# Efficient Quantum Modular Arithmetics for the ISQ Era

Parfait Atchade-Adelomou  MIT Media Lab - City Science Group, Cambridge, USA
<EMAIL_ADDRESS>Smart Society Research Group - La Salle - Universitat Ramon
Llull, Carrer de Sant Joan de La Salle, 42, 08022 Barcelona (Spain)
Lighthouse Disruptive Innovation Group, LLC, 7 Broadway Terrace, Apt 1,
Cambridge MA 02139, Middlesex County, Massachusetts (USA)
<EMAIL_ADDRESS>Saul Gonzalez  Lighthouse Disruptive
Innovation Group Europe, SL. Barcelona - Spain  saul.gonzalez@lighthouse-
dig.com

(November 2023)

###### Abstract

As we venture into the Intermediate-Scale Quantum (ISQ) era, the proficiency
of modular arithmetic operations becomes pivotal for advancing quantum
cryptographic algorithms. This study presents an array of quantum circuits,
each precision-engineered for modular arithmetic functions critical to
cryptographic applications. Central to our exposition are quantum modular
adders, multipliers, and exponential operators, whose designs are rigorously
optimized for ISQ devices. We provide a theoretical framework and practical
implementations in the PennyLane quantum software, bridging the gap between
conceptual and applied quantum computing. Our simulations validate the
efficacy of these methodologies, offering a strategic compass for developing
quantum algorithms that align with the rapid progression of quantum
technology.

KeyWords: quantum modular arithmetics, quantum adder, quantum multiplier,
quantum exponential operator, exponentiation, quantum circuits, Shor’s
algorithm, quantum Fourier transform, ISQ era

##  I Introduction

Quantum computing, with its transformative potential, is driving a paradigm
shift in computational methodologies. Classical operations foundational to
traditional algorithms undergo scrutiny and reinvention as they transition
into the quantum landscape. Modular arithmetic operators, specifically
addition and multiplication, are quintessential due to their ubiquitous
presence across classical and quantum algorithms. However, translating these
operations to quantum mechanisms introduces many challenges, primarily because
quantum operations necessitate reversibility.

A notable constraint in the quantum programming paradigm is the stipulation
for operations to be bijective when implemented Inplace. This poses tangible
impediments: for instance, iterative use of Outplace operators in an algorithm
demands auxiliary qubit registers for each iteration. The accumulating demand
for such registers can quickly surpass the qubit capabilities of contemporary
quantum devices. Given the centrality of modular arithmetic in seminal quantum
algorithms, including Shor’s factorization Shor (1994), there is a pressing
need to craft efficient, bijective modular operations.

This work embarks on an exploration and optimization of modular arithmetic
operators tailored for the impending Intermediate-Scale Quantum (ISQ) era
Arrazola and Xanadu (2023). Our endeavor is twofold: to dissect existing
quantum modular operators, unearthing their intricacies, and to propose their
fresh implementation, streamlined methodologies apt for the ISQ landscape. By
doing so, we aspire to fortify the bridge between theoretical quantum
constructs and their tangible applications, equipping neophytes and seasoned
quantum researchers with state-of-the-art modular arithmetic tools.

The organization of this manuscript is crafted to facilitate a comprehensive
understanding of the research presented. Section II articulates the impetus
and intrinsic motivations guiding this inquiry. Section IV constructs the
conceptual framework, defining the terminology and foundational principles
critical to the quantum modular arithmetic operators discourse. Section III
contextualizes our work within the existing scholarly landscape, critically
examining preceding studies on quantum modular operators. The methodology,
with its rigorous and detailed formulation, is expounded in Section V. After
the theoretical exposition, Section VI validates our theoretical constructs
with empirical evidence and provides a repository of executable code optimized
for quantum computational platforms. The manuscript reaches its denouement in
Section VII, where we encapsulate our seminal contributions and delineate
prospective trajectories for future scholarly exploration.

##  II Motivation and Problem Statement

Quantum computing, while heralding unparalleled computational prowess, is
still in its embryonic phase and is riddled with challenges. Among these is
the constrained availability of qubits in nascent quantum architectures
Atchade-Adelomou (2022). This context provides the foundation for our
exploration.

The academic and research landscapes are replete with groundbreaking work Shor
(1999, 1994); Pérez-Salinas  _et al._ (2020); Cerezo  _et al._ (2021);
Adelomou  _et al._ (2020a); Atchade-Adelomou and Larson (2023a); Gonzalez-
Bermejo  _et al._ (2022); Atchade-Adelomou and Larson (2023b) in quantum
algorithms and their applications Atchade-Adelomou  _et al._ (2021); Consul-
Pacareu  _et al._ (2023); Atchade-Adelomou and Alonso-Linaje (2022); Alonso-
Linaje and Atchade-Adelomou (2021); Adelomou  _et al._ (2020b, 2023); Atchade-
Adelomou  _et al._ (2020). However, despite this abundance, there is a
conspicuous gap: a comprehensive synthesis of optimal algorithms, particularly
in modular arithmetic. Furthermore, even amongst those that touch upon the
subject, there needs to be more literature that delineates the most efficient
implementations of modular arithmetic operators Shor (1999); Rines and Chuang
(2018). This observed lacuna, juxtaposed with the paramount significance of
modular arithmetic in quantum algorithms, particularly in foundational ones
like Shor’s, accentuates the pressing need for a systematic investigation into
this domain.

Driven by this palpable gap and a commitment to pragmatism and efficiency, we
focus on devising modular arithmetic operators meticulously crafted for the
ISQ era. The quantum milieu of today necessitates algorithms that judiciously
economize on qubit usage while embodying resilience and extensibility. By
optimizing the utility of extant, bounded qubit resources, such algorithms
should be congruent with contemporary quantum architectures and adaptable to
the more sophisticated quantum machines on the horizon.

We aim to address this literature void by proposing optimal modular arithmetic
operators and detailing their efficient implementations. While the
implementations herein are tailored for PennyLane Bergholm  _et al._ (2022),
lauded for its flexibility and ease of integration, our methods’ principles
are designed with cross-platform compatibility. We intend to furnish the
quantum research community with an authoritative resource on modular
arithmetic that can be readily adapted and applied within diverse quantum
computing frameworks, thereby fostering innovation in numerous quantum
domains.

##  III Work Context

Quantum computing, though nascent, has seen a surge of innovations tailored to
harness the singular capabilities of quantum architectures. Central to this
ecosystem is modular arithmetic, a linchpin in quantum cryptographic endeavors
and quintessential to Shor’s integer factorization algorithm Shor (1999).

As the field edges closer to the ISQ era, the imperative to scrutinize and
refine existing algorithms grows stronger. Modular exponentiation Dimitrov
_et al._ (1998); Van Meter and Itoh (2005); Hong  _et al._ (1996) exemplifies
this, melding classical and quantum paradigms Gordon (1998); Marwaha  _et al._
(2013). Noteworthy architectures like the Vedral-Barenco-Ekert (VBE) Vedral
_et al._ (1996) and the Beckman-Chari-Devabhaktuni-Preskill (BCDP) Beckman
_et al._ (1996) have set benchmarks. However, designs like Gossett’s Gossett
(1998) and Beauregard’s approach Beauregard (2003), championing pure quantum
arithmetic Ruiz-Perez and Garcia-Escartin (2017); Alonso-Linaje and Xanadu
(2023); Guillermo (2023), underscore the diversity and richness of the field.

In the rapidly evolving landscape of ISQ computing, our objective is clear-
cut: to craft modular arithmetic operators that resonate with current research
and are poised for future advancements. While our solutions are intricately
crafted upon the PennyLane framework, they are not bound by it. We have
designed our algorithms to transcend platform-specific constraints, ensuring
they can be effortlessly transposed to other quantum computing environments
like Qiskit Wille  _et al._ (2019). The forthcoming sections will articulate
the nuances of our modular arithmetic approach, juxtaposing it with existing
paradigms and underscoring its pivotal role in the onward march of the ISQ
era.

##  IV Preliminaries

This section lays the foundational terminology and concepts pivotal to our
study of quantum modular arithmetic operators. We establish the encoding
scheme used throughout this work, which is critical for consistency in our
theoretical and practical applications. Discussing the ISQ era contextualizes
our study within the current quantum computing landscape. Following this, we
precisely classify operators into Inplace and Outplace categories, each with
distinctive operational characteristics crucial for quantum algorithm design.
We introduce our operators with formal definitions that serve as a precursor
to the in-depth discussions in later sections.

###  IV.1 Notation and Qubit Encoding

In alignment with the encoding standards of the PennyLane quantum computing
framework Bergholm  _et al._ (2022), we adopt the conventional big-endian
format James (1990).

######  Definition 1 (Qubit Encoding in PennyLane).

PennyLane adheres to a big-endian qubit encoding convention, aligning with
standard practices in quantum computation frameworks. Within this convention,
$a_{0}$ is designated as the most significant bit (MSB), consistently
represented at the topmost or leftmost in quantum circuit diagrams, which
corresponds to the highest order of magnitude in the binary representation. In
contrast, $a_{n-1}$ is the least significant bit (LSB), placed at the
bottommost or rightmost, representing the lowest order of magnitude. This
encoding ensures coherence with PennyLane’s infrastructure and promotes
uniformity in implementing quantum algorithms.

For illustration, consider the integer $a$ expressed in binary form as
$a=a_{0}\cdot 2^{n}+a_{1}\cdot 2^{n-1}+\ldots+a_{n}\cdot 2^{0}$. When
representing the quantum state corresponding to the decimal number $6$ within
PennyLane, the binary equivalent $110$ is denoted as
$\ket{a_{0}a_{1}a_{2}}=\ket{110}$. This notation maps the binary
representation such that $a_{0}$ is the MSB and $a_{n}$ is the LSB, reflecting
a left-to-right orientation in circuit schematics.

This encoding scheme is essential for adequately interpreting quantum
registers and analyzing computational results, necessitating its comprehension
for an accurate understanding of the quantum operations presented herein.
Users employing alternative platforms, such as Qiskit, should adjust the qubit
ordering accordingly to maintain the integrity of the computational results.

###  IV.2 ISQ era and Operator Classification

With the advent of the ISQ era Arrazola and Xanadu (2023), a refined
comprehension of quantum operators gained paramount importance.

######  Definition 2 (Intermediate-Scale Quantum (ISQ) Era).

The ISQ era bridges the gap between the Noisy Intermediate-Scale Quantum
(NISQ) phase and the advent of fully fault-tolerant quantum computing. ISQ
devices, benefiting from nascent quantum error correction techniques, enable
extended quantum circuit depths, albeit within a limited qubit framework. This
period emphasizes the fine-tuning of quantum algorithms to resonate with these
newfound capabilities, heralding a new epoch of reliable quantum computations.

Our study categorizes these operators into two fundamental types—Inplace and
Outplace—which are instrumental for executing quantum arithmetic operations.
These classifications facilitate a structured approach to algorithmic design
and underpin the advancement of quantum computational methodologies.

######  Definition 3 (Inplace Operator).

An operator is termed Inplace if it applies a function
$f:\mathbb{N}\rightarrow\mathbb{N}$ directly to the input qubit register
without the need for additional output space. For the associated quantum
operator $U_{f}$, the transformation is described as
$U_{f}\ket{x}=\ket{f(x)}$, where the output $f(x)$ overwrites the original
input $x$ in the same register. The feasibility of an Inplace operator hinges
on the bijectivity of the function $f$, which guarantees the reversibility of
the operation, a requirement for quantum computations.

An illustrative example within this context is the modular addition by a
constant $k$ modulo $N$, defined as $Add_{\text{in}}(k,N)$, which computes
$\ket{x}\rightarrow\ket{x+k\mod N}$ using the same qubits that initially
represented $\ket{x}$.

######  Definition 4 (Outplace Operator).

An Outplace operator for a function $f:\mathbb{N}\rightarrow\mathbb{N}$, and
its corresponding quantum operator $U_{f}$, operates on two distinct
registers. It preserves the original input register $|x\rangle$ intact and
conveys the function’s output to an auxiliary register set to $|0\rangle$,
resulting in the transformation
$U_{f}|x\rangle|0\rangle=|x\rangle|f(x)\rangle$. This design allows for the
execution of non-bijective functions within a quantum framework.

For example, the operator $Add_{\text{out}}(k,N)$ showcases this architecture
by computing
$Add_{\text{out}}(k,N)\ket{a}\ket{0}\rightarrow\ket{a}\ket{a+k\mod N}$, where
the auxiliary register initially in the state $|0\rangle$ is employed to hold
the output of the modular addition.

The operators analyzed in this study are:

  * •

$Add_{\text{in}}(k,N)$: An Inplace modular adder for the classical constant
$k$, operating modulo $N$.

  * •

$Add_{\text{out}}(k,N)$: Its Outplace counterpart for modular addition.

  * •

$Add_{\text{out}}(N)$: An Outplace adder for dual quantum variables.

  * •

$Add_{\text{in}}(N)$: The Inplace version for two quantum variables.

  * •

$Mult_{\text{out}}(k,N)$: Outplace modular multiplier with the classical
constant $k$, modulo $N$.

  * •

$Mult_{\text{in}}(k,N)$: Inplace modular multiplier for the classical variable
$k$, modulo $N$.

  * •

$Mult_{\text{out}}(N)$: Outplace modular multiplier for two quantum variables.

  * •

$Exp_{\text{out}}(a,N)$: An Outplace exponential operator.

By distinctly listing each operator and specifying its type, we ensure clarity
and provide a detailed overview that serves as a foundation for the quantum
computational principles discussed subsequently.

##  V Implementation of Quantum Modular Operators

The quantum modular operators, while foundational, enable a myriad of quantum
algorithms, especially those focusing on cryptographic tasks and number-
theoretic challenges.

###  V.1 Fundamental Auxiliary Operators

Our journey in crafting optimal modular arithmetic operators frequently
intersects with the necessity for auxiliary quantum operators. These operators
simplify our primary implementations and enhance their efficiency and
adaptability, especially in the context of the ISQ era. In this section, we
spotlight these indispensable components:

  * •

Quantum Fourier Transform (QFT) Coppersmith (2002): The QFT is a linchpin for
in-place qubit addition. Its transformation, as represented in equation (1),
illuminates its relevance in the set of operators we delineate:

$\displaystyle
QFT\ket{j_{1}}\ket{j_{2}}...\ket{j_{n}}\rightarrow\frac{1}{2^{n/2}}\Bigg{[}$
$\displaystyle(\ket{0}+\exp^{2\pi i0.j_{n}})$
$\displaystyle\otimes(\ket{0}+\exp^{2\pi i0.j_{n-1}j_{n}})$
$\displaystyle\otimes\ldots$ $\displaystyle\otimes(\ket{0}+\exp^{2\pi
i0.j_{1}j_{2}...j_{n}})\Bigg{]}.$ (1)

  * •

Quantum Fourier Adder Alonso-Linaje and Xanadu (2023); Ruiz-Perez and Garcia-
Escartin (2017): Central to quantum modular arithmetic is the $Sum(k)$
operator, which facilitates the addition of a classical constant $k$ to a
quantum register state $\ket{a}$. This operation leverages the transformative
capabilities of the Quantum Fourier Transform (QFT) to transmute the addition
into a sequence of controlled phase shifts. Initially, the quantum state
undergoes the QFT, mapping the computational basis to a superposition where
each component’s phase is modulated by $k$ (Eq. (2)). Subsequently, the
inverse QFT (QFT-1) is applied, effectively synthesizing the augmented quantum
state $\ket{a+k}$. The operation is mathematically represented as:

$\begin{split}\ket{a}\xrightarrow{\text{QFT}}&\frac{1}{2^{n/2}}\left(\sum_{p=0}^{2^{n}-1}e^{2\pi
iap/2^{n}}\ket{p}\right)\\\
&\xrightarrow{Sum(k)}\frac{1}{2^{n/2}}\left(\sum_{p=0}^{2^{n}-1}e^{2\pi
i(a+k)p/2^{n}}\ket{p}\right)\\\
&\xrightarrow{\text{QFT}^{-1}}\ket{a+k}.\end{split}$ (2)

The circuit implementation of this adder is illustrated in Fig. (1), which
integrates the QFT with controlled phase rotations to enable quantum addition,
as elaborated in Eq. (2).

QFTSum(k)QFT-1$\ket{a}$$\ket{a+k}$

Figure 1: Visualization of the integer quantum adder operator.

Figure 2: Schematic of the Inplace Modular Quantum-Classical Adder. Commencing
with $\psi_{a}=\ket{0}\ket{a}\ket{0}$, the circuit employs a dual auxiliary
qubit system to facilitate underflow control, with one qubit corresponding to
the MSB and another to the LSB. Upon addition of the negative modulus $-N$ and
subsequent comparison, if the condition $a+k-N<0$ is met, the terminal qubit
is activated, yielding state $\psi_{b}=\ket{a+k-N}\ket{b}$ for the definition
of $b$, refer to Eq. (4)). A conditional restoration of $N$ follows for states
satisfying $a+k-N<0$, effectuating the state $\psi_{c}=\ket{a+(b-1)N}\ket{b}$.
The final addition of $k$ to the conditioned state $\ket{a+(b-1)N}\ket{0}$
culminates in the modular sum $\psi_{d}=\ket{a+k\mod N}$. Incorporating the
QFT within this process facilitates the efficient computation of addition and
subtraction within the Fourier space, illustrating the sophisticated interplay
between classical and quantum computational paradigms.

The ensuing segments delve deeper into these auxiliary operators’ circuit
designs and functionalities, elucidating their synergy with the primary
modular arithmetic operations central to our discourse.

###  V.2 Inplace Modular Quantum-Classical Adder

Central to our exploration is the Inplace Modular Quantum-Classical Adder,
represented by the equation (3) and the figure (2).

$Add_{\text{in}}(k,N)\ket{a}\rightarrow\ket{a+k\mod N}.$ (3)

This operator facilitates the modular addition of a constant $k$ to a qubit
$\ket{a}$ that resides in the Fourier basis. Given its foundational role,
several other modular arithmetic operators are derived, including sums,
multiplications, and even the exponential.

To construct this operator, we introduce the variable $b$, determined as:

$b=\begin{cases}0&\text{if }a+k\geq N,\\\ 1&\text{if }a+k<N.\end{cases}$ (4)

Here, $b$ indicates if a modular adjustment is required:

  * •

$b=0$: The sum surpasses the modulus, necessitating subtraction by $N$.

  * •

$b=1$: The sum remains within modular constraints, and no adjustment is
needed.

With $b$ defined, our objective is to design a circuit effecting the
transformation $f(\ket{a}):=\ket{a+k+(b-1)N}$. This circuit (detailed in Fig.
(2)) comprises three primary steps:

  1. 1.

Phase 1: Assess if subtraction by $N$ is required. If so, utilize an auxiliary
qubit to transition to the state $\ket{a+(b-1)N}$.

  2. 2.

Phase 2: Clean the auxiliary qubit.

  3. 3.

Phase 3: Add $k$ to the resultant state from Phase 1, post-auxiliary qubit
cleaning.

For this work, it is essential to highlight that all operations and
representations are considered within the standard computational basis.

###  V.3 Outplace Modular Quantum-Classical Adder

The Outplace Modular Quantum-Classical Adder (see Fig. (3)), denoted as
$Add_{\text{out}}(k,N)$, is defined by the transformation in equation (5).

$Add_{\text{out}}(k,N)\ket{a}\ket{0}\rightarrow\ket{a}\ket{a+k\mod N}.$ (5)

Figure 3: Schematic of the Outplace Modular Quantum-Classical Adder Operator.
The operation initiates with the quantum register $\psi_{a}=\ket{a}\ket{0}$,
where $\ket{a}$ encodes the quantum representation of the integer to which the
classical constant $k$ will be added. Through the controlled application of
the $Add_{\text{in}}(N)$ operators, the system transitions to the intermediate
state $\psi_{b}=\ket{a}\ket{a}$. Subsequently, the circuit performs the
modular addition, progressing to the state $\psi_{c}=\ket{a}\ket{a+k\mod{N}}$,
thereby encapsulating the sum in the second register.

Leveraging the previously defined $Add_{\text{in}}(k,N)$ operator, the
operational sequence of $Add_{\text{out}}(k,N)$ can be depicted as in equation
(6):

$\ket{a}\ket{0}\xrightarrow{\text{Copy}}\ket{a}\ket{a}\xrightarrow{Add_{\text{in}}(k,N)}\ket{a}\ket{a+k\mod
N}.$ (6)

Initially, the state $\ket{a}\ket{0}$ is duplicated. Following this, the
$Add_{\text{in}}(k,N)$ operator is employed to yield the desired outcome.

###  V.4 Inplace Modular Quantum-Quantum Adder

The Inplace Modular Quantum-Quantum Adder (see Fig. (4)), symbolized as
$Add_{\text{in}}(N)$, performs the transformation outlined in equation (7):

$Add_{\text{in}}(N)\ket{a}\ket{b}\rightarrow\ket{a}\ket{a+b\mod N}.$ (7)

The operator’s operational efficacy in isolation necessitates transposing
input states into the Fourier basis via the Quantum Fourier Transform (QFT).
However, when this operator is integrated with our
$\text{Add}_{\text{in}}(k,N)$ in a composite quantum circuit, the initial QFT
and the concluding inverse QFT ($QFT^{-1}$) operations are unnecessaries. This
optimization reduces the quantum resource requirements associated with Fourier
basis transformations. Adopting this integrated approach, we capitalize on the
standard configuration of the computational basis to implement controlled
additions, where each qubit $\ket{a_{i}}$ in the register $\ket{a}$
contributes proportionally to its binary significance, as clarified in
Equation (8).

$c_{\ket{a_{i}}}-Add_{\text{in}}(2^{n-i-1},N)\ket{b},$ (8)

wherein $c_{a_{i}}$ designates the control derived from qubit $\ket{a_{i}}$.

The associated quantum circuit is showcased in Fig. (4).

Figure 4: Schematic of the Inplace Modular Quantum-Quantum Adder Circuit. The
process begins with the state $\psi_{a}=\ket{a}\ket{b}$, where both $\ket{a}$
and $\ket{b}$ are quantum representations of the integers to be combined. The
circuit executes the addition Inplace, such that the second register
transitions to $\psi_{b}=\ket{a}\ket{a+b\mod{N}}$, encapsulating the sum
within the existing qubit structure. The Inplace operation preserves the
initial quantum register $\ket{a}$, while the second register is updated to
reflect the modular addition.

###  V.5 Outplace Modular Quantum-Quantum Adder

The Outplace Modular Quantum-Quantum Adder, referred to as
$Add_{\text{out}}(N)$, is characterized by the transformation described in
equation (9).

$Add_{\text{out}}(N)\ket{a}\ket{b}\ket{0}\rightarrow\ket{a}\ket{b}\ket{a+b\mod
N}.$ (9)

The schematic reveals a white-and-black diagonal control point that initiates
a sequence of controlled gates (refer to Fig. (5)). The control point depicted
employs a series of controlled gates, with each gate’s action scaled down by
powers of two by the binary sequence of the input qubits. This ensures a
precise manipulation of the quantum states, which is crucial for the intended
computational process.

This operator is conceptually constructed upon the $Add_{\text{out}}(k,N)$
operator, harnessing the Inplace modular adder method. The evolution of states
through this operator is represented in equation (10):

$\ket{a}\ket{b}\ket{0}\xrightarrow{\text{Copy}}\ket{a}\ket{b}\ket{a}\xrightarrow{Add_{\text{in}}(N)}\ket{a}\ket{b}\ket{a+b\mod
N}.$ (10)

For a visual depiction of the circuit effectuating this transformation, refer
to Fig. (5).

Figure 5: Depiction of the Outplace Modular Quantum-Quantum Adder Operator.
The circuit commences with two quantum registers in the state
$\psi_{a}=\ket{a}\ket{b}\ket{0}$, with $\ket{a}$ and $\ket{b}$ representing
the quantum states of the integers to be added. The operator facilitates the
modular addition, transitioning to the final state
$\psi_{b}=\ket{a}\ket{b}\ket{a+b\mod{N}}$, which embodies the result
$\ket{a+b\mod{N}}$ stored in the third register.

###  V.6 Outplace Modular Quantum-Classical Multiplier

The Outplace Modular Quantum-Classical multiplier, denoted as
$Mult_{\text{out}}(k,N)$, is defined by the equation (11).

$Mult_{\text{out}}(k,N)\ket{a}\ket{b}=\ket{a}\ket{b+ka\mod N}$ (11)

To construct this operator, we transition the qubit $\ket{b}$ into the Fourier
basis. Subsequently, controlled summations are applied, represented as
$c_{\ket{a_{i}}}Add_{\text{in}}(k2^{n-i-1},N)\ket{b}$.

The representation of this transformation within a quantum circuit is
showcased in Fig. (6).

Figure 6: Depiction of the Outplace Modular Quantum-Classical Multiplier
Operator. The operation commences with the state $\psi_{a}=\ket{a}\ket{b}$,
which, through the multiplier, transitions to
$\psi_{b}=\ket{a}\ket{b+ka\mod{N}}$. This transition embodies the modular
multiplication by a classical constant $k$, pivotal in the computational steps
of the Inplace Quantum-Classical Multiplier.

###  V.7 Inplace Modular Quantum-Classical Multiplier

The Inplace Modular Quantum-Classical Multiplier (see Fig. (7)), denoted as
$Mult_{\text{in}}(k,N)$, is defined by the transformation in equation (12):

$Mult_{\text{in}}(k,N)\ket{a}=\ket{ka\mod N}.$ (12)

The operator is implementable exclusively when $k$ possesses a modular inverse
relative to $N$. This constraint is vital since the operator is not
universally bijective. For illustration, given $N=8$, one discerns that
$Mult_{\text{in}}(4,8)\ket{2}=\ket{0\mod 8}$ and
$Mult_{\text{in}}(4,8)\ket{4}=\ket{0\mod 8}$. The primary requisite for this
operator’s instantiation is the existence of a modular inverse $k^{-1}$ for
$k$, guaranteeing its bijectiveness.

With the preconditions met and using an auxiliary register, one can deploy the
$Mult_{\text{out}}(k)$ operator twice to achieve the transformation delineated
in equation (15):

$\displaystyle\ket{a}\ket{0}$
$\displaystyle\xrightarrow{Mult_{\text{out}}(k,N)}\ket{a}\ket{ka}$ (13)
$\displaystyle\xrightarrow{\text{SWAP}}\ket{ka}\ket{a}$ (14)
$\displaystyle\xrightarrow{{Mult_{\text{out}}(k^{-1},N)}^{\dagger}}\ket{ka}\ket{0}.$
(15)

Figure 7: The Inplace Modular Quantum-Classical Multiplier Circuit initiates
with the state $\psi_{a}=|a\rangle|0\rangle$, transitioning to the
intermediate state $\psi_{b}=|a\rangle|ka\rangle$. Upon verification that $k$
and $N$ are coprime, the circuit implements the transformation to
$\psi_{c}=|ka\rangle|a-k^{-1}ka\rangle$, culminating in the final output state
$\psi_{d}=|ka\mod N\rangle$, which reflects the desired Inplace quantum
modular multiplier.

###  V.8 Outplace Modular Quantum-Quantum Multiplier

The Outplace Modular Quantum-Quantum Multiplier (see Fig. (8)), represented as
$Mult_{\text{out}}(N)$, is characterized by the transformation in equation
(16).

$Mult_{\text{out}}(N)\ket{a}\ket{b}\ket{0}=\ket{a}\ket{b}\ket{ab\mod N}.$ (16)

The underlying mechanism for this operator involves interchanging the third
register in the sequential execution of the operations
$c_{a_{i}}-Mult_{\text{out}}(2^{n-i-1},N)(\ket{b})$.

Figure 8: Schematic Illustration of the Outplace Modular Quantum-Quantum
Multiplier Operator. The process initiates with the state
$\psi_{a}=\ket{a}\ket{b}\ket{0}$, evolving to the outcome
$\psi_{b}=\ket{a}\ket{b}\ket{ab\mod{N}}$, thereby executing the multiplication
operation. This operator is a quintessential example of quantum-quantum
interaction, efficiently facilitating the multiplication of two quantum states
within a modular framework.

###  V.9 Inplace Modular Quantum-Quantum Multiplier

The objective of the Inplace Modular Quantum-Quantum Multiplier, denoted by
$Mult_{\text{in}}(N)$, is to execute the transformation described in Eq. (17).

$Mult_{\text{in}}(N)\ket{a}\ket{b}\rightarrow\ket{a}\ket{ab\mod N}.$ (17)

To conceive this operator, inspiration can be drawn from the strategy
delineated for in-place quantum-classic multiplication, as specified in Eq.
(12).

The procedure for the Inplace Modular Quantum-Quantum multiplier unravels as a
series of state transformations. Commencing with the initial state of
$\ket{a}\ket{b}\ket{0}\ket{0}$, where we encapsulate the product $ab$ in the
third register. This step engenders the intermediate state
$\ket{a}\ket{b}\ket{ab}\ket{a^{-1}}$, as depicted in the progression sequence
of Eq. (18).

Subsequent operations, as elucidated in the same sequence, transfer the
product from the auxiliary to the principal register, and further
transformations culminate in the final state $\ket{a}\ket{ab}\ket{0}\ket{0}$.

$\displaystyle\ket{a}\ket{b}\ket{0}\ket{0}$ (18)
$\displaystyle\quad\xrightarrow{\text{Multiplication and
Inverse}}\ket{a}\ket{b}\ket{ab}\ket{a^{-1}}$
$\displaystyle\quad\xrightarrow{\text{State
Swap}}\ket{a}\ket{ab}\ket{b}\ket{a^{-1}}$
$\displaystyle\quad\xrightarrow{\text{Modular
Correction}}\ket{a}\ket{ab}\ket{b-a^{-1}ab}\ket{a^{-1}}$
$\displaystyle\quad\xrightarrow{\text{Auxiliary
Reset}}\ket{a}\ket{ab}\ket{0}\ket{0}$

While the construction of the Inplace Quantum-Quantum Multiplier could
theoretically follow a methodology akin to that of the Inplace Quantum-Classic
Multiplier, such an endeavor has been pragmatically set aside. This decision
is attributed to the computational intricacies associated with the modular
inverse operation, as encapsulated by $U\ket{a}\ket{0}=\ket{a}\ket{a^{-1}\mod
N}$. The formulation of this inverse operator represents a significant
computational challenge that would necessitate a disproportionate allocation
of resources and algorithmic complexity, potentially overshadowing the
operator’s practical utility in the current quantum computing paradigm.

###  V.10 Outplace Modular Exponential Operator

Central to the machinery of Shor’s algorithm is the modular exponential
operator (see Fig. (9)), which undertakes the transformation delineated in Eq.
(19).

$Exp(a,N)\ket{x}\ket{1}\ket{0}\rightarrow\ket{x}\ket{a^{x}\mod N}\ket{0}.$
(19)

Considering this operator’s pivotal role in quantum algorithmic design, it is
imperative to architect its efficient and accurate realization. The circuit’s
elegance lies in its modular construction, which aligns with the architectural
constraints and operational paradigms of current quantum computational
frameworks.

The modular exponential operator, integral to executing Shor’s algorithm,
demands an effective implementation strategy. The foundation of our approach
to crafting this operator is expounded through the exploitation of the
exponentiation process’s binary decomposition, as shown in Eq. (20):

$a^{x}=a^{\sum_{i=0}^{n-1}x_{i}2^{n-i-1}}=\prod_{i=0}^{n-1}(a^{2^{n-i-1}})^{x_{i}}.$
(20)

In this algorithmic formulation, the $n$ necessary values $a^{2^{n-i-1}}\mod
N$ are precomputed, setting the stage for the subsequent construction of the
product over indices $i$ corresponding to $x_{i}=1$. The ensuing application
of the operators $Mult_{\text{in}}(k,N)$ composes the desired modular
exponentiation, as detailed in Eq. (20).

To further elucidate the modular exponential operator’s workings, we refer to
its corresponding quantum circuitry, depicted in Fig. 9. The circuit employs a
sequence of controlled modular multiplications, adeptly processing each qubit
of the state $\ket{x}$ to effectuate the prescribed transformation. This
systematic approach not only showcases algorithmic precision but also
underscores the adaptability and scalability of our design to the constraints
of near-term quantum devices.

Figure 9: Schematic Depiction of the Quantum Exponential Operator. Beginning
as depicted in the figure with the initial state
$\psi_{a}=\ket{x}\ket{1}\ket{0}$, the circuit culminates in the final state
$\psi_{b}=\ket{x}\ket{a^{x}\mod{N}}\ket{0}$, implementing the operation. The
transformation is achieved through an Inplace Modular multiplier operator that
marries quantum and classical computation in a seamless quantum circuit
execution.

##  VI Results

Our investigation culminates in the precise articulation of quantum modular
arithmetic operators, each with detailed implementation strategies and
performance metrics. Central to our exposition is the Outplace Exponential
Operator, whose intricacies are unraveled in Section (V.10), and whose
operational schema is depicted in Figure (9). This advanced operator,
essential for algorithms like Shor’s, is presented with a comprehensive and
pragmatic implementation guide, serving as an indispensable blueprint for
quantum computing practitioners.

The quintessence of our results is captured in Table 1, which synthesizes the
resource efficiency of the delineated operators. It quantifies their
computational demands regarding qubit count, auxiliary qubit, and algorithmic
depth. This critical appraisal enables a nuanced understanding of the
operators’ practical viability and scalability in the quantum computing
milieu.

Table 1: Resource assessment of quantum operators for modular arithmetic Operator | #qubits | Auxiliary qubits | Depth
---|---|---|---
$Add_{\text{in}}(k,N)$ | $O(n)$ | $O(1)$ | $O(n)$
$Add_{\text{out}}(k,N)$ | $O(n)$ | $O(1)$ | $O(n)$
$Add_{\text{in}}(N)$ | $O(n)$ | $O(1)$ | $O(n)$
$Add_{\text{out}}(N)$ | $O(n)$ | $O(1)$ | $O(n)$
$Mult_{\text{out}}(k,N)$ | $O(n)$ | $O(1)$ | $O(n^{2})$
$Mult_{\text{in}}(k,N)$ | $O(n)$ | $O(1)$ | $O(n^{2})$
$Mult_{\text{out}}(N)$ | $O(n)$ | $O(1)$ | $O(n^{2})$
$Exp(a,N)$ | $O(n)$ | $O(n)$ | $O(n^{3})$

In tandem with the theoretical insights, we democratize access to our
implementations through the open-source release of our code, as delineated in
Section (Code). The accessibility of our codebase invites replication,
validation, and further innovation, fueling the advancement of quantum
computational research.

Our work lays the cornerstone for next-generation quantum algorithms. The
synergy between rigorous implementation guidance and readily available code
catalyzes a broader engagement with quantum computing, lowering the threshold
for entry into this specialized domain. Such contributions are pivotal,
particularly as we navigate the complexities and possibilities of the quantum
computing renaissance.

In adapting the methodologies and quantum circuits detailed in this study
thoroughly examined, such as Qiskit, we strongly recommend that users
carefully consider the encoding and representation of qubits. The present work
utilizes a big-endian encoding scheme, which must be taken into account to
guarantee the correct operation and translatability of our implementations
across diverse quantum computing ecosystems.

##  VII Conclusion

Our research delineates the pivotal role of quantum operations in advancing
computational arithmetic, marking a cornerstone in the ISQ era Arrazola and
Xanadu (2023). We have thoroughly examined a spectrum of quantum operators,
ranging from adders to the modular exponential operator, which is fundamental
to executing advanced quantum algorithms. Our analysis maps the current state
of the art and dissects the intricate challenges inherent in scaling quantum
computations.

The implications of this work extend beyond mere technical advancements,
necessitating a reevaluation of quantum arithmetic’s foundational principles.
The clear insights presented here offer a vantage point on the trajectory of
quantum computation, foreshadowing a new wave of innovations poised to
redefine the limits of computational performance. As the quantum paradigm
shifts, the strategic insights garnered are indispensable for effectively
leveraging quantum mechanics in computational sciences.

In quantum computation, the exponential operator introduced here is paramount,
particularly within the subroutine of Shor’s algorithm. It is meticulously
designed to function with a $3n$-qubit system, striking a balance between
computational efficiency and the constraints of current quantum processors.
This optimization emphasizes practicality, albeit at the expense of direct
reversibility, which is a calculated concession given the nascent stage of
quantum technologies.

Our exponential operator, by initializing its inputs in superposition using
Hadamard gates and then applying the inverse Quantum Fourier Transform
($QFT^{-1}$) to the identical qubits, is effectively converted into the
period-finding circuit integral to Shor’s algorithm as shown in Fig. (10).

The adoption of the Iterative Quantum Phase Estimation (IQPE) Dobšíček  _et
al._ (2007) technique, as corroborated by extant literature Willsch  _et al._
(2023), suggests a significant reduction of qubit resources. In particular,
the application of IQPE is projected to streamline the qubit requirement from
$3n$ down to $2n+2$, optimizing the quantum circuitry for modular arithmetic
operations within a more resource-constrained quantum computational framework.

This convergence of an advanced exponential operator design with the precision
of phase estimation algorithms indicates a sophisticated approach toward
achieving potential advantage. The focus on the exponential operator does not
diminish the collective significance of the suite of quantum operators
discussed; instead, it exemplifies the critical innovations necessary for
advancing the field of quantum algorithms.

Figure 10: Transformation of the Exponential Operator into the Period-Finding
Shor (1994) Component for Shor’s Algorithm. This diagram delineates the
adaptation of the exponential operator to form the crux of the period-finding
subroutine. The input qubits are initially prepared in a superposition state
by applying Hadamard gates, setting the stage for phase estimation. This is
succeeded by the deployment of the inverse Quantum Fourier Transform
($QFT^{-1}$) across these qubits. A measurement post-$QFT^{-1}$ extraction
finalizes the period-finding sequence, which is instrumental in the
factorization process inherent to Shor’s algorithm Shor (1994).

## Code

The Python code implementing the seven modular operators discussed herein is
available in the accompanying GitHub repository. The code provides the
foundational algorithms enabling in-depth exploration and adaptation of the
operators. The code can be accessed at
https://github.com/pifparfait/Efficient-Quantum-Modular-Arithmetics.

## Acknowledgements

The authors thank Guillermo Alonso de Linaje for the discussions and
consideration during the experiments.

## Compliance with Ethics Guidelines

Funding: This research received no external funding. Institutional review:
This article contains no studies with human or animal subjects. Informed
consent: Informed consent was obtained from all participants in the study.
Data availability: Data sharing is not applicable. No new data were created or
analyzed in this study. Data sharing does not apply to this article.

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#  Interference and Coverage Analysis of mmWave
Inter-Vehicle Broadcast with Directional Antennas ††thanks: This work is
supported in part by the NSF award 1827211.

Tianyi Zhang, Hongwei Zhang, Zhibo Meng  Department of Electrical and Computer
Engineering, Iowa State University
<EMAIL_ADDRESS>
###### Abstract

Thanks to the availability of large bandwidth and high-gain directional
antennas at the millimeter-wave (mmWave) bands, mmWave communications have
been considered as one of the primary solutions to meet the high data rates
needs in vehicular networks. Unicast in mmWave vehicle-to-vehicle (V2V)
communications has been well-studied, but much less attention has been paid to
V2V broadcast which is required by many V2V applications such as active
safety. To fill the gap, this paper systematically investigates mmWave V2V
broadcast by considering the unique properties of mmWave signal propagation in
V2V environments as well as the impacts of directional antennas and
interference. Based on widely-accepted, high-fidelity system models, we
mathematically analyze the receiver-side signal-to-interference-plus-noise-
ratio (SINR) and broadcast coverage, and we study the impacts of blockage,
inter-vehicle distance, vehicle density and beam pattern. Through
comprehensive numerical analysis, we find out that, instead of a single unique
optimal beamwidth, there exists an optimal range of beamwidth, in which the
beamwidths have similar performance and can maximize the coverage. We also
find out that the selection of carrier sensing range plays an important role
as it highly influences the performance of the whole vehicular networks. Our
analysis provides unique insight into mmWave V2V broadcast, and it sheds light
on designing effective V2V broadcast protocols.

##  I Introduction

Motivation & Related work. Unicast in MmWave vehicular networks has been
heavily investigated in the literature. In particular, Verdone [1] studies the
base station (BS) to vehicle communications at the 60GHz band and analyzes the
outage probability. In [2], the authors consider the interference from a
nearby BS for vehicle-to-infrastructure communications and show that the
interference mainly comes from the BSs on the same street. The article in [3]
characterizes the interference from the side lanes in both highway and urban
scenarios. Based on measurements, the authors estimate the signal-to-noise
ratio (SNR) and perform the capacity analysis. Giordani et al. [4] present a
stochastic model for characterizing the beam coverage and connectivity
probability of a mmWave automotive network. The results show that stable
connectivity requires both accurate beam alignment between vehicles and
satisfactory signal quality.

Many investigations [5, 6, 7] have been carried out for vehicular networks in
the sub-6GHz band, and studies such as [8, 9] have considered efficient
broadcast using directional antennas. There have also been studies [10, 11]
that investigate mmWave broadcasting schemes for vehicle networks, but those
studies do not consider the impact of interference. To the best of our
knowledge, no study has focused on the performance of mmWave V2V broadcast
with directional antennas and inter-vehicle interference. Yet a basic
requirement in vehicular networks is for a vehicle to share its status with
close-by surrounding vehicles [12]. To this end, V2V unicast with directional
mmWave communications tends not to be the best solution when facing high
mobility and the need of exchanging huge amount of inertia, LiDAR, radar,
video, and other sensing data, and broadcast can help vehicles exchange data
more efficiently while lowering the chance of connection disruption due to
beam misalignment.

Contributions. To gain insight into mmWave V2V broadcast, we perform the
associated coverage and interference analysis and make the following
contributions:

  * •

We analyze broadcast coverage and SINR in mmWave V2V networks with directional
antennas using high-fidelity system models. We derive the equations to
calculate receiver-side SINR and broadcast coverage, and we analyze the
impacts of blockage, inter-vehicle distance, vehicle density, and beam
pattern.

  * •

We investigate the relation between the mmWave beam pattern and broadcast
coverage. Results show that there exists an optimal beamwidth range in which
the beamwidths have similar performance and can maximize the coverage. The
optimal beamwidth range varies with vehicle densities. The range is smaller
when the vehicle network is sparser, so it’s more critical to choose the right
beamwidth in sparser networks. This result provides new insight into optimal
beam pattern selection.

  * •

We find that the broadcast coverage is subject to the selection of carrier
sensing range. A too large or too small carrier sensing range lowers the
network performance. Hence, there is an open problem of how to optimally
select the carrier sensing range while taking into consideration the unique
characteristics of mmWave, directionality of antennas, and high mobility of
vehicles.

  * •

We validate the analytical results through simulation studies.

Organization. The paper is organized as follows: Section II introduces the
system model. The analysis of blockage probability, SINR, and broadcast
coverage is given in Section III, while the numerical and simulation results
are discussed in Section IV. We conclude with key findings in Section V.

##  II System Model

###  II-A V2V Broadcast Scenario

We consider a 2D 3-lane road section with width $W$ as shown in Fig. 1,

Figure 1: Top view of V2V broadcast scenario

where each vehicle periodically broadcasts its state (e.g., location and
speed) to neighboring vehicles. This traffic model has been used by 3GPP to
compare the performance of different technical options in the highway scenario
[13],111Note that the analytical methodology and insights of this paper can be
extended to other multi-lane scenarios. and Guo et al. [14] have shown that,
for free-flow vehicle traffic, the spatial distribution of vehicles along a
highway lane is Poisson. Therefore, we assume that vehicles are in the middle
of their lanes and follow Poisson distributions, with an average spatial
distribution density of $\lambda_{i}$ vehicles per meter along lane $i$. Like
the setup in [3, 15], each vehicle equips with the same antennas, including
one directional antenna pointing to the dead front and another pointing to the
dead astern. Both antennas are at the center of the rooftop. 222Putting
antennas at lower parts (e.g., bumpers) of vehicles does not work well for
broadcast and hence is not considered in this study.  Similar to 3GPP Release
15 [13], two vehicle types are defined as follows:

  * •

Passenger vehicle with high antenna position: length $l_{1}$, width $w_{1}$,
height $h_{1}$, antenna height $h_{ant1}$.

  * •

Truck/bus: length $l_{2}$, width $w_{2}$, height $h_{2}$, antenna height
$h_{ant2}$.

The percentage of truck/bus among all the vehicles is $r_{tall}$.

###  II-B Path Loss Model

Blockage effects play an important role in mmWave channels, and we adopt the
following well-established log-distance path loss model [16, 17]:

$\begin{split}PL_{k}=10\alpha_{PL_{k}}\log_{10}{\sqrt{d_{x}^{2}+d_{y}^{2}}}&+\beta_{PL_{k}}\\\
&+15\sqrt{d_{x}^{2}+d_{y}^{2}}/1000,\end{split}\vspace*{-0.07in}$ (1)

where $PL_{k}$ is the path loss between the transmitter and receiver in dB
when there are $k$ blockers between them, $d_{y}$ is the y-axis displacement
between the centers of the transmitter and receiver vehicles in the direction
$y$ that is parallel to the vehicle moving direction, and $d_{x}$ is the
x-axis displacement that is perpendicular to $y$ (see Fig. 1).
$\alpha_{PL_{k}}$ and $\beta_{PL_{k}}$ are the model parameters with
$\alpha_{PL_{k}}$ being the path loss exponent. Both $\alpha_{PL_{k}}$ and
$\beta_{PL_{k}}$ vary with the number of blockers $k$ (e.g., see Table II in
Section IV). The last term in (1) is the atmospheric attenuation at 60 GHz.
One strength of the aforementioned model is that it is based on real-world
measurement of mmWave vehicular communication channels, and all of its
parameters can be instantiated based on the real-world measurement data from
[16, 17], thus ensuring the high-fidelity of our study.

###  II-C Antenna Pattern

We adopt the well-established “cone-plus-sphere” model [18] to capture the
antenna radiation pattern, as shown in Fig. 2. The antenna radiation pattern
is modeled with

Figure 2: Cone-Plus-Sphere antenna pattern model

a single cone-shaped beam as the main lobe, and a sphere around the antenna,
which take into account the presence of side-lobes. Besides modeling
directional communications, this model enables effective evaluation of the
impact of side-lobes on interference, as shown in [18].

$G_{1}$ and $G_{2}$ are the gains for the main and side lobes, given by

$\begin{cases}G_{1}=\frac{2}{1-\cos{(\alpha/2)}+k(1+\cos{(\alpha/2)})}\\\
G_{2}=kG_{1}\end{cases},\vspace*{-0.03in}$ (2)

where $\alpha$ is the antenna directivity angle and $k\in(0,1)$ is the power
ratio between the side lobe and the main lobe. In this model, the total
transmit power doesn’t change with the antenna beamwidth, so it allows us to
compare the coverage and interference in different bandwidths in a fair
condition.

###  II-D Probability of transmission

To reflect real-world situations, here we consider the MAC layer of IEEE
802.11bd, an enhanced V2X standard based on IEEE 802.11p. To characterize the
channel access behavior, we adopt the model of semi-Markov process with
absorbing state, which has been proposed in [19] to model V2V broadcast,
instantiate the model with 802.11bd parameters as shown in Table I in Section
IV. In particular, given a node and all the other nodes within its carrier
sensing range, we use the following two probabilities derived in [19]:

  * •

probability $p_{t}$ that a node transmits at a time instant;

  * •

probability $p_{c}$ that more than one node transmit concurrently.

$p_{t}$ and $p_{c}$ capture the impacts of both carrier sensing and broadcast
message intervals. Note that the impact of carrier sensing on the spatial
distribution of concurrent transmitters and interference in 802.11bd is
similar to that of interference-oriented scheduling in C-V2X [20]. Thus the
insight gained in this study also sheds light on C-V2X networks, even though
the detailed study of C-V2X is beyond the scope of this paper.

##  III Interference and Coverage Analysis

###  III-A Blockage Probability

As vehicles are assumed to follow Poisson distributions along individual
lanes, the distribution of trucks/buses also follows a Poisson distribution
with an average spatial distribution density of $\lambda_{i}r_{tall}$ along
lane $i$. _When the transmitter and receiver are on different lanes_ ,
blockers can be on the transmitter’s and receiver’s lane as well as any of the
lanes in between. For the lanes of the transmitter and receiver, the length of
the area that the blocker can be placed on the y-axis is
$d_{L1}=\frac{w_{2}}{2}\frac{d_{y}}{d_{x}}$. As the length on each lane is the
same and considering the property of Poisson distributions, the total number
of blockers in this case is the same as the case where all the blockers are on
one lane and their spatial distribution density is the sum of the vehicle
spatial distribution densities of the lanes of the transmitter and receiver.
Thus the probability of having $k$ blockers on the same lane as the
transmitter or receiver is

$P_{block,1}(k)=\frac{(r_{tall}d_{L1}(\lambda_{tx}+\lambda_{rx}))^{k}}{k!}e^{-r_{tall}d_{L1}(\lambda_{tx}+\lambda_{rx})},\vspace*{-0.05in}$
(3)

where $\lambda_{tx}$ and $\lambda_{rx}$ are the average spatial distribution
densities of vehicles alone the lanes of the transmitter and receiver.

For the lanes between the transmitter and receiver, the length of the area
that the blocker can be placed on the y-axis is
$d_{L2}=w_{2}\frac{d_{y}}{d_{x}}+l_{2}$. The total number of blockers in this
case is the same as the case where all the blockers are on one lane and their
distribution density is $r_{tall}\sum_{i\in\mathcal{L}_{2}}\lambda_{i}$, with
$\mathcal{L}_{2}$ denoting the lanes between that of the transmitter and
receiver. The probability of having $k$ blockers on $\mathcal{L}_{2}$ is

$P_{block,2}(k)=\frac{(r_{tall}d_{L2}\sum_{i\in\mathcal{L}_{2}}\lambda_{i})^{k}}{k!}e^{-r_{tall}d_{L2}\sum_{i\in\mathcal{L}_{2}}\lambda_{i}}.\vspace*{-0.05in}$
(4)

_When the transmitter and receiver are on the same lane_ , blockers can only
locate between the transmitter and the receiver. Hence the length of the area
that the blocker can be placed on the y-axis is $d_{y}-l_{2}$.

Thus the probability of having $k$ trucks/buses between a transmitter and
receiver is

$P_{tall,k}=\begin{cases}\sum_{i=0}^{k}P_{block,1}(i)P_{block,2}(k-i),&d_{x}\neq
0\\\
\frac{(r_{tall}(d_{y}-l_{2})\lambda_{tx})^{k}}{k!}e^{-r_{tall}(d_{y}-l_{2})\lambda_{tx}},&d_{x}=0\end{cases}.\vspace*{-0.06in}$
(5)

Then, the probability that the signal from the transmitter to receiver is not
blocked is

$P_{b,0}=P_{tall,0}+(1-P_{tall,0})r_{tall}^{2},\vspace*{-0.05in}$ (6)

which includes the probabilities that 1) no truck/bus is between the
transmitter and receiver, 2) there is at least one truck/bus in between but
both the transmitter and receiver are trucks/buses so the signal isn’t
blocked. Similarly, the probability that the signal from the transmitter to
receiver is blocked by one blocker is

$P_{b,1}=P_{tall,1}(1-r_{tall}^{2}).\vspace*{-0.05in}$ (7)

This happens when there is one truck/bus in between and the transmitter and
receiver are not both trucks/buses.

###  III-B Receiver-Side Interference

Given a transmitter and receiver with the x-axis and y-axis displacement of
$d_{x}$ and $d_{y}$ respectively, when there are $k$ blockers, the received
signal strength is as follows:

$P_{r,k}(d_{x},d_{y})=\frac{P_{t}G_{t}G_{r}}{PL_{k}},\vspace*{-0.05in}$ (8)

where $P_{t}$ is the transmission power, $G_{t}$ and $G_{r}$ are the antenna
gains at the transmitter and receiver side respectively, whose values are
$G_{1}$ or $G_{2}$, depending on the relative positions $(d_{x},d_{y})$ of
this pair of nodes. $PL_{k}$ is the path loss in magnitude. The received
signal strength is

$P_{r}(d_{x},d_{y})=P_{b,0}P_{r,0}+P_{b,1}P_{r,1},\vspace*{-0.06in}$ (9)

which include the cases when there is no blocker or only 1 blocker. When there
is more than one blocker, the received signal strength is too weak and can be
ignored, which will be shown in section IV-A. Here we assume that:

  * •

All the transmitters have the same transmission powers.

  * •

All the transmitters and receivers have the same beam patterns, so the signals
will be transmitted through either two main lobes or two side lobes.

  * •

All the transmitters are using the front antennas while the receivers are
using the rear antennas. This will reduce the interference and increase the
probability that the transmitters and the receivers are communicating using
the main lobes on both sides.

To analyze the impact of interference, we consider the MAC layer of IEEE
802.11bd. In 802.11bd, if carrier sensing is able to prevent concurrent
transmissions by vehicles within the carrier sensing range $r_{E}$ from one
another, then the minimum distance between concurrent transmitters would be
the carrier sensing range. However, the nature of the CSMA/CA MAC of 802.11bd
is such that, with probability $p_{c}$, vehicles within the carrier sensing
range will transmit concurrently. Therefore, interference may be generated in
two ways:

  * •

Primary interference $I_{o}$ that comes from concurrent transmitters
$\mathcal{T}_{o}$ assuming that carrier sensing was able to prevent concurrent
transmissions by vehicles within carrier sensing range from one another.

  * •

Secondary interference $I_{c}$ that comes from the concurrent transmitters
$\mathcal{T}_{c}$ other than those causing $I_{o}$, that is,
$\mathcal{T}_{o}\cap\mathcal{T}_{c}=\emptyset$.

In what follows, we first analyze $I_{o}$. If without considering the carrier
sensing mechanism, the interferers (i.e., concurrent transmitters
$\mathcal{T}_{o}$) would follow a Poisson distribution with an average spatial
distribution density of $\lambda_{i}p_{t}$ along each lane $i$. Thus the
average distance between two neighbor interferers on the same lane is
$\frac{1}{\lambda_{i}p_{t}}$. Considering the property of Poisson
distribution, for one interferer, its nearest neighbor interferer can come
from any one of the lanes, so the average distance between two neighbor
interferers along the y-axis would be
$\frac{1}{p_{t}\sum_{i\in\mathcal{L}_{All}}\lambda_{i}}$, where
$\mathcal{L}_{All}$ denotes all the lanes and
$\sum_{i\in\mathcal{L}_{All}}\lambda_{i}$ is the sum of the average spatial
distribution densities on $\mathcal{L}_{All}$. With the carrier sensing
mechanism working between interferers, however, other nodes are not allowed to
transmit within the carrier sensing range $r_{E}$ of an interferer. Thus the
carrier sensing range $r_{E}$ needs to be added to the distance between two
neighbor interferers if without carrier sensing. The lane width $W$ is pretty
small when compared with the distance between two neighbor interferers along
the y-axis so we can ignore the effects caused by different lanes and use
$r_{E}+\frac{1}{p_{t}\sum_{i\in\mathcal{L}_{All}}\lambda_{i}}$ as the average
distance between two neighbor interferers. Then, the distance from the $k$th
nearest interferer to the receiver can be approximated as

$l_{I_{k},R}=l_{S,R}+k(r_{E}+\frac{1}{p_{t}\sum_{i\in\mathcal{L}_{All}}\lambda_{i}}),\vspace*{-0.05in}$
(10)

where $l_{S,R}$ is the distance between the transmitter and the receiver.
Accordingly, the total interference caused by the nodes outside the carrier
sensing range is

$I_{o}=\sum_{k=1}^{\infty}I_{r}(l_{I_{k},R}),\vspace*{-0.07in}$ (11)

where $I_{r}(l_{I_{k},R})$ is the received signal power from the $k$th nearest
interferer to the receiver. $I_{r}(l_{I_{k},R})$ is computed as

$\displaystyle I_{r}(l_{I,R})=$ $\displaystyle
P_{0}P_{r}(0,l_{I,R})+P_{W}P_{r}(W,\sqrt{l_{I,R}^{2}-W^{2}})+$ (12)
$\displaystyle P_{2W}P_{r}(2W,\sqrt{l_{I,R}^{2}-(2W)^{2}}),$

and it takes into account the fact that both the transmitter and receiver have
equal probabilities of being on different lanes. $P_{0}=\frac{1}{3}$ is the
probability that the x-axis distance between the transmitter and the receiver
is 0, which means they are on the same lane; $P_{W}=\frac{4}{9}$ is the
probability that the x-axis distance between the transmitter and the receiver
is the lane width $W$, which means they are on the neighbor lanes;
$P_{2W}=\frac{2}{9}$ is the probability that the x-axis distance between the
transmitter and the receiver is $2W$, which means there is one lane between
the lanes of the transmitter and receiver.

Now, we analyze the secondary interference $I_{c}$. The interferers
$\mathcal{T}_{c}$ that generate the secondary interference are not influenced
by carrier sensing and follow a Poisson distribution with an average spatial
distribution density of $\lambda_{i}p_{c}$ along lane $i$. Accordingly,
similar to the analysis of $I_{o}$, the total secondary interference is

$I_{c}=\sum_{k=1}^{\infty}I_{r}(\frac{k}{p_{c}\sum_{i\in\mathcal{L}_{All}}\lambda_{i}}),\vspace*{-0.05in}$
(13)

where the function $I_{r}(.)$ is as defined in (12).

###  III-C Broadcast Coverage

Inter-vehicle broadcast needs to be reliable for mission-critical applications
such as active safety, thus the receiver-side SINR needs to be above a certain
threshold $\gamma_{th}$ to ensure a reliable broadcast. To capture the
performance of a V2V broadcast, we define the _broadcast coverage_ as the
number of receivers whose SINRs are no less than a threshold $\gamma_{th}$.
Besides capturing the number of receivers that can reliably receive a
broadcast message, broadcast coverage reflects the size of the area in which a
broadcast is reliable, which can be computed as the number of receivers
divided by the vehicle distribution densities along traffic lanes. To decide
the SINR threshold $\gamma_{th}$ to be used in this study, we consider a
target packet error rate (PER) of $1\times 10^{-5}$ as suggested by the ITU
[21]. Then, from [22], we derive the required SINR as 23dB for 802.11bd with
256QAM and 5/6 coding rate.

With the interference formulas (11) and (13), we can derive as follows the
minimum required receiver-side signal power, denoted by $P_{r,th}$, in order
for a receiver to be considered “within a broadcast coverage” (i.e., having an
SINR no less than $\gamma_{th}$):

$P_{r,th}=(N+I_{o}+I_{c})\gamma_{th},\vspace*{-0.05in}$ (14)

where $N=-174+10log_{10}B+NF$ is the noise, $B$ is the bandwidth, $NF$ is the
noise figure.

Then, we can calculate the farthest distance $D_{r,th}$ from a transmitter to
its receiver such that the receiver-side signal power is no less than the
threshold $P_{r,th}$ using the inverse function of Equation 9. In particular,
assuming $d_{x}=k\times W$, $k=0,1,2$, (i.e., with $k$ lanes separation on
road), $d_{y}$ can be derived and we denote it as $P_{r,k}^{-1}(P_{r,th})$.

Then, counting all the receivers within $D_{r,th}$ distance from the
transmitter on all the lanes, the expectation of the number of receivers whose
SINRs are no less than a threshold $\gamma_{th}$ can be expressed as follows:

$\displaystyle E[N]=$
$\displaystyle\frac{1}{3}\lambda_{1}(P_{r,0}^{-1}(P_{r,th})+P_{r,1}^{-1}(P_{r,th})+P_{r,2}^{-1}(P_{r,th}))+$
(15)
$\displaystyle\frac{1}{3}\lambda_{2}(P_{r,0}^{-1}(P_{r,th})+2P_{r,1}^{-1}(P_{r,th}))+$
$\displaystyle\frac{1}{3}\lambda_{3}(P_{r,0}^{-1}(P_{r,th})+P_{r,1}^{-1}(P_{r,th})+P_{r,2}^{-1}(P_{r,th})).$

##  IV Numerical and Simulation Results

Here we first perform numerical analysis in specific real-world settings, and
then validate the analytical insight via simulation studies. In particular, we
consider a typical scenario suggested by the 3GPP release 15 [13] and papers
[17, 23]. Unless mentioned otherwise, the values of parameters used in our
analysis are as shown in Tables I, II and III. Note that the key objective of
this study is to distill insight into the behavior of mmWave inter-vehicle
broadcast with directional antennas; the insight will help develop effective
broadcast protocols, but the detailed study of specific protocols is beyond
the scope of this work.

Frequency $f$ |  60GHz
---|---
Transmission power $P_{t}$ |  23dBm
Beamwidth $\alpha$ |  30 ∘
Side-main lobe ratio $k$ |  0.1
Carrier sensing range $r_{E}$ |  50m
Bandwidth $B$ |  20MHz
Noise figure $NF$ |  6dB
Passenger vehicle size $l_{1}\times w_{1}\times h_{1}$ |  5m$\times$2m$\times$1.6m
Passenger vehicle antenna height $h_{ant1}$ |  1.6m
Truck/bus size $l_{2}\times w_{2}\times h_{2}$ |  13m$\times$2.6m$\times$3m
Truck/bus antenna height $h_{ant1}$ |  3m
Lane width $W$ |  3.2m
Number of lanes $n$ |  3
Percentage of bus/truck $r_{tall}$ |  0.1
Packet generation interval |  0.1s
Packet length |  200bytes
Symbol duration |  6.4$\mu$s
Transmission latency |  100$\mu$s
TABLE I: System parameters |  Lane 1 |  Lane 2 |  Lane 3
---|---|---|---
Low |  0.05 |  0.07 |  0.10
Intermediate |  0.07 |  0.10 |  0.13
High |  0.09 |  0.13 |  0.17
TABLE II: Vehicle density (vehicles/m) in different traffic conditions # blockers |  0 |  1 |  2
---|---|---|---
$\alpha_{PL}$ |  1.77 |  1.71 |  0.635
$\beta_{PL}$ |  70 |  78.6 |  115
TABLE III: Path loss model parameters

Figure 3: The impact of different numbers of blocker

Figure 4: The impacts of beamwidth and vehicle density

Figure 5: Receivers’ SINR CDF with different beamwidths

###  IV-A Impact of Blockage

Before analyzing broadcast coverage, the impact of blockage needs to be
discussed. From Fig. 5, we can see that one blocker will reduce the received
power by 8dB, while two blockers will bring more than 25dB extra path loss in
V2V communication. Due to this, we assume that the received signal power of a
link that has two or more blockers is so small that it can be ignored.

###  IV-B Broadcast Coverage

As mentioned in section III-C, the coverage of a transmitter can be reflected
by the number of its receivers whose SINRs are larger than a threshold
$\gamma_{th}$. The results of the numbers of receivers with SINR $>$ 23dB for
different beamwidths and vehicle densities are presented in Fig. 5.

The figure shows that both the beamwidth and the vehicle density influence the
broadcast coverage. For beamwidth, there exists an optimal range in which the
coverage is the largest or very close to the largest. The optimal range
changes with the vehicle density and the associated probability of blockage.
When the beamwidth is less than the optimal range, the number of receivers
with SNR $>$ 23dB decreases as fewer nodes are pointed at by the main lobe;
when the beamwidth is too large, the coverage also shrinks because the same
total transmission power is spent in unnecessarily broader directions which in
turn reduces the receiver-side signal power and generates more interference.
In addition, the number of receivers with SINR$>$23dB is pretty small when the
beamwidth is 360 degrees, which reflects the poor result of using the
omnidirectional beam pattern.

For the vehicle density, it can influence both the number of receivers and the
optimal range of the beamwidth. Notice that due to the carrier sensing
mechanism, the density of interferers doesn’t vary with the vehicle density.
The change of vehicle density mainly affects the number of potential receivers
of the transmitter and the blockage probability. The median of the optimal
beamwidth range increases with the vehicle density, which means for denser
vehicle networks, the best strategy is to use a wider beam to cover more
nearby nodes rather than using a narrower beamwidth to cover farther away
nodes. Focusing on the maximum number of receivers with SINRs no less than the
threshold for different vehicle densities, an interesting finding is that a
sparser vehicular network tends to have a higher upper bound on the broadcast
coverage. This is because, in a sparser network, there is a higher probability
that the signal can be sent to the receivers on the adjacent lanes of the
transmitter without any blockage.

To further understand what happens in the optimal beamwidth ranges, we show in
Fig. 5 the empirical CDF of the top 100 receiver-side SINRs for different
beamwidths when the traffic density is “intermediate” (see Table II). We see
that the distributions of SINR under different beamwidths are almost the same,
so all the beamwidths in the optimal beamwidth range can be considered to have
similar performance. Fig. 5 shows that the range of optimal beamwidths tends
to increase with decreasing vehicle density. Hence, it’s more critical to
choose the right beamwidth when the vehicle network is sparser, and, in a
denser vehicle network, it tends to be easier to choose a beamwidth that falls
into the optimal beamwidth range.

Figure 6: The impact of carrier sensing range

Figure 7: The impacts of beamwidth and vehicle density (simulation)

Figure 8: The impact of carrier sensing range (simulation)

###  IV-C Impact of Carrier Sensing Range

The impact of carrier sensing range $r_{E}$ is shown in Fig. 8. The number of
receivers with SINR no less than the given threshold varies with the carrier
sensing range, and there exists an optimal value for this range. As shown by
the blue line in Fig. 8, when the picked carrier sensing range is less than
the optimal value, the interferers will be too close to the receiver so the
SINR will decrease. However, if the picked carrier sensing range is too large,
as shown by the yellow line in Fig. 8, the per-broadcast coverage doesn’t
increase but the density of transmitters in the networks decreases, so the
overall data rate decreases. An interesting topic for future study will be to
find the solutions that can decide on-the-fly the optimal carrier sensing
range in real-world settings.

###  IV-D Simulation Validation of Analytical Results

To validate the analytical results, we simulate the network as specified by
Tables I, II, and III. Fig. 8 shows the impacts of beamwidth and vehicle
density on broadcast coverage, and Fig. 8 shows the impact of carrier sensing
range. We see that the analytical insight as shown by Fig. 5 is validated by
Fig. 8, so is the analytical insight of Fig. 8 validated by Fig. 8.

##  V Conclusion

Based on high-fidelity models and industry standards, we have investigated
mmWave V2V broadcast interference and coverage, and have studied the impacts
of system parameters such as vehicle density and carrier sensing range. Key
findings are as follows: 1) There exists an optimal beamwidth range in which
the beamwidths have the similar and best performance. The range is influenced
by the vehicle density, and it is smaller when the network is sparser; this
implies that optimal beamwidth selection is more important in a sparser
network, and it is relatively easier to find the beamwidth that maximizes the
broadcast coverage in a denser network. 2) Vehicle density influences the
median of the optimal beamwidths, in addition to their range. For a denser
network, it’s better to use a larger beamwidth to cover more nearby nodes
while, for a sparser network, choosing a narrower beamwidth to cover more far-
away nodes is preferred. The density also impacts the upper bound on broadcast
coverage. With less blockage, sparser vehicle networks tend to have higher
coverage upper bound. 3) Interference control is important for mmWave V2V
broadcast, and the optimal selection of carrier sensing range is important as
it impacts inter-vehicle interference as well as the concurrency and
throughput of V2V broadcast. These findings provide unique insight into
optimal beamwidth selection in mmWave V2V networks, and they also demonstrate
the importance of interference control in mmWave V2V broadcast. An interesting
future work is to develop mmWave V2V broadcast protocols based on these
findings.

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