I can only implement stabilizer
protocol [49] while type II could be a fully universal quantum computation. In
the space-time diagram, as shown in figure 10, a dot inside a Ket notation
represents a local process with a Type II operation, and a dot represents a
Type I operation.

Figure 10: Noisy event or communication can be indicated by a cross while type
II operation can be indicated by a dot inside a Ket notation.

###  II-I Examples

In this section, we will show some examples of how space-time diagrams can be
used to represent quantum protocols. The first example is of a distributed
CNOT gate as shown in the space-time diagram 11(a). The second example is of
entanglement distillation (see figure 11(b)), a LOCC protocol, where out of
$n$ copies of bipartite pure system $\psi$, one can generate copies of EPR
pair at a rate $r$. The third example (see figure 11(c)) is a protocol that
demonstrates Nielsen’s theorem, a simple criterion for the equivalence of
bipartite pure states under LOCC.

(a) Distributed CNOT

(b) Entanglement distillation

(c) Nielsen theorem

Figure 11: (a) Space-time diagram for the distributed CNOT gate as proposed in
[50]. (b) Entanglement distillation protocol using only LOCC
$\ket{\psi}^{\otimes n}\xrightarrow{}\ket{\text{EPR}}^{\otimes[r.n]}$. (c)
Space-time diagram showing the diagrammatic representation of Nielsen’s
theorem [51]. Nielsen’s theorem gives a simple criterion for the equivalence
of bipartite pure states under LOCC. In the diagram, the network of 5 nodes
$V=\\{P_{1},P_{2},P_{3},P_{4},P_{5}\\}$ is divided into two subnetworks
$V_{1}=\\{P_{1},P_{2},P_{3}\\},V_{2}=\\{P_{4},P_{5}\\}$ making it a bipartite
state. Within subnetworks $V_{1}$ and $V_{2}$, they are allowed to do quantum
communications, but across the $V_{1}$ and $V_{2}$, they are only allowed to
do classical communication. Nielsen’s theorem gives a criteria on how final
state $\ket{\phi}$ can be reached starting from initial state
$\ket{\psi}_{V_{1}V_{2}}$.

##  III Quantum information scrambling protocol in a quantum network

(a) A node $R$, marked yellow, wants to scramble its quantum information
across the quantum network. $D$ denotes the quantum data center node.

(b) Network after the information scrambling protocol.

(c) Network after the malicious party $E$, marked by red, gets access to a
subset of the network.

(d) A circuit representation of information scrambling protocol similar to
Figure 2. The external system R’ is entangled with R, which helps track the
network’s information dynamics.

(e) Space-time diagram for quantum information scrambling protocol where node
$R$ wants to protect secret quantum information by scrambling across the
network. Nodes $P_{1}$ to $P_{N}$ represent other nodes in the network while
$D$ represents the quantum data center, which purifies $P_{1}$ to $P_{N}$ if
necessary. All the local operations are of type II nature as they are from
random unitary operations, and the infinity label indicates that quantum
entanglement among nodes is assumed to be free. The future light cone of node
$R$ can be obtained by following casual paths from $R$.

Figure 12:  Quantum information scrambling protocol in different
representations. To get information scrambling, the casual future of the node
$R$ must be the entire network. $V_{E}$ indicates the nodes accessed by a
malicious party, while $V_{B}$ indicates the rest of the network. $D$ denotes
the quantum data center node.

In this section, we propose a novel protocol for distributed quantum networks
inspired by the phenomenon of quantum information scrambling. Additionally,
this protocol demonstrates the diagrammatic approach introduced in the earlier
section to investigate the quantum information spreading in a network setting.
The central concept of the protocol revolves around scrambling quantum
information of one particular node $R$ across the quantum network to achieve
the following two objectives. Firstly, to reconstruct the quantum information
of node $R$, any external malicious entity denoted as $E$ would need to gain
access to a substantial subset of the network since the information is not
within $R$ anymore. In the most adverse scenario, accomplishing this would
necessitate $E$ to breach the entire quantum network. Secondly, even if $E$
were to somehow access the requisite network size, the protocol’s construction
allows for implementing measures that render the decoding process
exponentially complex for $E$, thus making it hard to reconstruct $R$’s
information. Therefore, this protocol plays with both information-theoretic
and complexity-theoretic arguments to improve the security of a quantum
network.

To a certain extent, one could perceive quantum information scrambling as the
strongest manifestation of information spreading in a network. All nodes at
the end of the protocol are causally related to the node $R$. This feature,
that the future of the node $R$ is the entire network, is a necessary
condition for scrambling quantum information. We will now formulate the
protocol mathematically. At the start of the protocol, we divide the network
$V$ into two subsets, $V_{R}=\\{R\\}$ and $V_{A}=V\setminus V_{R}$ described
by the quantum state $\rho_{V_{A}}$. In the next subsection, we will go into
the protocol details. For now, suppose the scrambling protocol is implemented
successfully. After some time following the protocol, suppose the malicious
party $E$ got access to some subset of nodes, which we denote by $V_{E}$ while
the rest of the network is then given by $V_{B}=V\setminus V_{E}$. Thus, we
have a Hilbert space of $n=|R|+|V_{A}|=|V_{E}|+|V_{B}|$ qubits partitioned as:

$\mathcal{H}=R\otimes V_{A}=V_{E}\otimes V_{B}$ (2)

and a unitary map transforming them as:

$U_{RV_{A}}:R\otimes V_{A}\xrightarrow{}V_{E}\otimes V_{B}$ (3)

If the initial state is $\rho_{R}\otimes\rho_{V_{A}}$, then after the
information scrambling protocol $U$, the final quantum state is
$\rho^{\prime}=U\rho U^{\dagger}$. The quantum state of a malicious party,
$\rho_{E}$ would be described by tracing out the rest of the network one
doesn’t have access to $\Tr_{V_{B}}\rho^{\prime}$. Having done this, we can
now precisely formulate the problem as follows:

  * •

What is the minimum size of $|V_{E}|$ necessary for malicious party E to get
access in order to reconstruct the R’s quantum information?

  * •

Suppose malicious party E has access to this minimum necessary size but
doesn’t know the unitary $U$, i.e., how the information was scrambled. How
many queries does $E$ need to make on different nodes to learn how the
information was scrambled?

Interestingly, it turns out that the minimum size $|V_{E}|$ depends upon the
purity, a measure of entanglement entropy, of the system $V_{A}$. It is
possible to change the purity of $V_{A}$ by entangling it with some other node
$D$, which we refer to as a quantum data center. Quantum data centers could
serve various purposes in a network. In our context, it primarily functions as
a node that, at the protocol’s beginning, helps change the purity of $V_{A}$
by entangling it. The entire setup of the quantum information scrambling
protocol, with quantum network graph representation, quantum circuit
representation, and space-time diagrammatic representation, is shown in Figure
12.

###  III-A Description of the protocol

In this subsection, we describe how a scrambling unitary $U(t)$ of size
$2^{|V|}\times 2^{|V|}$ can be implemented in a quantum network setting. A
detailed analysis will be done in a subsequent section. A trivial way to
implement this would be if the node $R$, which wants to scramble the quantum
information, has $|V|$ qubits. $R$ performs the $U(t)$ locally and then
selects $|P_{i}|$ qubits randomly and teleport it to each node $P_{i}$ in a
network $V$. However, adopting such a strategy could compromise security since
it centralizes all computational activities within a single node. A better
method would be implementing the scrambling unitary $U(t)$ in a distributed
fashion.

A scrambling unitary $U(t)$ does not necessarily have any inherent local
structure. To account for the locality, for our protocol, we restrict it to be
local in neighboring nodes, which are connected by edges. In a quantum network
setting, it is reasonable to have all the quantum operations to be local in
nodes. A unitary local in neighboring nodes can be made local in nodes by
consuming entanglement under LOCC [52]. For example, if a node $A(\in V)$
wants to send an unknown quantum state with dimension $d$ to another node
$B(\in V)$, all they need to have is share a bipartite maximally entangled
state $\ket{\phi_{d}}^{AB}:=\sum_{i=1}^{d}\ket{ii}^{AB}/\sqrt{d}$. Thus, we
assume that the network is of $G_{\phi}^{\infty}$, meaning all the nodes
connected by an edge have an infinite supply of maximal EPR pairs at the
beginning of the protocol. Let $|P_{i}|$ be the number of qubits allocated by
node $P_{i}$ for the protocol. It is reasonable to assume that the node
$P_{i}$ can have much more qubits than just $|P_{i}|$, say at least
$|P_{i}|+\text{max}(|P_{i+1}|,|P_{i-1}|)$. This condition guarantees that node
$P_{i}$ can perform the non-local computation via LOCC, making the entire
protocol local in individual nodes.

As a simple example, let’s consider a network $V\in G_{\phi}^{\infty}$ with
three nodes $P_{1},P_{2},\&P_{3}$ , see Figure 13, with sizes satisfying
$|P_{1}|<|P_{2}|<|P_{3}$. This configuration implies that each node $P_{1}$,
$P_{2}$ and $P_{3}$ has at least
$|P_{1}|+|P_{2}|,|P_{2}|+|P_{3}|,|P_{2}|+|P_{3}|$ qubits respectively. We
consider a scrambling circuit with two Haar random unitaries $U_{P_{1}P_{2}}$
and $U_{P_{2}P_{3}}$, each sized at $2^{|P_{1}|+|P_{2}|}$ and
$2^{|P_{2}|+|P_{3}|}$ respectively, applied sequentially as depicted in Figure
13. Clearly, in the circuit representation, $U_{P_{1}P_{2}}$ and
$U_{P_{2}P_{3}}$ are local in neighboring nodes. We now map this circuit
representation into a quantum network using only LOCC protocols. The initial
setup involves a $G_{\phi}^{\infty}$ network. At time $t_{0}$, the objective
is to implement $U_{P_{1}P_{2}}$ at the local node level, either on node
$P_{1}$ or $P_{2}$. The first step is to flip a coin among $P_{1}$ and
$P_{2}$, which can be done without any communication cost by consuming
entanglement. Furthermore, coin toss introduces randomness in the protocol,
making it challenging for potential attackers to breach it. Assuming $P_{1}$
is chosen, $P_{2}$ teleports their qubits to $P_{1}$ through a classical
channel by sending $2|P_{2}|$ classical bits. Since $P_{1}$ possesses at least
$|P_{1}|+|P_{2}|$ qubits, $P_{1}$ can then execute the $U_{P_{1}P_{2}}$
unitary of size $2^{|P_{1}|+|P_{2}|}$. After the unitary operation, $P_{1}$
randomly selects $|P_{2}|$ qubits to teleport them back to $P_{2}$. This
process ensures the local implementation of $U_{P_{1}P_{2}}$ at the level of
individual nodes. At time $t_{1}$, the goal is to implement the
$U_{P_{2}P_{3}}$ unitary. Following a similar approach as with
$U_{P_{1}P_{2}}$, this can also be achieved locally. The entire protocol is
diagrammatically represented in Figure 13 where the type II notation is used
to represent non-trivial operations such as $U_{P_{1}P_{2}}$ and
$U_{P_{2}P_{3}}$. Figure 14 shows a more complicated setup.

(a)

(b)

(c)

Figure 13: (a) A graph $G_{\phi}^{\infty}$ with three nodes where $P_{1}$
$P_{2}$ and $P_{2}$ $P_{3}$ share unlimited EPR pairs. (b) A circuit with
three nodes where unitary $U_{P_{1}P_{2}}$ is local in nodes $P_{1}$ and
$P_{2}$ while unitary $U_{P_{2}P_{3}}$ is local in nodes $P_{2}$ and $P_{3}$.
(c) Space-time diagram showing a protocol where the unitary $U_{P_{1}P_{2}}$
and $U_{P_{2}P_{3}}$ are converted into being local in individual nodes using
quantum teleportation.

(a)

(b)

(c)

(d)

(e)

Figure 14: (a), (b) and (c) shows the information flow from a node $R$ in a
quantum network $G_{\phi}^{\infty}$. $D$ denotes a quantum data center. The
dotted line with $D$ indicates the possibility of sharing multiparty quantum
resources among the network and data center. (d) shows an example of a
possible quantum circuit. $U_{P_{2}P_{4}}$ is local in node $P_{2}$ and
$P_{4}$ as they are connected by an edge as shown in (a). (e) shows a space-
time diagram where all non-local computations of the circuit (d) are made
local in nodes using the technique of Figure 13.

###  III-B Analysis of the protocol

In this subsection, we primarily consider the scrambling unitary of Haar
scrambling nature, which refers to a unitary $U$ chosen randomly from the Haar
measure. We will consider other kinds of scrambling in Section III-C, where we
focus on efficiency issues. However, important insights can be drawn from Haar
scrambling as we expect the late-time values of entropy and mutual information
for any scrambling unitary $U(t)$ to match those of Haar random unitaries.

One can implement Haar scrambling unitary $U(t)$ in the quantum network
following the framework outlined in the subsection III-A. Once again, we have
a unitary map:

$U_{RV_{A}}:R\otimes V_{A}\xrightarrow{}V_{E}\otimes V_{B}$ (4)

where $U_{RV_{A}}$ is a Haar scrambling unitary local in nodes connected by
edges and $R$, $V_{A}$, $V_{B}$, and $V_{E}$ are subnetworks as described in
relation 2. For the quantum information theoretic analysis, it is often easier
to track the quantum information of $R$ in the network by considering an
external reference state $R^{\prime}$ which is perfectly entangled with $R$.
Furthermore, we can have a quantum data center $D$ which purifies the
subnetwork $V_{A}$. Thus, the total initial state is
$\ket{R^{\prime}R}\ket{V_{A}D}$, and the state after the information
scrambling protocol is:

$\ket{\psi}_{R^{\prime}DV_{E}V_{B}}=U_{RV_{A}}\otimes
I_{R^{\prime}D}\ket{R^{\prime}R}\ket{V_{A}D}$ (5)

.

####  III-B1 Without quantum data center $D$

We first consider the case without a quantum data center, i.e., $V_{A}$ is in
a pure state. After applying the Haar scrambling unitary, it can be shown that
(see [53]) the output pure state in $\mathcal{H}_{V_{E}V_{B}}$ is likely close
to a maximally entangled state if $\frac{|V_{E}|}{|V_{B}|}<<1$, more precisely
for the output bipartite Hilbert space
$\mathcal{H}_{E}\otimes\mathcal{H}_{V_{B}}$, we get

$\int
dU\left|\left|\rho_{V_{E}}-\frac{Id}{|V_{E}|}\right|\right|_{1}\leq\sqrt{\frac{|V_{E}|^{2}-1}{|V_{E}||V_{B}|+1}}$
(6)

where $||.||_{1}$ is the $L_{1}$ norm and $\rho_{V_{E}}$ describes the quantum
state accessed by malicious party $E$ in the network. When $|V_{B}|$ is
significantly larger than $|V_{E}|$, the typical deviation of $\rho_{V_{E}}$
from maximally mixed state is extremely small. For example, consider the case
when the size of $V_{B}$ exceeds the system in the control of $V_{E}$ by just
ten qubits, then the typical deviation of $V_{E}$’s quantum state from the
maximal mixed state is upper bounded by $2^{-5}$. And, $V_{E}$ cannot carry
any significant information about the scrambled information of $R$. Therefore,
as a consequence of equation 6, the malicious party $E$ will need access to at
least half the network’s size to get any information of $R$.

This result can be intuitively understood as follows. For the pure case,
making any arbitrary partition on a network $V$ after information scrambling,
if one partition can reconstruct the information of $R$, then the other
partition cannot. This follows simply from quantum no-cloning theorem which
states that no operation can make copies of an arbitrary unknown quantum
state. If both set is able to reconstruct the state $\rho_{R}$, then no-
cloning theorem is violated. This also immediately gives the bound that for
$E$ to reconstruct $\rho_{R}$, $E$ will need to have access to at least half
the size of the entire network, even though the nodes which form the set can
be completely arbitrary. More concretely, for $V_{E}$ to be able have any
chance to hack the network, the lower bound on the size of $V_{E}$ should be
$|V_{E}|=\frac{|V|+1}{2}$, where $|V|$ is the size of the network.

It is possible to provide a formal security analysis using the language of
quantum secret sharing. This follows from the decoupling theorem [35], which
provides a unifying framework for the theory of quantum error correction and
many other significant results of quantum Shannon theory. For the decoupling
theorem to work, the Haar random unitary encoding must be scrambling, implying
the possibility of quantum error correction [54, 35]. On the other hand, the
recoverability requirement of a quantum secret-sharing scheme is a sufficient
and necessary condition for quantum error correction. Thus, all quantum
secret-sharing schemes are also quantum error-correcting codes. A quantum
threshold secret sharing scheme is denoted by $((k,n))$ where an arbitrary
quantum secret is divided into $n$ parties such that any $k$ or more parties
can perfectly reconstruct the secret. In comparison, any $k-1$ or fewer
parties have no information about the secret. Thus, for this pure case, we get
a quantum secret sharing scheme similar to
$\left(\left(\frac{|V|+1}{2},|V|\right)\right)$. We plan to further explore
this relationship between Haar scrambling and quantum secret sharing in the
future. Because Haar scrambling has inherent randomness, it might not be
possible to get the exact threshold scheme. Instead, the scheme might be of
ramp secret sharing nature.

####  III-B2 With quantum data center $D$

We will now discuss the case with quantum data center $D$, which, by
entangling with $V_{A}$ can purify it. Therefore, $V_{A}$ is in a mixed
quantum state as it is part of a pure state $\ket{V_{A}D}$. The final quantum
state after the scrambling protocol is
$\ket{\psi}_{R^{\prime}DV_{E}V_{B}}=U_{RV_{A}}\otimes
I_{R^{\prime}D}\ket{R^{\prime}R}\ket{V_{A}D}$. Then, from the conservation of
information under unitary evolution, we get:

$I(R^{\prime}:R)=I(R^{\prime}:(V\backslash\\{D\\}=V_{B}V_{E}))+I(R^{\prime}:D)$
(7)

where $I(X:Y)$ is the quantum mutual information between $X$ and $Y$.

For the case, when $V_{B}V_{E}$ is maximally mixed,
$I(R^{\prime}:V_{B}V_{E})=0$ and all the information about $R$ should now be
in $D$ as $I(R^{\prime}:R)=I(R^{\prime}:D)$. Therefore, when $V_{B}V_{E}$ is
maximally mixed, even if the malicious party $E$ has access to all the nodes
of the network $V$ except for the quantum data center $D$, $E$ can have no
information about $R$. From equation 6, this condition is achieved as soon as
$|D|>\frac{|V|}{2}$. If the size of $D$ is equal to $\frac{|V|-1}{2}$, $E$
will need to get access to entire subnetwork $V\backslash\\{D\\}$ to get the
information of $R$. If $0<|D|<\frac{|V|-1}{2}$, then in contrast to the pure
case of $V_{A}$, the necessary size of $V_{E}$ to hack the information of $R$
is greater than $\frac{|V|+1}{2}$. Therefore, by changing the purity
(entanglement entropy) of $V_{A}$ and the size of quantum data center $D$, it
is possible to get other security protocols than the one obtained for the pure
case.

Furthermore, it is also possible to construct other interesting protocols
using a powerful quantum data center. Suppose $V_{A}$ is maximally entangled
with data center $D$. Then, after the information scrambling protocol, $D$ can
reconstruct the information of $R$ by just having access to a subnetwork of
size $|R|$ from $V\backslash\\{D\\}$. One can use these features to error-
correct quantum nodes in a network or improve security by quickly hiding
information if a malicious party $E$ tries to access it. We plan to explore
these features in the future.

####  III-B3 Achieving the two criteria

Now, we will discuss the two criteria we wanted to achieve with the Haar
scrambling protocol. The first criterion was that the minimum size of
$|V_{E}|$, necessary for malicious party E to get access to reconstruct the
R’s quantum information, should be large. In the previous two subsections, we
showed that changing the purity of $V_{A}$ makes it possible to achieve it.
Even when $V_{A}$ is pure, $E$ will need access to half the network to hack
the information of $R$. If $E$ is maximally entangled with quantum data center
$D$, then $E$ must access the entire subnetwork $V\backslash\\{D\\}$ to get
any information out. We may also already start with $V_{A}$ being maximally
mixed from the start. In this case, $E$ will need to access the entire network
$V$ to get the information of $R$. Therefore, by changing the purity of the
initial party $V_{A}$, the information scrambling protocol achieves the first
feature we wanted.

The second criterion is related to decoding the information of $R$ that was
scrambled across the network. Suppose malicious party E does have access to
this minimum necessary size but doesn’t know how the information was
scrambled. In this case, $E$ must query quantum nodes to learn the scrambling
unitary $U(t)$. If $E$ needs to make many queries, it will provide a security
advantage in that $E$ cannot decode information of $R$. [55] showed that if
$U_{t}$ is a $t$ doped Clifford circuit, it is possible to learn the circuit
using only $\mathcal{O}(\text{poly}(n)\exp(t))$ queries where $n$ is the depth
of the circuit. The scrambling circuit is a fully chaotic quantum circuit with
all local events requiring $t$ gates. Hence, $E$ must make exponential,
$\exp(M)$ queries to the network where $M$ is the total number of local events
in the protocol. Therefore, the Haar scrambling protocol achieves the second
criterion, offering the advantage that even if the information of $R$ is
within the grasp of the malicious party E, they cannot decode it.

###  III-C Efficient construction

Scrambling unitaries do not necessarily have to be perfect Haar unitaries;
even simple operators can exhibit excellent scrambling properties. Haar-random
and unitary-design states demonstrate nearly maximal entanglement,
specifically for $t\geq 2$. This implies that the states obtained from
$t$-design circuits [56] are information-theoretically indistinguishable from
Haar scrambling circuits up to the $t$-th moment [55]. Since we employed an
information-theoretic argument to derive ramp secret sharing schemes from Haar
scrambling, $t$-design circuits are already sufficient and come with the
benefit that the whole protocol can be constructed in polynomial time.
Nevertheless, it is necessary to take caution; otherwise, we may fail to
attain the desired second feature. For example, if the scrambling protocol is
composed of only Clifford gates, then $E$ can learn the scrambling unitary
$U(t)$ with only $\mathcal{O}(\text{poly}(n))$ queries where $n$ is the depth
of the protocol. Therefore, it is necessary to have at least $n$ number of $t$
gates for a $n$ depth protocol, such that it will take an exponential,
$\mathcal{O}(\exp(n))$, queries by $E$ to decipher the information of $R$.

Scrambling in a quantum network also depends on the network topology of the
underlying graph $G=(V,E)$. Information scrambling can be considered a quantum
counterpart of a broadcast algorithm. The time complexity of the scrambling
protocol is then lower bounded by the time complexity of the broadcast
algorithm, which is of order $\mathcal{O}(\text{Diam}(G))$ for dirty topology,
and of order $\mathcal{O}(|E|)$ for clean topology, where $\text{Diam}(G)$ is
the diameter of the graph. One suitable design for the network design is the
hyperbolic network design. In Euclidean networks, the number of nodes
increases only polynomially as the distance from the center grows. However, in
a hyperbolic network, the number of nodes increases exponentially with
distance from the center. This allows the spread of information quickly in the
network.

##  IV Conclusion

By presenting the diagrammatic visualization approach, motivated by the Hasse
diagram and Lamport diagrams, we contribute to the understanding and practical
implementation of large-scale quantum networks. These techniques shed light on
information flow dynamics, synchronization, error monitoring, resource
allocation, and security considerations within the network. In addition, we
have proposed a novel quantum information scrambling protocol where malicious
parties would need to access a significant fraction of the network to retrieve
the information scrambled by some node.

Several intriguing open questions remain to be addressed in our study.
Firstly, a deeper investigation into the mathematical structure of space-time
diagrams is necessary, with potential insights to be drawn from quantum tensor
networks[57], and classical distributed computing [42]. Equally important is
exploring the practical utility of space-time diagrams across various domains,
including synchronization, the construction of fault-tolerant quantum
networks, quantum error propagation and correction, and quantum resource
tracking. For example, one can explore what else a quantum data center can
achieve in terms of constructing fault-tolerant quantum networks and enhancing
security in the network. A quantum network’s topology shapes a network’s
information dynamics, necessitating an in-depth analysis of its relationship
with quantum information scrambling protocols. Lastly, the ongoing
achievements of quantum information theory in many-body systems [58, 59],
presents a promising avenue for the development of novel protocols in
distributed quantum networks, making it a captivating area for future
exploration.

## Acknowledgment

The research is part of the Munich Quantum Valley, which is supported by the
Bavarian state government with funds from the Hightech Agenda Bayern Plus.
C.Deppe additionally acknowledge the financial support by the Federal Ministry
of Education and Research (BMBF) in the programs with the identification
numbers: 16KISK002, 16KISQ028, 16KISQ038, 16K1S1598K, 16KISQ077, 16KISQ093,
16KISR027K.

## References

  * [1] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” _Science_ , vol. 362, no. 6412, p. eaam9288, 2018.
  * [2] R. Bassoli, H. Boche, C. Deppe, R. Ferrara, F. H. P. Fitzek, G. Janssen, and S. Saeedinaeeni, _Quantum Communication Networks_. Springer Cham, 2021.
  * [3] R. Van Meter, _Resource Management and Multiplexing_. John Wiley & Sons, Ltd, 2014, ch. 13, pp. 253–264.
  * [4] D. Cuomo, M. Caleffi, and A. S. Cacciapuoti, “Towards a distributed quantum computing ecosystem,” _IET Quantum Communication_ , vol. 1, no. 1, pp. 3–8, jul 2020.
  * [5] R. Van Meter and S. J. Devitt, “The path to scalable distributed quantum computing,” _Computer_ , vol. 49, no. 9, pp. 31–42, 2016.
  * [6] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” _Theoretical Computer Science_ , vol. 560, pp. 7–11, dec 2014.
  * [7] A. Singh, K. Dev, H. Siljak, H. D. Joshi, and M. Magarini, “Quantum internet- applications, functionalities, enabling technologies, challenges, and research directions,” 2021.
  * [8] A. K. Ekert, “Quantum cryptography based on bell’s theorem,” _Phys. Rev. Lett._ , vol. 67, pp. 661–663, Aug 1991.
  * [9] P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, “A quantum network of clocks,” _Nature Physics_ , vol. 10, no. 8, pp. 582–587, jun 2014.
  * [10] D. Gottesman, T. Jennewein, and S. Croke, “Longer-baseline telescopes using quantum repeaters,” _Phys. Rev. Lett._ , vol. 109, p. 070503, Aug 2012. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.109.070503
  * [11] M. Caleffi, M. Amoretti, D. Ferrari, D. Cuomo, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, “Distributed quantum computing: a survey,” 2022.
  * [12] A. Yimsiriwattana and S. J. Lomonaco Jr, “Distributed quantum computing: A distributed shor algorithm,” in _Quantum Information and Computation II_ , vol. 5436. SPIE, 2004, pp. 360–372.
  * [13] Z. Zhang and Q. Zhuang, “Distributed quantum sensing,” _Quantum Science and Technology_ , vol. 6, no. 4, p. 043001, 2021.
  * [14] F. L. Buessen, D. Segal, and I. Khait, “Simulating time evolution on distributed quantum computers,” _Physical Review Research_ , vol. 5, no. 2, apr 2023.
  * [15] M. Pompili, S. L. Hermans, S. Baier, H. K. Beukers, P. C. Humphreys, R. N. Schouten, R. F. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse _et al._ , “Realization of a multinode quantum network of remote solid-state qubits,” _Science_ , vol. 372, no. 6539, pp. 259–264, 2021.
  * [16] S. Daiss, S. Langenfeld, S. Welte, E. Distante, P. Thomas, L. Hartung, O. Morin, and G. Rempe, “A quantum-logic gate between distant quantum-network modules,” _Science_ , vol. 371, no. 6529, pp. 614–617, 2021.
  * [17] C. Simon, “Towards a global quantum network,” _Nature Photonics_ , vol. 11, no. 11, pp. 678–680, 2017.
  * [18] P. C. Humphreys, N. Kalb, J. P. Morits, R. N. Schouten, R. F. Vermeulen, D. J. Twitchen, M. Markham, and R. Hanson, “Deterministic delivery of remote entanglement on a quantum network,” _Nature_ , vol. 558, no. 7709, pp. 268–273, 2018.
  * [19] N. Sangouard, C. Simon, H. De Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” _Reviews of Modern Physics_ , vol. 83, no. 1, p. 33, 2011.
  * [20] K. Azuma, S. Bäuml, T. Coopmans, D. Elkouss, and B. Li, “Tools for quantum network design,” _AVS Quantum Science_ , vol. 3, no. 1, p. 014101, mar 2021.
  * [21] R. Bassoli, H. Boche, C. Deppe, R. Ferrara, F. H. P. Fitzek, G. Janssen, and S. Saeedinaeeni, _Quantum Communication Networks, 1_. Springer, 2021, vol. 23.
  * [22] R. Van Meter and S. J. Devitt, “Local and distributed quantum computation,” _arXiv preprint arXiv:1605.06951_ , 2016.
  * [23] R. Van Meter, T. D. Ladd, A. G. Fowler, and Y. Yamamoto, “Distributed quantum computation architecture using semiconductor nanophotonics,” _International Journal of Quantum Information_ , vol. 8, no. 01n02, pp. 295–323, 2010.
  * [24] J. Avron, O. Casper, and I. Rozen, “Quantum advantage and noise reduction in distributed quantum computing,” _Physical Review A_ , vol. 104, no. 5, p. 052404, 2021.
  * [25] C. Qiao, Y. Zhao, G. Zhao, and H. Xu, “Quantum data networking for distributed quantum computing: Opportunities and challenges,” in _IEEE INFOCOM 2022-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_. IEEE, 2022, pp. 1–6.
  * [26] T. Häner, D. S. Steiger, T. Hoefler, and M. Troyer, “Distributed quantum computing with qmpi,” in _Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis_ , 2021, pp. 1–13.
  * [27] R. Beals, S. Brierley, O. Gray, A. W. Harrow, S. Kutin, N. Linden, D. Shepherd, and M. Stather, “Efficient distributed quantum computing,” _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , vol. 469, no. 2153, p. 20120686, 2013.
  * [28] C. Cicconetti, M. Conti, and A. Passarella, “Resource allocation in quantum networks for distributed quantum computing,” in _2022 IEEE International Conference on Smart Computing (SMARTCOMP)_. IEEE, 2022, pp. 124–132.
  * [29] R. Parekh, A. Ricciardi, A. Darwish, and S. DiAdamo, “Quantum algorithms and simulation for parallel and distributed quantum computing,” in _2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS)_. IEEE, 2021, pp. 9–19.
  * [30] F. Shahandeh, A. P. Lund, and T. C. Ralph, “Quantum correlations and global coherence in distributed quantum computing,” _Phys. Rev. A_ , vol. 99, p. 052303, May 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.99.052303
  * [31] J. Chen, “Review on quantum communication and quantum computation,” in _Journal of Physics: Conference Series_ , vol. 1865, no. 2. IOP Publishing, 2021, p. 022008.
  * [32] S. Xu and B. Swingle, “Scrambling Dynamics and Out-of-Time Ordered Correlators in Quantum Many-Body Systems: a Tutorial,” 2022.
  * [33] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, “Chaos in quantum channels,” _Journal of High Energy Physics_ , vol. 2016, no. 2, feb 2016.
  * [34] A. Nahum, S. Vijay, and J. Haah, “Operator spreading in random unitary circuits,” _Physical Review X_ , vol. 8, no. 2, apr 2018.
  * [35] P. Hayden, M. Horodecki, A. Winter, and J. Yard, “A decoupling approach to the quantum capacity,” _Open Systems & Information Dynamics_, vol. 15, no. 01, pp. 7–19, mar 2008.
  * [36] M. M. Wilde, _Quantum Information Theory_. Cambridge University Press, 2013.
  * [37] K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, “Verified Quantum Information Scrambling,” _Nature_ , vol. 567, no. 7746, pp. 61–65, 2019.
  * [38] T. Schuster, B. Kobrin, P. Gao, I. Cong, E. T. Khabiboulline, N. M. Linke, M. D. Lukin, C. Monroe, B. Yoshida, and N. Y. Yao, “Many-Body Quantum Teleportation via Operator Spreading in the Traversable Wormhole Protocol,” _Phys. Rev. X_ , vol. 12, no. 3, p. 031013, 2022.
  * [39] Y. Sekino and L. Susskind, “Fast scramblers,” _Journal of High Energy Physics_ , vol. 2008, no. 10, pp. 065–065, oct 2008.
  * [40] A. Bhattacharyya, L. K. Joshi, and B. Sundar, “Quantum information scrambling: from holography to quantum simulators,” _The European Physical Journal C_ , vol. 82, no. 5, may 2022.
  * [41] S. Surya, “The causal set approach to quantum gravity,” _Living Reviews in Relativity_ , vol. 22, no. 1, sep 2019.
  * [42] L. Lamport, “Time, clocks, and the ordering of events in a distributed system,” _Commun. ACM_ , vol. 21, no. 7, p. 558–565, jul 1978. [Online]. Available: https://doi.org/10.1145/359545.359563
  * [43] V. K. Garg, _Elements of distributed computing_. John Wiley & Sons, 2002.
  * [44] H. Attiya and J. Welch, _Distributed computing: fundamentals, simulations, and advanced topics_. John Wiley & Sons, 2004, vol. 19.
  * [45] A. D. Kshemkalyani and M. Singhal, _Distributed computing: principles, algorithms, and systems_. Cambridge University Press, 2011.
  * [46] K. Thulasiraman and M. N. Swamy, _Graphs: theory and algorithms_. John Wiley & Sons, 2011.
  * [47] D. P. Rideout and R. D. Sorkin, “Classical sequential growth dynamics for causal sets,” _Physical Review D_ , vol. 61, no. 2, p. 024002, 1999.
  * [48] C. Howe, X. Wang, and A. Anwar, “Robust and efficient quantum communication,” in _Proceedings of the 2023 International Workshop on Quantum Classical Cooperative_ , 2023, pp. 13–16.
  * [49] V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, “The resource theory of stabilizer quantum computation,” _New Journal of Physics_ , vol. 16, no. 1, p. 013009, jan 2014.
  * [50] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, “Optimal local implementation of nonlocal quantum gates,” _Phys. Rev. A_ , vol. 62, p. 052317, Oct 2000. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.62.052317
  * [51] M. A. Nielsen, “Conditions for a class of entanglement transformations,” _Phys. Rev. Lett._ , vol. 83, pp. 436–439, Jul 1999. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.83.436
  * [52] A. S. Cacciapuoti, M. Viscardi, J. Illiano, and M. Caleffi, “Entanglement distribution in the quantum internet: Knowing when to stop!” _arXiv preprint arXiv:2307.05123_ , 2023.
  * [53] P. Hayden and J. Preskill, “Black holes as mirrors: quantum information in random subsystems,” _Journal of High Energy Physics_ , vol. 2007, no. 09, pp. 120–120, sep 2007.
  * [54] H. Imai, J. Müller-Quade, A. C. A. Nascimento, P. Tuyls, and A. Winter, “An information theoretical model for quantum secret sharing,” _Quantum Info. Comput._ , vol. 5, no. 1, p. 69–80, jan 2005.
  * [55] L. Leone, S. F. E. Oliviero, S. Lloyd, and A. Hamma, “Learning efficient decoders for quasi-chaotic quantum scramblers,” 2023.
  * [56] D. Gross, K. Audenaert, and J. Eisert, “Evenly distributed unitaries: On the structure of unitary designs,” _Journal of Mathematical Physics_ , vol. 48, no. 5, may 2007.
  * [57] J. Biamonte and V. Bergholm, “Tensor networks in a nutshell,” 2017.
  * [58] M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, “Random quantum circuits,” _Annual Review of Condensed Matter Physics_ , vol. 14, no. 1, pp. 335–379, mar 2023.
  * [59] E. H. Lieb and D. W. Robinson, “The finite group velocity of quantum spin systems,” _Communications in Mathematical Physics_ , vol. 28, no. 3, pp. 251–257, Sep. 1972.


11institutetext: BASIRA Lab, Istanbul Technical University, Istanbul, Turkey
22institutetext: Computing, I-X and Department of Computing, Imperial College
London, London, UK
22email<EMAIL_ADDRESS>
33institutetext: Freie Universität Berlin, Berlin, Germany
44institutetext: Zuse Institute Berlin, Berlin, Germany
44email<EMAIL_ADDRESS>
# Predicting Shape Development: A Riemannian Method

Doğa Türkseven 11 Islem Rekik 1122 0000-0001-5595-6673 Christoph von Tycowicz
44 0000-0002-1447-4069 Martin Hanik 3344 0000-0002-7120-4081

###### Abstract

Predicting the future development of an anatomical shape from a single
baseline observation is a challenging task. But it can be essential for
clinical decision-making. Research has shown that it should be tackled in
curved shape spaces, as (e.g., disease-related) shape changes frequently
expose nonlinear characteristics. We thus propose a novel prediction method
that encodes the whole shape in a Riemannian shape space. It then learns a
simple prediction technique founded on hierarchical statistical modeling of
longitudinal training data. When applied to predict the future development of
the shape of the right hippocampus under Alzheimer’s disease and to human body
motion, it outperforms deep learning-supported variants as well as state-of-
the-art.

###### Keywords:

Shape development Prediction Regression Riemannian manifold.

##  1 Introduction

Shapes of anatomical structures are of considerable medical interest, and they
are encountered particularly often in the analysis of medical images. Studies
have shown that they should be modeled as elements of curved manifolds—shape
spaces—instead of ordinary Euclidean space [18]. Thus, it is imperative to
develop methods for such spaces when working on problems involving anatomical
shapes.

A particularly interesting and relevant task is the prediction of the future
development of a shape—“How will an anatomical structure look like after a
certain amount of time has passed?” This question is of great interest as
shapes of anatomical structures are often correlated with (states of)
diseases; see, e.g., [18, 6, 16]. Predicting how a shape will develop in the
future could thus play a significant role in diagnosis and prevention as well
as aid physicians in their choice of treatment.

One important example is the relation between the shape of the hippocampus and
Alzheimer’s disease, as previous studies have shown that the former can be
used to discriminate between Alzheimer’s and normal aging [11]. Thus, if the
longitudinal development of the hippocampus’ shape can be predicted, a better
prognosis can be achieved; and with Alzheimer’s improved early diagnosis can
have a serious positive impact [19].

To uncover developmental trends in populations, longitudinal studies that
involve repeated observations of individuals play an essential role. The
variability in such data can be distinguished as cross-sectional (i.e.,
between individuals) and longitudinal (i.e., within a single individual over
time). The latter is highly correlated, violating the independence assumption
of standard statistical tools like mean-variance analysis and regression, thus
requiring inferential approaches that can disentangle cross-sectional and
longitudinal effects. For shape-valued data, another challenge that warrants
attention is that curved spaces lack a global system of coordinates. Assessing
differences in longitudinal trends requires a notion of transport between
tangent spaces to spatially align subject-specific trajectories. For
manifolds, parallel transport has been shown to provide highly consistent
transports [14, 16] with improved sensitivity over other methods.
Specifically, in [5] the adequateness of geodesic subject-wise models for the
progression of subcortical brain structures in Alzheimer’s disease and the
potential for prognosis via parallel transport of individual trends has been
shown.

To assess longitudinal and cross-sectional shape variation jointly,
hierarchical statistical models pose an adequate and very flexible framework
[12]. In recent years, various generalizations of such models to manifold-
valued data have been proposed based on probabilistic [3] and least-squares
theoretic [15, 17, 13] formulations. These approaches account for the inherent
interrelations by describing each subject with its own parametric
spatiotemporal model—most prominently geodesics. Additionally, the subject-
specific trends are assumed to be perturbations of a population-average trend,
which is referred to as the “fixed” effect and is often of primary interest.

Population-level analysis apart, a few studies have focused on shape
development prediction for individuals. A deep learning pipeline for
predicting longitudinal bone shape changes in the femur to diagnose knee
osteoarthritis was introduced in [6]. This approach utilizes a spherical
encoding to map a 3D point cloud of the bone into a 2D image. It thereby
relies on the assumption that the femora are (approximately) star-shaped.
While the proposed pipeline produces accurate predictions of future bone
shape, anatomies in general (e.g., hippocampi) are not star-shaped prohibiting
a spherical encoding. A further limitation of the method is that it needs 3
separate observations for its prediction—a requirement that strongly hinders
early diagnosis.

A varifold-based learning approach for predicting infant cortical surface
development has been proposed in [20]. The method uses regression on varifold
representations [9] to learn typical shape changes and uses the latter to
predict from a single baseline. Although the method partly uses the curved
space of diffeomorphisms, it only employs it for pre- and post-processing. It
further requires the user to set several data-dependent parameters as optimal
as possible (i.e., two data-dependent kernel sizes, a weighting between terms
in the loss of the regression, the concentration of time points, and the
number of neighbors to be considered in a nearest neighbor search), leading to
a relatively high entry-barrier for users.

In this work, we propose a novel method based on hierarchical statistical
modeling to predict shape evolution. In contrast to previous geodesic
approaches [5] that relied on hand-picked reference individuals, we provide a
data-driven approach for learning shape progression. Representing subject-wise
trends as geodesics in shape space allows us to learn from longitudinal
observations while respecting within-subject correlations. After training on
whole trajectories, the prediction requires only a single shape making it
applicable when only the baseline observation of the longitudinal development
is given. Conceptually, our prediction is very simple as it only consists of a
parallel translation of the initial velocity of the mean trend and a
subsequent evaluation of a geodesic. It thus provides a high degree of
interpretability. Furthermore, no data-dependent parameters need to be set,
which makes the method very user-friendly.

To validate our approach, we apply it to two real-world problems. We use it to
predict the development of the shape of the right hippocampus under
Alzheimer’s. To the best of our knowledge, this is the first approach that
applies longitudinal statistical modeling to the prediction of shape
developments from a single baseline. We also use our method to predict human
body motion as an example of an application in which large shape changes
occur. In both applications, we outperform state-of-the-art by a considerable
margin. Our method even slightly outperforms deep-learning-enriched variants.
Upon acceptance of the paper, we will make our source code public in order to
make our method easily accessible.

##  2 Method

Figure 1: Depiction of the shape prediction method in a shape space $M$.

We first recall the necessary basics from Riemannian geometry and geometric
statistics; a good reference on them is [18].

A Riemannian manifold is a differentiable manifold $M$ together with a
Riemannian metric $\langle\cdot,\cdot\rangle_{p}$ that assigns a
smoothly111Whenever we say “smooth” we mean “infinitely often differentiable”.
varying scalar product to every tangent space $T_{p}M$. The metric also yields
a (geodesic) distance function $d$ on $M$. A further central object is the
Levi-Civita connection $\nabla$ of $M$. Given two vector fields $X,Y$ on $M$,
it is used to differentiate $Y$ along $X$; the result is again a vector field,
which we denote by $\nabla_{X}Y$. With a connection one can define a geodesic
$\gamma$ as a curve without acceleration, i.e.,
$\nabla_{\gamma^{\prime}}\gamma^{\prime}=0$, where
$\gamma^{\prime}:=\frac{\scriptsize{\textnormal{d}}}{\scriptsize{\textnormal{d}}t}\gamma$.
It is a fundamental fact that each element of $M$ has a so-called normal
convex neighborhood $U$ in which any two points $p,q\in U$ can be joined by a
unique length-minimizing geodesic $[0,1]\ni t\mapsto\gamma(t;p,q)$ that does
not leave $U$. Since it is the solution of a second-order differential
equation, $p$, and $\gamma^{\prime}(0)$ determine $\gamma$ completely. Indeed,
for every $p\in U$ there are the Riemannian exponential $\exp_{p}:T_{p}M\to U$
and logarithm $\log_{p}:=\exp^{-1}:U\to T_{p}M$ with $\exp(v):=q$ such that
$\gamma^{\prime}(0;p,q)=v$ and $\log_{p}(q)=\gamma^{\prime}(0;p,q)$.

A further important fact is that in a Riemannian manifold, one usually cannot
identify tangent spaces with each other. Therefore, tangent vectors must be
transported explicitly along curves between points—the so-called parallel
transport. This process depends on the chosen path; however, in $U$, we can
always transport along the geodesic that connects the origin and destination.
Therefore, whenever we speak of parallel-translating a vector $v$ from some
$p\in U$ to $q\in U$ transport along the geodesic from $p$ to $q$ is meant;
the resulting vector (which is in $T_{q}M$) is denoted by $\Pi_{p}^{q}(v)$.

Whenever we work with data, means are of interest. Given $q_{1},\dots,q_{n}\in
M$, their Fréchet mean is the minimizer of the Fréchet variance
$F(p):=\sum_{i=1}^{n}d(p,q_{i})^{2}$.

Interestingly, the set of all geodesics $G(U):=\\{\gamma:[0,1]\to U\ |\
\gamma\textnormal{ geodesic}\\}$ in $U$ can also be given the structure of a
Riemannian manifold imposing a functional-based Riemannian metric [17]. As a
consequence, we can compute Fréchet means of geodesics in $G(U)$ if they are
sufficiently localized; we assume the latter throughout this work.

We are now ready to introduce our model for shape prediction, which we will
use for hippocampi. For this let $M$ be any shape space that is a Riemannian
manifold and $U\subseteq M$ a normal convex neighborhood consisting of the
shapes of interest to us. Given a shape $p^{*}\in U$ observed at time $t_{0}$
our goal is to predict its future form $q^{*}\in U$ at time point $t_{1}$. For
simplicity of exposition, we assume in the following that $t_{0}=0$ and
$t_{1}=1$. This can always be achieved when the times are viewed relative to
an interval that contains them.

Our fundamental assumption is that $p^{*}$ develops along a geodesic through
$U$, i.e., there is $\gamma_{p^{*}}\in G(U)$ such that the longitudinal
development of $p^{*}$ at time $t$ is given by $\gamma_{p^{*}}(t)$. Several
works have shown that this is often an adequate choice when modeling shape
developments in the medical context [16, 18]. Then, since geodesics are
determined by their starting point and initial velocity, we need to find
$\gamma_{p^{*}}^{\prime}(0)$; because then

$q^{*}=\exp_{p^{*}}(\gamma_{p^{*}}^{\prime}(0)).$ (1)

In other words, to predict the development of any shape $p^{*}\in U$ that is
of interest to us we need to approximate a vector field on $U$ that encodes
the direction and speed of change that $p^{*}$ undergoes until time 1.

In the following, we propose an approach that infers this vector field from
data. Assume $N$ shapes similar to $p^{*}$ (i.e., close to $p^{*}$ in $U$),
which are expected to show analogous progression and are observed at a
possibly varying number of time points, hence, yielding (training) data
$(t^{(j)}_{i},q_{i}^{(j)})\in[0,1]\times U$, for $i=1,\dots,n_{j}$ and
$j=1,\dots,N.$ Now, using geodesic regression [10] we can approximate the
individual trajectories $\gamma^{(1)},\dots,\gamma^{(N)}\in G(U)$ of the
training shapes. Utilizing the manifold structure of $G(U)$ from [13], the
(Fréchet) mean geodesic $\overline{\gamma}$ of
$\gamma^{(1)},\dots,\gamma^{(N)}$ can be computed. Note that
$\overline{\gamma}(0)$ and $\overline{v}:=\overline{\gamma}^{\prime}(0)\in
T_{\overline{\gamma}(0)}M$ can be interpreted as the mean starting point and
the average initial velocity of the trajectories of the training data,
respectively. The geodesic $\overline{\gamma}$ is further the central fixed
effect that describes the data in a geodesic hierarchical model [15, 17].

The fact that the training shapes are close to $p^{*}$ suggests that the
parallel transport of $\overline{v}$ to $p^{*}$ is a good approximation of our
target $\gamma_{p^{*}}^{\prime}(0)$. We thus propose to use the approximation
$\gamma_{p^{*}}^{\prime}(0)\approx\Pi^{p^{*}}_{\overline{\gamma}(0)}(\overline{v})$
in (1). Being a comparatively simple approach, our experiments show that it
can be a very good choice. The processing pipeline is shown in Fig. 1.

##  3 Experiments

###  3.1 Data and Methodology

Datasets: To evaluate our model we applied it to two data sets. The first was
shape data of right hippocampi derived from 3D label fields provided by the
Alzheimer’s Disease Neuroimaging Initiative222adni.loni.usc.edu (ADNI). The
ADNI database contains, amongst others, 1632 brain MRI scans with segmented
hippocampi. From them, we assembled three distinct groups: subjects with
Alzheimer’s (AD), Mild Cognitive Impairment (MCI), and cognitive normal (CN)
controls. The groups contained data from 86, 201, and 116 subjects,
respectively; for each subject, there were three MR images taken
(approximately) six months apart. Correspondence of the surfaces (2280
vertices, 4556 triangles) was established in a fully automatic manner by
registering extracted isosurfaces using the functional map–based approach of
[8]. As the final preprocessing step, all meshes were aligned using
generalized Procrustes analysis.

The second data set was taken from Dynamic FAUST [2], which is publicly
available and contains the motion data of 10 subjects. The data is given as
triangle meshes in correspondence (subject-wise and between subjects). For
each subject, we used three meshes (6890 vertices, 13776 triangles) that
constitute a raising of the left leg from the initial part of the “one leg
loose” pattern.

Shape Space: We used the differential coordinate model (DCM) shape space from
[21]. The DCM space works with triangular mesh representations and allows for
explicit and fast computations.

Comparison Methods:

Figure 2: GAN-enhanced prediction method.

For the ADNI experiment, we used the following comparison methods. The first
was the varifold-based (Varifold) method from [20]. To test whether
incorporating a deep neural network improves the prediction, we also tested
the following variation of our proposed method on the ADNI data: We tried both
a generative adversarial network (GAN) and a cyclic GAN (cGAN) to learn a
correction $w_{p^{*}}\in T_{\overline{\gamma}(0)}M$ of $\overline{v}$. The
idea was that shapes from different regions in $U$ might show systematic
differences in their development. Both the GAN and cGAN are designed to map
the coordinate vector333We denote the coordinate representation in
$\mathbb{R}^{d}$ of a tangent vector $v$ w.r.t. a fixed but arbitrary basis by
$[v]$. of the cross-sectional difference
$[\log_{\overline{\gamma}(0)}(p^{*})]\in\mathbb{R}^{d}$ to the correction
vector $[w_{p^{*}}]\in\mathbb{R}^{d}$. They were trained on the baselines
$\\{q_{1}^{(j)}\ |\ j=1,\dots,N\\}$ using three-fold cross validation. We then
used
$\gamma_{p^{*}}^{\prime}(0)\approx\Pi^{p^{*}}_{\overline{\gamma}(0)}(\overline{v}+w_{p^{*}})$
in (1), i.e., the prediction then became

$q^{*}_{\textnormal{GAN}}:=\exp_{p^{*}}(\Pi^{p^{*}}_{\overline{\gamma}(0)}(\overline{v}+w_{p^{*}})).$

The method using the GAN is illustrated in Fig. 2.

The generators in the GAN and cGAN networks consisted of four linear layers,
with dropout layers and ReLU activation functions between them; the
discriminators were composed of two linear layers without dropout. Both the
GAN and cGAN used the sum of the binary cross entropy and the difference
$\|\Pi_{\gamma^{(j)}(0)}^{\overline{\gamma}(0)}(v^{(j)})-\overline{v}-w_{p^{*}}\|_{\overline{\gamma}(0)}$
(with the norm that is induced by the Riemannian metric) as loss; for the cGAN
the standard forward cycle consistency loss was also added. When referencing
results obtained with the GAN and cGAN while using the DCM space as shape
space, we use the notations (DCM+GAN) and (DCM+cGAN), respectively.

To evaluate how important the DCM space is to the prediction, we replaced it
with the flat point distribution model (PDM) space from [7].

Finally, to further differentiate the behavior of our proposed and the
varifold-based approach, we tested how well the regressed geodesics in the DCM
space and the space of diffeomorphisms approximate the data. We report for all
groups the averages of the mean vertex-wise error (MVE) between the mesh
belonging to $\gamma^{(j)}(1)$ and the corresponding data mesh of
$q^{(j)}_{2}$ after rigid alignment.

In the experiment with the dynamic FAUST data, we compared our method against
the varifold-based method. We did not use these deep-learning refinements for
the FAUST data since only ten subjects were available. (Note that an advantage
of our approach is also that it can handle such small datasets.)

Software: The computations in the DCM and PDM space were performed in
Morphomatics v1.1 [1]. The GAN and cGAN were implemented in PyTorch. During
training, we used ADAM optimizer for both the generators and the
discriminators. All computations involving varifolds were performed in
Deformetrica 4.3.0rc0 [4], where we used the “landmarks” option as attachment
type for a fair comparison since all meshes were in correspondence. In the
ADNI experiment, we used the L-BFGS algorithm in Deformetrica, and gradient
ascent or the Faust data as L-BFGS had stability problems there.

Parameter Settings: In the ADNI experiment (where measurements are in
millimeters), we used Varifold with a smoothing kernel width of $2.5$, a
deformation kernel width of $5$, $t_{0}=0.5$, a concentration of time points
of 1, and the number of points used from the cloud was set to $25$. The GAN
and cGAN used a dropout probability of 0.2; the learning rates of both the
generators and discriminators were set to 0.0001. For each fold, we trained
the network with 400 epochs. In the FAUST experiment (where measurements are
in meters), we used Varifold with a smoothing kernel width of $0.1$, a
deformation kernel width of $0.1$, $t_{0}=0.5$, a concentration of time points
of 1; the number of points used from the cloud was also set to $25$.

All parameters were determined through exploration. All results were obtained
through three-fold cross-validation.

Evaluation Measures: To assess the effectiveness of our model, we employed the
MVE by calculating how much (after rigid alignment) each vertex deviates from
its corresponding ground truth vertex. Moreover, to compare the DCM and PDM
spaces, we contrasted the MVE (which can be viewed as the intrinsic distance
of PDM space) with the geodesic distance $d$ (GD) of DCM space.

Table 1: Comparison of (variants of) our and the varifold-based prediction method w.r.t. average mean vertex-wise error (MVE) and average geodesic distance (GD).  Metric | Group | DCM | PDM | DCM+GAN | DCM+cGAN | Varifold
---|---|---|---|---|---|---
MVE | AD | $\mathbf{0.70\pm 0.07}$ | $0.71\pm 0.02$ | $0.73\pm 0.02$ | $0.75\pm 0.02$ | $1.23\pm 0.14$
MCI | $\mathbf{0.64}\pm 0.05$ | $\mathbf{0.64\pm 0.03}$ | $0.67\pm 0.03$ | $0.75\pm 0.04$ | $1.27\pm 0.02$
CN | $\mathbf{0.71\pm 0.01}$ | $0.72\pm 0.08$ | $0.74\pm 0.07$ | $0.80\pm 0.05$ | $1.33\pm 0.06$
GD | AD | $\mathbf{15.51\pm 0.88}$ | $37.50\pm 0.79$ | $16.20\pm 0.60$ | $16.42\pm 0.47$ | $17.63\pm 1.97$
MCI | $\mathbf{15.23\pm 0.87}$ | $34.55\pm 1.81$ | $15.91\pm 0.89$ | $16.22\pm 0.50$ | $18.36\pm 0.25$
CN | $\mathbf{16.44\pm 0.22}$ | $38.39\pm 4.44$ | $17.35\pm 1.66$ | $18.35\pm 1.27$ | $19.07\pm 1.13$

###  3.2 Results

Table 2: Regression fidelity in terms of MVE. Group | AD | MCI | CN
---|---|---|---
DCM | $\mathbf{0.16\pm 0.06}$ | $\mathbf{0.17\pm 0.16}$ | $\mathbf{0.18\pm 0.18}$
Varifold | $0.68\pm 0.58$ | $0.62\pm 0.47$ | $0.73\pm 0.74$
Figure 3: Prediction comparison for a hippocampus from the AD group. Upper
left: base shape at $t=0$; upper right: the same hippocampus at $t=1$, i.e.,
after one year (the ground truth for our prediction); bottom left: varifold-
based prediction; bottom right: our prediction. The colors encode the vertex-
wise differences to the ground truth according to the color map (0 mm  2.25
mm).

The results of our prediction comparison for the hippocampi are shown in Table
1. Our proposed method outperformed the other methods in all categories. Even
though the proposed method is a relatively simple approach, it not only
performed better than GAN methods but was also faster and did not involve
hyperparameter tuning. Also, differently structured GANs did not improve the
results by a noticeable amount. Furthermore, we can see that using the DCM
space is superior to using the PDM space as its results are close w.r.t. MVE
(the intrinsic PDM distance) but significantly worse w.r.t. the GD of the DCM
space. Note that the magnitude of the prediction errors of our method is
probably close to the resolution of the scanner. However, the longitudinal
changes in shape over the course of one year are also only small.

In comparison to the varifold-based approach, the proposed method achieves
superior results. The difference in prediction is visualized in Fig. 3 using a
hippocampus from the AD group for which the MVE of our method was very close
($\approx 0.69$) to its mean MVE. The MVE of the varifold-based method was
slightly smaller ($\approx 1.03$) than its MVE.

A reason why our approach works better than the varifold-based one could be
the difference in approximation power of geodesic regression: Results for the
fitting quality of regression in DCM and the space of diffeomorphisms are
shown in Table 2. Clearly, the approximation is better when the DCM space is
used, thus, demonstrating an improved fidelity of DCM geodesics over
diffeomorphic representations.

Figure 4: Prediction of human body motion for FAUST dataset. From left to
right: Baseline, ground truth, and predicted shapes using our and the Varifold
method [20].

The results for the FAUST dataset reinforce the above findings. Our proposed
method (DCM) achieves an MVE of $0.03\pm 0.003$ and a GD of $0.54\pm 0.03$;
Varifold has an MVE of $0.07\pm 0.008$ and a GD of $1.16\pm 0.04$. Fig. 4
shows the predictions for subject 500021. While the result of our method is
close to the ground truth, the human is barely recognizable in the varifold-
based prediction. A reason is the high inter- and intra-subject variability in
the motion. Since point trajectories from different subjects are mixed to
produce the varifold prediction, relatively unlikely leg configurations are
obtained. Moreover, the formation of a male face in the varifold case
highlights the advantage of a differential encoding of shape changes (i.e.
tangent vectors) together with a consistent transport over combinations of
absolute configurations.

##  4 Conclusion

In this paper, we proposed a novel method for predicting shape development
based on hierarchical statistical modeling in Riemannian shape spaces. It
outperformed state-of-the-art in two experiments by a clear margin.
Furthermore, it performed better than deep learning–supported variants when
predicting the future development of the hippocampus shape. Our approach is
thus a good fit for shapes whose progression follows geodesics (such as
hippocampi) and who are well captured by population-average trends. As the
latter assumption will not always be valid, it is still a promising approach
to incorporate deep learning: Whenever the development depends strongly on the
individual characteristics of the baseline shape, we expect that deep learning
methods can be used to find adjustments to our prediction (direction) that
take the dependence into account. A path for future work is thus to test this
hypothesis on further anatomical structures.

####  4.0.1 Acknowledgements

This work was partially funded by grants from the European H2020 Marie
Sklodowska-Curie action (grant no. 101003403) and the Scientific and
Technological Research Council of Turkey under the TUBITAK 2232 Fellowship for
Outstanding Researchers (no. 118C288)444https://basira-lab.com/normnets/ &
https://basira-lab.com/reprime/. We are grateful for the funding by
DFG555Deutsche Forschungsgemeinschaft (DFG) through Germany’s Excellence
Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project
ID: 390685689) and BMBF666Bundesministerium für Bildung und Forschung (BMBF)
through BIFOLD - The Berlin Institute for the Foundations of Learning and Data
(ref. 01IS18025A and ref 01IS18037A).

Data collection and sharing for this project was funded by the
ADNI777adni.loni.usc.edu (National Institutes of Health Grant U01 AG024904)
and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is
funded by the National Institute on Aging, the National Institute of
Biomedical Imaging and Bioengineering, and through generous contributions from
the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery
Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb
Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli
Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated
company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen
Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson
Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co.,
Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies;
Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier;
Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian
Institutes of Health Research is providing funds to support ADNI clinical
sites in Canada. Private sector contributions are facilitated by the
Foundation for the National Institutes of Health (www.fnih.org). The grantee
organization is the Northern California Institute for Research and Education,
and the study is coordinated by the Alzheimer’s Therapeutic Research Institute
at the University of Southern California. ADNI data are disseminated by the
Laboratory for Neuro Imaging at the University of Southern California.

## References

  * [1] Ambellan, F., Hanik, M., von Tycowicz, C.: Morphomatics: Geometric morphometrics in non-Euclidean shape spaces (2021), https://morphomatics.github.io/
  * [2] Bogo, F., Romero, J., Pons-Moll, G., Black, M.J.: Dynamic FAUST: Registering human bodies in motion. In: IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) (Jul 2017)
  * [3] Bône, A., Colliot, O., Durrleman, S.: Learning the spatiotemporal variability in longitudinal shape data sets. Int. J. Comput. Vis. 128(12), 2873–2896 (2020)
  * [4] Bône, A., Louis, M., Martin, B., Durrleman, S.: Deformetrica 4: an open-source software for statistical shape analysis. In: International Workshop on Shape in Medical Imaging. pp. 3–13. Springer (2018)
  * [5] Bône, A., Louis, M., Routier, A., Samper, J., Bacci, M., Charlier, B., Colliot, O., Durrleman, S., Initiative, A.D.N.: Prediction of the progression of subcortical brain structures in Alzheimer’s disease from baseline. In: Graphs in Biomedical Image Analysis, Computational Anatomy and Imaging Genetics: First International Workshop, GRAIL 2017, 6th International Workshop, MFCA 2017, and Third International Workshop, MICGen 2017, Held in Conjunction with MICCAI 2017, Québec City, QC, Canada, September 10–14, 2017, Proceedings 1. pp. 101–113. Springer (2017). https://doi.org/10.1007/978-3-319-67675-3_10
  * [6] Calivá, F., Kamat, S., et al., A.M.: Surface spherical encoding and contrastive learning for virtual bone shape aging. Med. Image Anal. 77, 102388 (2022). https://doi.org/10.1016/j.media.2022.102388
  * [7] Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models-their training and application. Comput. Vis. Image Underst. 61(1), 38–59 (1995). https://doi.org/10.1006/cviu.1995.1004
  * [8] Ezuz, D., Ben-Chen, M.: Deblurring and denoising of maps between shapes. Comput. Graph. Forum 36(5), 165–174 (2017). https://doi.org/10.1111/cgf.13254
  * [9] Fishbaugh, J., Durrleman, S., Prastawa, M., Gerig, G.: Geodesic shape regression with multiple geometries and sparse parameters. Med. Image Anal. 39, 1–17 (2017)
  * [10] Fletcher, P.T.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2013). https://doi.org/10.1007/s11263-012-0591-y
  * [11] Gerardin, E., Chételat, G., et al., M.C.: Multidimensional classification of hippocampal shape features discriminates Alzheimer’s disease and mild cognitive impairment from normal aging. NeuroImage 47(4), 1476–1486 (2009). https://doi.org/10.1016/j.neuroimage.2009.05.036
  * [12] Gerig, G., Fishbaugh, J., Sadeghi, N.: Longitudinal modeling of appearance and shape and its potential for clinical use. Med. Image Anal. 33, 114–121 (2016). https://doi.org/10.1016/j.media.2016.06.014
  * [13] Hanik, M., Hege, H.C., von Tycowicz, C.: A nonlinear hierarchical model for longitudinal data on manifolds. In: 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2022)
  * [14] Lorenzi, M., Pennec, X.: Geodesics, parallel transport & one-parameter subgroups for diffeomorphic image registration. Int. J. Comput. Vis. 105(2), 111–127 (2013)
  * [15] Muralidharan, P., Fletcher, P.T.: Sasaki metrics for analysis of longitudinal data on manifolds. In: Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition. pp. 1027–1034. IEEE (2012). https://doi.org/10.1109/CVPR.2012.6247780
  * [16] Nava-Yazdani, E., Hege, H.C., Sullivan, T.J., von Tycowicz, C.: Geodesic analysis in Kendall’s shape space with epidemiological applications. J. Math. Imaging Vis. 60, 549–559 (2020). https://doi.org/10.1007/s10851-020-00945-w
  * [17] Nava-Yazdani, E., Hege, H.C., von Tycowicz, C.: A hierarchical geodesic model for longitudinal analysis on manifolds. J. Math. Imaging Vis. 64(4), 395–407 (2022)
  * [18] Pennec, X., Sommer, S., Fletcher, P.T. (eds.): Riemannian Geometric Statistics in Medical Image Analysis. Academic Press, London (2020). https://doi.org/10.1016/C2017-0-01561-6
  * [19] Rasmussen, J., Langerman, H.: Alzheimer’s disease - why we need early diagnosis. Degenerative Neurological and Neuromuscular Disease 9, 123–130 (2019)
  * [20] Rekik, I., Li, G., Lin, W., Shen, D.: Predicting infant cortical surface development using a 4d varifold-based learning framework and local topography-based shape morphing. Med. Image Anal. 28, 1–12 (2016)
  * [21] von Tycowicz, C., Ambellan, F., Mukhopadhyay, A., Zachow, S.: An efficient Riemannian statistical shape model using differential coordinates. Med. Image Anal. 43, 1–9 (2018). https://doi.org/10.1016/j.media.2017.09.004