Efficient reversible NOR gates and their mapping in optical computing domain

Abstract

Reversible logic is a computing paradigm in which there is a one to one mapping between the input and the output vectors. Reversible logic gates are implemented in an optical domain as it provides high speed and low energy computations. In the existing literature there are two types of optical mapping of reversible logic gates: (i) based on a semiconductor optical amplifier (SOA) using a Mach–Zehnder interferometer (MZI) switch; (ii) based on linear optical quantum computation (LOQC) using linear optical quantum logic gates. In reversible computing, the NAND logic based reversible gates and design methodologies based on them are widely popular. The NOR logic based reversible gates and design methodologies based on them are still unexplored. In this work, we propose two NOR logic based n-input and n-output reversible gates one of which can be efficiently mapped in optical computing using the Mach–Zehnder interferometer (MZI) while the other one can be mapped efficiently in optical computing using the linear optical quantum gates. The proposed reversible NOR gates work as a corresponding NOR counterpart of NAND logic based Toffoli gates. The proposed optical reversible NOR logic gates can implement the reversible boolean logic functions with a reduced number of linear optical quantum logic gates or reduced optical cost and propagation delay compared to their implementation using existing optical reversible NAND gates. It is illustrated that an optical reversible gate library having both optical Toffoli gate and the proposed optical reversible NOR gate is superior compared to the library containing only the optical Toffoli gate: (i) in terms of number of linear optical quantum gates when implemented using linear optical quantum computing (LOQC), (ii) in terms of optical cost and delay when implemented using the Mach–Zehnder interferometer.

Introduction

Reversible logic is emerging as a promising computing paradigm among the emerging technologies. Reversible logic has applications in quantum computing, quantum dot cellular automata, optical computing, etc. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Reversible logic also has applications in power-efficient nanocomputing [12], [13], [14]. In reversible logic there exists a unique one to one mapping between the input and output vectors. The unused outputs are used to maintain the reversibility of reversible circuits and are referred to as the garbage outputs. The inputs that are regenerated at the outputs are not considered as the garbage outputs [15]. The constant inputs in the reversible circuits are called the ancilla inputs.

A photon can provide unmatched high speed and can store the information in a signal of zero mass. These properties of photon have attracted the attention of researchers to implement the reversible logic gates in all optical domain. The optical implementation of reversible logic gates could be useful to overcome the limits imposed by conventional computing, and is also considered as an implementation platform for quantum computing [16], [17], [18], [19], [6], [7], [8], [9], [10], [11]. In the existing literature there are two types of optical mapping of reversible logic gates: (i) based on the semiconductor optical amplifier (SOA) using the Mach–Zehnder interferometer (MZI) switch [20], [3], [21]; (ii) based on linear optical quantum computation (LOQC) using linear optical quantum logic gates [6], [7], [8], [9], [10], [11].

In the existing literature the most widely used implementation of reversible logic gates and the reversible boolean functions are the implementations using NAND logic. This is due to the lack of research in the direction of NOR logic based reversible logic gates and functions. In this work, we propose two NOR logic based n-input and n-output reversible gates one of which can be efficiently mapped in optical computing using the Mach–Zehnder interferometer (MZI) while the other one can be mapped efficiently in optical computing using the linear optical quantum gates. The first reversible NOR gate is called as a Mach–Zehnder interferometer based reversible NOR gate (MZI-RNOR), and the second reversible NOR gate is called as a linear optical quantum computing based reversible NOR gate (LOQC-RNOR). The proposed optical reversible NOR gates are useful for NOR logic based implementation of reversible boolean functions. The proposed MZI-RNOR gate can implement the reversible boolean functions with reduced optical cost and propagation delay compared to the implementation of reversible boolean functions using optical reversible NAND gates (NAND logic based reversible gates is all optical Toffoli gate) implemented using the MZI switch. The proposed LOQC-RNOR can implement the reversible boolean functions with a reduced number of linear optical quantum logic gates compared to the implementation of reversible boolean functions implemented using linear optical quantum reversible NAND gates (NAND logic based reversible gates is linear optical quantum Toffoli gate). As the proposed optical reversible NOR gates are n-input and n-output optical reversible NOR based counterpart of NAND logic based n-input and n-output Toffoli gate, thus we have also illustrated the optical design of the n-input and n-output Toffoli gate. In this work, the optical cost of a reversible logic gate is defined as the number of MZI switches used in its all optical implementation [22]. We illustrated the advantages of proposed optical reversible NOR gates in terms of optical cost and delay by implementing the 13 standard boolean functions [23]. The 13 standard boolean functions proposed in [23] can represent all possible 256 combinations of three variable boolean functions. It is illustrated that an optical reversible gate library having both optical Toffoli gate and the proposed optical reversible NOR gate is superior compared to the library containing only the optical Toffoli gate: (i) in terms of number of linear optical quantum gates when implemented using linear optical quantum computing (LOQC) and (ii) in terms of optical cost and delay when implemented using the Mach–Zehnder interferometer.

The paper is organized as follows: the basics of optical computing and reversible logic in optical domain is presented in Section 2; Section 3 illustrates the proposed NOR logic based reversible gates and their mapping in the optical domain; Section 4 provides the comparison of proposed NOR logic based reversible gates with NAND logic based reversible gates mapped in the optical domain, while the conclusions are provided in Section 5.