Quantum computing with bits made of electrons on a helium surface
We describe a quantum computer based on electrons supported by a helium film and localized laterally by small electrodes. Each quantum bit (qubit) is made of combinations of the ground and first excited state of an electron trapped in the image potential well at the surface. Mechanisms for preparing...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2003
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| Цитувати: | Quantum computing with bits made of electrons on a helium surface / A.J. Dahm // Физика низких температур. — 2003. — Т. 29, № 6. — С. 659-662. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1288722018-01-15T03:04:48Z Quantum computing with bits made of electrons on a helium surface Dahm, A.J. 3-й Международный семинар по физике низких температур в условиях микрогравитации We describe a quantum computer based on electrons supported by a helium film and localized laterally by small electrodes. Each quantum bit (qubit) is made of combinations of the ground and first excited state of an electron trapped in the image potential well at the surface. Mechanisms for preparing the initial state of the qubit, operations with the qubits, and a proposed readout are described. This system is, in principle, capable of 10⁵ operations in a decoherence time. 2003 Article Quantum computing with bits made of electrons on a helium surface / A.J. Dahm // Физика низких температур. — 2003. — Т. 29, № 6. — С. 659-662. — Бібліогр.: 4 назв. — англ. 0132-6414 PACS: 03.67.Lx, 73.20.-r http://dspace.nbuv.gov.ua/handle/123456789/128872 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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3-й Международный семинар по физике низких температур в условиях микрогравитации 3-й Международный семинар по физике низких температур в условиях микрогравитации |
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3-й Международный семинар по физике низких температур в условиях микрогравитации 3-й Международный семинар по физике низких температур в условиях микрогравитации Dahm, A.J. Quantum computing with bits made of electrons on a helium surface Физика низких температур |
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We describe a quantum computer based on electrons supported by a helium film and localized laterally by small electrodes. Each quantum bit (qubit) is made of combinations of the ground and first excited state of an electron trapped in the image potential well at the surface. Mechanisms for preparing the initial state of the qubit, operations with the qubits, and a proposed readout are described. This system is, in principle, capable of 10⁵ operations in a decoherence time. |
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Dahm, A.J. |
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Dahm, A.J. |
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Quantum computing with bits made of electrons on a helium surface |
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Quantum computing with bits made of electrons on a helium surface |
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Quantum computing with bits made of electrons on a helium surface |
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Quantum computing with bits made of electrons on a helium surface |
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Quantum computing with bits made of electrons on a helium surface |
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quantum computing with bits made of electrons on a helium surface |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2003 |
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3-й Международный семинар по физике низких температур в условиях микрогравитации |
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Quantum computing with bits made of electrons on a helium surface / A.J. Dahm // Физика низких температур. — 2003. — Т. 29, № 6. — С. 659-662. — Бібліогр.: 4 назв. — англ. |
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Физика низких температур |
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AT dahmaj quantumcomputingwithbitsmadeofelectronsonaheliumsurface |
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2025-07-09T10:08:22Z |
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2025-07-09T10:08:22Z |
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1837163565088243712 |
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Fizika Nizkikh Temperatur, 2003, v. 29, No. 6, p. 659–662
Quantum computing with bits made of electrons
on a helium surface
A.J. Dahm
Department of Physics, Case Western Reserve University
Cleveland, OH 44106-7079, USA
E-mail: ajd3@cwru.edu
Received December 19, 2002
We describe a quantum computer based on electrons supported by a helium film and localized
laterally by small electrodes. Each quantum bit (qubit) is made of combinations of the ground and
first excited state of an electron trapped in the image potential well at the surface. Mechanisms for
preparing the initial state of the qubit, operations with the qubits, and a proposed readout are de-
scribed. This system is, in principle, capable of 105 operations in a decoherence time.
PACS: 03.67.Lx, 73.20.–r
1. Introduction
In a quantum computer two stationary states of a
quantum system are identified with classical bits 0
and 1. Each quantum bit (qubit) is made up of a su-
perposition of these two quantum states. The state of
the nth qubit
� = a 0 + b 1 ; �a�2 + �b�2 = 1 . (1)
In the majority of operations an = 0, 1 or 2–1/2.
A quantum computer is a superposition of all qubit
states. A simple superposition is a product of indivi-
dual qubit states. The general state of a quantum com-
puter is an entangled state, a state that cannot be
made of a product of individual qubits. An example is
the state
2–1/2 ( 01 + 10 ). (2)
One method of producing an entangled state is by us-
ing a controlled NOT (CNOT) gate. This gate con-
sists of a control bit and a target bit. If a control bit
is 0 the target bit is unchanged, while if the control
bit is 1 the components of the target bit 0 and 1
change to 1 and 0 , respectively. A CNOT gate is de-
scribed below. An example of an entangling operation
is with the first bit as the control bit
2–1/2( 0 + 1 )1 = 2–1/2( 01 + 10 ) . (3)
The general state is written as
� = � j
j
jx� ,
j
���
j
�2 = 1, (4)
where each basis vector xj is one of the 2n permu-
tations of the zeros and ones representing separate
qubits. In the final state one of these basis vectors is
the answer to the calculation. The entangled state
collapses when one of the qubits is read. For the most
general algorithms all qubits must be read out simul-
taneously.
We describe here a proposed quantum computer
that uses laterally confined electrons on the surface of
a liquid helium film as qubits and describe operations
with these qubits including a simultaneous readout. A
full description of quantum computing is beyond the
scope of this paper [1]. This concept was first intro-
duced by Platzman and Dykman [2].
2. Electrons on helium
Electrons are bound to the surface of liquid helium
by the dielectric image potential. A repulsive Pauli
potential that can be represented to a good approxima-
tion as V = � for z < 0 prevents them from penetrating
into liquid helium. The hydrogenic-like potential for
z > 0 is
V = –�e2/4� 0z; � = (
– 1)/4(
+ 1) , (5)
© A.J. Dahm, 2003
where z is the coordinate normal to the surface, and
is the dielectric constant of helium. The energy levels
form a Rydberg spectrum, En = – R/n2. The parame-
ters for liquid 3He are �3 = 0.00521, R3 = 0.37 meV,
and the effective Bohr radius is aB = 10.2 nm. The ave-
rage separations of the electron from the surface are
z = 15.3 and 61 nm for the ground and first excited
state, respectively. The transition frequency between
the ground and first excited state is 70 GHz. These
transitions can be shifted with a Stark field applied
normal to the surface. The potentials are shown in
Fig. 1 for applied electric fields F > 0, F = 0 and
F < 0.
3. Design of the computer
We identify the ground and first excited states of
these electrons with the states 0 and 1 , respectively.
In order to address and control the qubits each elec-
tron must be localized laterally. This will be accom-
plished by locating electrons above microelectrodes
(posts) that are separated by about 1 �m. The elec-
trons will be separated from the posts by an � 1 �m
thick helium film. Lateral confinement results from
the image potentials of the posts and the potential ap-
plied on the posts. The electron will be in the ground
state for lateral motion.
A schematic of posts and electrons for a four-qubit
system is shown in Fig. 2. A voltage applied to a given
post controls the Stark field for the corresponding
electron. An array of posts with leads has been fabri-
cated.
A schematic of our cell is shown in Fig. 3. The
plates form the top and bottom of an enlarged wave-
guide that transmits sub-mm radiation to the elec-
trons. Superconducting micro-bolometers will be lo-
cated at the top of the guide to detect electrons that
are allowed to escape from the posts. A tunnel-diode
electron-emission source will be located above the
electron detectors. Electrons will be loaded onto the
film through a hole in the detector chip, and one elec-
tron will be trapped over each post by image and ap-
plied potentials. The helium film thickness will be
measured with a capacitor made of metal strips depos-
ited on the ground plane. The system will be operated
at 10 mK to increase coherence times of the qubit
states.
4. Operations
Data input: The operation would normally begin
with all qubits in the ground state. Then initial data
660 Fizika Nizkikh Temperatur, 2003, v. 29, No. 6
A.J. Dahm
V
m=2
m=1
100 GHz
70 GHz
F < 0
F > 0
F = 0
z
Fig. 1. The potentials and energy levels with and without
an electric field F applied normal to the surface. The
ground (m = 1) and excited (m = 2) energy levels are in-
dicated schematically for each potential. The potential for
an extracting field F < 0 is also shown as a dashed line.
Fig. 2. The geometry of a four-qubit system with electrons
above the microstructure and the helium film. The draw-
ing is not to scale. Optimal dimensions are d � h � 1 �m.
Control potentials Vn are applied on the microelectrodes.
Fig. 3. Schematic of the cell. The upper plate includes de-
tectors used in the readout. The lower plate includes the
posts and is covered with the helium film. The electrons
float over the posts about 10 nm above the surface of the
helium film.
will be input by preparing each qubit in some admix-
ture of states 0 and 1 by Stark shifting individual
qubits into resonance with microwave radiation for a
predetermined time. The state of the qubit n will be
�
n
= cos (
n
/2) 0 – i sin (
n
/2) 1 , (6)
where
n = ��n, � = eErf 1 2z /� is the Rabi fre-
quency, Erf is the strength of the rf field, and �n is the
time the nth qubit is in resonance with the microwave
field.
Quantum gates: In general, computations will be
implemented by applying pulses of radiation to inter-
acting qubits. We illustrate a potential operating
mode of the system by describing two qubits operated
as a SWAP gate. The interaction is the Coulomb inter-
action between neighboring electrons. The dipolar
component of the direct interaction potential between
qubits i and j is
V(z
i
, z
j
) � (e2/8� 0d
3)(z
i
– z
j
)2, (7)
where d is the electron separation, and zi is the sepa-
ration of the ith electron from the helium surface.
Start with one qubit in the state 0 and the other in
the state 1 . Next apply the same Stark fields to both
qubits so that the states 01 and 10 would be degen-
erate. In this condition the system will oscillate be-
tween the two states at a frequency given by the in-
teraction energy, which in first order is given by
e2aB
2/4� 0d
3. This frequency is � 108 Hz for a sepa-
ration of 1 �m. By leaving the electric fields in this
condition for one half cycle of this oscillation, the
two qubits will swap states. It will be difficult to
tune neighboring qubits to precisely identical Stark
shifts, and in practice we may sweep the Stark shift
of one qubit through resonance with a neighboring
qubit. In this case, the final state of the qubit will de-
pend on the rate at which the electric field is swept
through the resonance condition.
A two-qubit CNOT gate can be operated as follows.
The energy for the target bit to make a transition de-
pends whether the separation of the electrons is in-
creased or decreased in the transition. The transition
frequency is
�������
�
� (e2/8� 0d
3h)(z2 – z1)
2, (8)
respectively, when the control bit is in the 1 or 0
state. Here ��� is the frequency in the absence of in-
teractions, h is Planck’s constant, and subscripts refer
to the ground and excited states. By applying radia-
tion at one of the frequencies, a transition will or will
not occur depending on the state of the control bit.
The control bit is Stark shifted out of resonance.
Readout. For the general case the states of all
qubits must be read within the time scale is set by the
plasma frequency � 100 GHz. We describe here our
initial proposal for a destructive readout pending re-
search into other schemes. We will apply a short,
� 1 ns, ramp of an extracting electric field to all
qubits. The potential for a fixed value of extracting
field is shown in Fig. 1. The tunneling probability is
exponential in the time-dependent barrier height and
width. All electrons in the upper 1 state will tunnel
through the barrier within a short period of time when
this probability becomes sufficiently large. For this
extracting field the tunneling probability will be neg-
ligibly small for electrons in the ground 0 state. After
the ramp is removed the remaining electrons will be in
the ground state. Subsequently, an extracting field
sufficiently large so that electrons in the ground state
will tunnel from the surface [3] will be applied se-
quentially to each post. A 0 will be registered for
each electron detected by the transition-edge
bolometer and a 1 for those states that are empty.
5. Decoherence
T1: Logic operations must be accomplished in less
time than it takes for the interactions of the qubit
with the environment to destroy the phase coherence
of the state functions. For electrons on 4He the life-
time of the excited state 1 is limited by interactions
with ripplons and coupling to phonons in the bulk li-
quid. These processes are discussed in detail by Dyk-
man et al. [4]. The electron-ripplon coupling Hamil-
tonian is
H er = eE
�
�, (9)
where E� is the normal component of the electric
field that includes both the applied field and varia-
tions in the helium dielectric image field due to sur-
face distortions, and � is the amplitude of the surface
Quantum computing with bits made of electrons on a helium surface
Fizika Nizkikh Temperatur, 2003, v. 29, No. 6 661
Fig. 4. Energy level schematic. The integer m labels the
hydrogenic-like states, and the integer l labels lateral
states, which represent harmonic oscillator states or Lan-
dau levels in a magnetic field. The hatched region indi-
cates the band of plasma oscillations associated with each
level.
height variation. The average rms thermal fluctuation
of the surface is
�
T
= (k
B
T/�)1/2 � 2�10–9 cm. (10)
The transition from the excited to ground state re-
quires a ripplon with a wave vector � aB
–1. For a sin-
gle electron on bulk helium a radiationless transition
occurs with the energy absorbed by electron
plane-wave states for motion parallel to the surface
and momentum absorbed by ripplons. For this case a
calculation of T1 yields
T1
1� � ��(�T/a
B
)2, (11)
where �� is the transition frequency. At T = 10 mK,
�T/aB � 10–3, and T1 � 10 �s.
For electrons confined by posts, the lateral states
are harmonic oscillator states of the image potential
well of the posts. These are separated in energy by
��1 � �(eE�/4� 0mh)1/2 � 1 K for E� = 500 V/cm.
For interacting electrons there is a band of plasma os-
cillations associated with each harmonic oscillator
level, which for a crystal array with a separation of
1 �m has a bandwidth that is 300 mK. The frequency
�1 can be tuned so the transition 1 � 0 is in-
commensurate in energy with the excitation to the
plasmon band of any harmonic level and suppresses
this channel for decay. This is illustrated in Fig. 4.
The separation of energy levels for lateral motion can
also be accomplished with the application of a mag-
netic field.
Decay can occur with the emission of two ripplons
with opposite wavevectors and the excitation of a har-
monic level nearest in energy to the 1 state plus
plasmons. This yields a value of T1 � 1 msec.
The dominant T1 decay mechanism is the emission
of a phonon into the bulk liquid with the excitation of
harmonic energy levels. The coupling is through
phonon-induced modulation of the image potential of
an electron and leads to a decay time of � 30 �sec.
T2: The phase of the wave-function varies a [U(t) +
+ e�(t)]t/�, where U and � are the energy of a state
and the electrostatic potential, respectively.
Dephasing occurs due to a decay of the phase differ-
ence between the two qubit states 1 and 0 . The dom-
inant dephasing mechanism is estimated to be Johnson
noise in the micro-electrodes. For a 25 � resistor at T
= 1 K attached to the posts, the dephasing time is esti-
mated to be T2 � 100 �sec. A two-ripplon scattering
process modulates the energy of the states and leads to
a value T2 � 10 msec.
Acknowledgements
The authors wish to acknowledge Mark Dykman
and John Goodkind for helpful conversations. This
work was supported in part by NSF grant
EIS-0085922.
1. A review of concepts for quantum computers is given in
Fortschr. Phys. 48, Issue 9–11, (2000).
2. P.M. Platzman and M.I. Dykman, Science 284, 1967
(1999).
3. G.F. Saville and J.M. Goodkind, Phys. Rev. A50, 2059
(1994).
4. M.I. Dykman, P.M. Platzman, and P. Seddighrad, to
be published.
662 Fizika Nizkikh Temperatur, 2003, v. 29, No. 6
A.J. Dahm
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