IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 85 Opertions Algorithms on Quntum Computer Moyd A. Fhdil, Ali Foud Al-Azwi, nd Smmer Sid Informtion Technology Fculty, Phildelphi University, Ammn, Jordn Informtion Technology Fculty, Phildelphi University, Ammn, Jordn Al-Mnsour University College Bghdd-Irq Abstrct A quntum computer is device for computtion tht mkes direct use of distinctively quntum mechnicl phenomen, such s superposition nd entnglement, to perform opertions on dt. In clssicl (or conventionl) computer, the mount of dt is mesured by bits, in quntum computer; it is mesured by quntum of bits (qubits). This work describes, nd implements the universl quntum logic gtes, the terminology is introduced with two well known quntum gtes, the quntum NOT, nd the quntum XOR gtes. The NOT nd XOR gtes hve lredy been described in clssicl reversible logic. This work gives the implementtion of the quntum circuits, such s the Quntum Hlf-Adder Circuits, which consists of quntum control control not gte (CCNot) nd quntum control not gte (CNot). Also, it describes nd implements the Quntum Full-Adder Circuits, which consists of two Quntum Hlf-Adder Circuits nd one control not gte (CNot). Depending on the quntum circuits, one cn implement the quntum bsic rithmetic opertions such s (ddition, subtrction, multipliction nd division). After ech implementtion, the computtionl complexity of ech step is clculted. Key words: Quntum computing, Quntum Logic Gtes, Truth Tble, quntum mechnicl, qubits, quntum mechnicl, superposition.. Quntum Computer A quntum computer is device for computtion tht mkes direct use of distinctively quntum mechnicl phenomen, such s superposition nd entnglement, to perform opertions on dt. In conventionl computer, the mount of dt is mesured by bits; in quntum computer, it is mesured by qubits. The bsic principle of quntum computtion is tht the quntum properties of prticles cn be used to represent nd structure dt, nd tht devised quntum mechnisms cn be used to perform opertions with this dt. In quntum mechnics, the stte of physicl system (such s n electron or photon) is described by n element of mthemticl object clled Hilbert spce. A clssicl computer hs memory mde up of bits, where ech bit holds either one or zero. The device computes by mnipulting those bits, i.e. by trnsporting these bits from memory to (possibly suite of) logic gtes nd bck. A quntum computer mintins set of qubits. A qubit cn hold one, or zero, or superposition of these. A quntum computer opertes by mnipulting those qubits, i.e. by trnsporting these bits from memory to (possibly suite of) quntum logic gtes nd bck. A clssicl computer opertes on 3 bit register. At given time, the stte of the register is determined by single string of 3 bits, such s "". This is usully expressed by sying tht the register contins single string of 3 bits. A quntum computer, on the other hnd, cn be in stte which is mixture of ll the clssiclly llowed sttes. The prticulr stte is determined by 8 complex numbers. In quntum mechnics nottion we would write: Q> = > + b > + c > + d > + e > + f > g > + h > Where, b, c, d, e, f, g, nd h re complex. A complex number (α+βi) is clled (complex vlued) mplitude, nd ech probbility ( α 2 + β 2 ) is the bsolute squre of the mplitude, becuse it equls α+ βi 2. The probbilities must sum to [, 2, 3]. 2. Trnsformtion Mtrices [4, 9] It is often the cse in quntum computtion tht we know wht the opertion we wnt to perform is, in terms of its effect on the stte vector of our system, but we don t know how to express the opertion s mtrix. In this short note, we demonstrte how this cn be done quite esily. Let s consider simple exmple: the truth tble for the controlled-not gte is Mnuscript received Jnury 5, 2. Mnuscript revised Jnury 2, 2.
86 IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 to express this s mtrix opertor so tht we cn pply it esily to our qubit sttes. In the usul lnguge, the CNOT gte trnsforms our bsis vectors in the following wy: > >; > >; > >; > > cn be trnslted into mtrix form s follows: or, in vector form So we hve tht The key to deducing the mtrix which performs these trnsformtions is to pprecite how we multiply mtrices together. It is most convenient to express this in component nottion, where write nd so The b i is the components of the new vector, the i re the components of the originl vector, nd the M ij re the elements of the mtrix opertor M. In this nottion i lbel the rows of M nd j lbels the columns. Expnding the summtion, yields Applying this to the exmple of the CNOT gte, gives nd so on. Other gtes re eqully esy. For the Hdmrd gte: nd since b = =, we must hve M =. Similrly, we find tht M22 = M34 = M43 =, with ll other Mij =. We write the mtrix down s: We hve nd for the more complex exmple of A useful wy of looking t this is to lbel the rows nd columns of the mtrix with the bsis sttes they correspond to. You simply lbel ech row nd column with the bsis sttes in the usul order. The row lbels lbel the bsis sttes of the new vector; the column lbels lbel the bsis sttes of the old vector. So, for exmple:
IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 87 We hve c > + c > + c > + c >. Q> = > + > + > + > 3. Quntum Bits Representtion In this section, we describe how the bits re represented in quntum computer. 3.. One-Quntum Bits [6, 7, 8, 9] A qubit cn exist in n rbitrry superposition stte, mesurement on it will lwys find it in one of the two eigensttes, > or >, ccording to the mesurement postulte of quntum mechnics. Q> = > + > > = Q>= >=,, = =, > = > = =, this is superposition mtrix for Dirc > > = =, this is superposition mtrix for Dirc > 3.2. Two-Quntum Bits [6, 7, 8, 9] If we hve more thn one qubit in our quntum system, we cn express its stte in terms of product eigensttes. For exmple, two-qubit system hs the bsis sttes, >, >, >, >, which re the quntum nlogs of the input lines in the truth tble for clssicl logic gte. Unlike clssicl bits however, two or more qubits cn interfere with one nother, creting mcroscopiclly coherent superposition, of the form 3.3. Representtion of the QuBit In this section, we will explin how Dirc representtions convert to the superposition representtion to del it s quntum bits (QuBit) in quntum computers. Below, we illustrte our lgorithm for convert Dirc qubits representtions to the superposition qubits representtion. Algorithm convert - qubit s Dirc representtion ( Q D ). - n= number of bits. - qubit s superposition mtrix representtion (Q B ). Step: set the superposition mtrix (Q B ) by zero ccording to 2 n Step2: let t=. k= Step3: Determine the position (t) of the superposition of the mtrix. Step4: compute this position (t) by the forloop s below
88 IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 For i=n downto do Begin t= t+qd[i] *k; k=k*2; End; hve lredy described the NOT nd XOR gtes in clssicl reversible logic. A stright-forwrd quntum generliztion of these gtes is the unitry mtrices. Step5: set the vlue of position (t) by one in QB like Q B [t] =. Step6: Output the superposition mtrix (Q B ). Step7: Finish. Below, we illustrte our lgorithm for converting superposition qubits representtions to the Dirc qubits representtion. Algorithm 2 deconvert - qubit s superposition mtrix representtion (Q B ). - n= number of qubits qubit s Dirc representtion ( QD). Step: let t=. Step2: find the position (t) in the superposition of the mtrix (Q B ) by the repet until loop below. Repet t=t+; Until Q D [t]=; Step3: compute the Dirc mtrix (Q D ) by the forloop below For i=n downto do Begin Q D [i] =t mod 2; t=t div 2; End; Step4: Output the Dirc mtrix (Q D ). Step5: Finish. 4. Quntum Not nd XOR Gtes [8, 9, ] The problem of universlity cn be posed for quntum computtion s well, in sking whether rbitrry unitry opertions cn be broken down into simpler ones. Similr to clssicl logic, quntum logic gtes exist tht operte on hndful of qubits t time, nd tht re ble to simulte rbitrry unitry opertions. Note tht it is property of unitry mtrices tht product of two of them remins unitry; hence product of unitry logic gtes will lso be unitry. Before we describe the universl quntum logic gtes, we introduce the terminology with two well known quntum gtes, the NOT, nd the XOR gtes. We Which describes the evolution of the two product eigensttes of the one-qubit NOT gte, nd the four products eigensttes of the two-qubit XOR gte. Note tht unlike their clssicl counterprts, these quntum gtes cn trnsform superposition sttes s well. For exmple, opertions of the form, U NOT : c > + c > c > + c >, re lso possible with the quntum NOT gte, we will describe the quntum Not Gte nd quntum XOR Gte with detils nd lgorithms in the next section. 4.. One Input Quntum Bit Gte The behvior of quntum Not Gte is:- But the unitry mtrix of quntum Not Gte is:- Unot = This is the block digrm of quntum Not Gte is:- X> Not Here, we multiply the unitry mtrix of quntum Not Gte with > tht represent with superposition or with > tht is represented with superposition to get the output of this gte. = X> i.e Unot > = >
IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 89 Bellow, we illustrte our lgorithm of quntum Not gte:- Algorithm 3 Quntum Not Gte qubit s Dirc representtion ( Q D ). quntum Not gte result. Step: set the unitry mtrix trnsformtion Step2: represent the sequence of qubit s one dimension (Q B [i]) by using lgorithm 4. convert. Step3: compute the quntum Not of qubit by multiplying The U QNot mtrix by Q B N QB = Q B * U QNot. Step4: Output the (N QB ). Step5: Finish. 4.2. Two Input Quntum Bit Gte When we use two input quntum Bit in gte, this cse is clled controlled not (CNOT) gte, this process is equivlent to XOR gte in clssicl computer, below we present the unitry mtrix of quntum XOR Gte or quntum controlled Not Gte (CNOT) with its coefficients. controlled Not Gte with > is represented in superposition Controlled Not Gte with > tht is represented in superposition Controlled Not Gte with > tht is represented in superposition Controlled Not Gte with > tht is represented in superposition = i.e. Unot > = > UQNot = U cnot > = > = U cnot > = > = Ucnot = =
9 IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 U cnot > = > = This is the block digrm of quntum Controlled Not Gte when it hs the two inputs ( x>, y> ) nd we cn get from it two outputs ( x>, x> y> ). x> Y> CNO Now, we illustrte our lgorithm of quntum Control Not gte:- Algorithm 4 Quntum Control Not Gte x> x Y> 4.3. Three Input Quntum Bits Gte This section, represents nd describes the three inputs of quntum bits, which re used in controlled CNOT (CCNOT), below we present the unitry mtrix of quntum controlled CNot Gte with its coefficients. It is lso clled controlled controlled NOT (CCNOT) or clled Toffoligte. The superposition of this qubits, is represented in the next description: Gte with > tht is represented in superposition ( ) T U ccnot > = > qubit s Dirc representtion ( Q D ). quntum control Not gte result. Gte with > tht is represented in superposition ( ) T U ccnot > = > Step: set the unitry mtrix trnsformtion U CNot = Step2: represent the sequence of qubit s one Dimension(Q B [i]) by using lgorithm 4. convert. Step3: compute the quntum control Not of qubit by multiplying The U CNot mtrix by Q B CN QB = Q B * U CNot. Step4: Output the (CN QB ). Step5: Finish. Gte with > tht is represented in superposition ( ) T U ccnot > = > Gte with > tht is represented in superposition ( ) T U ccnot > = > Gte with > tht is represented in superposition ( ) T
IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 9 U ccnot > = > Gte with > tht is represented in superposition ( ) T U ccnot > = > Gte with > tht is represented in superposition ( ) T U ccnot > = > Gte with > tht is represented in superposition ( ) T U ccnot > = > Below, the block digrm of quntum CCNot Gte represents: -Three inputs: ( X, Y, Z ). -Three outputs: ( X, Y, X Y Z ). Step: set the unitry mtrix trnsformtion Step2: represent the sequence of qubit s one dimension(q B [i]) by using lgorithm 4. convert. Step3: compute the quntum control contronot of qubit by Multiplying The U CCNot mtrix by Q B CCN QB = Q B * U CCNot. Step4: Output the (CCN QB ). Step5: Finish. U CCNot = 5. Quntum Circuits Gte Representtion In this section, we describe the quntum circuits gte tht is represented by Hlf Adder quntum circuit nd Full Adder quntum circuit. X Y CCNOT X Y 5.. Hlf Adder Quntum Circuits We design the Quntum Hlf-Adder Circuits, which consists of quntum control control not gte (CCNot) nd quntum control not gte (CNot), below is the structure design of Quntum Hlf-Adder Circuits with its tble. Z X Y Z Now, we illustrte our lgorithm of quntum Control Control Not gte:- X X CCNOT CNOT X Sum Algorithm 5 Quntum Control Control Not Gte Y Toffoli Crry qubit s Dirc representtion ( Q D ). quntum control Not gte result. This Circuit represents the three inputs ( X, X, Y ) nd three outputs ( X, Sum, Crry ).
92 IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 The result of the Hlf-Adder:- Sum = X X Crry = Crry Y = X X Y Let Y = zero Crry = X X See the tble below X X Sum Crry Below, we illustrte our lgorithm of Hlf Adder Quntum Circuits:- Algorithm 6 Quntum Hlf-Adder Circuit Two qubit s Dirc representtion ( X >, X >). Two qubit result (Sum,Crry). Step: let Y = Step2: compute the output of Quntum Control Control Not by using lgorithm 4.5 to get the crry crry = X X Y > Step3: compute the output of Quntum Control Not by using lgorithm 4.4 to get the sum Sum = X X The result of the Hlf-Adder one (AH):- Sum (S ) = X X Crry (C ) = X X Y Let Y = zero Crry = X X The result of the Hlf-Adder two (AH2):- Sum (S ) = B S Crry (C ) = B S Y Let Y = zero Crry = B S The result of the full-adder:- Sum (S ) = B S Crry = C C Below, we illustrte our lgorithm of Full Adder Quntum Circuits:- Algorithm 7 Quntum Full-Adder Circuit Step4: output the result of Quntum Hlf-Adder represented by Sum nd Crry. Step5: Finish. Two qubit s Dirc representtion ( X >, X >). Two qubit result (Sum,Crry). 5.2. Full Adder Quntum Circuits We design the Quntum Full-Adder Circuits, which consists of two Quntum Hlf-Adder Circuits nd one control not gte (CNot), below is the structure design of Quntum Full-Adder Circuits. Step: let Y = Step2: compute the output of Quntum Hlf-Adder (HA) by using lgorithm 4.6 to get Sum (S ) = X X crry (C ) = X X Y > Step3: compute the output of Quntum Hlf-Adder2 (HA2) by using lgorithm 4.6 to get
IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 93 Sum (S ) = B S crry (C ) = B S Y > Step4: compute the output of Quntum Control Not by using lgorithm 4.4 to get Sum = S Crry = C C Step5: output the result of Quntum Full-Adder represented by Sum nd Crry. Step5: Output the result. Step6: Finish. 6.2. Quntum Subtrction Opertion The quntum subtrction opertion cn be performed by converting ech number into sequence of qubit. Then tke 2's complement of the second number. This process is performed by using our lgorithm of Quntum Subtrction Opertion Algorithm 9 Quntum Subtrction Opertion Step6: Finish. 6. Quntum Arithmetic Opertions on Quntum Computer Two deciml numbers The result of subtrction s deciml number. This section describes nd represents how we cn design nd implement the bsic rithmetic opertions to quntum bsic rithmetic opertions to be suitble for use in the quntum computer, these opertions re (Addition, Subtrction, multipliction nd Division). 6.. Quntum Addition Opertion The quntum ddition opertion cn be performed by converting ech number into sequence of qubit then dding these two sequences by using our lgorithm of Quntum Addition Opertion Algorithm 8 Quntum Addition Opertion Two deciml numbers The result of Addition s deciml number. Step: Convert the first deciml number into dirc representtion A=,, 2,......, n Step2: Convert the second deciml number into Dirc Representtion B= b,, b 2,......, b n Step3: compute the ddition process by Quntum Full-Adder By using lgorithm 4.7. Process : Step: Convert the first deciml number into dirc representtion A=,, 2,......, n Step2: Convert the second deciml number into dirc representtion B= b,, b 2,......, b n Step3: convert the second dirc number into 2 S complement representtion B = B + Step4: compute the ddition process by Quntum Full-Adder By using lgorithm 4.7. Step5: convert the deciml number. Step6: Output the result. Step7: Finish. dirc result of ddition into 6.3. Quntum Multipliction Opertion The multipliction opertion cn be performed by converting ech number into sequence of qubit then multiplying these two sequences by using our lgorithm of Quntum Multipliction Opertion Step4: convert the dirc result of ddition into deciml number.
94 IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 Algorithm Quntum Multipliction Two deciml numbers The result of multipliction s deciml number. Process : Step: Convert the first deciml number into dirc representtion A=,, 2,......, n Step2: Convert the second deciml number into dirc representtion B= b,, b 2,......, b n Step3: set C = Step4: compute the multipliction process by Following: Step4.: if dirc digit = then compute the ddition process by Quntum Addition By using lgorithm 4.8. Step4.2: Shift A to the left by zero ccording to cse of Dirc of B. Step4.3: goto to the Step4.. Step5: convert the dirc result of Multipliction into deciml number. Step6: Output the result. Step7: Finish. 6.4. Quntum Division Opertion The division opertion cn be performed by converting ech number into sequence of qubit then dividing these two sequences by using our lgorithm of Quntum Division Opertion Algorithm Quntum Division Opertion Two deciml numbers The result of division s deciml number without reminder. 7. Conclusions Step: Convert the first deciml number into dirc representtion A=,, 2,......, n Step2: Convert the second deciml number into dirc representtion B= b,, b 2,......, b n Step3: set C = Step4: compute the division process by following Setp4.: do while A > = B Step4.2: compute the subtrction process by Quntum subtrction By using lgorithm 4.9. Step4.3: increment C by one: C=C+ Step5: convert the dirc result of Division into deciml number. Step6: Output the result. Step7: Finish. This work introduces the quntum computer nd computtion principles nd properties. It gives the implementtion of the quntum circuits, such s the Quntum Hlf-Adder Circuits, which consists of quntum control control not gte (CCNot) nd quntum control not gte (CNot). Also, it describes nd implements the Quntum Full-Adder Circuits, which consists of two Quntum Hlf-Adder Circuits nd one control not gte (CNot). The steps of implementtion of quntum bsic opertions (ddition, subtrction, multipliction division) with its lgorithms re n essentil step towrd building the hrdwre nd softwre of quntum computer. Reference [] Wikipedi, the free encyclopedi, Quntum Computer, Center For Quntum Computtion (CQC), 25. [2] Jcob West, The Quntum Computer, Published in Scientific of Americ, 2. [3] Elenor Rieffel & Wolfgrg Polk, An Introduction to Quntum Computer for Non-Physicists, www.rieffelpl.xerox.com& & www.polk.pl.xerox.com, 998. [4] John Preskill, Relible Quntum Computers, Cliforni Institute of Technology,997.
IJCSNS Interntionl Journl of Computer Science nd Network Security, VOL. No., Jnury 2 95 [5] Mohmmd Inytullh Bbri, Shkell Ahmd, Sheerz Ahmed, Iftikhr Ahmed Khni, nd Bshir Ahmd, Implementing Dimensionl-View of 4X4 Logic Gte/Circuit for Quntum Computer Hrdwre using Xylinx, JSSST Vol. 9 No. 5, December 28. [6] A.D. Mnzno & L. Steinberg, Ide of Quntum Computtion, Center for Quntum Computtion(CQC), 999. [7] Dvid Deutsch & Artur Ekert, Mchines, Logic nd Quntum Physics, Center for Quntum Computtion (CQC), University of Oxford, 999. [8] Xinln Zhou, Dbbie W. Leung & Isc L. Chung, Methodology for quntum logic gte construction, Stnford university, IBM Reserch nd Stnford university, 2. [9] Ashok Muthukrishnn, Clssicl nd Quntum Logic Gtes, Rochester Center for Quntum Informtion (RCQI), 999. [] Vipin Mishr, Developing Innovtive Progrms for Quntum Computers nd Algorithms for Interprettion of Complex Clcultions, Amity University, Amity Institute of Nnotechnology, Project Guide, 26.