On construction of quantum logical gate based on ESR

Abstract

A quantum computer is a computation device operated by means of quantum mechanical phenomena. There are many candidates that are being pursued for physically implementing the quantum computer.The quantum logical gate based on the electron spin resonance (ESR) was studied in ref. [3]. In this paper, we discuss a construction of Controlled-Controlled-NOT (CCNOT) gate by using the nonrelativistic formulation of ESR.

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Vestn. Sam. Gos. Techn. Un-ta. Ser. Fiz.-mat. nauki. 2013. Issue 1 (30). Pp. 297-304 Complex Systems, Quantum Mechanics, Information Theory UDC 004.38:530.145; MSC: 81P68; 81P10, 94C99 ON CONSTRUCTION OF QUANTUM LOGICAL GATE BASED ON ESR K. Mayuzumi1 , N. Watanabe1 , I. V. Volovich2 1 Science University of Tokyo, Noda City, Chiba 278-8510, Japan. 2 Steklov Mathematical Institute, Russian Academy of Sciences, 8, Gubkina st., Moscow, 119991, Russia. E-mails: watanabe@is.noda.tus.ac.jp, volovich@mi.ras.ru A quantum computer is a computation device operated by means of quantum mechan- ical phenomena. There are many candidates that are being pursued for physically im- plementing the quantum computer. The quantum logical gate based on the electron spin resonance (ESR) was studied in ref. [3]. In this paper, we discuss a construction of Controlled-Controlled-NOT (CCNOT) gate by using the nonrelativistic formulation of ESR. Key words: Electron Spin Resonance (ESR), quantum computer, quantum logical gates, Feynman gates, Controlled-Controlled NOT (CCNOT). 1. Introduction. In classical computer, there exist inevitable demerits for discussing logical gates. One of the demerits is an irreversibility of logical gates, that is the AND and the OR gates. This property causes to the restriction of computational speed for the classical computer. There are several kinds of approaches for avoiding these demerits. One of these approaches is proposed by Feynman [1]. He proved that every logical gates can be constructed by combining with only two reversible gates, i.e., the NOT and the Controlled-NOT (CNOT) gates. There are several approaches for realizing quantum logical gates. One of those approaches is the study by means of nuclear magnetic resonance (NMR). Quantum logical gate based on NMR is performed by controlling the nuclear spin under the additve magnetic fields from the environments. However, it might be difficulty to make the logical gate of NMR using a large number of quantum bits (qubits) because of the weakness of the spin-spin interactions among the nuclears. Our study uses ESR to construct Feynman gates that has NOT gate and CNOT gate, CCNOT gate. As quantum logical gate based on NMR, quantum gate based on ESR is performed by controlling the electron spin under the additive magnetic Kenichiro Mayuzumi, Graduate Student, Dept. of Information Sciences. Noboru Watanabe, Professor, Dept. of Information Sciences. Igor' V. Volovich (Dr. Sci. (Phys. & Math.), Corresponding member of RAS), Head of Dept., Dept. of Mathematical Physics. 297 K. M a y u z u m i, N. W a t a n a b e, I. V. V o l o v i c h fields from the environments. By employing Ising model, Ohya, Volovich and Watanabe constructed in [3] both NOT and CNOT gates based on ESR. In this paper, we construct the CCNOT gate in order to complete Feynman gates and universal quantum gates based on ESR. In general, any unitary operation on n qubits can be described by composing single qubit and CNOT gates. Unfortu- nately, no straightforward method is known to implement all these gates resisting errors. On the other hand, a discrete set of gates can be used to perform quantum computation in an error-resistant fashion. To perform fault-tolerant quantum computation, we consider discrete set of gates which are Feynman gates. 2. NOT gate based on ESR. In this section, we explain the NOT gate based on ESR. It is one of Feynman gates, which includes CNOT and CCNOT gate, and has been constructed [2]. First of all, let us consider one particle case. Let H be C2 with its canonical basis u+ = | = 1 0 , u- = | = 0 1 B(H) be the set of all bounded operators on H and B(H)sa {A B(H); A = A}, where A is the adjoint of A defined by A u, v = u, Av for any u, v H. B(H)sa has the basis x = 0 1 1 0 , y = 0 -i i 0 , z = 1 0 0 -1 , which are called Pauli spin matrices and I = 1 0 0 1 is an identity matrix on H. That is, = {x, y, z} is an orthogonal basis of B(H)sa with the scalar product i, j = 1 2 tr ij, j {x, y, z}. Let S = (Sx, Sy, Sz) be a spin (angular momentum) operator of electron, where Si = 1 2i is a component of spin operator of electron in the direction of i-axis (i = x, y, z). We denote unit vectors of x, y, z axis by ex, ey, ez and S is the spin vector given by S = (Sx, Sy, Sz) = Sxex + Syey + Szez. Let us consider two magnetic fields B0 and B1. B0 is a static magnetic field given by B0 = B0ez in the z direction and B1 is a rotating magnetic field given by B1(t) = B1(ex cos t + ey sin t) with frequency in the xy plain, where B0 and B1 are certain constants due to the magnetic fields. If B(t) is a magnetic vector defined by B(t) = B1(t) + B0, 298 On construction of quantum logical gate based on ESR then one has dS dt = S B(t) = B1(Sx cos t + Sy sin t) + B0Sz. Let | = 1 0 and | = 0 1 be spin vectors related to spin up and spin down, respectively. Let us take an initial state (0) = a0| + b0| = a0 b0 , then state vector at time t is denoted by (t) = a(t)| + b(t)| = a(t) b(t) , where a(t), b(t) C are satisfying |a(t)|2 + |b(t)|2 = 1. Let Schrodinger equation in one particle be i (t) t = -S B(t)(t) = -[B1(Sx cos t + Sy sin t) + B0Sz](t) where B0, B1, are arbitrary constants, A solution of the Schrodinger equation is given by (t) = e-itSz eit((+B0)Sz+B1Sx) (0), which means time evolution. In particular, we see the resonance condition + B0 = 0, that is, (t) = eiB0tSz eitB1Sx (0). Based on the above results, we reconstruct the Not gate based on ESR. If we take t = t1 such that B0t1 2 = B1t1 2 = 2 , then (t1) = 0 1 1 0 (0) = b0u+ + a0u-. It means that this gate is performed as the NOT gate based on ESR. Let UNOT (t) eiB0tSz eiB1tSx be a unitary operator expressing the NOT gate based on ESR. Quantum channel denoting the NOT gate based on ESR is defined by NOT(t1)( · ) UNOT (t1)( · )U NOT (t1). For the initial state |(0) (0)| at time 0, the output state of NOT(t1) is obtained by NOT(t1)(|(0) (0)|) = |(t1) (t1)|. 3. CNOT gate based on ESR. In this section, we introduce the CNOT gate based on ESR. Let us consider N particle systems to treat the Controlled Not 299 K. M a y u z u m i, N. W a t a n a b e, I. V. V o l o v i c h gate. Let e1, e2, e3 be unit vectors of x, y, z axis, respectively, and let S(1), . . . , S(N) be spin vectors of N electrons such as S(i) = (S (i) 1 , S (i) 2 , S (i) 3 ) = S (i) 1 e1 + S (i) 2 e2 + S (i) 3 e3. The spin operators satisfy the following commutation relations [S(p) , S (q) ] = ipq 3 =1 S(q) , where = +1 -1 and pq is a certain constant. Let us consider a Hamiltonian operator for N particle systems given by H(N) B3 N i=1 S (i) 3 + B1 N i=1 S (i) 1 f(t) + N i,j=1 JijS (i) 3 S (j) 3 , where f(t) is a certain function, for example f(t) = cos t and Jij is a coupling constant with respect to i-th spin and j-th spin. S (i) k is embedding Sk into i-th position of N tensor product. S (i) k = I · · · Sk · · · I (k = 1, 2, 3). Let us take a Hamiltonian H(N) as a Ising type interaction, that is H(N) B3 N i=1 S (i) 3 + N i,j=1 JijS (i) 3 S (j) 3 . If N = 2 then one can denote H(2) = B3(S3 I + I ~S3) + J(S3 S3) + B0(I I), where B0, B3 and J are determined by a certain phase parameter . Let us take u+, u-, v+, v- as u+v+ = 1 0 0 0 , u+v- = 0 0 1 0 , u-v+ = 0 1 0 0 , u-v- = 1 0 0 1 . Let (0) be an initial state vector given by (0) = a0u+ v+ + b0u+ v- + c0u- v+ + d0u- v- = a0 b0 c0 d0 (a0, b0, c0, d0 C). 300 On construction of quantum logical gate based on ESR For the initial state vector (0), if J = 2, B3 = -andB0 = 1 2 are hold, then the state vector at time t is expressed by (t) = e-itH(2) (0) = e-it(B3(S3I+I ~S3)+J(S3S3)+B0(II)) (0) = eit(S3I) eit(I ~S3) e-2t(S3 ~S3) e- 1 2 t(II) (0). If we take t = t1 such that 2t = ( 2 pulse) then one can denote the matrix form U(t1) of eit(S3I)eit(I ~S3)e-2t(S3 ~S3)e- 1 2 t(II) by U(t1) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 . Next we construct a unitary operator UH(t) related to a Hadamard transformation based on ESR. Let us define UH(t) by UH(t) = e-i2t(I ~S2) , where 2 is a certain phase parameter. Then we have e-i2t(I ~S2) = cos( 2t 2 )(I I) - 2i sin( 2t 2 )(I ~S2). For the initial state vector (0), the state vector at time t is expressed by (t) = UH(t)(0) = e-i2t(I ~S2) (0). If we take t = t2 such that 2t 2 = 4 2 pulse , then one can denote the matrix form UH(t2) of e-i2t2(I ~S2) UH(t2) = 1 0 0 1 1