# Efficient Multiparty Quantum Key Distribution over Quantum Networks

Samuel Oslovich1, Bing Wang1, Walter O. Krawec1, Kenneth Goodenough2  1School
of Computing, University of Connecticut, Storrs, CT, USA  2Manning College of
Information & Computer Sciences, University of Massachusetts, Amherst, MA, USA

###### Abstract

Multiparty quantum key distribution (QKD) is useful for many applications that
involve secure communication or collaboration among multiple parties. While it
can be achieved using pairwise QKD, a more efficient approach is to achieve it
using multipartite entanglement distributed over quantum networks that connect
the multiple parties. Existing studies on multipartite entanglement
distribution, however, are not designed for multiparty QKD, and hence do not
aim to maximize secret key generation rate. In this paper, we design efficient
strategies for multiparty QKD over quantum networks. For 3-party QKD, we
derive closed-form expressions for analyzing key distribution over quantum
networks. We then use it to develop an efficient strategy for 3-party QKD by
packing multiple stars that connect the 3 parties. For the general form of
$N$-party QKD, we develop an approach that packs multiple trees to connect the
$N$ parties, while directly incorporating the estimated key rates on network
paths. Extensive evaluation of our strategies, in both grid and random graphs,
under a wide range of settings, demonstrates that our schemes achieve high key
rate, which degrades gracefully when increasing the number of parties.

##  I Introduction

Quantum key distribution (QKD) leverages the principles of quantum mechanics
to allow two parties, Alice and Bob, to establish a secret key that is
information theoretically secure [1]. Multiparty QKD, also called conference
key agreement, extends pairwise QKD to allow multiple parties to establish a
common information-theoretic secret key. It is useful for many applications,
including secure multiparty communication, distributed cryptographic
applications, and quantum secure multi-party computation [2]. When the parties
are at geographically distributed locations, long-distance quantum networks,
as envisioned in the “quantum internet” [3, 4, 5], provide a mechanism to
connect the multiple parties using intermediate quantum repeaters.
Specifically, the usage of quantum repeaters addresses the issue of the
exponential decay in the rate as a function of distance, allowing higher
entanglement generation rates, and hence higher key generation rate. While
multiparty QKD can be achieved through pairwise QKD, which first establishes
pairwise secret keys and then uses secure communication to establish a common
key among all the parties, this process is inefficient [6]. A more efficient
alternative is distributing multipartite entanglement directly to the multiple
parties over a quantum network, which allows the parties to leverage the
multipartite entangled state, to achieve a higher key rate.

Existing experimental studies [7, 8] have demonstrated the feasibility of
multiparty QKD using multipartite entanglement distribution over small-scale
quantum networks. Efficient multiparty QKD over large-scale quantum networks,
however, faces many challenges. These challenges are due to limitations of
near-term technologies, such as lossy and noisy quantum channels, limited
quantum memory with short lifetime, and probabilistic entanglement swapping
operations. To improve the key rate of multiparty QKD, an important building
block is efficient multipartite entanglement distribution over quantum
networks. While this problem has been explored in recent studies [9, 10, 11,
12], their designs do not target multiparty QKD. Specifically, the studies in
[9, 10] aim to minimize the number of consumed entangled pairs under idealized
conditions, with no consideration of loss and noise. The study in [11]
considers a more realistic noisy quantum network setting, and develops a
multi-objective routing algorithm considering both entanglement generation
rate and fidelity. The study in [12] develops multipath routing techniques to
improve entanglement throughput, with no consideration of quantum link
fidelity, which is important for quantum network applications, especially QKD
where key rate is affected heavily by fidelity.

In this paper, we develop efficient multipartite entanglement distribution
strategies with the aim of maximizing key rate for multiparty QKD. Our
approach incorporates the various constraints in near-term quantum networks
and is applicable to general quantum network topologies. We start with the
simplest form of multiparty QKD, 3-party QKD. In this setting, we derive
closed-form expressions for estimating errors required for obtaining key rate.
We then use them to develop an efficient strategy for 3-party QKD by packing
multiple star topologies that connect the three parties. For the general form
of $N$-party QKD, we develop an approach that packs multiple tree topologies
to connect the $N$ parties, while directly incorporating the estimated key
rates on network paths into the design. While our approach also leverages
multipath routing as in [12], the multipartite entanglement distribution
strategy that we design differs significantly from that in [12] in that our
design explicitly considers fidelity and has the goal of maximizing key rate
for multiparty QKD.

We evaluate our proposed approach in both grid and random graphs under a wide
range of settings. The evaluation results show that our proposed approach
achieves high key rate, and degrades gracefully when increasing the number of
parties. Furthermore, it significantly outperforms several baselines.
Specifically, in the settings we explore, we find that our dynamic multi-tree
algorithm achieves up to $223\%$ higher key rate compared to a fixed multi-
tree algorithm.

The rest of the paper is organized as follows. In Section II, we present
background material on multiparty QKD and network model. In Sections III and
IV, we present our designs for 3-party and $N$-party QKD, respectively. In
Section V, we present our evaluation results. In Section VI, we briefly review
related work. Last, Section VII concludes the paper.

##  II Background and Network Model

In this section, we briefly overview the multipartite QKD protocol that is
used in this study, and present the network model.

QubitEntanglementMeasurementABCDE

(a) BSM at node $B$. Entanglement fusion at node $C$.

ABCDE

(b) Resulting 4-GHZ state shared by $A$, $C$, $D$, and $E$.

Figure 1: Illustration of BSM (entanglement swapping) and entanglement fusion.
(a) Node $B$ performs BSM on its two qubits, resulting a shared Bell state
between $A$ and $C$. Node $C$ performs entanglement fusion, resulting in a
shared 4-GHZ state between $A$, $C$, $D$, and $E$. (b) The resulting shared
4-GHZ state.

###  II-A $N$-party QKD Protocol

We use the $N$-BB84 protocol in [13], which is a $N$-partite variant of the
BB84 protocol [14]. This $N$-partite protocol considers $N$ parties, denoted
as Alice $A$, and a set of Bobs, $B_{1},\ldots,B_{N-1}$. The goal of the
protocol is to establish a common secret key among all $N$ parties,
$A,B_{1},\ldots,B_{N-1}$. In the rest of the paper, we refer to $A$ as the
leader, since it leads the error correction among the $N$ parties (see below).

Just like the standard BB84 protocol, this protocol contains two stages:
quantum information processing stage and classical information processing
stage. In the quantum information processing stage, the protocol starts with
distributing multipartite entangled $N$-GHZ states,
$\ket{N\text{-GHZ}}=\frac{1}{\sqrt{2}}(\ket{0}^{\otimes N}+\ket{1}^{\otimes
N})$ over the quantum channel to the $N$ parties. All parties perform local
measurements, either in $Z$ or $X$ basis, based on a preshared key, on their
respective quantum systems. The basis choice is biased so that the $X$ basis
is chosen with probability $p<1/2$. The basis choice will therefore consume
$MH(p)$ bits of pre-shared key every time the protocol is run, where $M$ is
the number of rounds and $H(p)=-p\log p-(1-p)\log(1-p)$ is the binary entropy
function. This key is then refreshed after each protocol run and, so long as
the noise is low enough, there will be more secret key bits produced then
consumed [13]. In our evaluations, we consider only the asymptotic key-rate,
in which case we may simply take $p$ to be arbitrarily close to zero, and, so
$H(p)$ is also arbitrarily close to zero.

In the classical information processing stage, the parties reveal a random
sample of the collected data over the authenticated classical channel for
parameter estimation. Specifically, this process estimates $Q_{A,B_{i}}$,
i.e., the $Z$ basis quantum bit error rate (QBER) between $A$ and $B_{i}$,
$i=1,\ldots,N-1$, as well as $Q_{X}$, i.e., the $X$ basis QBER across all $N$
parties. At this point, the raw keys held by the $N$ parties are partially
correlated and partially secret. In order to correct the errors in the raw
keys, $A$ performs pairwise error correction with each $B_{i}$ to create
shared raw keys among the $N$ parties. At last, through privacy amplification,
the shared raw key is turned into shared secret key for the $N$ parties. The
length of the secret key depends on the error rates (obtained using parameter
estimation) and the desired level of security.

Let $r$ be the key rate, i.e., the length of the secret key divided by the
number of $N$-GHZ states distributed among the $N$ parties. The study in [13]
presents a finite-key expression for $r$, which converges to the following
asymptotically

$r=1-H(Q_{X})-\max_{i}H(Q_{A,Bi})\ ,$ (1)

where $H(\cdot)$ is the binary entropy function. In this paper, we focus on
asymptotic results, and use the above asymptotic key rate above. Later, we
will also be interested in the _effective key rate_ which is defined to be the
number of secret key bits divided by the total timesteps. This can be derived
from the above, by multiplying with a suitable scaling factor based on the
number of GHZ states per time step.

###  II-B Network Model

The above $N$-party BB84 protocol requires $N$-GHZ states to be distributed to
the $N$ parties before running the protocol. We assume the $N$-GHZ states are
distributed using a quantum network that connects the $N$ parties. In the
following, we present the network model.

Consider a quantum network that contains a set of nodes and edges. Among them,
the $N$ nodes that will perform $N$-party QKD are referred to as end users or
terminal nodes, denoted as $T_{1},\ldots,T_{N}$, and the rest are quantum
repeaters. As in [15, 16, 17], we assume that the nodes have synchronized
clocks and the network operates in _rounds_. Each edge in this network
represents a fiber link for qubit transmission, which can be noisy and lossy.
We assume that all nodes have a quantum memory capable of storing a single
qubit for every adjacent node. In addition, this qubit can only be stored for
a single network round due to short quantum memory coherence time [18], after
which it must be discarded. The memory may also be noisy, causing the state to
decohere with a certain probability.

Each network round is divided into three _phases_ , for completing the quantum
information processing stage of $N$-party QKD over the quantum network. After
a sufficient number of rounds, classical post-processing is conducted among
all the end users to obtain secret key. We next briefly describe these three
phases and then classical post-processing.

Phase 1. This phase distributes link-level bipartite entanglement over each
link in the network. Specifically, each pair of connected nodes (quantum
repeaters), $u$ and $v$, attempts to share half of an entangled pair with each
other over the fiber link. This process succeeds with probability $p$, which
decays exponentially with distance [19, 20, 21]. In addition, due to noise in
the link, memory system, or both, even when entanglement is established
successfully, between $u$ and $v$, we assume a depolarizing channel with
parameter $\gamma_{u,v}$. That is, the entanglement may depolarize and become
a completely mixed state. Thus, at the end of this phase, with probability
$p$, each pair of neighbors $u,v$ share the state

$\rho_{u,v}=\gamma_{u,v}\ket{\Phi^{+}}\bra{\Phi^{+}}+(1-\gamma_{u,v})\frac{I}{4}\,,$
(2)

where $\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right)$ and
$I$ is the identity matrix. Otherwise, with probability $1-p$, those neighbors
share the vacuum state (i.e., they share no state). We assume nodes are able
to determine whether they have a vacuum or not, while they cannot determine if
the state is the desired Bell state $\ket{\Phi^{+}}$ or a completely mixed
state. Note that $p$ can vary for different links in a network due to
differences in quantum hardware.

Phase 2. In this stage, a routing algorithm is executed (see Sections III and
IV) to create $N$-party entanglement among the end users. Specifically, the
routing algorithm will determine a routing tree, with end users as leaf nodes
or interior nodes of the tree, and has quantum repeaters as the remaining
interior nodes. The interior nodes in the tree will perform $k$-GHZ projective
measurements if $k$ nodes are connected to it. If successful, this will create
a long-distance $k$-GHZ state among the $k$ nodes. In the special case where
an interior node is a user, it will perform an entanglement fusion operation
to retain entanglement with the resulting state. Entanglement fusion
operations are similar to $k$-GHZ projective measurements, but allows for the
measuring party to retain a qubit of the entanglement state [22]. Let $q$
denote the probability that the $k$-GHZ state among the $k$ nodes is created
successfully, $k\geq 2$. Note that $q$ can also vary for each node in the
network due to differences in quantum hardware. Henceforth, for simplicity, we
consider homogeneous $q$; our approaches in §III and §IV can be used for both
homogeneous and heterogeneous $q$.

A branch in the tree is a network path. For a path
$P=(u_{1},u_{2},\ldots,u_{\ell})$ with $\ell$ repeaters, each repeater,
$u_{i}$, will perform a Bell state measurement (BSM) or entanglement swapping
on its respective half of the shared state. Since each link can only be used
once, these network paths cannot have shared links. Then with probability
$q^{\ell}$, nodes $u_{1}$ and $u_{\ell}$ share a state of the form:

$\rho_{u_{1},u_{\ell}}=\gamma_{P}\ket{\Phi^{+}}\bra{\Phi^{+}}+\left[1-\gamma_{P}\right]\frac{I}{4}\,,$
(3)

where $\gamma_{P}=\Pi_{i=1}^{\ell-1}\gamma_{u_{i},u_{i+1}}$, and
$\gamma_{u_{i},u_{i+1}}$ is the depolarizing parameter for link
$(u_{i},u_{i+1})$. An example that illustrates the above BSM and entanglement
fusion is shown in Fig. 1.

Phase 3. In this phase, with entanglement shared among the $N$ users, each
user conducts the quantum portion of the $N$-QKD protocol (see §II-A).
Specifically, each party measures its qubit following the preshared classical
key sequence. At the end of this phase, all remaining qubits in the system are
discarded, as we assume quantum memories can only store qubits for a single
network round. The network then repeats the above three phases for the next
round.

Post-processing. When the network has been running for a sufficient number of
rounds, post-processing is executed. Here, error correction and privacy
amplification are run among all end users, generating secret key material. As
we are interested only in the asymptotic performance in this paper, we assume
perfect error correction and use asymptotic key rates for the $N$-party QKD
protocol (see §II-A). Our main performance metric is the final secret key
generation rate (key rate) among end users, namely Eq. (1).

##  III 3-Party QKD

In this section, we consider the simplest case of $N$-party QKD when $N=3$. We
first present 3-GHZ state distribution and then a multipath routing algorithm
to maximize the key rate among the 3 parties.

###  III-A 3-GHZ State Distribution

$\gamma_{A}$$\gamma_{B_{1}}$$\gamma_{B_{2}}$$A$$B_{1}$$B_{2}$$u$

(a)

$\gamma_{A}$$\gamma_{B_{1}}$$A$$B_{1}$$B_{2}$

(b)

Figure 2: 3-GHZ state distribution.

After Phase 1 of link-level bipartite entanglement distribution, the goal of
Phase 2 is to establish a 3-GHZ state among the three terminal nodes. In
general, this can be achieved as shown in Fig. 2a. Specifically, a center
node, $u$, has a path with each of the three terminal nodes, and on each path,
link-level bipartite entanglement has been established successfully. Then for
each path between a terminal node and $u$, the quantum repeaters along the
path performs BSM to establish an end-to-end entanglement between the terminal
node and $u$. After that, $u$ performs a 3-GHZ projective measurement so as to
create a 3-GHZ state among the three terminal nodes.

A special case of the above is that the center node $u$ overlaps with a
terminal node, forming the graph in Fig. 2b. In this case, the overlapping
terminal node will perform an entanglement fusion measurement to create a
3-GHZ state among the three terminal nodes.

Consider the setting in Fig. 2a. Denote the path from a terminal node $T$ to
the center $u$ as $P=(T,v_{1},\ldots,v_{k},u)$, where $v_{i}$ is a quantum
repeater. Then $T$ and $u$ share a state as in Eq. (3). Let $\gamma_{A}$ be
the parameter for the path between user $A$ and $u$. Similarly, let
$\gamma_{B_{i}}$ denote the parameters for the path between user $B_{i}$ and
$u$, $i=1,2$. In the special case in Fig. 2b, we have $\gamma_{T}=1$.

Estimating error rates. To estimate the key rate in Eq. (1), we need to
estimate the $Z$ basis error rate $Q_{A,B_{i}}$ and $X$ basis error rate
$Q_{X}$. For the 3-party case, we derive them as

$Q_{A,B_{i}}=\frac{1-\gamma_{A}\gamma_{B_{i}}}{2}\ ,$ (4)
$Q_{X}=\frac{1-\gamma_{A}\prod_{i}\gamma_{B_{i}}}{2}\ .$ (5)

The derivation is based on the stabilizer formalism and is found in the
Appendix. The estimates of $Q_{A,B_{i}}$ and $Q_{X}$ will be used in routing
and leader selection in the next subsection.

###  III-B Routing for 3-party QKD

Given the above, the routing for 3-party QKD is to find the optimal center so
that the number of secret keys is maximized. This involves solving two
problems: (i) for a given center node, find the optimal leader, $A$, so that
the key rate is maximized, and (ii) find the optimal center node so that the
key rate for the optimal leader is maximized.

####  III-B1 Leader Decision

Given the paths to a center, we can compute the key rate for that center and
which terminal node should be the leader. The optimal leader for a given
center node is the leader that results in the highest key rate. To determine
the leader, we first consider each terminal node as the leader $A$, and
compute $\max_{i}H(Q_{A,B_{i}})$ for our choice of leader. Note that we do not
need to compute the entire key rate for each possible leader, since $Q_{X}$ is
independent of leader choice. To find the leader that yields the highest key
rate, we choose $A$ such that $\max_{i}H(Q_{A,B_{i}})$ is minimized. This
results in the maximum key rate for that center as

$r_{\max}=1-H(Q_{X})-\min_{A}\max_{i}H(Q_{A,B_{i}})$ (6)

####  III-B2 Expected key rate for a given center

We wish to choose the center that will yield the highest key rate. Using the
asymptotic key rate equation (6), we can determine how many keybits are
distributed per noisy 3-GHZ state. However since our link generation and
entanglement swapping operations are probabilistic, it is not guaranteed that
a noisy 3-GHZ state will be distributed. To account for the probabilistic
nature of our network, we obtain expected key rate per round. To accomplish
this we compute the probability that our noisy 3-GHZ state has been
distributed. Specifically, this probability depends solely on whether our
entanglement swapping operations were successful (since this is in Phase 2,
after link-level entanglement establishment in Phase 1). The number of
entanglement swapping operations performed is proportional to the lengths of
the paths to our center node. Consider the path from terminal $T_{i}$ (which
can be $A$, $B_{1}$ or $B_{2}$) to the center. Let the length of this path be
$\ell_{i}$. Establishing end-to-end entanglement along this path requires
$\ell_{i}-1$ entanglement swapping operations. This results in terminal
$T_{i}$ sharing entanglement with the center node with probability
$q^{\ell_{i}-1}$. We can repeat this for the set of terminals, $\\{T_{i}\\}$,
connected to the center, with an additional $q$ for the 3-GHZ measurement (or
entanglement fusion) at the center node. Summarizing the above, the expected
effective key rate for the given center is

$r_{t}=q\left[\prod_{i\in
T}q^{\ell_{i}-1}\right]\left[1-H(Q_{X})-\min_{A}\max_{i}H(Q_{A,B_{i}})\right]$
(7)

####  III-B3 Center Selection

We find the optimal center node using a method similar to the shortest-star
algorithm in [11]. Intuitively the paths from each terminal to the best center
will be the paths that yield the highest key rate. We will refer to such paths
as the _shortest paths_. Additionally the paths to our center node must be
disjoint due to entanglement being a resource that can only be used once. This
means that our paths to the best center will be the shortest possible disjoint
paths. Knowing this, we can consider each node as a center and if shortest
disjoint paths exist we can compute its key rate. This results in the
algorithm described below.

  * 1.

Each terminal finds all shortest paths to each other node in the network.

  * 2.

Each node in the network is considered as the center choice. If disjoint paths
to that center do not exist, the node is ignored. Otherwise the expected key
rate, $r_{t}$, of that center is computed (see Eq. (7)).

  * 3.

The node that yielded the highest key rate is chosen as the center of our
star.

(a)
(b)

Figure 3: (a) Running the star algorithm results in finding one star. (b)
Greedily packing stars by repeatedly running the star algorithm until no more
star is found yields two stars.

Running the above algorithm results in a _star_ with the three terminal nodes
at the tip of the star. In the special case where a terminal node is the
center of our _star_ , our graph is a path with the remaining two terminals
located at each end. One example is shown in Fig. 3a. Notably we can see that
remaining entanglement exists in the residual graph after finding our star in
Fig. 3a. If we then remove all edges used in our star, and re-run the star
algorithm we are able to find another star as seen in Fig. 3b. We repeat this
process of greedily packing stars until no more can be found. Henceforth, we
refer to the above algorithm as multi-star algorithm.

##  IV n-Party QKD

TerminalRepeaterNetwork LinkDistribution
Path$T_{1}$$T_{2}$$T_{3}$$T_{4}$$R_{1}$$R_{3}$$R_{6}$$R_{5}$$R_{4}$$R_{2}$

(a) Multipartite entanglement distribution via a center node (star).

$T_{1}$$T_{2}$$T_{3}$$T_{4}$$R_{1}$$R_{3}$$R_{6}$$R_{5}$$R_{4}$$R_{2}$

(b) Multipartite entanglement distribution via a tree.

Figure 4: Difference between star distribution and tree distribution
visualized. (a) 4-GHZ state distributed via a central node highlighted in
blue. The links that make up the paths to the center node are highlighted in
blue. (b) 4-GHZ distributed via a tree. The edges of the tree are colored in
red.

###  IV-A $N$-GHZ State Distribution

For general $N$-GHZ state distribution, similar to the 3-GHZ case described in
Section III, the goal after Phase 1 is to establish an $N$-GHZ state amongst
our $N$ terminal nodes. As shown in [9], one way to distribute an $N$-GHZ
state in a network setting is through a tree. Specifically, a tree may consist
of leaf nodes (degree of 1), path nodes (degree of 2), and branch nodes
(degree of $\geq$3). In Fig. 4b, we see that $T_{1},T_{2},$ and $T_{4}$ are
leaf nodes, $T_{3}$ is a path node, and $R_{1}$ is an branch node. If our tree
consists of a single branch node and no terminal path nodes, we have the same
case as the star described in Section III-B. This can be seen in Fig. 4a,
where $R_{3}$ is the only branch node and will perform a 4-GHZ projective
measurement. Otherwise, each path node will perform Bell state measurements
(entanglement swapping) to establish bipartite entanglement between branch
nodes and between leaf and branch nodes. Branch nodes will perform $k$-GHZ
projective measurements to create a $k$-GHZ state between the connected nodes.
Repeating this process will result in a $N$-GHZ state shared among the $N$
terminal nodes. Similar to the 3-party case, a special case occurs when an
branch or path node is also a terminal node. In this case, the terminal node
will perform an entanglement fusion operation resulting in a $(k+1)$-GHZ state
where the terminal node retains an entangled qubit. This can be seen in Fig.
4b, where $T_{3}$ is a path node, and $T_{3}$ will perform entanglement
fusion.

###  IV-B Routing for N-party QKD

(a) Initial snapshot of $G$. (b) Subgraph $G_{1}$. (c) Min. spanning tree of
$G_{1}$. (d) Outputted Steiner Tree

Figure 5: Illustration of Steiner tree approximation algorithm used for
routing for $N$-party QKD. (a) Given a snapshot of our graph $G$, first we
find the shortest path between each pair of terminals. (b) Next construct the
subgraph of $G$, where each edge is the shortest path between terminals. (c)
Find the minimum spanning tree of $G_{1}$. (d) Replace each edge in $G_{1}$ by
its corresponding shortest path.

While the routing problem for 3-party QKD is finding stars with terminal nodes
at the tips of the stars (see Section III-B), this may no longer be the best
strategy for the general $N$-party case. For example, if there are (say) 5
terminals in a grid topology, it is not even possible to connect the nodes via
a central node since each node has at maximum a degree of 4. Instead, we must
find the best tree to distribute entanglement, where terminal nodes may be
interior nodes (i.e., path or branch nodes) in our tree.

Classically finding the minimum cost tree that connects $N$ terminals is an
NP-Hard problem known as the Steiner Tree problem [23]. In the classical
setting, the cost of the tree is determined by the sum of the weights of the
paths that make up the tree. However, in our case the key rate also depends on
the branch nodes, as different $k$-GHZ projective measurements/entanglement
fusion operations (depending on the degree of the branch node) will result in
different noisy states. Determining how branch nodes affect the fidelity, and
ultimately the key rate of the final system is not trivial. Since we were
unable to determine an analytical expression, we modeled these GHZ projective
measurements and entanglement fusion operations using density matrices in our
simulation. From these density matrices we were able to obtain $Q_{A,B_{i}}$
and $Q_{X}$ for a given leader, $A$. Leader selection is the same as before,
i.e., we iterate through all choices of $A$ and select the best one (see
§III-B). As before, given a tree and the key rate for that tree we can
calculate the expected key rate in a network round. To determine the
probability that entanglement swapping is successful we count the number of
non-leaf nodes in our tree, $n$. We then compute $q^{n}$, the probability that
all swaps are successful. This yields an equation similar to Eq. (7). In the
special case where the tree is a star, we can use similar expressions to the
ones previously shown in Eq. (4) and (5).

The study in [12] presents multipath routing algorithms for distributing
$N$-GHZ states, utilizing a Steiner tree approximation algorithm to find the
best tree to distribute entanglement. We chose a similar approach and used the
same approximation algorithm [23]. However, there is a key difference between
our implementations of the approximation algorithm. The study in [12] assumes
the fidelity of all links to be perfect, and focuses on routing for optimizing
entanglement rate. As such, their approximation algorithm weights paths only
by the number of links that they contain. Our design weights edges and paths
proportional to their key rate, allowing us to find an approximate tree that
is better suited for QKD.

Our design based on the Steiner tree approximation algorithm[23] consists of
the following five main steps (see illustration in Fig. 5):

  * 1.

Find the shortest path between each pair of terminals. Construct the subgraph
$G_{1}$ where each vertex is a terminal, and each edge is the shortest path
between these terminals, where the weight of each edge in G1 corresponds to
the negative key rate of the associated shortest path.

  * 2.

Find the minimum spanning tree of $G_{1}$. Since our edges are weighted by the
negative key rate, the minimum spanning tree will connect terminals by paths
with maximal key rates.

  * 3.

Construct the subgraph $G_{S}$ where each edge of the minimum spanning tree is
replaced by the shortest path. If $G_{S}$ is a Steiner tree, then we are done
(i.e., we can skip steps 4 and 5). This is the case for the example
illustrated in Fig. 5.

  * 4.

Find the minimum spanning tree of $G_{S}$.

  * 5.

Delete edges in the minimum spanning tree until the only leaves are terminals.

(a)
(b)

Figure 6: Running the Steiner tree approximation algorithm results in finding
one tree in (a). Greedily packing trees by repeatedly running the
approximation algorithm until no trees are found yields two trees in (b).

After running our algorithm and finding a tree to distribute an $N$-GHZ state
there may still be residual entanglement. One example is shown in Fig. 6a. We
can once again repeatedly remove all edges forming our tree from the graph and
repeatedly run our routing algorithm until no more trees are found.
Henceforth, we refer to the above algorithm as multi-tree algorithm.

Figure 7: Key rate results for 3-party QKD in a random graph: multi-star
versus multi-tree algorithm.

The multi-tree algorithm for the general $N$-party setting can be used for
3-party case. However, since the multi-tree algorithm uses an approximation
algorithm to find trees, while the multi-star algorithm is specifically
designed for the 3-party case, the multi-star algorithm can significantly
outperform the multi-tree algorithm for the 3-party case. One example is in
Fig. 7, which shows the key rates obtained by the two algorithms in a random
graph for 3-party QKD (see more details on evaluation settings in §V-A). It
shows the results for two settings: $p=q=0.85$ and $p=q=0.95$, while varying
$\gamma$ parameter for a link from 0.97 to 1 (see Eq. (2)). We see a large gap
between the multi-star algorithm and the multi-tree algorithm. This is due to
the multi-star algorithm finding more stars, and stars with higher key rates.
For $\gamma=1$, the multi-star algorithm finds 0.3 more stars/trees per round
on average than the multi-tree algorithm. In our evaluation (§V), for ease of
exposition, we mark all the results using our approach as “multi-tree", which,
for 3-party QKD, are obtained using the “multi-star" algorithm due to its
significantly better performance than the general the multi-tree algorithm.

##  V Performance Evaluation

(a) b (Bet) layout. (b) d (Dalet) layout. (c) g (Giml) layout. Parties $D$,
$E$, and $F$ incrementally added for $N=4,5,6$.

Figure 8: Placement of terminal nodes in a $7\times 7$ grid.

In this section, we evaluate our proposed approaches for multi-party QKD using
extensive simulations. We first present the evaluation setup, and then the
results.

###  V-A Evaluation Setup

We consider two different choices of routing strategies. The first choice is
between fixed routing and dynamic routing. In fixed routing, the routing
algorithm is ran once prior to link-level entanglement being established. This
results in a fixed star/tree being chosen; and if a single link in the pre-
chosen path fails to generate entanglement, then entanglement distribution
will fail in that round. In dynamic routing, in each round, the routing
algorithm is ran on a snapshot of the graph, where link-level entanglement
success is assumed to be global knowledge. The second choice is between
single-tree vs. multi-tree routing. In single-tree routing, the routing
algorithm finds a single tree for distributing entanglement, while in multi-
tree routing, trees/stars are greedily packed as described in Sections III-B
and IV-B. In the rest of this section, for ease of exposition, we refer to
multi-star and multi-tree algorithms both as multi-tree; if it is for 3-party
QKD, then the results are obtained using the multi-star algorithm.

The fixed strategies are presented as a baseline to see how dynamic strategies
that operate based on global information of link-state success (after Phase 1)
can greatly improve the key rate. Additionally single-tree vs. multi-tree
routing is intended to highlight how utilizing as many resources as possible
can improve the key rate.

Both grid and random graph topologies are examined. The grid topologies were
chosen to gain insights on how varying different parameters effects the key
rate. Additionally a random graph topology was chosen to see how the routing
algorithms perform in a more realistic network topology.

We chose to vary several parameters and examine how they effected the key
rate. For simplicity, we consider homogeneous settings where each link/node in
the network has the same parameter; our approach can be applied to
heterogeneous settings. The settings we examined for these parameters are as
follows: link-level entanglement success probability $p$ is set to 0.85 or
0.95, entanglement swapping success probability $q$ is also set to 0.85 or
0.95, and channel noise parameter $\gamma$ is varied from 0.97 to 1.0, with
step size of 0.005. The number of parties, $N$, is varied from 3 to 6.

(a) b (Bet) Layout (b) d (Dalet) Layout (c) g (Giml) Layout

Figure 9: Results for various 3-party routing schemes for three layouts in
7$\times$7 grid, $p=q=0.85$.

Figure 10: Key rate when $N$ varies from 3 to 6 for dynamic multi-tree scheme
$p=q=0.85$, incremental g (Giml) layout in 7$\times$7 grid.

###  V-B Results for Grid Topology

We explored two grid topologies: 7$\times$7 and 11$\times$11\. For each of
them, we examined 3 different 3-party layouts. Based on the locations of the
terminal nodes, we refer to them as b (Bet), d (Dalet), and g (Giml) layouts
due to their similarities to letters in the Phoenician alphabet. Fig. 8 shows
these three layouts in the $7\times 7$ grid; the layouts in the $11\times 11$
grid are in a similar manner. For both In 7$\times$7 and 11$\times$11 grids,
we further incrementally added terminals to the g (Giml) layout to form the
setup for $N=4$, 5, and 6.

We next present the results for the 7$\times$7 grid. In Fig. 9, we compare how
our different 3-party routing algorithms perform when $p=q=0.85$; the results
for other settings show similar trend. As expected, both the single-tree and
multi-tree dynamic outperform the baselines of single-tree and multi-tree
fixed respectively. We also see that the b (Bet) layout yields higher key
rates than the d (Dalet) and g (Giml) layouts. This is due to how much closer
the terminals are located in the b (Bet) layout when compared to the others.
Interestingly, we see that the single-tree dynamic algorithm outperforms the
multi-tree fixed algorithm. This is unique to the grid topology as when
examining the random graph we see that multi-tree fixed algorithm outperforms
the single-tree dynamic algorithm. This is partially due to the limited degree
of the grid network, which makes packing trees difficult. The multi-tree fixed
algorithm is able to find the maximum number of possible trees (4 trees),
however the path lengths of these packed trees become very long, resulting in
very long key rates for the additional trees. This significantly reduces the
impact that the multi-star algorithm can have. When increasing $p$ and $q$ in
a grid setting, all three layouts follow the same trends seen in the random
graph (see Section V-C).

As the number of parties increases, naturally we would expect the key rate to
diminish. Interestingly, for the incremental g (Giml) layout we see in Fig. 10
that 4, 5, and 6 party QKD have very similar key rates, with a large gap
between the 3-party and 4-party case. The closeness of the 4, 5, and 6 party
key rates due to the similarity of the trees found. The large gap between the
3 and 4 party cases can be partially explained by the non-optimality of our
Steiner tree approximation algorithm (see Section III-B). Additionally, this
gap is exacerbated in the grid setting due to the limited connectivity between
nodes. We later see that with richer connectivity, as is the case for the
random graph, that the degradation for increasing $N$ is much less
significant.

The results for 11$\times$11 grid show similar trends albeit with
significantly lower key rates due to the increased distance between nodes. In
fact for values of $\gamma$ below 0.98, all 3-party schemes in the
11$\times$11 grid result in a key rate of 0. This performance only worsens as
$N$ increases.

(a) $p=0.85$, $q=0.85$. (b) $p=0.95$, $q=0.85$. (c) $p=0.85$, $q=0.95$. (d)
$p=0.95$, $q=0.95$.

Figure 11: Results for 3-party QKD (averaged over 5 random graphs) when
varying $p$ and $q$.

(a) $p=0.85$, $q=0.85$. (b) $p=0.95$, $q=0.85$. (c) $p=0.85$, $q=0.95$. (d)
$p=0.95$, $q=0.95$.

Figure 12: Results for $N$-party QKD under dynamic multi-tree scheme (averaged
over 5 random graphs) when varying $p$ and $q$, for $N=3$, 4, 5, and 6.

###  V-C Results for Random Graph

We examined 5 random graphs and averaged their results. Each graph is
generated by placing 50 nodes randomly in a unit square. Two nodes are
connected by a link if their distance is less than a radius of 0.3. These
random graphs have a significantly higher average degree than our grid graphs
(9.752 vs. roughly 4). Our expectation is that this would increase the
performance of the multi-tree routing, as the higher degree allows for more
trees to exist.

Fig. 11 plots the average key rate for 3-party QKD in the random graphs for
various values of $p$ and $q$. Compared to 3-party QKD for the three grid
topologies in Fig. 9, we indeed see much higher key rate for the random
graphs. For instance, in Fig. 11a, the multi-tree dynamic algorithm achieves a
maximum key rate of over 4, compared to the maximum in the grid setting in
Fig. 9a of just over 0.9.

In Fig. 11, increasing $p$ has the most impact on fixed routing algorithms due
to their reliance on specific links being established to distribute
entanglement. Similarly, we see that increasing $q$ has the largest impact on
the dynamic routing algorithms. Once again this makes sense due to $q$ and
$\gamma$ being the only parameters in our key rate equation for dynamic
routing. We also see that when comparing the effects of increasing $p$ and $q$
that they impact the multi-tree routing algorithms significantly more than
single-tree. Intuitively this is expected since each found tree will
independently be impacted by the increase of $p$ or $q$. As expected when
increasing both $p$ and $q$ we see the largest increase in the key rate for
both the fixed and dynamic multi-tree routing algorithms.

We next compare the results of the different algorithms for multiparty QKD. In
Fig. 11 for 3-party, compared to the baseline fixed single-tree algorithm, the
dynamic single-tree algorithm results in a key rate increase of $11\%$ to
$54\%$ for the various values of $p$ and $q$. This increase is also seen in
the dynamic multi-tree compared to the fixed multi-tree algorithm, resulting
in a key rate increase of $25\%$ to $148\%$. As we increase $N$, we see even
larger percentage increase. For instance, for $N=4$, 5, and 6 parties, dynamic
multi-tree algorithm leads to $137\%$, $191\%$, $223\%$ higher key rate than
fixed multi-tree algorithm when $p=q=0.85$ (figure omitted). This is because
as the number of parties increases, the trees that are found consist of more
links, which degrades the performance of fixed routing schemes exponentially
in both $p$ and $q$, whereas our dynamic schemes degrade only in $q$.

Figure 13: Histogram showing the average number of trees found in each round
versus $N$ when $p=0.85$ or $0.95$, and $q=0.85$.

Fig. 12 plots the key rate for dynamic multi-tree algorithm when increasing
$N$ in the random graphs. We see that there is a clear gap between each
plotted key rate in Fig. 12, in contrast to the grid topology where 4, 5, and
6 parties achieved similar key rates. The incremental addition of nodes to our
random graph has significantly less overlap than our incremental grid setting,
which causes the trees found for increasing $N$ to be different and contain
more nodes. Once again we see a large gap between the key rates as $N$
increases from 3 to 4 due to the non-optimality of our approximation
algorithm. When varying $p$ and $q$ for increasing $N$ we see similar trends
exhibited in the 3-party case. We see that increasing $p$ has little effect on
the dynamic routing schemes. As seen in the 3-party case, when increasing $q$,
the key rate increases significantly. Notably increasing $q$ decreases the gap
between the 3 and 4 party key rates.

Fig. 13 plots the histogram of the average number of trees found as $N$
increases, where $q=0.85$ and $p$ is 0.85 or 0.95. We see that the average
number of trees found tends to decrease when increasing $N$. However, since we
are not optimally packing trees, it is not always the case that increasing $N$
will result in less trees found, as we see is the case for $N=6$. As expected,
when increasing $p$ we see that more trees are found. When more links exist in
the graph, there are more possible trees that can be found. Finding trees is
independent of $q$, which only is taken into account when attempting to
distribute entanglement across trees.

##  VI Related Work

The works that are closest to ours are [9, 10, 11, 12], which are on efficient
multipartite entanglement distribution in quantum networks. As described in
§I, the strategies that they design are not for multiparty QKD. As such, their
optimization goals do not match that of multiparty QKD. Our design of multi-
star packing for 3-party QKD and multi-tree packing for $N$-party QKD are
inspired by the above existing works. On the other hand, we explicitly
consider key rate in our design to optimize secret key generation rate for the
multiple parties.

The multipath routing strategies that we develop for multiparty QKD are
broadly related to the extensive literature on routing in quantum networks
(e.g., [24, 15, 25, 16, 26, 27]). These studies however focus on pairwise Bell
pair distribution, instead of multipartite entanglement distribution to
multiple parties. Specifically, these strategies use global or local
information of the network, use single or multiple network paths, and are
designed to improve bipartite entanglement distribution rate. Routing
multipartite entanglement to multiple parties that is studied in this work is
significantly more challenging than routing bipartite entanglement to two
parties. Our designed routing strategies leverage global information of the
network; designing strategies that only use local information is left as
future work.

The design and security analysis of multiparty QKD have been studied
extensively (see survey in [2] and the references within). These studies
mainly focus on the key generation rates of a conference key agreement
protocol, given an entangled GHZ state of particular fidelity. They do not,
however, consider distributing multipartite entanglement or multiparty key
establishment in the context of quantum networks, which is the focus of this
study.

##  VII Conclusion

In this paper, we developed efficient strategies for multiparty QKD over
quantum networks. We designed strategies for both 3-party and general
$N$-party QKD by incorporating various constraints in near-term quantum
network technologies into the design to optimize secret key generation rate.
Using extensive evaluation in both grid and random graphs under a wide range
of settings, we demonstrate that our schemes achieve high key rate, and
degrade gracefully when increasing the number of parties.

##  Appendix A Derivation of Error Rates for 3-party QKD

We now derive how to obtain $Q_{AB_{i}}$ and QX, given a star graph in a
network setting.

_Depolarizing Channel:_ We first redefine the depolarizing channel in its
general form for an $n$-qubit state $\rho$, being transmitted across channel
$i$. We will need this for modelling the noise in our system. Typically the
trace is omitted when defining the depolarizing channel (see Eq. 2) since the
trace of a density matrix of a state is always 1. However it is useful here
since we will be distributing the depolarizing channel over partial states.

$\mathcal{E}_{i}(\rho)=\gamma_{i}\rho+(1-\gamma_{i})tr(\rho)\frac{\mathbb{I}}{2^{n}}$
(8)

Where $\gamma_{i}=\frac{4F_{i}-1}{3}$, and $F_{i}$ is the fidelity of the
state relative to the ideal state $\rho$.

_Stabilizer Formalism:_ We next briefly overview the stabilizer formalism,
which proves useful for modelling our noisy 3-GHZ state and taking expectation
values. A quantum state $\ket{\psi}$ is said to be stabilized by the operator
$\mathcal{U}$ if the following equation holds:

$\mathcal{U}\ket{\psi}=\ket{\psi}\ .$

The stabilizer formalism allows us to describe quantum states by the operators
that stabilize them [28]. We can then use a subgroup of the Pauli group to
describe a quantum state. For the $N$-qubit state $\ket{\psi}$ stabilized by
the group $\mathcal{G}$ we can describe the density matrix, $\rho$, of
$\ket{\psi}$ in the following way.

$\outerproduct{\psi}{\psi}=\rho=\frac{1}{2^{n}}\sum_{P\in\mathcal{G}}P\ .$

For example, the stabilizer group for the Bell state $\ket{\Phi^{+}}$ is
$\mathcal{G}=\\{II,XX,ZZ,-YY\\}$, where $\mathcal{U}\mathcal{U}$ =
$\mathcal{U}\otimes\mathcal{U}$. We can then describe the density matrix as
follows

$\outerproduct{\Phi^{+}}{\Phi^{+}}=\frac{1}{4}\sum_{P\in\mathcal{G}}P=\frac{1}{4}(II+XX+ZZ-
YY)\ .$

_Error Rates for 3-party QKD:_ As mentioned in §III, 3-GHZ distribution for
3-party QKD is over a star topology using the aforementioned depolarizing
noise model (see Eq. (8)).

When modelling the noise in our star, we choose to model the noise on the
outermost qubits (equivalent to modelling the noise on the center qubits). We
can then perform a perfect 3-GHZ projective measurement on the center qubits.
After local correction this results in a noisy 3-GHZ state where the
depolarizing channel of each corresponding path is applied to each qubit of
the 3-GHZ state. This is shown in the following equation:

$\rho=\mathcal{E}_{A}(\mathcal{E}_{B_{1}}(\mathcal{E}_{B_{2}}(\outerproduct{\text{3-GHZ}}{\text{3-GHZ}})))$
(9)

Using the stabilizer representation of the 3-GHZ state, we can apply the
depolarizing channel and simplify (due to the trace often being zero),
resulting in the final noisy 3-GHZ state shown below:

$\rho=\frac{1}{8}(III+\gamma_{A}\gamma_{B_{1}}\gamma_{B_{2}}XXX+\gamma_{B_{1}}\gamma_{B_{2}}IZZ\\\
+\gamma_{A}\gamma_{B_{1}}ZZI+\gamma_{A}\gamma_{B_{2}}ZIZ-\gamma_{A}\gamma_{B_{1}}\gamma_{B_{2}}YYX\\\
-\gamma_{A}\gamma_{B_{1}}\gamma_{B_{2}}XYY-\gamma_{A}\gamma_{B_{1}}\gamma_{B_{2}}YXY)$
(10)

For the $N$-party QKD protocol that we use in this paper (see §II-A), as
defined in [13], $Q_{X}$ is the probability that the total number of
$\ket{-}$’s is odd when measuring in the $X$ basis (Note that measuring a pure
$N$-GHZ state in the $X$ basis always yields an even number of $\ket{-}$’s.).
Similarly $Q_{A,B_{i}}$ is the probability that $A$ and $B_{i}$ have the same
measurement outcome when measuring in the $Z$ basis. That is,

$Q_{AB_{i}}=\frac{1-\langle Z_{A}Z_{B_{i}}\rangle}{2}$ (11)
$Q_{X}=\frac{1-\langle X^{\otimes N}\rangle}{2}$ (12)

We can then compute $Q_{AB_{i}}$ and $Q_{X}$ by taking the expectation from
our noisy 3-GHZ state. Recall that the expectation of an observable, $O$, with
respect to system $\rho$, is defined as $\langle O\rangle=tr(\rho O)$.
Applying this we see that most of our terms can be dropped due to the trace of
the (non-identity) pauli-operators being 0. For example, when taking $\langle
XXX\rangle$, the only term that will have a non-zero trace is the $XXX$ term
with coefficient $\gamma_{A}\gamma_{B_{1}}\gamma_{B_{2}}$. This results in the
$Z$ and $X$ basis error rates shown in Eq. (4) and (5). Additionally, these
equations hold for the $N$-GHZ case, as long as the graph distributing the
state is a star.

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# Stereographic Projections for Designs on the Sphere

Linda M. Haines CONTACT Linda M. Haines. Email<EMAIL_ADDRESS>Department of Statistical Sciences, University of Cape Town, Private Bag X3,
7700, Rondebosch, South Africa.

###### Abstract

This paper is concerned with the use of the stereographic projection in order
to map the sphere points of designs for regression models in three variables
and design space the unit ball onto the two-dimensional stereogram. Details of
the projection and its attendant stereogram are given. Stereograms which
represent the spherical component of designs with points on the sphere and the
centre of the unit ball, including the central composite and the Box-Behnken
designs, are introduced. In addition, an example in which stereograms are used
to elucidate the geometric isomorphism of a suite of designs generated by an
exchange algorithm is presented.

###### keywords:

Stereographic projection; stereogram; quadratic polynomial regression model;
unit ball; geometric isomorphism

##  1 Introduction

Designs over the unit ball which comprise a finite set of support points have
been reported sporadically in the literature as, for example, in the papers of
Bose and Draper (1959), Hardin and Sloane (1991), Gilmour (2006), Dette and
Grigoriev (2014), Hirao et al. (2015) and Radloff and Schwabe (2023). The
designs are not straightforward to construct and, to compound matters, the
three-dimensional prototypes, which help to convey the innate structure of the
designs, are not easy to represent on paper. The aim of the present study
therefore is to introduce the stereographic projection as a valuable tool, in
research and pedagogically, for representing the spherical components of
designs in three independent variables over the unit ball in two dimensions.
The paper is organized as follows. The notion of the stereographic projection
and the construction of the attendant stereogram are presented in Section 2.
Illustrative examples relating to the quadratic polynomial model in three
variables are then provided in Section 3. Some general comments on the
relevance of the use of the stereographic projection in experimental design
are presented in Section 4.

##  2 The Stereographic Projection

Consider first a perfect terrestrial globe which is centred at the origin with
the North-South axis oriented vertically and the equatorial plane
perpendicular to that axis. The stereographic projection maps points and
circles on the three-dimensional globe onto the two-dimensional equatorial
plane, with the resulting construct termed a stereogram. In particular,
consider a point lying on the surface of the globe in the northern hemisphere.
Then a line can be drawn from that point to the South Pole and the point of
intersection of the line with the equatorial plane indicated on the stereogram
with a solid circle, as shown in Figure 1(a). Similarly, a line can be drawn
from a point on the surface of the globe in the southern hemisphere to the
North Pole and its intersection with the equatorial plane denoted for clarity
by an open circle, as in Figure 1(b). In addition, parallels, that is lines of
latitude, are projected onto the equatorial plane as circles centred at the
origin and meridians, that is lines of longitude, as straight lines through
the origin. These constructs can be represented by a polar net, as shown in
Figure 1(c). Great circles through two points on the surface of the sphere can
also be projected onto the equatorial plane, specifically as arcs with
diagonally opposite end points touching the equator. Selected meridians and
great circles are shown on the stereogram in Figure 1(d).

(a) Northern Hemisphere

(b) Southern Hemisphere

(c) Meridians and Parallels

(d) Meridians and Great Circles

Figure 1: Stereographic projections for points (a) in the northern hemisphere
and (b) in the southern hemisphere. Stereograms, looking down the North-South
axis, of (c) meridians and parallels and (d) meridians and great circles.

Three-dimensional directions are often indicated on the stereogram and, for
example, the vectors $(1,1,1)$ and $(-1,1,0)$ are denoted compactly by $[111]$
and $[\bar{1}10]$. To give a sense of the orientation of the equatorial plane,
selected directions are included in Figure 1(d). Finally, note that a small
circle drawn on the surface of the globe can be projected as a circle onto the
stereogram but the centres of the two circles do not necessarily correspond.
Further details on the stereographic projection are given, for example, in the
books on crystallography by Borchardt-Ott (2011), De Graef and McHenry (2012)
and Hammond (2015).

The stereographic projection can now be adapted to accommodate designs in
three dimensions on design space the unit ball with centre at the origin and
support points on the boundary and at the centre of the ball. Specifically,
consider the perfect terrestrial globe described above as a unit sphere with a
right-handed coordinate system specified by the coordinates $x_{1},x_{2}$ and
$x_{3}$. Then the point $(0,0,1)$ can be taken taken to coincide with the
North Pole, the point $(0,1,0)$ with the intersection of a line in the
direction $[0,1,0]$ with the equator and the $(x_{1},x_{2})$-plane with the
equatorial plane. The support points of a design on the surface of the sphere
can be projected onto the equatorial plane, that is the stereogram, by
invoking the mapping
$(x_{1},x_{2},x_{3})\mapsto\big{(}\displaystyle\frac{x_{1}}{1+|x_{3}|},\frac{x_{2}}{1+|x_{3}|}\big{)}$.
Projections onto the stereogram of the appropriate meridians and great circles
which connect the design points then follow.

##  3 Examples

###  3.1 Subset Designs

To fix ideas, consider the subset designs for the quadratic polynomial model
with three independent variables over the unit ball ntroduced into the
literature in the paper by Gilmour (2006). The support points of the designs
lie on the surface and at the centre of the unit ball and are taken from the
sets specified by

$S_{0}=(0,0,0),~{}S_{1}=(\pm 1,0,0)_{\mathrm{\small
cycle}},~{}S_{2}=(\pm\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}},0)_{\mathrm{\small
cycle}},~{}S_{3}=(\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}})$

which comprise $1,6,12$ and $8$ points respectively. The subscript ‘cycle’
indicates that the three coordinates of the points can be moved cyclically.
More specifically, subset designs can be expressed as a linear combination of
these sets of points, that is as
$c_{0}S_{0}+c_{1}S_{1}+c_{2}S_{2}+c_{3}S_{3}$, where $c_{0}>0$ and the
remaining coefficients, $c_{1},c_{2}$ and $c_{3}$, are greater than or equal
to zero and are chosen in such a way that the moment matrix is nonsingular.
From a symmetry perspective, the sets of points $S_{0},S_{1},S_{2}$ and
$S_{3}$ correspond to the centre point and, for a cube inscribed in the unit
sphere, to the intersection of the six normals to the faces of the cube with
the sphere, to the intersection of the 12 lines from the origin through the
mid-points of the edges of the cube with the sphere, and to the eight vertices
of the cube itself, respectively. It thus follows that the points on the unit
sphere of a subset design with equal weights assigned to the subsets
$S_{1},S_{2}$ and $S_{3}$ belong to the point group $O_{h}$.

(a) Central Composite Design

(b) Box-Behnken Design

(c) Rotatable Design

(d) Approximate $E$-optimal Design

Figure 2: Stereograms of the sphere points of (a) the central composite
design, (b) the Box-Behnken design, (c) the rotatable design
$S_{0}+S_{1}+2S_{2}$ and (d) the approximate $E$-optimal design
$\frac{9}{17}S_{0}+\frac{3}{17}S_{1}+\frac{5}{17}S_{3}$. In (c) and (d) the
areas of the solid circles are proportional to the relative weightings of the
points on the sphere.

The central composite design over design space the unit ball is the subset
design $S_{0}+S_{1}+S_{3}$ and the stereographic projection of the 14 design
points on the sphere yields the stereogram shown in Figure 2(a). Similarly,
the Box-Behnken design is the subset design $S_{0}+S_{2}$ and the stereogram
of the 12 points on the sphere is shown in Figure 2(b). Note that the points
in $S_{1},S_{2}$ and $S_{3}$ are coloured in the stereograms in black, blue
and red, respectively, in Figure 2 and that the complementary relationship
between the two designs is clearly illustrated by the stereograms. Subset
designs with different coefficients allocated to the sets of points
$S_{0},S_{1},S_{2}$ and $S_{3}$ can also be represented compactly on a
stereogram by using colour-coding and points with areas proportional to their
relative weightings. For example, the subset design $S_{0}+S_{1}+2S_{2}$ is
rotatable and the different allocations of points to the two sets $S_{1}$ and
$S_{2}$ is highlighted in the stereogram in Figure 2(c). In addition, the
approximate $E$-optimal design for the quadratic polynomial model in three
variables on the unit ball derived by Dette and Grigoriev (2014) is the subset
design $\frac{9}{17}S_{0}+\frac{3}{17}S_{1}+\frac{5}{17}S_{3}$ and thus a
weighted form of the central composite design. The stereogram of the sphere
points of the design and their relative weighting is shown in Figure 2(d).
Note that the design itself is rather curious in that more than half of the
weight is placed on the centre point.

###  3.2 An Application

Gilmour and Trinca (2012), in Example 2 of their paper, considered optimal
designs which accommodate inference for a full quadratic polynomial model in
three independent variables with support points on the surface and at the
centre of a ball with radius 3. For clarity, their results are reported here
for the unit ball. The authors focussed on nonsingular designs with 18 support
points taken, with replacement, from the $27$ individual points belonging the
subsets $S_{0},S_{1},S_{2}$ and $S_{3}$ of the previous example. More
specifically, they considered, inter alia, DS-optimal designs which minimise
the volume of the confidence region for the parameters of the model with the
intercept treated as a nuisance parameter and (DP)S-optimal designs which
minimise the DS-criterion multiplied by an appropriate quantile of the
$F$-distribution raised to the power equal to the number of parameters of
interest, in this case nine. It can in fact be easily shown that over $3\times
10^{10}$ possible designs must be enumerated in order to identify those
designs which are strictly DS\- and (DP)S-optimal. As a consequence, Gilmour
and Trinca (2012) used a heuristic, the exchange algorithm documented in, for
example, Atkinson et al. (2007), to search for designs which are optimal or
near-optimal.

The question now arises as to how many strictly DS\- and (DP)S-optimal designs
are generated by the exchange algorithm and, if more than one, how the
resultant designs are related. This question in turn impacts on the
performance of the algorithm. To this end, the exchange procedure of Gilmour
and Trinca (2012) was run five million times in the programming language gauss
(2022) and the requisite optimal designs which were distinct, that is non-
coincident, identified. Eight distinct DS-optimal designs with two centre
points and $1,080$ distinct (DP)S-optimal designs, with $216$ including a
single centre point and $864$ two centre points, were so generated. The
designs were then examined in the context of isomorphism and, since the
factors $0,1,\frac{1}{\sqrt{2}}$ and $\frac{1}{\sqrt{3}}$ are quantitative, in
that of geometric isomorphism (Cheng and Ye 2004). Specifically, two designs
are said to be geometrically isomorphic if the designs can be obtained, one
from the other, by rotations and reflections within the design space. In the
present example, the isomorphic properties of the designs are determined
solely by the spherical points and the actions of rotation and reflection can
therefore be mapped, albeit indirectly, onto the stereograms. It is crucial to
observe here that a design which is geometrically isomorphic to a DS-optimal
or a (DP)S-optimal design is itself necessarily DS-optimal or (DP)S-optimal.

The DS-optimal design given in Table 2 of Gilmour and Trinca (2012) comprises
16 points on the sphere, none of which are replicated, and two centre points,
and coincides with one of the eight distinct DS-optimal designs generated by
the exchange algorithm. The stereogram of the spherical points of the design
is presented in Figure 3(a), with the great circles introduced as guidelines
for the placement of the points. It is immediately clear from the figure that
a further three distinct designs can be generated from the design by repeated
rotations through $90^{\circ}$ about the $x_{3}$ axis and the associated
stereograms are shown in Figures 3(b), 3(c) and 3(d). A further four designs
can then be obtained by reflecting the designs shown in Figure 3 through the
plane perpendicular to the $x_{3}$ axis, that is the $(x_{1},x_{2})$-plane.
The resultant eight designs so obtained are geometrically isomorphic and
therefore DS-optimal. These isomorphic designs therefore account for the eight
distinct DS-optimal designs generated by the exchange algorithm.

(a) Design of Gilmour and Trinca (2012)

(b) Rotate (a) through $\displaystyle-\frac{\pi}{2}$

(c) Rotate (b) through $\displaystyle-\frac{\pi}{2}$

(d) Rotate (c) through $\displaystyle-\frac{\pi}{2}$

Figure 3: Stereograms of exact DS-optimal designs with (a) the design of
Gilmour and Trinca (2012) and (b), (c) and (d) the designs from successive
anti-clockwise rotations through 90 degrees about the $x_{3}$-axis, starting
with design (a).

The problem of identifying (DP)S-optimal designs which are geometrically
isomorphic from the 1,080 optimal designs generated by the exchange algorithm
is a little more nuanced than that relating to the DS-optimal designs.
Consider first the design reported in Table 2 of Gilmour and Trinca (2012)
which is easily identified as one of the 216 distinct (DP)S-optimal designs
with a single centre point generated by the exchange algorithm.

This $18$-point design is based on one centre point and on a set of nine
distinct points on the sphere, specifically the set

$B=\left\\{(100),(\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}),(0,\pm\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}},0,\pm\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}},0)\right\\},$

termed here the basis set, with each point replicated twice except for the
point $(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ which occurs once. A
stereogram of the 17 points on the sphere of the design is presented in Figure
4(a), with the point occurring once indicated in red.

Two distinct designs can be obtained by rotation twice through $180^{\circ}$
about the axis defined by the vector $[1,1,1]$ and are geometrically
isomorphic to the original design of Gilmour and Trinca (2012) and therefore
(DP)S-optimal. To illustrate, the stereograms of the two resultant designs are
shown in Figures 4(b) and 4(c). As an aside, the designs can also be
constructed by cycling the original axes $(x_{1},x_{2},x_{3})$ of Gilmour and
Trinca (2012) twice as $(x_{2},x_{3},x_{1})$ and $(x_{3},x_{1},x_{2})$. Eight
geometrically isomorphic designs can now be generated from each of the three
designs displayed in Figures 4(a), (b) and (c) by rotation through
$90^{\circ}$ about the $x_{3}$-axis and reflection about an appropriate plane
to yield a total of $3\times 8=24$ designs. The design of Gilmour and Trinca
(2012) is therefore representative of a class of 24 geometrically isomorphic
designs.

(a) Gilmour and Trinca Design

(b) Design 1 Isomorphic to (a)

(c) Design 2 Isomorphic to (a)

(d) A design not isomorphic to (a)

Figure 4: Stereograms of the spherical components of selected (DP)S-optimal
designs: (a) The design of Gilmour and Trinca (2012), with one point of $B$
occurring once, (b) and (c) designs which exhibit different patterns of
points, but which are geometrically isomorphic to (a), and (d) a design, not
isomorphic to (a), with one point, $(1,0,0)$, of $B$ occurring once. Points
which occurs once are indicated in red.

A cursory examination of the results from the exchange algorithm indicated
that designs with one centre point and any one of the nine points in the basis
set $B$ occurring once and not twice are (DP)S-optimal. In terms of geometric
isomorphism, consider now a design constructed directly from that of Gilmour
and Trinca (2012) but with the the point
$(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ now replicated twice and the
point $(1,0,0)$ occurring once. For clarity, a stereogram of the design is
shown in Figure 4(d), with the point occurring once indicated in red. It is
clear that the design so constructed is not geometrically isomorphic to the
design of Gilmour and Trinca (2012). However, by using arguments based on
rotations and reflections, it is straightforward to show that the design
represents a class of 24 isomorphic (DP)S-optimal designs. Furthermore, the
same result holds for all designs constructed with one of the points in the
set $B$ occurring once. There are therefore nine classes of designs, each
comprising 24 geometrically isomorphic designs, and thus a total of $9\times
24=216$ designs. This result is in accord with the fact that $216$
(DP)S-optimal designs with one centre point were identified from multiple runs
of the exchange algorithm.

Consider now the $864$ (DP)S-optimal designs with two centre points generated
from multiple runs of the exchange algorithm. A cursory examination of the
results indicated that the points on the sphere of the design comprise two
distinct points from the basis set $B$ chosen to occur once and the seven
remaining points in $B$ replicated twice. The $\binom{9}{2}=36$ designs so
defined were then constructed independently and shown to be (DP)S-optimal. It
now follows from arguments similar to those presented for the (DP)S-optimal
designs with one centre point that each of the designs so constructed
represents a class of $24$ geometrically isomorphic designs, thereby yielding
$36\times 24=864$ designs which are (DP)S-optimal and accounting for the
designs generated by the exchange algorithm.

##  4 Discussion

The focus of the present study is on the use of the stereographic projection
within the context of experimental design in order to map points on the three-
dimensional unit sphere onto the stereogram in two dimensions. The stereogram
gives a clear visualisation of the spatial arrangement of the points and can
be manipulated in order to emulate rotations of, and reflections about, the
main axes. Stereograms for subset designs for the quadratic polynomial model
in three independent variables over the unit ball, including the central
composite and Box-Behnken designs, are presented in order to fix ideas. An
example of the effective use of the stereogram is then introduced and is based
on a suite of designs generated numerically by an exchange algorithm.
Specifically, it is shown that classes of designs which are geometrically
isomorphic can be identified by a judicious choice of operations on the
stereogram. More broadly, generating optimal designs by invoking a heuristic
invariably yields large numbers of similar designs and the present example
reveals how such designs can be inter-related and, thereby, classified.

There are, in fact, many ways of representing an arrangement of points on the
three-dimensional sphere in two dimensions, that is on paper, other than the
stereogram. Thus, most simply, a drawing of the sphere with the points added
or a sketch of the solid with vertices defined by the points can be used but
these approaches are tedious to implement and not helpful in comparative
studies. Alternatively, since a set of points on the sphere defines a convex
polytope, the points can be represented by a 3-vertex-connected planar graph,
that is a polyhedral graph, but the sense of the spatial arrangement is then
lost. There are, in addition, projections, other than the stereographic
projection, such as the orthographic and the gnomonic projections, which can
be invoked but these distort the symmetry and, in the latter case, project the
points onto an infinite plane tangential to the sphere. On balance therefore,
none of the approaches discussed here offers the ease of use, the sense of the
spherical space and the flexibility provided by the stereogram, as is required
within the context of experimental design.

Finally, it would be remiss not to mention the use of computer-generated
three-dimensional models which represent points and solids with vertices on
the sphere and which are available in, for example, the programming language
Mathematica (2022). Such models can be rotated in space and thereby examined
but it is nevertheless not easy to identify the inbuilt symmetry. The question
also arises as to whether points on the four-dimensional sphere can be
represented as three- or two-dimensional objects. There are ways of doing
this, as for example with a Schlegel diagram, but these are complicated to
interpret and not immediately useful in design.

## Acknowledgements

The author would like to thank the University of Cape Town and the National
Research Foundation (NRF) of South Africa, grant (UID) 119122, for financial
support. Any opinion, finding and conclusion or recommendation expressed in
this material is that of the author and the NRF does not accept liability in
this regard.

## References

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  * Bose and Draper (1959) Bose, R. C. and N. R. Draper (1959).  Second order rotatable designs in three dimensions.  The Annals of Mathematical Statistics 30, 1097–1112.
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  * Hirao et al. (2015) Hirao, M., M. Sawa, and M. Jimbo (2015).  Constructions of $\Phi_{p}$-optimal rotatable designs on the ball.  Sankhyā: The Indian Journal of Statistics, Series A 77, 211–236.
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# Selberg’s Central Limit Theorem for Dirichlet $L$-functions of $q$-aspect

Madhuparna Das  Department of Mathematics, University of Exeter, Stocker Rd,
Exeter EX4 4PY, UK<EMAIL_ADDRESS>
(Date: May 2021)

###### Abstract.

We give a simple proof of Selberg’s central limit theorem for Dirichlet
$L$-functions for $q$-aspect by following the method established by Radiwiłł
and Soundararajan.

###### 2010 Mathematics Subject Classification:

11M06

##  1\. Introduction

Selberg observed that [4, 5] the logarithm of the Riemann zeta function
behaves like a Gaussian random variable with mean 0 and variance
$\frac{1}{2}\log\log|t|$. Later in 2017, Radziwiłł and Soundararajan [1] gave
a new and elegant proof of this result known as Selberg’s central limit
theorem. Later P. Hsu and P. Wong [3] proved this result for the Dirichlet
$L$-functions of level aspect by following the method of [1]. We have proved
[2, 6] this result for automorphic $L$-functions of level aspect for
$L$-functions of degree 2 and 3 associated with the primitive holomorphic cusp
forms and Hecke-Maass cusp forms respectively.

In this paper, we give a simple proof of Selberg’s central limit theorem for
the Dirichlet $L$-function of $q$-aspect by using the technique of [1]. Let
$L(s,\chi)$ be a Dirichlet $L$-function given by

$\displaystyle L(s,\chi)=\sum_{n}\frac{\chi(n)}{n^{s}}$

where $\chi$ is a primitive Dirichlet character modulo $q$, for $\Re(s)>1$.

Let $q$ be the conductor of $L(s,\chi)$. Since the subconvexity bound [7] for
the Dirichlet $L$-functions has already been proven we can get all the
necessary tools we need to prove our result. We need to keep in mind that we
will be handling the character sums so orthogonality relations of the
Dirichlet character is a very useful tool here.

######  Theorem 1.

Let $V$ be a fixed real number. Let $\chi$ be a primitive Dirichlet character
modulo $q$ and $q\asymp Q$. Then for all $t$,

$\displaystyle\frac{1}{\phi(q)}meas\left\\{{\chi(\bmod
q)}:\log|L(\frac{1}{2}+it,\chi)|\geq V\sqrt{\frac{1}{2}\log\log
Q}\right\\}\sim\frac{1}{\sqrt{2\pi}}\int_{V}^{\infty}e^{-u^{2}/2}du.$

We start with the setup of the proof then we will get into the details. Notice
that we are proving this result for the $q$-aspect so in this case $q\asymp Q$
and $t$ is fixed. In the first step, let us take away the problem from the
critical line by proving that for a suitable choice of $\sigma>\frac{1}{2}$
$L(\frac{1}{2}+it,\chi)$ and $L(\sigma+it,\chi)$ are typically close to each
other.

######  Proposition 1.

Let $q$ be the conductor and $q\asymp Q$. Then for any $\sigma>\frac{1}{2}$ we
have

$\displaystyle\sum_{\chi(\bmod{q})}\left|\log|L(\frac{1}{2}+it,\chi)|-\log|L(\sigma+it,\chi)|\right|\ll\left(\sigma-\frac{1}{2}\right)\log
Q.$

Fix the parameters

$\displaystyle W=(\log\log\log q)^{4},\quad X=q^{(\log\log\log q)^{2}},\quad
Y=q^{\left(\frac{1}{\log\log
q}\right)^{2}},\quad\sigma_{0}=\frac{1}{2}+\frac{W}{\log q}.$

where $q\asymp Q$ and sufficiently large so $W\geq 3$.

In the next step we consider the auxiliary series given by

$\displaystyle\mathcal{P}(s,\chi)=\mathcal{P}(s,\chi;X)=\sum_{2\leq n\leq
X}\frac{\Lambda(n)\chi(n)}{n^{s}\log n}$

where $\Lambda(n)$ is the Von-Mangoldt function. We determine the distribution
of the auxiliary series $\mathcal{P}(f,s)$ by computing its moments.

######  Proposition 2.

As $q\asymp Q$ the distribution of $\Re(\mathcal{P}(\sigma_{0}+it))$ is
approximately normal with mean 0 and variance $\frac{1}{2}\log\log Q$.

It is now obvious that if we can connect the $L$-function with the auxiliary
series $\mathcal{P}(s,\chi)$ that will prove Theorem 1.

To establish this connection we use the mollification technique. Consider the
Dirichlet polynomial $M(s,\chi)$ given by

$\displaystyle M(s,\chi)=\sum_{n}\frac{\mu(n)a(n)\chi(n)}{n^{s}}$

where $a(n)$ is given by

$\displaystyle a(n)=\begin{cases}1&\text{if $n$ is composed only primes below
$X$ and has at most $100\log\log Q$ primes below $Y$,}\\\ &\text{and at most
$100\log\log\log Q$ primes between $Y$ and $X$.}\\\ 0&\text{otherwise
}.\end{cases}$

By the definition of $a(n)$ it takes the value 0 except when $n\leq
Y^{100\log\log q}X^{100\log\log\log q}<Q^{\epsilon}$. It is now evident that
$M(s,\chi)$ is short Dirichlet polynomial. In the next step, we establish the
connection between $M(s,\chi)$ and $\mathcal{P}(s,\chi)$.

######  Proposition 3.

For the primitive Dirichlet character $\chi(\bmod{q})$

$\displaystyle M(\sigma_{0}+it)=(1+o(1))\exp(-\mathcal{P}(\sigma_{0}+it)).$

It remains to prove that the mollifier and the $L$-function are inverse to
each other.

######  Proposition 4.

Let $\chi$ be a primitive Dirichlet character modulo $q$,

(1)
$\displaystyle\sum_{\chi(\bmod{q})}\left|1-L(\sigma_{0}+it,\chi)M(\sigma_{0}+it,\chi)\right|^{2}=o(1)$

so as $q\asymp Q$ we have

$\displaystyle L(\sigma_{0}+it,\chi)M(\sigma_{0}+it,\chi)=1+o(1).$

Now we prove our main theorem. We prove the propositions in later sections.

Proof of Theorem 1: Recalling Proposition 4, it typically says that
$L(\sigma_{0}+it,\chi)\approx M(\sigma_{0}+it,\chi)^{-1}$. By Proposition 3 we
know that
$M(\sigma_{0}+it,\chi)^{-1}\approx\exp(\mathcal{P}(\sigma_{0}+it,\chi))$ and
by Proposition 2 we can conclude that $\log|L(\sigma_{0}+it,\chi)|$ is
normally distributed with mean 0 and variance $\frac{1}{2}\log\log Q$. Finally
with the help of Proposition 1 we deduce that $\log|L(\sigma_{0}+it,\chi)|$
and $\log|L(\frac{1}{2}+it,\chi)|$ have the same distribution if we set
$\sigma=\sigma_{0}$ under the estimate $W^{2}=o(\sqrt{\log\log q})$. This
completes the proof. ∎

##  2\. Proof of Proposition 1

Let $\chi$ be the primitive Dirichlet character and $q$ be the conductor with
$q\asymp Q$. We set

$\displaystyle
G(s,\chi)=\left(\frac{\pi}{q}\right)^{-(s+a)/2}\Gamma\left(\frac{s+a}{2}\right)$

where

$\displaystyle a=a(\chi)=\begin{cases}0&\text{ if }\chi(-1)=1\\\ 1&\text{ if
}\chi(-1)=-1.\end{cases}$

Consider the Stirling approximation $\Gamma(\sigma+it)\sim
e^{\frac{1}{2}\pi|t|}|t|^{\sigma-\frac{1}{2}}\sqrt{2\pi}$.

$\displaystyle\left|\log\frac{G(\sigma+it,\chi)}{G(\frac{1}{2}+it,\chi)}\right|$
$\displaystyle=$
$\displaystyle\left|\log\frac{\left(\frac{\pi}{q}\right)^{-(\sigma+it+a)/2}\Gamma\left(\frac{\sigma+it+a}{2}\right)}{\left(\frac{\pi}{q}\right)^{-(\frac{1}{2}+it+a)/2}\Gamma\left(\frac{\frac{1}{2}+it+a}{2}\right)}\right|$
$\displaystyle\ll$ $\displaystyle\left(\sigma-\frac{1}{2}\right)\log q.$

Note that we are working with the $q$-aspect here so we can fix $t$ and ignore
the term inside the logarithm. By putting the Stirling’s approximation we
conclude the above equation. To prove proposition 1 consider the complete
Dirichlet $L$-function $\xi(s,\chi)=G(s,\chi)L(s,\chi)$. Recall Hadamard’s
factorization formula, there exist constant $A$ and $B$ with
$\Re(B(\chi))=-\sum_{\rho\in Z_{\chi}}\frac{1}{\rho}$ such that

$\displaystyle\xi(s,\chi)=e^{A+Bs}\prod_{1-\frac{s}{\rho}e^{s/{\rho}}}$

where we denote the set of non-trivial zeros by $Z_{\chi}$. Thus we have

$\displaystyle\log\left|\frac{\xi(\frac{1}{2}+it,\chi)}{\xi(\sigma+it,\chi)}\right|=\sum_{\rho\in
Z_{\chi}}\log\left|\frac{\frac{1}{2}+it-\rho}{\sigma+it-\rho}\right|$

which implies that

(2)
$\displaystyle\sum_{\chi(\bmod{q})}\left|\log\left|\frac{\xi(\frac{1}{2}+it,\chi)}{\xi(\sigma+it,\chi)}\right|\right|\leq\sum_{\rho\in
Z_{\chi}}\log\left|\frac{\frac{1}{2}+it-\rho}{\sigma+it-\rho}\right|$

Suppose $\rho=\beta+i\gamma$ is a zero o $L(s,\chi)$, if $|t-\gamma|\geq 2$ we
have (as argued in [1])

$\displaystyle\sum_{\rho\in
Z_{\chi}}\log\left|\frac{\frac{1}{2}+it-\rho}{\sigma+it-\rho}\right|\ll\frac{\sigma-\frac{1}{2}}{1+(t-\gamma)^{2}}.$

Combine this with equation (2) and the fact that the number of zeros of
$L(s,\chi)$ in the box $k\leq|t-\gamma|\leq k+1$ concludes the proposition. ∎

##  3\. Proof of Proposition 2

We prove this proposition by restricting the sum to primes and then compute
moments. We can restrict the sums to primes because for the primes $p$,
$p^{k}$ terms contribute $O(1)$ for $k\geq 3$. For the terms involving $p^{2}$
we can ignore them as argued in [3].

Consider the auxiliary series

$\displaystyle\mathcal{P}_{0}(\sigma_{0}+it)=\mathcal{P}_{0}(\sigma_{0}+it,\chi;X)=\sum_{p\leq
X}\frac{\chi(p)}{p^{\sigma_{0}+it}}$

We study the moments by obtaining a mean value estimate to prove Proposition
2.

######  Lemma 1.

Suppose that $k$ and $l$ are non-negative integers with $X^{k+l}\ll q\asymp
Q$. Then if $k\neq l$

$\displaystyle\sum_{\chi(\bmod{q})}\mathcal{P}_{0}(\sigma_{0}+it)^{k}\overline{\mathcal{P}_{0}(\sigma_{0}+it)}^{l}\ll\phi(q)Q$

If $k=l$, we have

$\displaystyle\sum_{\chi(\bmod{q})}\left|\mathcal{P}_{0}(\sigma_{0}+it)\right|^{2k}=k!\phi(q)(\log\log
Q)^{k}+O_{k}(\phi(q)(\log\log Q)^{k-1+\epsilon})$

###### Proof.

Write $\mathcal{P}_{0}(s)^{k}=\sum_{n}\frac{a_{k}(n)\chi(n)}{n^{s}}$ where

$\displaystyle
a_{k}(n)=\begin{cases}\frac{k!}{\alpha_{1}!\cdots\alpha_{r}!}&\text{ if
}n=\prod_{j=1}^{r}p_{j}^{\alpha_{j}},p_{1}<\ldots<p_{r}<X,\sum_{j=1}^{r}\alpha_{j}=k.\\\
0&\text{ otherwise }.\end{cases}$

Therefore,

$\displaystyle\sum_{\chi(\bmod q)}\left|\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\frac{a_{k}(n)\chi(n)}{n^{s}}\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\frac{a_{l}(n)\bar{\chi}(n)}{n^{s}}\right|$
$\displaystyle=$
$\displaystyle\sum_{\chi(\bmod{q})}\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\frac{a_{k}(n)a_{l}(n)}{n^{2\sigma_{0}}}\chi(n)\bar{\chi}(n)+O\left(\sum_{\chi(\bmod{q})}\sum_{\begin{subarray}{c}n\neq
m\\\ (n,q)=1\\\
(m,q)=1\end{subarray}}\frac{a_{k}(n)a_{l}(m)\chi(n)\bar{\chi}(m)}{(mn)^{\sigma_{0}}}\right)$
$\displaystyle=$
$\displaystyle\sum_{\chi(\bmod{q})}\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\frac{a_{k}(n)a_{l}(n)}{n^{2\sigma_{0}}}\chi(n)\bar{\chi}(n)+O\left(\phi(q)\sum_{\begin{subarray}{c}n\neq
m\\\ m\equiv
n(\bmod{q})\end{subarray}}\frac{a_{k}(n)a_{l}(m)}{(mn)^{\sigma_{0}}}\right)$

We calculate the diagonal term for $m=n$ and off-diagonal term for $m\neq n$.
For the diagonal term $a_{k}(n)a_{l}(n)$ is 0 by definition for $k\neq l$. For
the off-diagonal term we have

$\displaystyle\phi(q)\sum_{m\neq n}a_{k}(n)a_{l}(m)\ll X^{k+l}\ll\phi(q)Q.$

We conclude the first part of the lemma.

For $k=l$ we have the diagonal term

$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{P}_{0}(\sigma_{0}+it)|^{2k}dt=\sum_{\chi(\bmod{q})}\sum_{n}\frac{a_{k}(n)^{2}|\chi(n)|^{2}}{n^{2\sigma_{0}}}+O\left(\sum_{\chi(\bmod{q})}\sum_{\begin{subarray}{c}n\neq
m\\\ (n,q)=1\\\
(m,q)=1\end{subarray}}\frac{a_{k}(n)a_{k}(m)\chi(m)\chi(n)}{(mn)^{\sigma_{0}}}\right)$

For $n$ not being square free the diagonal term contribute

$\displaystyle k!\sum_{\begin{subarray}{c}p_{1},\ldots,p_{r}\leq X\\\
(p_{j},q)=1\end{subarray}}\frac{|\chi(p)|^{2}}{(p_{1}\cdots
p_{k})^{2\sigma_{0}}}=k!\left(\sum_{p\leq
X}\frac{1}{p^{2\sigma_{0}}}\right)^{k}=k!(\log\log q)^{k}+O_{k}((\log\log
q)^{k-1})$

Recalling the definition of $X$ we conclude the proof. ∎

Proof of Proposition 2: By Lemma 1 for any odd $k$ we have

$\displaystyle\sum_{\chi(\bmod{q})}\left(\Re(\mathcal{P}_{0}(\sigma_{0}+it))\right)^{k}=$
$\displaystyle\sum_{\chi(\bmod{q})}\frac{1}{2^{k}}\left(\mathcal{P}_{0}(\sigma_{0}+it)+\overline{\mathcal{P}_{0}(\sigma_{0}+it)}\right)^{k}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{k}}\sum_{l=0}^{k}\binom{l}{k}\sum_{\chi(\bmod{q})}(\mathcal{P}_{0}(\sigma_{0}+it))\overline{(\mathcal{P}_{0}(\sigma_{0}+it)}^{k-l}\ll\phi(q)Q$

Observe that it is impossible to have $l=k=l$ for any odd $k$, we have for all
even $k$, $l=k-l=k/2$ and again with the help of Lemma 1 we obtain,

$\displaystyle\frac{1}{\phi(q)}\sum_{\chi(\bmod{q})}\left(\Re(\mathcal{P}_{0}(\sigma_{0}+it))\right)^{k}=\frac{1}{2^{k}}\binom{k}{k/2}\left(\frac{k}{2}\right)!(\log\log
Q)^{\frac{k}{2}}+O_{k}\left((\log\log Q)^{\frac{k}{2}-1+\epsilon}\right)$

The above equation matches with the moments of Gaussian random variable with
mean 0 and variance $\frac{1}{2}\log\log Q$. ∎

##  4\. Proof of Proposition 3

As we have decomposed $\mathcal{P}(s)$ in [2] we are going to use the same
technique here. So we have

$\displaystyle\mathcal{P}_{1}(s,\chi)=\sum_{2\leq n\leq
Y}\frac{\Lambda(n)\chi(n)}{n^{s}\log n}$
$\displaystyle\mathcal{P}_{2}(s,\chi)=\sum_{Y<n\leq
X}\frac{\Lambda(n)\chi(n)}{n^{s}\log n}$

We further set

$\displaystyle\mathcal{M}_{1}(s,\chi)=\sum_{0\leq k\leq 100\log\log
Q}\frac{(-1)^{k}}{k!}\mathcal{P}_{1}(s)^{k}$
$\displaystyle\mathcal{M}_{2}(s,\chi)=\sum_{0\leq k\leq 100\log\log\log
Q}\frac{(-1)^{k}}{k!}\mathcal{P}_{1}(s)^{k}$

######  Lemma 2.

For the primitive Dirichlet character $\chi$ modulo $q$

$\displaystyle|\mathcal{P}_{1}(\sigma_{0}+it)|\leq\log\log Q$ (3)
$\displaystyle|\mathcal{P}_{2}(\sigma_{0}+it)|\leq\log\log\log Q$

Moreover,

$\displaystyle\mathcal{M}_{1}(\sigma_{0}+it)=\exp(-\mathcal{P}_{1}(\sigma_{0}+it))(1+O(\log
Q)^{-99})$ (4)
$\displaystyle\mathcal{M}_{2}(\sigma_{0}+it)=\exp(-\mathcal{P}_{2}(\sigma_{0}+it))(1+O(\log\log
Q)^{-99})$

###### Proof.

For the first assertion of the lemma,

$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{P}_{1}(\sigma_{0}+it)|^{2}$
$\displaystyle=$ $\displaystyle\sum_{\chi(\bmod{q})}\left(\sum_{2\leq
n_{1}=n_{2}\leq
Y}\frac{\Lambda(n_{1})\Lambda(n_{2})\chi(n_{1})\bar{\chi}(n_{2})}{(n_{1}n_{2})^{\sigma_{0}}\log
n_{1}\log n_{2}}\right)+O\left(\sum_{\chi(\bmod{q})}\sum_{2\leq n_{1}\neq
n_{2}\leq
Y}\frac{\Lambda(n_{1})\Lambda(n_{2})\chi(n_{1})\bar{\chi}(n_{2})}{(n_{1}n_{2})^{\sigma_{0}}\log
n_{1}\log n_{2}}\right)$ $\displaystyle=$
$\displaystyle\sum_{\chi(\bmod{q})}\left(\sum_{2\leq n_{1}=n_{2}\leq
Y}\frac{\Lambda(n_{1})\Lambda(n_{2})}{(n_{1}n_{2})^{\sigma_{0}}\log n_{1}\log
n_{2}}\right)+O\left(\phi(q)\sum_{\begin{subarray}{c}2\leq n_{1}\neq n_{2}\leq
Y\\\ n_{1}\equiv
n_{2}(\bmod{q})\end{subarray}}\frac{\Lambda(n_{1})\Lambda(n_{2})}{(n_{1}n_{2})^{\sigma_{0}}\log
n_{1}\log n_{2}}\right)$ $\displaystyle\ll$ $\displaystyle\phi(q)\log\log Q$

Since we know that $q\asymp Q$ by recalling the definition of $Y$ we conclude
the above equation. Similarly, we have

$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{P}_{2}(\sigma_{0}+it)|^{2}\ll\phi(q)\log\log\log
Q$

and the first part of the lemma follows. For the second part we simply apply
the Stirling formula.

If $|z|\leq k$ then by Stirling formula

$\displaystyle\left|e^{z}-\sum_{0\leq K\leq 100k}\frac{z^{k}}{k!}\right|\leq
e^{-99k}$

For the second part we simply apply the Stirling formula by taking
$z=-\mathcal{P}_{1}(\sigma_{0}+it)$ and $k=\log\log q$ and (2) holds. ∎

Now it remains to establish the connection between $M(s,\chi)$ and
$\mathcal{P}(s,\chi)$. To do so in a similar way we decompose $M(s,\chi)$ as
well. But if we observe the definition of $M(s,\chi)$ we see that we need to
decompose $a(n)$ first.

$\displaystyle a_{1}(n)=\begin{cases}1&\text{if $n$ has at most $100\log\log
q$ prime factors with all $p\leq Y$ }\\\ 0&\text{otherwise }.\end{cases}$
$\displaystyle a_{2}(n)=\begin{cases}1&\text{if $n$ has at most
$100\log\log\log q$ prime factors with all $Y<p\leq X$ }\\\ 0&\text{otherwise
}.\end{cases}$

Therefore we have

$\displaystyle M(s,\chi)=M_{1}(s)M_{2}(s)$ $\displaystyle
M_{1}(s,\chi)=\sum_{n}\frac{\mu(n)a_{1}(n)\chi(n)}{n^{s}}$ $\displaystyle
M_{2}(s,\chi)=\sum_{n}\frac{\mu(n)a_{2}(n)\chi(n)}{n^{s}}$

######  Lemma 3.

For the primitive Dirichlet character $\chi$ modulo $q$ we have

$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{M}_{1}(\sigma_{0}+it)-M_{1}(\sigma_{0}+it)|^{2}\ll\phi(q)(\log
q)^{-60}$
$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{M}_{2}(\sigma_{0}+it)-M_{2}(\sigma_{0}+it)|^{2}\ll\phi(q)(\log\log
q)^{-60}$

###### Proof.

First we define

$\displaystyle\mathcal{M}_{1}(s,\chi)=\sum_{n}\frac{b(n)\chi(n)}{n^{s}}$

where $b(n)$ satisfies the following properties.

  1. (1)

$|b(n)|\leq 1$ for all $n$.

  2. (2)

$b(n)=0$ unless $n\leq Y^{100\log\log Q}$ has only prime factors below $Y$.

  3. (3)

$b(n)=\mu(n)a_{1}(n)$ unless $\Omega(n)>100\log\log Q$ or, $p\leq Y$ s.t
$p^{k}|n$ with $p^{k}>Y$.

Set $c(n)=(b(n)-\mu(n)a_{1}(n))\chi(n)$, then we have

$\displaystyle\sum_{\chi(\bmod{q})}|\mathcal{M}_{1}(\sigma_{0}+it)-M_{1}(\sigma_{0}+it)|^{2}$
$\displaystyle\ll$
$\displaystyle\phi(q)\sum_{n_{1}=n_{2}}\frac{|c(n_{1})\overline{c(n_{2})}|}{(n_{1}n_{2})^{\sigma_{0}}}+\sum_{n_{1}\neq
n_{2}}\frac{|c(n_{1})\overline{c(n_{2})}|}{(n_{1}n_{2})^{\sigma_{0}}}$
$\displaystyle\ll$ $\displaystyle\phi(q)(\log q)^{-60}$

∎

Since we are dealing with the $q$-aspect here for the Dirichlet $L$-functions
we need to normalize with $\phi(q)$ because of the orthogonality relation of
the Dirichlet characters.

Proof of Proposition 3: It follows from (2) that

$\displaystyle\mathcal{M}_{1}(\sigma_{0}+it)=\exp(-\mathcal{P}_{1}(\sigma_{0}+it))(1+O(\log
Q)^{-99})$

and by (2) we can write

$\displaystyle(\log Q)^{-1}\ll|\mathcal{M}_{1}(\sigma_{0}+it)|\ll\log Q.$

Combining these two equations we get

$\displaystyle M_{1}(\sigma_{0}+it)=$
$\displaystyle\mathcal{M}_{1}(\sigma_{0}+it)+O((\log Q)^{-25})$
$\displaystyle=$ $\displaystyle\exp(-\mathcal{P}_{1}(\sigma_{0}+it))(1+O(\log
Q)^{-20})$

Similarly, we have

$\displaystyle M_{2}(\sigma_{0}+it)=$
$\displaystyle\mathcal{M}_{2}(\sigma_{0}+it)+O((\log\log Q)^{-25})$
$\displaystyle=$
$\displaystyle\exp(-\mathcal{P}_{2}(\sigma_{0}+it))(1+O(\log\log Q)^{-20})$

Recall the decomposition of $\mathcal{M}(f,s)$ and $\mathcal{P}(f,s)$, by
multiplying these estimates we obtain

$\displaystyle
M(\sigma_{0}+it)=\exp(-\mathcal{P}()\sigma_{0}+it)(1+O(\log\log)^{-20})$

completes the proof of the proposition. ∎

##  5\. Proof of Proposition 4

We do not follow the same method established in [1] to prove this proposition.
The proof of this proposition relies on [8].

Consider the approximate functional equation for Dirichlet $L$-function [9],
we have

$\displaystyle L(\sigma_{0}+it,\chi)=\sum_{\begin{subarray}{c}n\leq q\\\
q\asymp Q\end{subarray}}\frac{\chi(n)}{n^{\sigma_{0}+it}}+O(q^{-\frac{1}{2}})$

Then

$\displaystyle\sum_{\chi(\bmod{q})}\sum_{1\leq n\leq
q^{\epsilon}}\frac{\chi(n)}{n^{\sigma_{0}+it}}\sum_{1\leq m\leq
q^{\epsilon}}\frac{\mu(m)a(m)\chi(m)}{m^{\sigma_{0}+it}}+O(q^{\frac{1}{2}+\epsilon})$

From the orthogonality relation of the Dirichlet characters and by the
definition of $a(n)$ the above equation is $\ll\phi(q)\log
q+O(q^{\frac{1}{2}+\epsilon})$.

Expanding (1) we get

$\displaystyle\sum_{\chi(\bmod{q})}|1-L(\sigma_{0}+it,\chi)M(\sigma_{0}+it,\chi)|^{2}$
(5) $\displaystyle\sim$
$\displaystyle\sum_{\chi(\bmod{q})}|L(\sigma_{0}+it,\chi)M(\sigma_{0}+it,\chi)|^{2}-\phi(q)\log
q+O(q^{\frac{1}{2}+\epsilon})+o(1)$

In order to prove Proposition 4, we have to prove that the first term of (5)
is $\sim\phi(q)\log q$. We prove this using the argument given in [8].
Consider the integral

$\displaystyle I_{\chi}=\int|L(s,\chi)M(s,\chi)|^{2}\Phi(t)dt$

where $\Phi(t)$ is smooth, $\Phi(t)\geq 0$ with

$\displaystyle\hat{\Phi}(1)=\int_{-\infty}^{\infty}\Phi(t)dt>0$

and

$\displaystyle(1+|t|)^{j}\Phi^{(j)}(t)\ll\left(1+\frac{|t|}{T}\right)^{-A}$

for any $j\geq 0$ and any $A\geq 0$ the implied constant depending on $j$ and
$A$.

Averaging over the characters as argued in [8] we have

$\displaystyle\frac{1}{\phi(q)}\sum_{\chi(\bmod{q})}I_{\chi}\sim c\text{
}\hat{\Phi}(1)$

As we know that there exist no perfect mollifier (in which case we have $c=1$)
and $\hat{\Phi}(1)\asymp T$ where $(\log q)^{6}\leq T\leq(\log q)^{A}$, we can
conclude that the first term of (5) is $\sim\phi(q)\log q$, completes the
proof. ∎

######  Remark.

In [8], the authors have set $G(s,\chi)=L(s,\chi)+\lambda L^{\prime}(s,\chi)$
for $\lambda=1/r\log\mathbf{q}$ and put $G(s,\chi)$ in the integration
$I_{\chi}$. Due to the mollifier we have set in our paper we can put
$L(s,\chi)$ insted of $G(s,\chi)$ in the integration $I_{\chi}$.

##  6\. Conclusion

In this paper, we have proved Selberg’s central limit theorem for Dirichlet
$L$-functions of $q$-aspect. Since we have the sub-convexity bound for
Dirichlet $L$-functions of $q$-aspect, we have all the necessary tools we need
to prove this result. Notice that to prove our main result, we heavily rely on
the orthogonality relation of the Dirichlet character. The proof for
$q$-aspect is almost similar to the proof of level aspect but we have proved
the last step (or Proposition 4) differently.

One can prove this result for quadratic Dirichlet $L$-functions twisted by the
primitive Dirichlet character $\chi$ for $q$-aspect. It is the most
generalized (as well as difficult) possible case to prove Selberg’s central
limit theorem for $GL(3)$ $L$-functions of $q$-aspect. We don’t have much
information on the shifted convolution problem for higher degree
$L$-functions, so the proof for these cases can not be seen easily.

Moreover, one can prove the independence of the Dirichlet $L$-functions for
$q$-aspect associated with distinct primitive Dirichlet characters. The proof
technique is similar to given in [3, 6].

## Acknowledgement

I would like to express my gratitude to my primary supervisor, Dr Gihan
Marasingha, who guided me throughout this project.

## References

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