On construction of quantum logical gate based on ESR Текст научной статьи по специальности «Физика»

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 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.

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.

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. Unfortunately, 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

TL be C2 with its canonical basis u+ = | t) = ( I ),«- = U) = ( I ) B(H)

be the set of all bounded operators on TL and B('H)sa = {A € B(TL);A = A*}, where A* is the adjoint of A defined by

(A*u, v) = (u, Av) for any

B (H)sa has the basis <rx = o ) ’ = ( i 0* ) ’ ^ = ( 0 -1 ) ’ which

are called Pauli spin matrices and I = ^ J ^ is an identity matrix on TL. That is, a = {ax, ay, az} is an orthogonal basis of B(TL)sa with the scalar product

(cruCTj) = j € {x, y, z}.

Let S = (Sx,Sy,Sz) be a spin (angular momentum) operator of electron, where Si = \(Ti 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 e^, ey, ez and S is the spin vector given by

S = (Sxi, Sy, Sz) = Sxex + Syey + Szez.

Let us consider two magnetic fields Bo and B\. Bo is a static magnetic field given by

Bo = Boez

in the 2 direction and B\ is a rotating magnetic field given by

B\(t) = B\(ex cos out + ey sin Lot)

with frequency uj in the xy plain, where Bo and B\ are certain constants due to the magnetic fields. If B(t) is a magnetic vector defined by

B(t) = B\{t) + Bo,

then one has

dS

— = S x B(t) = Bi(Sx cos wi + Sy sinwi) + BqSz.

Let | t) = ^ q ^ and | -J,) = ^ ^ be spin vectors related to spin up and spin

down, respectively. Let us take an initial state 0(0) = ao| t) + &o| ^

then state vector at time t is denoted by

0CO = a(t)\ t> + 6(^)1 I) = (

where a(t), b(t) € C are satisfying |a(i)|2 + |6(i)|2 = 1.

Let Schrodinger equation in one particle be

i ^= —S x mm = — [Bi(Sx cos out + Sy sin out) + BoSz]ip(t)

where Bq,Bi,oj are arbitrary constants, A solution of the Schrodinger equation is given by

0(t) = e~^tSZeit((uJ+Bo)Sz+B1Sx)^0^

which means time evolution. In particular, we see the resonance condition

ui + Bq = 0,

that is, 0(i) = elBotSzettBlSxip(0). Based on the above results, we reconstruct the Not gate based on ESR. If we take t = t\ such that

Bot\ B\t\ it 2 _ 2 _ 2 ’

then

ip(ti) = (^ i J) 0(°) = b0u+ + aou-.

It means that this gate is performed as the NOT gate based on ESR. Let UNor(t) = eiB0tszeiB-itsx kg a unitaiy operator expressing the NOT gate based on ESR. Quantum channel denoting the NOT gate based on ESR is defined by

AwoT(ti)( •) = UNoT(ti)( ■ )U^OT(ti).

For the initial state |0(O)) ("0(0)1 at time 0, the output state of is obtained

by

^*NOT(t!) (10(0)) (0(0) |) = |0(tl))(0(tl)|.

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

gate. Let el, e"2, e3 be unit vectors of x, y, z axis, respectively, and let S'W,..., S(N) be spin vectors of N electrons such as

sw = (S^,S^,S^) = sf ex + sfe* + sfeli.

The spin operators satisfy the following commutation relations

[S^,S<j3q)]=i6pqJ2^S^,

7=1

where ea/37 = | ^ and 5pq is a certain constant. Let us consider a Hamiltonian operator for N particle systems given by

H (N) = Ba

i= 1

i= 1

*J=1

where f(t) is a certain function, for example f(t) = cos out and Jij is a coupling

(i)

constant with respect to i-th spin and j-th spin. Sk is embedding Sk into i-th position of N tensor product.

(A: = 1,2,3).

Let us take a Hamiltonian H(W) as a Ising type interaction, that is

N N

3(E^) + E

H(iv) = B-

i= 1 = l

If N = 2 then one can denote

H(2) = B3(S3 CX) / + / eg) S3) + J(S3 CX) S3) + Bo(I CX) /),

where Bq,B3 and J are determined by a certain phase parameter uj. Let us take u+,v,-,v+,v- as

u+®v+

(1 \

0

0

V 0 )

, U+<2>V-

(0 \

0

W

U-

(0 \

1

0

V 0 )

U-&V-

(1 \

0

0

V1 /

Let 0(0) be an initial state vector given by

0(0) = aou+ <g> v+ + bo%i+ <g> V- + cqU- <g> v+ + doU- <g> V-

/ ao \

bo

co

V d0 J

(ao, bo, Co, do € C).

For the initial state vector 0(0), if J = 2oj,Bs = — wand£>o = are hold, then the state vector at time t is expressed by

0(t) = e~itH^i){ 0)

= f>-'<‘KB3(s3(&I+I(&S3)+J(S3<&S3)+B0(I<&I))

_ eujt(S3®I) eujt(I® S3) e~2ujt(S3 (S1S3) g- §wt(iW) ^(Q)

If we take t = t\ such that 2out = tt (§ pulse) then one can denote the matrix form U$(t{) of QiMKs3®I)eiujt(I<2>S3)e-2u)t(S3<)<>S3)e-\ut(I®I)

/ 1 0 0 0 \ 0 10 0 0 0 10 Vo 0 0 -1 /

Next we construct a unitary operator £/#(i) related to a Hadamard transformation based on ESR. Let us define £/#(i) by

UH(t) = e-i"2t(/®S2)) where UJ2 is a certain phase parameter. Then we have

e-iuj2t(i<g>s2) _ 0 /) _ 2isin(^|^)(/ <S> S2).

For the initial state vector 0(0), the state vector at time t is expressed by

0(t) = t/ff(t)0( 0) = e-iW2t(/^2)0(O).

If we take t = t2 such that

0J2t 7T /7T \

— =4 V 2 ^ eJ’

then one can denote the matrix form Un(t2) of e-tuJ2t2(i®s2)

л/2 V 1 1

Thus unitary operator UcNOT(t 1 +2іг) related to the CNOT gate can be reconstructed by the combination of f7$(ti) and Unit2) as

UcNOTifl + 2іг)

= UH{t2)*U<s>{t{)UH{t2)

— ЄІШ2І2(І®32)еішіі(3з®1)еішіі(І®Зз)е~2ішіі(3з®3з)e~ ^ішіі(ШІ)Є~ІШ2І2(Ш32)

It means that this unitary operator UcNOT(t 1 + 2І2) is performed as CNOT (Controlled-NOT) gate based on ESR. Quantum channel denoting the CNOT gate based on ESR is defined by

A-CNOT(t1+2t2)( ') = UCNOT{tl + 212){ ■ )UcNOT{t 1 + 2h)-

For the initial state |0(O))(0(O)| at time 0 , the output state of ^c'NOT(t1+2t2) obtained by

AcwoT(ti+2t2)(l0(o))(0(°)l) = |0(*i + 2*2)>(0(*i + 2i2)|.

4. CCNOT gate based on ESR. This section shows how to construct CCNOT gate, which is our main result, based on ESR. Let us consider some gates to treat the CCNOT gate. First of all, we construct Controlled-phase gate based on ESR. If Hamiltonian has N = 3 then one can denote

3) = ^ + S3 ^ + S3 + J( 1,2)(<53 ® <§3 ® I) + J(2,3)(-^ ® ^3 ® S3)

+BQ{I eg) I eg) /),

where Bo, Bs, andJ(i;2), J(2,3) are determined by a certain phase parameter uj. Let •0(0) be an initial state vector given by

•0(0) = aou+ <g> u+ <g> u+ + boU- <g>u+ (&u+ + cou+ <g>u- <g>u+

+doU- <g> U- <g> u+ +

&0U+ <S> U+ <S> U~ + f0U- <g) U+ <g) U- + <7oW+ &U-&U-+hoU- <g> U- + <gm_.

For the initial state vector 0(0), if Bo = j, Bs = =f- and </(1,2) = ^(2,3) = w are hold, then operator [/$ related to the state vector at time i is expressed by

U$(t) = ei^(s3)+s3)+s3))-i“<s3®s3®l+l®s3®s3)-i^(l®l®l)

If we take ti,t2 such that out 1 = —^2^2 = § then operator Us of Controlled-phase is denoted by

Us(ti+2t2) =

( 1 0 0 0 \

0 10 0

0 0 10

V 0 0 0 i J

/(

Second, we reconstruct a CNOT gate of three particles, employing Ising model. We denote unitary operator UqNOT related to a CNOT gate of three particles by use of operator U$(t) of time evolution. Let us define U^NOT by

UcnotW = e-^8^ U$(t)ei2“5t(-S i2)-43))e~^43)

If we take ts,t4,t5 such that ousts = §, out4 = 7r, = §, then one has

UcNOT^h +U+ 215) =

Then one can denote the matrix form U^NOT(2ts + £4 + 2t$) by

UcNOT^t 3 + £4 + 2t$) —

/ 1 0 0 0 \

0 10 0

0 0 0 1

V 0 0 1 0 )

I.

Next we denote operator UQNOT(t) of CNOT gate that the role of control and target are changed. Let us define UQNOT(t) by

U*NOT(t) =

If we take ts,t4,t5 such that ousts = £, out4 = 7r, = £, then one has

?(2) o(3)x

?(3)

?(1)

_ e-iuj3t3S,

U^(t4)ei2M6t^S3>'S3>^e~i0J6t6S3> ei0J3t3Sl-

-,(2) c,(3)

(3)

?(1)

and the matrix form of U^NOT(2t3 + £4 + 2ts) is obtained by

^CNOr(^3 + ^4 + 2ts) —

/ 1 0 0 0 \

0 0 0 1

0 0 10

V 0 1 0 0 )

I.

It means that this unitary operator U^,NOT(2t^ + £4 + 2ts) is performed as CNOT gate of three particles.

Third, we construct a SWAP gate based on ESR. The SWAP gate swaps two qubits. Let us define a unitary operator UswAp(t) by

Usw Apifits + 3t4 + 615) =

Ucnot(,^3 + ti + 2t§)lJq ^ Qrp(2t% + ti + 2i5) X UqnOt(^3 + £4 + 2i5)

/ 1 0 0 0 \

0 0 10

0 10 0

V 0 0 0 1 /

Thus the unitary operator UswAp(Gts + 3i4 + 6ts) related to the SWAP gate can be reconstructed by the combination of U^NOT. It means that this unitary operator is performed as SWAP gate based on ESR.

We reconstruct the CCNOT gate based on ESR using by Controlled-phase and SWAP gate.

\Qo)

\Qi)

|g2> ~ H- S

St

s-h-

Fig. CCNOT

If Ufj is operator related to a Hadamard gate of three particles then it is

denoted by e-^3*3^ . The unitary operator IJccnot related to the CCNOT gate can be reconstructed by the combination of U^NOT ,Us , Ufj and IJswap as

UcCNOT^ti + 6t2 + 14i3 + 6t4 + 12*s)

= UsWAp(6h + 3i4 + 6i5)[/^(i3)(/ (g) Us(t\ + 2i2))

UcNOT(2ts + £4 + 2t5)3UQNOT(2ts + £4 + 2*5)

(/ (g) E/s(ti + 2t2))UcNOT(2ts + £4 + 2ts)3 (/ <g> Us(t\ + 2t2))Ufj(ts)

and the matrix form of £/ccwoT(3ti+6t2+i4t3+6t4+i2tB) is obtained by

/10000000\ 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

\00000010/

UcCNOT^ti + 6i2 + 14i3 + 6t4 + 12*5) =

This unitary operator Uccnot is performed as CCNOT gate based on ESR. Quantum channel denoting the CCNOT gate based on ESR is defined by

A*ccnot( ' )

= C^CCWOT(3tl+6t2 + 14t3+6t4 + 12t5)( • ) X UccNOT^tl +6i2 + 14i3+6i4 + 12i5).

For the initial state |0(O))(,0(O)| at time 0, the output state of ^-ccNOT(3t1+6t2+ut3+6t4+i2t6) obtained by

A,

CCNOT

mmm\)

— \tp(3ti + 6i2 + 14t3 + 6t4 + 12ts)) (0(3ii + 6t2 + 14t3 + 6i4 + 12t5)|.

REFERENCES

1. R. P. Feynman, “Quantum mechanical computers” // Optics News, 1985. Vol. 11. Pp. 11-20.

2. C. P. Slichter, Principles of Magnetic Resonance / Springer Ser. Solid-State Sci. Vol. 1. Berlin, Heidelberg: Springer, 1992.

3. M. Ohya, I. V. Volovich, N. Watanabe, “Quantum logical gate based on ESR” / In: Quantum, Information. Vol. 3. River Edge, N.J.: World Sci. Publ., 2001. Pp. 143-156.

Original article submitted 21/1/2013; revision submitted 26/11/2013.