††thanks: Current Address: Electrical Engineering and Computer Sciences,
University of California, Berkeley, California 94720, USA

# Momentum relaxation effects in 2D-Xene field effect device structures

Anirban Basak  Pratik Brahma  Bhaskaran Muralidharan  Department of Electrical
Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076,
India<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

We analyze the electric field driven topological field effect transition on
2D-xene materials with the addition of momentum relaxation effects, in order
to account for dephasing processes. The topological field effect transition
between the quantum spin Hall phase and the quantum valley Hall phase is
analyzed in detail using the Keldysh non-equilibrium Green’s function
technique with the inclusion of momentum and phase relaxation, within the
self-consistent Born approximation. Details of the transition with applied
electric field are elucidated for the ON-OFF characteristics with emphasis on
the transport properties along with the tomography of the current carrying
edge states. We note that for moderate momentum relaxation, the current
carrying quantum spin Hall edge states are still pristine and show moderate
decay with propagation. To facilitate our analysis, we introduce two metrics
in our calculations, the coherent transmission and the effective transmission.
In elucidating the physics clearly, we show that the effective transmission,
which is derived rigorously from the quantum mechanical current operator is
indeed the right quantity to analyze topological stability against dephasing.
Exploring further, we show that the insulating quantum valley Hall phase, as a
result of dephasing carries band-tails which potentially activates parasitic
OFF currents, thereby degrading the ON-OFF ratios. Our analysis sets the stage
for realistic modeling of topological field effect devices for various
applications, with the inclusion of scattering effects and analyzing their
role in the optimization of the device performance.

##  I Introduction

Electric field driven topological phase switching between a topological and a
trivial phase can be harnessed in the form of topological field effect
transistors (TFET) [1, 2, 3]. The pursuit of such a transition in two-
dimensional (2D) materials has seen significant activity [4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14] and experimental progress [15, 16, 3] recently. In 2D-Xene
materials, where “X” refers to Si, Ge, Sb, etc., a perpendicular electric
field can trigger a phase change between quantum spin Hall (QSH) phase and
quantum valley Hall (QVH) phase [17]. The quantized conductance of the
protected dissipationless current carrying edge states of the QSH phase and
the insulating QVH phase can be used as ON and OFF states respectively.
Previous analysis of Various 2D field effect transistor (FET) structures show
promising ON-OFF currents which suggest that they can be used in high-speed
and low-power applications [10, 18, 12, 19, 11].
There has been tremendous interest in both 2D and 3D topological insulator
(TI) device structures with some promising results [8, 5, 20, 10, 7, 21, 22,
23, 24, 25, 26, 27, 28, 29] in terms of the FET operation. In recent years,
experiments have demonstrated the capacity of $Bi_{2}Se_{3}$ layers to change
the conductance of its surface in response to applied electric field [1, 4],
although the presence of bulk states reduce the effectiveness of such device
structures to be used as an FET device [1, 30, 31]. Few methods can be used to
reduce the number of bulk states so that the surface conduction dominates [32,
33] and other methods reduce the film thickness to introduce a gap between
Dirac cones to improve conductivity [34, 35]. Other device structures like
gate all around or a conventional MOSFET with depletion region can also be
realized with this material [36, 37]. However, TFETs are yet to demonstrate
better performance than tunnel FETs to be a viable alternative to the existing
technologies [6, 9].

Figure 1: Device schematics: (a) A typical representation of the nanoribbon
FET structure, with gate all around configuration. (b) Top view and side view
of the buckled hexagonal lattice structure of a typical 2D-Xene channel. (c) a
representative phase diagram of silicene in the presence of an applied
perpendicular electric field. In (b) the green dashed rectangular box shows
one unit cell and the green letters represent the sub-lattice sites marked by
A and B.

There is considerable interest in buckled 2D-Xene materials [2, 38] with
respect to engineering the QSH-QVH phase transition [39, 17, 29] with the
application of a perpendicular electric field. With appropriate gating
structures, this field effect can be used as a topological transistor, with
the switching between the ON and OFF states achieved by an applied electric
field. While the QSH edge states are themselves considered dissipationless, a
clear understanding of the character of this transition will necessitate the
actual power considerations from a device engineering perspective.
Most works which analyze the topological robustness of various phases focus on
coherent effects, which include the addition of impurities via either random
disorder potential [40, 38, 41] or via self consistent fields [42, 43]. The
consideration of finite temperature fluctuating impurities [44, 45, 46, 47]
and hence near elastic scattering, which are non-coherent effects, is almost
absent in this context [48]. These fluctuating impurities can systematically
degrade the phase coherence of electrons and give rise to dephasing, which in
the parlance of quantum transport effectuates a transition from the coherent
ballistic regime to the diffusive regime. The Keldysh non-equilibrium Green’s
function (NEGF) technique allows one to study this transition [46, 45, 44,
47], via the inclusion of such dephasing mechanisms. A very common dephasing
mechanism at finite temperature is that of local fluctuating impurities that
give rise to momentum relaxation, and it can be rigorously shown that
increasing the magnitude of the impurity potential leads to the transition
from quantized conductance to Ohm’s law [47].
The object of this paper is to delve into the topological field effect
transition in 2D-Xene materials with quantum transport models in the coherent
ballistic regime and the non-coherent regime with dephasing effects [46, 45,
47] resulting from momentum relaxation processes [44, 49, 50, 51, 52, 53, 54,
48]. The transition between the QSH and the QVH states is analyzed in detail
using the NEGF technique [45, 55]. Details of the transition with applied
electric field are elucidated for the ON-OFF characteristics, with emphasis on
the transport properties. A deeper analysis of the stability of the current
carrying QSH edge states as well as the nature of the switching process is
considered in detail. We note that for moderate momentum relaxation, the edge
states are still pristine and show moderate decay with propagation.
We further elucidate the physics in the non-coherent limit by introducing two
possible metrics: a) the coherent transmission and the b) effective
transmission. We show that although the coherent transmission seems like an
intuitive quantifying metric when transitioning from the coherent limit into
the non-coherent limit, the effective transmission that is rigorously derived
from the current operator paints a realistic picture when it comes to
evaluating the stability of the topological edge states. We elucidate that
although the coherent transmission spectrum shows a significant decrease in
transmission under elastic dephasing, the effective transmission in such cases
remain almost close to unity, preserving the QSH property and demonstrating
that the device can withstand a lot of edge defects and still maintain its ON-
current. Further, we also point out the important aspect of band-tail
formation [56, 57], which can lead to band-gap narrowing effects in the QVH
phase, a remarkable effect that can arise even in the absence of in-elastic
scattering effects, which can lead to a degradation in the performance by
enhancing the OFF current. The methods detailed in this paper can also be
extended to include various effects of quantum dot formation and the related
electron-electron interactions [42, 58], which typically case phase
randomization without momentum relaxation [44], leading toward realistic
modeling of topological field effect devices for various applications.
The organization of this article is as follows: In Sec. II, we describe the
the structure of our device, the formalism and the transport calculations
using the Keldysh NEGF technique. In Sec. III we discuss the bandstructure,
local density of states, transmission spectra, and the transport properties of
our device. In Sec. IV we summarize our work.

##  II Device set-up and formalism

###  II.1 Hamiltonian

The device structure can be visualized as a single layer of silicene placed on
top of an oxide as schematized in Fig. 1. The buckled hexagonal lattice as
shown in Fig. 1 is a quasi-2D structure, where if seen from top the atoms
seems to be arranged in a hexagonal manner on a 2D plane. However, due to the
nature of the bonds and the bond angle, each atom is deviated from this common
plane by a distance of $+l_{s}$ or $-l_{s}$, i.e., the two planes of atoms are
separated by a total distance of $2l_{s}$. To maintain maximum structural
stability, the nearest neighbours are displaced in opposite directions. There
are four atoms in one unit cell of hexagonal lattice, arranged in lattice
sites in ABAB fashion, starting from bottom-center (A) towards top-center (B)
traversing clockwise through two more atoms (B and A).

We start with a generalized Hamiltonian of buckled hexagonal lattice in tight-
binding representation [17, 22],

$\displaystyle H=-t\sum\limits_{\langle
i,j\rangle\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+i\frac{\lambda_{SO}}{3\sqrt{3}}\sum\limits_{\langle\langle
i,j\rangle\rangle\sigma\sigma^{\prime}}\nu_{ij}c_{i\sigma}^{\dagger}\sigma_{\sigma\sigma^{\prime}}^{z}c_{j\sigma^{\prime}}+i\lambda_{R1}(E_{z})\sum\limits_{\langle
i,j\rangle\sigma\sigma^{\prime}}c_{i\sigma}^{\dagger}(\bm{\sigma}\times\mathbf{\hat{d}_{ij}})_{\sigma\sigma^{\prime}}^{z}c_{j\sigma^{\prime}}$
$\displaystyle-i\frac{2}{3}\lambda_{R2}\sum\limits_{\langle\langle
i,j\rangle\rangle\sigma\sigma^{\prime}}\mu_{i}c_{i\sigma}^{\dagger}(\bm{\sigma}\times\mathbf{\hat{d}_{ij}})_{\sigma\sigma^{\prime}}^{z}c_{j\sigma^{\prime}}-l_{s}E_{z}\sum\limits_{i\sigma}\mu_{i}c_{i\sigma}^{\dagger}c_{i\sigma},$
(1)

where $c_{i\sigma}^{(\dagger)}$ represents annihilation (creation) operator of
a spin $(\sigma=\uparrow/\downarrow)$ spin at the site $i$, $\lambda_{SO}$,
$\lambda_{R1}(E_{z})$ and $\lambda_{R2}$ represent the intrinsic spin-orbit
coupling, the first Rashba spin-orbit coupling and the second Rashba spin-
orbit coupling respectively. For the intrinsic spin-orbit coupling term,
$\nu_{ij}=+1$, if the next nearest neighbour hopping is anti-clockwise and
$\nu_{ij}=-1$, if the next nearest neighbour hopping is clockwise. And for the
perpendicular electric field term and the second Rashba spin-orbit coupling
term, $\mu_{i}=+1$ for lattice site A and $\mu_{i}=-1$ for lattice site B. The
vector $\mathbf{d_{ij}}$ is the distance between the site $i$ and $j$. The
$l_{s}$ and $E_{z}$ are the buckle height and perpendicular electric field
respectively [17].
The first term in the Hamiltonian that of an ordinary hexagonal lattice like
graphene and gives rise to the Dirac cones at the well known $K$ and
$K^{\prime}$ points. The intrinsic spin-orbit coupling term, $\lambda_{SO}$ ,
opens the gap between the Dirac cones at $K$ and $K^{\prime}$ points [59]. The
third term is the first Rashba spin-orbit coupling, $\lambda_{R1}(E_{z})$,
which is quite negligible for low electric field near the Dirac points. The
second Rashba spin-orbit coupling term, $\lambda_{R2}$, although not zero, is
quite small. The presence of electric field removes the valley degeneracy
between site A and B, and thus with the application of electric field we see a
separation of spins into two distinct valleys. For silicene, the tight-binding
hopping parameter $t=1.6eV$. In our case, we take a large
$\lambda_{SO}=0.32eV$, to have a pronounced energy gap to work with. We also
take $\lambda_{R1}(E_{z})=0eV$ and $\lambda_{R2}=0.06eV$. The buckling height
is set at $l_{s}=0.23\mathring{A}$.
Our work is centered around the topological transition between the QSH state
and the QVH state, with the application of the electric field $E_{z}$ as
schematized in Fig. 1. The spin valley polarized metal (SVPM) state occurs in
the intermediate state between the two phases in such a topological field
switching. In order to analyze the nature of this switching process, we
utilize the NEGF formalism, which we now detail below.

###  II.2 The Keldysh NEGF formalism

The analysis of transport for our set up is performed via the Keldysh NEGF
formalism and the Landauer-Büttiker formalism. The Hamiltonian matrix, $[H]$,
is constructed with the tight-binding matrix elements, $[\alpha]$ and
$[\beta]$, where $[\alpha]$ represents the Hamiltonian of a single column of
unit cells lying along the transverse direction of the device and $[\beta]$
represents the overlap matrix between two such columns [51, 49]. In a device
connected with two leads, the starting point of the calculation is the energy
resolved retarded Green’s function of the device region in its matrix
representation given by:

$[G^{r}(E)]=[(E+i\eta)I-H-U-\Sigma_{L}-\Sigma_{R}-\Sigma_{s}]^{-1},\\\ $ (2)

where, $[I]$ is an identity matrix of the dimensions of the device
Hamiltonian, $\eta$ is a small damping parameter, $U$ is the random impurity
potential, $[\Sigma_{L(R)}]$ and $[\Sigma_{s}]$ are respectively the self-
energy matrices of the contacts $L/R$ and the scattering self energy matrices
representing the momentum dephasing processes respectively. The contact self-
energies are calculated using the recursive Green’s function technique where
any geometry can be accommodated [45] and the scattering self energy is
evaluated using the self-consistent Born approach to be described below. In
the definitions to follow, the energy argument is assumed implicit.

Figure 2:  Bandstructures of (a) the QSH, (b) the SVPM and (c) the QVH phases
of a pristine silicene nanoribbon. The QSH phase shows the existence of a
spin-degenerate edge state. The SVPM and QVH phase shows the up-spin and down-
spin bands separating into two different valleys under the application of the
external electric field. No edge states can be found around the Fermi energy
in the case of the QVH phase. The SVPM phase represents a boundary phase
between the conducting QSH phase and the insulating QVH phase.

Using the self energies, the broadening matrix, $\Gamma$, of the contacts
$L,R$ and the scattering processes are defined as,

$\displaystyle[\Gamma_{L,R}]$
$\displaystyle=i[\Sigma_{L,R}-\Sigma_{L,R}^{\dagger}]$ (3)
$\displaystyle[\Gamma_{s}]$
$\displaystyle=i[\Sigma_{s}-\Sigma_{s}^{\dagger}].$ (4)

The spectral function, $A$, is obtained from (2) as,

$\displaystyle[A(E)]=i[G^{r}-G^{a}]=[G^{r}][\Gamma_{L}+\Gamma_{R}+\Gamma_{s}][G^{a}].$
(5)

The diagonal elements of the spectral function are related to the local
density of states (LDOS). We also require the correlator, which is related to
the lesser Green’s function given by

$[G^{<}]=[G^{r}][\Sigma^{<}_{L}+\Sigma^{<}_{R}+\Sigma^{<}_{s}][[G^{a}],$ (6)

where $\Sigma^{<}_{L(R)}=-i\Gamma_{L(R)}f_{L(R)}$ represent the lesser (in-
scattering) self energy of the left (right) contact, with
$f_{L(R}=f(E-\mu_{L(R)})$ is the Fermi-Dirac distribution in the left (right)
contact described via their respective electrochemical potentials
$\mu_{L(R)}$. The quantity $\Sigma^{<}_{s}$ that represents the lesser self
energy of the scattering process is evaluated self-consistently, as will be
detailed below. The retarded self energy and the lesser self energy for the
momentum dephasing process is given below from (2) and (6),

$\displaystyle\Sigma^{r}_{s}(i,j)=$ $\displaystyle\bar{D}(i,j)G^{r}(i,j)$ (7)
$\displaystyle\Sigma^{<}_{s}(i,j)=$ $\displaystyle-i\bar{D}(i,j)G^{<}(i,j),$
(8)

from which, we can define the in-scattering function
$[\Sigma^{in}_{s}]=-i[\Sigma^{<}_{s}]$ and the electron correlation function
$[G^{n}]=-i[G^{<}]$. These quantities are determined by the dephasing matrix,
$\bar{D}(i,j)$, which is given by

$\displaystyle\bar{D}(i,j)=$
$\displaystyle\innerproduct{U_{s}(i)}{U^{*}_{s}(j)}.$ (9)

We define the dephasing matrix that determines the correlation between the
random scattering potentials $U_{s}$, distributed along the sites of the
device region is defined as

$\displaystyle\bar{D}(i,j)=$ $\displaystyle d_{m}\delta_{ij},$ (10)

where $d_{m}$ is the strength of momentum relaxation, which will be added by
hand to quantify the extent of dephasing. This is a homogeneous model with
uniformly distributed, elastic, and spatially uncorrelated interactions,
resulting in a diagonal form of $\bar{D}(i,j)=d_{m}\delta_{ij}$. This model
discards the off-diagonal elements of the Green’s function, thus relaxing both
the phase and momentum of quasiparticles in the nanowire. [46, 45]. The
quantity $d_{m}$ typically represents the magnitude squared of the fluctuating
scattering potentials. In this work, the parameter $d_{m}$ is modulated so
that by gradually increasing it, one can transition from the coherent
ballistic limit to the diffusive limit. This model can also be extended to
include non-local fluctuations via the spatial correlations of the impurity
potentials [46, 45].
Our solution for the self-energies for the dephasing process is based on the
self-consistent Born approximation [45], in which we solve (7) and (8) self
consistently with (2) and (6) respectively to obtain the full solution of the
retarded and the lesser Green’s function following the desired numerical
convergence [47]. Following this, we obtain the current operator [45] as

$I^{op}_{L}={\bf{Re}}\left({\bf{Tr}}\left[([\Sigma^{in}_{L}][A])-([\Gamma_{L}][G^{n}])\right]\right).$
(11)

In order to quantify the effect of dephasing, inspired by the Landauer-
Büttiker form, it is quite tempting to define a coherent transmission,
$T_{coh}$, as a measure of electronic channels through which electrons can
travel ballistically from left lead to right lead without encountering any
scattering in the channel. It is obtained from (3) and (2) as,

$\displaystyle T_{coh}={\bf{Re}}({\bf{Tr}}(\Gamma_{L}G^{r}\Gamma_{R}G^{a})),$
(12)

where ${\bf{Tr}}$ stands for the matrix trace operation, and ${\bf{Re}}$
stands for the real part of the complex number obtained. However, using the
current operator, we can define yet another transmission, which we term as the
effective transmission, $T_{eff}$ as

$\displaystyle
T_{eff}=\frac{1}{f_{L}-f_{R}}{\bf{Re}}\left({\bf{Tr}}\left[([\Sigma^{in}_{L}][A])-([\Gamma_{L}][G^{n}])\right]\right),$
(13)

that represents the net number of channel for electrons to travel from left
lead to right lead after encountering any number of scattering in the channel.
We must note that in the presence of dephasing, the current operator, as noted
in (11) cannot be written in the Landauer-Büttiker form [48, 47]. Due to this,
we will establish in the upcoming section that the latter definition makes
better sense as a transmission metric to analyze channels under dephasing.

##  III Results

###  III.1 Bandstructure

Let us first consider the bandstructures of the three phases involved, namely,
the QSH phase, the SVPM phase and the QVH phase, as pointed out in the
schematic of the phase transition in Fig. 1, and qualitatively understand the
role of the perpendicular electric field $E_{z}$ in effectuating the
transitions. The hexagonal arrangements of atoms produces Dirac cones at the
six high symmetry points in reciprocal space in the Brillouin zone. Since we
have a zigzag termination of atoms along the longitudinal direction, only two
$K$, $K^{\prime}$ symmetry points fall within the first Brillouin Zone. In the
absence of spin-orbit coupling, the two Dirac cones touch each other at $K$
and $K^{\prime}$ symmetry points.
The intrinsic spin-orbit coupling causes the bulk states of the two Dirac
cones at the $K$ and $K^{\prime}$ symmetry points to move away from the
$\Gamma$ point by an equal amount and also open a gap. The energy gap induced
depends on the strength of the intrinsic spin-orbit coupling. The effect of
the second Rashba spin-orbit coupling is a relationship between the transverse
spin and crystal momentum, which manifests as change in band-structure
depending on the crystal momentum. This change is different for different
crystal momentum, i.e., different band number or longitudinal wavenumber. This
change is seen more prominently in the QVH phase when the spin-up or down edge
states feature a band curvature which makes their bands move towards or away
from the bulk conduction or valence band edges in the vicinity of the $K$ and
$K^{\prime}$ points.
Since the orbital momentum in 2D hexagonal lattices can have two opposite
orientations, the electrons at $K$ and $K^{\prime}$ symmetry points can
experience two equal but opposite magnetic moments. When the electron spin is
coupled to this orbital momentum through spin-orbit coupling, the spin-valley
degeneracy between the pairs ($K^{\prime}\uparrow$,$K\downarrow$) and
($K^{\prime}\downarrow$,$K\uparrow$) is lifted and a gap appears between their
energy bands at $K$ and $K^{\prime}$ points. Due to the nature of this effect,
the electrons with the same spin but moving clockwise or counterclockwise
within the hexagonal lattice will experience opposite magnetic moments.
Our simulations on the bandstructure of the three phases are based on a
silicene nanoribbon structure twenty unit cells wide and are summarized in
Fig. 2. The red and blue bands represent energy bands for the up-spin and the
down-spin electrons respectively. We can see that for
$|E_{z}|<\lambda_{SO}/l_{s}$, we have the QSH phase, where we have two edge
states with spin degeneracy but of opposite directions as shown in Fig. 2. The
separation between the upper (conduction) and lower (valance) bulk bands in
our simulation is about $0.723eV$ which is just a little over
$2\lambda_{SO}=0.64eV$.
With the application of a perpendicular electric field, electrons on all next
nearest neighbouring sites gain an equal amount of energy, whereas those on
the nearest neighbouring sites gain an opposite amount. Since, spin-orbit
coupling also only works on the next nearest neighbouring sites, these two
terms have a combined effect on the bands, shifting them vertically up or
down, depending on the spin ($s$) and valley ($k$) of the electrons. Without
non-zero spin-orbit coupling, the effect of the perpendicular electric field
will not result in the separation of the bands.
For small $E_{z}$ ($|E_{z}|<\lambda_{SO}/l_{s}$), the bands of up or down-spin
will either come close together or move away from the Fermi energy. This
causes the bands to separate into up-spin and down-spin bands. At the same
time, for the edge states, the point at which the two opposite spin bands
cross each other at the Fermi level separates in wavenumber, giving rise to
two bands crossing the Fermi level at different $k$-points. This continues
until $|E_{z}l_{s}|=\lambda_{SO}$, when, for a particular spin, the
preexisting bulk gap from the spin-orbit effect vanishes due to the effect of
the electric field at either the $K$ or the $K^{\prime}$ point and the two
crossings of the two opposite spin edge states merge at either the $K$ or the
$K^{\prime}$ point at the Fermi energy. Thus, for
$|E_{z}|=\lambda_{SO}/l_{s}$, we have the SVPM phase, where the up and down-
spin bands are separated into two valleys of opposite wavenumber as shown in
Fig. 2. The SVPM is a metallic state where the Fermi level lies touching both
conduction and valance bands and serves as the intermediate phase where the
gap begins to open.
For large $E_{z}$ ($|E_{z}|>\lambda_{SO}/l_{s}$), the overall effect of the
electric field dominates and it pushes both up and down-spin bands away from
the Fermi energy. The edge spin bands also move away from each other along
with the bulk bands and a band-gap opens. Thus, at higher electric fields
$|E_{z}|>\lambda_{SO}/l_{s}$, we have the QVH phase as shown in Fig. 2. The up
and down spin bands are still separated, but now there is an energy gap
between the upper (conduction) bands and lower (valence) bands. The Hall
conductance in such case will only depend on the electrons at one of the two
valleys. A phase diagram is shown in Fig. 1 to depict the various phases of
the material with respect to the applied perpendicular electric field
($E_{z}$).

###  III.2 Coherent Transport

Figure 3: LDOS profiles of the (a) QSH, (b) SVPM, and (c) QVH phases of at the
Fermi energy. The QSH phase has two conducting edge states of different spins,
moving along opposite directions. The SVPM phase shows the onset of the phase
transition of the current carrying edge states into bulk bound states. The QVH
phase does not have any current carrying states and only has some states
spilled over from the two contacts.  Figure 4:  Transmission spectra in the
coherent regime $d_{m}=0$ for the QSH, the SVPM, and the QVH phases. The two
opposite spin edge states produce a conductance of $G_{0}$ at the Fermi energy
for the QSH phase, whereas the absence of such states produces zero
transmission at the Fermi energy for the QVH phase. The SVPM phase is where
the transmission at Fermi energy falls to zero and a gap starts to open.
Figure 5:  Current-field $I_{ds}$ vs $E_{z}$ characteristics of Topological
FET with an applied bias $V_{ds}=1V$ at $300K$ for different strengths of
momentum dephasing in the log-scale. The inset shows the same in the linear
scale only for the QSH phase. The changes in $E_{z}$ result in changes in the
topological phases, which are indicated in the figure. The device shows a
transistor-like behaviour where QSH and QVH represent the ON and OFF currents
respectively.

Figure 6:  Transmission spectra under dephasing: (a) coherent transmission
$T_{coh}$ and (b) the effective transmission $T_{eff}$ of the QSH phase with
varying strengths of dephasing $d_{m}$. For identical dephasing strengths, the
effective transmission $T_{eff}$ demonstrates more realistic profile in
comparison with $T_{coh}$, indicating that the edge states are quite robust
against dephasing. (c) LDOS of the QSH phase at Fermi energy with the momentum
dephasing strength of $d_{m}=10^{-2}t^{2}$. The decay of the different spin
edge states demonstrates that the up-spin (down-spin) states are moving in the
(counter) clockwise direction.

The local density of states (LDOS) in the device region of our structure is
shown is Fig. 3. The left and right leads are connected to the central part
through the self-energies $\Sigma^{r}_{L,R}$ defined via the NEGF formalism,
and account for the quasi-continuum contribution of the electronic states from
the left and right leads. The device Hamiltonian determines where these states
will be present in the central part. The density of states (DOS) at the Fermi
energy for the QSH phase reveals the presence of two edge-states along the
edges of nanoribbon as shown in Fig. 3. Despite there being no bulk states in
this phase, current carrying states are due to the presence of edge states.
The edge states here consists of one up-spin (red) and one down-spin (blue)
edge states on each of the two edges of the material. Interestingly, the up-
spin states move clockwise along the edges and the down-spin states move
counter-clockwise. Since, there is no relaxation or decay of states at this
point they overlap perfectly and are seen here together in purple shade in
Fig. 3.
While transitioning from the QSH phase to the SVPM phase the two edge states
are maintained with gradual decay in their LDOS magnitude. The SVPM phase, as
shown in Fig. 3, is a boundary phase between the QSH and the QVH phase, where
the edge states completely vanish. The QSH and the QVH states thus represent
the ON and OFF states of the FET device respectively. In the QVH phase, at the
Fermi energy, the only contribution to the very small LDOS comes from the two
leads and is localised in the small area near both leads as shown in Fig. 3.
In this phase, there are almost no current carrying states in the channel and
the nanoribbon acts as a band insulator. The transmission characteristics of
the device for different phases is shown in Fig. 4. In the QSH phase (red-dash
lines), there are two edge current carrying channels of opposite spins, with a
total conductance $G_{0}$, where $G_{0}=2e^{2}/h$ is the conductance quantum.
This remains with increasing electric fields, until the transition to the SVPM
phase (green dashed lines), where the gap opens up. In the QVH phase, the
device remains non-conductive for a range of energies above and below the
Fermi energy (blue dash lines), due to the absence of any edge states.

###  III.3 Relaxation effects

The dephasing model considered here is based on (10), which represents
spatially uncorrelated scattering potentials. We can view this in momentum
space as something that couples all momentum states and hence results in
momentum relaxation. Due to this, we can also account for incoherent
backscattering using this model. We use the current formula in (11) to
evaluate the current drain current-electric field $I_{ds}-E_{z}$
characteristics of the device in Fig. 5, when a source drain voltage of
$V_{ds}=1V$ is applied. Here, $I_{ds}$ is the terminal current evaluated using
(11) at either the left or the right contact renamed as the source and drain
respectively, when a voltage is symmetrically applied such that
$\mu_{L}=-\mu_{R}=qV_{ds}/2$. The plots in this section include the coherent
limit and the non-coherent limit with varying amounts of momentum relaxation
$d_{m}$ added.
The $I_{ds}-E_{z}$ characteristics show that the device can be made to undergo
phase transitions by applying the perpendicular electric field and turning it
on or off by controlling the gate voltage, much like in a FET device. The
current through the device in the ON-state QSH phase remains almost the same,
only decreasing a bit with the advent of the SVPM phase and then decreasing
rapidly in the OFF-state QVH phase. The device acts as a depletion mode FET,
where the conductance falls with increasing magnitude of the gate voltage.
There is an asymmetry in the current characteristics between the positive and
negative gate bias. This is caused as a result of the non-zero buckling height
of the 2D channel and the perpendicular electric field applied to the device
through the gate.
With the introduction of momentum relaxation by varying the parameter $d_{m}$,
the edge states undergo scattering and the overall conductance is bound to
decrease. This is seen in the $I_{d}-E_{z}$ plot, as the decreasing current
for QSH region with increasing $d_{m}$. The coherent transmission $T_{coh}$
for different strengths of momentum relaxation, as shown in Fig. 6, depicts
that the edge states are indeed encountering scattering and the pristine
states decay with increasing dephasing strengths. The LDOS plot with a
dephasing strength of $d_{m}=10^{-2}t^{2}$ shows that the left going up-spin
state and the right going down-spin state of the top edge and vice-versa for
the bottom edge decays via edge scattering the further it goes into the
channel as shown in Fig. 6. However, the effective transmission $T_{eff}$ for
different dephasing strengths, as shown in Fig. 6, shows that even with
dephasing present in the form of momentum relaxation, the pristine nature of
the QSH current carrying states are more likely to be preserved and will
conduct close to the ballistic limit with a conductance of $G_{0}$. This
indicates the suppression of back-scattering in the edge states as these edge
states are only allowed to conduct in a particular direction for a particular
spin. And since, there is no nearby state with the same spin and opposite
electron velocity where they can jump to, these edge states are topologically
protected, up to a certain degree of dephasing. This aspect has been very well
captured via the metric $T_{eff}$, that is rigorously derived from the current
operator. It is also clear that merely using a transmission metric $T_{coh}$
inspired by the Landauer like form, is insufficient to capture the topological
stability of the QSH phase. The effective transmission gives a more realistic
picture of the actual current flow process as it is derived from the net
current operator flowing into a terminal and encompasses all the scattering
effects through in-scattering and electron correlation operators.

Figure 7:  band tails: The effective transmission spectrum $T_{eff}$ of the
QVH phase with varying dephasing strengths. We note that at larger values of
dephasing, a distinct band tail effect is seen. Despite having a bandgap, the
scattering states introduce a small amount of transmission within the gap,
which in turn will affect the OFF current and hence reduce the ON-OFF ratio.

We now consider the OFF-state described by the QVH phase. In any device, the
ON-OFF ratio must be maintained high in order to ensure a reliable
performance. Although, the QVH phase is supposed to be insulating due to the
presence of a bandgap of $0.723eV$, a small current is observed in addition to
the leakage current expected after the introduction of momentum dephasing, as
seen in Fig 5. The magnitude of this parasitic leakage current seems to
increase with the increasing dephasing strength. Since the absence of almost
any electronic states in the bandgap region, all method of current conduction
is exhausted save for a very negligible amount of tunnelling. Looking back at
the current operator (11), we note that the total current can be composed as a
sum of the regular ballistic current and the scattering current. The
scattering current indeed includes a DOS contribution
($A_{s}(E)={\bf{Tr}}\left([G^{r}][\Gamma_{s}][G^{r}]\right)$) and an electron
correlation component
($G^{n}_{s}(E)={\bf{Tr}}\left(-i[G^{r}][\Sigma^{<}_{s}][G^{r}]\right)$) that
are non-zero. These scatterer induced states [57] can be connected to the lead
energy levels, which are also above the same bandgap as the channel. Thus, we
notice that a non-zero effective transmission is present in the QVH phase in
presence of scattering relaxation, as shown in Fig 7, which is well related to
the band-tail formation, whose role is detrimental to the device performance
[57]. This transmission is larger close to the band-edges where the effect of
such a level broadening is greater and falls off towards the middle of the
gap.
The rapid fall of the device current in QVH regime clearly indicates a sharp
transition of the device from ON-state to OFF-state and hence points toward
the feasibility of a good transistor device. Apart from the ON-OFF ratios
analyzed here, it is also important to evaluate the sub-threshold slope
characteristic of the transition. Although the sub-threshold slope is an
important parameter for any transistor, it is difficult to analyze it without
a detailed understanding of the gate electrostatics. As it depends on many
parameters, they can be adjusted to control the sub-threshold slope.
Topological switching of the kind analyzed in this paper has possible unique
electrostatic signatures [14, 16] that may signal a smaller sub-threshold
slope. We leave the detailed analysis of the subthreshold characteristics as
well as the role of detailed device electrostatics and scattering for a
detailed future work.

##  IV Conclusion

This paper dealt with a detailed analysis of the electric field driven
topological field effect transition between the QSH phase and the QVH phase in
2D-Xene based FET structures using the Keldysh non-equilibrium Green’s
function technique. Details of the transition with applied electric field were
elucidated for the ON-OFF characteristics, with emphasis on the transport
properties. A deeper analysis of the stability of the edge states as well as
the nature of the switching process was considered in detail. We showed that,
for moderate momentum relaxation, the current carrying QSH edge states are
still pristine and show moderate decay with propagation. We also defined a
metric, the effective transmission that is rigorously derived from the quantum
mechanical current operator which paints a realistic picture when it comes to
evaluating the stability of the topological edge states. Further, we also
pointed out the important aspect of band-tail formation, which can lead to
band-gap narrowing effects in the QVH phase, a remarkable effect that can
arise even in the absence of in-elastic scattering effects that can lead to a
degradation in the performance by enhancing parasitic OFF currents. The
methods detailed in this paper can also be extended to include various other
important effects like quantum dot formation and the related electron-electron
interactions, which typically cause phase randomization without momentum
relaxation, leading toward realistic modeling of topological field effect
devices for various applications.

###### Acknowledgements.

The authors acknowledge Tillmann Kubis for insightful discussions. The
research and development work undertaken in the project under the Visvesvaraya
Ph.D Scheme of the Ministry of Electronics and Information Technology (MEITY),
Government of India, is implemented by Digital India Corporation. This work is
also supported by the Science and Engineering Research Board (SERB),
Government of India, Grant No. STR/2019/000030, the Ministry of Human Resource
Development (MHRD), Government of India, Grant No. STARS/APR2019/NS/226/FS
under the STARS scheme.

## Data availability statement

The data generated and/or analysed during the current study are not publicly
available for legal/ethical reasons but are available from the corresponding
author on reasonable request.

## References

  * Steinberg  _et al._ [2010] H. Steinberg, D. R. Gardner, Y. S. Lee, and P. Jarillo-Herrero, Nano Letters 10, 5032 (2010).
  * Kara  _et al._ [2012] A. Kara, H. Enriquez, A. P. Seitsonen, L. Lew Yan Voon, S. Vizzini, B. Aufray, and H. Oughaddou, Surface Science Reports 67, 1 (2012).
  * Gilbert [2021] M. J. Gilbert, Communications Physics 4, 70 (2021).
  * Liu and Ye [2011] H. Liu and P. D. Ye, Applied Physics Letters 99, 052108 (2011).
  * Chang  _et al._ [2012] J. Chang, L. F. Register, and S. K. Banerjee, Journal of Applied Physics 112, 124511 (2012).
  * Ionescu and Riel [2011] A. M. Ionescu and H. Riel, Nature 479, 329 (2011).
  * Fu  _et al._ [2014] H.-H. Fu, J.-H. Gao, and K.-L. Yao, Nanotechnology 25, 225201 (2014).
  * Akhavan  _et al._ [2014] N. D. Akhavan, G. Jolley, G. A. Umana-Membreno, J. Antoszewski, and L. Faraone, Journal of Applied Physics 116, 084508 (2014).
  * Sarkar  _et al._ [2015] D. Sarkar, X. Xie, W. Liu, W. Cao, J. Kang, Y. Gong, S. Kraemer, P. M. Ajayan, and K. Banerjee, Nature 526, 91 (2015).
  * Vandenberghe and Fischetti [2017] W. G. Vandenberghe and M. V. Fischetti, Nature Communications 8, 14184 (2017).
  * Simchi  _et al._ [2018] H. Simchi, M. Simchi, M. Fardmanesh, and F. M. Peeters, Journal of Physics: Condensed Matter 30, 235303 (2018).
  * Xu  _et al._ [2019a] Y. Xu, Y.-R. Chen, J. Wang, J.-F. Liu, and Z. Ma, Phys. Rev. Lett. 123, 206801 (2019a).
  * Yang  _et al._ [2020] J.-E. Yang, X.-L. Lü, and H. Xie, New Journal of Physics 22, 103018 (2020).
  * Nadeem  _et al._ [2021] M. Nadeem, I. Di Bernardo, X. Wang, M. S. Fuhrer, and D. Culcer, Nano Letters 21, 3155 (2021).
  * Tao  _et al._ [2015] L. Tao, E. Cinquanta, D. Chiappe, C. Grazianetti, M. Fanciulli, M. Dubey, A. Molle, and D. Akinwande, Nature Nanotechnology 10, 227 (2015).
  * Collins  _et al._ [2018] J. L. Collins, A. Tadich, W. Wu, L. C. Gomes, J. N. B. Rodrigues, C. Liu, J. Hellerstedt, H. Ryu, S. Tang, S.-K. Mo, S. Adam, S. A. Yang, M. S. Fuhrer, and M. T. Edmonds, Nature 564, 390 (2018).
  * Ezawa [2012a] M. Ezawa, Phys. Rev. Lett. 109, 055502 (2012a).
  * Zheng  _et al._ [2020] J. Zheng, Y. Xiang, C. Li, R. Yuan, F. Chi, and Y. Guo, Phys. Rev. Applied 14, 034027 (2020).
  * Ishida and Liebsch [2020] H. Ishida and A. Liebsch, Phys. Rev. Research 2, 023242 (2020).
  * Qian  _et al._ [2014] X. Qian, J. Liu, L. Fu, and J. Li, Science 346, 1344 (2014).
  * Hsieh  _et al._ [2012] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature Communications 3, 982 (2012).
  * Ezawa [2014] M. Ezawa, New Journal of Physics 16, 065015 (2014).
  * Sun and Singh [2017] J. Sun and D. J. Singh, Journal of Applied Physics 121, 064301 (2017).
  * Zhu  _et al._ [2019] H. Zhu, C. A. Richter, S. Yu, H. Ye, M. Zeng, and Q. Li, Applied Physics Letters 115, 073107 (2019).
  * Hu  _et al._ [2018] G. Hu, Y. Zhang, L. Li, and Z. L. Wang, ACS Nano 12, 779 (2018).
  * Vali  _et al._ [2015] M. Vali, D. Dideban, and N. Moezi, Physica E: Low-dimensional Systems and Nanostructures 69, 360 (2015).
  * Liu  _et al._ [2015] Q. Liu, X. Zhang, L. B. Abdalla, A. Fazzio, and A. Zunger, Nano Letters 15, 1222 (2015).
  * Xu  _et al._ [2019b] Y. Xu, Y.-R. Chen, J. Wang, J.-F. Liu, and Z. Ma, Phys. Rev. Lett. 123, 206801 (2019b).
  * Fan  _et al._ [2016] Y. Fan, X. Kou, P. Upadhyaya, Q. Shao, L. Pan, M. Lang, X. Che, J. Tang, M. Montazeri, K. Murata, L.-T. Chang, M. Akyol, G. Yu, T. Nie, K. L. Wong, J. Liu, Y. Wang, Y. Tserkovnyak, and K. L. Wang, Nature Nanotechnology 11, 352 (2016).
  * Hor  _et al._ [2009] Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, Phys. Rev. B 79, 195208 (2009).
  * Cho  _et al._ [2011] S. Cho, N. P. Butch, J. Paglione, and M. S. Fuhrer, Nano Letters 11, 1925 (2011).
  * Checkelsky  _et al._ [2011] J. G. Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. 106, 196801 (2011).
  * Bansal  _et al._ [2012] N. Bansal, Y. S. Kim, M. Brahlek, E. Edrey, and S. Oh, Phys. Rev. Lett. 109, 116804 (2012).
  * Taskin  _et al._ [2012] A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Phys. Rev. Lett. 109, 066803 (2012).
  * Thalmeier and Akbari [2020] P. Thalmeier and A. Akbari, Phys. Rev. Research 2, 033002 (2020).
  * Zhu  _et al._ [2013] H. Zhu, C. A. Richter, E. Zhao, J. E. Bonevich, W. A. Kimes, H.-J. Jang, H. Yuan, H. Li, A. Arab, O. Kirillov, J. E. Maslar, D. E. Ioannou, and Q. Li, Scientific Reports 3, 1757 (2013).
  * Satake  _et al._ [2018] Y. Satake, J. Shiogai, K. Fujiwara, and A. Tsukazaki, Phys. Rev. B 98, 125415 (2018).
  * Jana and Muralidharan [2021] K. Jana and B. Muralidharan, ArXiv: 2107.13318 (2021), 2107.13318 .
  * Ezawa [2012b] M. Ezawa, New Journal of Physics 14, 033003 (2012b).
  * Black-Schaffer and Balatsky [2012] A. M. Black-Schaffer and A. V. Balatsky, Phys. Rev. B 85, 121103 (2012).
  * Groth  _et al._ [2014] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New Journal of Physics 16, 063065 (2014).
  * Rischau  _et al._ [2016] C. W. Rischau, A. Ubaldini, E. Giannini, and C. J. van der Beek, New Journal of Physics 18, 073024 (2016).
  * Balram  _et al._ [2019] A. C. Balram, K. Flensberg, J. Paaske, and M. S. Rudner, Phys. Rev. Lett. 123, 246803 (2019).
  * Golizadeh-Mojarad and Datta [2007] R. Golizadeh-Mojarad and S. Datta, Phys. Rev. B 75, 081301 (2007).
  * Datta [1997] S. Datta,  _Electronic Transport in Mesoscopic Systems_ (Cambridge University Press, 1997).
  * Danielewicz [1984] P. Danielewicz, Annals of Physics 152, 239 (1984).
  * Camsari  _et al._ [2020] K. Y. Camsari, S. Chowdhury, and S. Datta, The non-equilibrium green function (negf) method (2020), arXiv:2008.01275 [cond-mat.mes-hall] .
  * Duse  _et al._ [2021] C. Duse, P. Sriram, K. Gharavi, J. Baugh, and B. Muralidharan, Journal of Physics: Condensed Matter 33, 365301 (2021).
  * Sharma  _et al._ [2016] A. Sharma, A. Tulapurkar, and B. Muralidharan, IEEE Transactions on Electron Devices 63, 4527 (2016).
  * Sharma  _et al._ [2017] A. Sharma, A. A. Tulapurkar, and B. Muralidharan, Phys. Rev. Applied 8, 064014 (2017).
  * Sharma  _et al._ [2018] A. Sharma, A. A. Tulapurkar, and B. Muralidharan, Applied Physics Letters 112, 192404 (2018).
  * Singha and Muralidharan [2018] A. Singha and B. Muralidharan, Journal of Applied Physics 124, 144901 (2018).
  * Lahiri  _et al._ [2018] A. Lahiri, K. Gharavi, J. Baugh, and B. Muralidharan, Phys. Rev. B 98, 125417 (2018).
  * Sriram  _et al._ [2019] P. Sriram, S. S. Kalantre, K. Gharavi, J. Baugh, and B. Muralidharan, Phys. Rev. B 100, 155431 (2019).
  * Meir and Wingreen [1992] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).
  * Overhauser [1989] A. W. Overhauser, Applied Physics Letters 54, 2490 (1989).
  * Sarangapani  _et al._ [2019] P. Sarangapani, Y. Chu, J. Charles, G. Klimeck, and T. Kubis, Phys. Rev. Applied 12, 044045 (2019).
  * Muralidharan  _et al._ [2008] B. Muralidharan, L. Siddiqui, and A. W. Ghosh, Journal of Physics: Condensed Matter 20, 374109 (2008).
  * Konschuh  _et al._ [2010] S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 82, 245412 (2010).


††thanks: All authors contributed equally to this work.

# Quantum Accelerated Causal Tomography:
Circuit Considerations Towards Applications

Tamal Acharya<EMAIL_ADDRESS>QWorld Association.  Purdue University,
West Lafayette, USA.  Akash Kundu<EMAIL_ADDRESS>QWorld Association.  Joint
Doctoral School, Silesian University of Technology,
Akademicka $2$A, $44-100$ Gliwice.  Institute of Theoretical and Applied
Informatics, Polish Academy of Sciences, Bałtycka $5$, $44-100$ Gliwice.
Aritra Sarkar<EMAIL_ADDRESS>QWorld Association.  Quantum Machine
Learning group, QuTech,
Delft University of Technology, Delft, The Netherlands

###### Abstract

In this research we study quantum computing algorithms for accelerating causal
inference. Specifically, we investigate the formulation of causal hypothesis
testing presented in [Nat Commun 10, 1472 (2019)]. The theoretical description
is constructed as a scalable quantum gate-based algorithm on qiskit. We
present the circuit construction of the oracle embedding the causal hypothesis
and assess the associated gate complexities. Our experiments on a simulator
platform validates the predicted speedup. We discuss applications of this
framework for causal inference use cases in bioinformatics and artificial
general intelligence.

causal inference, causal hypothesis, error probability, process distance

††preprint: AIP/123-QED

##  I Introduction

Despite the huge success of machine learning (ML) algorithms based on deep
neural network, these systems are inscrutable black-box models. This hamper
users’ trust in the system and obfuscated the discovery of algorithmic biases
arising from flawed generated processes that are prejudicial to certain inputs
(e.g. racial discrimination). Explainable artificial intelligence (XAI)
focuses on human understanding of the decision from the learned solution, as
white-box models. These models provide results that are understandable for
domain experts, thus providing transparency, interpretability and
explainability. XAI algorithms provide a basis for justifying decisions,
tracking and thereby verifying them, improving the algorithms, and exploring
new facts. There has been relatively slow progress in XAI, despite realizing
the importance of it as we increasingly automate critical systems. Early
advances on XAI were based on symbolic reasoning systems and truth maintenance
systems. To achieve causal-reasoning, rule-based learning and logic-based
inference systems were proposed. Methods to address inherent opaque modern
methods like deep learning based neural networks and genetic algorithm include
layer-wise relevance propagation and local interpretability. There exists
other ML algorithms (e.g. decision trees, Bayesian classifiers, additive
models) that generate interpretable models i.e. the model components (e.g.,
weight of a feature, a path in a decision tree, or a specific rule) can be
directly inspected to understand the predictions. However, these models are
not general and scalable to compete with the adoption and impact of neural
networks. On the other hand, symbolic reasoning systems were abandoned owing
to the difficultly in scaling these systems for large number of parameters.

The capability of quantum computation allows us to scale symbolic reasoning
models by encoding the classical rules as a superposition of quantum states or
processes. Thus, quantum mechanics provide enhanced ways to identify causal
links. Certain quantum correlations can be used to infer classical causal
relationships [1, 2]. This could overcome the apprehension of existing
classical approaches being pursued in XAI.

In this article, we will explore how we can distinguish quantum processes by
their causal structure. Specifically, we study the construction proposed in
[3] towards a quantum circuit implementation on Qiskit. In doing so, we
uncover: (i) the implementation aspects of the causal oracle, (ii) the gate
and qubit complexity of the full algorithm, and (iii) the dependence of the
error probability on the set of processes/hypothesis being tested. While the
current technology readiness level of quantum systems prevent us from
embedding this causal reasoning within a broader application framework, in
principle it is a general technique and will be useful in XAI pipelines [4, 5]
in particular.

The remaining article is organized as follows. In Section [II] we briefly
review the specifics of causal reasoning and some of the well studied
techniques. A discussion on the basic concepts and quantum advantage of causal
hypothesis testing is given in Section [III]. In the following Section [IV] we
describe the problem formulation; containing the main findings of the article.
Here we define a model implementation in qiskit followed by a correction
factor introduced in the error probability based on our empirical results;
which we call practical error probability. Finally, we conclude the article by
discussing some potential use cases in Section V.

##  II Overview of causal inference

Causal inference refers to an intellectual discipline that considers the
assumptions, study designs, and estimation strategies that allow researchers
to draw conclusions on the cause-effect relationships between data. The
dominant perspective on causal inference in statistics has philosophical
underpinnings that rely on consideration of counterfactual states. In
particular, it considers the outcomes that could manifest given exposure to
each of a set of treatment conditions (dynamics of a specific causal
variable). Causal effects are defined as comparisons between these potential
outcomes. Causal inference consists of a family of statistical methods whose
purpose is to answer the question of why something happens. Standard
approaches in statistics, such as regression analysis, are concerned with
quantifying how changes between two variables are associated, with no
directional sense. In contrast to that, causal inference methods are used to
determine whether changes in a variable X cause changes in another variable Y,
or vice-versa. If X is causally related with Y, then Y’s change can (at least
partially) be explained in terms of X’s change.

###  II.1 Challenges of performing causal inference

Causal models are based on the idea of potential outcomes. The fundamental
problem for causal inference is that, for any individual unit (a physical
object at a particular point in time), we can observe only one of the
potential outcomes, either Y(1) (under treatment) or Y(0) (under no
treatment). As we observe the value of the potential outcome under only one of
the possible treatments111A treatment is an action that can be applied or
withheld from that unit., namely the treatment actually assigned, hence the
potential outcome under the other treatment is missing.

The inadequacy of the causal models is due to their failure to include
relevant spatial and structural information in a way that does not render the
model non-explanatory, unmanageable, or inconsistent with basic assumptions of
causal graph theory. Making valid causal inferences is challenging because it
requires high-quality data and adequate statistical methods. One prominent
example of common non-causal methodology is the erroneous assumption of
correlative properties as causal properties. Correlated phenomena may not
carry inherent causation. Regression models are designed to measure variance
within data relative to a theoretical model. This suggests that there is
nothing in the data that presents high levels of covariance have any
meaningful relationship among them. This suggests absence of a proposed causal
mechanism with predictive properties or a random assignment of treatment. The
presupposition that two correlated phenomena are inherently related is a
logical fallacy, known as spurious correlation.

  * •

Causation does not imply association

For example, we want to compare the impact of academic degree on income of a
middle aged individual. The person might have attended the academic degree or
might not have. To calculate the causal effect of going to academic degree we
need to compare the output in both situations which is not possible. This
dilemma is a fundamental problem of causal inference. The challenge in causal
inference is that we do not observe both potential outcomes, we only observe
one. Another example is described by the situation that, the black cat ran
under the fence and I tripped and fell over. We would have tripped anyway. It
had nothing to do with the cause of the cat running under the fence.
Association is not causation.

  * •

Correlation does not prove causality

Causal inference is usually a missing data problem [7] and we tend to make
assumptions to make up for the missing values. An example is the correlation
between people eating ice cream and people drownings. Causal inference would
indicate that eating ice cream effects drownings. The actual correlation is
between the season (summer) and these otherwise unrelated things. In this case
the missing data is the season. Another example is the correlation between
higher SAT scores and a greater number of books in the house of the student
taking the tests. Causal inference would imply that the number of books
directly affect the SAT scores when in reality they are both effected by
something else (in this case most likely a higher average intelligence in the
household). Causal inference is the process of drawing a conclusion about a
causal connection based on the conditions of the occurrence of an effect.

The main difference between causal inference and inference of association is
that the prior analyzes the response of the effect variable when the cause is
changed, and latter inferences about the strength of association between
variables are made using a random bivariate sample of data drawn from the
population of interest. Another problem of causal inference is that the cause
and the effect may have occurred by chance rather than by intention.

The causal effect of a drug on systolic blood pressure, one month after the
drug dosage vs no exposure to the drug dosage can be represented as a
comparison of systolic blood pressure that would be measured at the time of
given exposure to the drug with the systolic blood pressure that would be
measured at the same point in time in the absence of exposure to the drug. The
challenge for causal inference is that we are not generally able to observe
both of these states: at some point in time when we are measuring the
outcomes, each individual either has had drug exposure or has not. From the
data perspective, there are some open problems to review regarding the great
potential of learning causality with data. Such as the study of heterogeneous
groups, learning causality with imbalanced data, and learning causality with
complex variables.

###  II.2 Classical techniques in causal inference

Causal inference is conducted via the study of systems where the measure of
one variable is suspected to affect the measure of another. Causal inference
is conducted with regard to the scientific method. The first step of causal
inference is to formulate a falsifiable null hypothesis, which is subsequently
tested with statistical methods. Frequentist statistical inference is the use
of statistical methods to determine the probability that the data occur under
the null hypothesis by chance. Bayesian inference is used to determine the
effect of an independent variable.

Common frameworks for causal inference include the causal pie model
(component-cause) [8], Pearl’s structural causal model (causal diagram and do-
calculus) [9], structural equation modeling, and Rubin’s causal model
(potential-outcome) [10].

The most frequently used causal models can be grouped into two kinds: causal
Bayesian networks and structural equation models (which are distinct but
closely related). Causal graph models combine mathematics and philosophy: the
mathematical elements are Directed Acyclic Graphs (DAGs) and probability
theory (with focus on conditional independence); the philosophical elements
are assumptions about the relationship between causation and probability [11].

###  II.3 Algorithmic machine learning

An alternative approach to causal inference based on algorithmic generative
models is currently gaining popularity. In Ref.[12] describes the process of
performing causal deconvolution using this technique. This paper talks about
the different generating mechanisms by which complex data is produced. The
authors introduced a universal, unsupervised and parameter-free model-oriented
approach based upon the seminal concept of algorithmic probability that
decomposes an observation into its most likely algorithmic generative sources.
It uses a causal calculus to infer model representations. This novel framework
is being evaluated by using the ability of the methods to separate data from
observations that is being produced by discrete dynamical systems such as
cellular automata and complex networks. This research suggests that cellular
automata offers an optimal testbed as they are discrete dynamical systems
which are able to illustrate an algorithm’s inner workings because of their
visual nature.

The approach is similar to the Data Generating Process (DGP) in statistics
which posits that each set of observations is just a snapshot that we see and
is just one instance of the many that could have been possible. This approach
also tells us that each and every data point is being generated from a data
generating process which can be taken as coming from some distribution of data
and is just a sample of the total population. We try to understand the DGP
while analyzing that data and try to find out what could have been those
sources that generated those data. The research introduces algorithmic
probability and the coding theorem. The deconvolution algorithm breaks a
dataset into groups that do not share certain features, i.e. essentially
causal clustering and algorithmic partition by probable generative mechanism.
These methods are conceptually different from traditional clustering and
partition in machine learning approaches.

Ultimately the purpose of this method is to distinguish components that are
generated similarly from those that are generated differently. In information
theoretic terms, the question the method answers is, what are the elements
(e.g. nodes or edges) that can break a network into the components that
maximize their algorithmic information content, i.e., those elements that
preserve the information about the underlying programs generating the data. It
basically preserving maximum information while removing the redundant
components. This new conceptual framework involving parameter-free methods as
a way to tackling the problem of information decomposition by means of causal
deconvolution based on the fundamental theory of algorithmic probability which
defines optimal inductive inference.

Figure 1: All possible causal relations for 3 variables. Grey shaded blocks
have causal loops (and are typically not considered). Green shaded set of
blocks indicate a case of causal hypothesis testing.

###  II.4 Quantum computation and algorithmic information

The synergy between quantum computation and algorithmic information has been
studied extensively in [13]. Two main directions were explored, that can be
applied for causal inference.

In [14] a global/objective view is presented, which involves quantum automata
for algorithmic information. A framework for causal inference based on
algorithmic generative models is developed. This technique of quantum-
accelerated experimental algorithmic information theory (QEAIT) can be
ubiquitously applied to diverse domains. Specifically for genome analysis, the
problem of identifying bit strings capable of self-replication is presented. A
new quantum circuit design of a quantum parallel universal linear bounded
automata (QPULBA) model is developed. This enables a superposition of
classical models/programs to be executed, and their properties can be
explored. The automaton prepares the universal distribution as a quantum
superposition state which can be queried to estimate algorithmic properties of
the causal model.

In [15] a local/subjective view is presented, which involves universal
reinforcement learning in quantum environments. This theoretical framework can
be applied for automated scientific modeling. A universal artificial general
intelligence formalism is presented that can model quantum processes. The
developed quantum knowledge seeking agent (QKSA) is an evolutionary general
reinforcement learning model for recursive self-improvement. It uses resource-
bounded algorithmic complexity of quantum process tomography (QPT) algorithms.
The cost function determining the optimal strategy is implemented as a
mutating gene within a quine. The utility function for an individual agent is
based on a selected quantum distance measure between the predicted and
perceived environment.

These techniques motivates our research in this paper. Unlike
quantum/classical data driven machine learning, these primitives of QEAIT and
QKSA preserves the explanatory power of the model the learning converges to by
exploring the space of programs on an automata model. The QPULBA model (in
QEAIT) and QPT algorithms (in QKSA) can be generalized to a causal oracle and
causal tomography respectively.

##  III Causal hypothesis testing

A canonical approach in causal inference is to formulate different hypotheses
on the cause–effect relations and test them against each other. This technique
is typically used when there is some knowledge of the characteristic of the
phenomenon that is being tested. The complete search-space of directed graphs
between events grows exponentially. An exhaustive example of 3 variable causal
relations is shown in Fig. [1]. In causal hypothesis testing (CHT) a subset of
these graphs are considered as the set of hypothesis being tested against each
others. For the example, in Fig. [1], we can consider, two hypothesis (shaded
in green):

  * •

A causes B, C is independent

  * •

A causes C, B is independent.

###  III.1 Quantum advantage in classical causal hypothesis testing

Quantum information enables a richer spectrum of causal relations that is not
possible to access via classical statistics. Most research in this direction
is towards exploring causality in the quantum context [16, 17, 18, 19, 20,
21]. Our focus in this work is specifically using the quantum formulation to
provide a computational advantage with respect to a classical technique on
classical data.

This problem is studied extensively in [3]. The research analyzes the task of
identifying the effect of a given input variable. The constructed quantum
strategy reduces the error probability by an exponential amount, doubling the
decay rate of the error probability with the number of accesses to the
relevant variables. This decay rate is the highest achievable rate allowed by
quantum mechanics, even if one allows for exotic setups where the order of
operations is indefinite. The key ingredient of the quantum speedup is the
ability to run multiple equivalent experiments in a quantum superposition.
This can be used for the detection of a causal link between two variables, and
in the identification of the cause of a given variable, as well.

The set of allowed causal relationships depends on the physical theory, which
determines which maps can be implemented by physical processes. In classical
physics, cause-effect relations are typically represented by conditional
probability distributions, while, in quantum theory, they are described by
quantum channels, i.e., completely positive trace-preserving maps transforming
density matrices of system A into density matrices of system B.

##  IV Problem formulation

Contrary to the general case considered in [3], our formulation is towards an
implementable quantum circuit that is demonstrated on the Qiskit
statevector_simulator. This formulation is detailed here. We consider an equal
number of input and output variables to implement a quantum circuit. Thus, we
describe our input variables (causes) through the set
$C=\\{c_{0},c_{1},\dots,c_{k-1}\\}$, while the output variables (effects) are
the denoted by the set $E=\\{e_{0},e_{1},\dots,e_{k-1}\\}$.

$|C|=|E|=k,$ (1)

where $k$ denotes the total number of causes/effects. We will also consider
the simplification that all variables are of equal length $d=|c_{i}|=|e_{j}|$,
where $c_{i}\in C$ and $e_{j}\in E$. Thus each variable has $2^{d}$ states.
Furthermore, we maintain the simplification that each effect is a permutation
of only one cause, i.e. the map from $C$ to $E$ is a bijective function, one-
to-one correspondence, or invertible function.

###  IV.1 Model implementation

The practical implementation is decomposed into three following steps:

  * •

$U_{\textrm{in}}^{\textrm{A}},U_{\textrm{in}}^{\textrm{B}}$, represents the
initialization of the system A and B respectively.

  * •

$U_{\textrm{per}}$, decomposes into parallel strategies which represents the
possible ways to partition input into pairs. The number of qubits in the
reference ($N_{\textrm{ref.}}$) is formulated as follows

$N_{\textrm{ref.}}=\lceil\textrm{log}_{2}(kd)\rceil,$ (2)

where $kd=N_{\textrm{A}}$ = $N_{\textrm{B}}$ represents the dimension of
system A and B respectively.

  * •

$U_{\textrm{orc}}$, represents an unknown process that induces a causal
relation between input and output.

The qubit requirement of our model grows as

$2N_{A}+\lceil\log(N_{A})\rceil.$ (3)

The model is illustrated through an implementable quantum circuit in Fig. [4]
in Appendix.

Figure 2: Illustration of number of controlled Bell unitary required to
implement the $U_{\textrm{per}}$ in respect with number of linearly
independent state $r$ (left hand side figure) and with the dimension of the
subsystem (right hand side figure).

In the previous work one of the innovations introduced was to entangle the
inputs for the parallel strategy instead of initializing with tensor product
state which reduces the exponential measurement resource requirement by
correlating the basis. But the cost of implementing the entangled
initialization was not analysed. Through our model implementation we are able
to show that the operations required to implement the $U_{\textrm{per}}$ grows
exponentially with the dimension of the of the subsystem as illustrated in
Fig. [2] but linear in respect with the linearly independent states i.e. $r$.

###  IV.2  Practical error probability

Figure 3: On the left hand side we illustrate the of variation of practical
case error probability in respect with oracle parameter $\theta$. It should be
noted that the default hypothesis is maximally mixed whereas the alternative
hypothesis is CRY($\theta$). The $x$-axis indicates the variation of the angle
in CRY($\theta$). On the right hand side we present the distance between the
two hypothesis with $\theta$ when a class of distance measures are considered.

In quantum case the coherent superposition of equivalent configuration gives
the error probability [3]

$p_{\textrm{err}}=\frac{r}{2d^{N}}\left(1-\sqrt{1-r^{-2}}\right)\xrightarrow[]{r>>1}\frac{1}{4rd^{N}},$
(4)

where $r$ is the number of linearly independent states. Ref.[3] suggests that
the Eq.(4) gives us the limiting case error probability. However for specific
case of causal hypothesis testing the error probability should be dependent on
the two specific hypothesis. As an example if
$U_{\textrm{orc}}=U_{\textrm{orc}}^{\textrm{alter}}$ where
$U_{\textrm{orc}}^{\textrm{alter}}$ is the alternate hypothesis then
$p_{\textrm{err}}=1$.
From the numerical illustrations we introduce a correction factor proportional
to the process distance between the two oracles. Hence the modified Eq.(4)
takes the form

$p_{\textrm{err}}^{\textrm{prac}}=\frac{r}{2d^{N}}\left(1-\sqrt{1-r^{-2}}\right)\Delta\left[U_{\textrm{orc}},U_{\textrm{orc}}^{\textrm{alter}}\right]\xrightarrow[]{r>>1}\frac{\Delta\left[U_{\textrm{orc}},U_{\textrm{orc}}^{\textrm{alter}}\right]}{4rd^{N}},$
(5)

###  IV.3 Numerical results

To obtain numerical results we consider the setup represented in Fig. [4]
where the dimension of the subsystems are same. Throughout the numerical
investigation we choose the oracles as follows.
$U_{\textrm{orc}}=\mathbb{I}^{\otimes
N_{\textrm{A}}}\otimes\mathbb{I}^{\otimes N_{\textrm{B}}}$ and as for the
alternate hypothesis we choose

$U_{\textrm{orc}}^{\textrm{alter}}(\theta)=\begin{cases}\lvert
0\rangle^{\otimes N_{\textrm{A}}}\otimes\mathbb{I}^{\otimes
N_{\textrm{B}}},\\\ \\\ \lvert 1\rangle^{\otimes
N_{\textrm{A}}}\otimes\textrm{RY}(\theta)^{\otimes
N_{\textrm{B}}}.\end{cases}$ (6)

We have utilized IBM open-source quantum computer simulator qiskit and
simulated the above mentioned implementation for
$N_{\textrm{A}}=N_{\textrm{B}}=4$ qubits using statevector_simulator. The main
motivation behind the choice of CRY$(\theta)$ as the alternate hypothesis is
by varying the parameter $\theta$ one can go from non-correlated to fully
correlated state.

In left hand side subfigure of Fig. [3] we investigate the variation of error
probability (see Eq.(4)). It can be seen that the error probability varies as

$p_{\textrm{err}}=\begin{cases}p_{\textrm{err}}^{\textrm{min}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\;\theta=(2m+1)\pi,\\\
p_{\textrm{err}}^{\textrm{max}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\;\theta=2m\pi,\\\
p_{\textrm{err}}^{\textrm{min}}<p_{\textrm{err}}<p_{\textrm{err}}^{\textrm{max}}\;\;\;\;\;\;\;\;\textrm{otherwise},\end{cases}$

where $m=0,1,2,\ldots$. The difference in minimum error probability and the
limiting case probability that is given in ref.[3] is constant at

$\lvert p_{\textrm{err}}-p_{\textrm{err}}^{\textrm{min}}\rvert\sim 0.194.$ (7)

This suggests that the error probability in Eq.(4) requires a slight
modification as per the practical implementation. This modification is
reflected through the distance between the oracles under consideration i.e.
$\Delta\left[U_{\textrm{orc}}^{\textrm{alter}},U_{\textrm{orc}}\right]$. Based
on this in Eq.(5) we introduce the practical case error probability.

Next we move forward to observe the variation of process distance in respect
with oracle parameter. On the right hand side subfigure of Fig. [3] we
illustrate the distance between $U_{\textrm{orc}}$ and
$U_{\textrm{orc}}^{\textrm{alter}}(\theta)$ in respect with $\theta$. For this
investigation we have considered three distance measures

  1. 1.

Trace distance which is defined by

$\Delta=\frac{1}{2}\textrm{Tr}\left(\rho_{\textrm{orc}}-\rho_{\textrm{orc}}^{\textrm{alter}}\right),$
(8)

where $\rho$ represents the Choi representation of the unitary.

  2. 2.

Bures distance which defined as

$\Delta=2\left(1-\sqrt{F(\rho_{\textrm{orc}},\rho_{\textrm{orc}}^{\textrm{alter}})}\right),$
(9)

where $F$ quantifies the process fidelity between the Choi representation of
oracles.

  3. 3.

Hilbert-Schmidt distance which is defined by

$\Delta=\textrm{Tr}\left[\left(\rho_{\textrm{orc}}-\rho_{\textrm{orc}}^{\textrm{alter}}\right)^{2}\right].$
(10)

It can be seen that the Bures and Hilbert-Schmidt distance coincides at
$\theta=2m\pi$, where $m=0,1,2,\ldots$.

##  V Conclusion

In this article, we presented a quantum circuit formulation of causal
tomography, as formulated in [3]. This was implemented using the open source
quantum programming and simulation platform qiskit. To this end, we studied
stricter bounds of error probability relative to the process distance between
the causal hypotheses. We also elucidate the pragmatic gate complexity of the
causal tomography entangled pair indexing. As in [3] it is shown that the
quantum advantage holds for generalized probabilistic theory so the practical
case that we present here is optimal in any scenario.

Our motivation for this project is driven by the increasing focus on causal
inference in machine learning. Besides monitoring the information flow in
future quantum communication networks, as discussed in [3], causal tomography
is crucial for understanding the bounds of general intelligence and for
bioinformatics use cases. In our future work, we aim to apply our developed
quantum accelerated causal tomography framework for these applications.

###### Acknowledgements.

This project was initiated under the QIntern 2021 project “Reinforcement
Learning Agent for Quantum Foundations”. Therefore we would like to thank the
organizers of the program and QWorld Association. We would also like to thank
QIndia for providing us with a collaboration platform for the ideas motivating
this research. AK was partially supported by the Polish National Science
Center (NCN) under the grant agreement $2019/33/B/ST6/02011$.

##  Appendix A Circuit Implementation

Here we provide an illustration of the circuit that has been implemented to
obtain the numerical results. Th dimension of the subsystem is kept at $4$ due
to the hardware limitations.

Figure 4: The illustration of the model implementation. Where we also show the
decomposition of $U_{\textrm{per}}$ for $N_{\textrm{A}}=N_{\textrm{B}}=4$.

##  Appendix B Poster

Here attach the poster which has been presented in the Causalworlds: The
interface between quantum and relativistic causality, foundations and
practicalities. The link to the conference https://causalworlds.ethz.ch/.

Figure 5: The poster.

## References

  * Ried  _et al._ [2015] K. Ried, M. Agnew, L. Vermeyden, D. Janzing, R. W. Spekkens, and K. J. Resch, A quantum advantage for inferring causal structure, Nature Physics 11, 414 (2015).
  * Fitzsimons  _et al._ [2015] J. F. Fitzsimons, J. A. Jones, and V. Vedral, Quantum correlations which imply causation, Scientific reports 5, 1 (2015).
  * Chiribella and Ebler [2019] G. Chiribella and D. Ebler, Quantum speedup in the identification of cause–effect relations, Nature communications 10, 1 (2019).
  * Lavin  _et al._ [2021] A. Lavin, H. Zenil, B. Paige, D. Krakauer, J. Gottschlich, T. Mattson, A. Anandkumar, S. Choudry, K. Rocki, A. G. Baydin, _et al._ , Simulation intelligence: Towards a new generation of scientific methods, arXiv preprint arXiv:2112.03235 (2021).
  * Maruyama [2021] Y. Maruyama, Categorical artificial intelligence: the integration of symbolic and statistical ai for verifiable, ethical, and trustworthy ai, in  _International Conference on Artificial General Intelligence_ (Springer, 2021) pp. 127–138.
  * Note [1] A treatment is an action that can be applied or withheld from that unit.
  * Guo  _et al._ [2020] R. Guo, L. Cheng, J. Li, P. R. Hahn, and H. Liu, A survey of learning causality with data: Problems and methods, ACM Computing Surveys (CSUR) 53, 1 (2020).
  * Rothman and Greenland [2005] K. J. Rothman and S. Greenland, Causation and causal inference in epidemiology, American journal of public health 95, S144 (2005).
  * Pearl  _et al._ [2000] J. Pearl _et al._ , Models, reasoning and inference, Cambridge, UK: CambridgeUniversityPress 19 (2000).
  * Imbens and Rubin [2015] G. W. Imbens and D. B. Rubin,  _Causal inference in statistics, social, and biomedical sciences_ (Cambridge University Press, 2015).
  * Spirtes  _et al._ [2000] P. Spirtes, C. N. Glymour, R. Scheines, and D. Heckerman,  _Causation, prediction, and search_ (MIT press, 2000).
  * Zenil  _et al._ [2019] H. Zenil, N. A. Kiani, A. A. Zea, and J. Tegnér, Causal deconvolution by algorithmic generative models, Nature Machine Intelligence 1, 58 (2019).
  * Sarkar [2022] A. Sarkar, Applications of quantum computation and algorithmic information: for causal modeling in genomics and reinforcement learning, (2022).
  * Sarkar  _et al._ [2021a] A. Sarkar, Z. Al-Ars, and K. Bertels, Estimating algorithmic information using quantum computing for genomics applications, Applied Sciences 11, 2696 (2021a).
  * Sarkar  _et al._ [2021b] A. Sarkar, Z. Al-Ars, H. Gandhi, and K. Bertels, Qksa: Quantum knowledge seeking agent–resource-optimized reinforcement learning using quantum process tomography, arXiv preprint arXiv:2112.03643 (2021b).
  * Costa and Shrapnel [2016] F. Costa and S. Shrapnel, Quantum causal modelling, New Journal of Physics 18, 063032 (2016).
  * Giarmatzi [2019] C. Giarmatzi, A quantum causal discovery algorithm, in  _Rethinking Causality in Quantum Mechanics_ (Springer, 2019) pp. 125–150.
  * Javidian  _et al._ [2021] M. A. Javidian, V. Aggarwal, F. Bao, and Z. Jacob, Quantum entropic causal inference, in  _Quantum Information and Measurement_ (Optical Society of America, 2021) pp. F2C–3.
  * Bai  _et al._ [2021] G. Bai, Y.-D. Wu, Y. Zhu, M. Hayashi, and G. Chiribella, Quantum causal unravelling, arXiv preprint arXiv:2109.13166 (2021).
  * Bai  _et al._ [2020] G. Bai, Y.-D. Wu, Y. Zhu, M. Hayashi, and G. Chiribella, Efficient algorithms for causal order discovery in quantum networks, arXiv preprint arXiv:2012.01731 (2020).
  * Bavaresco  _et al._ [2021] J. Bavaresco, M. Murao, and M. T. Quintino, Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination, Physical review letters 127, 200504 (2021).