Mach-Zehnder Interferometer Based All Optical Reversible NOR Gates

Transcription

1 2012 IEEE Computer Society Annual Symposium on VLSI Mach-Zehnder Interferometer Based All Optical Reversible NOR Gates Saurabh Kotiyal, Himanshu Thapliyal and Nagarajan Ranganathan Department of Computer Science and Engineering University of South Florida Tampa, FL Abstract Reversible logic has promising applications in dissipation less optical computing, low power computing, quantum computing etc. Reversible circuits do not lose information, and there is a one to one mapping between the input and the output vectors. In recent years researchers have implemented reversible logic gates in optical domain as it provides high speed and low energy computations. The reversible gates can be easily fabricated at the chip level using optical computing. The all optical implementation of reversible logic gates are based on semiconductor optical amplifier (SOA) based Mach-Zehnder interferometer (MZI). The Mach-Zehnder interferometer has advantages such as high speed, low power, easy fabrication and fast switching time. In the existing literature, the NAND logic based implementation is the only implementation available for reversible gates and functions. There is a lack of research in the direction of NOR logic based implementation of reversible gates and functions. In this work, we propose the NOR logic based all optical reversible gates referred as all optical TNOR gate and all optical PNOR gate. The proposed all optical reversible NOR logic gates can implement the reversible boolean logic functions with reduced optical cost and propagation delay compared to their implementation using existing all optical reversible NAND gates. The advantages in terms of optical cost and delay is illustrated by implementing 13 standard boolean functions that can represent all 256 possible combinations of three variable boolean function. I. 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]. Reversible circuits does not lose information while performing the computations. A reversible circuit must comprise of reversible logic gates. Reversible logic also has applications in power-efficient nanocomputing [8], [9], [10]. 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 a reversible circuits and are referred as the garbage outputs. The inputs that are regenerated at the outputs are not considered as the garbage outputs [11]. 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 all optical implementation of reversible logic gates could be useful to overcome the limits imposed by conventional computing, and is also considered as implementation platform for quantum computing [12], [13], [14], [15], [16]. In the recent years, researchers have implemented several reversible logic gates in optical computing domain such as Feynman gate, Toffoli gate, Peres gate and Modified Fredkin gate. The all optical implementation of reversible logic gates can be achieved using semiconductor optical amplifier (SOA) based Mach-Zehnder interferometer (MZI) optical switches. The Mach-Zehnder interferometer based implementation of reversible logic gates provides significant advantages such as high speed, low power, fast switching time, and ease in the fabrication [17], [3], [18]. In the existing literature, the NAND logic based implementation of reversible logic gates and reversible boolean functions are the only available implementations. 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 novel all optical reversible NOR logic gates referred as all optical TNOR gate and all optical PNOR gate. The proposed TNOR and PNOR gates are useful for NOR logic based implementation of reversible boolean functions. The proposed all optical reversible NOR logic gates can implement the reversible boolean functions with reduced optical cost and propagation delay compared to the implementation of reversible boolean function using all optical reversible NAND gates (NAND logic based reversible gates are all optical Toffoli gate and all optical Peres gate). The optical cost of a reversible logic gate is defined as the number of MZI switches used in its all optical implementation. We illustrated the advantages of proposed all optical reversible NOR gates in terms of optical cost and delay by implementing the 13 standard boolean functions[19]. The 13 standard boolean functions proposed in [19] can represent all possible 256 combinations of three variable boolean functions. The paper is organized as follows: the basics of existing all optical reversible logic gates is presented in Section II; Section II illustrate the proposed all optical TNOR gate; Section IV illustrate the proposed all optical PNOR gate; The optical cost and delay analysis of proposed reversible NOR gates are discussed in Section V, while the conclusions are provided in Section VI /12 $ IEEE DOI /ISVLSI

2 II. BASICS OF ALL OPTICAL REVERSIBLE LOGIC The Mach-Zehnder interferometer (MZI) based optical switch is widely used to implement reversible logic gates [3], [17], [18], [20]. The design of all optical MZI switch is shown in Fig. 1(a). The all optical MZI switch can be designed using 2 Semiconductor optical amplifier (SOA-1, SOA-2) and two couplers(c-1, C-2). The operating principle of MZI based all optical switch can be explained as follows: In MZI switch, there are two inputs ports A and B and two output ports called as bar port and cross port, respectively as shown in Fig.1(a). At the input ports, the optical signal coming at port B is considered as the control signal (λ 2 ), and the optical signal coming at port A is considered as incoming signal( λ 1 ). The working of a MZI can be explained as: (i) when there is an incoming signal at port A and the control signal at port B then there is a light present at the output bar port and no light is present at the output cross port, (ii) in the absence of control signal at input port B and there is a incoming signal at input port A, then the outputs of MZI are switched and results in the presence of light at the output cross port and no light at the bar port. We consider no light or absence of light as the logic value 0. The above behavior of MZI based all optical switch can be written as boolean functions having inputs to outputs mapping as (A, B) to (P=AB, Q = A B), where A(incoming signal), B(control signal) are the inputs of MZI and P(Bar Port), Q(Cross Port) are the outputs of MZI, respectively. The block diagram of MZI based all optical switch is shown in Fig. 1(b). the optical cost and the delay (Δ) of MZI based all optical switch is considered as unity. generated at Q will be complement of input B that is Q= B. A Feynman gate can be implemented using 2 MZI based all optical switch, 2 beam combiner (BC) and 2 beam splitter (BS) in all optical reversible computing [3]. As the working of the beam combiner (BC) is to simply combines the optical beams while the beam splitter simply splits the beams into two optical beams, hence researchers do not consider them in the optical cost and the delay calculations [21], [22]. Figure 2(a) and 2(b) shows the block diagram and the all optical implementation of the Feynman gate. As the Feynman gate can be implemented using 2 MZI based optical switches thus the optical cost of Feynman gate is considered as 2. In the all optical implementation of the Feynman gate, two MZIs switches works in parallel thus the delay of the optical Feynman gate is considered as 1Δ. (a) Feynman gate (b) All optical implementation of the Feynman gate Fig. 2. Feynman gate and its all optical implementation (FG: Feynman gate, MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter) (a) Semiconductor optical amplifier based Mach-Zehnder interferometer (SOA: Semiconductor Optical Amplifier, C: Coupler) Fig. 1. (b) Mach-Zehnder interferometer Mach-Zehnder interferometer (MZI) based all optical switch A. All optical Feynman gate The Feynman gate (FG) is a 2 inputs and 2 outputs reversible gate. It has the mapping (A, B) to (P=A, Q= A B) where A, B are the inputs and P, Q are the outputs, respectively. The Feynman is also referred as the Controlled- NOT gate (CNOT) as when the input A=1 then the output B. All Optical Toffoli Gate The Toffoli gate is a 3 inputs and 3 outputs reversible gate. The inputs to outputs mapping of a Toffoli gate is (A,B,C) to (P = A, Q = B, R = A.B C), where A, B, C are the inputs and P, Q, R are the outputs, respectively [3]. The Toffoli gate is one of the most popular reversible logic gate as it is known to be universal for implementing the reversible boolean logic. A Toffoli gate can works as a NAND gate when the value at input C=1. When C=1, the outputs of the Toffoli gate transform as P = A, Q = B, R = A.B 1=A.B. Figure 3(b) shows the Toffoli gate working as a NAND gate when the value of input signal C is set to one. An all optical Toffoli gate can be implemented using 3 MZI based all optical switches, 1 beam combiner (BC) and 4 beam splitters [3]. Figure 3(a) and Fig. 3(c) shows the block diagram and all optical implementation of Toffoli gate, respectively. The optical cost of Toffoli gate is considered as 3 as the Toffoli gate can be implemented using 3 MZI based all optical switches. The Toffoli gate has a delay 208

3 of 2Δ as two MZI switches out of three MZI switches work in parallel. (a) Peres gate (a) Toffoli gate (b) NAND gate implementation using Toffoli Gate (c) All optical implementation of the Toffoli gate (b) All optical implementation of the Peres gate Fig. 4. Peres gate and its all optical implementation (PG: Peres gate, MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter) Fig. 3. Toffoli gate and its all optical implementation (TG: Toffoli gate, MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter) C. All Optical Peres Gate The Peres gate is a 3x3 reversible logic gate with the inputs to outputs mapping as (A,B,C) to(p = A, Q = A B, R = A.B C), where A, B, C are the inputs and P, Q, R are the outputs respectively [18]. An all optical Peres gate can be implemented using 4 MZI based switches, 5 beam splitters (BS) and 3 beam combiners (BC). Figure 4(a) and Fig. 4(b) shows the block diagram and the all optical implementation of a Peres gate, respectively. The optical cost of Peres gate is considered as 4 as the all optical implementation of Peres gate requires 4 MZI based switches. The delay of Peres gate is 2Δ as two MZI switches works in parallel with the two other MZI switches. III. PROPOSED ALL OPTICAL TNOR GATE The proposed all optical TNOR gate can work as a replacement of existing NAND based all optical Toffoli gate. The TNOR gate can perform NOR based implementation of reversible boolean functions in optical computing domain with reduced optical cost and delay. The all optical TNOR gate is a 3x3 reversible logic gate having inputs to outputs mapping as (A, B, C)to(P = A, Q = B, R =(A+B) c),wherea,b,c are the inputs and P, Q, R are the outputs, respectively. Figure 5(a) and Fig. 5(c) show the block diagram and the all optical implementation of all optical TNOR gate, respectively. The truth table of all optical TNOR gate is shown in Table I. The all optical TNOR gate will work as a NOR gate when the value of input signal C is 1. When C=1, the outputs of the TNOR gate transforms as P = A, Q = B and R =(A+B) 1 =A + B. The all optical TNOR gate working as a NOR gate is shown in Fig. 5(b). The all optical TNOR gate can be implemented using 2 MZI based switches, 4 beam splitters (BS) and 2 beam combiners (BC). The optical cost of TNOR gate is considered as 2, since its optical implementation requires 2 MZI based switches. The all optical TNOR gate has delay of 1Δ as in its optical design two MZI switches works in parallel. TABLE I TRUTH TABLE OF ALL OPTICAL TNOR GATE A B C P = A Q = B R =(A + B) C IV. PROPOSED ALL OPTICAL PNOR GATE The proposed all optical PNOR gate is 3 inputs and 3 outputs optical reversible gate with the inputs to outputs mapping as (A,B,C) to(p = A, Q = A B, R =(A + B) C), where A, B, C are the inputs and P, Q, R are the outputs, respectively. The proposed all optical PNOR gate can be used as the replacement of existing NAND based all optical Peres gate. The truth table of all optical PNOR gate is shown in Table II. An all optical PNOR gate can be implemented using 4 MZI based switches, 7 beam splitters (BS) and 3 beam combiners (BC). Figure 6(a) and Fig. 6(b) show the block diagram and the all optical implementation of PNOR gate, respectively. The optical cost of PNOR gate is considered as 4 as its optical implementation requires 4 MZI based switches. The delay of all optical PNOR gate is 1Δ, as in its optical design four MZI switches works in parallel. 209

4 (a) All optical TNOR gate (b) NOR gate implementation using TNOR Gate (c) All optical implementation of the TNOR gate (a) PNOR gate Fig. 5. TNOR gate and its all optical implementation (TNORG: TNOR gate, MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter) TABLE II TRUTH TABLE OF ALL OPTICAL PNOR GATE A B C P = A Q = A B R =(A + B) C V. OPTICAL COST AND DELAY ANALYSIS OF REVERSIBLE NOR GATES The proposed all optical TNOR gate and all optical PNOR gate can provide NOR based implementation of reversible boolean functions. In the existing literature, the all optical Toffoli gate and all optical Peres are used for NAND based implementation of reversible boolean functions. The proposed all optical reversible NOR gates have advantages compared to the existing all optical reversible NAND gates in terms of optical cost and delay. The Table III shows the optical cost and delay analysis of existing all optical reversible NAND gates and the proposed all optical reversible NOR gates. The proposed all optical TNOR gate (Optical cost=2, Delay=1Δ) has reduced optical cost and delay compared to the existing all optical Toffoli gate that has optical cost of 3 and delay of 2Δ. The proposed PNOR gate (Optical cost=4, Delay=1Δ) which is the NOR logic based counterpart of the Peres gate (Optical cost=4, Delay=3Δ) shows the significant reduction in delay with the same optical cost. In the existing literature, researchers have proposed 13 standard boolean functions that can represent all 256 possible combinations of three variable boolean function[19]. To illustrate the advantages of proposed all optical reversible (b) All optical implementation of the PNOR gate Fig. 6. PNOR gate and its all optical implementation (PNORG: PNOR gate, MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter) TABLE III OPTICAL COST AND DELAY OF ALL OPTICAL REVERSIBLE NAND LOGIC GATES AND PROPOSED ALL OPTICAL REVERSIBLE NOR LOGIC GATES Optical Cost Delay Feynman Gate [3] 2 1Δ Toffoli Gate [3] 3 2Δ Peres Gate [18] 4 2Δ Proposed TNOR Gate 2 1Δ Proposed PNOR Gate 4 1Δ NOR gates in terms of optical cost and delay, 13 standard boolean functions are implemented using the proposed gates as well as the existing all optical reversible NAND gates. The comparison study is shown in Table IV. The table IV provides the optical cost and delay analysis of all 13 standard boolean functions using (i) existing reversible NAND logic gates (all optical Toffoli gate), (ii) proposed all optical reversible NOR gates (all optical TNOR gate), (iii) using existing and proposed all optical reversible logic gates (all optical Toffoli gate and proposed all optical TNOR gate). The implementation of 13 standard boolean functions using proposed all optical TNOR gates has 20.57% improvement in terms of total optical cost, and 50.52% improvement in terms of total delay compared to the implementation using existing all optical Toffoli gate. The improvement using the combination of all optical Toffoli gate and the proposed all optical TNOR gate shows the 37.32% improvement in terms of optical cost and 26.80% improvement 210

5 in terms of delay compared to the implementation using only all optical Toffoli gate. For illustration purpose, implementation of a complex standard boolean function no. 13 F 13 = ABC + Ā BC + A B C + ĀB C is shown in Fig. 7. Fig. 7(a) shows the NAND based implementation of standard boolean function no. 13 using existing all optical Toffoli gate, where the all optical Feynman gates are used to generate the copy of inputs and complementing the inputs. The all optical implementation shown in Fig. 7(a) requires 3 all optical Feynman gates and 11 all optical Toffoli gates. The optical cost of NAND based implementation of standard function F 13 = ABC + Ā BC + A B C + ĀB C using existing all optical Toffoli gate and all optical Feynman gate can be computed as Optical Cost(TG) (F 13 )= Optical cost of Feynman gate * No. of all optical Feynman gate + Optical cost of all optical Toffoli gate * Optical cost of Toffoli gate = 3*2+11*3=39. The delay of all optical implementation of function no. 13 can be computed as 11Δ. The NOR based all optical implementation of function no. 13 using the proposed all optical TNOR gate is shown in Fig. 7(b). The all optical implementation shown in Fig. 7(b) requires 3 all optical Feynman gates and 11 proposed all optical TNOR gates. The optical cost of implementation using proposed all optical TNOR gate can be computed as Optical Cost(TNORG) (F 13 )= 3*2+11*2=28. The implementation using existing all optical TNOR gate has delay of 6Δ. The all optical implementation of standard boolean function no. 13 using the combination of all optical Toffoli gate and the proposed all optical TNOR gate is shown in Fig.7(c). Here the combination of all optical Toffoli gate and proposed all optical TNOR gate represents that 1) At the appropriate places to realize the NAND functions the all optical Toffoli gate will be used and 2) At the appropriate places to realize the NOR functions the proposed all optical TNOR gate will be used. Fig.7(c) requires 4 all optical Toffoli gates and 6 proposed all optical TNOR gates. The optical cost of standard boolean function implemented using combination of all optical Toffoli gate and proposed all optical TNOR gate can be computed as Optical Cost(TG+TNORG) (F 13 )= 4*3+6*2=24. From Fig.7(c), the delay can be computed as 11Δ. The all optical implementation of standard boolean function no. 13 (F 13 = ABC +Ā BC +A B C +ĀB C) using proposed all optical TNOR gate (Optical cost =28, Delay = 6Δ) has significant reduction in terms of optical cost and delay compared to the all implementation using existing all optical Toffoli gate (Optical cost =39, Delay = 11Δ). Also the all optical implementation using combination of all optical Toffoli gate and proposed all optical TNOR gate (Optical cost =24, Delay = 11Δ) shows the significant improvement in terms of optical cost compared to the all optical implementation using only the existing all optical Toffoli gate (Optical cost =39, Delay = 11Δ). In all optical reversible domain the all optical Peres gate does not have the minimal optical cost among existing 3X3 all optical reversible gates, thus the comparison study of implementation using all optical Peres gate is not shown in this work. (a) All optical implementation using existing all optical Toffoli gate (b) All optical implementation using proposed all optical TNOR gate (c) All optical implementation using existing all optical Toffoli gate and proposed all optical TNOR gate Fig. 7. All optical implementation of standard boolean function no. 13 (F = ABC + Ā BC + A B C + ĀB C) VI. CONCLUSION In this work, we have proposed two new all optical reversible NOR gates for NOR based implementation of reversible boolean functions. The first all optical TNOR gate can work as a replacement for existing all optical Toffoli gate, while the second all optical PNOR gate can work as a replacement of all optical Peres gate. It is illustrated that the proposed all optical reversible NOR gates have significant advantages over existing reversible NAND gates in terms of optical cost and delay. Thus, the proposed new optical reversible NOR gates can be beneficial for minimizing the optical cost and delay of the optical reversible circuits. All the optical reversible gates are functionally verified at the logic level using Verilog HDL. This is done by creating a Verilog library of Mach-Zehnder interferometer, beam combiner and beam splitter. In conclusion, this work advances the state of the art of reversible optical circuits by providing NOR logic based reversible gates as an alternative to NAND logic based reversible gates. REFERENCES [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. New York: Cambridge Univ. Press, [2] X. Ma, J. Huang, C. Metra, F.Lombardi, Reversible gates and testability of one dimensional arrays of molecular QCA, J. Elect. Testing, vol. 24, no. 1-3, pp , jan

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