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 FIRST QUARTER 2021

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