Counterfactual Universal Logic Gates Nang Paing Saw, Fakhar Zaman, Junaid ur Rehman, and Hyundong Shin Department of Electronic Engineering, Kyung Hee University, Yongin-si, 17104 Korea Email: [email protected] Abstract—In this paper, we develop universal logic gates such as NAND and NOR gate using optical devices and chained Alice X quntuam Zeno (CQZ) effect which lead us to the counterfac- tual implementation of the universal logic gates without prior Logic Gate Z Charlie entanglement. We consider that the two binary inputs Alice (X) and Bob (Y); and the output Charlie (Z) of the countfactual logic Bob Y gate are the distinct parties and demonstrate the counterfactual Channels implementation of 2-inputs/1-output universal logic gates. Then, we generalize these results for n-inputs/1-output counterfactual Fig. 1. Basic model of logic gate. We considered the three distinct parties universal logic gates. Alice, Bob and Charlie where Alice and Bob equips the binary inputs X and Y of the logic gate and Charlie has the output Z. I. INTRODUCTION A logic gate is an electronic device to implement basic channel.2 The CQZ gate transforms the vertically (V) polarized operation such as AND, OR, XOR, etc. and is the fundamental input photon as function of absorptive object as follows: element of digital era of electronics, i.e., millions of logic ( V if AO = 0; gates are used in a microprocessor. Implementation of logic j i (1) H if AO = 1; gates [1], [2] is an important concept in studying electronics j i since various laws and theorems of Boolean algebra can be where AO = 0 (1) denotes the absence (presence) of the implemented with a complete set of logic gates. Given two or absorptive object and V(H) denotes the vertical (horizontal) j i more binary inputs, the logic gate carries out specified logical polarized photon respectively. Recently, the counterfactual 1 function and gives only one binary output. Among the logic implementation of classical logic gates [15] has been demon- gates, the universal logic gates (NAND and NOR gates) [3], strated. Though the desired output of logic gate is achieved, but [4]–the set of logic gates that can be used individually or it increases the complexity of the circuit by using attenuators, together to realize all of the Boolean switching functions– phase shifters etc in addition to modified CQZ gate. In this have their own importance in digital electronics. This property paper, we demonstrate that CQZ gate itself works as logic is called “Functional Completeness”. Regarding this property, gates. Our new scheme reduces the circuit complexity and the entire microprocessor can be designed using only NAND enhances the probability of success of the protocol. The rest or (and) NOR gates. In addition, the universal logic gates are of the paper is structured as follows: first we demonstrate 2- also popular in designing Transistor-Transistor Logic (TTL) input counterfactual universal logic (CUL) gates in Section II and Complementary Metal Oxide Semiconductor Transistor followed by n-input CUL gates in Section III. (CMOS) logics. We consider the distributed computation model where the II. COUNTERFACTUAL UNIVERSAL LOGIC GATES binary inputs of a 2-input logic gate are possessed by the Fig. 2 shows the schematics of counterfactual universal logic remote parties say Alice (X) and Bob (Y) as shown in gates to implement a Boolean function where binary inputs Fig. 1. In order to compute the output of a logic gate at X and Y are encoded as absence (0) or presence (1) of the third party Charlie (Z), Alice and Bob need to transmit their absorptive objects. Here, the combination of CQZ gate and binary inputs to Charlie through a classical channel. In this optical oscillators (OC) work as counterfactual logic gate unit. article, we consider the same scenario and demonstrate the The Boolean function is implemented counterfactually on the implementation of universal logic gates by using chained two distinct inputs X and Y; and the output is generated at third quantum Zeno (CQZ) gate but no physical particle is found party Charlie in terms of the polarization of the photon. Charlie in the transmission channel. starts the protocol by throwing his V polarized photon towards Counterfactual communication [5], [6] is a surprising phe- the OC2 and inputs the photon in CQZ gate. In each inner nomenon in quantum mechanics [7]–[10]. It takes account into cycle, the photon first interacts with Alice’s absorptive object the logic of CQZ gate and interaction free measurement [11], counterfactually followed by Bob’s absorptive object. After [12], thereby enabling the transfer of information between the 2 sender and receiver but no physical particles traveling over the The CQZ gate is the nested version of the quantum Zeno gate [13] (see Fig. 2 in Reference [14]). The schematic of the CQZ is based on polarization of the input photon. Here we use only CQZ gate as the input photon is always 1Logic gates like NOT gate are 1-input/1-output gates. V polarized. In the rest of the article, we refer CQZ gate as CQZ gate. Alice X TABLE I TRUTH TABLE OF UNIVERSAL LOGIC GATES I (σx) MR1 Charlie V Alice Bob AO1 | i Cases NOR NAND X Y Z = X + Y Z = X · Y Case 1 0 0 1 1 AO2 I (σx) Z Charlie Case 2 0 1 0 1 Case 2 1 0 0 1 OC CQZ Gate OC MR2 1 2 Case 2 1 1 0 0 I (σx) Conterfactual Bob Y Logic Gate Unit Channels Unless the photon is absorbed by the Charlie’s absorptive Fig. 2. Schematics of counterfactual NOR (NAND) gate. Charlie starts the object with probability protocol by throwing his V polarized photon towards the optical oscillator (OC) and inputs the photon in CQZ gate. Alice and Bob encode their binary M inputs in the presence or absence of their respective absorptive object (AO). Y N λ = 1 sin2 (iθ ) sin2 θ : (5) In each inner cycle, the photon first interacts with Alice’s absorptive object 01 − M N followed by Bob’s absorptive object. At the end of the protocol, Charlie i=1 applies I (σx) operator on the output photon for NOR (NAND) gate where I is the single qubit identity operator and σx denotes the Pauli x operator, where θN = π= (2N), the counterfactual NOR gate respectively. outputs 0 after M outer cycles. j i • Case 3: (1 0) & Case 4: (1 1) As the photon first interacts with Alice’s absorptive object M outer and N inner cycles, the polarization of the photon (counterfactually), if X=1, the inner cycles are non- depends only on the value of X, Y and Boolean function. blocking for outer cycles regardless the state of Y. The state evolution of the photon is similar to case 2 where A. Counterfactual NOR Gate = . The output of the counterfactual NOR j mi1Y j mi01 gate is 0 for x=1 (independent of y) unless the photon is Table I shows the truth table of NOR gate corresponding to j i four possible binary inputs. The counterfactual implementation absorbed by the Alice’s absorptive object with probability λ = λ of all four cases is explained as follows: 1Y 01. From the above discussion, we conclude that the counterfac- • Case 1: (0 0) tual NOR gate outputs 1 if both the binary inputs are Low When both the binary inputs are low, the inner cycles of j i and 0 vice versa with certainty under the asymptotic limits the CQZ gate act as blocking for the outer cycles. The j i of M and N. For the finite values of M and N, the average state of the photon in mth ( M) outer cycle is given as ≤ probability that the photon is not discarded is j mi00 1 1 = cosm−1 θ (cos θ 1 + sin θ 0 ) ; (2) X X j mi00 M M j i M j i P = pXYλXY; (6) X=0 Y=0 where θ = π= (2M), H = 0 and V = 1 . The M j i j i j i j i state of the photon collapses back to the initial state after where pXY is the probability mass function of 00; 01; 10; 11 . f g M outer cycles and the counterfactual NOR gate outputs 1 , unless the photon is discarded in the counterfactual B. Counterfactual NAND Gate j i communication with probability In the previous subsection, we briefly explained the working 2M of the counterfactual NOR gate with two binary inputs. Two- λ00 = cos θM ; (3) inputs counterfactual NAND gate works similar as counterfac- tending to 1 as M . tual NOR gate. The only difference is that Alice and Bob need ! 1 • Case 2: (0 1) to apply σx operator before and after the counterfactual logic For X=0 and Y=1, Alice allows the photon to pass but gate unit on their respective binary inputs. At the end of the Bob blocks the transmission channel by introducing an protocol, Charlie simply applies σx operation on his output absorptive object. In this case, the inner cycles act as non- photon to complete the counterfactual NAND gate operation. blocking for outer cycles and each outer cycle rotates the Due to σx operation before the counterfactual logic gate unit, state of the photon by an angle of θM . After m outer the state evolution of the photon corresponding to the inputs X=Y=i differ from counterfactual NOR gate where i 0; 1 . cycles, the CQZ gate transforms the state of the photon 2 f g to φ = ; (7) j mi00 j mi11 = cos (mθ ) 1 + sin (mθ ) 0 : (4) φ = ; (8) j mi01 M j i M j i j mi11 j mi00 Alice1 X1 30 1.01.0 1.0 I (σx) 0.91.0 MR1 0.9 0.9 0.9 AO1 25 0.80.9 0.8 0.8 0.8 Alice2 X2 I (σx) 0.7 0.7 Counterfactual 0.7 20 0.60.7 MR2 AO2 Logic Z Charlie 0.6 Gate Unit 0.6 OC 0.50.6 0.5 M 0.5 15 0.5 0.40.5 0.4 AOn 0.4 Channels 0.30.4 MR 0.3 2 10 0.3 I (σx) 0.9 0.20.3 0.20.3 X 0.2 Alicen n 0.2 0.10.2 5 0.1 Fig.