IEEE Circuits and Systems Magazine - Q1 2021 - 30

Moreover, although any kind of DNA strand can be synthesized using biological methods, the preparative step
requires a long time in practice, as well as the preparation of inputs [109].
Although the DNA manipulation required in the
models has already been realized in the laboratory and
the procedures have been implemented in practice,
some defects exist in the procedures, thus hindering
practical use. This is an example of the application of
nonconventional arithmetic to DNA-based computing.
The RRNS has been applied to overcome the negative
effects caused by the defects and instability of the biochemical reactions and errors in hybridizations for
DNA computing [113]. Applying the RRNS 3-moduli set
" 2 n - 1, 2 n + 1, 2 n + 1 , to the DNA model in [114], as discussed in Section II-B, leads to one-digit error detection.
The parallel RRNS-based DNA arithmetic improves the
reliability of DNA computing while at the same time simplifies the DNA encoding scheme [113].
F. Quantum Computing
QC cannot stand alongside DNA computing or any other
type of classical computation referred to so far in this

H

1
√2

1 1
1 -1

S

(a)
T

1
0

0
j

0
1

1
0

(b)

1
0
0 expjπ /4

X

(c)

paper. Classical computing operates over bits, and even
DNA-based computing refers only to the substrate on
which computation over these bits is performed. The
number of logical states for an n-bit representation is
2 n, and Boolean operations on the individual bits are
sufficient to realize any deterministic transformation. A
quantum bit (qubit), by contrast, typically a microscopic
unit, such as an atom or a nuclear spin, is a superposition
of basis states, orthogonal and typically represented by
0 and 1 . In Dirac notation, also known as bra-ket notation, a ket such as x refers to a vector representing
a state of a quantum system. A qubit, represented by
the vector x , corresponds to a linear combination, the
superposition of the basis vectors with coefficients a
and b defined in a unit complex vector space called the
Hilbert space ^C 2 h (39).
x = a 0 + b 1 ; ; a ; 2 + ; b ; 2 = 1.

(39)

Regarding measurement, the superposition a 0 + b 1
corresponds to 0 with probability ; a ; 2 and 1 with
probability ; b ; 2.
Common simple qubit gates are represented in
Fig. 20. Since operations on a qubit preserve the norm
of the vectors, the gates are represented by 2 # 2 unitary matrices. Some algebraic properties, such that
H = ^ X + Z h / 2 and S = T 2, are useful to correlate some
of these quantum gates.
A two-qubit system can be represented by a vector in
the Hilbert space C 2 7 C 2, with , denoting the tensor
product, which is isomorphic to C 4 . Thus, the basis of
C 2 7 C 2 can be written as:

(d)

0 7 0, 0 7 1, 1 7 0, 1 7 1
0
j

Y

-j
0

0 1
1 -1

Z

(e)

(f)

Figure 20. Single qubit gates: symbols and unitary matrices.
(a) Hadamard gate. (b) Phase gate. (c) r/8 gate. (d) Pauli-X
gate. (e) Pauli-Y gate. (f) Pauli-Z gate.

and a 7 b is often expressed as a b or ab . Generally, the state of an n-qubit system is expressed by
n
(40) in C 2 .
Y=

/

b ! {0, 1} n

(a)

1

0

0

0

0

1

0

0

0

0

0

1

0

0

1

0

(b)

Figure 21. CNOT: matrix and circuit representation. (a) Control qubit in the top, target qubit in the bottom. (b) Matrix representation.
30

IEEE CIRCUITS AND SYSTEMS MAGAZINE

cb b ;

/ ;cb ;2 = 1

(40)

b

with the state of an n-particle system being represented
in a 2 n- dimensional space. The exponentially large dimensionality of this space makes quantum computers
much more powerful than classical analogue computers, the state of which is described only by a number
of parameters proportional to the size of the system.
By contrast, 2 n complex numbers are required to keep
track of the state of an n-qubit system.
If the qubits are allowed to interact, then the closed
system includes both qubits together, meaning that
the qubits are entangled. When two or more qubits are
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