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Quantum computingEquivalent circuits of quantum teleportation and swapMasatsugu Sei Suzuki and Itsuko S. SuzukiDepartment of Physics, SUNY at Binghamton(Date: January 04, 2021)Quantum circuit is a sort of electric circuits (such as Wheatstone bridge and laddercircuit), which one may study in the class of circuit analysis (electricity and magnetism).For complicated circuits (such as network), it is essential to make use of the theorems ofcircuit analysis (such as Thevinin theorem and Norton theorem). So, the circuit becomesmuch more simplified by using the corresponding equivalent circuits. It may be true forquantum computing. We can also apply various kinds of techniques (based on the quantummechanics) to the quantum circuits (such as the quantum teleportation and SWAP). Theequivalent circuits can be used for the simplification of quantum circuits.In a Website, we find a very interesting article on the discussion on the equivalence ofquantum computer circuit between quantum teleportation and the SWAP circuit. It issurprising for one that the SWAP circuit is literally equivalent to the quantum teleportation.The title: From Swapping to teleporting with Simple Circuit Moves;https://algassert.com/post/1628. In the introduction of this article, we found the followingexciting statements. “We are going to prove that quantum teleportation works. Not bycarefully considering how it affects input states, but by starting with a circuit that obviouslymoves a qubit from one place to another and then applying simple obviously-correcttransformations until we end up with the quantum teleportation circuit.”We also had an excellent opportunity to listen to a series of lectures on the quantumcomputer, in a web site (in Japanese). In the second lecture (Quantum teleportation, doneby Eisuke Abe, Keio University on November 15, 2009), the quantum circuit for thequantum teleportation circuit is discussed in the association with the SWAP circuit. Wewere very impressed with a possible equivalence of the quantum circuits between thequantum teleportation and the SWAP. Note that unfortunately, these lectures were done inJapanese.Here we will show that the quantum circuit of the SWAP circuit is essentiallyequivalent to that of the quantum teleportation. This lecture note is mainly written in orderto reproduce the content of lectures given by Eisuke Abe. In other words, there is nothingnew in this note. Nevertheless, we think that the content of this lecture note may be veryuseful to undergraduate students and graduate students who want to know about theprinciple of the quantum entanglement [quantum teleportation among three people Alice(A), Bob (B), and Charlie (C)]. The information of Charlie (or state) is delivered to Bob,immediately after the Bell state shared with Alice and Charlie is observed by Alice.Here we discuss the quantum circuits of the quantum teleportation and the swap basedon the lecture. We show the similarity of the quantum circuits between the quantumteleportation and SAP. In this note, we first discuss the fundamental circuits in quantumcomputer, in particular, various kinds of equivalent circuits.1

1.Quantum bitsIn quantum computing, a qubit or quantum bit is the basic unit of quantuminformation—the quantum version of the classical binary bit physically realized with atwo-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one ofthe simplest quantum systems displaying the peculiarity of quantum mechanics. Examplesinclude: the spin of the electron in which the two levels can be taken as spin up z 0and spin down z 1 ; or the polarization of a single photon in which the two states canbe taken to be the and the horizontal polarization x and the vertical polarization y . Ina classical system, a bit would have to be in one state or the other. However, quantummechanics allows the qubit to be in a coherent superposition of both states simultaneously,a property which is fundamental to quantum mechanics and quantum computing. Here weuse the Dirac notation, the eigenkets 0 and 1 ; 1 0 z , 0 0 1 z . 1 https://en.wikipedia.org/wiki/QubitThe quantum circuits consist of various kinds of quantum gate (unitary operators, Paulioperators, and so on). These gates are reversible in the processes. Here we discuss thequantum circuits of the quantum teleportation and swap, as typical examples.2.Quantum NOT: Pauli spin 1/2 operators 0 1 X corresponds to the Pauli matrix; X ˆ x . We note that 1 0 X a a ,X 0 0 1 ,X 1 1 0 . 0 i Y corresponds to the Pauli matrix ˆ y . i 0 Y a ( 1)a i a . 1 0 Z corresponds to the Pauli matrix ˆ z , 0 1 2

Z a ( 1)a a ,Note thatX 2 Y 2 Z 2 1.Commutation relations:3.[ X , Y ] 2iZ ,[Y , Z ] 2iX ,[Z , X ] 2iYHadamard gateHadamard gate is expressed by H in the quantum circuit.Hadamard gate H.Fig1.1 1 1 1Uˆ x H ( ˆ z ˆ x ) ; 2 1 1 2H 0 0 12,H 1 0 12.In general, we haveH a 1 11( 1)ab b [ 0 ( 1)a 1 ] , 2 b 02H 0 1 1 b ,2 b 0H1 1 1( 1)b b . 2 b 0Note thatH2 1HXH Z ,HYH Y ,X a a a 1 ,HZH X ,Y a ( 1)a i a ,3Z a ( 1)a a .

X, Y, and Z are the Pauli matrices of 2x2. The detail of the properties is presented in theAPPENDIX.3.1.Proof of HXH ZHXH a HX1 1 ( 1)ab b2 b 01 1( 1)ab X b 2 b 01 1 H( 1)ab b 2 b 01 H[ 0 ( 1)a 1 ]21 H[ 1 ( 1) a 0 ]21 ( 1)a H[ 0 ( 1) a 1 ]2 ( 1)a H 2 a H ( 1)a aorHXH a ( 1)a a Z a .3.2Proof of HYH Y4

1 1( 1) ab bHYH a HY 2 b 01 1( 1) ab Y b H 2 b 01 1( 1)( a 1)b iH b 2 b 01 11 1( 1)bc c i ( 1)( a 1)b 2 b 02 c 01 i ( 1)( a 1)b bc c2 b,c11 i[1 ( 1)a ] 0 [1 ( 1)a ] 12211 [ 0 1 ] ( 1)a [ 0 1 ]223.3Proof of HZH XHZH a HZ H1 1( 1)ab b 2 b 01 1( 1)ab Z b 2 b 01 1( 1)( a 1)b H b 2 b 011 1( a 1) b 1 ( 1) ( 1)bc c2 b 02 c 01 ( 1)( a c 1)b c2 b ,c 11 [1 ( 1) a ] 0 [1 ( 1) a ] 12211 [ 0 1 ] ( 1) a [ 0 1 ]22 aorHZH a a X a .4.Controlled-U gate5

The controlled-U gate is defined bya b a U a b ,withFig.2Control bit:a ,Target bit:b ,Controlled U gate. Control ( a ). Target ( b ).We have another type of control-U gate where the target and the control are different fromthat in the above control-U gate.Fig.3Controlled X-gate. a : control. b : target. X a b .6

5.CNOT gateThe controlled NOT gate (CNOT) is a quantum logic gate that is an essentialcomponent in the construction of a gate-based quantum computer. It can be used toentangle and disentangle EPR states. Any quantum circuit can be simulated to an arbitrarydegree of accuracy using a combination of CNOT gates and single qubit rotations.https://en.wikipedia.org/wiki/Controlled NOT gateHere we use C12 as the CNOT gate.aaControlTargetbFig.4b aControlled Not gate (CNOT). a : control. b : target.a b a X a b a b a 1 0C12 0 00 0 0 0 0 0 ,0 0 1 0 1 0 where a : control. b : target.C217

aa bTargetControlbbCNOT gate: b : control. a : target.Fig.5a b a b b .where b : control. a : target.The matrix of C12 is 1 0C21 0 0000100100 1 .0 0 ((Note))H0000Fig.6Encode of the Bell state 00 using CNOT gate with control ( H 0 ) andtarget ( 0 .H 0 111[ 0 1 ] . 00 0 0 1 1 (Bell state).2228

Using the CNOT the input state of two unentangled qubits can be changed into anentangled state.Input:(110 1 ) 0 .22Output:C12 [6.11110 1 ) 0 ] 00 0 0 1 1 .2222SWAP:Dirac exchange operatora b a b a a b a b a b b a b b a b b aNotea a 0.Dirac exchange operator is defined by1Pˆ (1ˆ σˆ1 σˆ 2 ) ,2in terms of the Pauli operators. 1 0 0 0001001000 0 .0 1 The SWAP circuit is expressed by a circuit9

Fig.7(a)SWAP circuit.which is equivalent toabFig.7(b)b12aCircuit equivalent to SWAP circuit.using the CNOT circuits. The proof for this is given as follows.C12C21C12 a b C12C21 a b a C12 a b a b a C12 b b a b b a b b awhere b b 0 .7.Controlled Z gateThe controlled Z-gate is expressed by a circuit,10

Controlled Z gate with control ( a ) and target ( b . Z a b ( 1) ab b .Fig.8(a)aC bT a Z a b ( 1)ab a b ,which is equivalent to the upside-down circuit, where C is control and T is target.Controlled Z gate with control ( b ) and target ( a , which is equivalent toFig.8(b)Fig.8(a)sinceaT bC (Z b a ) b ( 1) ab a b .Here we note thatZ a b ( 1) ab b ,8.8.1Z b ( 1)b b .Base transformation for the Bell statesEncode of Bell state: x y xyEncode of Bell circuit is obtained using the combination (H-CNOT) circuits.11

Encode of Bell state xy . Base transformation ( x y xy by H-Fig.9CNOT gate.The transformationx y xycan be proved as follows.Step A:Application of H-gateHx y (H x ) y1 1( 1) xb b y 2 b 01 [ 0 y ( 1) x 1 y2 where1 11H x ( 1) xb b [ 0 ( 1) x 1 ] 2 b 02Step B:Application of CNOT gate12

C1211[ 0 y ( 1) x 1 y [ 0 y 0 ( 1) x 1 y 1221 [ 0 y ( 1) x 1 y2 xyor xy 1[ 0 y ( 1) x 1 y .2wherey 0 y,y 1 yor more directly, we have1 1( 1) xb C12 b y 2 b 01 1 ( 1) xb b y b 2 b 01 [ 0 y 0 ( 1) x 1 y 1 ]21 [ 0 y ( 1) x 1 y ]2C12 H1 x y xyNote that the Bell states (4 states) are defined by xy 1[ 0 y ( 1) x 1 y ,2where13

8.2 00 1 11 0 , [ 0 0 1 1 ] 22 0 1 01 0 11 1 , [ 0 1 1 0 ] 22 1 0 10 1 11 0 ,[ 0 0 1 1 ] 22 0 1 11 0 11 1 . [ 0 1 1 0 ] 22 1 0 Decode of Bell states: xy x yThe decode of the Bell state is obtained from the combination CNOT-H;Fig.10Decode of Bell state xy . Base transformation ( x y xy by HCNOT gate.1 xy [ 0 y ( 1) x 1 y2Step A.CNOT gate ( C12 )14

C121[ 0 y 0 ( 1) x 1 y 1 ]21 [ 0 y ( 1) x 1 y ]21 [ 0 ( 1) x 1 ] y ]2 (H x ) y xy Step B.H gateH(H x ) y (H 2 x ) y x ysince H 2 1 .9.Equivalent quantum circuitsHere, we discuss several equivalent-circuits. These equivalent circuits are essential tothe simplification of quantum circuits for the quantum teleportation and the swap circuit.9.1Equivalent quantum circuits-1: Controlled X - CNOTThe following two circuits are equivalent.Fig.11(a)Quantum circuit-1C12 X 2 a b C12 a b 1 a a b 115

whereb b 1This circuit is equivalent toFig.11(b)Quantum circuit equivalent to Fig.11(a).X 2C12 a b X 2 a a b a a b 19.2Equivalent quantum circuits-2The following two circuits are equivalent.Fig.12(a)Quantum circuit-2C12 Z 2 a b ( 1)b C12 a b ( 1)b a a b16

Fig.12(b)Quantum circuit equivalent to Fig.12(a).Z1Z 2C12 a b Z1 Z 2 a a b ( 1)a a b a a b ( 1)b a a b9.3Equivalent quantum circuits-3The following two circuits are equivalent.Fig.13(a)Quantum circuit-3.C12 X 1 a b C12 a 1 b a 1 a b 117

Fig.13(b)Quantum circuit equivalent to Fig.13(a).X 1 X 2C12 a b X 1 X 2 a a b a 1 a b 19.4Equivalent quantum circuits-4The following two circuits are equivalent.Fig.14(a)Z-gate and CNOT. Quantum circuit-4. 1: control. 2: target. Input: a b .Output: ( 1) a a a b .Z1C12 ( a b ) Z1 a a b ( 1) a a a b18

Fig.14(b)CNOT and Z-gate. Quantum circuit equivalent to the quantum circuit -4 ofFig.14(a). 1: control. 2: target. Input: a b . Output: ( 1) a a a b .Figs.14(a) and (b) are equivalent.C12 Z1 ( a b ) ( 1) a C12 a a ( 1) a a a b9.5Equivalent quantum circuits-5The following two circuits are equivalent.Fig.15(a)Controlled Z gate. Quantum circuit-5. Control gate ( a ) and target gate( b ). 1: control ( a. 2: target ( b ) . Input: a b . Output:( 1) ab a b .a Z a b a ( 1) ab b ( 1)ab a b19

Fig.15(b)Controlled Z gate. Control gate ( b ) and target gate ( a ).Z b a b ( 1) ab a b .Figs.15(a) and 15(b) are equivalent. CZ gateis non-local, independent of the choice of control and target.9.6Equivalent quantum circuit-6The following two circuits are equivalent.Fig.16(a)Quantum circuit-6, with 4 H-gates and 1 CZ-gate. Control gate ( a ) andtarget gate ( b ). Input: a b . Output: a b b .This quantum circuit is equivalent to the following circuit.ab20

Fig.16(b)CNOT gate. Quantum circuit equivalent to Fig.16(a). Control gate ( b ) andtarget gate ( a ). The input: Input: a b . Output: a b b .Step-1:H a 1 1 ( 1) aa ' a ' ,2 a ' 0H a H b H b 1 1 ( 1)bb ' b '2 b ' 01 1 ( 1)aa ' bb ' a ' b '2 a ',b ' 0Step-2:C12 ( H a H b ) 1 1 ( 1)aa ' bb ' C12 a ' b '2 a ',b ' 01 1 ( 1)aa ' bb ' a ' b ' a '2 a ',b ' 01 [ 0 0 0 ( 1)b 0 1 02 ( 1)a 1 0 1 ( 1)a b 1 1 1 ]1 [ 0 0 ( 1)b 0 12 ( 1)a 1 1 ( 1)a b 1 0 ]or1C12 ( H a H b ) { 0 [ 0 ( 1)b 1 ]2 ( 1) a b 1 [ 0 ( 1) b 1 ]}1 { 0 [ 0 ( 1)b 1 ]2 ( 1) a b 1 [ 0 ( 1)b 1 ]}1 [ 0 ( 1)a b 1 ] [ 0 ( 1)b 1 ]2 H a b H bStep-3:21

H2 a b H2 b a b bsince H 2 1 .10.10.1Quantum teleportationPrinciple of quantum teleportationQuantum teleportation. The detail of this Figure is given in the section ofQuantumteleportation http://bingweb.binghamton.edu/ suzuki/QuantumMechanicsFiles/103 Quantum teleportation.pdfFig.17 CZ X 0 1 : CCthe state of Charlie Z 0 Z 1 0 1 X 0 X 1 1 0 0 122

XZ C X 0 X 1 1 0 xy : 00ABBell state (EP pair; Alice and Bob) 1 11 0 [0 0 1 1 22 0 1 (the Bell state shared by Alice and Bob)It is shown that C 00AB11 00 CA B 01 CA X B2211 10 CA Z B 11 CA XZ 2211 xy CA X y Z x B2 x , y 0 where C 00AB 0 0 1 1 0 1 2 0 2 0 1 0 which is equal to23B

00 0 0 0 0 0 00000 00 00 2 002 2 002 where 00 01 10CACACA B X Z 0 1 1 0 1 0 2 0 2 0 0 1 BB 0 0 0 1 1 1 2 1 2 0 0 0 0 1 1 0 1 0 2 0 2 0 0 1 24

11 CCA XZ 00ABB 0 0 0 1 1 1 2 1 2 0 0 0 11 00 CA B 01 CA X B2211 10 CA Z B 11 CA XZ 2211 xy CA X y Z x B2 x , y 0 Suppose that Alice measures the state x ' y 'CABshared with Charlie. Immediately, thesystem collapses, and Bob can measure the stateX yZ x B 00C,AB 1 1 xy2 x, y 0 1 1 xy2 x , y 0CACA ( Z x X y )( X y Z x ) BImmediately after Alice measures the Bell state xyxyCA10.2 B. So, Bob gets the state BB.Quantum circuit of quantum teleportation (I)25CA, the system collapses into

Fig.18Quantum teleportation circuit (Alice, Bob, and Charlie). The box B denotesthe generation of the Bell state. The box M denotes the measurementFig.19Equivalent quantum circuit of quantum teleportation, with encode of Bellstate 00 and decode of the Bell state.Fig.20The quantum circuit of quantum teleportation. Within the green box, the Hgate can be shifted to the right, while the X-gate can be shifted to the left.The output is not influenced by these shifts.26

After shifting the Hadamard gate to the right in Charlie-channel, equivalently, we haveFig.21The circuit as input with the states a (Charlie), 0 (Alice), and 0 (Bob).We now discuss the response of this circuit when the states a (Charlie), 0 (Alice),and 0 (Bob) are given as input from the right.Fig.22Quantum teleportation circuit. The output after step-G is1( 00 CA 01 CA ) a B . Immediately after Alice measures one of the2Bell states ( 00CA, 01CA), the state of the system collapses, leading thestate of Bob into the state a B . In other words, the state aCof Charlie istransferred to Bob as the state a B .We will show that this quantum circuit is similar to the circuit of SWAP, by each step.Step-A:27

aC 0A 0(as input)BStep-B:1a ( 0 1 ) 021 [a 0 0 a 1 0]2a (H 0 ) 0 Step C:C2311[ a ( 0 0 ) a ( 1 0 )] [a 0 0 a 1 1]22Step-D:11C12 [( a 0 ) 0 ( a 1 ) 1 ] [ a a 0 a a 1 1 ]22Step-E:11[ a [ a X a 0 a 1 X a 1 1 ] [ a [ a a a 1 a 1 1 ]22 a H 0 a ]wherea a 1 X a 0 X a 1 2X aH 0 2H 0X a H 0 HZ a 0 H 0Step-F:H a H 0 a 1[ 0 ( 1)a 1 ] H 0 a228

whereH a 1[ 0 ( 1) a 1 ]2Step-G:1[ 0 H 0 Z 0 a ( 1) a 1 H 0 Z 1 a21[ 0 H 0 a ( 1)a 1 H 0 ( 1)a a 21 [ 0 H 0 a 1 H 0 a21 [ 0 0 0 1 1 0 1 1 ) a21 0 0 1 11 0 1 1 0) a a222211 00 a 01 a22oroutput 1( 00 01 ) a .2( 1)2 a 1 , and the Bell states are defined aswhere 00 1 11 0 [ 0 0 1 1 ] 22 0 1 01 0 11 1 , [ 0 1 1 0 ] 22 1 0 29

Finally, Alice can measure either the state 00CAor 01CAwith the probability of 50 %.After the measurement by Alice, the state collapses. As a result, Bob measures the statea , independent of her choice of the state.11.More simplified circuit of quantum teleportation (I)We start with the quantum circuit of quantum teleportation which is previous derived.Fig.23(a)The same figure as Fig.22Fig.23(b)The change of the location of the input from the position A to B. The inputat the position A is a C H 0 A 0 B .The input is expressed by the state, aaC H 0AC 0A 0 B . The state at the line B is given by 0 B . The circuit after the line B can be rewritten as30

Fig.23(c)The change of location between H and X withing the green box. The elementH shifts to the left, while the element X shifts to the right, without any effecton the output.There are H and CX within enclosed by the green box. It is clear that the output at the lineF is not influenced by the shift of H to the left and the shift of CX to the right within thegreen box. The output at the line G is expressed by the stateFig.23 (d)Quantum circuit of quantum teleportation. The output at the line G is given1( 00 01 CA ) a B . Immediately after Alice measures the BellbyCA2states (either 00CAor 01CA), the state of Bob collapses to the state a B .12.Swap circuit12.1 Original swap circuitWe consider the following SWAP circuit. Is this another quantum teleportation? In theinput, the states of Bob and Charlie are 0 B and C , respectively. Alice is not involvedin the process. There is also no classical communication, unlike the quantum teleportation.31

Fig.24 (a)SWAP circuit. Alice is not involved in this process. The result isindependent of the choice of the state of Alice.Fig.24(b)Equivalent circuit of SWAP, where the SWAP is replaced by CNOTs.32

The same circuit as Fig.26(b), with input ( a for Charlie, for Alice b ,Fig.25and 0 for Bob).Step-A:Step-B:C12 a b 0 a a b 0C23 a a b 0 a a b a bStep-C:C12 a a b a b a a a b a b a b a ba a 0whereStep-D:C23 a b a b a b b a b a b aStep-E:C31 a b a a a b a 0 b a33

Fig.26In SWAP, the circuit elements E1 , E2 , and E3 can be replaced by thecorresponding equivalent circuits.12.2 Use of the equivalent circuitsThe following two circuits are equivalent.Fig.27Controlled Z gate. Z a b ( 1) ab b .Z a b ( 1) ab b ,leading to ( 1) ab a bwhich is equivalent to a circuit (controlled Z gate)34

Fig.28Z b a ( 1) ab a . CZ (controlled Z) is nonlocal, independent of the choiceof the control and the target.Z b a ( 1) ab aFig.29leading to ( 1) ab a baaba bca b cThe circuit of Fig.31 is equivalent to that in Fig.32.a a b a b cwhich is equivalent to35

Fig.30The circuit of Fig.32 is equivalent to that in Fig.31.sincea a b X a b c a a b a b c ,whereX 1 c c c 1X a b c c a b12.3Modified swap circuit (which is equivalent to the original SWAP36

Fig.31(a)The staring quantum circuit (which is the same as Fig.28.which is equivalent to the following circuit.The elements E1 and E3 are replaced by thecorresponding equivalent circuits as follows.Fig.31 (b)The element E2 is replaced by the corresponding equivalent circuit as follows.Fig.31 (c):Controlled Z is nonlocal in the elements E4. Even if the location of thecontrol and target is changed, the role of CZ remains unchanged.The element E4 is replaced by the corresponding equivalent circuit as follows.Fig.31(d)37

It is obvious thatFig.31(e):For the H and CX in the green box, with on change of the circuit, H can beshifted to the left, while CX can be shifted to the right within the box.is equivalent toFig.31(f):The circuit after shifting H and CX within the green box. The input: a forCharlie. The input: 0 for Bob. For simplicity, we choose the input H 0for Alice. This input is arbitrary. H 0Fig.32A 1[02The equivalent quantum circuit for the SWAP.38A 1 A] .

We check the result after each step.Step-A:a 0 0Step-B:a Za 0 0 a 0 0(input)(the control Z gate is actuallynecessary)Step-C:1[ a ( 0 1 ) 0 ]21 [ a00 a10 ]2a H 0 0 Step-D:11a C23 ( 00 10 ) a ( 00 11 )22Step-E:11( a, a 0 a, a 1 1 )C12 ( a 00 a11 ) 221( a, a 0 a, a 1 ) 2Step-F:12(H a a 0 H a a 1 )1( 1)aa ' [ a ' a 0 a ' a 1 ] 2 a'11 0 a 0 0 a 12211 ( 1) a 1 a 0 ( 1) a 1 a 122 Step-G:39

110 a X a 0 0 a 1 X a 1 12211 ( 1)a 1 a X a 0 ( 1)a 1 a 1 X a 1 12211 0 a a 0 a 1 a 22211 ( 1)a 1 a a ( 1)a 1 a 1 a 22211 0 a a 0 a 1 a2211 ( 1)a 1 a a ( 1)a 1 a 1 a22Fig.33Step-H:110 a Z 0 a 0 a 1 Z 0 a2211 ( 1) a 1 a Z a ( 1) a 1 a 1 Z a2211 0 a a 0 a 1 a2211 ( 1) 2 a 1 a a ( 1) 2 a 1 a 1 a2211 0 a a 0 a 1 a2211 1 a a 1 a 1 a2240

Step-I:11H 0 a a H 0 a 1 a2211 H 1 a a H 1 a 1 a2211 [H 0 H 1 ] a a [H 0 H 1 ] a 1 a2211 0 a a 0 a 1 a2211 0 [a a 1 ] a22Here we note that111a a 1 ] X a[0 1]222 X aH 0 H 0sinceX a 1 0 a 1 .Xa 0 a ,Using the relationsHXH Z ,H 2 1,we get( HXH )a ( HXH )( HXH ).( HXH ) HX a HorZ a ( HXH ) a HX a H ,or41

HZ a X a H .Thus, we haveX a H 0 HZ a 0 H 0 .So the output of the above quantum circuit is0 H 0 a .This is the equivalent circuit of swap. The final result is as follows.Fig.34SWAP. The output: 0 C , H 0 A , and a B . The input: aC, H 0 A , and0 B.Even if the box with green can be removed from the circuit, both the input and outputremain unchanged.Fig.35No change occurs to the circuit upon the removal of the circuit elementenclosed by the green box.Note that42

Checking the possibility of removing H-Z-H gate.Fig.36Input:a H 0Step-A:a H2 0 a 0Step-B:a Za 0 a 0Step-C:a H 0So that the output is the same as the input. If we use the input a H 0 , the H-Z-H gatecan be removed.Fig.37The output after the step E;1[ 002CA 01Alice measures one of the Bell states ( 00CACA] a . Immediately after, 01CAcollapse into the state a . The output after the step F; 0Input:43), the state of BobC H 0A a B.

1a ( 0 1 ) 021 [ a 0 0 a 1 ) 0 ]2a H 0 0 Step-A:11C23[ a 0 0 a 1 ) 0 ] [ a 0 0 a 1 ) 1 ]220 0 1 1 a 2 a 00where 00AB 0 0 1 ) 1(Bell state)2Step-B:1C12 [ a 0 0 a 1 ) 1 ]21[ a a 0 a a 1 ) 1 ] 2Step-C:1H a [ a 0 a 1 1 ]21 ( 1)aa ' a ' [ a 0 a 1 ) 1 ]2 a'1 [ 0 ( 1)a 1 [ a 0 a 1 ) 1 ]211 0 a 0 0 a 1 ) 12211 ( 1) a 1 a 0 ( 1) a 1 a 1 ) 12244

Step-D:110 a X a 0 0 a 1 ) X a 1 12211 ( 1) a 1 a X a 0 ( 1)a 1 a 1 ) X a 1 12211 0 a a 0 a 1 ) a 1 12211 ( 1) a 1 a a ( 1)a 1 a 1 ) a 1 12211 [ 0 a a 0 a 1 ) a2211 ( 1) a 1 a a ( 1)a 1 a 1 ) a22Step-E:110 a Z 0 a 0 a 1 )Z 0 a2211 ( 1)a 1 a Z a ( 1)a 1 a 1 ) Z a2211 0 a a 0 a 1 ) a2211 ( 1)2 a 1 a a ( 1)2 a 1 a 1 ) a2211 0 a a 0 a 1 ) a2211 1 a a 1 a 1 ) a22where110 ( a a 1 ) a 1 ( a a 1 ) a2211 0 H 0 a 1 H 0 a22 H 0 H 0 a 1[ 002CA 01CA] a45

Note that1H 0 H 0 [( 0 1 ) ( 0 1 )]21 [0 0 0 1 1 0 1 1]20 1 1 01 0 0 1 1 [ ]2221 [ 00 CA 01 CA ]2Step-F:H2 0 H 0 a 0 H 0 aor1[ 0 [ 0 1 ] a211 0 0 a 0 1 ] a220 H 0 a Note that1( a a 1 ) H 0 .213.SummaryUsing the equivalent circuits of the quantum computer, it is shown that the SWAPcircuit is equivalent to the quantum circuit of the quantum teleportation. The equivalentcircuit for the quantum teleportation and the SWAP is obtained as46

Equivalent circuit both for the quantum teleportation and the SWAP.Fig.38Immediately after one of the Bell states ( 00CAor 01CA) shared with Alice and Charlie,is measured by Alice, the system, collapses. Bob observes the state a B , which is the samestate which Charlie has as an input ( aC. Note that a is any linear combination of qubits0 and 1 .REFERENCESC. Bernhardt, Quantum Computing for Everyone (MIT Press, 2019).B. Zygelman, A First Introduction to Quantum Computing and Information (Springer,2018).M.A. Nielsen and I.L. Chuang, Quantum Computation, 10th Anniversary Editions(Cambridge, 2010).N.D. Mermin, Quantum Computer Science: An Introduction (Cambridge, 2007).M.L. Bellac, Quantum Physics (Cambridge, 2006).Eisuke Abe, Class of Quantum Computer #2: Quantum Teleportation (November, 15,2009), Keio University. https://www.youtube.com/watch?v mose-W49uF8(in Japanese).APPENDIX-AFundamental propertiesa 0ora 1a a 0,,a a 1a 0 a,a 1 a .( a (a 1) (a a) 1 1 .a 1 (a 1) 1 a 0 a .47

H a 1 1( 1) ab b , 2 b 0H a 1( 1) aa ' a ' 2 a'H 1(X Z) .21 ( 1)aa ' Z b a '2 a'1( 1)( a b ) a ' a ' 2 a' H a bZ bH a HZ b H a H 2 a b a b .X a a a 1 .Y a ( 1)a i a ( 1) a i a 1 .Z a ( 1)a a .X a b a b .Z a b ( 1) ab b .H 2 1.HXH Z ,HYH Y ,HZH X .HX a H Z a , HY a H ( )a Y a ,HZ a H X a .48

111a a 1 ] X a[0 1]222 X aH 0 HZ a 0. H 0 1[0 1]2C12 a b a b a .X 2 Y 2 Z 2 1.[ X , Y ] 2iZ ,Kronecker product:[Y , Z ] 2iX ,[Z , X ] 2iY .H 0 H 0 ( H H )( 0 0 ) .H 1 1 1 ,2 1 1 1 0 . 0 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 H H 1 1 1 1 0 .0 0 000 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 .H 0 H 0 2 1 1 1 1 0 2 1 1 1 1 1 0 1 49

Pauli matrix: 0 i Y , i 0 0 1 X , 1 0 X 0 1 , 1 0 Z . 0 1 X 1 0 ,Y 0 i 1 , Y 1 i 0 ,Z 0 0 ,Z 1 1 . 1 0 , 0 0 1 . 1 Qubits:APPENDIX BB-1 Another example of quantum teleportation (II)Here we consider the following simple quantum circuit (as a simple example). We findthis example from a book [C. Bernhardt, Quantum Computing for Everyone (MITPress, 2019)].Quantum teleportation. Alice, Bob, and CharlieFig.B1 C a 0 b 1 ,(the initial state of Charlie)50

00AB 1[02A 0B 1 A 1 B](Bell state between Alice and Bob)Quantum circuit corresponding to Fig.B1.Fig.B2Step-A: C 00AB [a 0 b 1 ]C 1( 00 11 ) AB21[a 000 a 011 b 100 b 111 ]21 [a 00 0 a 01 1 b 10 0 b 11 1 ]2 Step-B:CNOT between channels 1 (Charlie) and 2 (Alice):1[aC12 00 0 aC12 01 1 bC12 10 0 bC12 11 1 ]21 [a 00 0 a 01 1 b 11 0 b 10 1 ]21 [a 0 00 a 0 11 b 1 10 b 1 01 ]21 [a 0 ( 00 11 ) b 1 ( 10 01 )]2where we use the relation C12 ( a 1 b 2 ) a 1 a b512

Step-C:Hadamard gate in the channel-1 (Charlie)1aH 0 ( 00 11 ) 1bH 1 ( 10 01 ]2211 a( 0 1 ) ( 00 11 ) b( 0 1 ) ( 10 01 ]221111 00 a 0 01 a 1 10 a 0 11 a 122221111 01 b 0 00 b 1 11 b 0 10 b 1222211 00 [a 0 b 1 ] 01 [a 1 b 0 ]2211 10 [a 0 b 1 ] 11 [a 1 b 0 ]22whereH 0 1[ 0 1 ],2H1 1[0 1]2Alice now measures her two particles in the standard basis. She will get one of 00 ,01 , 10 , and 11 , each with probability 1/4.If she gets 00 , Bob’s qubit will jump to state:a 0 b 1 .If she gets 01 , Bob’s qubit will jump to state:a 1 b 0 .If she gets 10 , Bob’s qubit will jump to state:a 0 b 1If she gets 11 , Bob’s qubit will jump to statea 1 b 0 .Alice and Bob want Bob’s qubit to be in the state a 0 b 1 . It is almost there, but notquite. To sort things out, Alice has to let Bob know which of the four possible situationshe is in. She sends Bob two classical bits of information, 00, 01, 10, or 11, correspondingto the results of her measurements, to let him know. These bits of information can be sentin any way, by text (by classical communication).52

If Bob receives 00 from Alice, he knows that his qubit is in the correct form a 0 b 1and so does nothing.Z 0 X 0 [a 0 b 1 ] a 0 b 1 .If Bob receives 01 from Alice, he knows that his qubit is a 1 b 0 . He applies thegate X to it.Z 0 X 1[a 1 b 0 ] aX 1 bX 0 a 0 b 1.If Bob receives 10 from Alice, he knows that his qubit is a 0 b 1 . He applies thegate Z to it.Z 1 X 0 [a 0 b 1 ] aZ 0 bZ 1 a 0 b 1.If Bob receives 11 from Alice, he knows that his qubit is a 1 b 0 . He applies the gateZX to it.Z 1 X 1[a 1 b 0 ] Z [aX 1 bX 0 ] Z [a 0 b 1 ]. a 0 b 1In every case, Bob’s qubit ends in state a 0 b 1 , the original state of the qubit that Alicewanted to teleport. It is important to note that there is only one qubit in state a 0 b 1 atany point during the process. Initially, Alice has it. At the end Bob has it, but as the nocloning theorem tells us, we cannot copy, so only one of them can have it at a time. It isalso interesting to observe that when Alice sends her qubits through her circuit Bob’s qubitinstantaneously jumps to one of the four states. He has to wait for Al

1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.

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