Resonant-tunneling solid state NMR quantum computer.
A.V. Tsukanov, A.A. Larionov, K.A. Valiev
Institute ofPhysics and Technology, RAS, 1 17218 Moscow, Russia
ABSTRACT.
A novel solid state quantum computer is discussed. Nuclear spins-qubits are the basic elements of quantum register,
while single electron resonant transfer is used to obtain complex many-qubit gates. The electron's quantum dynamics
analysis shows the possibility of individual addressing in large registers. Planar and ensemble architectures are also
proposed.
ENTRODUCTION.
In the quantum computing field there are a lot of different proposals on the physical implementation of the
quantum computer's architecture. The enormous progress of the up-to-date technology allows us to invent the new
schemes for quantum algorithm's realization. Nevertheless, there is no a large-scale working device that we may call a
"quantum computer". It would be more correct to speak about some elementary steps toward the profound
understanding ofthe fundamental properties of such hypothetical devices.
After Feynman consideration of a very promising idea of quantum computation possibility, a lot of suggestions
of quantum computer realization appeared. What we can say about the most probable candidate for the construction
base of the quantum computers? Today we have experimentally realized prototypes of such devices according to
different theoretical approaches. One of these variants is bulk ensemble NMR quantum computer. However its
computational resources are limited and it's clear that such quantum computer realization with the number of
elementary cells (qubits) much more than 30 is difficult. And to demonstrate the real power of quantum computers we
need at least hundreds qubits.
Proposing the idea of solid-state NMR quantum computer was the way out in this situation. From the
technological point of view solid state quantum computers are considered to be the most perspective variants. The idea
based on creating artificial multispin systems in semiconductor structures with individual addressing to any qubit was
suggested in [1] and discussed in [2]. It is based on creating artificial multi-spin system and using individual addressing
to the different spins-qubits. For this purpose it is suggested to use silicon-based structure of MOS4ype, where donor
atoms of stable phosphorus isotope 31P are implanted to the thin layer of spinfree silicon isotope 28Si at the defmite
depth. Donor atoms replace silicon ones in knots of crystal lattice. Such a donor has shallow impurity states, which have
a large magnitude of the effective Bohr radius and nuclear spin 1=1/2. Every donor atom with nuclear spin in
semiconductor structure is supposed to place regularly with adequate accuracy under its "own" control metallic gate
(gate A), which is partitioned off from the silicon surface by thin dielectric layer (for example, silicon oxide with the
thickness of about some nanometers). A-gates form linear regular structure of arbitrarily length with the period 1 (Fig.
1). Changing the electrical potential of J-gates, which are placed between A-gates, allows us, redistributing electron
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density between neighbor donors, to control the degree of overlapping of electron wave functions, which are localized
on the neighbor donors a and b, and to control the constant of exchanging interaction J, and the constant of scalar
interaction of their nuclear spins 'a as well. It is supposed that with the help of the electric field, induced by Agates,
one can change the distribution of electron density near the nuclei in the ground state, choosing individually,
correspondingly, each donor atom nuclear spin resonance frequency, which is determined by the hyperfine interaction
with its electron spin. This allows to fulfill quantum operations by the way of selective reaction of resonance radiofrequency pulses on nuclear spins ofgiven donors.
Figure 1 . The scheme of semiconductor NMR quantum computer. D — thickness of the silicon layer on the conducting substrate. A
and J — gates on the dielectric layer ofthickness d.
Experiments in this field are very complicated, but the base for phosphorus qubits in a silicon quantum
computer has already been demonstrated [3]. However, there are a lot difficulties connected with technology and
problem of decoherence to create even though quantum computer prototype. Main disadvantages of Kane scheme are
the next;
1. Small characteristic size of quantum register cells and corresponding distances between them, which are
determined by the dependence ofthe electron wave function overlapping integral J on effective Bohr radius.
2. Available interqubit interaction only between close neighbors and absence ofthis interaction between arbitrary
qubits ofthe quantum register.
3. Presence ofthe large amount ofcontrolling metallic gates.
4. Existence of the unwanted nuclear spin-qubit decoherence channel determined by the interaction with the
environment by means ofthe impurity atom's electron.
RESONANT-TUNNELING SOLiD STATE NMR QUANTUM COMPUTER.
In our work we propose a novel resonant tunnel solid-state NMR quantum computer scheme, where significant
disadvantages of Kane model are overcome (Fig. 2).
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There a donor nuclear spins embedded in 3D semiconductor quantum dots are qubits and electron plays role
the information carrier. The electron ground-state in the dot coincides with the donor level. The structure composed of
such dots could be considered as a quantum register. Calculation process is similar to that described in the well-known
Kane's scheme of solid-state NMR quantum computer. Such quantum computer should also be classified as a solid-state
NMR QC but it differs from that proposed by Kane. The main difference arises from dynamics rather than calculation
part. Both devices use the electron transfer to interchange the information encoded in nuclear spins between them. In
the Kane's scheme it was supposed to realize by suitable manipulating on the gate voltages. Thus a set of consequent
operations should be performed to transfer ancilla's electron qubit from one nucleus to any other. But in the resonant-
tunneling NMR QC together with the mechanism analogous to mentioned above the other way exists that should be
distinguished from the procedure proposed by Kane. The point is based on the resonant electron transfer owed to
specific features of electron spectrum in the periodic potential (resonant tunneling). It is possible to leave only one
valent electron in the considering cell, when other electrons must be displaced from this area. Similar single electron
manipulations are discussed by K. Likharev [7].
It is worth of noting that using only one electron to be the connection between two nuclei significantly
decrease decoherence processes in quantum computer due to electron-phonon interaction. Also, owing to other valent
electrons absence metallic gates do not influence them (including other donors' electrons), that is why considered
scheme is more simple to control.
Between all qubits-nuclei in the register the only one has a different resonant frequency owing to the hyperfme
interaction (which also depends on gate voltage) with its electron while there are no electron on the others. That is why
individual addressing to the qubit is realized and the other qubits remain undisturbed when rf pulse of the quantum
protocol is made. Possible improvement of this scheme consists in using several quantum registers with the single
electron in each of them. Individual addressing in such approach is achieved by turning on complementary magnetic
field gradient.
Figure 2. The scheme of double-cell semiconductor structure of the resonant-tunneling NMR quantum computer,
'cell 3Onm, I IOOnm, c 2Onm.
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133
When excited the electron spreads over the structure and may, in principle, directly interact with all donors that
in this case to be thought equivalent. So one can realize the tmnsfer between remote qubits at once as a single operation.
Consider a single electron in the system ofN quantum dots separated by finite potential. Each independent isolated QD
has a well-localized ground-state and an excited state near the barrier top (Fig. 3). When dots are coupled and only one
electron is inserted, the electron spectrum consists of N-degenerate ground state (if the dots are identical) and the
excited bound, consisting ofN sublevels. It comes from the fact the excited energy levels of isolated dots coincide the
electron transfer between dots removes the degeneracy if tunneling is sufficiently large. Since the groundstates are
supposed to be isolated each from other there is no coupling and tunneling may be neglected. The energy differences
between these states are defmed by differences in the dot sizes. From the other hand the potential barriers must be rather
transparent for excited levels. So the excited states depend strongly on the tunneling energy. New states wave functions
have even or odd-parity and spread over the structhre. It is not nesessary to analize the total energy spectrum of the
system. we just need to know that two bounds ground and excited - exist.
_____
-—_2______ -—
Energy
U
—E
C __
- ____ -
_______
— El
A
B
Figure 3. Energy diagram oftwo coupling quantum dots.
If the electron is localized in the ground-state of one dot, it can stay in this state for extremely long time since
tunneling is suppressed. To translate the electron from initial state to groundstate of other dot an electromagnetic field
with appropriate resonant frequency should be applied. As it was demonstrated by Openov [5], the resonant
electromagnetic field pulse forces the electron to shuttle between groimdstates in the symmetric doubledot structure.
The electron at the initial moment located in the dot A may be moved by the laser perturbation from the ground
localized state to the excited one, where it immediately becomes spread over both dots, and then it may be moved by the
same perturbation acting on both dots to the ground localized state of the other dot. In [5} it was shown that if laser
pulse has amplitude E0 -' 10 ÷100L , duration 7 109s and resonance offset
" iO ÷lO6eV the value of the
probability of the considering transfer is close to 1. It is worth noting that owing to the duration of the laser pulse the
electron may be distributed on several dots simultaneously providing complex many-qubit operations. To account for
such phenomena in a common way the second..quantized approach has been applied. The results clearly demonstrate the
main features of electron's behavior in such a system (electron oscillations, etc.). Our numerical modeling has shown
the possibility of the resonant transfer of the electron between remote quantum dots [8]. We have analyzed such
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dynamics of the electron's behavior under laser perturbation in the structure consisted on large amount of nonequivalent
quantum dots As the results we can formulate individual addressing mode parameters. The possibility of this mode is
the demonstration of the possibility of normal quantum computer functioning in the case of large registers. Our
assumption includes qwintum register description in terms of secondary quantization. From this point of view one can
ignore an apparent structure of the eleclroifs wave function in the separate dot. That is why this approach is fruitful
owing to the fact that it can be used to describe quantum dots of any geometry. The only requirement consists of two
energy levels existence in each dot as well as the vicinity of the excited level to the top of the potential barrier. The
consequence ofthis feature is the possibility to describe correctly not only linear quantum register, but two dimensional
array of quantum dots as well. This occasion helps us to explore so called "planar architecture" of resonant-tunnel solid
state NIMR quantum computer. Iflarge amount of equivalent registers is conducted uniformly then this architecture type
is called ensemble. The advantage ofthis variant is the simple procedure ofthe final quantum state measurement, which
is similar to the analogous one in liquid bulk ensemble NMR quantum computers.
The quantum calculations are performed as follows. The single qubit operations are the same of the Kane
scheme. The two-qubit operations implementation, for example CNOT gate, evolves by the next way.
Operation
After initialization
Swap operation between A-qubit and
the electron
Resonant transfer ofthe electron from
the nucleus A to the nucleus B
CNOT-gate between the electron and
B-qubit
Resonant transfer oftbe electron from
the nucleus B to thenucleus A
Swap operation between the electron
and A-qubit
Aqubit's state
1A
B-qubit's state
Electron's state
'F13
10>
0>
'i's
The electron spin state is conserved during the electron's
transport
JO>
'PB
CNOT NA, B>
-
The electron spin state is conserved during the electron's
transport
CNOT 'PA, 'PB>
'PB
1°>
At the first step the information encoded into nuclear spin of the donor impurity is transmitted to the electron
spin by the SWAP gate. Then the electron is transferred to the other dot by resonant laser pulse. Here the required
CNOT gate is made and the electron's spin state is changed under the influence of the nuclear spin state of the qubit B.
Finally, the electron is transferred to the initial dot A and transmits its spin state to the nuclear spin A.
One of the most important problems in quantum computing is the problem of the fmal state measurement.
Unfortunately, the absence of the energy level anticrossing effect prevents using it to measure the final state, as it was
made in [9]. We suppose recent suggestion [10] to be the promising to solve this problem. This scheme can be used not
only for single spin state measurements, but for spin system state measurement as well. The measurement concept
consists in several steps: one has to prepare a suitable auxiliary system (ancilla) at an arbitrary moment of time, turn on
the interaction between the ancilla and the studied system, leave them for their joint unitary evolution, turn off the
interaction at an arbitrary moment of time, and detect the state of the ancilla. When one wants to measure any qubit's
state, SWAP operation between that nucleus spin-qubit and the electron must be made. After that electron spin state
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135
measurement with the help of detecting definite spin polarization electron current completes the measurement process
by the denoted way. Our proposal consists in using resonant transfer of the electron, which carries quantum information
from one qubit to another. This electron will also be used to transmit fmal state of some qubit to the detector. Detailed
theoretical discussion of electrical methods of electron and nuclei spin states measurement in semiconductor structures
with the presence of different noises using single electron transistor (SET) was made in [1 11. Among the other electrical
detecting methods are the next: using heterostructure spin filter with ferromagnetic barrier [12] and the method of
magnetic resonance force microscopy [13].
Simple analisys gives us the excited-states energies ofN coupled identical dots:
'Tn
Sn =s0+2Vcos— n=1... N
N+1
so - the excited-state energy of isolated dot. The energy difference between adjacent sublevels depends on
both its positions in the spectrum and the number of dots N: Ann+i = e —n+1
In the case of N>>1 for sublevels
closed to the bound edges we have
.
A12
AN1N = 4Vsin
3r
.
2V
sin
2(N+1)
3'T —
2(N+1)
N2
and for central zone
A
[][1+1
.
=4Vsin
2t
V
2(N+1)
N
The bound width depends on the tunneling matrix element V: A1N = 4V Sjfl!L—! 4V . The qualitative
estimates for number of dots N follow from the restriction imposed on the value of detuning by resonant conditions
S << Ann+i
n=
Setting V '- 1O2eV, 8 - 1OeV that can be realised by up-to-date optical techniques, we get N iO for
[1 and N 102 for n 1, n ' N . Consequently for the gate voltage we have SU iO ÷ lO V .
[2]
We adopt the
Kane's gate geometry but, naturally, for essentially different aimes. There are two ways to realize our proposal. First
described above implies the gate voltages act on N - 2 dots. But in order to complete the electron transfer between two
dots one can operate with only these dots. Starting from the register state where all dots are off resonanse with
electromagnetic field we apply the gate voltage to the two required dots and turn them on resonanse. However, in reality
we deal with non-ideal QD's, and some difficulties will arise from the coupling the excited levels of individual dots
together. To resolve this problem the set of gates may be used again. It is not a good complication relative to the QC
architecture for many reasons. Moreover, it is not nesessary that for an array of such dots "fitted" to identical
(A = E0d,, where P20 - the laser field amplitude, ci, - the optical dipole matrix element in the i-th dot) will be equal. As
e
an example for N=3 there are three excited sublevels with energies e0,
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V.
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In this system we have
=
1M for any excited state. It is easily to show if these states are used as the
transport levels the electron transfer between central dot and others has a very small probability.
For these reasons it will be helpftill to consider the next variants of the electron transfer, combining the set of
gates, the resonant tunneling and the field-induced electron transition. An array of QD's that have a smalldifferences in
their linear sizes is proposed here as a potential candidate for quantum register design. The set of gates is applied only
to control the resonant tunneling.
Since each dot exhibits the individual optical resonant frequency, the electron transport is realized as the
sequential process.
Initially only one of dots (sayj-th) containing the electron in its ground-state is on resonance with laser field
and the transition between ground and excitedstates is described by Rabi oscillations. Namely, we deal with the case
of typical two-level oscillations. If the gates are switched on during the resonant r - pulse the electron is drown to the
excited bound and spreads over the structure. Further we apply r pulse with appropriate frequency to drive the
electron to the ground-state of other dot. This scheme requires all gates to be turned on simultaneously (parallel dot
connection).
If the gates are switched off the electron will be found in the excited state ofj-th dot. To bring the electron
into adjacent dot we must align the excited levels of this dots together. It may be performed by applying the gate
voltage to one of these dot (with time duration). Next we drive the electron into the ground-state of adjacent dot by
resonant if - pulse or bring it along array by gating.
The procedure presented above allows one's to carry out the electron transfer between two arbitrary dots in
QD's array. We have seen this transfer is possible as soon as the interface control including the set of gates is
performed. It was shown there are at least two methods to realize the electron transfer and to make the system work as
the quantum register for quantum calculations scheme for resonant-tunneling solid-state NMR quantum computer. The
results may be qualitatively applied to any QD system where the electron dynamics exhibits quasi-one-dimensional
character.
We keep in mind the fact that the nuclear spins are the natural qubits the nuclear magnetic resonance technique
is rather developed to be applied to the simplest qubit manipulations. In addition to the very useful experimental base,
the silicon nanotechnology seems to be a very promising tool for the quantum computer's embodiment. Combining both
techniques we will be able to operate with more ordinary things rather than with somewhat unrealistic and unrealizable
models.
CONCLUSION.
Let us pay attention on the main features of the proposed variant of solid state NMR quantum computer. The
original architecture has more simple basic elements owing to the single electron usage. It is very important in
numerical simulation and computational process conduction. The significant occasion is considerable decoherence
decrease owing to collective excitations absence, lower phonon influence and metallic gate noise influence decrease.
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The proposed architecture has all advantages of quantum computers based on quantum dots, but it has one more further
because of its basic elements are natural qubits - nuclear spins which are enough insulated from the environment. The
consequence of the last fact is high stability of the quantum register state. Not the least feature is the existence of
controllable interaction between any qubit ofthe register. Because ofnuclear spins are separated from each other so far
that one can ignore even indirect interaction between them, generation of unwanted states during computational process
usual for solid state NMR quantum computer does not take place. The stated feature is principal since it gives an
occasion to describe quantum registers with large amount of qubits almost without limits. The presumed action on the
electron, which transfers it from one qubit to another, does not influence the spin part of the electron's wave function.
That is the reason of decoherence decrease and it helps to describe separately and, consequently, more adequately the
computational process and the process of electron transfer. Doubtless advantage of the considering architecture is its
resourcefulness regarding to applying of measurement methods of the fmal (and intermediate if necessary) state. With
the help of single electron necessary quantum information may be transferred to the detector, which can by no way be
connected with the register. Finally, the possibility of electron transfer, which forms the effective interaction between
any qubits, allows to enlarge significantly distinctive scale of the elementary cell of the register. This is an important
fact from the technological point of view.
REFERENCES.
1. Kane B.E. "A Silicon-based Nuclear Spin Quantum Computer." Nature, 393, 133 (1998).
2. Valiev K. A., Kokin A. A., Larionov A. A., Fedichkin L. E. "Nuclear magnetic resonance spectrum of 31P
donors in silicon quantum computer." Nanotechnology, 11, 392 (2000).
3. O'Brien L.J., Schofield S.R., Simmons M.Y., Clark R.G., Dzurak A.S., Curson N.J., Kane B.E., McAlpine
N.S., Hawley M.E., Brown G.W. "Towards the fabrication of phosphorus qubits for a silicon quantum
computer." Phys Rev B 64, R161401 (2001).
4. Larionov A. A., Fedichkin L.E., Valiev K.A. "A silicon-based nuclear magnetic resonance (NMR) quantum
computer using resonant transfer of a single electron for the inter-qubit interaction." Nanotechnology, 12, 536
(2001).
5. Openov L. A. "Resonant electron transfer between quantum dots." Phys Rev B 60, 8798 (1999).
6. Berman G. P., Kamenev D. I., Tsifrmnovich V. I. "Perturbation approach for a solid-state quantum
computation." LANL e-print quant-ph/01 10069.
7.
Likharev K. "Dragging single electrons." Nature, 2001,410, pp.53 1-533.
8. Larionov A.A. "Resonant electron transfer between remote nonequivalent quantum dots." Quantum Computers
& Computing, 2002, to be published
9. Larionov A.A., Fedichkin L.E., Kokin A.A., Valiev K.A. "Nuclear magnetic resonance spectrum of 31P donors
in silicon quantum computer." Nanotechnology, 11, Th 4, 2000, pp. 392-396;
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