Knotted Pictures of Hadamard Gate and CNOT Gate

Commun. Theor. Phys. (Beijing, China) 51 (009) pp. 967 97 c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 6, June 15, 009 Knotted Pictures of Hadamard Gate and CNOT Gate GU Zhi-Yu 1 and QIAN Shang-Wu 1 Physics Department, Capital Normal University, Beijing 100037, China Physics Department, Peking University, Beijing 100871, China (Received July 7, 008) Abstract This paper obtains the knotted pictures of Hadamard gate and CNOT gate in terms of surgical operations described in knot theory. PACS numbers: 0.10.Kn, 03.65.Ud, 03.67.Hk Key words: two qubit quantum logic gate, surgical operations in knot theory, knotted picture 1 Introduction In our previous articles we have found knotted pictures of the four Bell bases and the m GHZ states and the knotted picture of Pauli operators acting on the four Bell bases, [1 7] furthermore we have obtained the knotted picture of unitary transformation applied to single particle quantum state [8] and the knotted picture of the unknown quantum state in the process of quantum teleportation. [9] Since any unitary evolution can be accomplished via universal quantum logic gates, [10] we have further studied the single qubit quantum logic gate U(α, φ) encountered in the discussion of quantum teleportation. [11] The matrix expression of the single qubit gate U(α, φ) is ( ) cos α i exp( iφ) sinα. i exp(iφ) sinα cos α When we take α = arccosa, φ = π/, then we have U A = U (arccosa, π/) and we can obtain single qubit state φ A by simply operating U A on the basic vector A, i.e. U A A = cosα A i exp(iφ) sinα A = a A + b A = φ A. (1) In terms of the surgical operations described in the Ref. [11], we have obtained the knotted picture of the single qubit gate U A, i.e. {U A } {U A } = D A tc A. () Where c A is the cutting operation operated on { φ A }, the knotted picture of the single particle quantum state φ A, t is the twisting operation, and D A is the denominator operation. In the process of quantum teleportation, suppose we choose the quantum channel state to be Φ + 1, then before any measurement, the entire system, comprising Alice s unknown particle A and the EPR pair (particle 1 and particle ), is in a pure product state φ A Φ + 1. The process of teleportation composes of two parts: Alice s measurement on particle A and particle, abbreviated as M(A, ); Bob s measurement on particle 1, abbreviated as M(1). Actually this process is a process of re-entanglement, it is equivalent to the expansion of product state φ A Φ + 1 in terms of the four Bell bases Bell i A, for which Bell 1 A = Φ+ A, Bell A = Φ A, Bell 3 A = Ψ+ A, Bell 4 A = Ψ A, i.e. 4 φ A Φ + 1 = Bell i A φ i 1. (3) i=1 In order to get the knotted picture of the process of teleportation, we must get the knotted pictures of the measurements M(1) and M(A, ). M(1) can be carried out by the single qubit gate U(α, φ), in Ref. [11] we have already obtained the knotted picture {U A }, which is shown by Eq. (). For the measurement M(A, ) we need to use Hadamard gate and two qubit CNOT gate. This paper will discuss the knotted pictures of these two quantum logic gates. In Sec. we shall discuss Hadamard gate and CNOT gate and the obtainment of quantum states Bell i A by the combined application of Hadamard gate and CNOT gate on the four basic two qubit states,,, and. In Sec. 3 we shall give the knotted pictures of these basic two qubit states, they are trivial knots: two concentric oriented circles, and gives the process of trivialization of the knotted picture of Bell i A, i.e. finds the surgical operations such that after the applications of these surgical operations, the knotted picture of Bell i A becmes two concentric oriented circles. In Sec. 4 we shall compare the results obtained from Sec. with the results obtained from Sec. 3 to obtain the knotted pictures of the Hadamard gate and CNOT gate. Obtainment of Quantum state Bell i A.1 Hadamard Gate (i) Matrix expression The matrix expression of Hadamard gate is H = 1 ( 1 1 1 1 ). (4) (ii) Application of H on basis bectors The matrix expressions of basis vectors and are ( ) ( ) 1 0 =, =. (5) 0 1

968 GU Zhi-Yu and QIAN Shang-Wu Vol. 51 Application of H on and gives H = 1 ( ) ( ) 1 1 1 1 1 0 = 1 ( ) 1 = 1 ( + ), 1 H = 1 ( ) ( ) 1 1 0 1 1 1 = 1 ( ) 1 = 1 ( ). (6) 1 Since we only apply Hadamard gate to the controlled qubit hereafter we use the symbol H c instead of H to represent the Hadamard gate.. CNOT Gate (Controlled-NOT Gate) (i) Two qubit state CNOT gate is a two qubit gate, a two qubit state is represented by c, w, the left qubit is called controlled qubit, the right qubit is called worked qubit. The value of qubit is taken to be zero and one for and respectively. Therefore, for the four basic two qubit states we have = 0, 0, = 1, 0, = 0, 1, = 1, 1. (7) Bell bases Bell i A are two qubit states, in which A corresponds to controlled qubit, corresponds to worked qubit, hence we can rewrite Bell i A as Bell i cw. (ii) Action of CNOT gate The result of the action of CNOT gate on a two qubit state c, w is c, w c, c w, (8) where the operation means mod addition, i.e., 1 1 = 0. Hence, when the value of the controlled qubit equals to zero, the action of the CNOT gate does not change the value of the worked qubit, i.e., 0, w 0, 0 w = 0, w, hence we have CNOT = CNOT 0, 0 = 0, 0 0 = 0, 0 =, CNOT = CNOT 0, 1 = 0, 0 1 = 0, 1 =, (9) i.e., when the controlled qubit is, CNOT gate has no effect on the two qubit state c, w. On the contrary, when the value of the controlled qubit equals to one, the action of the CNOT gate does change the value of worked qubit, i.e., 1, w 1, 1 w, i.e., the value of worked qubit change from zero to one or from one to zero, therefore we have CNOT = CNOT 1, 0 = 1, 1 0 = 1, 1 =, CNOT = CNOT 1, 1 = 1, 1 1 = 1, 0 =. (10) The action of CNOT gate can be schematically represented by Fig. 1. Since CNOT gate only changes the value of the worked qubit, the value of worked qubit depends on the value of the controlled qubit, hence hereafter we use the notation CNOT w(c) instead of CNOT. Fig. 1 Schematic diagram of the action of CNOT gate..3 Combined Action of Hadamard Gate H c and CNOT w(c) Gate Now we shall consider the combined action of Hadamard gate and CNOT gate on the two qubit state c, w. Firstly we apply H c, then we apply CNOT w(c) gate to the resulted two qubit state H c c, w, i.e., we obtain CNOT w(c) H c c, w, the combined action of Hadamard gate H c and CNOT gate CNOT w(c) on the two qubit state c, w can be schematically represented by Fig.. Fig. Combined action of Hadamard gate and CNOT gate on the two qubit state c, w. From Eqs. (6) (8) we readily see CNOT w(c) H c cw = CNOT w(c) 1 ( + ) c w Similarly we have = CNOT w(c) 1 ( + ) cw = 1 ( + ) cw = Φ + cw = Bell 1 cw. (11) CNOT w(c) H c cw = 1 ( ) cw = Φ cw = Bell cw, (1) CNOT w(c) H c cw = 1 ( + ) cw = Ψ + cw = Bell 3 cw, (13) CNOT w(c) H c cw = 1 ( ) cw

No. 6 Knotted Pictures of Hadamard Gate and CNOT Gate 969 = Ψ cw = Bell 4 cw. (14) We can introduce an unified expression c, w i to represent four basic two qubit states cw, cw, cw, and cw, i = 1 corresponds to cw, i = corresponds to cw, i = 3 corresponds to cw, i = 4 corresponds to cw. Using the unified expressions c, w i and Bell i cw we can write Eqs. (11) (14) with the following one equation: CNOT w(c) H c c, w i = Bell i cw. (15) Using the language of knot theory, we can write the Eq. (15) in the following form: {CNOT w(c) }{H c }{ c, w i } = { Bell i cw}. (16) where {CNOT w(c) } and {H c } are the knotted pictures of CNOT w(c) gate and H c gate respectively, { c, w i } and { Bell i cw} are the knotted pictures of the basic two qubit states c, w i and the Bell bases Bell i cw respectively. We have already found { Bell i cw} in Ref. [], which are four oriented links of the linkage 4 1 in knot theory. Since {CNOT w(c) } depends on the controlled qubit, hence we must consider {CNOT w( c )} and {CNOT w( c )} for the cases c and c separately. 3 Trivialization of Knotted Picture of Bell i cw 3.1 Knotted Picture of Four Basic Two qubit States c, w i In Ref. [9] we have pointed out that for single qubit state, the knotted pictures of two basic states and are the circles with counter clockwise and clockwise orientations respectively, naturally we shall use two disconnected circles to represents four basic two qubit states c, w i. We let the inner circle represents the controlled qubit, and let the outer circle represent the worked qubit, counter clockwise orientattion corresponds to the qubit with value zero, i.e., whereas the clockwise orientation corresponds to the qubit with value one, i.e.. Thence we readily obtain the knotted pictures { c, w i } of the four basic two qubit states shown in Fig. 3. Fig. 3 states. Knotted pictures of the four basic two qubit 3. Application of Cutting and Glue Operations The knotted picture of basic state is represented by { }, which is a circle with counter clockwise orientation, in Fig. 5 we use the notation T CR to represent trivial circle of radius R with counter clockwise orientation, the action of the cutting operator c = c left c right on the trivial circle T CR, i.e. { }, will yield two arcs with opposite directions P as shown in Fig. 4. Fig. 4 Action of the cutting operator on the trivial circle T CR. The action of the glue operator g = g left g right on P will restore to { }, as shown in Fig. 5, hence the operation g is the inverse operation of c, i.e. cg = I (identity), or c = g 1, g = c 1. Fig. 5 Action of glue operator on P.

970 GU Zhi-Yu and QIAN Shang-Wu Vol. 51 3.3 Application of twisting and Untwisting Operations The action of twisting operator t on two parallel vectors with the same direction P will yield a tangle T(, ) with two entrances and two exits as shown in Fig. 6. The repeated action of untwisting operator u on +P, i.e. u (+P), will restore to P as shown in Fig. 9. Fig. 9 Repeated action of untwisting operator on +P. Fig. 6 Action of twisting operator on two parallel vectors with the same direction. The action of untwisting operator u on T(, ), i.e. ut(, ), will restore to P as shown in Fig. 7, hence the operation u is the inverse operation of t, i.e. tu = I (identity), or t = u 1, u = t 1. 3.4 Application of Denominator Operation The action of denominator D on T(, ) is schematically shown in Fig. 10, D = g left g right = (c left c right ) 1 = c 1. Fig. 7 Action of untwisting operator on T(,). Fig. 10 Action of denominator operator on T(,). Fig. 8 Repeated action of twisting operator on two parallel vectors with the same direction. The repeated action of twisting operator t on P, i.e. t P, will yield a double arc coil with same direction +P, as shown in Fig. 8. 3.5 Trivialization of { Bell 1 A } = { Φ+ A } and { Bell 3 A } = { Ψ+ A } Now we shall use the surgical operations to trivialize the knotted picture { Bell 1 A } = { Φ+ A }, i.e. we shall apply the surgical operations to the 4 1 linkage { Bell 1 A } = { Φ+ A } = { Φ + cw }, such that it will become the two concentric circles, i.e. the knotted picture of the basic two qubit state. In Refs. [1,] we have already given the knotted pictures of the four Bell bases, which are shown in Fig. 11. Fig. 11 Correspondences of 4 1 linkages with the four Bell bases.

No. 6 Knotted Pictures of Hadamard Gate and CNOT Gate 971 The process of application of suitable surgical operations on { Φ + A } is shown in Fig. 1. Fig. 1 Process of application of surgical operations cu D on { Φ + A }. Firstly we apply the cutting operation c = c c A, after this operation the knotted picture { Φ + A } becomes two double arc coils (+P) up = up double arc coil and (+P) down = down double arc coil. Secondly we apply u to the A resulted (+P) up and (+P) down, after this operation we obtain two P for (+P) up and (+P) down respectively. Finally we apply D = D A D to the resulted two P, after this operation we obtain { A } = { cw }. Hence we have Similarly we have From Eq. (16) we have and D A D u Ac c A { Φ + A } = D A D u Ac c A { Bell 1 A} = { A }. (17) D A D u Ac c A { Ψ + A } = D A D u Ac c A { Bell 3 A} = { A }. (18) {CNOT w( c )}{H c }{ c, w 1 } = {CNOT w( c )}{H c }{ cw } = { Bell 1 cw} = { Φ + cw }, (19) {CNOT w( c )}{H c }{ c, w 3 } = {CNOT w( c )}{H c }{ cw } = { Bell 3 cw} = { Ψ + cw }. (0) From comparison of Eq. (17) with Eq. (19) or from comparison of Eq. (18) with Eq. (0), we can obtain {CNOT w( c )} and {H c }. 3.6 Trivialization of { Bell A } = { Φ A } and { Bell 4 A } = { Ψ A } Similar to Eq. (17) we can get D A D u Ac c A { Φ A } = { A }. (1) Similar to Eq. (18) we can get D A D u Ac c A { Ψ A } = { A }. () From Eq. (16) we have and {CNOT w( c )}{H c }{ c, w } = {CNOT w( c )}{H c }{ cw } = { Bell cw} = { Φ cw }, (3) {CNOT w( c )}{H c }{ c, w 4 } = {CNOT w( c )}{H c }{ cw } = { Bell 4 cw} = { Ψ cw }. (4) For comparison of Eq. (1) with Eq. (3) to get {CNOT w( c )}, we must further apply the inversion operator τ w = τ to the worked qubit of { cw } such that it becomes { cw }, i.e. τ D A D u Ac c A { Φ A }= τ { A }= { A }. (5) Now we can obtain {CNOT w( c )} by comparison of Eq. (3) with Eq. (5). 4 Obtainment of {H c }, {CNOT w( c )}, and {CNOT w( c )} 4.1 Obtainment of {H c } and {CNOT w( c )} From Eq. (17) we obtain { Φ + A } = (D A D u Ac c A ) 1 { A } = c 1 A c 1 (u A) 1 D 1 D 1 A { A} = D A D t Ac c A { A } ; (6)

97 GU Zhi-Yu and QIAN Shang-Wu Vol. 51 or D c D w t cw c wc c { cw } = { Φ + cw }. (7) Comparing Eq. (7) with Eq. (19) we readily obtain {CNOT w( c )}{H c } = D c D w t cwc w c c. (8) Since {H c } only applies on the controlled qubit, hence we have {H c } = D c c c. (9) Obviously the remained operation D w t cwc w related to the worked qubit is {CNOT w( c )}, i.e. {CNOT w( c )} = D w t cwc w. (30) Hence {H c } contains two operations, the first operation is cutting operation acting upon the knotted picture of the two qubit state, then after the finish of the operation {CNOT w( c )}, the second operation denominator acts upon the resulted picture. 4. Obtainment of {CNOT w( c )} From Eq. (5), noting that τw 1 = τ w, we get { Φ A } = (τ D A D u Ac c A ) 1 { A } Hence we have = c 1 A c 1 (u A) 1 D 1 D 1 A τ 1 { A} = D A D t Ac c A τ { A } = D A D t Ac τ c A { A }. (31) { Φ cw } = D c D w t cwc w τ w c c { cw }. (3) Comparing Eq. (3) with Eq. (3), we readily obtain CNOT w( c ){H c } = D c D w t cwc w τ w c c. (33) Since we already know that {H c } = D c c c, hence we get CNOT w( c ) = D w t cwc w τ w. (34) Thus we have successfully found {H c }, {CNOT w( c )}, and {CNOT w( c )}, i.e. the knotted pictures of quantum gates H c and CNOT w(c), these pictures should be used for describing the knotted picture of the measurement M(A,), i.e. Alice s measurement on particle A and particle in the process of teleportation. References [1] S.W. Qian and Z.Y. Gu, J. Phys. A: Math. Gen. 35 (00) 3733. [] S.W. Qian and Z.Y. Gu, Commun. Theor. Phys. 37 (00) 659. [3] S.W. Qian and Z.Y. Gu, Commun. Theor. Phys. 38 (00) 41. [4] S.W. Qian and Z.Y. Gu, Commun. Theor. Phys. 39 (003) 15. [5] Z.Y. Gu and S.W. Qian, Commun.Theor.Phys. 39 (003) 41. [6] S.W. Qian and Z.Y. Gu, Commun. Theor. Phys. 41 (004) 01. [7] Z.Y. Gu and S.W. Qian, Commun. Theor. Phys. 41 (004) 531. [8] Z.Y. Gu and S.W. Qian, Commun. Theor. Phys. 49 (008) 65. [9] Z.Y. Gu and S.W. Qian, Commun. Theor. Phys. 49 (008) 1163. [10] A. Barenco, et al., Phys. Rev. A 5 (1995) 3457. [11] Z.Y. Gu and S.W. Qian, Commun. Theor. Phys. 51 (009) 769.