IEEE Circuits and Systems Magazine - Q1 2021 - 23

Nonconventional arithmetic is fundamental not only in the design of efficient
systems, but also to accommodate new computing paradigms and mitigate
intrinsic disadvantageous characteristics of emergent technologies.
the paradigm can be applied to the different representations discussed in this section.

C o = ^ A 5 Bh / C 5 A / B = A / C 5 B / C 5 A / B
(37)
= A/C +B/C + A/ B

III. Arithmetic: New Technologies and
Alternative Computing Paradigms
As referred to in the previous sections, although arithmetic units and processors have been designed in
CMOS-based ASICs and Field Programmable Gate Arrays (FPGAs), nonconventional number representations are also fundamental for developing computing
systems based on new technologies and circuits, such
as superconductor devices [79], integrated nanophotonics [94], nanoscale memristor devices and hybrid
memories [98], [105], and invertible logic devices [78].
Moreover, new paradigms for designing computers with
emergent technologies also cross nonconventional
arithmetics to cover a wide range of applications. This
section introduces not only those emergent technologies but also two of the most representative new computing paradigms, QC and DNA-based computing [80],
[81]. This survey considers both the technology and the
new computing paradigms from the perspective of computer arithmetic.
The lowest limit for the energy required to process
one bit of information is E b = k B # T # ln 2, where k B is
the Boltzmann constant and T the temperature in Kelvin (approximately 3 # 10 -21 Joule for a temperature of
25cC) [82]. However, it has been shown that this energy
can be reduced, achieving even near-zero energy dissipation, by using reversible gates [82]. Reversible logic
requires not only the reverse mode operation but also a
one-to-one mapping between inputs and outputs.
Reversible gates have been implemented in nanotechnologies. Some of the most representative reversible
gates are presented in Fig. 12 [81]. The Feynman gate (FG),
also known as the Controlled-NOT (CNOT) gate, is a 2 # 2
reversible gate described by the equations P = A and
Q = A 5 B, where A and B operate as control and data
bits, respectively. The Toffoli gate (TG) is a 3 # 3 reversible logic gate, with P = A, Q = B and R = C 5 ^ A / B h . A
1-bit Full Adder (FA) can be built with a single Haghparast
and Navi gate (HNG) by considering A, B, C i and D = 0.
The carry out ^C o h bit that results from the addition of A,
B and C i is easily computed by (37), since the result of the
XOR of three terms is equal to the logic OR whenever the
number of terms that assume a value of one is different
from two, a situation that never occurs in (37).

Efficient reversible circuits have been implemented
on QCA-based technologies [80]. Binary information is
represented by means of Quantum Dots (QDs), semiconductor particles a few nanometers in size having
optical and electronic properties that are central to
nanotechnology [84]. When illuminated by ultraviolet
light, an electron in the QD can be excited to a state
of higher energy, while for a semiconducting quantum
dot, an electron can transit from the valence band to
the conductance band. Reversible QCA-based gates
have been applied to design arithmetic units [85],
which can be implemented, for example, on DNA molecular structures [86], [87] or on quantum computing devices [88]. Moreover, reversible gates and QCA

FIRST QUARTER 2021

A

P=A
Feynman
Gate

B

Q=A⊕B
(a)

A

B

P=A

Toffoli Gate

C

Q=B

R = AB ⊕ C
(b)

A
B
C

P=A
Haghparast
and
Navi Gate

Q=B
R=A⊕B⊕C
S = (A ⊕ B) C ⊕ AB ⊕ D

D
(c)

Figure 12. Examples of reversible gates. (a) FG (CNOT).
(b) TG. (c) HNG.

IEEE CIRCUITS AND SYSTEMS MAGAZINE

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IEEE Circuits and Systems Magazine - Q1 2021

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