CMOS Position-Based Charge Qubits: Theoretical Analysis of Control and Entanglement

Abstract

In this study, a formal definition, robustness analysis and discussion on the control of a position-based semiconductor charge qubit are presented. Such a qubit can be realized in a chain of coupled quantum dots, forming a register of charge-coupled transistor-like devices, and is intended for CMOS implementation in scalable quantum computers. We discuss the construction and operation of this qubit, its Bloch sphere, and relation with maximally localized Wannier functions which define its position-based nature. We then demonstrate how to build a tight-binding model of single and multiple interacting qubits from first principles of the Schrödinger formalism. We provide all required formulae to calculate the maximally localized functions and the entries of the Hamiltonian matrix in the presence of interaction between qubits. We use three illustrative examples to demonstrate the electrostatic interaction of electrons and discuss how to build a model for many-electron (qubit) system. To conclude this study, we show that charge qubits can be entangled through electrostatic interaction.

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CMOS Position-Based Charge Qubits:
Theoretical Analysis of Control and
Entanglement
ELENA BLOKHINA1(Senior Member, IEEE), PANAGIOTIS GIOUNANLIS*,1 (Member, IEEE),
ANDREW MITCHELL3, DIRK R. LEIPOLD2(Member, IEEE), and ROBERT BOGDAN
STASZEWSKI1,2 (Fellow, IEEE)
1School of Electrical and Electronic Engineering, University College Dublin, Dublin 4, Ireland (e-mail: elena.blokhina@ucd.ie)
2Equal1 Labs, Fremont, CA 94536, USA (e-mail: dirk.leipold@equal1.com)
3School of Physics, University College Dublin, Belfield, Dublin 4, Ireland (e-mail: andrew.mitchell@ucd.ie)
∗Corresponding author: Panagiotis Giounanlis (e-mail: panagiotis.giounanlis@ucd.ie).
This work was supported by Science Foundation Ireland under Grant 14/RP/I2921.
ABSTRACT In this study, a formal definition, robustness analysis and discussion on the control of
a position-based semiconductor charge qubit are presented. Such a qubit can be realized in a chain of
coupled quantum dots, forming a register of charge-coupled transistor-like devices, and is intended for
CMOS implementation in scalable quantum computers. We discuss the construction and operation of this
qubit, its Bloch sphere, and relation with maximally localized Wannier functions which define its position-
based nature. We then demonstrate how to build a tight-binding model of single and multiple interacting
qubits from first principles of the Schrödinger formalism. We provide all required formulae to calculate
the maximally localized functions and the entries of the Hamiltonian matrix in the presence of interaction
between qubits. We use three illustrative examples to demonstrate the electrostatic interaction of electrons
and discuss how to build a model for many-electron (qubit) system. To conclude this study, we show that
charge qubits can be entangled through electrostatic interaction.
INDEX TERMS CMOS technology, charge qubit, position-based qubit, electrostatically controlled qubit,
single-electron devices, tight-binding formalism, Schrödinger formalism, entanglement, entanglement
entropy, Bloch sphere.
I. INTRODUCTION
Quantum computing is a still-emerging paradigm that utilizes
the fundamental principles of quantum mechanics such as
superposition and entanglement. The range of complex prob-
lems from mathematics, chemistry and material science that
could be solved with quantum computing is immense [1]–[3].
A quantum bit (qubit) is the basic unit of quantum informa-
tion, typically comprising a nanoscale quantum mechanical
two-state system. The qubits are extremely fragile and dif-
ficult to manipulate and read out, since all the operations
until the final read-out must be done non-destructively to
preserve the crucial property of quantum coherence. They
typically require extremely low, cryogenic temperatures to
operate in order to preserve their coherent superposition state.
Furthermore, quantum computation requires a fault-tolerant
manipulation of many coupled qubits, making the search
for robust qubits suitable for mass production a necessity.
In this regard, an appealing paradigm for scalable quantum
computation would exploit existing advanced semiconductor
technologies.
The quantum computer differs from the classical digital
computer in the sense that instead of using a binary digit
(bit) to represent Boolean logic states of ‘0’ or ‘1’, it uses
a qubit which can be in a superposition of quantum coun-
terpart states of |0iand |1i. Among a number of proposed
technologies for realizing quantum computation, one can
highlight the following ones. Possibly the most promising
approach from the point of view of decoherence time is
based on trapped ions [4]. It has been recently demonstrated
that trapped ions and photons can achieve a large number
of entangled qubits [5]. Unfortunately, they are difficult to
manipulate and are perceived as rather unsuited for a very
large-scale integration (although the research into trapped
ions is actively ongoing). Solid-state methods, on the other
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Blokhina et al.: CMOS Position-Based Charge Qubits: Theoretical Analysis of Control and Entanglement
SD
VDS VG
CG
Tunn el ing
Barrier
Tunn el ing
Barrier
Island
FIGURE 1. Schematic diagram of a single-electron transistor (SET), a device
allowing a single injected electron to be manipulated via the tunneling barriers
by controlling voltages applied at the source (S), drain (D) and gate (G)
terminals.
hand, rely on the usage of collective physical phenomena
of macroscopic quantum states as, for instance, observed
in superconductors or in superfluids [6]. Integrated, on-chip
solutions based on superconductors (employing Josephson
junctions) are currently the dominant technology with the
number of qubits ranging from 19 to 72 and operating at
extremely low temperatures of 15 mK. There are many other
promising theoretical proposals based on topological quan-
tum computation by braiding Majorana fermions or other
non-Abelian quasiparticles in condensed matter systems [7].
However, none of these proposals have been experimentally
realized to date. Some further discussion on state-of-the-art
of qubits can be found in [8].
Over recent years, there has been ongoing research on
semiconductor qubits [8]–[16] due to their promising com-
patibility with batch fabrication and enormous progress
in CMOS fabrication technologies. Inspired by previous
and very recent works on charge-based semiconductor
qubits [8], [10], [11], [16]–[18] and spin-based semicon-
ductor qubits [19]–[21], here we discuss the feasibility of
realizing a semiconductor charge qubit in the CMOS tech-
nology. Our proposed charge semiconductor qubit originates
from a fundamental device known as a single-electron device
(SED) [22]–[25]. This device allows to precisely control
and manipulate individual electrons. To further facilitate the
electron transport and control, the SED can be refined into
a single-electron transistor (SET), see Fig. 1. Multiple gate
extensions will enable controllable movement of individual
electrons as quantum dot as their superposition and entangle-
ment [8], [9], [16], [26]. A qubit based on the SET is also
referred to as a ‘charge qubit’, and was very actively studied
between the early 90s and mid-noughties (see, for instance,
Ref. [27]).
However, concerns about the decoherence time and charge
noise [11], resulted in a period of relative inactivity in this
field, compared with other technologies. Very recently, in-
terest in charge qubit devices has resurged, in part because
previous problems are now mitigated by new technological
advances, especially the fine nanometer-scale feature size of
CMOS lithography and short propagational delay afforded by
the cut-off frequency (fT) reaching almost a terahertz.
Our research is motivated by recent, significant efforts
made to advance quantum qubits and quantum gates imple-
mented in semiconductor and, in particular, CMOS technolo-
gies, with a number of very recent studies reporting silicon
quantum dots [19]–[21], [28]–[30] and support electron-
ics [16], [31]–[37]. These studies demonstrate a more practi-
cal view on quantum computing, highlighting the feasibility
of large-scale fully integrated quantum processors, where
many qubits will be controlled by the means of conventional
electronic circuitry. The presented study is particularly rel-
evant to such implementations of quantum processors [16],
[38] which may become dominant architectures in the future.
In support of the scientific and technological feasibility of
the proposed charge qubit, the purity of silicon in modern
high-volume commercial nanometer-scale CMOS has dra-
matically increased since the previous wave of realizations
of the charge qubit [39]. This has been driven by increas-
ingly sophisticated CMOS lithographic processes with ultra-
high switching speed of devices as well as improved defect
tolerances, required to achieve ever increasing densities of
functioning transistors (tens of millions per mm2) on micro-
processor chips. Hence, the semiconductor qubit under study
exploits this highly refined CMOS manufacturing processes,
using ultra-pure silicon, precise control of doping, and high
control of the interface between silicon and silicon dioxide.
Semiconductor/silicon qubits have been studied both the-
oretically and experimentally in the literature, taking into
account spin, valley and orbital degrees of freedom [40], [41].
The manipulation of various quantum states of such systems
is based on the control of external electrical or magnetic
fields to achieve desired qubit operations [42]. Most of these
works rely on the manipulation of either spin or both the
spin and charge of a particle (hybrid qubits), and usually
are restricted to the analysis of a single double-quantum-dot
(DQD) or three quantum-dots [17], [41], [43], [44].
As aforementioned, a large number of various charge
qubit implementations have also been reported in the liter-
ature. Briefly outlining some implementations, we note that
charge qubits implemented as DQDs based on a Joseph-
son circuit [45], semiconductor charge qubits fabricated in
AlaAS/GaAs [46], a possibly large-scale implementation
of charge-based semiconductor quantum computing [47].
Among the most recent studies, Ref. [28] introduces high-
fidelity single-qubit gates. Charge qubits in van der Waals
heterostructures are discussed in [48]. Relevant to charge
qubits, an electron localization due to Coulomb repulsion is
investigated along with the time evolution of quantum states
in the presence of charge noise [28], [30], [45], [46], [49].
In this work, we focus our attention on a system based on
the electrostatic manipulation of single-electron semiconduc-
tor charge qubits implemented in modified structures based
on FDSOI 22 nm technology [16]. We extend the methodol-
ogy, commonly applied to a double quantum dot system, to
the case of multiple-particle qubits, each having an arbitrary
number of energy states, interacting electrostatically.
We also investigate entanglement between two or more
interacting qubits by the use of the Von Neumann entan-
glement entropy. Other approaches to measure entanglement
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Blokhina et al.: CMOS Charge Qubits: Theoretical Analysis of Control and Entanglement
between two qubits have been suggested in the literature,
for example with the use of a correlator or the concept of
concurrence [50], [51]. In this study, we are interested to
provide a proof of entanglement between an arbitrary number
of qubits or DQDs. However, we should mention that the
measurement of entanglement entropy is an open problem
and cannot be achieved in a straightforward way, especially
for multi-particle systems [52].
Lastly, we should point out that this study, even though
it is focused on charge qubits, can be extended to spin or
hybrid qubits, which are currently perceived as the leading
trend in CMOS. Specifically, particles with information en-
coded in spin also need to be moved around with positional
precision in a quantum processor. From this point of view,
each particle would also undergo, at some point during the
computations, some type of shuttling across different QDs to
reach another part of the processor and to carry the quantum
information. The body of theoretical work on this so-called
spin-bus architecture is extensive and has also been reported
experimentally [43]. The presented formalism in this study
extents the description from a DQD cell to a multiple-QD
cell (and even multiple multi-QD cells).
This paper is organized as follows. Section II presents the
statement of the problem, describing a chain of transistor-
like QD devices implementing a quantum register. Section III
proceeds with the formal development of a charge qubit
whose quantum logic states are defined by the detection of
an electron in a specific quantum dot of the quantum register.
That section provides the rigorous definition of the qubit
and shows how to construct maximally localized functions.
In addition, we discuss the robustness of the charge qubit
and possible methods to control the angles of its Bloch
sphere. Section IV is dedicated to the derivation from first
principles of the tight-binding model for this system. We
show how to obtain the Hamiltonian matrix elements and
solve the relevant equations for one qubit, and go on to extend
the procedure for multiple charge qubits interacting elec-
trostatically. Selected cases of interest, and comparison of
the tight-binding formalism with the Schrödinger equation,
are presented. Finally, Section V defines the Von Neumann
entanglement entropy SNin terms of the reduced density
matrix in the context of charge qubits. This is used to demon-
strate entanglement between states and the importance of
the Coulomb electrostatic interaction on the entanglement of
charge qubits.
II. DESCRIPTION OF THE SYSTEM UNDER STUDY
The key building block of the proposed semiconductor
charge qubit can be realized in CMOS fully depleted silicon-
on-insulator (FDSOI) technology [53] and is shown in
Figs. 2(a) and (b). It resembles a transistor and comprises two
depleted silicon dots separated by a silicon channel, which
acts as a tunneling barrier whose potential energy is con-
trolled electrostatically by the gate. Each dot acts as a single
quantum dot (QD). When the barrier separating the QDs is
very high, quantum mechanical tunneling is exponentially
FIGURE 2. (a) A representative example of a CMOS transistor-like
silicon-on-insulator device that serves as a coupled quantum-quantum dot
system. (b) Charge qubit formed by two coupled quantum dots, each of which
acting as a quantum dot (QD). (c) QDs in series forming a quantum register.
(d) Block diagram of the system showing an injector and a detector. The
injection of an electron is performed through an injector on the left quantum
dot of the register whilst the readout is carried out on the right quantum dot by
using a single-electron detector. The double-QD (DQD) system forms a charge
qubit. The potential function U(x, t)appearing in the Hamiltonian of the
system is controlled by the voltages applied at the terminals of the structure,
with the barrier UB(t)varying in time in the most general case. When in a
coherent state, an electron injected into such a system can tunnel quantum
mechanically through the barrier between the two quantum dots. The electron
exists in a superposition of left and right quantum quantum dot states
described by its wavefunction. Measurement of the electron position causes
wavefunction collapse (it is a destructive/projective measurement); the
electron is found to be in either the left or right quantum dot with probabilities
related to the wavefunction density (repeated independent measurements
yield the left and right position probabilities). Physical device parameters are
given in Table 1.
suppressed and the QDs are effectively decoupled. A single
electron injected into the system is then trapped in either left
or right dot and the quantum state has a very long lifetime.
By lowering the barrier, a single electron can tunnel between
the left and right dots in the double QD (DQD) device. The
potential barrier UB(t)between the two dots, controlled by
the voltage applied at the gate of the device, can vary with
time in the most general case, and hence allows control over
the electronic tunnelling in the DQD (see Fig. 2(d)). To
complete the structure, one adds an injector (a device which
is able to inject a single electron into one quantum dot) and
a detector (a device which is able to detect an electron at the
same or the other quantum dot). This geometry allows one to
define a charge qubit. We assume that the state of the qubit,
as a closed system, can be expressed as a superposition of
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Blokhina et al.: CMOS Position-Based Charge Qubits: Theoretical Analysis of Control and Entanglement
Well
Well
Well
Injector/
Detector
Injector/
Detector
Barrier
Barrier
Detector/
Injector
Imposer
(Gate) Detector/
Injector
Channel BoX
Wafer
Imposer
(Gate)
Imposer
(Gate) Injector/Detector
Injector/Detector
Wafer
Insulator
Injector/
Detector
Imposer Imposer Injector/
Detector
(a)
(b)
Imposer
voltage sweep
(c) Injector/
Detector
Imposer Imposer Injector/
Detector
BoX
Silicon Channel
FIGURE 3. (a) Schematic 3D structure containing gates (imposers), silicon
channel, buried oxide (BoX) and wafer. (b)Finite element method (FEM)
simulations of the electrostatically shaped potential energy as a function of the
coordinate along the structure. By manipulating the electric potential applied at
the gates, one can achieve desired potential energy profiles; It can be seen
that the potential energy can be approximated by an equivalent piece-wise
linear function. The charge qubit can be defined using two potential energy
wells separated by a barrier. (c) One can achieve a desired potential energy
profile along the coordinate to facilitate or decrease tunneling of an electron
between adjacent dots.
eigenstates.
As this study aims to support the design of a quantum pro-
cessor in FDSOI 22-nm CMOS technology, finite-element
method (FEM) simulations of 2D and 3D structures have
been carried out using semiconductor, electromagnetic and
Schrödinger-Poisson simulators of COMSOL Multiphysics
using the dimensions, materials and dopant concentrations
of that technology over temperatures 2–70 K to support the
model presented in this study. From these simulations, the
potential energy of an electron is calculated. The potential
energy on the surface of the silicon channel along the sym-
metry line is then used in the tight-biding model. Both the
semiconductor simulation of the modulation of the conduc-
tion band by the applied electric field and electromagnetic
simulators (the penetration of the electric field in the channel)
return consistent results, showing the freeze-out of the chan-
nel and the depth of the quantum wells forming along the
structure and controlled by the potential applied at the gates
(imposers). The schematic 3D structure is shown in Fig. 3(a)
and it contains gates (imposers) made of a stack of SiO2,
high-εdielectric and heavily doped polysilicon, thin silicon
channel, buried oxide (BoX) and thick wafer. An additional
insulating coat is deposited on the top of the structure. Some
minor effects, such as trapped gate charges, are also taken
into account. At the beginning (and also at the end) of
the structure, the devices serving as injectors/detectors are
connected.
A representative example of the electron’s potential energy
on the surface of the silicon channel of a quantum regis-
ter with three dots, as obtained from FEM simulations, is
shown in Fig. 3(b). In such a geometry, there exists such
a combination of gate voltages that causes potential energy
“barriers” to be formed under the imposers and “wells” to
be formed in between the imposers. The minimum of the
potential energy is conventionally placed at 0 meV. In a
typical scenario, barriers of 2 to 4 meV are formed when
a sub-threshold voltage is applied at the imposers. The re-
sulting potential energy can be effectively approximated by
an equivalent piece-wise linear function. The Schrödinger
equation with a piece-wise linear potential energy can be
solved to find a set of eigenfunctions and eigenenergies. We
use three different methods to ensure that the eigenenergies
are consistent. Any DQD in a register can be viewed a charge-
based qubit. However, the term position-based qubit has also
been used [47], [54].
The electron’s potential energy can be controlled by ap-
plying appropriate potential at the imposer terminals. The
imposer voltages allow DC bias voltage and pulses of indi-
vidually controlled magnitude and duration to be applied to
control the barrier height separating the pairs of neighboring
DQDs. Figure 3(c) shows how the potential energy changes
when we sweep the potential at one of the imposer over the
range from 300 to 350 meV. One can note that the relative
height of the barrier separating dot 2 and dot 3 decreases,
facilitating tunneling between the two quantum dots.
III. FORMAL DEFINITION AND ELECTROSTATIC BLOCH
SPHERE CONTROL OF A CHARGE QUBIT
The parameters of our investigated DQD system are given
in Table 1 and correspond to the device shown in Fig. 2(d).
Here, eand medenote the electron charge and mass, respec-
tively, 2Lis the length of the DQD device, bwis the length
of the barrier separating the two quantum dots, and UBis
the barrier potential. The left and right dots themselves are at
potential ULand UR. These chosen parameters correspond
to 22-nm FDSOI CMOS. For the sake of completeness, all
variables are concentrated in one table, although xdand zd
will be later defined in Fig. 7 for separated DQD structures.
When coupled to other similar QDs in a chain, as shown
in Fig. 2(c), the dynamics of an injected electron can be
manipulated on a larger scale. Such an array of QDs is similar
to a charge-coupled device (CCD) and allows formation of a
quantum register.
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Blokhina et al.: CMOS Charge Qubits: Theoretical Analysis of Control and Entanglement
TABLE 1. System parameters for a two-quantum dot qubit(s).
e1.6·10−19 C
m∗
e0.2·m0and 1.08·m0
L80 nm
bw0.25 L
E0~2/2m∗
eL2= 0.006meV
UL=UR300 E0= 1.8 meV
UB120 E0= 0.72 meV
xd, zd2L
A qubit is generally defined as an isolated quantum system
that has two distinctive quantum states (denoted |0iand |1i)
controlled via various technical means (for example, by ap-
plying electric or/and magnetic fields). In the case of a qubit
implemented through two coupled quantum quantum dots
(i.e. a DQD), electrostatic or electromagnetic fields facilitate
occupancy (Rabi) oscillations between the two states. The
spectral theorem guarantees that two different eigenvalues of
the system’s Hamiltonian, expressed as a Hermitian matrix,
have orthogonal normalized eigenstates, which is essential
for the operation of a qubit [55]. The fidelity is preserved
provided the quantum system remains effectively isolated
from any decohering environment over the timescales of the
experiment. In this section, we aim to show the feasibility
of charge qubits and introduce their formal definition. To
analyze the eigenstates of a two-dot or multi-dot system,
we begin with the time-independent Schrödinger formalism.
Later we will show an extension to a time-dependant Hamil-
tonian and multi-particle case. The Schrödinger equation is
written as follows:
ˆ
H|ψj(x)i=Ej|ψj(x)i(1)
where ˆ
H is the Hamiltonian operator for the system, |ψj(x)i
is an eigenstate labelled by index j, and Ejis its corre-
sponding energy. For a time-independent Hamiltonian, the
wavefunction dynamics can be obtained in the Schrödinger
picture simply from |Ψj(x, t)i=e−iEjt/~|ψj(x)i, where i
is the imaginary unit.
Consider the simplest case with only two energy levels,
E0with corresponding wavefunction |ψ0i, and E1with
wavefunction |ψ1i. At any given time t, the state of a qubit
can be represented in terms of the superposition,
|ψi=c0|ψ0i+c1|ψ1i(2)
where c0,c1are the probability amplitudes of each eigen-
state in the |ψibasis, with |α|2+|β|2= 1 to preserve the
normalization of the wavefunction.
However, the eigenstates |ψ0ior |ψ1ido not typically
correspond to states of a single electron in a DQD that are
physically detectable by means of a standard electrostatic
(charge) detector, as used in the proposed device. Instead, we
change the basis and write |ψiin terms of the detectable qubit
states |0iand |1i[8],
|ψi=c0|0i+c1|1i ≡ cos θ
2|0i+eiϕ sin θ
2|1i(3)
where |c0|2+|c1|2= 1, and the angles ϕ∈[0,2π)and
θ∈[0, π]define the so-called Bloch sphere representation
of a qubit. Since the QDs are physical quantum quantum
dots with spatial extent along the lateral x-axis as shown in
Fig. 2(d), the states |0iand |1iare associated with time-
independent wavefunctions |φL(x)iand |φR(x)i, defined
such that |φL(x)imaximizes the electronic occupancy of
the left quantum dot and |φR(x)imaximizes the electronic
occupancy of the right quantum dot. A prescription to de-
termine these functions is given below. First, note that the
coefficients aand bare generally complex-valued, following
from orthonormality as,
c0=h0|ψi,Z
D
φ∗
Lψdxand c1=h1|ψi,Z
D
φ∗
Rψdx
(4)
where x∈ D denotes the entire domain of existence of an
electron injected into the two-quantum dot system with finite
walls. Hence, Eqs. (3)–(4) define formally a charge qubit.
We emphasize that the new orthonormal states |φL(x)iand
|φR(x)iare a linear combination of the original eigenstates
|ψ0(x)iand |ψ1(x)i. This is not surprising since the physical
system considered has only two energy levels, and so only
two orthogonal wavefunctions are available to construct the
new basis. This formalism can be straightforwardly general-
ized to multi-quantum dot and multi-level system.
In general, the basis transformation is linear since the
Schrödinger equation is a linear differential equation, and
takes the form,
|φζ=L,Ri=X
j=0,1
Uζj |ψji(5)
where Uζj = [ ˆ
U]ζj are elements of a unitary matrix ˆ
U. Since
ˆ
U†ˆ
U=ˆ
I for a unitary matrix and the original eigenstates
satisfy hψi|ψji=δij , the states of the new basis are guaran-
teed to be orthonormal and the antisymmetry of the fermionic
wavefunction is preserved (this is referred to as a ‘canonical
transformation’). Here, the dagger symbol denotes Hermitian
conjugation, and δij is the Kronecker delta symbol.
Of course, there exist an infinite number of unitary ma-
trices ˆ
U that satisfy Eq. (5). To complete the definition of
the charge qubit, we must find the specific representation that
satisfies an additional constraint – namely, that |φL(x)imax-
imizes the occupancy of the left quantum dot, and |φR(x)i
maximizes the occupancy of the right quantum dot.
It should be emphasized here that to faithfully model the
physical DQD device, the quantum quantum dots do not have
infinite potential walls. Although the probability of locating
the electron in either quantum dot is relatively high, there is
a finite probability that the electron can exist in classically
forbidden regions, such as in the barrier region between the
quantum dots, or outside of the device entirely, leading to a
loss of quantum information from the system. This motivates
us to define the “robustness” of the charge qubit and to
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estimate its non-ideality. In particular, we wish to construct a
basis that minimizes any such non-ideality.
We start by recalling that the total probability of locating
an injected electron across the entire domain of existence is
exactly unity (this is the physical property responsible for
wavefunction normalization). Therefore,
Z
D|ψ|2dx=Z
c.f.L & c.f.R
|ψ|2dx+Z
B
|ψ|2dx+
Z
wL
|ψ|2dx+Z
wR
|ψ|2dx= 1
(6)
where ‘c.f.L’ and ‘c.f.R’ stand for the classically forbidden
regions outside the left and right quantum dots, ‘B’ stands
for the controllable barrier region separating the two quantum
dots, and wLand wRare respectively the left and right
quantum dots themselves (see Fig. 2(d)).
We define the probability of locating an electron in quan-
tum dot ζ=Lor Ras pwζ. From Eq. (3) it then follows
that,
pwζ=Z
wζ|c0|2|φL|2+|c1|2|φR|2+c0c∗
1φLφ∗
R+c.c.dx
(7)
where ‘c.c’ denotes complex conjugate. We recognize that
φL(x)and φR(x)are not vanishing at the barriers but have
(exponentially) decaying tails that constitute the dominant
source of non-ideality of the charge qubit. A measure of the
non-ideality (or residual error factor) is then = 1 −pwL−
pwR.
The localized-state basis of Wannier functions [56], [57]
is defined such as to maximize the probability to locate an
electron in the relevant quantum dot. Using formula (5) for
ζ=Lor R, we write:
{(x, φL(x)) : x∈ D} and {(x, φR(x)) : x∈ D}
max Z
wζ
|φζ|2dx =
max Z
wζ
(Uζ0·ψ0+Uζ1·ψ1)(U∗
ζ0·ψ∗
0+U∗
ζ1·ψ∗
1)dx
(8)
This optimization problem allows one to find the matrix
elements Uζj performing the basis transform.
For a DQD qubit comprising two quantum quantum dots
with the parameters of Table 1, one can straightforwardly
determine the maximally localized basis functions, as shown
in Fig. 4. Figure 4(a) shows the probability density for an
electron in eigenstate |ψ0(x)ior |ψ1(x)ias a function of
position along the x-axis, highlighting how the eigenbasis
is typically delocalized over the entire device. By contrast,
Fig. 4(b) shows the probability density for an electron in
the maximally localized (Wannier) basis |φL(x)ior |φR(x)i,
FIGURE 4. Representation of qubit states in the proposed DQD device. (a)
Probability density of and electron in eigenstates of the DQD, | |ψ0(x)i |2and
| |ψ1(x)i |2as a function of position x/L. (b) Corresponding maximally
localized functions | |φL(x)i |2and | |φR(x)i |2.
demonstrating suitability as a charge qubit basis. Note how-
ever that even in the maximally localized basis, there is ap-
preciable tunneling amplitude inside the classically forbidden
barrier region.
The full control of the Bloch sphere requires one to be able
to change both angles, θand ϕ. However, it is easy to see that
an equilibrium system in the eigenfunction representation is
characterised by a fixed angle θwith the angle ϕprecessing at
the frequency of occupancy oscillations δω = (E1−E0)/~
where the energy levels E0and E1are associated with the
states |ψ0iand |ψ1i. Angle θcan be adjusted by dynamically
modulating the potential function UB(x)as quantum dot
adjusting the bottoms of the potential quantum dots UwL
and UwR. However, once the system is in equilibrium (i.e.,
FIGURE 5. Rotation of angle θboth in the eigenfunction basis {|ψ0i,|ψ1i}
and position basis {|0i,|1i} (i.e. the Bloch sphere). Angle θcan be adjusted
by dynamically manipulating the potential function UB(x)as quantum dot as
by setting the bottoms of the potential quantum dots UwLand UwR.
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Blokhina et al.: CMOS Charge Qubits: Theoretical Analysis of Control and Entanglement
UB(x)is high enough), θdoes not change. The eigenfunction
representation of the two-quantum dot system is shown in
Fig. 5 (see the left column) where we show the effect of the
potential function variation on the angle θ. The state vector
describing such a system precession with the frequency δω
along the paths shown in those figures is defined by the height
of the barriers separating the quantum dots.
It is interesting to note that in the position-based repre-
sentation (5), the Bloch sphere and the original trajectories
are transformed as shown in the middle column of Fig. 5,
with both angles, θand ϕ, being functions of time even
in the equilibrium case. It is straightforward to obtain the
expressions for the angles in explicit form:
cos2θ
2=1
2+|c0||c1|cos(δωt)
sin2θ
2=1
2− |c0||c1|cos(δωt)
ϕ=arctan 2|c0||c1|sin(δωt)
|c0|2− |c1|2
(9)
Here, according to Eq. (2),|c0|and |c1|represent the prob-
ability amplitudes of the eigenstates ψ0and ψ1, and the
frequencies ω0=E0/~and ω1=E1/~are associated with
those states.
IV. DERIVING TIGHT-BINDING MODEL OF INTERACTING
QUBITS FROM SCHRÖDINGER FORMALISM
The Schrödinger formalism, both time-dependent and time-
independent, allows one to capture the dependence of the
wavefunction on spatial coordinates from first principles.
However, it becomes increasingly inconvenient to use when
handling multiple interacting particles or more complex
structures. In this section, we show how to derive a tight-
binding model directly from the Schrödinger equation, al-
lowing us to model single and multiple electrons in such
structures easily and effectively. The tight-binding model
is often used in systems where localized Wannier orbitals
constitute a good basis for quantum tunneling of electrons in
a periodic potential [18], [58]–[60]. In bulk materials treated
within the tight-binding formalism, the long-range Coulomb
interaction is often assumed to be screened, but in quantum
confined nanostructures as studied here, the electron-electron
interactions are crucial and must be considered explicitly.
The aim here is to show how the parameters appearing in the
model will be related directly to the geometry of the system
and to the maximally localized Wannier functions introduced
in the previous section.
A. MULTI-PARTICLE FORMALISM IN APPLICATION TO
THE STUDIED SYSTEM
For convenience, we will use the first quantization formalism.
The Hamiltonian of Ninteracting particles in one dimension
contains the kinetic energy operator −P~2
2m∗∇2
k, the po-
tential energy operator PU(xk)and the interaction energy
PUC
kj due to the Coulomb force between electrons. Hence,
the Hamiltonian operator is written as follows:
ˆ
H=
N
X
k=1 −~2
2m∗
k∇2
k+Uk(xk, t)+
N
X
k>j=1
g·e2
4πεeff|xk−xj|
(10)
where εeff is the effective dielectric constant and gis a
coefficient accounting for screening effects [61], eis the
magnitude of the electronic charge and m∗
kis the (effective)
mass of the kth particle. It must be noted that if equation (10)
is used in a one-dimensional case, the correction for di-
mensionality for the electric potential should be taken into
account. The time-dependent Schrödinger equation is then
written in terms of the N-particle eigenstate wavefunctions
|Ψji≡|Ψj(x1, . . . , xN, t)ias follows:
i~∂|Ψji
∂t =ˆ
H|Ψji(11)
As usual, N-particle wavefunctions |Ψ(x1, . . . , xN, t)i
can be represented in terms of a linear combination
of N-particle basis states |ψ(x1, . . . , xN, t)i, which are
themselves constructed as the tensor product of the M
single-particle states of each particle i,|ψ(xi, t)i=
PM
ni=1 c(i)
ni(t)

φ(i)
ni(xi)E, according to:
|ψ(x1, . . . , xN, t)i=|ψ(x1, t)i ⊗ . . . ⊗ |ψ(xN, t)i=
M
X
n1=1
c(1)
n1(t)

φ(1)
n1(x1)E⊗... ⊗
M
X
nN=1
c(N)
nN(t)

φ(N)
nN(xN)E
(12)
In the combined Hilbert space H(⊗N ), we introduce the basis

φ(⊗N):
n{φ(⊗N)
k}:

φ(1)
n1, ..., φ(N)
nNE=
=

φ(1)
n1E⊗. . . ⊗

φ(N)
nNE, n1, .., nN= 1, ..., M o(13)
where k= 1, ..., M Nwith MNproviding the total number
of such basis functions. The Schmidt decomposition theorem
states that all states in the combined Hilbert space can be
expressed as a linear combination of these tensor product
states, so we write:
|Ψi=
MN
X
k=1
ck

φ(⊗N)
kE(14)
In the basis of 

φ(⊗N)
kE, the Hamiltonian matrix is not
diagonal, and so in general we have finite matrix elements
of the type
Hmn =Dφ(⊗N)
m

ˆ
H

φ(⊗N)
nE(15)
Eq. (11) then implies the following equation for the time-
evolution of the coefficients,
i~dcm
dt=X
n
Hmncn(16)
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Equation (16) is a linear system of ordinary differen-
tial equations, which can be solved analytically for a
time-independent Hamiltonian or numerically for a time-
dependent commuting Hamiltonian:
c(t) = c0e−i
~Rt
0ˆ
H(τ)dτ (17)
where c(t)is a vector containing the probability ampli-
tudes ciand c0is a vector of initial conditions subjected to
the usual normalization constraint. If one deals with a time-
dependent non-commuting Hamiltonian, the Dyson series
can be used to calculate it numerically.
In addition to the formalism stated above, the postulates of
our model are as follows:
An electron, injected into a double-quantum dot cell
(this could be straightforwardly extended to a multi-
quantum dot arrangement) is confined to that cell, even
if it interacts with other electrons.
We will consider one or a set of interacting double-
quantum dot cells, each containing an electron that can
occupy the two lowest energy levels. Hence, in for-
mula (12) we take into account only two basis functions
for each electron (for example, maximally localised φL
and φRfor each particle).
The electron’s time-dependent wavefunction becomes
Ψ(x, t) = c0(t)φL(x) + c1(t)φR(x). When a system
of Ninteracting electrons is considered, their individual
wavefunctions are combined using formula (12).
The applications of these postulates can be easily under-
stood by the example of one electron in a double-quantum
dot cell, see Fig. 6. If the electron is actualized in the left
quantum dot wL, its state is associated with the maximally
localized function φL. Hence, the wavefunction of the elec-
tron when ‘firmly’ detected in wLat a given instance of time
is |0i= 1 ·φL(x) + 0 ·φR(x). The actualization of the
electron in the right quantum dot wRis associated with the
maximally localized function φR. Hence, the wavefunction
of the electron when ‘firmly’ detected in wRat a given
instance of time is |1i= 0 ·φL(x)+1·φR(x).
Equating the maximally localized functions to the actu-
alization of the electron can be seen as an approximation,
but it is held to a high degree of accuracy. Indeed, since the
functions φLand φRare the solution to the maximization
problem (7), the instantaneous probability of locating the
electron, for instance, in wLis
pwL=Z
wL
[c∗
0φ∗
L+c∗
1φ∗
R][c0φL+c1φR]dx≈
|c0|2Z
wL
φ∗
LφLdx≈ |c0|2Z
D
φ∗
LφLdx=|c0|2
(18)
which is the probability amplitude of φL.
In the next section, we consider the application of this
general formalism to three cases of interest. The underlying
feature of the studied system is that electrons, while interact-
ing through Coulomb force, stay confined to their respective
double-quantum dot cells. We will see that combining the
equations obtained from first principles with the postulates
we formulated is essentially the tight-binding model.
B. ONE ELECTRON IN A DOUBLE-QUANTUM-DOT:
QUBIT
We return to the basic definition of the qubit introduced in
Section III and analyze it using the framework proposed in
Section IV-A with all expressions simplified for the one-
particle case. The charge qubit is shown in Fig. 6 where
the double-quantum dot structure, the building block of the
charge qubit, is represented as a symbolic cell with an elec-
tron actualized either in the left or in the right quantum dot.
We have already calculated the maximally localized Wannier
functions for this system, φL(x)and φR(x), see Fig. 4. The
Hamiltonian in the matrix form becomes (not including the
Coulomb interaction for the moment since we deal with one
electron):
H11 =hφR|ˆ
H|φRi=−~2
2m∗hφR|φ00
Ri+hφR|U(x)|φRi
H22 =hφL|ˆ
H|φLi=−~2
2m∗hφL|φ00
Li+hφL|U(x)|φLi
H12 =hφR|ˆ
H|φLi=−~2
2m∗hφR|φ00
Li+hφR|U(x)|φLi
H21 =hφL|ˆ
H|φRi=−~2
2m∗hφL|φ00
Ri+hφL|U(x)|φRi
(19)
where the double apostrophe symbol denotes the second
derivative with respect to coordinate. Hence, knowing the
system’s geometry and the potential function U(x), it is
possible to calculate the functions φL,R and then the matrix
elements. The calculation of the localized functions φL,R is
required only once in the case of time-independent system.
In the case of weakly perturbed systems, one can assume
that the maximally localised functions are not ‘disturbed’ too
much, and the time-dependent dynamics are expressed in the
probability amplitudes. We note that in the case of cells with
symmetrical functions U(x), as the one shown in Fig. 2(d),
H12 =H21. As an illustration, the matrix entries calculated
for our particular geometry are given in Table 2. For conve-
nience, we normalize the energy of electrons by the quantity
E0=~2/(2m∗
eL2), so the entries of the Hamiltonian matrix
are expressed in units of E0.
TABLE 2. System parameters for a two-quantum dot qubit.
H11 = 9.4·E0H12 = 0.23 ·E0
H21 = 0.23 ·E0H22 = 9.4·E0
Hence, for a two-level system of Fig. 6 representing the
electrostatic qubit, we have the following model. The state
when the electron is actualised in wLis denoted as |0i ≡
|φLiwhile the state when it is actualized in the right quantum
dot — as |1i≡|φRi. The time evolution of the states is
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described by equation (16) written in terms of the probability
amplitudes c= (c1, c0)T:
|Ψi=c1|1i+c0|0i(20)
The Hamiltonian matrix in that equation becomes:
H=Ep1th,01
th,10 Ep2(21)
It is conventional to use the following notation H11 =Ep1,
H22 =Ep2,H12 =th,01 and H21 =th,10. The off-diagonal
terms th,01 and th,10 are known as tunnelling or hopping
terms. In this form, the charge qubit is no different than any
other quantum two-state system, and hence it will display all
the expected features. We note that the matrix (21) is the
fundamental building block for the Hamiltonian matrix of
many-particle systems as will be shown later.
Figure 6 plots the electron occupancy oscillation curves in
quantum dot wL(green curve) and quantum dot wR(blue
curve) in the double-quantum dot structure. We used the
parameters from Table 1 and expressions (19) to calculate
the entries of matrix (21). Then, the set of equations (16)
was solved to find the coefficients c0(t)and c1(t)and the
probabilities |c0|2and |c1|2associated with the occupancy of
the quantum dots. For comparison, the direct solution of the
Schrödinger equation is marked by circles and squares, and
it provides exactly the same probabilities. As expected, the
occupancy of wLand wRare in anti-phase. In the case of one
electron, it can be localized only in one quantum dot. Hence,
when the occupancy of wLreaches unity, the occupancy of
wRmust be zero. In other words, the normalization condition
is preserved.
C. TWO INTERACTING QUBITS
Now we examine the extension of the formalism to many-
particle systems where different DQDs are only electrostat-
ically coupled, with no wavefunction overlap, and where
dissipation is not considered. As an illustration, we will study
12
Occupancy of well wL
12
Occupancy of well wR
Double-well
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●
1 2 3 4 t[ns]
0.2
0.4
0.6
0.8
1.0
Pwell1(t)
●Schrödinger Formalism
●Schrödinger Formalism
Tight Binding
Tight Binding
P(t) Occupancy of wLOccupancy of wR
0
1
FIGURE 6. Charge qubit representation: double-quantum dot potential is seen
as a cell where the actualisation of the electron in quantum dot wL
corresponds to state |0iand the actualisation of the electron in quantum dot
wRcorresponds to |1i. Such a qubit will display the occupancy oscillations
between the quantum dots reflected in oscillating probabilities PwL(t)and
PwR(t)as functions of time. The continuous lines show the probabilities
calculated using the equation (16) with the matrix elements (19) while the
squares and circles show the probabilities calculated directly from the
Schrödinger equation.
FIGURE 7. Schematic structures of two and three interacting qubits (i.e.,
double-quantum dot cells with an electron injected in each of the cells).
FIGURE 8. Two interacting qubits, each containing a single electron:
occupancy oscillations evolving in time of the electron in double-quantum dot
αexpressed as probabilities Pα
|1iand Pα
|0ito locate the particle in the left or
in the right quantum dot in the presence of interaction with the electron in
double-quantum dot β. The scheme on the left shows the least and most
probable states of the particles.
in detail two and three interacting electrons, as illustrated
in Fig. 7. We will use the superscript αto denote the first
double-quantum dot cell where particle 1 is injected and the
superscript βto denote the second double-quantum dot cell
where particle 2 is injected. For two electrons interacting
through Coulomb force, each confined to their respective
double-quantum dots, the wavefunction is written as:
|Ψi=X
nα=1,0X
nβ=1,0
cnαnβ

n(α)
αn(β)
βE=
=c11 
1α1β+c10 
1α0β+c01 
0α1β+c00 
0α0β
(22)
where the states |0αi,|1αi,
0βand 
1βare associated
with the maximally localised functions φα
L,φα
R,φβ
Land φβ
R,
respectively.
The wavefunction contains possible detectable states of the
two-particle system. For instance, the probability of finding
particle αin the right quantum dot and particle βalso in
the right quantum dot of their respective cells is |c11|2,
etc. The Hamiltonian matrix accommodates the electrostatic
interaction by including the terms due to the electrostatic
interaction and tunnelling terms for both particles by com-
bining formulae (10) and (15).
H=H(β)⊗I+I⊗H(α)=




Ep11 tβ
h,10 tα
h,10 0
tβ
h,01 Ep22 0tα
h,10
tα
h,01 0Ep33 tβ
h,10
0tα
h,01 tβ
h,01 Ep44





(23)
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FIGURE 9. Three interacting qubits: evolution of occupancy oscillations of
particle 1 in double-quantum dot αexpressed as probabilities Pα
|1iand Pα
|0i
to locate the particle in the left or in the right quantum dot of the structure in
the presence of interaction with particle 2 in double-quantum dot βand
particle 3 in double-quantum dot γ. The scheme on the left shows the least
and most probable configurations of the particles.
where Hαand Hβdenote the matrices including the
Coulomb interaction for the first and second particles and I
is the 2×2identity matrix.
System of equations (16), solved for the two-particle case,
allows one to look at different configurations of electrons in
two-quantum dot cells as a function of geometry, potential
and strength of interaction. For example, Fig. 8 shows an
analog of occupancy oscillations from Fig. 6 for particle 1
in its respective double-quantum dot cell αin the presence
of electrostatic interaction with particle 2 in double-quantum
dot β(refer to Fig. 7(a) for the geometry and the arrangement
of double-quantum dot cells). We begin with the configu-
ration where particles 1 and 2 are both located in the left
quantum dots of their respective double-quantum dot cells.
Due to the repelling action of interaction, the occupancy
oscillations are disturbed, and it is less likely to localize
particle 1 in the right quantum dot of double-quantum dot
α. Figure 8 also shows the least and the most probable
configurations of such a system.
D. THREE AND MORE INTERACTING QUBITS.
GENERATING THE WAVEFUNCTIONS AND MATRICES
FOR AN ARBITRARY NUMBER OF PARTICLES
The generalization of the formalism, allowing an extension
of the model to a multi-particle system, can be derived from
these examples as follows. The wavefunction is expressed
as a tensor product defined in the combined Hilbert space
H(⊗N ):
|Ψi=X
nα=1,0X
nβ=1,0
.. . X
nω=1,0
cnαnβ...nω(t)

n(α)
αn(β)
β. . . n(ω)
ωE
(24)
and the corresponding Hamiltonian is constructed as follows:
ˆ
H=ˆ
H(ω)⊗ˆ
I⊗. . . ... ⊗ˆ
I+. . . +ˆ
I⊗. . .ˆ
I⊗ˆ
H(α)(25)
Expressions (24) and (25) allow us to assemble and solve
the matrix equation (16) for an arbitrary number of charge
qubits of a given geometry. As an example, Fig. 9 visualizes
a simulation for a three-electron system, where we plot an
analog of occupancy oscillations for three interacting qubits.
As expected, the occupancy oscillations are disturbed even
more compared to Fig. 8 due to the presence of one additional
particle and its repelling action. This approach can be easily
automated and used to simulate interacting qubits and quan-
tum gates in an environment compatible with circuit design
and simulations.
V. VON NEUMANN ENTANGLEMENT ENTROPY IN THE
CONTEXT OF SEMICONDUCTOR CHARGE QUBITS
To conclude this study, we shall also discuss whether it is
possible to entangle two charge qubits when they interact
electrostatically through the means of the Coulomb force. If
entanglement is feasible for such qubits, this would mean
that conventional quantum computing operations could be
implemented on charge qubits. We recognize that there ex-
ist a number of methods to investigate entanglement. For
example, the correlator as defined in Ref. [50] is a direct
derivation of the Bell inequality between two-qubits (DQDs).
Both the Von Neumann entropy and this correlation function
will result in the same conclusion of such a system in terms
of simulation results. In [51], the concept of concurrence is
defined for a pair of qubits. As a measure of entanglement
in this study, we will use the Von Neumann entropy. It is a
bipartile quantity requiring to divide the original quantum
system into two sub-parts. The aim of the analysis is to
understand whether the sub-parts of the original system are
in separable states or not.
A. SYSTEM OF ONE CHARGE QUBIT
To understand how the entropy of entanglement works, let
us start with a single qubit in a double-quantum dot system.
For a bipartition of the single qubit system, we will use the
following notation (this will help us in avoiding confusion
with the notation introduced previously for the multi-electron
case). We will denote the situation when the electron is in the
left quantum dot of the double-quantum dot system as sub-
part ‘a’, and, likewise, for the right quantum dot as sub-part
‘b’. Subsequently, the state of one qubit can be conveniently
written in the form similar to the multi-particle formalism we
introduced before:
|ψi=c10 
1a0b+c01 
0a1b(26)
Here, c10(t)and c01 (t)are the probability amplitudes in the
localized position basis. This form should be interpreted as
follows: 0ameans that there is no electron in sub-part aof
the system (i.e., in the left quantum dot) and 1ameans that
the electron is present in sub-part aof the system. The same
applies when the index bis used. For obvious reasons, the
states 
1a1b,
0a0bor similar are not normally possible, as
that would make the system physically corrupted.
To describe the Von Neumann entanglement entropy, we
will use the density matrix formalism. In the position basis,
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noting equation (26), we write the density operator for the
full system (no bipartition applied yet) in the following form:
ˆρab =|ψihψ|=
=c10c∗
10 
1a0b1a0b
+c01c∗
01 
0a1b0a1b

(27)
Equivalently, the density matrix operator can be written in a
matrix representation, known as the density matrix ρ, with
elements ρab = [ρ]ab as follows,

1a0b
0a1b
ρ=1a0b

0a1b
|c10|20
0|c01|2(28)
By dividing the system into two parts, aand b, we can
now introduce the reduced density operators ˆρaand ˆρbvia
the partial trace as follows:
ˆρa=0b
ˆρab 
0b+1b
ˆρab 
1b
ˆρb=h0a|ˆρab |0ai+h1a|ˆρab |1ai(29)
which describe the state of each sub-part, tracing out the
complement. For example, in the matrix form, the reduced
density matrix of sub-part atakes the following form, which
is equivalent to the reduced density matrix
|0ai |1ai
ρa=h0a|
h1a||c01|20
0|c10|2(30)
Then, the Von Neumann entanglement entropy SNis defined
as follows:
SN=−Tr(ˆρaln ˆρa) = −Tr(ˆρbln ˆρb)(31)
While the entanglement entropy is not particularly useful
for one electron or one qubit, it can provide some interesting
insights to its meaning. As an illustration, the Von Neumann
entropy applied to one qubit in a superposition state is vi-
sualized in Fig. 10. One can clearly see that the entropy is
FIGURE 10. Von Neumann entanglement entropy calculated for two states of
a single charge qubit. Entanglement entropy is at maximum when the
probability of occupying each of the states is the same. Vice versa,
entanglement entropy is zero when the electron actualises in one of the
quantum dots.
maximal, SN= ln 2, when the qubit is found in one of the
two states: (
1a0b+
0a1b)/√2or (
1a0b−
0a1b)/√2.
Recalling that we are operating in the localized basis, this su-
perposition of the localized states corresponds to the situation
when the electron is maximally delocalized. We also note that
in the localized basis a qubit displays occupancy oscillations
although the period can vary by several orders-of-magnitude
as a function of UB. Hence, the Von Neumann entanglement
entropy is generally a function of time.
B. SYSTEM OF TWO INTERACTING QUBITS
Having understood the Von Neumann entropy applied to
the simplistic case of one qubit, we can now apply it to
two interacting qubits. We will now return to our standard
notation for double-quantum dots as used in Sec. IV-C. In this
case, sub-part ‘α’ will denote an electron in the first double-
quantum dot and sub-part ‘β’ will denote an electron in the
second double-quantum dot. The wavefunction is the same as
given by formula (22):
|Ψi=c11 
1α1β+c10 
1α0β+c01 
0α1β+c00 
0α0β
(32)
The density operator becomes:
ˆρ=c11 c∗
11 
1α1β1α1β
+c10c∗
10 
1α0β1α0β
+
c01c∗
01 
0α1β0α1β
+c00c∗
00 
0α0β0α0β

(33)
Equivalently, the density matrix can be written as follows:

1α1β
1α0β
0α1β
0α0β
ρ=1α1β

1α0β

0α1β

0α0β





|c11|2000
0|c10|20 0
0 0 |c01|20
000|c00|2




(34)
Applying a bipartition to the system, we obtain the reduced
density matrices. For example, following Eq. (29), the re-
duced density matrix for sub-part αcan be written as follows:
|0αi |1αi(35)
ρα=h0α|
h1α||c01|2+|c00 |20
0|c10|2+|c11 |2(36)
We note here that the entries of the reduced density matrix
can be easily understood. In the matrix above, the first non-
zero entry gives the probability of finding sub-part αof the
quantum system in state |0i, while the second non-zero entry
gives the probability of finding it in state |1i, regardless of
the state of the second electron. These entries, as a matter of
fact, were plotted in the graphs of Fig. 8. As a final step, the
Von Neumann entanglement entropy is calculated using the
same formula (31).
Figure 11 shows the entanglement entropy SN(t)calcu-
lated for two qubits (each containing an electron) interacting
electrostatically though the Coulomb’s force. The Hamil-
tonian operator is given by expression (10), and we note
that we use the effective screening constant gto account
for possible screening effects in solid-state structures [61].
VOLUME 12, 2018 11

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Blokhina et al.: CMOS Position-Based Charge Qubits: Theoretical Analysis of Control and Entanglement
FIGURE 11. Von Neumann entanglement entropy SN(t)of two
electrostatically interacting qubits as a function of time calculated at
electrostatic screening coefficient g= 0.1.
FIGURE 12. Initial Von Neumann entanglement entropy SN(t)of two
electrostatically interacting qubits as a function of time calculated at
electrostatic screening coefficient g= 1.
Fig. 11 also shows the occupancy oscillations experienced by
sub-part α(i.e., the electron in the first double-quantum dot)
in the presence of the second electron when the electrostatic
interaction between them is rather weak (g= 0.1). In this
case, as was discussed earlier, the electron experiences large-
amplitude occupancy oscillations, with ∼100% probability
to be eventually found in the left quantum dot and a bit less
in the right quantum dot. Reciprocally, there are instances of
time when the electron is delocalised, and the entanglement
entropy reaches its maximum. For comparison, Fig. 12 shows
FIGURE 13. Von Neumann entanglement entropy SN(t)of two
electrostatically interacting qubits as a function of time over a longer time
stretch calculated at electrostatic screening coefficient g= 1.
FIGURE 14. (a) Time to reach maximum entanglement SN(t)as a function of
the tunneling probability thfor a fixed value of the screening coefficient g= 1.
(b) Time to reach maximum entanglement SN(t)as a function of the screening
coefficient gfor a fixed value of the tunneling probability th= 0.33E0.
an example of strong interaction between the two qubits
(g= 1), whilst Fig. 13 shows the same example for a longer
time stretch. The occupancy oscillations are disturbed, and
the electron αis initially found mostly in the left quantum dot
due to the strong electrostatic repulsion. As a consequence,
entanglement entropy decreases. However, the dynamics fol-
12 VOLUME 12, 2018

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Blokhina et al.: CMOS Charge Qubits: Theoretical Analysis of Control and Entanglement
FIGURE 15. (a) Von Neumann entanglement entropy SN(t)between two
single-electron registers interacting electrostatically via Coulomb interaction.
Each line consists of two DQDs which corresponds to two qubits, denoted as
qubit#1 and qubit#2; (b) Time to reach maximum entanglement SN(t)as a
function of the tunneling probability thin a given time duration, tmax = 10 ns.
low the same pattern (with a different frequency), when
one visualizes the same plot for a longer time period. We
conclude that if the interaction between the two qubits is
strong, it leads to the localisation of electrons and hence it
reduces their entanglement.
To conclude this section, the time to reach maximum
entanglement between the two qubits is plotted in Fig. 14(a)
as a function of the tunneling probability thand for a fixed
value of the screening coefficient g= 1. It is evident that the
higher the hopping term the shorter the time needed. Lastly,
we also plot in Fig. 14(b) the time to reach maximum entan-
glement between the two qubits as a function of the screening
coefficient g, for a fixed value of the tunneling probability
th= 0.33E0. The higher the screening coefficient the longer
it takes for the system to reach the maximum entanglement.
C. SYSTEM OF TWO INTERACTING REGISTERS
In this section, we will examine a system of two coupled reg-
isters each consisting of four QDs, as depicted in Fig. 15(a).
We will assume one particle in each line interacting electro-
statically via the Coulomb force with one particle on the other
line. In this case, sub-part ‘α’ denotes the first line and sub-
part ‘β’ denotes the second line. Each line includes four dots
(or two DQDs) and can be seen as a qudit. The wavefunction
of the system is:
|Ψi=X
nA=0#1,1#1 ,0#2,1#2 X
nB=0#1,1#1 ,0#2,1#2
cnAnB

n(A)
An(B)
BE
(37)
where we again assume four quantum states for each particle.
The reduced density matrix for sub-part αcan be written
as follows:
ˆρα=D0β
#1

ˆρ

0β
#1E+D1β
#1

ˆρ

1β
#1E+D0β
#2

ˆρ

0β
#2E+D1β
#2

ˆρ

1β
#2E
(38)
Let us now assume the maximally entangled Bell state [62],
[63]:
|Φi=1
√2
0α0β+
1α1β (39)
which is defined generally between any two systems, αand
β. Writing this state in the basis of our system, we get
|Φi=1
√2

0α
#10β
#1E+

0α
#20β
#2E+

0α
#10β
#2E+

0α
#20β
#1E+
+

1α
#11β
#1E+

1α
#21β
#2E+

1α
#11β
#2E+

1α
#21β
#1E
(40)
and
ˆρα=trβ(hΦ| |Φi) = 1
2I4(41)
From (31) we can calculate SN= 2 ln 2.
In Fig. 15(a), the Von Neumann entanglement entropy
SN(t)is plotted as defined between the two single-electron
registers interacting electrostatically via the Coulomb inter-
action. Each line consists of two DQs which correspond to
two qubits, denoted as qubit #1 and qubit #2. Interestingly,
it is visible that an almost maximally entangled state can be
achieved in this case (∼2 ln 2) for the selected parameters.
In principle, the maximum entanglement is harder to achieve
as the spatial degrees of freedom for each particle increases.
Finally, the maximum entanglement achieved in the system
SNas a function of the tunneling probability th, in a given
time duration tmax = 10[ns], is plotted in Fig. 15(b). As
the tunneling probability thincreases, the system can get
maximally entangled.
VI. DECOHERENCE AND FIDELITY OF CHARGE
QUBITS
Qubits based on nuclear spin, electron spin or combination
of charge-spin (hybrid) appear to be the best candidates for
quantum computing from the point of view of achievable
decoherence times [8], [12]. As reported in the literature,
charge-based qubits have much shorter decoherence times
(within a range of 50 ns to 1 µs). However, reasonably high
practical qubit fidelity can still be achieved by using very fast
state flip times as explained below. The 22-nm FDSOI CMOS
process used to develop the quantum structures considered
VOLUME 12, 2018 13

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Blokhina et al.: CMOS Position-Based Charge Qubits: Theoretical Analysis of Control and Entanglement
in this study has transition frequencies (fT) in the several
hundreds of gigahertz range [37]. Even with a 50 ns deco-
herence time, charge-based qubits could still perform over a
thousand operations per useful coherence duration. This is an
advantage that can compensate for the charge qubits’ shorter
decoherence time. The FDSOI process has a roadmap for a
12-nm feature node that should provide state flip times as
low as 10 ps and hence can facilitate high quality quantum
operation. We also note that the measurement/readout of
spin qubits is very challenging due to their need to operate
with narrowband microwave electromagnetic fields which are
inherently slow (tens of ns-level access time) and consume
rather high power (over 10mW per qubit) [15], [38], [64].
Consequently, being able to operate on charge qubits 2–
3 orders-of-magnitude faster can compensate for their 2–3
orders-of-magnitude decoherence time handicap.
Lastly, we should also mention that by choosing the state
and the physical parameters properly, the dephasing can be
largely suppressed in a quantum-dot array [28], [30], [44],
[49]. Therefore, we can conclude that the relatively short
decoherence time is not expected to prohibit the application
of charge qubits for quantum computing.
VII. CONCLUSIONS
This paper provides a formal definition, robustness analysis
and discussion on the control of a charge qubit intended for
semiconductor implementation in scalable CMOS quantum
computers. The construction of the charge qubit requires
maximally localized functions, and we show such functions
for double-quantum dot structures with dimensions corre-
sponding to a 22-nm FDSOI CMOS technology. We also
discuss how an individual qubit can be manipulated in terms
of the two angles of the Bloch sphere.
Based on the electrostatic nature of the qubit, we demon-
strate how to build a tight-binding model of one and multiple
interacting qubits from first principles of the Schrödinger
formalism. We provide all required formulae to calculate the
maximally localized functions and entries of the Hamiltonian
matrix in the presence of interaction between qubits. We
use four illustrative examples to demonstrate interaction of
electrons in three cases of interest and discuss how to build a
model for many-electron (qubit) system. Finally, we use the
Von Neumann entanglement entropy in the context of charge
qubits to show that the electrostatically interacting electrons
in these qubits can be entangled.
VIII. ACKNOWLEDGMENT
Elena Blokhina and Panagiotis Giounanlis contributed
equally to this work.
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