Hafiz M. H. Babu - Reversible and DNA Computing

The quantum cost of every 2 2 gate is the same. It can be easily assumed that 1 1 gate cost nothing, since it can always be included to arbitrary 2 2 gate that precedes or follows it. Thus, in first approximation, every permutation quantum gate will be built from 1 1 and 2 2 quantum primitives and its cost is calculated as a total sum of 2 2 gates used. All gates of the form 2 2 has equal quantum cost, and the cost is unity.

A 2 2 Feynman gate is also called CNOT. This gate is one through because it passes one of its inputs. Every linear reversible function can be built by using only 2 2 Feynman gates and inverters. Since this is a 2 2 gate, the quantum cost is 1. Quantum equivalent circuit of the Feynman gate is shown in Figure 1.9.

Figure 1.10shows the equivalent quantum realization of three input Toffoli gate. The cost of the Toffoli gate is five 2 2 gates, or simply 5. In Figure 1.10, is a square‐root of NOT gate and is its hermitian. Thus, creates a unitary matrix of NOT gate and = I (an identity matrix, describing just a quantum wire).

Figure 1.9Quantum cost calculation of Feynman gate.

Figure 1.10Quantum circuit of Toffoli gate.

Figure 1.11Quantum circuit of Fredkin gate.

Figure 1.12Quantum circuit of a Peres gate.

The Fredkin gate costs the same as the Toffoli gate. The Toffoli gate includes a single Davio gate, while the Fredkin gate includes two multiplexers. The quantum equivalent Toffoli gate is shown in Figure 1.10. Each dotted rectangles in Figure 1.11is equivalent to a 2 2 Feynman gate and so the cost is 1 for the particular case.

This gate can be realized with cost 4. It is just like a Toffoli gate but without the last Feynman gate from right. This is the cheapest realization of a complete (universal) 3 3 permutation gate. Figure 1.12shows the quantum realization of a Peres gate.

Maxwell's demon and Szilard's analysis of the demon at first suggested the connection between a single degree of freedom (one bit) and a minimum quantity of entropy. In the 1950s, this connection had been popularly interpreted to mean that computation must dissipate a corresponding minimum amount of energy during every elemental act of computation. Landauer later recognized that energy dissipation is only unavoidable when information is destroyed. Bennett and Toffoli first realized that a reversible computation, in which no information is destroyed, may dissipate arbitrarily small amounts of energy. The reversible circuits form the basic building block of quantum computers. This chapter presents some reversible gates. This chapter will help researchers/designers in designing higher complex computing circuits using reversible gates. It can further be extended toward the digital design development using reversible logic circuits, which are helpful in quantum computing, low‐power CMOS, nanotechnology, cryptography, optical computing, DNA computing, digital signal processing (DSP), quantum dot cellular automata, communication, and computer graphics.

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