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 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 | i (1) H if AO = 1, gates [1], [2] is an important concept in studying electronics | i since various laws and theorems of 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) | 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 . This property paper, we demonstrate that CQZ gate itself works as logic is called “”. 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 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 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. olwdb o’ bopieojc.A h n ftepooo,Charlie protocol, the of end object the absorptive At Alice’s with object. (AO). interacts absorptive object first applies binary Bob’s absorptive photon respective their by the encode their cycle, followed Bob of oscillator inner and absence optical Alice each . the In presence CQZ towards the in in photon photon inputs polarized the inputs V and his (OC) throwing by protocol 2. Fig. falfu ae sepanda follows: as explained is cases four all implementation counterfactual of The inputs. binary possible four Gate NOR function. Counterfactual Boolean and A. Y X, of value the on only depends M respectively. I lc X Alice MR MR o Y Bob stesnl ui dniyoeao and operator identity qubit single the is • to corresponding gate NOR of table truth the shows I Table • 2 1 ue and outer h Q aeata lcigfrteotrcce.The cycles. outer the in for | photon blocking of the cycles as of inner act state the gate low, are CQZ inputs the binary the both When 0) (0 1: Case yls h Q aetasom h tt ftephoton the of state of the transforms angle to gate an CQZ by the the photon cycles, rotates the cycle outer of each non- state and as cycles act outer an cycles but for inner introducing the blocking pass case, by to this channel In photon object. transmission absorptive the the allows blocks Alice Bob Y=1, and X=0 For 1) (0 2: Case edn o1as 1 to tending probability with communication after state | initial the to back M collapses photon the of state where ψ 1 I I AO AO I ceaiso onefculNR(AD gate. (NAND) NOR counterfactual of Schematics ( ( i m σ σ | ( nestepoo sdsaddi h counterfactual the in discarded is photon the unless , ue ylsadtecutrata O aeoutputs gate NOR counterfactual the and cycles outer ψ x x σ 2 1 i ) ) x m 00 ) i | θ prtro h uptpoo o O NN)gt where gate (NAND) NOR for photon output the on operator ψ 00 Channels M m N cos = i = 01 ne yls h oaiaino h photon the of polarization the cycles, inner π/ o ( cos = M m (2 OC − ∞ → λ M 1 1 00 θ mθ m ) M , cos = th . | (cos M H ( i ) ≤

oi aeUnit Gate Logic CQZ Gate | Conterfactual = 1 θ 2 σ M M M i x OC i ( sin + | | ) V 0 θ eoe h Pauli the denotes | 1 i i M ue yl sgvnas given is cycle outer 2 i and , sin + θ M I mθ ( σ After . | x V ) M θ hri trsthe starts Charlie i M ) = | | 0 0 i x i Charlie Z | m 1 . ) operator, i , The . outer (4) (3) (2) of h tt vlto ftepoo orsodn oteinputs the to corresponding photon the X=Y= of evolution state operation. the gate NAND the to counterfactual of Due the end complete the to At applies photon inputs. simply binary Charlie respective protocol, their on need unit Bob Alice that is apply difference to only The counterfac- Two- gate. as inputs. NOR similar tual binary works two gate NAND with counterfactual gate inputs NOR counterfactual the of Gate NAND Counterfactual B. where is discarded not is photon the that probability counterfac- the and outputs that gate conclude NOR we tual discussion, above the From ntepeiu uscin ebifl xlie h working the explained briefly we subsection, previous the In • M | bobdb h lc’ bopieojc ihprobability with λ object absorptive Alice’s the where by absorbed 2 case to is gate The similar Y. is non- of photon are state | the the of cycles regardless evolution inner cycles state outer the for X=1, blocking object if absorptive Alice’s with (counterfactually), interacts first photon the As 0) (1 3: Case outputs where nestepoo sasre yteCalesabsorptive Charlie’s probability the with by object absorbed is photon the Unless 0 ψ i 1Y i and p m σ XY ifrfo onefculNRgt where gate NOR counterfactual from differ ievrawt etit ne h smttclimits asymptotic the under certainty with versa vice x σ i ae2 Case 2 Case 2 Case 1 Case Cases = 1Y x N prto eoetecutrata oi aeunit, gate logic counterfactual the before operation stepoaiiyms ucinof function mass probability the is | λ T θ prtrbfr n fe h onefcullogic counterfactual the after and before operator 0 | o h nt ausof values finite the For . = N RUTH 01 0 i λ i o = idpneto )uls h htnis photon the unless y) of (independent x=1 for . 01 | ψ after = Alice m = T X P 1 1 0 0 & BEOF ABLE i π/ 01 i Y | | M =1 = M φ φ ae4 11) (1 4: Case h upto h onefculNOR counterfactual the of output The . m m (2 | X=0 1 X  Bob ue cycles. outer i i 1 i Y 1 1 0 1 0 N 11 00 AL I TABLE U − fbt h iayipt r Low are inputs binary the both if ) Y=0 X NIVERSAL h onefculNRgate NOR counterfactual the , = = sin 1 = Z | | ψ ψ 2 p NOR σ m m XY ( Y + X x iθ 0 0 0 1 i i 00 11 M λ prto nhsoutput his on operation L M XY Charlie OGIC , , sin ) , and = Z 2 G NAND θ ATES { N N 00 0 1 1 1 X  h average the , , N · 01 Y i . { ∈ , 10 , 0 11 , 1 (8) (7) (6) (5) } } . . 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 0.4 Channels 0.30.4 MR2 0.3 I (σx) 10 0.9 0.20.3 0.20.3 0.2 Alicen Xn 0.2 0.10.2 0.1 5 0.00.1 Fig. 3. n-inputs/1-output counterfactual NOR (NAND) gate. 0.1 0.00.1 2 0.0 1 12 200 400 600 800 1,000 unless the photon is discarded in the counterfactual logic gate N unit with probability Fig. 4. Average success probability for n = 2 under the uniform distribution ζ00 = λ11, (9) of possible inputs as function of M and N.

ζ11 = λ00. (10) IV. CONCLUSIONS III. n-INPUTS/1-OUTPUT COUNTERFACTUAL UNIVERSAL LOGIC GATES In this paper, we presented a new scheme of n-inputs/1- output counterfactual universal logic gates by using CQZ Without a loss of generality, we consider the n-inputs logic gate. For n = 2, Alice and Bob can independently control gate as shown in Fig. 3. Similar to 2-inputs counterfactual the absorptive object and hence determining Charlie’s output logic gates, the photon interacts with each absorptive object without any physical particles traveling over the channel. in every inner cycle. For n-inputs counterfactual NOR gate, Though the process involves three parties, there is no need for the state evolution of the photon after m ( M) is given as ≤ preshared entanglement. Based on this gate design, it can be extended to build other counterfactual logic gates and thereby using the combination of these gates will help to prepare the χm = ψm , (11) | i00...0 | i00 implementation of other complex circuits. χm X1X2...Xn=006 ...0 = ψm 11 . (12) | i | i ACKNOWLEDGMENT unless the photon is discarded in the counterfactual logic gate This work was upheld by the National Research Foundation unit with probability of Korea (NRF) give financed by the Korea government

γ00...0 = λ00, (13) (MSIT) (No. 2019R1A2C2007037).

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