# Distributed Partial Quantum Consensus of Qubit Networks with Connected
Topologies

Xin Jin, , Zhu Cao, Yang Tang, , Jürgen Kurths  This work was supported in
part by the National Natural Science Foundation of China under Grant 62233005
and Grant 12105105, the Young Elite Scientist Sponsorship Program by Cast
under Grant YESS20220198, the Natural Science Foundation of Shanghai under
Grant 21ZR1415800, the Shanghai Sailing Program under Grant 23YF1409600 and
Grant 21YF1409800, the Fundamental Research Funds for the Central Universities
under Grant 2024SMECP06, and the startup fund from East China University of
Science and Technology under Grant YH0142214. (Corresponding authors: Zhu Cao,
Yang Tang.) Xin Jin is with the Key Laboratory of Smart Manufacturing in
Energy Chemical Process, Ministry of Education, East China University of
Science and Technology, Shanghai, 200237, China, and with the Research
Institute of Intelligent Complex Systems, Fudan University, Shanghai, 200433,
China. Zhu Cao and Yang Tang are with the Key Laboratory of Smart
Manufacturing in Energy Chemical Process, Ministry of Education, East China
University of Science and Technology, Shanghai 200237, China (e-mail:
<EMAIL_ADDRESS>tangtany@gmail.com).Jürgen Kurths is with the Potsdam
Institute for Climate Impact Research, 14473 Potsdam, Germany, and Institute
of Physics, Humboldt University of Berlin, 12489 Berlin, Germany.

###### Abstract

In this paper, we consider the partial quantum consensus problem of a qubit
network in a distributed view. The local quantum operation is designed based
on the Hamiltonian by using the local information of each quantum system in a
network of qubits. We construct the unitary transformation for each quantum
system to achieve the partial quantum consensus, i.e., the directions of the
quantum states in the Bloch ball will reach an agreement. A simple case of
two-qubit quantum systems is considered first, and a minimum completing time
of reaching partial consensus is obtained based on the geometric configuration
of each qubit.  Furthermore, we extend the approaches to deal with the more
general $N$-qubit networks. Two partial quantum consensus protocols, based on
the Lyapunov method for chain graphs and the geometry method for connected
graphs, are proposed. The geometry method can be utilized to deal with more
general connected graphs, while for the Lyapunov method, the global consensus
can be obtained.  The numerical simulation over a qubit network is
demonstrated to verify the validity and the effectiveness of the theoretical
results.

###### Index Terms:

quantum system, consensus, qubit network, connected graphs

##  I Introduction

Consensus problems have been studied in the field of distributed computing in
the early stage, and have wide applications in distributed control and
optimization of networked systems [1, 2, 3, 4]. In the past decade, the
quantum network has been an emerging concept both in theoretical studies and
physical implementations of distributed quantum computing[5], secure
communication [6], and quantum machine learning [7, 8, 9, 10, 11, 12]. In a
quantum network, quantum information is generated, processed, and stored
locally in quantum nodes, which are linked through communication channels
[13].

Seeking the quantum consensus protocol, which ensures the agreement in a
quantum network, has been a fundamental step in the quantum applications [14].
The pioneering work in quantum consensus has been made in [15], which defines
four types of quantum consensus states. Based on the classical gossip
algorithm, a quantum gossip consensus protocol is proposed, which can achieve
the symmetric state consensus. Two improved gossip-like consensus algorithms
are proposed in [16] to reach a purer output state and guarantee the symmetric
state consensus simultaneously. The connection between quantum consensus and
classical consensus is initially established by introducing an induced graph
in[17, 18]. Here, quantum dynamics are modeled using master equations in
Lindblad form, and interactions among quantum subsystems are described using
the swapping operator. This research gives necessary and sufficient conditions
for exponential and asymptotic (symmetric-state) consensus convergence for
undirected connected graphs.  Furthermore, in [19], it considers the average
consensus of quantum networks under the balanced directed graph. The result is
extended to switching interactions based on the properties of cut-balanced
graphs. However, the induced graph in [17, 19] is modeled by a $2^{N}\times
2^{N}$ dimensional Laplacian matrix, of which dimension exponentially
increases with the number of nodes in quantum networks.  The exponential
increase in the dimension of the Laplacian matrix of the induced graph will
induce a heavy computation burden as the number of qubits is large. The
converge rate of the consensus algorithm based on quantum consensus master
equations (QCMEs) is investigated in [20, 21, 22], and the result shows the
optimal convergence rate of quantum consensus (the second smallest eigenvalue
of the Laplacian matrix) can be obtained from the Laplacian matrix of the
partitions, which is not directly related to the Laplacian matrix of the whole
induced graph.  In [14], an open quantum network of qubits is considered with
sequential measurements described by a probabilistic Boolean network. Some
structure characteristics for induced Boolean networks are established from
the relationship between the measurement bias and the master equation.

It should be noted that the above works[16, 17, 19, 21, 22, 14] consider the
qubit network as a whole instead of from the perspective of a multi-agent
system. More specifically, the consensus protocol was proposed based on a
global operator performed on a tensor product state of the quantum network,
whose dimension is exponentially increasing with the network scale [16, 17,
19, 21, 22]. This is different from the distributed consensus protocol in
multi-agent systems, which only governs the local behavior of each agent. In
other words, the quantum consensus protocol in [16, 17, 19, 21, 22] cannot be
used in a fully distributed way. This makes the results less feasible in a
large-scale qubit network such as the quantum internet [23].

On the other hand, the dynamics of the quantum system in [16, 17, 19, 21, 22]
assume that the Hamiltonian is zero, which means that the local operation is
not considered for each quantum system.  Designing a Hamiltonian in the
quantum feedback control is an effective approach in quantum control systems
[24, 25, 26, 27]. The controllability of quantum systems is defined in [13].
The main idea is to design the Hamiltonian driving the quantum states to the
desired ones. Lyapunov functions are powerful tools for feedback control
design and asymptotic convergence analysis in classical control problems.
However, in quantum control systems, the measurement action usually destroys
the state being measured, which brings difficulties in designing the feedback
control based on Lyapunov functions directly [28, 29, 30].  A feedback design
and open-loop control strategy has been proposed in the existing research
[31]. Nevertheless, implementing this strategy in complex network systems,
such as multi-agent systems, poses significant challenges. This is due to the
fact that the control action of each agent is correlated with the interaction
of neighbors, which cannot be determined on the computer previously.

Motivated by the above discussions, we exploit the quantum consensus problem
from a multi-agent system perspective. The main difficulty of this approach is
to construct the local operation which is only based on the local information
of each quantum system. In this work, we assume that each qubit will interact
with its neighboring qubits through weak measurements. Such a measurement can
be made to determine the density matrix of quantum systems with high accuracy,
while slightly disturbing the quantum state itself [32]. The transition
between the pure quantum states can be completed by a unitary transformation
based on the local operator. Due to the entanglement of quantum states, it is
difficult to define the quantum consensus based on the classical counterpart.
Hence, by decoupling quantum states into the pure states and the maximum mixed
states based on the convexity property of the density matrix, we consider
reaching partial quantum consensus based on the local operation. The main
contributions of this work are summarized as follows:

1) A minimum time of completing the unitary transformation for a two-qubit
system is obtained based on the geometric configuration of the quantum state,
and the corresponding Hamiltonian is constructed for each qubit. Compared with
the minimum time of the unitary transformation for two-qubit systems in the
recent work [33], we only need to perform a single rotation which is more
efficient.

2) A partial quantum consensus protocol for $N$-qubit networks with chain
topologies is proposed by means of the Lyapunov analysis. The Lyapunov
function is designed based on the Euclidean distance between two qubits.
Compared with the quantum steering control based on Lyapunov approaches in
[28], we propose a quantum consensus protocol of a qubit network. The main
difficulty lies in deriving the Hamiltonian operator of each qubit since it is
both affected by quantum mechanics and network topologies.

3) A geometric approach is proposed which builds a connection between the
partial quantum consensus and consensus on the 2-sphere. Then, the partial
quantum consensus protocol with general connected graphs is constructed based
on the perpendicular relationship between the tangent direction and the
rotation axis of a point on the Bloch ball. Compared with the quantum
consensus protocols in [16, 17, 19, 21, 22], we consider the quantum consensus
problem from a multi-agent perspective and design a distributed consensus
protocol by only using the neighbors’ quantum state.

The remaining part of this paper is organized as follows. In Section II, the
preliminary knowledge and problem formulation are presented. The partial
quantum consensus of two-qubit systems is investigated in Section III. The
main results of partial quantum consensus of $N$-qubit network are shown in
Section IV. Section V gives the numerical simulations and the conclusion is
drawn in Section VI finally.

Notations: $\mathbb{R}^{N}$ and $\mathbb{R}^{{N}\times{N}}$ represent the
Euclidean vector space and real matrix space, where $N$ is a positive integer
number. $\mathbb{N}^{+}$ denotes the positive integer number. $\mathbb{R}^{+}$
represents the positive real number. $\mathbb{C}^{N}$ represents the $N$
dimensional complex vector space.
$\mathbf{1}_{N}=[1,...,1]^{\top}\in\mathbb{R}^{N}$. For a vector
$\mathbf{x}=[x_{1},...,x_{N}]^{\top}\in\mathbb{R}^{N}$, $\|\mathbf{{x}}\|$
denotes the Euclidean norm, which is defined as
$\|\mathbf{x}\|=\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{N}^{2}}$.
$\operatorname{tr}(\mathbf{A})$ represents the trace of matrix $\mathbf{A}$.
$\times$ represents the cross product of two vectors in Euclidean space.
$\mathbb{S}^{2}$ represents the 2-sphere defined as
$\mathbb{S}^{2}=\\{\mathbf{x}\in\mathbb{R}^{3}:\mathbf{x}^{{\top}}\mathbf{x}=1\\}$.
A three-dimensional special orthogonal group is denoted as
$\mathbb{SO}(3)=\\{\mathbf{R}\in\mathbb{R}^{3\times
3}:\mathbf{R}^{{\top}}\mathbf{R}=\mathbf{I}_{3},\text{det}(\mathbf{R})=1\\}$.
The set of three-dimensional skew symmetric matrices is denoted as
$\mathfrak{so}(3)=\\{\mathbf{S}\in\mathbb{R}^{3\times
3}:\mathbf{S}^{\top}=-\mathbf{S}\\}$. The two dimensional special unitary
group is denoted as $\mathbb{SU}(2)=\\{\mathbf{U}\in\mathbb{C}^{2\times
2}:\mathbf{U}^{{\dagger}}\mathbf{U}=\mathbf{I}_{2},\text{det}(\mathbf{U})=1\\}$.
$\text{Skew}(\cdot)$ represents a mapping from a vector
$\mathbf{x}=[x_{1},x_{2},x_{3}]^{\top}\in\mathbb{R}^{3}$ to a skew symmetric
matrix $\mathbf{X}=\left[\begin{matrix}0&x_{3}&-x_{2}\\\ -x_{3}&0&x_{1}\\\
x_{2}&-x_{1}&0\\\ \end{matrix}\right].$

##  II Preliminaries and Problem Formulation

###  II-A Quantum systems

1) Pure states and mixed states: In quantum mechanics, a quantum state is
represented as a unit vector $|\psi\rangle$ in a Hilbert space $\mathcal{H}$,
where $|\cdot\rangle$ is Dirac’s notation (called a ket). Its dual vector is
denoted as $|\psi\rangle^{\dagger}=\langle\psi|$ (called a bra). The inner
product $\langle\psi|\phi\rangle$ which maps a pair of vectors to a complex
number $C$. Particularly, we have $\langle\psi|\psi\rangle=1$. A quantum state
that can be represented by a unit vector $|\psi\rangle$ is called a pure
state. In an open quantum system, a state could be a mixture of a number of
pure states which is called the mixed state. Supposed that a quantum system is
in a state $|\psi_{i}\rangle$ with probability $p_{i}$ such that
$\sum_{i}p_{i}=1$ where $0<p_{i}\leq 1$, we can represent it by using the
density matrix

$\displaystyle\bm{\rho}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$ (1)

Mathematically, a density matrix is a self-adjoint operator
$\bm{\rho}=\bm{\rho}^{\dagger}$ and satisfies $\text{tr}(\bm{\rho})=1$.
Generally, a state in quantum systems can be represented by a density operator
either it is a pure state or a mixed state. For a pure state, the density
matrix can be expressed as $\bm{\rho}=|\psi\rangle\langle\psi|$, where
$\text{tr}(\bm{\rho}^{2})=1$. Note that if the state is not pure,
$\text{tr}(\bm{\rho}^{2})<1$.

Due to the convex property of the density matrix [34], we can express
$\bm{\rho}$ as a convex combination of other two states

$\displaystyle\bm{\rho}=P\frac{1}{2}\mathbf{I}_{2}+(1-P)\bm{\rho}^{\prime},$
(2)

where $P$ is a positive real number, $\mathbf{I}_{2}$ is the maximally mixed
state and $\bm{\rho}^{\prime}$ is the pure state.

2) Qubits and Quantum networks: The smallest unit of quantum information is a
qubit in a two-dimensional Hilbert space $\mathcal{H}_{2}$. Suppose that the
$|0\rangle$ and $|1\rangle$ are two-dimensional orthonormal basis. A qubit is
expressed as

$\displaystyle\mathbf{q}=a|0\rangle+b|1\rangle,$ (3)

where $a,b$ are two complex numbers such that $|a|^{2}+|b|^{2}=1$.

In this work, we consider a quantum network with $N$ qubits. It is worth
mentioning that the qubit in (3) can be interpreted as the spin-$\frac{1}{2}$
object. A spin-$\frac{1}{2}$ representation of a qubit expressed in the world
frame $\mathcal{W}$ is

$\displaystyle|\psi_{i}(\theta_{i},\phi_{i})\rangle^{\mathcal{W}}=\left(\begin{matrix}e^{-i\frac{\phi_{i}}{2}}\cos\frac{\theta_{i}}{2}\\\
e^{i\frac{\phi_{i}}{2}}\sin\frac{\theta_{i}}{2}\end{matrix}\right),$ (4)

where the polar angle $\theta_{i}$ and the azimuthal angle $\phi_{i}$ describe
the orientation of the spin in a three-dimensional space in Fig. 1A.

Each qubit related to a local body frame $\mathcal{B}_{i}$ is denoted as
$|\psi_{i}(\theta_{i},\phi_{i})\rangle^{\mathcal{B}_{i}}$. The relationship
between qubits in different frames can be expressed as
$|\psi_{i}(\theta_{i},\phi_{i})\rangle^{\mathcal{W}}=\mathbf{U}_{i}|\psi_{i}(\theta_{i},\phi_{i})\rangle^{\mathcal{B}_{i}},$
where $\mathbf{U}_{i}\in\mathbb{SU}(2)$ is a unitary matrix describing the
rotation from the body frame $\mathcal{B}_{i}$ to the world frame
$\mathcal{W}$. More generally,
$\bm{\rho}_{i}^{\mathcal{W}}=\mathbf{U}_{i}\bm{\rho}_{i}^{\mathcal{B}_{i}}\mathbf{U}_{i}^{\dagger}.$
For the presentation clarity, we omit the $\mathcal{W}$ if the state is
expressed in the world frame in the following.

The evolution of the spin-$\frac{1}{2}$ object in time is governed by the
$Schr\ddot{o}dinger\;equation$

$\displaystyle\frac{\text{d}}{\text{dt}}|\psi_{i}(t)\rangle=-i\mathbf{H}_{i}|\psi_{i}(t)\rangle,$
(5)

where $\mathbf{H}_{i}$ is a constant self-adjoint operator called the
Hamiltonian. Hence, we have

$\displaystyle|\psi_{i}(t)\rangle=\mathbf{U}_{i}(t)|\psi_{i}(0)\rangle,$ (6)

where $\mathbf{U}_{i}(t)=e^{-it\mathbf{H}_{i}}$ is a unitary transform.

Figure 1: A) A density matrix can be seen as a point on the Bloch ball and a
pure state lies on the sphere of the Bloch ball. B) The trajectories of two
qubits evolving on the Bloch ball.

###  II-B Graph Theory

The interaction among $N$ qubits in a network can be considered as a graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, where $\mathcal{V}=\\{1,...,N\\}$
denotes the node set and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$
denotes the edge set. A directed edge $(i,j)\in\mathcal{E}$ means that there
exists a communication link from node $i$ to node $j$. A graph is called
undirected if the edge $(i,j)\in\mathcal{E},i,j\in\mathcal{V}$ then
$(j,i)\in\mathcal{E}$, and it is connected if there exists at least one path
for any two nodes in the graph. An adjacency matrix associated with the graph
$\mathcal{G}$ is defined as $\mathbf{A}=\\{a_{ij}\\}\in\mathbb{R}^{N\times
N},i,j=1,...,N$, where $a_{ij}>0$ if $\\{j,i\\}\in\mathcal{E}$, and $a_{ij}=0$
otherwise. The node $j$ is node $i$’s neighbor if there is an edge connected
from node $j$ to node $i$. A neighboring set
$\mathcal{N}_{i}=\\{j\in\mathcal{V},(j,i)\in\mathcal{E}\\}$ consists of all
the neighbors of node $i$.

###  II-C Problem Formulation

In an $N$-qubit network, we express the state of each qubit as a density
matrix $\bm{\rho}_{i},i\in\mathcal{V}$. Based on the property of the density
matrix, we can present each qubits’ state in the general form

$\displaystyle{}\bm{\rho}_{i}=\frac{1}{2}\Big{(}\mathbf{I}_{2}+\mathbf{p}_{i}\cdot\vec{\bm{\sigma}}\Big{)},$
(7)

where $\mathbf{p}_{i}=[p_{1},p_{2},p_{3}]^{\top}\in\mathbb{R}^{3}$ is a vector
which satisfies $0\leq\|\mathbf{p}_{i}\|\leq 1$ and
$\mathbf{p}_{i}\cdot\vec{\bm{\sigma}}=p_{1}\bm{\sigma}_{1}+p_{2}\bm{\sigma}_{2}+p_{3}\bm{\sigma}_{3}$,
$\bm{\sigma}_{i},i=1,2,3$ are Pauli matrices defined as

$\displaystyle\bm{\sigma}_{1}=\left[\begin{matrix}0&1\\\ 1&0\\\
\end{matrix}\right],\bm{\sigma}_{2}=\left[\begin{matrix}0&-i\\\ i&0\\\
\end{matrix}\right],\bm{\sigma}_{3}=\left[\begin{matrix}1&0\\\ 0&-1\\\
\end{matrix}\right].$

Therefore, there is a one-to-one mapping between a density matrix and a point
in a three-dimensional unit ball (called Bloch ball) presented in Fig. 1A.

Note that the quantum state for each system $\bm{\rho}_{i}$ can be a mixed
state. According to (2), we can separate the state into a maximally mixed
state and a pure state. Based on (4), we can rewrite the pure state in the
following form,

$\displaystyle{}\bm{\rho}^{\prime}_{i}(\mathbf{u}_{i})=\frac{1}{2}\Big{(}\mathbf{I}_{2}+\mathbf{u}_{i}\cdot\vec{\bm{\sigma}}\Big{)},$
(8)

where
$\mathbf{u}_{i}=\left(\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i}\right)$
is a unit vector. From (8), we know that each pure state
$\bm{\rho}^{\prime}_{i}(\mathbf{u}_{i})$ lies on the sphere of the Bloch ball
shown in Fig. 1A. Thus, a pure state
$\bm{\rho}^{\prime}_{i}=|\psi_{i}\rangle\langle\psi_{i}|$ can be transferred
to another pure state
$\bm{\rho}^{\prime}_{j}=|\psi_{j}\rangle\langle\psi_{j}|$ through a unitary
transformation $\mathbf{U}_{ij}$, i.e.,
$\bm{\rho}^{\prime}_{j}=\mathbf{U}_{ij}\bm{\rho}^{\prime}_{i}\mathbf{U}^{\dagger}_{ij}$,
and it is also possible to construct a unitary transformation for each quantum
system $\bm{\rho}_{i}$ to drive the part of the pure state of each quantum
system to partial quantum consensus. More specifically, assuming that the
unitary transformation for the pure state $\rho_{i}^{\prime}$ in the part of
the mixed state $\rho_{i}$ is defined as $\mathbf{U}_{i}$, i.e.,
$\mathbf{U}_{i}\bm{\rho}^{\prime}_{i}\mathbf{U}_{i}^{\dagger}=\bar{\bm{\rho}}_{i}^{\prime}$,
where $\bar{\bm{\rho}}_{i}^{\prime}$ denotes the consensus pure state. Then,
by utilizing the unitary transformation to the mixed state $\rho_{i}$, we have

$\displaystyle{}\mathbf{U}_{i}\bm{\rho}_{i}\mathbf{U}_{i}^{\dagger}=$
$\displaystyle\mathbf{U}_{i}\left(P\frac{1}{2}\mathbf{I}_{2}+(1-P)\bm{\rho}^{\prime}_{i}\right)\mathbf{U}_{i}^{\dagger}$
$\displaystyle=$ $\displaystyle
P\frac{1}{2}\mathbf{I}_{2}+(1-P)\bar{\bm{\rho}}^{\prime}_{i}.$ (9)

From (II-C), we know that based on the unitary transformation
$\mathbf{U}_{i}$, the pure state $\bm{\rho}_{i}^{\prime}$ achieves the
consensus. This allows us to only design the unitary transformation for the
pure state $\bm{\rho}_{i}^{\prime}$ in the part of the mixed state
$\bm{\rho}_{i}$ to achieve the partial quantum consensus as defined in
Definition 1. The pure state $\bm{\rho}^{\prime}_{i}$ can be calculated as
$\bm{\rho}^{\prime}_{i}=\frac{1}{2}\Big{(}\mathbf{I}_{2}+\frac{\mathbf{p}_{i}}{\|\mathbf{p}_{i}\|}\cdot\vec{\bm{\sigma}}\Big{)},$
where $\mathbf{p}_{i}$ is defined in (7). The relationship between the quantum
state $\bm{\rho}_{i}$ and its related pure state $\bm{\rho}^{\prime}_{i}$ is
shown in Fig. 1A.

###### Definition 1

[17, 22] Supposed that a quantum network of $N$-qubit which is governed by
(5), the partial quantum consensus is achieved if all the quantum states point
in the same direction on the Bloch ball, i.e.,

$\displaystyle\bm{\rho}^{\prime}_{1}=\bm{\rho}^{\prime}_{2}=...=\bm{\rho}^{\prime}_{N}=\bar{\bm{\rho}}^{\prime}.$
(10)

where $\bar{\bm{\rho}}^{\prime}$ is the consensus state of all qubits.

In this problem, we consider a multi-agent system consisting of $N$ qubits,
where the state of each qubit is represented by a density matrix
$\bm{\rho}_{i}\in\mathbb{C}^{2\times 2}$, and the quantum state of the whole
qubit network is a product state. We assume that the local interaction between
qubits is made through the weak measurement [35, 32]. For the pure states, the
dynamics is governed by the $Schr\ddot{o}dinger\;equation$ (5). The main
contribution of this work is to design the interactive Hamiltonian
$\mathbf{H}_{i},\;i=1,...,N$, which is also constructing the unitary
transformation of each qubit to complete the partial quantum consensus.

In the interactive Hamiltonian of each qubit, the local information, which is
its own quantum state $\bm{\rho}^{\prime}_{i}$ and its neighbors’ quantum
states $\bm{\rho}^{\prime}_{j},j\in\mathcal{N}_{i}$, will be used. To obtain
the neighbors’ quantum states, we assume that each qubit can make a weak
measurement to its neighbors, while not destroying its neighbors’ state. It is
worth mentioning that this Hamiltonian $\mathbf{H}_{i}$ is similar to the
protocol in multi-agent systems, which specifies the information that each
qubit interacts with its neighbors. Hereafter, we will call the interactive
Hamiltonian $\mathbf{H}_{i},\;i=1,...,N$, in a qubit network as the _quantum
protocol_.

###### Remark 1

The quantum consensus algorithm is proposed based on QCMEs in [18, 20, 17,
19]. Based on a swapping operator, the reduced state synchronization is
achieved. However, this swapping operator is a global operation for the tensor
product state of qubit networks. Note that this operation is not suitable in
the scenarios when two nodes are located in different places which are far
away from each other such as quantum internet. To seek a fully distributed
quantum consensus algorithm, we aim to propose a local operator based on the
local information for each qubit to reach consensus. From this perspective, we
design the local interactive Hamiltonian of each qubit. In order to reach
quantum consensus, the state information of neighboring qubits should be
included in the Hamiltonian design of each subsystem. To facilitate the
theoretical analysis, we use the weak measurement assumption in this work,
which aims to achieve the local interaction in qubit networks while preserving
quantum coherence. Specifically, qubit $i$ can measure the density matrix
$\bm{\bm{\rho}}_{j}$ of qubit $j$ while not destroying the state
$\bm{\bm{\rho}}_{j}$ through the weak measurement. The details on weak
measurements are referred to [35].

Figure 2: The relationships among main results.

##  III Partial quantum consensus for two-qubit systems

In this section, we firstly consider the simple case where quantum networks
only have two qubits. The settling time is one of the important criteria in
the control task. In this part, we firstly give an explicit expression for the
minimum time that two qubits needed to complete the unitary transformation.
This problem can be seen as a stepping stone for the partial quantum consensus
protocol design for the quantum networks with more qubits. A diagram of the
main theoretical results is shown in Fig. 2.

For two-qubit systems, the quantum consensus in Definition 1 reduces to
$|\psi_{i}(\theta_{i},\phi_{i})\rangle=|\psi_{j}(\theta_{j},\phi_{j})\rangle,i=1,j=2$.
Let $\mathbf{s}_{i}(\theta_{i},\phi_{i})$ and
$\mathbf{s}_{j}(\theta_{j},\phi_{j})$ be the corresponding two points of the
qubits in the spherical coordinates. A direct idea is to construct a single
unitary transformation $\mathbf{U}_{ij}$ from
$\mathbf{s}_{i}(\theta_{i},\phi_{i})$ to
$\mathbf{s}_{j}(\theta_{j},\phi_{j})$. A unitary transformation
$\mathbf{U}_{ij}(t)\in\mathbb{SU}(2)$ can represent a rotation $\theta_{ij}$
of the spin around the axis $\mathbf{n}_{ij}$ in the following form,

$\displaystyle\mathbf{U}_{ij}(\mathbf{n}_{ij},\theta_{ij})=e^{-i\frac{\theta_{ij}}{2}\mathbf{n}_{ij}\cdot\vec{\bm{\sigma}}},$
(11)

where $\vec{\bm{\sigma}}$ are the Pauli matrices. We can compare (11) with
(6). The Hamiltonian can be designed as
$\mathbf{H}_{i}=\mathbf{n}_{ij}\cdot\vec{\bm{\sigma}}$, then the completing
time of the unitary transformation will be ${\theta_{ij}}$. The trajectories
of two qubits under the unitary transformation are shown in Fig. 1B. The
rotation axis $\mathbf{n}_{12}$ is designed based on the cross product of both
direction vectors of $\bm{\rho}^{\prime}_{1}$ and $\bm{\rho}^{\prime}_{2}$. It
shows that the two pure states can rotate to the same point (red marker) on
the Bloch ball.

###### Theorem 1

Suppose that a two-qubit system governed by (5). There exists a single unitary
transformation $\mathbf{U}_{ij}(\mathbf{n}_{ij},\theta_{ij})$ such that
$\mathbf{U}_{ij}(\mathbf{n}_{ij},\theta_{ij})|\psi_{i}(\theta_{i},\phi_{i})\rangle=\mathbf{U}_{ij}^{\dagger}(\mathbf{n}_{ij},\theta_{ij})|\psi_{j}(\theta_{j},\phi_{j})\rangle$.
Furthermore, the minimum time for completing the rotation is

$\displaystyle{}T=\frac{1}{2}\arccos(\mathbf{s}_{i}\cdot\mathbf{s}_{j}),$ (12)

where
$\mathbf{s}_{i}(\theta_{i},\phi_{i})=\left(\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i}\right)$
and
$\mathbf{s}_{j}(\theta_{j},\phi_{j})=\left(\sin\theta_{j}\cos\phi_{j},\sin\theta_{j}\sin\phi_{j},\cos\theta_{j}\right).$

###### Proof:

Since we know that there exists a one-to-one mapping from the qubit
$|\psi_{i}\rangle$ to the vector $\mathbf{s}_{i}(\theta_{i},\phi_{i})$ in the
spherical coordinate, the two vectors $\mathbf{s}_{i}(\theta_{i},\phi_{i})$
and $\mathbf{s}_{j}(\theta_{j},\phi_{j})$ can form a plane $\mathbb{P}$.
Hence, we can construct a rotation along the direction of the normal vector on
the plane $\mathbb{P}$ and the rotation angle is $\theta_{ij}$. The unitary
transformation of the rotation can be expressed as
$\mathbf{U}_{ij}(\mathbf{n}_{ij},\theta_{ij})=e^{-i\frac{\theta_{ij}}{2}\mathbf{n}_{ij}\cdot\vec{\bm{\sigma}}}$,
where

$\displaystyle\mathbf{n}_{ij}=$
$\displaystyle\;\mathbf{s}_{i}(\theta_{i},\phi_{i})\times\mathbf{s}_{j}(\theta_{j},\phi_{j})$
$\displaystyle=$
$\displaystyle\;\left[\begin{matrix}\sin\theta_{i}\cos\theta_{j}\sin\phi_{i}-\cos\theta_{i}\sin\theta_{j}\sin\phi_{i}\\\
\cos\theta_{i}\sin\theta_{j}\cos\phi_{j}-\sin\theta_{i}\cos\theta_{j}\cos\phi_{i}\\\
\sin\theta_{i}\sin\theta_{j}\cos\phi_{i}\sin\phi_{j}-\sin\theta_{i}\sin\theta_{j}\sin\phi_{i}\cos\phi_{j}\end{matrix}\right].$

The Hamiltonian for qubit $i$ and qubit $j$ can be determined as
$\mathbf{n}_{ij}$ and $-\mathbf{n}_{ij}$, respectively. Thus, the minimum time
for completing the unitary transformation is
$T=\frac{1}{2}\arccos(\mathbf{s}_{i}(\theta_{i},\phi_{i})\cdot\mathbf{s}_{j}(\theta_{j},\phi_{j}))$,
which completes the proof. ∎

###### Remark 2

The minimum time is similar to the concept of the finite-time, which both
achieve the control task in a settling time [36]. The difference is that the
finite-time controller is designed based on an analytic solution of
differential equations while the minimum completing time in this work is
achieved by constructing a unitary transformation based on the geometric
configuration of qubits on the Bloch ball. A similar problem is considered in
[33], which derives an analytical expression of the minimum time for
completing the unitary transformation from the initial pure states to the
target pure states. However, the designing unitary transformation is completed
by the $x-z-x$ rotations, and the minimum time is based on the assumption with
the unbound control. In this work, we only design the Hamiltonian with a one-
step unitary transformation, which is more straightforward and efficient.

##  IV Partial quantum consensus for $N$-qubit networks

It is obvious that a two-qubit system is quite particular. In this section, we
consider the more general case where the number of qubits is $N$ by extending
the approaches in the two-qubit case. Two different topology cases are
investigated, respectively. For the chain graph, the Hamiltonian for each
quantum system is designed by the Lyapunov approach and the global convergence
result is obtained. However, it is difficult to generalize the result to a
connected graph based on the Lyapunov approach. Thus, for the general
connected graph, we build the relationship of this problem with the consensus
on 2-sphere and design the Hamiltonian based on the geometric configuration of
the quantum state on the Bloch ball while the local partial quantum consensus
is achieved.

###  IV-A Lyapunov-based quantum protocol design

The chain graph is firstly considered in the following. This configuration of
quantum networks can be seen in the physical applications such as the linear
ion trap, which is used as a component of a trapped ion quantum computer.

###### Assumption 1

The topology of an $N$-qubit network is a chain graph.

For each qubit, the quantum protocol is given as

$\displaystyle\mathbf{H}_{i}=\mathbf{n}_{i}^{(p)}\cdot\vec{\bm{\sigma}},$ (13)

where $p=1,2,3$ and

$\displaystyle\mathbf{n}^{(p)}_{i}=\left\\{\begin{matrix}&\mathfrak{R}\left(i\langle\psi_{1}|\bm{\sigma}_{p}|\psi_{2}\rangle\right),\;i=1,\\\
&\mathfrak{R}\left(i\langle\psi_{i}|\bm{\sigma}_{p}|\psi_{i+1}\rangle\right)-\mathfrak{R}\left(i\langle\psi_{i-1}|\bm{\sigma}_{p}|\psi_{i}\rangle\right),\\\
&i=2,...,N-1,\\\
&-\mathfrak{R}\left(i\langle\psi_{N-1}|\bm{\sigma}_{p}|\psi_{N}\rangle\right),\;i=N.\end{matrix}\right.$

Based on the quantum protocol (13), we have the following result.

###### Theorem 2

Suppose that an $N$-qubit network is governed by (5), and Assumption 1 is
satisfied. When using (13), the partial quantum consensus can be achieved
globally.

###### Proof:

Consider the Lyapunov function as follows

$\displaystyle V=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{N-1}\langle\psi_{i}-\psi_{i+1}|\psi_{i}-\psi_{i+1}\rangle$
$\displaystyle=$ $\displaystyle
N-1-\frac{1}{2}\sum_{i=1}^{N-1}\mathfrak{R}(\langle\psi_{i}|\psi_{i+1}\rangle).$
(14)

By taking the derivative of $V$, we can obtain that

$\displaystyle\dot{V}=$
$\displaystyle\sum_{i=1}^{N-1}-\mathfrak{R}\left(i\langle\psi_{i}|\mathbf{H}_{i}|\psi_{i+1}\rangle\right)+\mathfrak{R}\left(i\langle\psi_{i}|\mathbf{H}_{i+1}|\psi_{i+1}\rangle\right)$
$\displaystyle=$
$\displaystyle-\mathfrak{R}\left(i\langle\psi_{1}|\mathbf{H}_{1}|\psi_{2}\rangle\right)$
$\displaystyle+\sum_{i=2}^{N-1}\mathfrak{R}\left(i\langle\psi_{i-1}|\mathbf{H}_{i}|\psi_{i}\rangle\right)-\mathfrak{R}\left(i\langle\psi_{i}|\mathbf{H}_{i}|\psi_{i+1}\rangle\right)$
$\displaystyle+\mathfrak{R}\left(i\langle\psi_{N-1}|\mathbf{H}_{N}|\psi_{N}\rangle\right)$
$\displaystyle=$
$\displaystyle-\sum_{p=x,y,z}\mathbf{n}_{1}^{(p)}\cdot\mathfrak{R}\left(i\langle\psi_{1}|\bm{\sigma}_{p}|\psi_{2}\rangle\right)$
$\displaystyle+\sum_{i=2}^{N-1}\sum_{p=x,y,z}\mathbf{n}_{i}^{(p)}\cdot\Big{(}\mathfrak{R}\left(i\langle\psi_{i-1}|\bm{\sigma}_{p}|\psi_{i}\rangle\right)$
$\displaystyle-\mathfrak{R}\left(i\langle\psi_{i}|\bm{\sigma}_{p}|\psi_{i+1}\rangle\right)\Big{)}$
$\displaystyle+\sum_{p=x,y,z}\mathbf{n}_{N}^{(p)}\cdot\mathfrak{R}\left(i\langle\psi_{N-1}|\bm{\sigma}_{p}|\psi_{N}\rangle\right)$
$\displaystyle=$ $\displaystyle\;W_{1}+W_{2}+...+W_{N},$ (15)

where
$W_{1}=-\sum_{p=x,y,z}\mathbf{n}_{1}^{(p)}\cdot\mathfrak{R}\left(i\langle\psi_{1}|\bm{\sigma}_{p}|\psi_{2}\rangle\right)$,
$W_{i}=\sum_{p=x,y,z}\mathbf{n}_{i}^{(p)}\cdot\Big{(}\mathfrak{R}\left(i\langle\psi_{i-1}|\bm{\sigma}_{p}|\psi_{i}\rangle\right)-\mathfrak{R}\left(i\langle\psi_{i}|\bm{\sigma}_{p}|\psi_{i+1}\rangle\right)\Big{)},\;i=2,...,N-1$,
$W_{N}=\sum_{p=x,y,z}\mathbf{n}_{N}^{(p)}\cdot\mathfrak{R}\left(i\langle\psi_{N-1}|\bm{\sigma}_{p}|\psi_{N}\rangle\right)$.

Then, substituting (13) into the above equation, we have $W_{i}\leq 0,\forall
i=1,2,...,N$. Hence, the trajectories of the $N$-qubit network converge to the
invariant set
$S=\\{|\psi\rangle\in\mathbb{C}^{2N}\;|\;W_{i}=0,i=1,2,...,N\\}$.

For the first term

$\displaystyle W_{1}=$
$\displaystyle-\sum_{p=x,y,z}\Big{(}\mathfrak{R}\left(i\langle\psi_{1}|\bm{\sigma}_{p}|\psi_{2}\rangle\right)\Big{)}^{2}$
$\displaystyle=$
$\displaystyle-\sin^{2}\frac{1}{2}\left(\theta_{1}-\theta_{2}\right)-\sin^{2}\frac{1}{2}\left(\phi_{1}-\phi_{2}\right)\cos^{2}\frac{1}{2}\left(\theta_{1}-\theta_{2}\right)$
$\displaystyle=$ $\displaystyle\;0,$

we know that $\theta_{1}=\theta_{2}+2k\pi$ and
$\phi_{1}=\phi_{2}+2k\pi,k=0,1,2,...$, which means that
$|\psi_{1}\rangle=|\psi_{2}\rangle$. Due to
$\langle\psi_{i}|\bm{\sigma}_{p}|\psi_{i}\rangle=0$,
$\forall\psi_{i}\in\mathbb{C}^{2}$, we can obtain that

$\displaystyle
W_{2}=-\sum_{p=x,y,z}\Big{(}\mathfrak{R}\left(i\langle\psi_{2}|\bm{\sigma}_{p}|\psi_{3}\rangle\right)\Big{)}^{2}=0,$

which follows that $|\psi_{2}\rangle=|\psi_{3}\rangle$. Repeating this
procedure, one has

$\displaystyle
W_{i}=-\sum_{p=x,y,z}\Big{(}\mathfrak{R}\left(i\langle\psi_{i}|\bm{\sigma}_{p}|\psi_{i+1}\rangle\right)\Big{)}^{2}=0,\forall
i=3,...,N,$

which follows that $|\psi_{3}\rangle=|\psi_{4}\rangle=...=|\psi_{N}\rangle$.
Hence, we can conclude that the only equilibrium point in the invariant set
$S$ is $|\psi_{1}\rangle=|\psi_{2}\rangle=...=|\psi_{N}\rangle$, which means
that the $|\psi_{i}(t)\rangle-|\psi_{j}(t)\rangle\rightarrow 0,i,j=1,...,N$ as
time goes infinity.

Next, we show that the Hamiltonian
$\mathbf{H}_{i}=\mathbf{n}_{i}^{(p)}\cdot\bm{\sigma}$ designed above can be
used in the local frame. Due to
$|\psi_{i}\rangle=\mathbf{U}_{i}|\psi_{i}\rangle^{\mathcal{B}_{i}}$, we have
$\frac{\text{d}}{\text{dt}}|\psi_{i}(t)\rangle^{\mathcal{B}_{i}}=-i\mathbf{H}_{i,\mathcal{B}_{i}}|\psi_{i}(t)\rangle^{\mathcal{B}_{i}}$,
where
$\mathbf{H}_{i,\mathcal{B}_{i}}=\mathbf{U}_{i}^{\dagger}\mathbf{H}_{i}\mathbf{U}_{i}$.
Recalling $\dot{V}$ in (IV-A), we can consider
$\langle\psi_{i}|\mathbf{H}_{i}|\psi_{j}\rangle$ and
$\langle\psi_{i}|\mathbf{H}_{j}|\psi_{j}\rangle$ in the body frame.

By substituting
$\langle\psi_{i}|=^{\mathcal{B}_{i}}\langle\psi_{i}|\mathbf{U}_{i}^{\dagger}$,
we have

$\displaystyle\langle\psi_{i}|\mathbf{H}_{i}|\psi_{j}\rangle=$
${}^{\mathcal{B}_{i}}\langle\psi_{i}|\mathbf{U}_{i}^{\dagger}\mathbf{U}_{i}\mathbf{H}_{i,\mathcal{B}_{i}}\mathbf{U}_{i}^{\dagger}\mathbf{U}_{j}|\psi_{j}\rangle^{\mathcal{B}_{j}}$
$\displaystyle=$
${}^{\mathcal{B}_{i}}\langle\psi_{i}|\mathbf{H}_{i,\mathcal{B}_{i}}|\psi_{j}\rangle^{\mathcal{B}_{i}},$
(16)

where
$|\psi_{j}\rangle^{\mathcal{B}_{i}}=\mathbf{U}_{i}^{\dagger}\mathbf{U}_{j}|\psi_{j}\rangle^{\mathcal{B}_{j}}$
denotes the $|\psi_{j}\rangle$ expressed in the local frame $i$. Similarly, we
can derive that

$\displaystyle\langle\psi_{i}|\mathbf{H}_{j}|\psi_{j}\rangle=$
${}^{\mathcal{B}_{i}}\langle\psi_{i}|\mathbf{U}_{i}^{\dagger}\mathbf{U}_{j}\mathbf{H}_{j,\mathcal{B}_{j}}\mathbf{U}_{j}^{\dagger}\mathbf{U}_{j}|\psi_{j}\rangle^{\mathcal{B}_{j}}$
$\displaystyle=$
${}^{\mathcal{B}_{j}}\langle\psi_{i}|\mathbf{H}_{j,\mathcal{B}_{j}}|\psi_{j}\rangle^{\mathcal{B}_{j}},$
(17)

where
${}^{\mathcal{B}_{j}}\langle\psi_{i}|=^{\mathcal{B}_{i}}\langle\psi_{i}|\mathbf{U}_{i}^{\dagger}\mathbf{U}_{j}$
denotes the $|\psi_{i}\rangle$ expressed in the local frame $j$. From (IV-A)
and (IV-A), the deriving procedure of the Hamiltonian $\mathbf{H}_{i}$ also
works in the local frame of each qubit, which means that the Hamiltonian
$\mathbf{H}_{i}$ can be used in the local frame $\mathcal{B}_{i}$ of each
qubit. ∎

The above quantum protocol (13) can also be used in an $2$-qubit system. For
each qubit, the Hamiltonian is given as

$\displaystyle\mathbf{H}_{i}=\mathbf{n}_{i}\cdot\vec{\bm{\sigma}},$ (18)

where

$\displaystyle\mathbf{n}_{i}=\left[\begin{matrix}\sin\frac{1}{2}(\phi_{i}+\phi_{j})\sin\frac{1}{2}(\theta_{i}-\theta_{j})\\\
-\cos\frac{1}{2}(\phi_{i}+\phi_{j})\sin\frac{1}{2}(\theta_{i}-\theta_{j})\\\
-\sin\frac{1}{2}(\phi_{i}-\phi_{j})\cos\frac{1}{2}(\theta_{i}-\theta_{j})\end{matrix}\right],i,j=1,2,$

where $\theta_{i}\in[0,\pi]$ and $\phi_{i}\in[0,2\pi)$.

In (18), the Hamiltonian is expressed in the world frame. We will also show
that (18) can be also used in the local body frame, i.e.,
$\mathbf{H}_{i}^{\mathcal{B}}=\mathbf{n}_{i}^{\mathcal{B}}\cdot\vec{\bm{\sigma}}$.
The result is concluded in the following corollary.

###### Corollary 1

Suppose that a two-qubit system is governed by (5), when using the partial
quantum protocol (18), the partial quantum consensus can be achieved globally.

###### Remark 3

In Theorem 2, a Hamiltonian is designed for each qubit to achieve the quantum
consensus based on Lyapunov-based approach. The physical realization of the
protocol (13) can be inspired from the experiments realizing quantum feedback
through measurement-based feedback and coherent feedback in physical settings
such as cavity QED [37], opto-mechanics [38], superconducting circuits [39],
and quantum dots [40]. The Lyapunov-based approach has been used to design the
quantum control feedback in [28], and to achieve the asymptotic convergence.
However, these works only consider the quantum state stabilization of a single
quantum system. Different from these results, we design a quantum protocol for
an $N$-qubit system. The main difficulty lies in the dynamics and the
Hamiltonians of qubits being coupled.

###### Remark 4

In the above quantum consensus, we assume that the quantum state is a product
state, and the weak measurement is performed on each subsystem. For the
general entangled state, the measurement performed on the state of each
subsystem will affect the state of other nodes. In fact, there are some
existing results focusing on this problem [41, 42]. In [43], a quantum
feedback control is designed based on the partial information extracted from
quantum weak measurements. Through the quantum feedback control, the coherence
and entanglement of quantum systems can be partially preserved. Inspired by
this technique, we can exploit the quantum consensus protocol while protecting
the coherence and entanglement of $N$-qubit systems. The detailed procedure is
shown in Appendix A. However, the result in [43] only considers the single
qubit system and the two-qubit system. In addition, the Hamiltonian design for
two-qubit systems is non-local. How to design the local Hamiltonian for
$N$-qubit systems to preserve the entanglement is still an open problem to be
solved.

###### Remark 5

Recently, the data-driven robust quantum control has been investigated in [44,
45, 7]. A robust quantum unitary transformation is computed based on a
gradient flow algorithm, and an optimal control strategy is find by a training
process based on the data sampled previously [45]. Compared with the data-
driven quantum control method in [45], the unitary transformation (13) is
designed by using local quantum states obtained by weak measurements without
using other additional information.

###  IV-B Geometry-based quantum protocol design

In this part, we consider the more general graph case where the topology is a
connected graph. There are some situations where the particles are trapped in
a lattice or a two-dimensional space.

In the connected graph case, it is difficult to derive the Hamiltonian to
guarantee that the invariant set $S$ consists of only the consensus point,
i.e., $|\psi_{1}\rangle=|\psi_{2}\rangle=...=|\psi_{N}\rangle$ using the
Lyapunov function in (IV-A). Recalling that in Theorem 1, we have derived the
rotation axis $\mathbf{n}_{12}$ of two qubits by using the cross product of
two direction vectors in three-dimensional spherical coordinates. The two
direction vectors correspond to the two quantum states on the Bloch ball. This
fact inspires us to investigate the quantum consensus protocol from a
geometric point of view.

For an agent whose state is a three-dimensional vector
$\mathbf{x}_{i}\in\mathbb{R}^{3}$ evolving on the 2-sphere, we obtain its
dynamics as follows,

$\displaystyle{}\dot{\mathbf{x}}_{i}=(\mathbf{I}_{3}-\mathbf{x}_{i}\mathbf{x}_{i}^{\top})\mathbf{u}_{i},$
(19)

where $\mathbf{u}_{i}$ is the velocity input and
$(\mathbf{I}_{3}-\mathbf{x}_{i}\mathbf{x}_{i}^{\top})$ is a projection which
projects the input $\mathbf{u}_{i}$ to the tangent space of the 2-sphere
located at $\mathbf{x}_{i}$. To achieve the consensus on the 2-sphere, i.e.,
$\mathbf{x}_{1}=...=\mathbf{x}_{N}$, we can construct the consensus protocol
of each agent as follows,

$\displaystyle\mathbf{u}_{i}=\sum_{j\in\mathcal{N}_{i}(t)}w_{ij}\mathbf{x}_{j},$
(20)

where $w_{ij}(\cdot)$ is the weighted function depending on the Euclidean
distance $\|\mathbf{x}_{i}-\mathbf{x}_{j}\|$. The protocol (20) can be
considered as a weighted sum of neighbors’ states which is projected onto the
tangent space of 2-sphere located at $\mathbf{x}_{i}$ [46].

Next, we build a connection between the partial quantum consensus to the
consensus on the 2-sphere.

###### Lemma 1

Suppose the qubit dynamics governed in (5). Let $\mathbf{x}_{i}$ be a vector
in the three-dimensional sphere. The dynamic equation (5) can be transformed
into the following form

$\displaystyle\dot{\mathbf{x}}_{i}=\left(\mathbf{n}_{i}\cdot\vec{\mathbf{J}}\right)\mathbf{x}_{i},$
(21)

where
$\mathbf{n}_{i}\cdot\vec{\mathbf{J}}=n_{i}^{(1)}\mathbf{J}_{1}+n_{i}^{(2)}\mathbf{J}_{2}+n_{i}^{(3)}\mathbf{J}_{3}$,
and

$\displaystyle\mathbf{J}_{1}=\left[\begin{matrix}0&0&0\\\ 0&0&-1\\\ 0&1&0\\\
\end{matrix}\right],\mathbf{J}_{2}=\left[\begin{matrix}0&0&1\\\ 0&0&0\\\
-1&0&0\\\ \end{matrix}\right],\mathbf{J}_{3}=\left[\begin{matrix}0&-1&0\\\
1&0&0\\\ 0&0&0\\\ \end{matrix}\right].$

###### Proof:

In (4), we know that $|\psi_{i}\rangle\in\mathbb{C}^{2}$ is a 2-dimensional
complex vector such that $\langle\psi_{i}|\psi_{i}\rangle=1$. According to the
dynamics (5), the evolution of $|\psi_{i}(t)\rangle$ is a unitary
transformation

$\displaystyle|\psi_{i}(t)\rangle=\mathbf{U}_{i}(t)|\psi_{i}(0)\rangle,$ (22)

where $\mathbf{U}_{i}(t)\in\mathbb{SU}(2)$ is a unitary matrix. Alternatively,
we can equivalently express the element in $\mathbb{S}^{2}$ in Cartesian
coordinate, i.e., $\mathbf{x}_{i}\in\mathbb{R}^{3}$ and
$\|\mathbf{x}_{i}\|_{2}=1$. In Cartesian coordinate, the rotation is denoted
as a three-dimensional orthogonal matrix $\mathbf{R}_{i}\in\mathbb{SO}(3)$,
i.e.,

$\displaystyle\mathbf{x}_{i}(t)=$
$\displaystyle\;\mathbf{R}_{i}(t)\mathbf{x}_{i}(0)$ $\displaystyle=$
$\displaystyle\;e^{\mathbf{n}_{i}\cdot\vec{\mathbf{J}}t}\mathbf{x}_{i}(0),$
(23)

where $\mathbf{n}_{i}\cdot\vec{\mathbf{J}}$ is the skew symmetric matrix
expanded by the basis $\vec{\mathbf{J}}$ in $\mathfrak{so}(3)$, and
$\mathbf{n}_{i}\in\mathbb{R}^{3}$ is the unit vector which represents the
rotation axis. From (IV-B), we have
$\dot{\mathbf{x}}_{i}(t)=\left(\mathbf{n}_{i}\cdot\vec{\mathbf{J}}\right)\mathbf{x}_{i}(t),$
which completes the proof. ∎

Based on Lemma 1 and there is a one-to-one mapping from
$|\psi_{i}\rangle\in\mathbb{C}^{2}$ to $\mathbf{x}_{i}\in\mathbb{S}^{2}$, we
can actually construct the rotation $\mathbf{R}_{i}$ by designing the rotation
axis $\mathbf{n}_{i}$ in (21), which is equivalent to determining the
Hamiltonian in (5). Inspired by the protocol (20), we can construct the
quantum consensus protocol below,

$\displaystyle\mathbf{H}_{i}=\mathbf{n}_{i}\cdot\vec{\bm{\sigma}},$ (24)

where
$\mathbf{n}_{i}=\;\mathbf{x}_{i}\times(\mathbf{I}_{3}-\mathbf{x}_{i}\mathbf{x}_{i}^{\top})\sum_{j\in\mathcal{N}_{i}}w_{ij}\mathbf{x}_{j}.$
Now, we can state the main theorem in the following.

###### Theorem 3

Suppose that an $N$-qubit network is described by (5) under a connected graph.
When using (24), if the initial quantum states are contained in the hemisphere
on the Bloch ball, then the partial quantum consensus is achieved. In
addition, the quantum protocol (24) can be also used in the local frame of
each qubit, i.e.,
$\mathbf{H}_{i}^{\mathcal{B}}=\mathbf{n}_{i}^{\mathcal{B}}\cdot\vec{\bm{\sigma}}$.

###### Proof:

We will establish the relationship between the dynamics (21) and (20). Since
$\left(\mathbf{n}_{i}\cdot\vec{\mathbf{J}}\right)$ is a real three-dimensional
skew symmetric matrix, we have
$\left(\mathbf{n}_{i}\cdot\vec{\mathbf{J}}\right)\mathbf{x}_{i}=\mathbf{n}_{i}\times\mathbf{x}_{i}$.
Recalling (21), one has

$\displaystyle\dot{\mathbf{x}}_{i}=\left(\mathbf{n}_{i}\cdot\vec{\mathbf{J}}\right)\mathbf{x}_{i}=\mathbf{n}_{i}\times\mathbf{x}_{i}.$
(25)

From (25), we know that the vectors $\mathbf{n}_{i}$, $\mathbf{x}_{i}$, and
$\dot{\mathbf{x}}_{i}$ are mutually orthogonal. By using (24) and based on the
fact that each qubit state $|\psi_{i}\rangle$ corresponds to a state
$\mathbf{x}_{i}(\theta_{i},\phi_{i})$, we can obtain that the dynamics of
$\mathbf{x}_{i}$ in (21) can be transformed into (19). Then, inspired by the
results in [46], we can conclude that the hemisphere is invariant and the only
stable equilibrium point of the system (20) is the consensus point in
$\\{\mathbf{x}\in\mathbb{R}^{3N}\;|\;\mathbf{x}_{i}=\mathbf{x}_{j},i,j=1,...,N\\}$.
Hence, the asymptomatic consensus is achieved under connected graphs such that
$\mathbf{x}_{i}(t)\rightarrow\mathbf{x}_{j}(t),i,j=1,...,N$ as
$t\rightarrow+\infty$ for all $\mathbf{x}_{i}(0)$ contained in the hemisphere.
Due to the one-to-one corresponding of $\mathbf{x}_{i}(t)$ and
$|\psi_{i}(t)\rangle$ in the Bloch ball, we know that the
$|\psi_{i}(t)\rangle\rightarrow|\psi_{j}(t)\rangle,i,j=1,...,N$ as
$t\rightarrow+\infty$ for all $|\psi_{i}(0)\rangle$ contained in the
hemisphere.

Figure 3: The difference value $T_{1}-T_{\min}$ between the settling time of
the protocol and the minimum time on the Bloch ball. The $y$-axis and $x$-axis
are defined as $\theta_{1,2}(0)=\theta_{1}(0)=\theta_{2}(0)$ and
$\Delta\phi_{12}(0)=\phi_{2}(0)-\phi_{1}(0)$ where the initial values of polar
angles and azimuthal angles are set as
$\theta_{1}(0)=\theta_{2}(0)\in(0,\frac{\pi}{4})$ and
$\phi_{1}(0),\phi_{2}(0)\in(0,\frac{\pi}{2})$. The settling time $T_{1}$ is
chosen as $T_{1}=\inf_{t>0}\\{t\in\mathbb{R}:V(t)<10^{-5}\\}$ and the minimum
time $T_{\min}$ is calculated by (12).  Figure 4: A qubit network with a chain
connected topology.

Next, we show that the designing rotation axis $\mathbf{n}_{i}$ in (24) can be
used in the local body frame $\mathcal{B}_{i}$, i.e.,

$\displaystyle{}\mathbf{n}_{i}^{\mathcal{B}_{i}}=\;$
$\displaystyle\mathbf{x}_{i}^{\mathcal{B}_{i}}\times(\mathbf{I}_{3}-\mathbf{x}_{i}^{\mathcal{B}_{i}}{\mathbf{x}_{i}^{\mathcal{B}_{i}}}^{\top})\sum_{j\in\mathcal{N}_{i}}w_{ij}\mathbf{x}_{j}^{\mathcal{B}_{i}},$
(26)

where
$\mathbf{x}_{i}^{\mathcal{B}_{i}}=\mathbf{R}_{i}^{\top}\mathbf{x}_{i}^{\mathcal{W}}$
and $\mathbf{x}_{j}^{\mathcal{B}_{i}}$ is the qubit $j$’s state measured in
the qubit $i$’s local frame $\mathcal{B}_{i}$. Recalling the dynamics (21) and
$\mathbf{R}_{i}\mathbf{x}_{i}^{\mathcal{B}_{i}}=\mathbf{x}_{i}^{\mathcal{W}}$,
where $\mathbf{R}_{i}$ is the rotation from the world frame $\mathcal{W}$ to
the body frame $\mathcal{B}_{i}$, one has

$\displaystyle{}\dot{\mathbf{x}}_{i}^{\mathcal{B}_{i}}=$
$\displaystyle\left(n_{i}^{(1)}\mathbf{R}_{i}^{\top}\mathbf{J}_{1}\mathbf{R}_{i}+n_{i}^{(2)}\mathbf{R}_{i}^{\top}\mathbf{J}_{2}\mathbf{R}_{i}+n_{i}^{(3)}\mathbf{R}_{i}^{\top}\mathbf{J}_{3}\mathbf{R}_{i}\right)\mathbf{x}_{i}^{\mathcal{B}_{i}}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{3}n_{i}^{(k)}\text{Skew}(\mathbf{R}_{i}^{\top}\mathbf{e}_{i}^{\mathcal{W}})\right)\mathbf{x}_{i}^{\mathcal{B}_{i}}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{3}n_{i}^{(k)}\mathbf{J}_{k}^{\mathcal{B}_{i}}\right)\mathbf{x}_{i}^{\mathcal{B}_{i}}$
$\displaystyle=$
$\displaystyle\;\mathbf{n}_{i}^{\mathcal{B}_{i}}\times\mathbf{x}_{i}^{\mathcal{B}_{i}},$
(27)

where we use the facts that
$\mathbf{R}_{i}^{\top}\mathbf{e}_{1}^{\mathcal{W}}=\mathbf{e}_{1}^{\mathcal{B}_{1}}$
to obtain the third equality. The $\mathbf{e}_{1}^{\mathcal{B}_{1}}$ is the
first column vector of the matrix $\mathbf{I}_{3}$. For (20), we can also
derive that

$\displaystyle{}\dot{\mathbf{x}}_{i}^{\mathcal{B}_{i}}=$
$\displaystyle\;\mathbf{R}_{i}^{\top}\dot{\mathbf{x}}_{i}$ $\displaystyle=$
$\displaystyle\;\mathbf{R}_{i}^{\top}\left(\mathbf{I}_{3}-\mathbf{R}_{i}\mathbf{x}_{i}^{\mathcal{B}_{i}}{\mathbf{x}_{i}^{\mathcal{B}_{i}}}^{\top}\mathbf{R}_{i}^{\top}\right)\mathbf{R}_{i}\sum_{j\in\mathcal{N}_{i}}w_{ij}\mathbf{R}_{i}^{\top}\mathbf{R}_{j}\mathbf{x}_{j}^{\mathcal{B}_{j}}$
$\displaystyle=$
$\displaystyle\;(\mathbf{I}_{3}-\mathbf{x}_{i}^{\mathcal{B}_{i}}{\mathbf{x}_{i}^{\mathcal{B}_{i}}}^{\top})\sum_{j\in\mathcal{N}_{i}}w_{ij}\mathbf{x}_{j}^{\mathcal{B}_{i}}.$
(28)

Hence, by combing (IV-B) and (IV-B), we can derive (26), which shows that the
consensus protocol (24) is not dependent on the coordinate frame. The proof is
completed. ∎

###### Remark 6

Most of the existing research on quantum consensus [16, 17, 19, 21, 22]
describes the interaction between qubits by the swapping operator, and the
symmetric quantum consensus and reduced state quantum consensus can be
achieved. However, the swapping operator is a quantum global operation which
cannot be implemented in a distributed manner. Thus, we design the Hamiltonian
for each quantum system only depending on the local information by using the
Lyapunov method and geometry method, respectively. In addition, we also show
that the proposed Hamiltonian for each quantum can be used in its own body
frame, which removes the requirement of a global frame for partial quantum
consensus of qubit networks.

###### Remark 7

In this section, we propose two partial quantum consensus protocols based on
the Lyapunov method and the geometry method, respectively. Both of them have
its benefits. The quantum protocol designed by the geometry method can be used
in more general graphs. However, the initial quantum states of qubit networks
should be contained in the hemi-sphere due to the geometry constraint on
$\mathbb{S}^{2}$, which implies that the consensus result is local. For a
particular chain connected graph, the consensus protocol can be designed by
the Lyapunov method and the global partial quantum consensus can be achieved
if initial quantum states are contained in the Bloch ball. It should be noted
that achieving partial quantum consensus globally with general connected
graphs is an open and challenging problem to be studied further.

##  V Simulation

In this section, we are firstly interested in evaluating the convergent
settling time of the Hamiltonian (18) compared with the minimum time
estimation in Theorem 1. Given a two-qubit system, we set the initial quantum
states of both qubits as $\theta_{1}(0)=\theta_{2}(0)\in(0,\frac{\pi}{4})$,
$\phi_{1}(0),\phi_{2}(0)\in(0,\frac{\pi}{2})$ and define
$\Delta\phi_{12}=\phi_{2}(0)-\phi_{1}(0)$. By using the Hamiltonian (18), the
function
$V=\frac{1}{2}\langle\bm{\psi}_{1}-\bm{\psi}_{2}|\bm{\psi}_{1}-\bm{\psi}_{2}\rangle$
will tend to zero as $t\rightarrow\infty$. In order to make the comparison
easy, we choose the settling time
$T_{1}=\inf_{t>0}\\{t\in\mathbb{R}:V(t)<10^{-5}\\}$ in the numerical
simulation. In Fig. 3, it shows the difference value of the settling time of
the Hamiltonian (18) compared with the minimum time estimation calculated by
$T_{1}-T_{\min}$. We observe that the settling time of the protocol (18) is
very close to the minimum time in most regions on the Bloch ball, especially
when the initial values of $\theta_{1}$ and $\theta_{2}$ are close to
$\frac{\pi}{2}$ (the points located at the equator of the Bloch ball). The
only exception is when $|\bm{\psi}_{1}\rangle(0)$ and
$|\bm{\psi}_{2}\rangle(0)$ are located near the north pole. In this case, we
find that the settling time of the protocol (18) is larger than the minimum
time. This is because the protocol (18) is designed based on the Euclidean
distance between two quantum states which is not always the shortest distance
on the Bloch ball.

Figure 5: The partial quantum consensus result under chain graphs. A) The
trajectories of $5$ qubits under chain graphs evolving on the Bloch ball. B)
The norm of error for the density matrix of quantum states under chain graphs.
C) The settling time $T_{2}$ of partial quantum consensus for qubit networks
with the different number of qubits. Figure 6: A 9-qubit network with
connected topologies.

Then, we simulate a five-qubit network with a chain graph. The initial states
located on the Bloch ball are shown in Fig. 4. The white ball denotes the
initial mixed quantum state and the black ball on the surface of the Bloch
ball denotes its part of the pure state. Then, by using the quantum protocol
(13), the trajectories of qubits evolving on the Bloch ball are shown in Fig.
5A and the norm of error for the density matrix of the pure state is shown in
Fig. 5B. It can be seen that all quantum states are located in the same
direction on the Bloch ball, which implies that the partial quantum consensus
is achieved. In Fig. 5C, it shows the settling time of the partial quantum
consensus for qubit networks with the different number of qubits. Let
$V(t)=\frac{1}{2}\sum_{i=1}^{N-1}\langle\bm{\psi}_{i}-\bm{\psi}_{i+1}|\bm{\psi}_{i}-\bm{\psi}_{i+1}\rangle$.
The settling time $T_{2}$ is defined as
$T_{2}=\inf_{t>0}\\{t\in\mathbb{R}:V(t)<10^{-2}\\}$. We can easily observe
that the settling time will not exponentially increase when the scale of
networks becomes larger.

Figure 7: The partial quantum consensus result under connected graphs. A) The
trajectories of $9$ qubits under connected graphs evolving on the Bloch ball.
B) The norm of error for the density matrix of quantum states under connected
topologies. C) The settling time $T_{3}$ of partial quantum consensus for
qubit networks with the different number of qubits.

Next, we consider an example of a 9-qubit network with a connected graph. The
topology graph is a $3\times 3$ grid shown in Fig. 6. The initial quantum
states are illustrated by the red arrow on the Bloch ball. By using the
protocol (24), the trajectories of each qubit evolving on the Bloch ball are
shown in Fig. 7A and the norm of error for the density matrix of the pure
state is shown in Fig. 7B. It is clear that the partial quantum consensus is
achieved.

In Fig. 7C, it shows the relationship between the settling time of the partial
quantum consensus and the scales of qubit networks. The settling time $T_{3}$
is defined as $T_{3}=\inf_{t>0}\\{t\in\mathbb{R}:V(t)<10^{-2}\\}$. Without
losing generality, we assume that the topology graph is a grid with $n\times
n$ nodes, where $n=2,3,4,...,10$ in the simulation. From Fig. 7E, we know that
the settling time increases when the scale of networks becomes larger. In
addition, by comparing the results in Fig. 5E, we can find that the settling
time of partial quantum consensus for connected graphs is much smaller than
chain graphs. This is reasonable since the algebraic connectivity of a grid
graph considered in the simulation is larger than the chain connected graph.
Note that the algebraic connectivity of network topology, i.e., the second
smallest eigenvalue of Laplacian, is a measure of convergence speed of
consensus algorithms.

We further compared the convergence performance of the proposed quantum
consensus protocols with QCMEs in [17, 19, 22]. Here, we consider a 3-qubit
network with a chain topology and a connected graph, respectively, to
facilitate the comparison of three kinds of consensus protocols. The composite
state of the multi-agent system is defined as
$\bm{\rho}=\bm{\rho}^{\prime}_{1}\otimes\bm{\rho}^{\prime}_{2}\otimes\bm{\rho}^{\prime}_{3}\in\mathbb{C}^{8\times
8}$, where $\bm{\rho}^{\prime}_{i},\;i=1,2,3$ is the pure state of each qubit.
Let the quantum average of the 3-qubit network be $\bar{\bm{\rho}}$. For the
chain graph, the edge set is given as
$\mathcal{E}_{l}=\\{(1,2),(2,1),(2,3),(3,2)\\}$. We will compare the
convergence rate of three different protocols in terms of the 2-norm of the
matrix $(\bm{\rho}-\bar{\bm{\rho}})$. The trajectory of the value of
$\|\bm{\rho}-\bar{\bm{\rho}}\|$ is shown in Fig. 7A. It can be seen that the
protocol (24) converges to the consensus state faster than QCMEs while the
protocol (13) converges slower than QCMEs in Fig. 7A. For the connected graph,
the edge set is given as
$\mathcal{E}_{c}=\\{(1,2),(2,1),(2,3),(3,2),(3,1),(1,3)\\}$. We will compare
the convergence rate of QCMEs and the protocol (24). Figure 7B shows that the
protocol (24) converge faster than QCMEs. It should be noted that we only
compare the converge rate numerically here. The rigorous comparison with the
theoretical analysis is worth studying in the future work.

Figure 8: A) Comparisons of the convergence rate under chain graphs. B)
Comparisons of the convergence rate under connected graphs. C) Plot of
coherence function $C_{xy}(\bm{\rho}_{1,2})$. D) Distance between
$\bm{\rho}_{1}$ and $\bm{\rho}_{2}$.

##  VI Conclusion

In this paper, a partial quantum consensus problem is considered for qubit
networks. Based on the convexity property of the density matrix, we construct
the unitary transformation for each quantum system to drive the part of pure
states to consensus. The two-qubit system is firstly studied by using the
geometry method. The minimum completing time for the unitary transformation is
derived based on the geometric configuration of the quantum state. To obtain
an asymptomatic consensus result, we design the Hamiltonian for each quantum
system based on the Lyapunov approach. Then, we extend the approaches to the
more general $N$-qubit network. For a chain topology case, the Hamiltonian of
each quantum system is designed based on the Lyapunov approach. For the
general connected graph case, we exploit a geometric method which builds the
relationship between partial quantum consensus and 2-sphere consensus problem.
Finally, a numerical simulation for 2-qubit and 9-qubit quantum networks is
demonstrated to show the effectiveness of theoretical results.

Future works may concentrate on three aspects. Firstly, this work focuses on
qubit networks in two-dimensional Hilbert space $\mathcal{H}_{2}$, the more
general problem is the quantum consensus on the Hilbert space with higher
dimensions such as $\mathcal{H}_{2^{n}},\;n>1$. Secondly, the complete quantum
consensus for qubit networks should be further investigated. The main idea is
to find a local Lindblad operator for each agent which only depends on the
information of the quantum state of itself and its neighbors. Thirdly, the
multi-agent deep reinforcement learning has shown a great potential in the
multi-agent framework where multiple agents learn to act optimally in a shared
environment. A typical limitation of the multi-agent deep reinforcement
learning (MADRL) algorithm is the long training time and high requirement of
the computation resource. Thus, the quantum multi-agent deep reinforcement
learning (QMADRL) is a promising direction which offers a novel method to deal
with this problem owing to its strong ability of parallel computing.

##  Appendix A Feedback control protocol design for coherence protection of
two-qubit systems

Due to the disturbance induced by weak measurements, we consider the system
dynamics represented by the following master equation [43],

$\displaystyle d\bm{\rho}_{i}=$
$\displaystyle-i\left[\mathbf{H}_{i},\bm{\rho}_{i}\right]dt+4\Gamma_{r}\mathcal{D}\left[\bm{\sigma}_{-}\right]\bm{\rho}_{i}dt$
$\displaystyle+\left(\Gamma_{\phi}+\Gamma_{z}\right)\mathcal{D}\left[\bm{\sigma}_{z}\right]\bm{\rho}_{i}dt+\sqrt{\eta_{z}\Gamma_{z}}\mathcal{H}\left[\bm{\sigma}_{z}\right]\bm{\rho}_{i}dW,$
$\displaystyle dy_{z}^{i}=$
$\displaystyle\left\langle\bm{\sigma}_{z}\right\rangle_{\bm{\rho}_{i}}dt+\frac{1}{2\sqrt{\eta_{z}\Gamma_{z}}}dW,\;i=1,2.$
(29)

In (A), the last three terms represent the influence of environmental
disturbances including relaxation and dephasing effects. The Hamiltonian
$\mathbf{H}_{i}$ aims to preserve the value of the coherence function
$C_{xy}(\bm{\rho}_{i})$ defined as
$C_{xy}(\bm{\rho}_{i})=\sqrt{\left\langle\bm{\sigma}_{x}\right\rangle_{\bm{\rho}_{i}}^{2}+\left\langle\bm{\sigma}_{y}\right\rangle_{\bm{\rho}_{i}}^{2}},\;i=1,2.$
The second equation in (A) is the measurement outcome. Next, by using
information obtained from weak measurements and the initial state, we design
the Hamiltonian $\mathbf{H}_{i}$ for two-qubit systems to protect the
coherence. The Hamiltonian adjusted by the time-varying control $\mu_{i}$ for
two-qubit systems can be designed as
$\mathbf{H}_{i}=\mu_{i}\left(\cos\phi_{ij}\bm{\sigma}_{x}+\sin\phi_{ij}\bm{\sigma}_{y}\right)$,
$\mu_{i}=\frac{\Gamma C_{xy}(\bm{\rho}_{i}(0))}{Y_{z}^{i}}$,
$\phi_{ij}=0.5\arctan\left(-\frac{\left\langle\bm{\sigma}_{x}\right\rangle_{\rho_{i}}}{\left\langle\bm{\sigma}_{y}\right\rangle_{\rho_{i}}}\right)+0.5\arctan\left(-\frac{\left\langle\bm{\sigma}_{x}\right\rangle_{\rho_{j}}}{\left\langle\bm{\sigma}_{y}\right\rangle_{\rho_{j}}}\right)$,
and
$Y_{z}^{i}=\frac{1}{t}\int_{0}^{t}\left\langle\bm{\sigma}_{z}(\tau)\right\rangle_{\bm{\rho}_{i}}d\tau+\frac{1}{2\sqrt{\eta_{z}\Gamma_{z}}t}\int_{0}^{t}dW,\;i,j=1,2,i\neq
j.$

To show the effectiveness of the control strategy, we choose
$\Gamma_{r}=\Gamma_{\phi}=10$, $\Gamma_{z}=0.1$,
$\Gamma=\Gamma_{r}+\Gamma_{\phi}+\Gamma_{z}$, and $\eta_{z}=1$. Under the
feedback control, the numerical result is shown in Figs. 7C-7D. In Fig. 7C,
the solid lines are trajectories of coherence functions of two qubits under
feedback control and the dash line is the target trajectories without feedback
control and decoherence. It shows that coherence values of two qubits tend to
the target trajectory of the initial coherence. This result shows that the
coherence of two qubits are protected effectively. In Fig. 7D, the Euclidean
distance between $\rho_{1}$ and $\rho_{2}$ is decreasing as time goes to
infinity. Due to the influence of environmental noises, the Euclidean distance
may not converge to zero precisely. Note that the consensus error is dependent
on the mean value of environmental noises. $\hfill\blacksquare$

## Acknowledgment

The authors would like to thank Wangli He and Zhixing Cao for helpful
discussions.

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