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43
ФИЗИКА
УДК 535.8
O. N. Gadomsky, G. V. Yakimov, I. A. Shchukarev
SELECTIVE EXCITATION OF QUBITS AND TRANSFER OF QUANTUM INFORMATION FROM ONE QUBIT TO ANOTHER1
Abstract.
Background. The proposed construction of a qubits as fragments of composite nanostructured materials with quasi-zero refractive index allows to realize the resonance energy transfer over long distances by selective excitation of one of the qubits using the continuous ultraviolet radiation.
Materials and methods. A new construction of the qubit made of composite material with quasi-zero-refractive index synthesized by us is represented. Qubit is a fragment of this material with one silver nanoparticle inside a cylinder with a base area n (28 nm)2 and a height of 56 nm, which corresponds to 3 % of weight content of the silver in the composite material with uniformly distributed nanoparticles with a radius of 2.5 nm.
Results. On the basis of the equations of motion for coupled quantum dipoles and integro-differential equations for the electric field, a nonlocal problem was solved, in which one qubit is excited by continuous radiation, and at the location of the other qubit the local field is induced. The distance between the qubits is 5 m. Quantum information is transmitted due to entanglement of quantum states of qubits.
Conclusions. The article shows that the system of two qubits, which are fragments of a composite material with a quasi-zero refractive index, is an ideal energy transporter from one qubit to another over long distances. The new construction of a qubit with nanofibers, allowing to implement selective excitation of qubits by external radiation, was represented.
Key words: composite materials with a quasi-zero-refractive index, Ag-nanoparticle cubit, quantum information, inversion and local dipole moments of cubits, entanglement of quantum states, transfer of the quantum information between cubits.
О. Н. Гадомский, Г. В. Якимов, И. А. Щукарев
СЕЛЕКТИВНОЕ ВОЗБУЖДЕНИЕ КУБИТОВ И ПЕРЕДАЧА КВАНТОВОЙ ИНФОРМАЦИИ ОТ ОДНОГО КУБИТА К ДРУГОМУ
Аннотация.
Актуальность и цели. Предлагается конструкция кубитов как фрагментов композитного наноструктурного материала с квазинулевым показателем преломления, которая позволяет реализовать резонансную передачу энергии на
1 This work was supported by the Ministry of Education and Science of the Russian Federation (Project № 14.Z50.31.0015).
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большие расстояния при селективном возбуждении одного из кубитов непрерывным ультрафиолетовым излучением
Материалы и методы. Представлена новая конструкция кубита из синтезируемого нами композитного материала с квазинулевым показателем преломления. Кубит представляет собой фрагмент этого материала с одной наночастицей серебра внутри цилиндра с площадью основания п • (28 нм) и высотой 56 нм, что соответствует 3 % весовому содержанию серебра в композитном материале при равномерном распределении наночастиц радиусом 2,5 нм.
Результаты. На основе уравнений движения для связанных квантовых диполей и интегродифференциального уравнения для напряженности электрического поля решена нелокальная задача, в которой один из кубитов возбуждается непрерывным излучением, а вместе расположения другого кубита индуцируется локальное поле. Расстояние между кубитами равно 5 м. Передача квантовой информации обусловлена перепутыванием (entanglement) квантовых состояний кубитов.
Выводы. Показано, что система из двух кубитов, представляющих собой фрагменты композитного материала, обладающего квазинулевым показателем преломления, является идеальным транспортером энергии от одного кубита к другому на большие расстояния. Представлена новая конструкция кубита с нановолокном, позволяющая реализовать селективное возбуждение кубитов внешним излучением.
Ключевые слова: композитный материал с квазинулевым показателем преломления, серебряная наночастица - кубит, квантовая информация, инверсные и локальные дипольные моменты в кубитах, перепутывание квантовых состояний, передача квантовой информации между кубитами.
Introduction
Quantum information related to the solution of some problems such as the transfer of quantum information at any distance, quantum teleportation, quantum computing in a quantum computer, quantum cryptography, decoherence problem [1-4]. The basis for the solution of these problems is the physical realization of a quantum bit (qubit) as a physical system, enabling them to realize the selective excitation without a noticeable influence of various random processes, to encode, to store, to transmit information from one qubit to another, to record and to erase information. In quantum communication systems, information can be transmitted by the physical transfer of a qubit - the information medium or by the teleportation of the quantum state of the qubit [5]. Qubit can be represented as a quantum system with two states |0) and |l) with energy Eq and £], respectively. These basic functions allow to submit the wave function of the qubit as follows:
T = a|l)expf -j + b|0)expf -^Eot j , |0) =
1 =
(1)
where a and b are complex coefficients of quantum superposition, depending on time in the field of excitation and satisfies the normality condition |a |2 + |b |2 = 1.
Excitation of qubits by the electromagnetic fields allows to realize all quantum states with different values of the inversion w as the difference between the probability of detection of the qubit in the excited and ground states (w = |a|2 - |b|2 ).
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This property of qubit is what fundamentally distinguishes it from the classical bits of information, where only two states are realized, for example, in magnetic film [1]. There are proposals of the physical realization of qubits, for example, using
nuclear or electron spins I = 2 [1, 2] or Ca2 ion composed of a single ionic crys-
tal [1]. There are other options of choice of qubit states [1, 2]. A large number of experiments are performed on the qubit represented as a single photon with two orthogonal polarizations. Systems of photonic and atomic qubits are investigated experimentally in a resonator in cavity QED [6].
In this article we propose the construction of a qubit based on silver nanoparticles (Ag-nanoparticle cubit), which is a fragment of a composite film made of composite material synthesized by our technology [7]. This fragment is a cylinder of glass or polymer 56nm in height and base radius of 28 nm (Fig. 1).The cylinder is dimensioned so that at 3 % of weight content of the silver in the composite material the average distance between nanoparticles of silver is equal to 28 nm. The optical properties of the layer of the composite material with silver nanoparticles with radius 2,5 nm are shown in [8, 9] using the experimental spectra of reflection and transmission in the wavelength range from 400 to 1200 nm. The theoretical description of these spectra is represented in our works [8-11], which proved that the synthesized materials have a quasi-zero refractive index and low absorption in the wavelength range of at least from 400 to 1200 nm. Excitation of qubits with proposed construction can be performed using the optical nanofibers, such as those described in [12], wherein excitation of photons in the optical fiber of glass with a facing of silver, was made by electron beam.
glass
Ag-Facing of
r-a^ofibers
External
electric
field
Fig. 1. Construction of Ag-nanoparticle qubit
The interaction between qubits by the optical circuit (Fig. 2) is described in this work using a system of equations for coupled quantum dipoles [11] and the integral-differential equation for the electric field at different points of observation both inside and outside of these nanoparticles in the far and near zone [13].We
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found a particular solution of these equations, which shows that the transfer of quantum information from one qubit to another can occur at arbitrary distances between qubits. This mechanism of transmission of quantum information by selective excitation of one of the qubits is significantly different from the quantum teleportation, where the two qubits simultaneously excited by pulsed field and the transfer of quantum information over long distances is a result of quantum interference of two qubits [5].
Fig. 2. Optical circuit of selective excitation of qubits by field of external radiation. - distance between qubits
1. The equation of motion for two Ag-nanoparticle cubits as composite parts
The electric field strength E(r, t )for different observation points for two qubits is defined by the following equation [13]:
d j (r', t - R / c)
E(r, t) = E J (r, t) + rotrot N ^^------------dV', (2)
j=1 R
where Ej (r, t) is the electric field strength of the external wave, R = |r'- r| is the
distance between the radius vector r' inside of spherical qubits and the observation point r , N is the concentration of free electrons in qubits, c is the speed of light in a medium surrounding the Ag-nanoparticle and the quantum-mechanical average d j = X j exp(-/Qt) of induced electric dipole moments of qubits obeys the equations for coupled quantum dipoles:
Xj = ~iAXj-J*jM2 E0j-^T7Xj■ (3)
W = { <X>0j - X/E0j ) - jn(Wj - wo). (4)
where Л = Qo - “ is the detuning from the resonance frequency ®o , ю is the frequency of the external field, do is the transition dipole moment of the free electrons in the Ag-nanoparticle cubit, wo is the inversion of the equilibrium state is
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equal to -1 , 1/72' is the width of the resonance at a frequency , wj is the inversion of qubits, E01 and E02 are a local fields without the factor exp(-mt). Values X1, X2 and W1 , W2 of qubits in according to sense of equations (3) and (4) can be represented by the coefficients of quantum superposition fiq, 61 and 02,62 of qubits as follows:
X1 = 2d0b*a1, X2 = 2d0b*a2 , W1 = О^2 - faf , W2 = -1^2 . (5)
When qubits are being excited by short pulses, the duration of which is much less T2', last term on the right-hand side of equations (3) and (4) can be dropped, resulting in the implementation of the following conservation laws during A = 0
|X1| + W12d02 = d02, |X21 + d2 = d2. (6)
2. Single-qubit transformations
The equations (3) and (4) allows us to define a single-qubit transformation when E0 j = E01, ie local fields are converted to the external field. Consider the
case of pulsed irradiation one of the qubits, for example, qubit 1 by external field. Suppose that the pulse duration is much less time T2'. Then for qubit 1 we have (omit the index 1):
X = -/AX-^w|d0|2 E0j, (7)
n
w = n (X*E0 j - XE0 j). (8)
The values X and w are defined as the quantum-mechanical average of the Pauli matrices 01,02,03 in the energy space with the help of the wave function (1). Herewith
( | Y) = a*b + ba = u , (¥|c2 | Y) = -i( ab - ba) = v,
(^|o3Y) = |a|2 -|b|2 = w , (9)
and the value X is defined as
X = d0(u -iv)exp(-mt) = 2d0ab expf --П-(1 -E0)t j.
(10)
The functions u and v change over time much more slowly than exp(-imt) and obey to the law of conservation
u2 + v2 + w2 = 1. (11)
The coefficients of superposition a and b can be represented as
a = \a\ei(?“ , b = \b\ei(^b . (12)
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Also these coefficients can be identified through the variables u, v, w
a = 2(W + \Ь\2) b = W + \Ь12 e^a b = 2(lN2 - W) a = N2 - W
U + IV
u - IV
|b|
(13)
The equations of motion for the variables u, v and w with accordance to equations (7) and (8) in the case of the exact resonance A = 0 take the following form:
u = 0, v = wxE) i, w = -xEt/v, (14)
where x = (2/ Й) |d0'|, d0'- the real part of the transition dipole moment d0 , E01 is the real amplitude of the external wave parallel to the vector d0'.
The pulse area can be represented as
t
©(t) = x \ E01 (t')dt'. (15)
Then the solution of (14) takes the following form:
u(t) = u(0), v(t) = w(0)sin 0(t) + v(0)cos 0(t) w(t) = -v(0)sin 0(t) + w(0)cos0(t),
(16)
where u(0), v(0), w(0) are values of functions u, v, w at the initial time t = 0 . If at the initial time w(0) = -1, than, according to the law (11), we have u(0) = v(0) = 0 . The pulse area (15) can be regarded as the angle of rotation of the Bloch vector p (u, v, w) in the energy space. For rectangular pulse © = XE0iT, where t- pulse width. At 0 = n we have w(t) = 1, i.e. the operation NOT is implemented , as
NOT\0) = 1, (17)
when the qubit is transferred from the ground to the excited state. Other quantum states of the qubit according to the equation (13) can be implemented by changing the pulse area 0 .
In a field of continuous radiation instead of equations (14) using (3), (4) for one qubit, when A^ 0 we obtain the following equations:
u = -Av - (u / T2 '), v = Au + XE0iw - (v / T2 '), w = -xE0iv -
f w +1 ^ V ~Т2Г J
. (18)
The stationary solution of this system of equations under the conditions u = v = w = 0 is:
u =
( AT2')(xE0iT2')
1 + (AT2')2 + (T2')2(xE01 )2
v = -
xE0iT2'
1 + (AT2')2 + (T2')2(xE0i )2
w = ■
1 + (AT2')2
1 + (AT2')2 + (T2')2(xE01)2 '
(19)
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When
(xEoI )2 (T2 ')2 << 1 (20)
according to the solution (19), a qubit inversion does not change and it is equal to wo =-1. In this case, under the influence of a low-intensity excitation only quantum information in the qubit will be changed which is associated with a phase of the qubit in accordance with formula (13) for the coefficients of the quantum superposition. In intensive field in case of violation of conditions (20) both amplitude and phase quantum information of the qubit will be changed, herewith, to define phase quantum information of the qubit, it is necessary to use interference measurements. Full definition of the state vector is called quantum state tomography [2].
3. Selective excitation of the qubits by the field of continuous radiation. Two-qubit transformation
Consider the stationary solution of equations (3), (4), under the conditions
X = 0, wj = 0. (21)
These conditions can be implemented by stationary excitation of one of the qubits, for example, qubit 2 by external continuous radiation. In this case of selective excitation of qubit 2 the local fields according to the equation (2) can be represented as follows:
E01 = X2®rN + X^jN, E02 = E01 + X1aRN + X2ajN , (22)
where E01 is the amplitude of the external wave, Cr and aj are external and internal geometric factors, which for spherical nanoparticles are calculated using the appropriate procedure for transitions from the volume integral to a surface by using the Green's theorem [14]. For spherical nanoparticles of small radius as such that k0a << 1, we have the following expression for the geometric factor aj :
4n4 aj =“3(1 + lk0a),
(23)
where k0 =(co / c )nM, c is the speed of light in vacuum, is the refractive index of the sphere surrounding the silver nanoparticles in the qubit. Diagonal tensor Cr was calculated for observation points outside the spherical nanoparticles and after appropriate calculations it takes the following form:
4nA ( _2_
nk03(n2 -1) l R3
2ik0 \ eik0R R2 J
aR =-
4nA
nk03(n2 ■
1)
R R
ikQ_
,2
k0
R
2 X
Jk0R
where
A = -n cos(k0na) sin(k0a) + cos(^a) sin(k0na)
(24)
(25)
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Let us define the refractive indexn by following [8], with the formula:
n2 - 1 _ 4n N0aeffq + Nmam
n2 + 2 3 1 —P( Noaeffq + Nmam )
where q is the number of free electrons in the silver nanoparticles, Nm , am are the concentration and the polarizability of the molecules in the matrix of the composite, N0 is the concentration of silver nanoparticles in the composite, aef is the
effective polarizability of the free electrons in the silver nanoparticles,
a f _------0-----, a is the quantum polarizability
JJ 1 — aTN a
a_
2d02 1
h wo — w—(i / T2')
(27)
P is the structural factor that takes into account the discrete distribution of nanoparticles within an imaginary sphere surrounding the observation point. We will explain the meaning of the structural factors in the assessment of the final results.
We consider the interaction of two qubits at a great distance between the qubits such that koR12 >> 1. In this case, in the external geometrical factor (24) is sufficient to take into account only the terms proportional to 1 / R, which corresponds, for example, the x - component of the tensor. In line with this we will consider only the x - component of the external and local fields in the equation (22). For a small radius of nanoparticles koa << 1 under the condition of exact resonance w_W) we have:
ax 2n(koa) (n + 2) eik0R12
aR _------2-----------e 0 12
3( n —1) koR12
2d0
W) _ 2n
a_—r— T', ko = —
h c к0
2
(28)
From equation (3) under the condition (7) we have the following equality
X1 _ —^1aEo1, X2 _ —W2aEo2 . (29)
Therefore, solving the system of algebraic equations (8), we obtain the following expression for the local field at the center of nanoparticles in the qubits 1 and 2:
Eq1 _— E
or
W2aaR>N
(1 + W10QtN )(1 + W2001tN ) — W1W2 (aRN a)
(3o)
T7x _ T7x
Eo2 _ Eoi
1 + W10aTN
(1 + W1001tN )(1 + W2 ooitN ) — W1W2 (aRN a)
(31)
Single-qubit operations describe the rotation of the vector p(u, v, w)of individual qubit, where the components of this vector is uniquely connected with the coefficients of the quantum superposition associated with (13). Two-qubit operations involve the interdependence of states of two qubits, a some kind of control of
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one qubit (controlled) by another (controlling). This interdependence requires
physical interaction between the qubits, which can be turned on at the time of the operation, or can be constant. It is important, as discussed in this article, to be able to selectively excite the individual qubits by the external field. With continuous irradiation of one of the qubits by resonant radiation the coefficients of quantum superposition of qubit can be expressed in terms of variables u,v and w of qubit (omit the index of qubit) according to the expressions:
Xexp(-iat) = 2d0ab* exp (-i^t) =
= d0(u - iv )exp (-iat) = -waE0 exp (-iat). (32)
From these expressions using (11) - (13) we obtain that:
2( w + lb I2), .2 2do b2 = waE0, u + iv (33)
a |2=w+ь |2, (34)
2(w + b 2)( a 2 - w) ,-(ф„ -фь) . 2 ' Yb; = u - ,v . №1 (35)
Using equations (33) and (34) we will define |b| and \a\, respectivetly,
through the u, v and w, and using (35) we will define the phase difference of the states | a) and |b) of the qubit.
4. Entanglement and resonance energy transfer in the system of two qubits with selective excitation of one of the qubits by the field of continuous radiation
Consider the property of the solution (30), (31), in which the entanglement of the qubit state allows to achieve the transfer of quantum information between qubits over long distances between qubits. The entanglement of quantum states of qubits in this decision reflected in the fact that the local fields at the location of qubits 1 and 2 depends on inversion of both 1 and 2 qubits. This means that the selective excitation of qubit 2 is detected at the location of the qubit 1 due to induction at the location of the qubit 1 the local field, and hence the local induced dipole
moment = -W10E0X1. The induction of a local field E01 is getting weaker with
the distance between the qubits, if qubits 1 and 2 are represented as silver nanoparticles in vacuum. Let us consider the qubits 1 and 2 as fragments of a composite material with a quasi-zero refractive index.
Consider the property of the solution (30), (31), wherein the induction of local moments at the location of qubits 1 and 2 not decreases with increasing distance between qubits, and vice versa, even increases. For this it is necessary to consider conditions under which part of the denominator in (30), (31)
(1 + wyoafN )(1 + W2aafN) = 0. (36)
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vanishes. For this it is necessary that one of the factors in (36) becomes zero. If this ratio is (1 + M>iOafN), it will lead to the fact that the induction of the local field at
the location of qubit 2 will become impossible, ie £02 will be zero. Therefore, to satisfy the equation (36), we assume that
(1 + W2aajN) = 0. (37)
from which we obtain the value of a inversion for qubit 2
w2 =
1
(4n /3) |a| N
(38)
The second value of W2 , which correspond to the solution of the equation Re(1 + W20jaN) = 0 when ^a << 1 gives the value W2 >> 1 that contradicts the physical significance of inversion.
Thus, the qubit 1, which is in the ground state W1 =-1 prior to the moment, when the excitation field, at the position of the qubit 2, will be turned on, can get quantum information which is defined by the inversion (38). When the equation (37) is true, a local induced dipole moment of the qubit 1 takes the following form:
X1x =a
2E0)IR12i
a 3k02
(39)
Local induced dipole moment of the 2nd qubit wherein, in accordance with the solution (31) and the condition (37) is equal to:
X2 x = -a
4 E0i (1 + w^j aN )^122
W1(° V)2
(40)
In the formulas (39), (40) takes into account only the terms of the external geometric factor (24), which are dominated for large distances between qubits. Formulas (39), (40) demonstrate the ability to transmit quantum information over long distances between the qubits, due to the effect of quantum teleportation. Inversion of qubit 2 is given by (38) and it is close to the value of 1, and inversion W1 of first qubit can take different values from -1 to +1, including W1 = 0.
As follows from (39), (40), the induced dipole moment of qubits 1 and 2 increases with increasing distance between qubits. However, there is a limit to the
values E0x1 and E0x2 of local fields in the center of qubits, which follows from the equation (4) for the inversions W1 and W2 . For stationary solutions of equations
under conditions (21), from (4) we obtain the limit of values of the fields £01 and
Ex :
£02 :
T7x
£01
(h / T2')(W! +1) W1 lal
T7x
£02
(h / T2')( w2 +1) W2 |a|
2
2
(41)
Consider qubits, as fragments of composite material with a quasi-zero refractive index. In [8] a formula for the refractive index of the material, which has a
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form (26), was obtained. Following [8], we obtain the following expression for the structure factor in (26):
P = —
N
^ 2n ^
Vл 0j
2 v 1
nM Z — ■
a
a
(42)
Where ra is the distance from the observation point at the center of an imaginary sphere to a silver nanoparticle with index a , nM is the refractive index of the matrix of the composite. When the weight of silver content in the composite is equal to 3% [7], the radius of silver nanoparticles a = 2.5 nm and concentration of
nanoparticles N0 = 0.4555 10 cm is uniform, we obtain that at the imaginary sphere whose radius is significantly smaller than the wavelength X0 = 300 nm, there are 26 silver nanoparticle. Then the structure factor в = 21.42 if the refractive index of the matrix composite пм = 1.49 . Let us define value |a| N when the refractive index of the composite becomes to zero. As follows from formula (26), this will happen if the following equation is true:
1 + N0aeffq + Nmam = 0 (43)
3 1 — P(N0aeffq + Nmam )
Then we find that
N |a| = -
1 PNmam + 3 Nmam
-(1 - PNmO m + NmOm ) Ы + PN(Vo - у
(44)
4n 3
where V0 =— a is the volume of spherical silver nanoparticles. For a = 2.5 nm
1
therefore inversion of 2nd qubit, according
and Nmam = 0.077, N lal ~ —
m m 1 1 (4n/3)
to formula (38) is about W2 ~—1, that is close to the equilibrium value.
Finally, let us calculate the maximum distance R12 between qubits, at which
there is a resonance transfer of the energy from qubit 1 to qubit 2. For this we will
use the maximum value of the local field £01, in accordance with the formula (41).
15 -1
When w = 1 and 1/ 72' = 10 s we obtain the value
T7X
E01
= 4.9 105SGSE and
the maximum distance between qubits R12 = 519 cm if the external field, which
selectively excite a qubit 2, is small or equal
Fx E01
= 10-6 SGSE.
Conclusions
So, in this article was shown that the system of two qubits, which are fragments of a composite material with a quasi-zero refractive index, is an ideal energy transporter from one qubit to another over long distances. A new construction of a
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qubit with nanofibers that allows to implement selective excitation of qubits by external radiation, was represented. The resonance transfer of energy from one qubit to another with selective excitation of one of the qubits by external radiation results in a change of qubits inversion and induction of local electric dipole transitions of qubit 1(qubit-observer) and the qubit 2 (qubit- inductor). Qubit 2 is being irradiated by external radiation and it is induces a local field at the location of the qubit-observer 1, located at a great distance from the qubit-inductor 2. This means that there is a transfer of quantum information that is obtained by applying the coefficients of quantum superposition by (13).
Resonance energy transfer between the qubits is described using the resulting solutions (30), (31) and equation of motion (2), (3), (4). Due to the entanglement of quantum states of qubits and coherence in the system of qubit caused by the zero refractive index of qubits, the local field at the location of qubit-observer
achieved an increase in 4.9 1011 times, at the distance between the qubits R\2 = 519 cm if inducing field at the location of the qubit-inductor is small. The law of conservation of energy in this case is represented as follows:
(E0 f ^ = йм»B,1-1 ^
X 15
where Vr is the volume of the quantization of the electromagnetic field, AQ is the solid angle in the direction of dipole radiation of qubit 1. If dipole radiation of qubit 1 is emitted in the direction of nanofibers attached to qubit 1 (fig.1), then
AQ = 0.2461 10-10sr and the electric field strength Eqi exciting the qubit 2, in accordance with the law of conservation of energy (45) is equal to small value Eqj = 3.14-10-6SGSE if VR = 1 cm3, w1 = 1, R12 = 519 cm .
Thus, the proposed construction of a qubits as fragments of composite nanostructured materials with quasi-zero refractive index allows to realize the resonance energy transfer over long distances by the selective excitation of one of the qubits using the continuous ultraviolet radiation. The proposed construction of qubits, in our opinion, can be applied to the implementation of quantum computing in a quantum computer, where the transmission of quantum information over long distances is not required. Herewith it is necessary to define logical operations NOT and CNOT under the influence of short radiation pulses, and this is what we will devote our next article.
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Гадомский Олег Николаевич доктор физико-математических наук, профессор, кафедра радиофизики и электроники, Ульяновский государственный университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: [email protected]
Якимов Георгий Вячеславович аспирант, Ульяновский государственный университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: [email protected]
Щукарев Игорь Александрович аспирант, Ульяновский государственный университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: [email protected]
Gadomskiy Oleg Nikolaevich Doctor of physical and mathematical sciences, professor, sub-department of radio physics and electronics,
Ulyanovsk State University
(42 Lva Tolstogo street, Ulyanovsk, Russia)
Yakimov Georgiy Vyacheslavovich Postgraduate student, Ulyanovsk State University (42 Lva Tolstogo street, Ulyanovsk, Russia)
Shchukarev Igor' Aleksandrovich Postgraduate student, Ulyanovsk State University (42 Lva Tolstogo street, Ulyanovsk, Russia)
УДК 535.8 Gadomsky, O. N.
Selective excitation of qubits and transfer of quantum information from one qubit to another / O. N. Gadomsky, G. V. Yakimov, I. A. Shchukarev // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. - 2015. - № 3 (35). - С. 112-124.
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