# Monitored Recurrence of a One-parameter Family of Three-state Quantum Walks

Martin Štefaňák  Department of Physics, Faculty of Nuclear Sciences and
Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19
Praha 1 - Staré Město, Czech Republic<EMAIL_ADDRESS>
###### Abstract

Monitored recurrence of a one-parameter set of three-state quantum walks on a
line is investigated. The calculations are considerably simplified by choosing
a suitable basis of the coin space. We show that the Polya number (i.e. the
site recurrence probability) depends on the coin parameter and the probability
that the walker is initially in a particular coin state for which the walk
returns to the origin with certainty. Finally, we present a brief
investigation of the exact quantum state recurrence.

##  1 Introduction

Quantum walks [1, 2, 3] are generalizations of classical random walks to
evolution of a quantum particle on a graph or a lattice. Similar ideas
appeared already in 1960’s in the works of Feynman and Hibbs on discretization
of the Dirac equation [4] and in the 1980’s in the work of Gudder on quantum
graphic dynamics [5, 6]. Quantum walks became versatile tools in various field
of physics and computation, for a recent review see [7].

One of the interesting questions in stochastic processes is the problem of
recurrence. For classical random walks, this was studied in detail by Polya
[8] who proved that in 1 and 2 spatial dimensions the balanced walk returns
with certainty, i.e. is recurrent, while for $d\geq 3$ the recurrence
probability (also called Polya number) is strictly less than unity, implying
transience. Nevertheless, walks on infinite line and square lattice are null-
recurrent, since the expected recurrence time is infinite. On the other hand,
irreducible walks on finite graphs are positive-recurrent since the expected
recurrence time is finite. In fact, for finite state space Kac shown [9] that
it is an integer.

Studying recurrence of quantum walks requires a detailed description of the
measurement process since it has a nontrivial effect on the wavefunction of
the quantum walker. To minimize the influence of measurement we have proposed
a scheme [10, 11] where the quantum walk evolves freely for a certain number
of steps, then a measurement detecting the presence of the walker at the
origin is performed, after which the process is restarted. In this concept the
recurrence or transience of a quantum walk can be decided from the behaviour
of the probability of finding the walker at the origin after $t$ steps
(without prior measurement) in the same way as for the classical random walk.
In fact, for classical random walks this scheme is equivalent to the Polya
approach, where we monitor the origin after each step, in the sense that the
walk is recurrent in one approach if and only if it is recurrent in the other
scheme. This stems from the fact that in the classical setting, the
fundamental role is played by probabilities and there is a direct relation
between the return probabilities without prior measurement and the first
return probabilities of the monitored walk. However, this does not hold
anymore in the quantum case. Due to the effect of measurement on the
wavefunction the two probabilities cannot be accessed in the same experiment
and they are not related to each other. Nevertheless, there exists a relation
between the return amplitudes of the unitary quantum walk and the first return
amplitudes of the monitored walk, as shown in [12, 13] for a much broader
setting of iterated unitary evolutions. Note that some particular examples
were investigated before the general theory was developed, e.g. the monitored
recurrence of the Hadamard walk on a line [14] which was shown to be
equivalent to the absorption of the walk on the half-line [15]. In the
monitored recurrence approach, the expected return time to the exact initial
state (state recurrence) of a finite system is an integer [12] as in the
classical case. This holds even for iterated open quantum evolutions [16, 17].
Note that in the coined quantum walks return to the initial position can be
understood as a subspace recurrence, since we are not interested in the exact
state of the quantum coin. For finite systems the expected return time for a
subspace recurrence is a rational number [13]. We point out that in the recent
years the study of recurrence was considerably broadened to open quantum
systems, quantum Markov chains and open quantum walks, see eg. [18, 19, 20,
21, 22, 23, 24].

In contrast to the classical case the two schemes for detecting recurrence
(restart after measurement vs monitored after each step) are not equivalent
for quantum walks. As an example, the Hadamard walk on a line is recurrent in
the restart scheme [10] while transient in the monitored case [14]. This was
also demonstrated in an optical experiment [25] where the quantum walk was
simulated by a weak laser pulse cycling in a time-delay loop, utilizing time-
multiplexing to encode the position and number of steps of the walk into the
time of detection of a photon. It was a considerable experimental challenge to
perform the local measurement at the origin without disturbing the rest of the
pulse sequence. The deterministic out-coupling from the loop was achieved by a
programmable fast-switching electro-optical modulator, which allowed to
address specific time slots corresponding to the walker being at the origin.

In the present paper we take the monitored approach to the recurrence problem
and apply it to the three-state quantum walks on a line with a particular one-
parameter family of coins [26]. We show that there is a unique initial coin
state for which the walk stays at the origin in the first step, so in this
case the walk is recurrent. We then choose this initial state as one of the
basis vectors in the coin space and complement the basis with two vectors from
orthogonal complement. This leads to significant reduction of the complexity
of the follow-up analytical calculations. We show that the subspace recurrence
probability is determined by the coin parameter and the probability that the
walker is initially in the unique recurrent state. For the exact quantum state
recurrence we provide a numerical investigation. In particular, we identify
initial states for which the state recurrence is greater than the subspace
recurrence, a paradoxical feature of monitored quantum evolution which was
already discussed previously in the literature [13].

The paper is organized as follows. Section 2 reviews the basic concepts and
tools for investigation of monitored recurrence. Section 3 is dedicated to the
study of site recurrence for a one-parameter family of three-state quantum
walks. In Section 4 we present a numerical investigation of the state
recurrence. We conclude and present an outlook in Section 5.

##  2 Monitored site recurrence of a quantum walk

We begin by a brief overview of the methods for studying recurrence developed
in [12, 13], adopted for a coined quantum walks with constant coin. Consider a
discrete time quantum walk on a lattice starting from the origin. The Hilbert
space is a tensor product of the position and the coin spaces

$\mathcal{H}=\mathcal{H}_{p}\otimes\mathcal{H}_{c}.$

We denote the evolution operator by $U$, it has the usual decomposition

$U=S\cdot(I_{p}\otimes C),$ (1)

where $S$ is the shift operator and $C$ is the coin. Return of the quantum
walker to the original site can be understood as a subspace recurrence [13],
since we are not interested in the internal state of the coin, only the
position of the quantum walker. Let us denote the orthogonal projector onto
the origin subspace as

$\Pi_{0}=|0\rangle\langle 0|\otimes I_{c}.$ (2)

The evolution of the monitored quantum walk is given by the operator

$\tilde{U}=(I-\Pi_{0})U,$

i.e. the walk continues if we do not find the walker at the origin. Normalized
state after $n$ steps of the monitored walk reads

$|\psi_{n}\rangle=\frac{1}{\sqrt{s_{n}}}\tilde{U}^{n}|\psi\rangle,\quad|\psi\rangle=|0\rangle\otimes|\psi_{c}\rangle,$

where $s_{n}$ is the survival probability until the $n$-th step

$s_{n}=\lVert\tilde{U}^{n}\psi\rVert^{2}.$

$s_{n}$ is the probability that the walker has not returned to the origin in
the first $n$ steps. Hence, the complement of the limiting value of $s_{n}$

$P(\psi)=1-\lim\limits_{n\to\infty}s_{n},$

corresponds to the probability that the walker ever returns, i.e. the site
recurrence probability (or Polya number). Alternatively, we can derive the
Polya number from the first return probabilities. Let us denote the
probability of first return to the origin after $n$ steps as $q_{n}$. Since
these events are mutually exclusive, the overall site recurrence probability
is given by the sum

$P(\psi)=\sum_{n=1}^{\infty}q_{n}.$

In case of iterated unitary evolution the first return probability after $n$
steps is given by

$q_{n}=\lVert\Pi_{0}U\tilde{U}^{n-1}\psi\rVert^{2}=\lVert
a_{n}\psi\rVert^{2},$

where we have introduced the first return amplitude operator (note that
$\Pi_{0}\psi=\psi$)

$a_{n}=\Pi_{0}U\tilde{U}^{n-1}\Pi_{0}.$

The Polya number of a quantum walk for a given initial state is therefore
given by

$P(\psi)=\sum_{n=1}^{\infty}\lVert a_{n}\psi\rVert^{2}.$

Determining the operators $a_{n}$ directly is rather difficult, however, they
can be related to the $n$-th step return amplitude operators without prior
monitoring, which we denote as

$\mu_{n}=\Pi_{0}U^{n}\Pi_{0}.$

We define the operator valued generating functions (for complex variable $z$
with $|z|<1$)

$\mu(z)=\sum_{n=0}^{\infty}\mu_{n}z^{n},\quad
a(z)=\sum_{n=1}^{\infty}a_{n}z^{n}.$

Note that $\mu(z)$ is also called the Stieltjes operator and the first return
generating function $a(z)$ is related to the Schur function $f(z)$ by

$a(z)=zf^{\dagger}(z),$

which is more extensively used in the literature [12, 13, 20, 23]. Introducing
the resolvents

$G(z)=\sum_{n=0}^{\infty}U^{n}z^{n}=(I-zU)^{-1},\quad\tilde{G}(z)=\sum_{n=0}^{\infty}\tilde{U}^{n}z^{n}=(I-z\tilde{U})^{-1},$

we see that the generating functions can be written in the form

$\mu(z)=\Pi_{0}G(z)\Pi_{0},\quad a(z)=z\Pi_{0}U\tilde{G}(z)\Pi_{0}.$ (3)

Using the resolvent identities

$G(z)-\tilde{G}(z)=zG(z)\Pi_{0}U\tilde{G}(z)=z\tilde{G}(z)\Pi_{0}UG(z),$

we can derive the renewal equations [12, 13]

$\mu(z)a(z)=a(z)\mu(z)=\mu(z)-\Pi_{0}.$ (4)

In the above equation all operators act on the origin subspace (see (2) and
(3)), i.e. they are of the form

$\mu(z)=|0\rangle\langle 0|\otimes\mu_{c}(z),\quad a(z)=|0\rangle\langle
0|\otimes a_{c}(z),$

where $\mu_{c}(z)$ and $a_{c}(z)$ act on the coin space. Thus, we can rewrite
(4) into operator equation on the coin space

$a_{c}(z)=I_{c}-\mu_{c}(z)^{-1}.$ (5)

Hence, we can express the first return generating function $a_{c}(z)$ from the
return generating function $\mu_{c}(z)$, which is easier to obtain. All
functions can be extended to the unit circle in the complex plane by the
radial limit. The site recurrence probability is expressed in terms of the
boundary values [12, 13] by

$P(\psi_{c})=\int\limits_{0}^{2\pi}\lVert
a_{c}(e^{it})\psi_{c}\rVert^{2}\frac{dt}{2\pi}=\langle\psi_{c}|R|\psi_{c}\rangle,$
(6)

where we have denoted the recurrence probability operator

$R=\int\limits_{0}^{2\pi}a_{c}^{\dagger}(e^{it})a_{c}(e^{it})\frac{dt}{2\pi}.$
(7)

In summary, the recipe to obtain the Polya number is to find the Stieltjes
operator $\mu_{c}(z)$, from (5) we find the first return generating operator,
construct the recurrence probability operator (7) and find its average value
for a given initial state of the coin $\psi_{c}$ (6).

The site recurrence probability of a two-state quantum walk on a line, where
the walker can move to the right or left in each step, was evaluated
explicitly in [12]. The unitary coin operator was parameterized in the
following way

$C=\begin{pmatrix}\rho&-\gamma\\\
\overline{\gamma}&\rho\end{pmatrix},\quad\rho=\sqrt{1-|\gamma|^{2}}.$ (8)

Since the considered walk is translationally invariant, one can use Fourier
transformation to diagonalize the step operator and evaluate the Stieltjes
operator with the formula

$\mu_{c}(z)=\int\limits_{0}^{2\pi}\frac{dp}{2\pi}(I_{c}-zU(p))^{-1}.$ (9)

Here $U(p)$ is the evolution operator (1) in the Fourier picture

$U(p)=S(p)\cdot C=\begin{pmatrix}e^{ip}&0\\\
0&e^{-ip}\end{pmatrix}\cdot\begin{pmatrix}\rho&-\gamma\\\
\overline{\gamma}&\rho\end{pmatrix},$

which is a multiplication operator. Using the substitution $e^{ip}=x$,
$dp=\frac{dx}{ix}$ the RHS of (9) is turned into an integral over the unit
circle in the complex plane, which can be evaluated with the residue theorem.
After some calculations it is found that the recurrence operator (7) for a
two-state walk is a multiple of identity, i.e. the Polya number does not
depend on the initial coin state, only on the quantum coin. Explicitly, the
site recurrence probability for a two-state quantum walk on a line with the
coin (8) is given by [12, 27]

$P=\frac{2\left(|\gamma|\sqrt{1-|\gamma|^{2}}-(1-2|\gamma|^{2})\arcsin|\gamma|\right)}{\pi|\gamma|^{2}}.$

##  3 Site recurrence of a three-state walk on a line

Let us now consider a three-state walk on a line, where the walker can move to
the right, stay, or move to the left. This corresponds to the standard basis
of the coin space $|R\rangle$, $|S\rangle$ and $|L\rangle$. We choose the
following one-parameter set of coins [26, 28]

$C=\begin{pmatrix}-\rho^{2}&\rho\sqrt{2(1-\rho^{2})}&1-\rho^{2}\\\
\rho\sqrt{2(1-\rho^{2})}&2\rho^{2}-1&\rho\sqrt{2(1-\rho^{2})}\\\
1-\rho^{2}&\rho\sqrt{2(1-\rho^{2})}&-\rho^{2}\end{pmatrix},\quad 0<\rho<1,$
(10)

which reduces to the 3x3 Grover matrix for $\rho=\frac{1}{\sqrt{3}}$. Before
turning to the derivation of the Polya number, we make the following
observation. Since the coin operator (10) is equal to its inverse, the initial
coin state

$|\alpha_{1}\rangle=C|S\rangle=\rho\sqrt{2(1-\rho^{2})}(|R\rangle+|L\rangle)+(2\rho^{2}-1)|S\rangle=\begin{pmatrix}\rho\sqrt{2(1-\rho^{2})}\\\
2\rho^{2}-1\\\ \rho\sqrt{2(1-\rho^{2})}\end{pmatrix},$

is mapped to $|S\rangle$. Hence, if we take $|\alpha_{1}\rangle$ as the
initial coin state, the walker is absorbed with certainty after one step, i.e.
for this initial state the site recurrence probability is one. Numerical
simulations indicate that for all coin states in the orthogonal complement to
$|\alpha_{1}\rangle$ the Polya number has the same value, which is less than
unity. Let us complete the orthonormal basis in the coin space by

$\displaystyle|\alpha_{2}\rangle$
$\displaystyle=\frac{2\rho^{2}-1}{\sqrt{2}}(|R\rangle+|L\rangle)-2\rho\sqrt{1-\rho^{2}}|S\rangle=\begin{pmatrix}\frac{2\rho^{2}-1}{\sqrt{2}}\\\
-2\rho\sqrt{1-\rho^{2}}\\\ \frac{2\rho^{2}-1}{\sqrt{2}}\end{pmatrix},$
$\displaystyle|\alpha_{3}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|R\rangle-|L\rangle)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
0\\\ -1\end{pmatrix}.$

The discussion above indicates that the recurrence probability operator should
be diagonal when expressed in the basis formed by
$\left\\{|\alpha_{1}\rangle,|\alpha_{2}\rangle,|\alpha_{3}\rangle\right\\}$.
From now on all matrices will be expressed in this basis. For this we will
utilize the transition matrices $T$ and $T^{\dagger}$, where

$T=\begin{pmatrix}\rho\sqrt{2(1-\rho^{2})}&2\rho^{2}-1&\rho\sqrt{2(1-\rho^{2})}\\\
\frac{2\rho^{2}-1}{\sqrt{2}}&-2\rho\sqrt{2(1-\rho^{2})}&\frac{2\rho^{2}-1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\end{pmatrix}.$

We denote the matrix $M$ in the $\alpha$ basis as ${}^{\alpha}M$.

Turning to the derivation of the Polya number, we begin by expressing the
Stieltjes operator through the Fourier transformation (9), where
${}^{\alpha}U(p)$ is given by

${}^{\alpha}U(p)$ $\displaystyle=T\cdot\begin{pmatrix}e^{ip}&0&0\\\ 0&1&0\\\
0&0&e^{-ip}\end{pmatrix}\cdot C\cdot T^{\dagger}$
$\displaystyle=\begin{pmatrix}2\rho^{2}-1&-2\rho\sqrt{1-\rho^{2}}\cos{p}&-2i\rho\sqrt{1-\rho^{2}}\sin{p}\\\
-2\rho\sqrt{1-\rho^{2}}&(1-2\rho^{2})\cos{p}&i(1-2\rho^{2})\sin{p}\\\
0&-i\sin{p}&-\cos{p}\end{pmatrix}.$

The resolvent in the Fourier space is equal to

${}^{\alpha}(I_{c}-zU(p))^{-1}$ $\displaystyle=\
\frac{1}{(z-1)(1+z(2-2\rho^{2}+z)+2\rho^{2}z\cos{p})}\times$
$\displaystyle\begin{pmatrix}u(z)-2\rho^{2}z\cos
p&2\rho\sqrt{1-\rho^{2}}z(\cos p+z)&2i\rho\sqrt{1-\rho^{2}}z\sin p\\\
2\rho\sqrt{1-\rho^{2}}z(z\cos p+1)&w(z)(z\cos p+1)&-izv(z)\sin p\\\
-2i\rho\sqrt{1-\rho^{2}}z^{2}\sin p&i(u(z)+z+1)\sin p&zv(z)\cos
p+w(z)\end{pmatrix},$

where to shorten the formula we have denoted

$u(z)=z^{2}(1-2\rho^{2})-1,\quad v(z)=1-2\rho^{2}+z,\quad
w(z)=z(2\rho^{2}-1)-1.$

The integral in (9) can be again evaluated with residues. We will utilize the
following integrals (with $n=1,0,-1$)

$\displaystyle{\cal
I}(n)=\int\limits_{0}^{2\pi}\frac{dp}{2\pi}\frac{e^{inp}}{(z-1)(1+z(2-2\rho^{2}+z)+2\rho^{2}z\cos{p})}$
$\displaystyle=\frac{1}{2\pi i}\oint\frac{x^{n}dx}{b(x)},$

where we have denoted

$b(x)=(z-1)(\rho^{2}z+(1+2(1-\rho^{2})z+z^{2})x+\rho^{2}zx^{2}).$

The roots of the equation $b(x)=0$ are

$x_{\pm}=\frac{2(\rho^{2}-1)z-z^{2}-1\pm\sqrt{\left(1+2(1-\rho^{2})z+z^{2}\right)^{2}-4\rho^{4}z^{2}}}{2\rho^{2}z},$

with $|x_{-}|>1$ for $|z|\leq 1$. Hence, for $n=0,1$ there is only the residue
at $x_{+}$ and we find

$\displaystyle{\cal I}(0)$ $\displaystyle={\rm
Res}\left(\frac{1}{b(x)},x_{+}\right)=\frac{1}{(z^{2}-1)g(z)},$
$\displaystyle{\cal I}(1)$ $\displaystyle={\rm
Res}\left(\frac{x}{b(x)},x_{+}\right)=\frac{(z+1)g(z)-1-2(1-\rho^{2})z-z^{2}}{2\rho^{2}z(z^{2}-1)g(z)},$

where we have used the notation

$g(z)=\sqrt{1+2(1-2\rho^{2})z+z^{2}}.$

For $n=-1$ there is an additional residue at $x=0$ and we obtain

$\displaystyle{\cal I}(-1)$ $\displaystyle={\rm
Res}\left(\frac{1}{xb(x)},x_{+}\right)+{\rm
Res}\left(\frac{1}{xb(x)},0\right)$
$\displaystyle=\frac{2\rho^{2}z}{(z^{2}-1)g(z)[(z+1)g(z)-1-2(1-\rho^{2})z-z^{2}]}+\frac{1}{\rho^{2}z(z-1)}.$

After some algebra, we express the Stieltjes operator in the form

$\displaystyle^{\alpha}\mu_{c}(z)$
$\displaystyle=\frac{1}{(z-1)g(z)}\begin{pmatrix}2(\rho^{2}-1)z-g(z)&\frac{\sqrt{1-\rho^{2}}(g(z)+w(z))}{\rho}&0\\\
\frac{\sqrt{1-\rho^{2}}z(g(z)-v(z))}{\rho}&\frac{w(z)(g(z)-v(z))}{2\rho^{2}}&0\\\
0&0&\frac{v(z)g(z)-1-z(z+2-4\rho^{2})}{2\rho^{2}}\end{pmatrix}.$ (11)

From (5) we find the first return generating operator

${}^{\alpha}a_{c}(z)=\begin{pmatrix}(2\rho^{2}-1)z&-2\rho\sqrt{1-\rho^{2}}-\frac{4\rho^{3}\sqrt{1-\rho^{2}}(z-1)}{g(z)-v(z)}&0\\\
-2\rho\sqrt{1-\rho^{2}}z&1-2\rho^{2}+\frac{2\rho^{2}(2\rho^{2}-1)(z-1)}{g(z)-v(z)}&0\\\
0&0&-\frac{j(z)}{2(\rho^{2}-1)}\end{pmatrix}.$

Here we have denoted

$j(z)=1-2\rho^{2}z+z^{2}+(z-1)g(z).$

For the product ${}^{\alpha}a^{\dagger}_{c}(z)\ ^{\alpha}a_{c}(z)$ we obtain a
diagonal matrix as expected

$^{\alpha}r(z)=^{\alpha}a^{\dagger}_{c}(z)\
^{\alpha}a_{c}(z)=\begin{pmatrix}|z|^{2}&0&0\\\
0&\frac{|j(z)|^{2}}{4(\rho^{2}-1)^{2}}&0\\\
0&0&\frac{|j(z)|^{2}}{4(\rho^{2}-1)^{2}}\end{pmatrix}.$ (12)

To determine the matrix of the recurrence probability operator in the $\alpha$
basis ${}^{\alpha}R$ we have to take $z=e^{it}$ and perform the integral (7).
The first diagonal entry in (12) is 1, and so is the result of the integral,
corresponding to the fact that for the $|\alpha_{1}\rangle$ state the walker
returns to the original site with certainty. Then we are left with evaluating

$Q=\int\limits_{0}^{2\pi}\frac{dt}{8\pi(\rho^{2}-1)^{2}}|j(e^{it})|^{2}.$

We have to carefully express $g(e^{it})$ and $j(e^{it})$. We find the
following

$g(e^{it})=\left\\{\begin{array}[]{cc}e^{\frac{it}{2}}\operatorname{sgn}{(\sin{t})}\sqrt{2(\cos{t}+1-2\rho^{2})},&\cos{t}\geq
2\rho^{2}-1\\\ \\\
-ie^{\frac{it}{2}}\sqrt{-2(\cos{t}+1-2\rho^{2})},&\cos{t}<2\rho^{2}-1\end{array}\right..$

Hence, for $t\in(0,\arccos(2\rho^{2}-1))\cup(2\pi-\arccos(2\rho^{2}-1),2\pi)$
, we obtain

$j(e^{it})=2e^{it}\left(\cos{t}-\rho^{2}+i\operatorname{sgn}{(\sin{t})}\sin(t/2)\sqrt{2(\cos{t}+1-2\rho^{2})}\right),$

which has a constant modulus square

$\displaystyle|j(e^{it})|^{2}=4(\rho^{2}-1)^{2}.$

The contribution to $Q$ is then proportional to the length of the interval and
is found to be

$Q_{1}=\frac{\arccos(2\rho^{2}-1)}{\pi}.$

Turning to the case $\cos{t}<-2\rho^{2}-1$, i.e. the interval
$t\in(\arccos(2\rho^{2}-1),2\pi-\arccos(2\rho^{2}-1))$, we find

$j(e^{it})=2e^{it}\left(\cos{t}-\rho^{2}+\sin(t/2)\sqrt{-2(\cos{t}+1-2\rho^{2})}\right).$

The resulting integral can be evaluated directly

$\displaystyle Q_{2}=$
$\displaystyle\int\limits_{\arccos{(2\rho^{2}-1)}}^{2\pi-\arccos(2\rho^{2}-1)}\frac{dt}{8\pi(\rho^{2}-1)}|j(e^{it})|^{2}$
$\displaystyle=$
$\displaystyle\int\limits_{\arccos{(2\rho^{2}-1)}}^{2\pi-\arccos(2\rho^{2}-1)}\frac{dt}{2\pi(\rho^{2}-1)}\left(\cos{t}-1+\sin(t/2)\sqrt{-2(\cos{t}+1-2\rho^{2})}\right)^{2}$
$\displaystyle=\rho\frac{2(1+2\rho^{2})\sqrt{1-\rho^{2}}-\rho(2+\rho^{2})\arccos(2\rho^{2}-1)}{\pi}.$

In summary, we find

$Q=Q_{1}+Q_{2}=\frac{2\rho\left(2\rho^{2}+1\right)\sqrt{1-\rho^{2}}+\left(1-4\rho^{2}\right)\arccos\left(2\rho^{2}-1\right)}{\pi\left(\rho^{2}-1\right)^{2}}.$
(13)

The matrix of the recurrence probability operator in the $\alpha$ basis is
then given by

${}^{\alpha}R=\begin{pmatrix}1&0&0\\\ 0&Q&0\\\ 0&0&Q\end{pmatrix}.$

This means that the Polya number for the three-state walk with the coin (10)
depends on the probability $p_{1}$ to be initially in the $|\alpha_{1}\rangle$
state

$p_{1}=|\langle\alpha_{1}|\psi_{c}\rangle|^{2},$

and the coin parameter $\rho$. The resulting formula is

$P(\psi_{c})=p_{1}+Q(1-p_{1}).$ (14)

The site recurrence probability thus ranges between 1 and $Q$ (13). In
particular, for the Grover walk corresponding to $\rho=\frac{1}{\sqrt{3}}$, we
find

$P(\psi_{c})=p_{1}+\frac{10\sqrt{2}-3\arccos{(-1/3)}}{4\pi}(1-p_{1})\stackrel{{\scriptstyle.}}{{=}}0.67+0.33p_{1}.$

We illustrate our results in Figures 1 and 2. Figure 1 shows the site
recurrence probability (14) in dependence of $\rho$ and $p_{1}$. In Figure 2
we display the value of $Q$ (13) as a function of the coin parameter $\rho$.

Figure 1: Density plot of the Polya number (14) of the three-state quantum
walk as a function of the probability $p_{1}$ and the coin parameter $\rho$.
Dashed lines show the contours of $P=k/10$ for $k=1,\ldots 9$.  Figure 2: $Q$
(13) as a function of the coin parameter $\rho$. $Q$ represents the minimal
value of the recurrence probability for the walk with a given parameter
$\rho$.

##  4 State recurrence of a three-state walk on a line

Let us now briefly comment on the state recurrence [12]. In this case we are
interested in the return to the exact initial state, so the orthogonal
projector (2) has the form

$\Pi_{0}=|0\rangle\langle 0|\otimes|\psi_{c}\rangle\langle\psi_{c}|.$

The recipe to determine the state recurrence probability $S(\psi_{c})$ is
similar to the site recurrence, but instead of the operator valued generating
functions we will deal with scalars. The matrix element of the Stieltjes
operator (11) with the initial coin state $|\psi_{c}\rangle$ yields the
generating function for return amplitudes without prior monitoring

$\mu_{\psi_{c}}(z)=\langle\psi_{c}|\mu_{c}(z)|\psi_{c}\rangle.$

The generating function for the first arrival amplitudes is then given by [12]

$a_{\psi_{c}}(z)=1-\frac{1}{\mu_{\psi_{c}}(z)}.$

Finally, the state recurrence probability is obtained by a formula analogous
to (7)

$S(\psi_{c})=\int\limits_{0}^{2\pi}\lvert
a_{\psi_{c}}(e^{it})\rvert^{2}\frac{dt}{2\pi}.$ (15)

In contrast to the site recurrence, which was tractable, the integrand in (15)
is usually a rather complicated function and we rely on numerical integration.
Below we present plots with numerical results for several initial states.

For Figure 3 we consider the basis states $|\alpha_{i}\rangle$ and plot the
state recurrence probability as a function of the coin parameter $\rho$. For
$|\alpha_{2}\rangle$ and $|\alpha_{3}\rangle$ the integral in (15) can be
evaluated analytically and we find that the state recurrence is equal to the
site recurrence. In fact, numerical simulations reveal that for the initial
state $|\alpha_{3}\rangle$ the walker always returns to the origin in the
$|\alpha_{3}\rangle$ state. However, for $|\alpha_{2}\rangle$ the situation is
more complicated - the walker returns to the original site in a superposition
of $|\alpha_{1}\rangle$ and $|\alpha_{3}\rangle$ with different weights in
every time step. Nevertheless, the state recurrence probability equals (13).
For $|\alpha_{1}\rangle$ the state recurrence probability is smaller than site
recurrence probability (which equals 1), except for the boundary cases of
$\rho=0,1$.

Figure 3: State recurrence probability for the state $|\alpha_{1}\rangle$
(blue dots) and $|\alpha_{2}\rangle$ and $|\alpha_{3}\rangle$ (orange dots) in
dependence on $\rho$.

Figures 4 and 5 focus on the three-state Grover walk
($\rho=\frac{1}{\sqrt{3}}$). In Figure 4 we consider the initial state
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{2}\rangle)$ and plot
$S$ as a function of the angle $\phi$. Note that for superpositions
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{3}\rangle)$ and
$\frac{1}{\sqrt{2}}(|\alpha_{2}\rangle+e^{i\phi}|\alpha_{3}\rangle)$ the angle
$\phi$ does not affect the state recurrence probability due to the block-
diagonal form of the Stieltjes operator (11). Figure 5 shows state recurrence
probability for superposition states
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{2}\rangle$,
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ and
$a|\alpha_{2}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ as a function of $a$.

Figure 4: State recurrence probability of the Grover walk ($\rho=1/\sqrt{3}$)
for the state
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{2}\rangle)$ as a
function of the angle $\phi$.  Figure 5: State recurrence probabilities of the
Grover walk ($\rho=1/\sqrt{3}$) for the states
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{2}\rangle$ (blue dots),
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ (orange dots) and
$a|\alpha_{2}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ (green dots) as a
function of $a$.

Finally, we point out that the three-state walk shows similar paradoxical
behavior as reported in [13] for a walk on a half-line or some 2D quantum
walks, where the state recurrence probability can be greater than the site
recurrence probability. In Figure 6 we consider the initial coin state

$|\psi_{c}\rangle=\sqrt{\frac{1-\rho^{2}}{2}}|R\rangle+\rho|S\rangle+\sqrt{\frac{1-\rho^{2}}{2}}|L\rangle=\rho|\alpha_{1}\rangle-\sqrt{1-\rho^{2}}|\alpha_{2}\rangle,$
(16)

which is one of the eigenvectors used in the construction of coin operator
[26]. The plot indicates that the site recurrence probability, depicted by the
orange curve, is smaller than the state recurrence probability up to
$\rho\approx 0.79$.

Figure 6: State recurrence (blue dots) versus site recurrence (orange curve)
for the initial coin state (16).

##  5 Conclusions

The presented results demonstrate that in contrast to the simple two-state
quantum walks the site recurrence of the three-state model depends on the
initial state. For the selected one-parameter family of coins we were able to
derive the site recurrence probability in closed form. Extension to an
arbitrary 3x3 unitary coin seems to be intractable due to the complexity and
large number of free parameters. Nevertheless, for all three-state walks there
will be an initial coin state with Polya number equal to one, namely the state
given by $C^{-1}|S\rangle$. Indeed, such state will remain on the initial
position after the first step, and thus is absorbed with certainty. It is an
open question if such behaviour of site recurrence is possible for models
without the staying put option in the shift operator, e.g. in quantum walks
driven by Wigner rotation matrices of order $2j+1$ studied in [29, 30] for the
case of half-integer $j$.

As discussed in Section 4, obtaining closed formulas for state recurrence
probability was possible only for particular initial coin states due to the
complexity of the involved integrals. The same applies to both site and state
recurrence for quantum walks on higher dimensional lattices, since in such
cases already the evaluation of the Stieltjes operator through
multidimensional Fourier transform cannot be easily reduced to calculation of
residues as in the one-dimensional case. Nevertheless, the formulas allow to
obtain approximations through numerical evaluation. It would be interesting to
find if there exists site recurrent initial conditions e.g. for 2D quantum
walks with Grover coin and its extensions. Additional twist can come from
considering coined quantum walks combined with some form of classical
randomness resulting in iterated open quantum evolution, e.g. quantum walks on
dynamically percolated lattices [31, 32, 33]. Recent extension of monitored
recurrence to arbitrary iterated closed operators on Banach spaces [20] and
quantum Markov chains [23] allows to investigate such scenarios.

MŠ is grateful for financial support from RVO14000 and ”Centre for Advanced
Applied Sciences”, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778, supported
by the Operational Programme Research, Development and Education, co-financed
by the European Structural and Investment Funds. On a personal note, I want to
express my gratitude to prof. Igor Jex for years of stimulating discussions
and mediation of a myriad of personal contacts with excellent researchers
worldwide. Live long and prosper!

## References

## References

  * [1] Y. Aharonov, L. Davidovich, and N. Zagury.  Quantum random walks.  Phys. Rev. A, 48:1687–1690, 1993.
  * [2] D. A. Meyer.  From quantum cellular automata to quantum lattice gases.  J. Stat. Phys., 85:551–574, 1996.
  * [3] E Farhi and S Gutmann.  Quantum computation and decision trees.  Phys. Rev. A, 58:915–928, 1998.
  * [4] R. P. Feynman and A. R. Hibbs.  Quantum mechanics and path integrals.  International series in pure and applied physics. McGraw-Hill, New York, 1965.
  * [5] S. P. Gudder.  Quantum graphic dynamics.  Found. Phys., 18:751–776, 1988.
  * [6] S. P. Gudder.  Quantum Probability.  Academic Press, New York, 1988.
  * [7] K. Kadian, S. Garhwal, and A. Kumar.  Quantum walk and its application domains: A systematic review.  Comput. Sci. Rev., 41, 2021.
  * [8] G. Pólya.  Uber eine aufgabe betreffend die irrfahrt im strassennetz.  Math. Ann., 84:149–160, 1921.
  * [9] M. Kac.  On the notion of recurrence in discrete stochastic processes.  Bull. Amer. Math. Soc., 53:1002, 1947.
  * [10] M. Štefaňák, I. Jex, and T. Kiss.  Recurrence and Pólya number of quantum walks.  Phys. Rev. Lett., 100:020501, 2008.
  * [11] M. Štefaňák, T. Kiss, and I. Jex.  Recurrence properties of unbiased coined quantum walks on infinite d-dimensional lattices.  Phys. Rev. A, 78:032306, 2008.
  * [12] F. A. Grünbaum, L. Velázquez, A. H. Werner, and R. F. Werner.  Recurrence for Discrete Time Unitary Evolutions.  Communications in Mathematical Physics, 320:543, 2013.
  * [13] J. Bourgain, F. A. Grünbaum, L. Velazquez, and J. Wilkening.  Quantum recurrence of a subspace and operator-valued schur functions.  Commun. Math. Phys., 329:1031–1067, 2014.
  * [14] T. Kiss, L. Kecskés, M. Štefaňák, and I. Jex.  Recurrence in coined quantum walks.  Phys. Script., T135:014055, 2009.
  * [15] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous.  One-dimensional quantum walks.  In STOC’01 Proceedings of the 33rd annual ACM symposium on Theory of computing, pages 37–49, 2001.
  * [16] P. Sinkovicz, Z. Kurucz, T. Kiss, and J. K. Asbóth.  Quantized recurrence time in unital iterated open quantum dynamics.  Phys. Rev. A, 91:042108, 2015.
  * [17] P. Sinkovicz, T. Kiss, and J. K. Asbóth.  Generalized Kac lemma for recurrence time in iterated open quantum systems.  Phys. Rev. A, 93:050101, 2016.
  * [18] C. F. Lardizabal and R. R. Souza.  On a Class of Quantum Channels, Open Random Walks and Recurrence.  J. Stat. Phys., 159:772, 2015.
  * [19] S. L. Carvalho, L. F. Guidi, and C. F. Lardizabal.  Site recurrence of open and unitary quantum walks on the line.  Quantum Inform. Process., 16:17, 2016.
  * [20] F.A. Grünbaum and L. Velázquez.  A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks.  Adv. Math., 326:352, 2018.
  * [21] I. Bardet, H. Bringuier, Y. Pautrat, and C. Pellegrini.  Recurrence and Transience of Continuous-Time Open Quantum Walks.  In C. DonatiMartin, A. Lejay, and A. Rouault, editors, Seminaire De Probabilites L, volume 2252, page 493. Springer International Publishing Ag, Cham, 2019.
  * [22] A. Dhahri and F. Mukhamedov.  Open quantum random walks, quantum Markov chains and recurrence.  Rev. Math. Phys., 31:1950020, 2019.
  * [23] F. A. Grunbaum, C. F. Lardizabal, and L. Velazquez.  Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules.  Ann. Henri Poincare, 21:189, 2020.
  * [24] T. S. Jacq and C. F. Lardizabal.  Homogeneous Open Quantum Walks on the Line: Criteria for Site Recurrence and Absorption.  Quantum Inform. Comput., 21:37, 2021.
  * [25] T. Nitsche, S. Barkhofen, R. Kruse, L. Sansoni, M. Štefaňák, A. Gábris, V. Potoček, T. Kiss, I. Jex, and Ch. Silberhorn.  Probing measurement-induced effects in quantum walks via recurrence.  Sci. Adv., 4:eaar6444, 2018.
  * [26] M. Štefaňák, I. Bezděková, and I. Jex.  Continuous deformations of the grover walk preserving localization.  Eur. Phys. J. D, 66:142, 2012.
  * [27] M. Sabri, E. Segawa, and M. Štefaňák.  Conditional limit measure of a one-dimensional quantum walk with an absorbing sink.  Phys. Rev. A, 98:012136, 2018.
  * [28] M. Štefaňák, I. Bezděková, and I. Jex.  Limit distributions of three-state quantum walks: The role of coin eigenstates.  Phys. Rev. A, 90:012342, 2014.
  * [29] T. Miyazaki, M. Katori, and N. Konno.  Wigner formula of rotation matrices and quantum walks.  Phys. Rev A, 76:012332, 2007.
  * [30] I. Bezděková, M. Štefaňák, and I. Jex.  Suitable bases for quantum walks with wigner coins.  Phys. Rev. A, 92:022347, 2015.
  * [31] B. Kollár, T. Kiss, J. Novotný, and I. Jex.  Asymptotic dynamics of coined quantum walks on percolation graphs.  Phys. Rev. Lett., 108:230505, 2012.
  * [32] B. Kollár, J. Novotný, T. Kiss, and I. Jex.  Percolation induced effects in two-dimensional coined quantum walks: analytic asymptotic solutions.  New J. Phys., 16:023002, 2014.
  * [33] B. Kollár, J. Novotný, T. Kiss, and I. Jex.  Discrete time quantum walks on percolation graphs.  Eur. Phys. J. Plus, 129:103, 2014.


#  A Joint Design for STAR-RIS enhanced NOMA-CoMP Networks: A Simultaneous-
Signal-Enhancement-and-Cancellation-based (SSECB) Design

Tianwei Hou, Jun Wang, Yuanwei Liu, Xin Sun,
Anna Li and Bo Ai This work is supported by the Fundamental Research Funds for
the Central Universities Foundation of China under Grant KWRC21005532.
(Corresponding author: Anna Li.)T. Hou, J. Wang and X. Sun are with the School
of Electronic and Information Engineering, Beijing Jiaotong University,
Beijing 100044, China ({email: twhou, wangjun1, xsun}@bjtu.edu.cn).B. Ai is
with the State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China (email: boai@bjtu.edu.cn).Y. Liu
and Anna Li are with School of Electronic Engineering and Computer Science,
Queen Mary University of London, London E1 4NS, U.K. (e-mail:
<EMAIL_ADDRESS>A.li@qmul.ac.uk).

###### Abstract

In this correspondence, a novel simultaneous transmitting and reflecting
(STAR) reconfigurable intelligent surfaces (RISs) design is proposed in a non-
orthogonal multiple access (NOMA) enhanced coordinated multi-point
transmission (CoMP) network. Based on the insights of signal-enhancement-based
(SEB) and signal-cancellation-based (SCB) designs, we propose a novel
simultaneous-signal-enhancement-and-cancellation-based (SSECB) design, where
the inter-cell interferences and desired signals can be simultaneously
eliminated and boosted. Our simulation results demonstrate that: i) the inter-
cell interference can be perfectly eliminated, and the desired signals can be
enhanced simultaneously with the aid of a large number of RIS elements; ii)
the proposed SSECB design is capable of outperforming the conventional SEB and
SCB designs.

###### Index Terms:

CoMP, NOMA, STAR-RIS, SSECB.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) stand as a potential solution of
spectral efficiency (SE) enhancement and coverage enhancement for the sixth-
generation (6G) wireless networks [1, 2]. The received signals can be
positively boosted or mitigated by appropriately controlling both the phase
shift and the amplitude of the RIS elements. Recently, researchers have shown
an increased interest in a novel concept of simultaneous transmitting and
reflecting RISs (STAR-RISs), which consider both the transmission and
reflection of the RIS elements. The discipline of STAR-RISs is the concept
that the incident wireless signals can be reflected within the half-space at
the same side of the RIS, whereas they can also be transmitted into the other
side of the RIS [3, 4].

In practice, the coordinated multi-point transmission (CoMP) technique is
adopted in the third generation partnership project (3GPP) for cellular
networks, where the SE and energy efficiency (EE) is limited by the impacts of
inter-cell interferences [5]. The scarce of SE and EE in 6G has stimulated the
development of advanced multiple access techniques, i.e., non-orthogonal
multiple access (NOMA) [6]. One of the main direction of the NOMA-CoMP
networks, namely NOMA-coordinated-beamforming (CB), received considerable
attention [7]. In NOMA-CB, the cell-edge user (CEU) and cell-center user (CCU)
are served by its desired BS, where zero-forcing based design was proposed for
mitigating the intra-cell and inter-cell interferences by sacrificing the
spatial diversity gain of the desired links [8, 9], which motivates us to
employ RIS for the NOMA-CoMP networks.

By considering the application of the reflected signals at the RIS elements, a
pair of main directions of the RIS enhanced networks were proposed, namely
signal-enhancement-based (SEB) and signal-cancellation-based (SCB) designs
[10].  By assuming that multiple waves are co-phased at the receivers, the
received signal can be significantly boosted [11]. However, both desired
signals and interferences increase with the aid of SEB designs, where the rate
ceiling occurs in the high-SNR regime. In contrast, aligning the reflected
signals for signal cancellation is another way for implementing RISs [12],
where the performance can be enhanced in the high-SNR regime, but the
enhancement of the desired signal is ignored. The above phenomenons motivate
us to simultaneously enhance and mitigate the desired signals and the
interferences, respectively. However, the research to date has tended to focus
on the SEB-only or SCB-only designs rather than simultaneous-signal-
enhancement-and-cancellation-based (SSECB) designs, which is capable of
beneficially enhancing the network performance in both the medium-low-SNR and
high-SNR regimes. Fortunately, with the aid of the STAR-RIS, the SSECB designs
become possible for implementing NOMA-CoMP networks.

However, the research of STAR-RIS enhanced NOMA-CoMP networks is technically
challenging. For example, on the one hand, the passive beamforming design of
STAR-RIS for the SSECB design is still an open issue. On the other hand, the
minimal required number of RIS elements is also worth estimating. Hence, in
order to provide the benchmark scheme of the STAR-RIS enhanced NOMA-CoMP
networks, the main contributions can be summarized as follows:

  * •

We first propose a novel RIS enhanced CoMP network by NOMA-CB, where each BS
serves its own users and RIS array. By assuming that each STAR-RIS elements
are capable of both transmitting and reflecting the incident signals, the
signals can be controlled in full-spaces.

  * •

We then propose a new SSECB design in the STAR-RIS enhanced NOMA-CoMP
networks, where the signal and interference can be boosted and eliminated by
the reflection and the transmission of the STAR-RIS elements for enhancing the
network’s performance. The minimal required number of RIS elements is derived
in the case of strong-line-of-sight (LoS) links.

  * •

Our simulation results illustrate that 1) The minimal required number of RIS
elements is mainly impacted by the distances and path-loss exponent of the
interference and transmission links; 2) Since the small-scale fading gains are
random variables, the inter-cell interference can be cancelled perfectly in
the case that the number of RIS elements is large enough; 2) The proposed
SSECB design is capable of outperforming the conventional SEB and SCB designs.

##  II System Model

Figure 1: System model of the STAR-RIS enhanced NOMA-CoMP networks.

The system model of the STAR-RIS enhanced NOMA-CoMP networks is shown in Fig.
1, where two cells are deployed for serving the CCUs and CEUs simultaneously
in downlink. In order to illustrate the performance enhancement by the STAR-
RIS enhanced SSECB design, the BSs equipped with a single transmitting antenna
is communicating with one CCU and one CEU each equipped with a single
receiving antenna by utilizing the power-domain NOMA techniques, which avoids
the impact of the active beamforming at the BSs and detection vectors at the
users.

Without loss of generality, it is assumed that the BS located in cell1 is BS1,
and the users located in cell1 treat the signals received from BS2 as
interferences, whereas the users located in cell1 treat the signals received
from BS1 as desired signals. The CCU and CEU located in cell1 are denoted by
CCU1 and CEU1, respectively.

In this correspondence, there are two STAR-RIS arrays, namely RIS1 and RIS2,
located at the intersection of two cells, each of which is employed with $L$
RIS elements with $L\geq 1$ for implementing the proposed CoMP communications.
In this Section, by properly controlling the transmission and reflection
amplitude coefficients and phase shifts of each RIS element, both the desired
and interference signals can be beneficially enhanced and mitigated,
respectively. Note that ${{\bf{H}}^{{T}}}$ represents the transpose of the
matrix $\rm\bf H$ in the rest of this correspondence.

###  II-A System Description of STAR-RIS enhanced NOMA-CoMP Networks

Owing to the strong scattering effect of the BS-user links, the small-scale
fading between BS1 and CCU1, which is the desired link, can be modelled by
Rayleigh fading channel $w_{1,1,c}$, where the probability density function
(PDF) is given by

${f}(x)={e^{-{{x}}}}.$ (1)

In this letter, we only focus on the impact of the path-loss, and hence the
large-scale fading of the desired links for CCU1 in BS1 can be considered as
Non-LoS (NLoS) link, which is given by

${\varepsilon_{1,1,c}}=d_{1,1,c}^{-\alpha_{1}},$ (2)

where $\alpha_{1}$ refers to the path-loss exponent of the desired link,
$d_{1,1,c}$ denotes the distance of the BS1-CCU1 links.

In practice, the users located in BS1 also detect the signals from BS2, which
can be modelled by

$I=w_{2,1,c}d_{2,1,c}^{-{\alpha_{4}}},$ (3)

where $\alpha_{4}$ refers to the path-loss exponent of the interference link,
$d_{2,1,c}$ refers to the distance, and $w_{2,1,c}$ refers to small-scale
fading gain of the BS2-CCU1 links. Note that the small-scale fading gain of
the interference links also modelled by Rayleigh fading channel. Note that the
proposed network in Fig. 1 is a general scenario, which can be readily
extended to the cellular networks with massive cells, where only received
interference in (3) changes.

In the proposed CoMP networks, for simplicity, we consider that one RIS arrays
and one BS are grouped, where RIS1 and RIS2 face to the selected BS1 and BS2,
respectively. Since the RIS elements are located at the cell edge, the
Rayleigh fading channels are adopted for modelling the NLoS of the BS1-RIS1
links as follows:

${{\bf{H}}_{1}}=\left[{\begin{array}[]{*{20}{c}}{{h_{1,R,1}}}\\\ \vdots\\\
{{h_{1,R,L}}}\end{array}}\right],$ (4)

where ${\rm\bf H}_{i}$ is a $(L\times 1)$ vector containing Rayleigh fading
channel gains of the BS-RIS links.

On the one hand, the RIS elements are capable of reflecting the received
signals to the users. Hence, similar to (4), the small-scale fading vector of
the reflected links, i.e., RIS1-CCU1, is defined by

${{\rm\bf{R}}_{1,c}}=\left[{\begin{array}[]{*{20}{c}}{{r_{1,c,1}}}&\cdots&{{r_{1,c,L}}}\end{array}}\right],$
(5)

where ${{\rm\bf{R}}_{1,c}}$ is a $(1\times L)$ vector representing Rician
fading channel gains with fading parameter $\mathcal{K}_{1}$. Correspondingly,
the channel gains of the RIS-user link is computed as:

$r_{l,c,l}=\sqrt{\frac{{{{\cal K}_{1}}}}{{{{\cal
K}_{1}}+1}}}r_{1,c,l}^{{\rm{LoS}}}+\sqrt{\frac{1}{{{{\cal
K}_{1}}+1}}}r_{1,c,l}^{{\rm{NLoS}}},$ (6)

where $r_{1,c,l}^{{\rm{LoS}}}$ denotes the LoS components, and
$r_{1,c,l}^{{\rm{NLoS}}}$ denotes the LoS and NLoS components.

The RIS arrays are co-located at the intersection of two cells, and hence the
large-scale fading channels of the BS1-RIS1 and RIS1-CCU1 can be considered as
LoS links, which is defined by

${\varepsilon_{1,{\rm{R}},c}}=d_{1,{\rm{R}}}^{-{\alpha_{2}}}d_{{\rm{R}},1,c}^{-{\alpha_{3,c}}},$
(7)

where $d_{1,{\rm{R}}}$ denotes the distance of the BS1-RIS1,
$d_{{\rm{R}},1,c}$ denotes the distance of the RIS1-CCU1, ${\alpha_{2}}$ and
${\alpha_{3,c}}$ refer to the path-loss exponents of the BS-RIS and the RIS-
CCU links, respectively.

On the other hand, the RIS elements are also capable of transmitting the
received signals to the users located in the nearby cells. Hence, RIS1
transmits the signal to the users located in cell2, and the small-scale fading
vector of the RIS1-CCU2 links can be defined by

${{\rm\bf{T}}_{1,2,c}}=\left[{\begin{array}[]{*{20}{c}}{{t_{1,2,c,1}}},&\cdots&,{{t_{1,2,c,L}}}\end{array}}\right],$
(8)

where ${{\rm\bf{T}}_{1,2,c}}$ is a $(1\times L)$ vector representing Rician
fading channel gains with fading parameter $\mathcal{K}_{2}$, which can be
written as

${t_{1,2,c,l}}=\sqrt{\frac{{{{\cal K}_{2}}}}{{{{\cal
K}_{2}}+1}}}t_{1,2,c,l}^{{\rm{LoS}}}+\sqrt{\frac{1}{{{{\cal
K}_{2}}+1}}}t_{1,2,c,l}^{{\rm{NLoS}}},$ (9)

where $t_{1,2,c,l}^{{\rm{LoS}}}$ and $t_{1,2,c,l}^{{\rm{NLoS}}}$ denote the
LoS and NLoS components, respectively. Similarly, the large-scale fading of
the BS1-RIS1-CCU2 is defined by

${\varepsilon_{1,{\rm{T}},c=}}d_{1,{\rm{R}}}^{-{\alpha_{2}}}d_{{\rm{R}},2,c}^{-{\alpha_{3,c}}},$
(10)

where $d_{{\rm{R}},2,c}$ represents the distance of the RIS1-CCU2 links.

##  III SSECB Beamforming Designs

In order to simultaneously control multiple RIS elements, and to implement the
proposed SSECB design, as well as based on the maturity of low-complexity
channel estimation algorithms [13, 14], we assume that the global CSIs are
perfectly known at the RIS controller.

The received signals at CCU1 is given by

$\displaystyle{y_{1,c}}$
$\displaystyle=\underbrace{\sqrt{{\varepsilon_{1,1,c}}}w_{1,1,c}p+\sqrt{{\varepsilon_{1,{\rm{R}},c}}}{{\bf{R}}_{1,1,c}}{{\bf{\Phi}}_{1,{\rm{R}}}}{{\bf{H}}_{1}}p}_{{\rm{Desired~{}Signal}}}$
(11)
$\displaystyle+\underbrace{\sqrt{{\varepsilon_{2,1,c}}}w_{2,1,c}p+\sqrt{{\varepsilon_{1,{\rm{T}},c}}}{{\bf{T}}_{1,2,c}}{{\bf{\Phi}}_{2,{\rm{T}}}}{{\bf{H}}_{2}}p}_{{\rm{Interference}}}+{N_{0}},$

where $p$ refers to the transmit power at the BS,
${{\bf{\Phi}}_{2,{\rm{T}}}}\buildrel\Delta\over{=}{\rm{diag}}\left[{{\beta_{2,{\rm{T,1}}}}{\phi_{2,{\rm{T,1}}}},{\beta_{2,{\rm{T,2}}}}{\phi_{2,{\rm{T,2}}}},\cdots,{\beta_{2,{\rm{T,}}L}}{\phi_{2,{\rm{T,}}L}}}\right]$
refers to both the transmitting phase shifts and transmitting amplitude
coefficients of RIS2. More specifically,
$\beta_{2,{\rm{T}},l}\in\left({0,1}\right]$ refers to the transmission
amplitude coefficient of RIS element $l$ in RIS2,
${\phi_{2,{\rm{T}},l}}=\exp(j{\theta_{2,{\rm{T}},l}}),j=\sqrt{-1},\forall
l=1,2\cdots,L$. ${\theta_{l}}\in\left[{0,2\pi}\right)$ denotes the
transmitting phase shift by RIS element $l$ in RIS2. Similarly,
${{\bf{\Phi}}_{1,{\rm{R}}}}\buildrel\Delta\over{=}{\rm{diag}}\left[{{\beta_{1,{\rm{R,1}}}}{\phi_{1,{\rm{R,1}}}},\cdots,{\beta_{1,{\rm{R,}}L}}{\phi_{1,{\rm{R,}}L}}}\right]$
represents the reflection phase shifts and reflection amplitude coefficients
of RIS1. In this correspondence, both RIS1 and RIS2 can be perfectly
controlled, where both the phase shifts and amplitude coefficients are
continuous. The additive white Gaussian noise (AWGN) is denoted by $N_{0}$
with variance ${\sigma^{2}}$.

Based on the insights from [3, 4], there is one obvious constraint on the
reflection and transmission amplitude coefficients, which is computed as:

$\beta_{1,{\rm{R}},l}^{2}{\rm{+}}\beta_{1,{\rm{T}},l}^{2}\leq 1.$ (12)

For simplicity, it is assumed that
$\beta_{1,{\rm{R}},l}^{2}{\rm{+}}\beta_{1,{\rm{T}},l}^{2}=1$ in the rest of
this correspondence.

To simultaneously accomplish signal enhancement and cancellation goals, the
passive beamforming at both RIS1 and RIS2 must be designed jointly. We first
focus our attention on cell1, and hence the design of the transmission at RIS2
is to mitigate the inter-cell interference of the users in cell1, which can be
expressed as follows:

$\displaystyle\min\sqrt{{\varepsilon_{2,1,c}}}w_{2,1,c}p+\sqrt{{\varepsilon_{2,{\rm{T}},c}}}{{\bf{T}}_{2,1,c}}{{\bf{\Phi}}_{2,{\rm{T}}}}{{\bf{H}}_{2}}p,$
(13a)
$\displaystyle\min\sqrt{{\varepsilon_{2,1,e}}}w_{2,1,e}p+\sqrt{{\varepsilon_{2,{\rm{T}},e}}}{{\bf{T}}_{2,1,e}}{{\bf{\Phi}}_{2,{\rm{T}}}}{{\bf{H}}_{2}}p,$
(13b) $\displaystyle{\rm{Subject~{}to}}~{}{\beta_{2,{\rm{T,}}l}}\leq 1,\forall
l{\rm{=}}1\cdots L,$ (13c)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\phi_{2,{\rm{T,}}l}}\in\left[{0,2\pi}\right),\forall
l{\rm{=}}1\cdots L,$ (13d)

where (13a) and (13b) are the interference mitigation objectives of CCU1 and
CEU1, respectively.

To achieve the ambitious objective in (13), we first generate the effective
transmission channel gain of the BS2-RIS2-CCU1 and BS2-RIS2-CEU1 links as
follows:

${\overline{\bf{H}}_{2,{\rm{T}}}}=\left[{\begin{array}[]{*{20}{c}}{{h_{2,{\rm{R}},1}}{t_{2,1,c,1}}}&\cdots&{{h_{2,{\rm{R}},L}}{t_{2,1,c,L}}}\\\
{{h_{2,{\rm{R}},1}}{t_{2,1,e,1}}}&\cdots&{{h_{2,{\rm{R}},L}}{t_{2,1,e,L}}}\end{array}}\right].$
(14)

Then, the diagonal matrix ${{\bf{\Phi}}_{2,{\rm{T}}}}$ is transformed into a
vector as follows:

${\overline{\bf{\Phi}}_{2,{\rm{T}}}}={\left[{{\beta_{2,{\rm{T,1}}}}{\phi_{2,{\rm{T,1}}}},\cdots,{\beta_{2,{\rm{T,}}L}}{\phi_{2,{\rm{T,}}L}}}\right]^{T}}.$
(15)

Then, we focus on the passive beamforming design at RIS2, and the objective in
(13) can be rewritten as

${\overline{\bf{H}}_{2,{\rm{T}}}}{\overline{\bf{\Phi}}_{2,{\rm{T}}}}={{\bf
I}_{2,1}},$ (16)

where ${{\bf I}_{2,1}}$ is a 2 $\times$ 1 vector, which represents the
received inter-cell interference at CCU1 and CCU2 with
${I_{2,1}}=\left[{\begin{array}[]{*{20}{c}}{-\sqrt{\frac{{{\varepsilon_{2,1,c}}}}{{{\varepsilon_{2,{\rm{T}},c}}}}}w_{2,1,c}}\\\
{-\sqrt{\frac{{{\varepsilon_{2,1,e}}}}{{{\varepsilon_{2,{\rm{T}},e}}}}}w_{2,1,e}}\end{array}}\right]$.
Thus, the transmission phase shifts and transmission amplitude coefficients
can be readily obtained by

${\overline{\bf{\Phi}}_{2,{\rm{T}}}}=\overline{\bf{H}}_{2,{\rm{T}}}^{-1}{{\bf{I}}_{2,1}}.$
(17)

Similarly, RIS1 transmits the signal received from BS1 to the users located in
cell2 for interference cancellation, and hence the rest of power can be
reflected to the CCU1 for signal enhancement. Then, the reflection phase
shifts and reflection amplitude coefficients at RIS1 can be given by

${\overline{\bf{\Phi}}_{1,{\rm{R}}}}={\left[{\beta_{1,{\rm{R,1}}}{\phi_{1,{\rm{R,1}}}},\cdots,\beta_{1,{\rm{R,}}L}{\phi_{1,{\rm{R,}}L}}}\right]^{T}},$
(18)

where ${{\phi_{1,{\rm{R,}}l}}}$ denotes the reflection phase shifts at the
$l$-th RIS element of RIS1. It is commonly assumed that the transmission and
reflection phase shifts at RIS elements are independent [3, 4]. Thus, the
reflection of the RIS is to maximal the received power at the CCU, and the
objective can be given by

$\displaystyle\max\sqrt{{\varepsilon_{1,1,c}}}w_{1,1,c}p+\sqrt{{\varepsilon_{1,{\rm{R}},c}}}{{\bf{R}}_{1,1,c}}{{\bf{\Phi}}_{1,{\rm{Rx}}}}{{\bf{H}}_{1}}p,$
(19) $\displaystyle{\rm{Subject~{}to}}~{}{\beta_{1,{\rm{R,}}l}}\leq 1,\forall
l{\rm{=}}1\cdots L,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\phi_{1,{\rm{R,}}l}}\in\left[{0,2\pi}\right),\forall
l{\rm{=}}1\cdots L,$

where ${\varepsilon_{1,1,c}}=d_{1,1,c}^{-{\alpha_{1}}}$ and
${\varepsilon_{1,R,c=}}d_{1,{\rm{R}}}^{-{\alpha_{2}}}d_{{\rm{R}},1,c}^{-{\alpha_{3}}}$
represent the large-scale fading of the BS1-CCU1 and the BS1-RIS1-CCU1 links,
respectively. $w_{1,1,e}$ and ${{\bf{R}}_{1,1,e}}$ denote the small-scale
channel gains of BS1-CCU1 and RIS1-CCU1, respectively. ${{\bf{\Phi}}_{1,{\rm
R}}}$ denotes the reflection diagonal matrix of the RIS1.

To solve the objective in (19), we then generate a reflection vector of RIS1
as follows:

${\overline{\bf{H}}_{1,{\rm{R,c}}}}=\left[{\begin{array}[]{*{20}{c}}{{h_{1,{\rm{R}},1}}{r_{1,{\rm{c}},1}}}&\cdots&{{h_{1,{\rm{R}},L}}{r_{1,{\rm{c}},L}}}\end{array}}\right].$
(20)

Then, the reflection phase shift of RIS1 can be obtained as:

$\widetilde{\bf{\Phi}}_{1,{\rm{R}}}^{T}=\arg\left({{{\overline{\bf{H}}}_{1,{\rm{R,c}}}}}\right)-\arg\left({w_{1,1,c}}\right),$
(21)

where $\arg()$ denotes the angle function. In practice, the phase shifts and
amplitude coefficients are discrete due to the hardware limitations, which can
be solved by a linear mapping based on an equivalent amplitude-coefficient
vector as well as phase-shift vector [12]. By doing so, the received signals
from the BS1-CCU1 and BS1-RIS1-CCU1 can be coherently boosted, and thus the
channel gain is given by

${\left|h\right|_{1,1,c}}=\sqrt{{\varepsilon_{1,{\rm{R}},c}}}\sum\limits_{l=1}^{L}\left|{{h_{1,{\rm{R}},l}}{r_{1,{\rm{c}},l}}\beta_{1,{\rm{R,}}l}}\right|+\sqrt{{\varepsilon_{1,1,c}}}\left|{w_{1,1,c}}\right|.$
(22)

The objectives of both RIS1 and RIS2 are concluded to avoid any confusion in
Table I.

TABLE I: OBJECTIVES OF RIS1 AND RIS2 RIS1 | Interference mitigation for CCU2 and CEU2
---|---
signal enhancement for CCU1 and CEU1
RIS2 | Interference mitigation for CCU1 and CEU1
signal enhancement for CCU2 and CEU2

Based on the upon joint passive beamforming designs in (17) and (21) at two
STAR-RIS arrays, the proposed SSECB design can be achieved, where the desired
signals can be enhanced, while the interference signals can be cancelled
simultaneously. Hence, the signal-to-interference-plus-noise-ratio (SINR) of
the CEU1 can be written as

$SIN{R_{1,e}}=\frac{{\left|h\right|_{1,1,e}^{2}p\gamma_{e}^{2}}}{{\left|h\right|_{1,1,e}^{2}p\gamma_{c}^{2}+{\rho^{2}}}},$
(23)

where ${\gamma_{e}^{2}}$ and ${\gamma_{c}^{2}}$ denote the power allocation
factors of the CEU and CCU with $\gamma_{e}^{2}+\gamma_{c}^{2}=1$,
respectively. Since the reflection at the RIS mainly focuses on the signal
enhancement for the CCUs, the signals received at the CEUs are superimposed,
and hence the channel gain of CEU1 can be given by
$\left|h\right|_{1,1,e}=\left|{\sqrt{{\varepsilon_{1,{\rm{R}},e}}}\sum\limits_{l=1}^{L}{{h_{1,{\rm{R}},l}}{r_{1,{\rm{e}},l}}\beta_{1,{\rm{R,}}l}}+\sqrt{{\varepsilon_{1,1,e}}}w_{1,1,e}}\right|$.

Based on the application of the SIC technique at the CCUs, who first detect
the signal of the CEU with the following SINR:

$SIN{R_{1,e\to
1,c}}=\frac{{\left|h\right|_{1,1,c}^{2}p\alpha_{e}^{2}}}{{\left|h\right|_{1,1,c}^{2}p\alpha_{c}^{2}+{\rho^{2}}}}.$
(24)

Then the CCUs can detect its own signal with the following signal-to-noise-
ratio (SNR):

$SN{R_{1,c}}=\frac{{\left|h\right|_{1,1,c}^{2}p\alpha_{c}^{2}}}{{{\rho^{2}}}}.$
(25)

######  Remark 1.

For simplicity, it is assumed that the channels are in the strong LoS
scenario, where all channel gains are one. Based on the proposed SSECB design
in (17), and in order to mitigate inter-cell interferences at both the CCUs
and CEUs, the following constraints on the number of RIS elements need to be
met:

$L>\max\left\\{{\sqrt{\frac{{{\varepsilon_{2,1,c}}}}{{{\varepsilon_{2,{\rm{T}},c}}}}},\sqrt{\frac{{{\varepsilon_{2,1,e}}}}{{{\varepsilon_{2,{\rm{T}},e}}}}}}\right\\},$
(26)

otherwise no solution exists.

Thus, the achievable rate of the CCU and CEU can be obtained by:

${R_{1,{\rm{c}}}}=\mathbb{E}\left\\{{{{\log}_{2}}\left({1{\rm{+}}SN{R_{1,c}}}\right)}\right\\},$
(27)

and

${R_{1,{\rm{e}}}}=\mathbb{E}\left\\{{{{\log}_{2}}\left({1{\rm{+}}SIN{R_{1,e}}}\right)}\right\\},$
(28)

where $\mathbb{E}\\{\\}$ represents the expectation function.

##  IV Numerical Results

In this section, numerical results for evaluating the proposed SSECB design in
the STAR-RIS enhanced NOMA-CoMP networks are presented. Note that in this
paper, the transmission bandwidth of the proposed network is set to $B=1$ MHz,
and the power of the AWGN is set to
${\rho^{2}}=-174+10{\rm{lo}}{{\rm{g}}_{10}}(B)$ dBm.  The paired users share
the power with the power allocation factors $\gamma_{e}^{2}=0.6$ and
$\gamma_{c}^{2}=0.4$ on the basis of NOMA protocol, which is a common
parameter setting in previous contributions with fixed-power allocation NOMA
[15].  Note that more sophisticated optimization on the power allocation
factors is capable of further enhancing the network’s performance, however,
the power allocation factor optimization is beyond the scope of this
correspondence.

Some detailed parameters are illustrated in Table II.

TABLE II: SIMULATION PARAMETERS The distances of the BS1-CCU1 and BS2-CCU2 | 30 m
---|---
The distances of the BS1-CEU1 and BS2-CEU2 | 60 m
The distances of the BS1-RIS1 and BS2-RIS2 | 70 m
The distances of the RIS1-CEU1 and RIS2-CEU2 | 15 m
The distances of the RIS1-CCU1 and RIS2-CCU2 | 50 m
The distances of the BS1-CCU2 and BS2-CCU1 | 120 m
The distances of the BS1-CEU2 and BS2-CEU1 | 90 m
The path-loss exponent of BS-user links | $\alpha_{1}=3$
The path-loss exponent of BS-RIS links | $\alpha_{2}=3$
The path-loss exponent of RIS-CEU links | $\alpha_{3,e}=2.4$
The path-loss exponent of RIS-CCU links | $\alpha_{3,c}=2.7$
The path-loss exponent of interference links | $\alpha_{4}=3.5$
The fading parameters |  ${\cal{K}}_{1}=2$ ${\cal{K}}_{2}=3$
Figure 2:  Minimal required number of RIS elements for the proposed SSECB
design in the STAR-RIS enhanced NOMA-CoMP networks, where the small-scale
channel gains are normalized to all one, and
$\alpha_{3,c}=\alpha_{3,e}=\alpha_{3}$ for simplicity.

_1) Minimal Required Number of RIS elements:_ The minimal required number of
RIS elements of each RIS array for implementing the proposed SSECB design is
evaluated in Fig. 2. It is observed that with the increases of the path loss
exponent, the minimal required number of RIS elements increases, which
indicates that the path-loss exponent of the BS-RIS and RIS-user links are
required to be higher than that of the interference links of the proposed
SSECB design. Furthermore, note that in the simulation results, it is also
assumed that the small-scale channel gains are also normalized to one for
simplicity, and hence, the number of RIS elements should be greater than the
minimal number of RIS elements in practice.

Figure 3:  The achievable rate of both the CCU and CEU versus the transmit
power with different number of RIS elements. The minimal number of RIS element
is obtained to $L=27$ for the selected parameters.

_2) Impact of the Number of RIS Elements:_ The achievable rate of both the CCU
and CEU in the proposed SSECB design are evaluated in Fig. 3. The number of
RIS elements is obtained by (26), where $L=27$ in the case of the selected
parameter setting.

By comparing the red solid curve and dashed curve, we can see that in the case
of $L=54$, which is two times of the minimal required number of RIS elements,
the high-SNR slope of the achievable rate is one, which illustrates that the
interferences received from nearby cells are mitigated perfectly. One can also
observe that the achievable rate ceilings occur in the case of $L=27$, which
is due to the fact that the number of RIS elements is not enough for
interference cancellation, and hence the inter-cell interference residue
exists. Observe that the performance gap between the proposed SSECB design and
the case without RIS increases, which also show the superiority of the
proposed SSECB design.

Figure 4: The achievable rate of both the CCU and CEU versus the number of RIS
elements with different passive beamforming designs. The transmit power of the
BS is set to $p=-50$ dBm.

_3) Impact of Different Passive Beamforming Designs:_ The achievable rate of
the paired NOMA users with different passive beamforming at RIS elements are
evaluated in Fig. 4. By aligning the reflected signals and desired signal
perfect coherent at the CCU or CEU, the dashed, green and black curves are
obtained for comparing the performance of a pair of SEB designs. We can see
that the achievable rate of the proposed SSECB design is higher than other
benchmark schemes, which illustrate that the proposed SSECB design is capable
of outperforming the other SEB and SEC designs.

##  V Conclusions

In this letter, firstly, previous contributions related to the RIS-enhanced
SEB and SCB designs are presented. Then, a novel STAR-RIS for the proposed
SSECB design was utilized. We designed a joint passive beamforming weight,
where the transmission and reflection of RIS elements are used for inter-cell
interference cancellation and signal enhancement, respectively, to provide a
practical design of the STAR-RIS enhanced NOMA-CoMP networks. Compared to the
previous SEB and SCB designs, our proposed STAR-RIS enhanced SSECB design has
superior performance on achievable rate. Further research should be undertaken
to add more TA and RA at the BS and users by jointly designing both the active
beamforming, passive beamforming, and detection vectors.  Another future
direction is to combine stochastic geometry tools for evaluation the impact of
location randomness of users [16].

## References

  * [1] M. D. Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and road ahead,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 11, pp. 2450–2525, Nov. 2020\.
  * [2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Commun. Mag._ , vol. 58, no. 1, pp. 106–112, Jan. 2020.
  * [3] Y. Liu, X. Mu, J. Xu, R. Schober, Y. Hao, H. V. Poor, and L. Hanzo, “STAR: Simultaneous transmission and reflection for 360∘ coverage by intelligent surfaces,” _Arxiv_ , vol. 2103.09104v2, pp. 1–1, Mar. 2021.
  * [4] J. Xu, Y. Liu, X. Mu, and O. A. Dobre, “STAR-RISs: Simultaneous transmitting and reflecting reconfigurable intelligent surfaces,” _Arxiv_ , vol. 2101.09663v2, pp. 1–1, Mar. 2021.
  * [5] 3GPP TR 36.819 V11.0.0, “Coordinated multi-point operation for LTE,” Sep. 2011.
  * [6] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, C. L. I, and H. V. Poor, “Application of non-orthogonal multiple access in LTE and 5G networks,” _IEEE Commun. Mag._ , vol. 55, no. 2, pp. 185–191, Feb. 2017.
  * [7] W. Shin, M. Vaezi, B. Lee, D. J. Love, J. Lee, and H. V. Poor, “Non-orthogonal multiple access in multi-cell networks: Theory, performance, and practical challenges,” _IEEE Commun. Mag._ , vol. 55, no. 10, pp. 176–183, 2017.
  * [8] W. Shin _et al._ , “Coordinated beamforming for multi-cell MIMO-NOMA,” _IEEE Commun. Lett._ , vol. 21, no. 1, pp. 84–87, 2017.
  * [9] D. Wan, R. Huang, M. Wen, G. Chen, F. Ji, and J. Li, “A simple multicarrier transmission technique combining transmit diversity and data multiplexing for non-orthogonal multiple access,” _IEEE Trans. Veh. Technol._ , Accepted for appear 2021.
  * [10] Y. Liu, X. Liu, X. Mu, T. Hou, J. Xu, Z. Qin, M. D. Renzo, and N. Al-Dhahir, “Reconfigurable intelligent surfaces: Principles and opportunities,” _Arxiv_ , vol. 2007.03435v2, pp. 1–1, 2021.
  * [11] T. Hou, Y. Liu, Z. Song, X. Sun, Y. Chen, and L. Hanzo, “Reconfigurable intelligent surface aided NOMA networks,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 11, pp. 2575–2588, Nov. 2020.
  * [12] T. Hou, Y. Liu, Z. Song, X. Sun, and Y. Chen, “MIMO-NOMA networks relying on reconfigurable intelligent surface: A signal cancellation-based design,” _IEEE Trans. Commun._ , vol. 68, no. 11, pp. 6932–6944, Nov. 2020\.
  * [13] Q. Wu and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” _IEEE Trans. Commun._ , vol. 68, no. 3, pp. 1838–1851, Mar. 2020.
  * [14] Z. Wan, Z. Gao, and M.-S. Alouini, “Broadband channel estimation for intelligent reflecting surface aided mmWave massive MIMO systems,” _Arxiv_ , vol. 2002.01629v1, pp. 1–1, Feb. 2020.
  * [15] T. Hou, Y. Liu, Z. Song, X. Sun, and Y. Chen, “Multiple antenna aided NOMA in UAV networks: A stochastic geometry approach,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1031–1044, Feb. 2019.
  * [16] Y. Zhu, G. Zheng, and K.-K. Wong, “Stochastic geometry analysis of large intelligent surface-assisted millimeter wave networks,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 8, pp. 1749–1762, Aug. 2020.