IEEE Circuits and Systems Magazine - Q1 2021 - 31

Machine Learning and Quantum-Resistant Cryptography are examples of
applications that can benefit the most from nonconventional arithmetic.

entangled, there exists a special connection between
them: the outcome of the measurement of one qubit is
correlated to the measurement of the other qubits. The
quantum version of the CNOT gate in Fig. 12(a) is an example of the interaction between 2 qubits that operate
on the same basis:
00 " 00 ,

01 " 01 , 10 " 11 , 11 " 10

The CNOT gate flips the state of the target (t) qubit according to the state of the first control (c) qubit. If the control qubit is set to 1 , then the target qubit is flipped; otherwise, nothing is done with the target qubit, which action
can be represented as: c t " c c 5 t . Fig. 21 provides
the CNOT gate matrix and circuit representations.
A quantum circuit that implements a 1-bit FA, acting
on the superposition of states, is presented in Fig. 22.
With the application of the equations of the classic reversible logic (Fig. 12) to the circuit in Fig. 22, the output equations for the full adder are achieved: P = A,
Q = B, S = A 5 B 5 C i , and C o = AB + AC i + BC i . Thus,
if, for example, the four qubits of the input are set to
the state ABC i X = 1010 , then the system goes, with
a probability equal 1, to a state that provides an output
PQSC o = 1001 .
Quantum gates and quantum computers have already been developed, for example, TGs based on quantum mechanics were successfully realized more than
ten years ago at the University of Innsbruck [115]. The
IBM Q systems are physically supported by the transmon qubit represented in Fig. 23. The transmon qubit
circuit corresponds to Josephson junctions, kept at a
very low temperature, shunted by an additional large
capacitance and matched by another comparably large
gate capacitance (Fig. 1 [116]). Although the transmon
qubit is closely related to the Cooper PairBox (CPB) qubit [117], it is operated at a significantly different ratio of
Josephson energy for the charging energy. It overcomes
the main weakness of the CPB by featuring an exponential gain in the insensitivity to charge noise [116]. IBM Q
processors are composed of transmon qubits that are
coupled and addressed through microwave resonators,
as depicted in Fig. 23. Several quantum computers have
been constructed based on that technology, the most
powerful and recent one being a 53-qubit system [118].
In the next section, applications of the nonconventional arithmetic, new technologies and sysFIRST QUARTER 2021

tems to emerging applications are discussed. While
QC algorithms have been derived to solve linear
systems of equations with low complexity [119], in
the next section, we refer the capacity of QC to attack the current security measures of cryptographic protocols.
IV. Applications in emerging areas
Postquantum cryptography and machine learning, in
particular DL, have been selected as case studies of
applications that use nonconventional arithmetic supported by the technologies and systems presented in
the previous sections. The criterion for selection was
based on the importance of these applications in emerging areas, particularly those that can benefit the most
from nonconventional arithmetic.
A. Deep Learning
A type of DL dominant in a broad spectrum of applications is the CNN [120], a class of the general Deep Neural Network (DNN). The CNN encompasses two main
different phases of operation: training and inference.
The training is an offline process, usually performed in
powerful servers, for learning features from (typically
massive) databases. The inference applies the trained
model to predict and classify according to the features
" learned " by the neural network. A CNN architecture
consists of layers of convolution, activation, pooling, and
fully connected neural networks [120]. The convolution
kernels, represented by the weight w in the general equation (41), are applied to identify features by also adding a scalar bias b to (41). For the activation, nonlinear

Figure 22. Quantum 1-bit FA: Diagram of a circuit using CNOT
and TG reversible gates-behavior simulated on the Quantum
Inspire QI platform [168] inputs: A, B, Ci, 0; outputs: P, Q, S, Co.

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