# Quantum Resonant Dimensionality Reduction and Its Application in Quantum
Machine Learning

Fan Yang  These authors contributed equally to this work.  Beijing Academy of
Quantum Information Sciences, Beijing 100193, China  State Key Laboratory of
Low-Dimensional Quantum Physics and Department of Physics, Tsinghua
University, Beijing 100084, China  Furong Wang  These authors contributed
equally to this work.  China North Vehcle Research Institute, Beijing 100072,
China  Xusheng Xu  State Key Laboratory of Low-Dimensional Quantum Physics and
Department of Physics, Tsinghua University, Beijing 100084, China  Pao Gao
Beijing Academy of Quantum Information Sciences, Beijing 100193, China  Tao
Xin  Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China  ShiJie Wei
<EMAIL_ADDRESS>Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Guilu Long<EMAIL_ADDRESS>State Key Laboratory of Low-
Dimensional Quantum Physics and Department of Physics, Tsinghua University,
Beijing 100084, China  Tsinghua National Laboratory for Information Science
and Technology, Beijing 100084, People’s Republic of China  Collaborative
Innovation Center of Quantum Matter, Beijing 100084, China

###### Abstract

Quantum computing is a promising candidate for accelerating machine learning
tasks. Limited by the control accuracy of current quantum hardware, reducing
the consumption of quantum resources is the key to achieving quantum
advantage. Here, we propose a quantum resonant dimension reduction (QRDR)
algorithm based on the quantum resonant transition to reduce the dimension of
input data and accelerate the quantum machine learning algorithms. After QRDR,
the dimension of input data $N$ can be reduced into desired scale $R$, and the
effective information of the original data will be preserved correspondingly,
which will reduce the computational complexity of subsequent quantum machine
learning algorithms or quantum storage. QRDR operates with polylogarithmic
time complexity and reduces the error dependency from the order of
$1/\epsilon^{3}$ to the order of $1/\epsilon$, compared to existing
algorithms. We demonstrate the performance of our algorithm combining with two
types of quantum classifiers, quantum support vector machines and quantum
convolutional neural networks, for classifying underwater detection targets
and quantum many-body phase respectively. The simulation results indicate that
reduced data improved the processing efficiency and accuracy following the
application of QRDR. As quantum machine learning continues to advance, our
algorithm has the potential to be utilized in a variety of computing fields.

##  I introduction

Machine learning has proven to be a remarkably powerful tool with broad
practical applications across various fields of science and engineering,
including finance Goodell et al. (2021); Ghoddusi et al. (2019), medical
science Erickson et al. (2017); Deo (2015), and the simulation of classical
and complex quantum systems. It has shown notable success in compressing high-
dimensional data, where near-term quantum devices may offer significant speed
enhancements Gujju et al. (2023). Given the potential quantum advantage of
quantum computing, numerous quantum machine learning (QML) algorithms have
been proposed Dunjko and Briegel (2018), such as quantum recommendation
systems (QRS) Kerenidis and Prakash (2016), quantum support vector machines
(QSVM) Rebentrost et al. (2014), quantum principal component analysis (QPCA)
Lloyd et al. (2014), and quantum neural networks Abbas et al. (2021); Beer et
al. (2020), among others.

In machine learning, dimensionality reduction (DR) is a valuable technique for
refining information and significantly decreasing data processing time. DR
involves compressing high-dimensional datasets into lower-dimensional
representations while preserving key information from the original dataset.
Principle component analysis (PCA) DR is the most representative example of
linear DR. It works by projecting the original data onto the subspace of the
covariance matrix with larger singular values while disregarding components
with smaller values. These smaller principal components often represent noise,
so dimensionality reduction can enhance the accuracy of subsequent data
processing. By applying kernel methods, kernel principal component analysis
(KPCA) extends the concept of PCA to enable nonlinear dimensionality
reduction.

In recent years, several quantum dimensionality reduction (QDR) methods have
been proposed. Here we focus on an important type, the PCA-based QDR
algorithms, which can efficiently remove redundant information by identifying
orthogonal principal components, reducing computational complexity, and
enhancing model performance, especially in high-dimensional datasets. Among
PCA-based QDR algorithms, the most renowned one is quantum principal component
analysis (QPCA) Lloyd et al. (2014), which uses density matrices to obtain
their eigenvalues and eigenstates. In QPCA, quantum amplitude estimation and
quantum phase estimation (QPE) are also necessary for preparing data vectors
after DR, but they demand significant quantum resources. Following QPCA, two
approaches can achieve QDR, one is QPE-based QDR Li et al. (2020); Yu et al.
(2019) and the other is variational quantum algorithm (VQA)-based QDR Liang et
al. (2020, 2020).QPE-based algorithms have reduced time complexity compared to
QPCA. They still face challenges related to the required number of ancilla
qubits and evolution time, posing difficulties for current state-of-the-art
quantum computers. In contrast, VQA-based algorithms are generally seen as
friendly for noisy intermediate-scale quantum (NISQ) devices, but its
complexity becomes increasingly challenging to analyze as the number of qubits
grows.

There are also many other types of quantum dimension reduction algorithms,
like based on autoencoders Romero et al. (2017); Huang et al. (2020),
A-optimal projection Duan et al. (2019); Pan et al. (2020), linear
discriminant analysis Yu et al. (2023); Cong and Duan (2016) and locally
linear embedding He et al. (2020); Sidhu (2019). These algorithms are based on
different data mining principles and are used in different data sets and
different demand scenarios. We don’t go to details here.

To address these challenges, we introduce a novel quantum dimensionality
reduction algorithm called quantum resonant dimensionality reduction (QRDR),
based on quantum resonant transitions (QRT) Li et al. (2019, 2021). By
employing QRT instead of QPE, the number of ancilla qubits is reduced from
$O(1/\epsilon+\log R)$ to $O(\log R)$ in QRDR that is independent of the
accuracy $\epsilon$, offering an advantage over other quantum DR algorithms
such as QPE-based methods Li et al. (2020); Yu et al. (2019). In contrast to
VQA-DR, our QRDR presents an exact time complexity. Comnpared to QPE-based DR
algorithm and QPCA algorithm, the time complexity of QRDR is reduced from the
order $1/\epsilon^{3}$ to the order $1/\epsilon$. We simulate the QRDR
algorithm numerically and evaluate its performance using quantum support
vector machines and quantum convolutional neural networks (QCNN). The results
demonstrate that our QRDR effectively reduces the dimension of quantum data,
enabling various quantum machine learning algorithms to process the data more
efficiently post-reduction.

The remainder of this paper is organized as follows: Sec. II provides the
framework of the algorithm and details the quantum resources consumed. Sec.
III presents the simulation results of applying QRDR in quantum support vector
machines and quantum convolutional neural networks. In Sec. IV, we compare our
algorithm with existing approaches and conclude with a summary of our
findings.

##  II The framwork of the algorithm

###  II.1 Principal component analysis dimension reduction and its quantum
expression

Principal component analysis for dimensionality reduction relies on linear
transformations involving the covariance matrix (or kernel matrix). In this
paper, we focus on the basic PCA method, leaving out advanced techniques such
as kernel methods and centering, which are detailed in references Li et al.
(2020); Yu et al. (2019).

The original data before DR can be represented as a matrix
$X\in\mathbb{R}^{M\times N}$, where the $i$th row of $X$ corresponds to the
$i$th sample, $\bm{x_{i}}=[x_{i}^{1},x_{i}^{2},\ldots,x_{i}^{N}]$, containing
$N$ features. The covariance matrix of this data, denoted as
$A\in\mathbb{R}^{N\times N}=X^{T}X$. The data undergoes singular value
decomposition (SVD) as follows

$X=\sum_{k=1}^{M}\sigma_{k}\bm{u_{k}}^{T}\bm{v_{k}},$ (1)

where $\bm{u_{k}}$ and $\bm{v_{k}}$ are singular vectors corresponding to
singular value $\sigma_{k}$. It is usually assumed that the singular values
are ordered from largest to smallest. The matrix $A$ has eigenvalue
decomposition

$A=\sum_{k=1}^{M}\sigma_{k}^{2}\bm{v_{k}}^{T}\bm{v_{k}}.$ (2)

By definition, $A$ is the Hermitian matrix and the $k$-th eigenvalue is
$\lambda_{k}=\sigma_{k}^{2}$ with eigenvector $\bm{v_{k}}$. When the dimension
is reduced from $N$ to $R$, the data matrix $Z\in\mathbb{R}^{M\times R}$ after
dimensionality reduction can be represented by the formula $Z=XV_{R}^{T}$. The
matrix $V_{R}$ is a projection matrix composed of the first $R$ singular
vectors, with each row $j$ of $V_{R}$ corresponding to the $j$th singular
vector $\bm{v_{j}}$. Consequently, the value of the $j$-th feature for $z_{i}$
is given by $z_{i}^{j}=\bm{x_{i}}\cdot\bm{v_{j}}^{T}$.

For the QDR, the input is a quantum state that encodes the information from
the original data

$|X\rangle=\sum_{i=1}^{M}\sum_{j=1}^{N}x_{i}^{j}|j\rangle|i\rangle=\sum_{i=1}^{M}\|x_{i}\||x_{i}\rangle|i\rangle,$
(3)

where $|i\rangle$ and $|j\rangle$ represent quantum computational basis
states, and $|x_{i}\rangle=\frac{1}{|x_{i}|}\sum_{j=1}^{N}x_{i}^{j}|j\rangle$
with the normalized coefficient
$|x_{i}|=\sqrt{\sum_{j=1}^{N}|x_{i}^{j}|^{2}}$. The output state of QDR is
given by

$|Z\rangle=\sum_{i=1}^{M}\sum_{j=1}^{R}z_{i}^{j}|j\rangle|i\rangle=\sum_{i=1}^{M}\|z_{i}\||z_{i}\rangle|i\rangle.$
(4)

The goal of QDR is to prepare the state $|Z\rangle$ as described in Eq.(4). In
this section, the overall normalization coefficients for the quantum states
are omitted.

###  II.2 Quantum Resonant Dimensionality Reduction Algorithm

We use quantum resonant transitions (QRT) to realize the dimensionality
reduction process, which includes designing the Hamiltonian and implementing
the time evolution operator. Here we take one of the $M$ samples, $N$
dimensional vector $\bm{x}$, as the original data. Our algorithm utilizes one
probe qubit to find the eigenstate, $n=\lceil\log_{2}N\rceil$ qubits to store
the state corresponding to the original data, and $r=\lceil\log_{2}R\rceil$
qubits to store the state after dimensionality reduction. The Hamiltonian is
defined as follows

$\displaystyle\mathcal{H}=$ $\displaystyle|0\rangle\langle
0|\otimes(|0\rangle\langle 0|^{\otimes r}-I_{2}^{\otimes r})\otimes
I_{2}^{\otimes n}+|1\rangle\langle 1|\otimes H_{p}$ (5)
$\displaystyle+\frac{c\pi}{2}\sigma_{y}\otimes B\otimes I_{2}^{\otimes n}.$

Here $c$ represents the resonant parameter, $\sigma_{y}$ is the Pauli matrix,
and $B=\sqrt{2^{r}}H_{d}^{\otimes r}$, where
$H_{d}=\frac{1}{\sqrt{2}}[1,1;1,-1]$ is the Hadamard matrix. The three terms
in Equation (5) correspond to the Hamiltonian of the probe qubit, the problem
Hamiltonian, and their interaction, respectively. Specifically, the second
term, $H_{p}$, can be expressed as follows

$H_{p}=H_{\lambda}\otimes I_{2}^{\otimes n}+I_{2}^{\otimes r}\otimes A,$ (6)

where $H_{\lambda}$ is a diagonal matrix that encodes the $R$ eigenvalues
$\lambda_{k}=\sigma_{k}^{2}$ of the covariance matrix $A$. The diagonal
elements of $H_{\lambda}$ are given by $h_{k}=-\lambda_{k}$. Given the
original data $\bm{x}$, the input state $|\psi\rangle_{in}$ is represented as
$|0\rangle|0\rangle^{\otimes r}|x\rangle$. This typically requires specialized
quantum devices such as quantum random access memory (QRAM) Giovannetti et al.
(2008) to implement. The state can also be expanded in terms of the
eigenvectors of $A$ as

$|\psi\rangle_{in}=|0\rangle|0\rangle|x\rangle=|0\rangle|0\rangle^{\otimes
r}\sum_{k=1}^{N}z^{k}|v_{k}\rangle,$ (7)

where $z^{k}=\langle v_{k}|x\rangle$. Setting $t=1/c$, the state after the
application of the Hamiltonian evolution operator $e^{-i\mathcal{H}t}$ is

$\displaystyle e^{-i\mathcal{H}t}|\psi\rangle_{in}$
$\displaystyle=|1\rangle\sum_{k=1}^{R}z^{k}|k\rangle|v_{k}\rangle$ (8)
$\displaystyle+|0\rangle\sum_{k^{\prime}=R+1}^{N}|0\rangle^{\otimes
r}z^{k^{\prime}}|v_{k^{\prime}}\rangle.$

It is assumed that $c$ is much smaller than the minimum difference between
eigenvalues of $H$, allowing us to neglect the effects of off-resonance. A
detailed analysis of the dynamics, errors, and time complexity will be
provided in the next section.

To obtain the low-dimensional state, the probe qubit must be collapsed to
state $|1\rangle$ and the second and third registers disentangled. It means
that we need to perform additional operations to make the quantum state evolve
from $|k\rangle|v_{k}\rangle$ to $|k\rangle|0\rangle$, for each $k$. We employ
the method proposed in Ref. Yu et al. (2019), which utilizes a reference state
$|\psi\rangle$ and its corresponding inverse unitary operator
$U_{\psi}^{-1}|\psi\rangle=|0\rangle^{\otimes n}$ to achieve this step.The
projection $\phi_{k}$ of the reference state and each eigenstate $v_{k}$ needs
to be measured. Then, apply the inverse unitary operator and controlled
rotations to the ancilla qubits and work qubits for different wave vectors
$k$. The final state is obtained in the second register is

$|\psi\rangle_{out}=|1\rangle\sum_{k=1}^{R}z^{k}|k\rangle|0\rangle^{\otimes
n}.$ (9)

According to the above derivation, we take $|X\rangle$ that defined in Eq. 3
as original data state, the input state can be expressed as

$|\Psi\rangle_{in}=|0\rangle|0\rangle^{\otimes
r}|X\rangle=|0\rangle|0\rangle^{\otimes
r}\sum_{i=1}^{M}\|x_{i}\||x_{i}\rangle|i\rangle.$ (10)

The evolution operator does not affect the serial number state $|i\rangle$, so
the state after applying the operator is

$|1\rangle\sum_{i=1}^{M}\sum_{k=1}^{R}z_{i}^{k}|k\rangle|v_{k}\rangle|i\rangle+|0\rangle|0\rangle^{\otimes
r}\sum_{i=1}^{M}\sum_{k^{\prime}=R+1}^{N}z_{i}^{k^{\prime}}|v_{k^{\prime}}\rangle|i\rangle,$
(11)

where $z_{i}^{k}=\langle v_{k}|x_{i}\rangle$. After measuring the probe qubit
and disentangling the second and third registers, the final state is
consistent with Eq. (4)

$|\Psi\rangle_{out}=|1\rangle\sum_{i=1}^{M}\|z_{i}\||z_{i}\rangle|i\rangle|0\rangle^{\otimes
n}=|1\rangle|Z\rangle|0\rangle^{\otimes n},$ (12)

which is the output state of the algorithm. The QRDR process can be summarized
in Alg.1

Inputs: Dataset state $|X\rangle$, covariance matrix $A$ and its first $R$
eigenvalues $\lambda_{k},k\in[1,R]$.

step 1: Prepare the initial state $|0\rangle|0\rangle^{\otimes r}|X\rangle$.

step 2: Set appropriate resonant parameter $c$.

step 3: Simulate the Hamiltonian $\mathcal{H}$ for a duration time of $t=1/c$.

step 4: Measure the probe qubit and proceed if the measurement result is
$|1\rangle$.

step 5: Disentangle the second and third registers using
$U_{\psi}^{-1}|\psi\rangle$.

Outputs: $|1\rangle|Z\rangle|0\rangle^{\otimes n}$.

Algorithm 1 Quantum Resonant Dimensionality Reduction

The different procedures for dimension reduction using QRDR, QPE-DR, and QPCA
are shown in Fig.fig:flow.

Figure 1: The procedures for dimension reduction using QRDR, QPE-DR, and QPCA
differ significantly. The main distinction between QRDR and QPE-DR lies in the
use of the Hamiltonian, which replaces the need for QPE and controlled
rotation operators in QRDR. In contrast, QPCA requires the use of QPE and
involves repeated measurements of auxiliary qubits.

###  II.3 Errors and Complexity of QRDR

In this section, we give the details of the key step of QRDR, implementing
time evolution operator $e^{-i\mathcal{H}t}$ Yu et al. (2019); Li et al.
(2020). We provide the specific procedures that how does the quantum state
evolve from Eq. (7) to Eq. (8). Assuming the input state is the eigenstate of
matrix $A$

$|\Psi\rangle_{in}=|0\rangle|0\rangle^{\otimes r}|v_{k}\rangle,$ (13)

where $k\leq R$. Without considering interaction term
$\frac{c\pi}{2}\sigma_{y}\otimes B\otimes I_{2}^{\otimes n}$, the other parts
of Hamiltonian are

$\mathcal{H^{\prime}}=|0\rangle\langle 0|\otimes(|0\rangle\langle 0|^{\otimes
r}-I_{2}^{\otimes r})\otimes I_{2}^{\otimes n}+|1\rangle\langle 1|\otimes
H_{p}.$ (14)

$|\Psi\rangle_{in}$ is the eigenstate of $\mathcal{H^{\prime}}$ with
eigenvalue $0$. For the second part of $\mathcal{H^{\prime}}$,
$|1\rangle|j\rangle|v_{k}\rangle$ are also the eigenstates with eigenvalues
$\lambda_{k}-\lambda_{j}$. If $j=k$, it has eigenvalues
$\lambda_{k}-\lambda_{j}=0$, which are same with $|0\rangle|0\rangle^{\otimes
r}|v_{k}\rangle$.

Firstly, we consider the dynamics between $|0\rangle|0\rangle^{\otimes
r}|v_{k}\rangle$ and $|1\rangle|j\rangle|v_{j}\rangle$ ($j,k\leq R$) because
of their same eigenvalues $0$. For each pair of $|0\rangle|0\rangle^{\otimes
r}|v_{k}\rangle$ and $|1\rangle|j\rangle|v_{j}\rangle$, $\mathcal{H}$ in this
subspace can be written as Wang (2016)

$\mathcal{H}_{kj}=\left[\begin{array}[]{cc}0&-id_{kj}\\\ id_{kj}^{*}&0\\\
\end{array}\right],$ (15)

where $d_{kj}=\frac{c\pi}{2}\langle 0|^{\otimes r}B|j\rangle\langle
v_{k}|I_{2}^{\otimes n}|v_{j}\rangle=\frac{1}{2}c\pi\delta_{kj}$.
$\delta_{kj}$ is $1$ if and only if $j=k$, and $0$ otherwise. It means that
resonant transitions just happens between $|0\rangle|0\rangle^{\otimes
r}|v_{k}\rangle$ and $|1\rangle|k\rangle|v_{k}\rangle$. So the state in Eq.
(13) after evolution operator $e^{-i\mathcal{H}t}$ is

$e^{-i\mathcal{H}t}|\Psi_{in}\rangle=\cos(\frac{c\pi
t}{2})|0\rangle|0\rangle^{\otimes r}|v_{k}\rangle+\sin(\frac{c\pi
t}{2})|1\rangle|k\rangle|v_{k}\rangle.$ (16)

In setting $t=1/c$, the state is $|1\rangle|k\rangle|v_{k}\rangle$ after this
step.

The other case is off-resonance that happens between
$|0\rangle|0\rangle^{\otimes r}|v_{k}\rangle$ and
$|1\rangle|j\rangle|v_{k^{\prime}}\rangle$. Here $j\neq k^{\prime}$, and in
this subspace $\mathcal{H}$ is a little different with Eq. (15)

$\mathcal{H}_{kjk^{\prime}}=\left[\begin{array}[]{cc}0&-id_{kjk^{\prime}}\\\
id_{kjk^{\prime}}^{*}&\lambda_{k^{\prime}}-\lambda_{j}\\\ \end{array}\right].$
(17)

Here $d_{kjk^{\prime}}=\frac{1}{2}c\pi\delta_{kk^{\prime}}$, so the resonant
transition just happens when $k=k^{\prime}$ and the amplitude of this small
off-resonance is

$\displaystyle b_{kjk^{\prime}}$
$\displaystyle=\frac{c\pi|d_{kjk^{\prime}}|}{\sqrt{(c\pi|d_{kjk^{\prime}}|)^{2}+|\lambda_{k^{\prime}}-\lambda_{j}|^{2}}}$
(18)
$\displaystyle=\frac{c\pi\delta_{kk^{\prime}}}{\sqrt{(c\pi\delta_{kk^{\prime}})^{2}+|\Delta_{jk^{\prime}}|^{2}}}$
$\displaystyle\leq\frac{c\pi\delta_{kk^{\prime}}}{|\Delta_{jk^{\prime}}|}=\mathcal{O}(c/\Delta_{min}),$

where $\Delta_{min}$ denotes the min erengy gap
$\Delta_{\min}=\min{|E_{i+1}-E_{i}|}$. For the other eigenstates that $k>R$,
they always fit $k\neq j$ and almost stay in original state with a small off-
resonance.

Therefore, considering the off-resonance error, the input state in Eq. (7)
after evolution operator $e^{-i\mathcal{H}/c}$ is

$\displaystyle|1\rangle\sum_{k=1}^{R}z^{k}\alpha_{k}|k\rangle|v_{k}\rangle+\sum_{k^{\prime}=R+1}^{N}|0\rangle|0\rangle^{\otimes
r}z^{k^{\prime}}\alpha_{k}^{\prime}|v_{k^{\prime}}\rangle$ (19)
$\displaystyle+\mathcal{O}(c/\Delta_{min})|\Psi\rangle_{e}.$

$|\Psi\rangle_{e}$ is the error state here, and
$\alpha_{k}\geq\sqrt{1-\sum_{j\neq k}b^{2}_{kjk}}$. Because the sum of
$1/n^{2}$ sequences is converged, it has

$\displaystyle|\alpha_{k}|^{2}$ $\displaystyle\geq 1-\sum_{j\neq
k}|b_{kjk}|^{2}$ (20) $\displaystyle\geq 1-(c\pi)^{2}\sum_{j\neq
k}1/|\Delta_{jk}|^{2}$ $\displaystyle\geq
1-(\frac{c\pi}{\Delta_{min}})^{2}\sum_{j\neq k}1/|j-k|^{2}$
$\displaystyle=1-\mathcal{O}(c^{2}/\Delta_{min}^{2}).$

Defining the fidelity as

$\epsilon=1-|\langle
Z|\Psi\rangle_{out}|^{2}=\mathcal{O}(c^{2}/\Delta_{min}^{2}),$ (21)

so that the query complexity for time evolution step is

$\displaystyle\mathcal{O}(d||\mathcal{H}||_{max}\mathrm{polylog}(MN)t)=$ (22)
$\displaystyle\mathcal{O}(d\|A\|_{max}\mathrm{polylog}(MN)/c)=$
$\displaystyle\mathcal{O}(d\|A\|_{max}\mathrm{polylog}(MN)/\Delta_{min}\epsilon^{0.5}).$

Here $d$ is the sparsity of $A$.

QRDR’s time complexity mainly consists of three parts: obtaining eigenvalues,
Hamiltonian evolution, and decoupling. Firstly, QRDR requires the eigenvalues
of matrix $A$. In previous work Yang et al. (2023), it has been proved that
the error $\epsilon$ is proportional to accuracy of eigenvalues, so the
complexity of obtaining the first $R$ eigenvalues is
$\mathcal{O}(Rd\|A\|_{max}\mathrm{polylog}(MN)/\epsilon)$. For the initial
state, it can be prepared in time $\mathcal{O}(\mathrm{poly}\log(MN))$
according to the QRAM Giovannetti et al. (2008). In step 4, the probability of
success is constant if the first $R$ eigenvalues are enough to reconstruct $A$
as
$\sum_{k=1}^{R}\sigma_{k}^{2}\approx\sum_{k^{\prime}=1}^{N}\sigma^{2}_{k^{\prime}}$.
Secondly, the query complexity for Hamiltonian evolution is
$\mathcal{O}(d\|A\|_{max}\mathrm{polylog}(MN)/\Delta_{min}\epsilon^{0.5})$.
Finally, the query complexity of decoupling corresponding to step 5 is
$\mathcal{O}(R^{1.5}d\|A\|_{max}\mathrm{polylog}(MN)/\epsilon)$ Yu et al.
(2019). We use the method in Ref.Low and Chuang (2017) to achieve the
Hamiltonian time evolution operator, and other approaches like VQAs-based
methods Yuan et al. (2019); Cirstoiu et al. (2020); Li and Benjamin (2017) are
also appliable.

Figure 2: The Schematic quantum circuits of simulating QRDR in quantum machine
learning. The two blue parts are the simulated result of quantum algorithms in
the classical computer. Three orange dotted line parts are used to calculate
precision, convergence rate, and classification accuracy. $|X\rangle$ and
$|Z\rangle$ are the quantum states of training data before and after QRDR.
Similarly, $|q\rangle$ and $|z_{q}\rangle$ correspond to the data to be
classified. $y_{q}$ is the predicted label of $y$.

In conclusion, the total time complexity of QRDR is

$\displaystyle\mathcal{O}($ $\displaystyle
Rd\|A\|_{max}\mathrm{polylog}(MN)/\epsilon+$ (23) $\displaystyle
d\|A\|_{max}\mathrm{polylog}(MN)/\Delta_{min}\epsilon^{0.5}+$ $\displaystyle
R^{1.5}d\|A\|_{max}\mathrm{polylog}(MN)/\epsilon).$

Under the condition that $R=\mathcal{O}(\mathrm{polylog}N)$ eigenvalues can
reconstruct the covariance matrix $A$, and $\Delta{\min}=O(\epsilon^{0.5}/R)$,
the time complexity is simplified to
$\mathcal{O}(d,\mathrm{polylog}(MN)/\epsilon)$. The following discussion of
QPE-based QDR and QPCA assumes the same conditions, where
$R=\mathcal{O}(\mathrm{polylog}N)$. QPE-based QDR Yu et al. (2019); Li et al.
(2020) solve the PCADR problem using quantum phase estimation, which can
prepare the eigenvalues in ancilla qubits. As shown in Fig. 1, it applies QPE
to data state $|x\rangle$ and obtained the state

$|\Psi_{1}\rangle=\Psi\sum_{i=1}^{M}\sum_{j=0}^{N}z_{i}^{j}|\hat{\sigma_{j}}\rangle|v_{j}\rangle|i\rangle.$
(24)

Then it uses control rotation operator like HHL method to remove the
eigenstate corresponding to the eigenvalue after the $R$-th eigenvalue and
applies the inverse operation of QPE, then the state is

$|\Psi_{2}\rangle=\sum_{i=1}^{M}\sum_{j=1}^{R}z_{i}^{j}|v_{j}\rangle|i\rangle.$
(25)

$|v_{i}\rangle$ is the $i$-th eigenstate of $XX^{\dagger}$. QRDR can acheive
the same steps by simulating the Hamiltonian in Eq. 5. The runtime of this
QPE-based DR algorithm is $\mathcal{O}(d,\mathrm{polylog}(MN)/\epsilon^{3})$.
The complexity of QRDR is $\mathcal{O}(d,\mathrm{polylog}(MN)/\epsilon)$ under
the assumption that $\|A\|_{\max}$ is $\mathcal{O}(1)$ order. In the
circumstances $\Delta{\min}=\mathcal{O}(\epsilon^{0.5}/R)$, our algorithm
provides the advantage of polynomial order in accuracy $\epsilon$ compared to
QPE-based QDR. Additionally, QPE requires an extra $\mathcal{O}(1/\epsilon)$
qubits to recover the eigenvalues, whereas QRDR does not require this extra
quantum register. QPCA Lloyd et al. (2014) and Resonant QPCA Li et al. (2021)
can reduce the dimension of the quantum state, but they must solve each of the
$R$ eigenvalues and $M$ samples one by one, as shown in Fig. 1, resulting in a
complexity of $\mathcal{O}(MR)$. Detailed results are shown in the Table 1.

Table 1: Comparison of the complexity of several different quantum dimensionality reduction algorithms with QRDR. | Time complexity | Space complexity
---|---|---
QRDR | $O(d,\mathrm{polylog}(MN)/\epsilon)$ | $O(\log R+\log(MN))$
QPE-DR Li et al. (2020); Yu et al. (2019) | $O(d,\mathrm{polylog}(MN)/\epsilon^{3})$ | $O(\log 1/\epsilon+\log R+\log(MN))$
QPCA Li et al. (2021); Lloyd et al. (2014) | $O(M\mathrm{polylog}(MN)/\epsilon^{3})$ | $O(\log(MN))$

##  III NUMERICAL SIMULATION RESULTS OF QRDR IN QUANTUM MACHINE LEARNING

In this section, we use the QRDR method to reduce the dimensionality of data
prior to its use in quantum machine learning algorithms. First, we assess the
precision of our quantum algorithm by comparing the results of quantum
dimensionality reduction with theoretical expectations. We then apply two
distinct quantum classification algorithms to measure the efficiency of QRDR.

There are two primary reasons for choosing classification as the subsequent
task: First, classification tasks allow for easy evaluation of data quality.
If the data resulting from QRDR cannot be effectively classified (e.g.,
through a decrease in final accuracy), it indicates that some useful
information may have been lost during the process. Second, accuracy can be
used to gauge the convergence rate of quantum machine learning algorithms,
which are two key performance indicators for many machine learning tasks. The
numerical simulation process is summarized in Fig. 2.

###  III.1 QRDR FOR QUANTUM SUPPORT VECTORS MACHINE

In this section, _Connectionist Bench (Sonar, Mines vs. Rocks) Data Set_ is
used to evaluate our QRDR algorithm. This dataset includes samples of mines
and rocks. First, we apply QRDR to reduce the dimensionality of the dataset.
Then, we use a quantum support vector machine (QSVM) to classify the data
before and after QRDR. The algorithm simulation flow in this section is
consistent with that in Fig. 2, with taking QSVM as the quantum classfier. The
main steps of QRDR were discussed in the previous section, while the steps of
QSVM are briefly outlined here.

The input of the quantum support vector machine (QSVM) is a data matrix
$A\in\mathbb{R}^{M\times N}$, containing $M$ distinct sample vectors
$\bm{x_{i}}$, each with $N$ parameters. Each $i$-th sample is assigned a label
$y_{i}$ to classify this sample. The objective of QSVM is to train a quantum
state in $N+1$ dimensions, enabling the classification of new vectors
$\bm{x^{\prime}}$ by determining their label $y^{\prime}$.

The core step of QSVM is to solve the following linear equation using a
quantum computer

$\left(\begin{array}[]{l}\eta_{0}\\\
\bm{\eta}\end{array}\right)=A^{-1}\left(\begin{array}[]{l}0\\\
\bm{y}\end{array}\right),A=\left(\begin{array}[]{cc}0&\overrightarrow{1}^{T}\\\
\overrightarrow{1}&\mathcal{K}+\gamma^{-1}I\end{array}\right).$ (26)

$\mathcal{K}$ is the kernel matrix with the element
$\mathcal{K}_{jk}=\bm{x}^{T}_{j}\cdot\bm{x}_{k}$.
$\bm{y}=\left(y_{1},\ldots,y_{M}\right)^{T}$ and
$\overrightarrow{1}=(1,\ldots,1)^{T}$. The hyperplane can be constructed by
$\bm{w}=\sum_{j=1}^{M}\eta_{j}\bm{x}_{j}$ and $w_{0}=\eta_{0}$. There are
several quantum algorithms available for solving this problem, such as the HHL
method Harrow et al. (2009), variational quantum linear solvers, and others
Patil et al. (2022); Huang et al. (2019). Once the state
$|\eta_{0},\bm{\eta}\rangle$ has been prepared, the label of a new sample,
$y_{i}$, can be determined by calculating the inner product of the state
$|\eta_{0},\bm{\eta}\rangle$ with the new sample state. This paper primarily
focuses on dimensionality reduction, so the QSVM component is not covered in
detail. For more comprehensive information, please refer to prior literature
Yang et al. (2024); Xu et al. (2024); Yang et al. (2023).

This dataset contains $M=208$ samples, each with $N=60$ features. First, the
unitary operator $e^{-i\mathcal{H}t}$ is directly applied to the states. The
input state is

$|\Psi_{in}\rangle=\sum_{k=1}^{M}\sum_{j=1}^{N}x_{k}^{j}|j\rangle|k\rangle.$
(27)

Figure 3: The error of QRDR with different $c$. Red points are simulating
results of QSVM in classical computer in $R=16$ and blue line is in $R=8$. The
dashed line is the result of a linear fit of error and $c$.

We set $R=16$ and varied $c$ proportionately from $0.001$ to $0.032$. After
applying QRDR to the input data, the simulation error is shown in Fig. 2. The
error defined in Eq. 21 represents the deviation between the numerical
simulation results of QRDR and the theoretical results of the classcial DR.

Simulation results indicate that the discrepancy between the QRDR output and
theoretical expectations aligns with the theoretical predictions based on the
parameter $c$. By calculating the linear fitting coefficients of $\log(1/c)$
and $\log(1/\epsilon^{0.5})$, we can evaluate whether the two parameters are
linearly correlated. For $R=16$, the correlation coefficient is 0.9835, while
for $R=8$, it is 0.9948. This strong linear correlation between $c$ and
$\epsilon^{0.5}$ confirms the accuracy of our theoretical analysis. Next, we
use the quantum data processed by QRDR and the original quantum data as
different input sets for the QSVM. By comparing the training results and
complexity of QSVM, we can evaluate the performance of QRDR.

The data is divided into 8 groups, each containing 26 samples, following the
standard K-fold cross-validation procedure. Seven sets of data are used as the
training set, while one set serves as the test set. Eight QSVM models are
trained, and the average accuracy across these models is taken as the final
accuracy. For the original data with $N=60$ features, the accuracy is 0.8625.
For $R=16$, the accuracy improves to 0.8937.

Next, we select 20 random samples as the validation set and use the remaining
data to train the parameters, repeating the process eight times to assess the
performance of QSVM with different reduced dimensions $R$ of input data. We
observe that QSVM performs very well with $R=8$. When using higher dimensions
like $R=16$ and $R=32$, there is only a marginal increase in accuracy. This
suggests that only around $8$ parameters are necessary to capture the features
of the samples in this dataset. The accuracy of QSVM is illustrated in Fig. 4.

The complexity of QSVM is $O(\log MN)$, where $N$ is the dimensionality and
$M$ is the number of samples Rebentrost et al. (2014); Li et al. (2015). If
the dimensionality after QRDR is set to $R=O(\log N)$, the complexity of QSVM
with QRDR becomes $O(\log M\log N)$, providing a speedup at this stage.

Figure 4: The accuracy of QSVM is presented with varying input data dimensions
$R$. The dots represent the average accuracy across multiple calculations,
while the upper and lower bounds of the error bars represent the maximum and
minimum accuracies observed across multiple calculations. Apart from $R=4$,
QSVM is effective at classifying the data

By simulating the QRDR and QSVM, the error rate of QRDR aligns with our
expectations, while the accuracy of QSVM improves and complexity decreases.
Our QRDR algorithm enhances the accuracy and speed of QSVM classification,
demonstrating its practical value. More details about QSVM, including its
workflow and parameter optimization process, can be found in prior work
Rebentrost et al. (2014); Yang et al. (2024).

###  III.2 QRDR FOR QUANTUM MANY BODY PHASE CLASSIFICATION

In this section, the quantum phase classification of 1-D Transverse-field
Ising model(TFIM) is performed to verify the practicability of our QRDR
algorithm.

1D TFIM is a fundamental toy model for quantum phase trasition, which consider
the nearest neighbor $ZZ$ interation as well as an extra x-direction tansverse
magnetic field in a spin-half chain, with hamiltonian

$\displaystyle
H=-J\sum_{i,i+1}\sigma^{z}_{i}\sigma^{z}_{i+1}+h\sum_{i}\sigma^{x}_{i}.$ (28)

This Hamiltonian is invariant under a global $Z_{2}$ transformation,
$\tau=\prod_{i}\sigma_{x}$. Because of the competition of the two non-commute
parts in Eq.(28), quantum phase transition appears at the critical point
$h/J=1$. In the region $h>J$, there is a unique ground state that preserve the
$Z_{2}$ symmetry and this system is in a disordered paramagnetic phase. In the
region $h<J$, $Z_{2}$ symmetry is spontaneously broken in the thermodynamic
limit, this system is in an ordered ferromagnetic phase.

We first use QRDR to reduce the dimension of quantum state, and then use a
QCNN model based on Ref. Wei et al. (2022) to classify the quantum state
before and after QRDR respectively. In this case, we simulate a complete
quantum convolutional neural network model, including a dimension reduction
layer, a convolutional layer, a pooling layer, and a full-connected layer, as
shown in Fig. 5.

Figure 5: The simplified quantum circuit of QRDR-QCNN algorithm. The initial
state $|X\rangle$ is prepared in register A by operator $O_{X}$, and QRDR
utilizes a probe qubit and state $|X\rangle$ to prepare state $|Z\rangle$
after DR in register B. Then, QCNN employs ancillary qubits to process state
$|Z\rangle$ with the linear combination of unitary operators $Q_{0}$,
$Q_{1}$,$\cdots$ $Q_{8}$ which can act as a group of convolution operator.

Operator $S$ is the ansatz to generate superpositon state in the ancillary
system. The ansatzSim et al. (2019) we used for the four ancillary qubits
contains 28 trainable parameters, and is shown in Fig. 6(a). For the
convolutional layer, the nine operators $Q_{0}$, $Q_{1}$,$\cdots$, $Q_{8}$
consist of the convolution operator. $Q_{k}$ is the tensor product of two of
the following three operators, $E_{1}$ ,$E_{2}$ and $E_{3}$. $E_{2}$ is the
identity matrix and $\mathbf{E_{3}}=\mathbf{E_{1}}^{\dagger}$, where

$E_{1}(i,j)=\begin{cases}1&(i=j+1),(i=0,j=2^{r})\\\ 0&Else.\\\ \end{cases}$
(29)

Figure 6: (a) Quantum circuit for realizing operator $S$. (b) Decomposition of
operator $E_{1}$ in the form of basic gates.

We primarily need to provide the decomposition method for $E_{1}$. $E_{1}$ can
be constructed using a combination of $O(r^{3})$ CNOT gates and Pauli $X$
gates, as illustrated in Fig. 6(b).

The function of pooling layer after the convolutional layer is to reduce the
spatial size so as to reduce the amount of parameters. Here we choose
$R_{y}(\theta=0)$. The entire QCNN network consists of 3 layers with a total
of 84 trainable parameterized quantum gates. The loss function is related to
the expectation value of the parametrized Hamiltonian
$e(p)=\langle{p}|\mathcal{H}|p\rangle$. This Hamiltonian consists of identity
operators $I$ and Pauli operators $\sigma_{z}$,

$\displaystyle\mathcal{H}=h^{0}I+\sum_{i}h^{i}\sigma_{z}^{i}+\sum_{i,j}h^{ij}\sigma_{z}^{i}\sigma_{z}^{j}$
(30)

where $h^{0},h^{i},h^{ij}$ are the parameters, and Roman indices $i,j$ denote
location of the qubit. On the other hand, the classical neural network is
composed of three fully connected layers, each containing 128 neurons. In both
models, we randomly divided a total of 200 data points into 160 training data
and 40 testing data. The training set utilized a batch size of 20, and the
training process was performed for 20 epochs. The optimizer used was Adam, and
the loss function employed was the cross-entropy loss function. The numerical
simulation is carried out on ’MindSore Quantum’ platform and the results
indicate that QRDR can improve both the performance of QCNN and CNN. Moreover,
QCNN with QRDR can extremly reduce the training parameter and computation
complexity, achieving higher accuracy. The simulation results are shown in
Fig. 7.

Figure 7: The performance of QRDR plus QCNN. The blue, orange, green, and red
curves denote the average accuracy of the QCNN with QRDR, QCNN, CNN with DR,
and CNN, respectively. The shadow areas of the corresponding color denote the
accuracy fluctuation range in the 10 times simulation results.

The Hamiltonian in Eq. (28) can be directly linked to a p-wave superconducting
Hamiltonian via the Jordan-Wigner transformation. In this context, two
dangling Majorana fermions can be found at the end of the chain under open
boundary conditions. This implies that our QRDR method can be used to
distinguish between different symmetry-protected topological phases as well as
genuine topological phases. Topological states, which are independent of local
perturbations, have fewer dominant components in the state vector, allowing
for natural compression of the state.

##  IV DISCUSSION

To summarize, we propose a novel quantum dimension reduction algorithm called
QRDR for preprocessing high-dimensional quantum data prior to the application
of quantum machine learning algorithms. In comparison to existing quantum DR
algorithms, such as QPE-based algorithms and VQA-based algorithms, it offers
distinct advantages in terms of computational complexity and qubit resources
by employing QRT. We employ QRDR alongside two different quantum machine
learning models, QSVM and QCNN, to evaluate our algorithm’s efficacy. Our
method effectively accomplishes two critical tasks: classifying underwater
detection targets and analyzing quantum many-body phases. The simulation
results demonstrate its effective reduction of dimensionality in quantum data,
thereby enhancing the performance and speed of subsequent algorithms. Given
these aforementioned advantages, QRDR is poised to play a significant role in
the future era of general quantum computing. Despite our reduction in quantum
resource consumption, achieving large-scale algorithm demonstrations on NISQ
devices still presents many challenges.

##  V Acknowledgements.

S.W. acknowledges the National Natural Science Foundation of China under
Grants No. 12005015 and Beijing Nova Program under Grants No. 20230484345; We
acknowledges the National Key Research and Development Program of China
(2017YFA0303700); Beijing Advanced Innovation Center for Future Chip (ICFC).
P. G. acknowledges the National Natural Science Foundation of China under
grant No. 12247164. T.X acknowledges the the National Natural Science
Foundation of China No.12275117, Guangdong Basic and Applied Basic Research
Foundation No.2022B1515020074.

##  VI Data availability statement

The data that support the findings of this study are available on Developer
(2021).

## References

  * Goodell et al. [2021] John W. Goodell, Satish Kumar, Weng Marc Lim, and Debidutta Pattnaik.  Artificial intelligence and machine learning in finance: Identifying foundations, themes, and research clusters from bibliometric analysis.  _Journal of Behavioral and Experimental Finance_ , 32:100577, 2021.  ISSN 2214-6350.  URL https://www.sciencedirect.com/science/article/pii/S2214635021001210.
  * Ghoddusi et al. [2019] Hamed Ghoddusi, Germán G. Creamer, and Nima Rafizadeh.  Machine learning in energy economics and finance: A review.  _Energy Economics_ , 81:709–727, 2019.  ISSN 0140-9883.  URL https://www.sciencedirect.com/science/article/pii/S0140988319301513.
  * Erickson et al. [2017] Bradley J. Erickson, Panagiotis Korfiatis, Zeynettin Akkus, and Timothy L. Kline.  Machine learning for medical imaging.  _RadioGraphics_ , 37(2):505–515, 2017.  URL https://doi.org/10.1148/rg.2017160130.  PMID: 28212054.
  * Deo [2015] Rahul C. Deo.  Machine learning in medicine.  _Circulation_ , 132(20):1920–1930, 2015.  URL https://www.ahajournals.org/doi/abs/10.1161/CIRCULATIONAHA.115.001593.
  * Gujju et al. [2023] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond.  Quantum machine learning on near-term quantum devices: Current state of supervised and unsupervised techniques for real-world applications, 2023.  URL https://arxiv.org/abs/2307.00908.
  * Dunjko and Briegel [2018] Vedran Dunjko and Hans J Briegel.  Machine learning and artificial intelligence in the quantum domain: a review of recent progress.  _Reports on Progress in Physics_ , 81(7):074001, jun 2018.  URL https://dx.doi.org/10.1088/1361-6633/aab406.
  * Kerenidis and Prakash [2016] Iordanis Kerenidis and Anupam Prakash.  Quantum recommendation systems.  _arXiv preprint arXiv:1603.08675_ , 2016.  URL https://arxiv.org/pdf/1603.08675.
  * Rebentrost et al. [2014] Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd.  Quantum support vector machine for big data classification.  _Physical Review Letters_ , 113:130503, Sep 2014.  URL https://link.aps.org/doi/10.1103/PhysRevLett.113.130503.
  * Lloyd et al. [2014] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost.  Quantum principal component analysis.  _Nature Physics_ , 10(9):631–633, 2014.  URL https://www.nature.com/articles/nphys3029.
  * Abbas et al. [2021] Amira Abbas, David Sutter, Christa Zoufal, Aurélien Lucchi, Alessio Figalli, and Stefan Woerner.  The power of quantum neural networks.  _Nature Computational Science_ , 1(6):403–409, 2021.  URL https://doi.org/10.1038/s43588-021-00084-1.
  * Beer et al. [2020] Kerstin Beer, Dmytro Bondarenko, Terry Farrelly, Tobias J Osborne, Robert Salzmann, Daniel Scheiermann, and Ramona Wolf.  Training deep quantum neural networks.  _Nature communications_ , 11(1):808, 2020.  URL https://www.nature.com/articles/s41467-020-14454-2.
  * Li et al. [2020] YaoChong Li, Ri-Gui Zhou, RuiQing Xu, WenWen Hu, and Ping Fan.  Quantum algorithm for the nonlinear dimensionality reduction with arbitrary kernel.  _Quantum Science and Technology_ , 6(1):014001, nov 2020.  URL https://dx.doi.org/10.1088/2058-9565/abbe66.
  * Yu et al. [2019] Chao-Hua Yu, Fei Gao, Song Lin, and Jingbo Wang.  Quantum data compression by principal component analysis.  _Quantum information processing_ , 18(8), 1 2019.  URL https://doi.org/10.1007/s11128-019-2364-9.
  * Liang et al. [2020] Jin-Min Liang, Shu-Qian Shen, Ming Li, and Lei Li.  Variational quantum algorithms for dimensionality reduction and classification.  _Physical Review A_ , 101:032323, Mar 2020.  URL https://link.aps.org/doi/10.1103/PhysRevA.101.032323.
  * Romero et al. [2017] Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik.  Quantum autoencoders for efficient compression of quantum data.  _Quantum Science and Technology_ , 2(4):045001, aug 2017.  URL https://dx.doi.org/10.1088/2058-9565/aa8072.
  * Huang et al. [2020] Chang-Jiang Huang, Hailan Ma, Qi Yin, Jun-Feng Tang, Daoyi Dong, Chunlin Chen, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo.  Realization of a quantum autoencoder for lossless compression of quantum data.  _Physical Review A_ , 102:032412, Sep 2020.  URL https://link.aps.org/doi/10.1103/PhysRevA.102.032412.
  * Duan et al. [2019] Bojia Duan, Jiabin Yuan, Juan Xu, and Dan Li.  Quantum algorithm and quantum circuit for a-optimal projection: Dimensionality reduction.  _Physical Review A_ , 99:032311, Mar 2019.  URL https://link.aps.org/doi/10.1103/PhysRevA.99.032311.
  * Pan et al. [2020] Shi-Jie Pan, Lin-Chun Wan, Hai-Ling Liu, Qing-Le Wang, Su-Juan Qin, Qiao-Yan Wen, and Fei Gao.  Improved quantum algorithm for a-optimal projection.  _Physical Review A_ , 102:052402, Nov 2020.  URL https://link.aps.org/doi/10.1103/PhysRevA.102.052402.
  * Yu et al. [2023] Kai Yu, Song Lin, and Gong-De Guo.  Quantum dimensionality reduction by linear discriminant analysis.  _Physica A: Statistical Mechanics and its Applications_ , 614:128554, 2023.  ISSN 0378-4371.  URL https://www.sciencedirect.com/science/article/pii/S0378437123001097.
  * Cong and Duan [2016] Iris Cong and Luming Duan.  Quantum discriminant analysis for dimensionality reduction and classification.  _New Journal of Physics_ , 18(7):073011, jul 2016\.  URL https://dx.doi.org/10.1088/1367-2630/18/7/073011.
  * He et al. [2020] Xi He, Li Sun, Chufan Lyu, and Xiaoting Wang.  Quantum locally linear embedding for nonlinear dimensionality reduction.  _Quantum Information Processing_ , 19:1–21, 2020.  URL https://doi.org/10.1007/s11128-020-02818-y.
  * Sidhu [2019] Gagan Sidhu.  Locally linear embedding and fmri feature selection in psychiatric classification.  _IEEE Journal of Translational Engineering in Health and Medicine_ , 7:1–11, 2019.  URL https://ieeexplore.ieee.org/document/8807145.
  * Li et al. [2019] Zhaokai Li, Xiaomei Liu, Hefeng Wang, Sahel Ashhab, Jiangyu Cui, Hongwei Chen, Xinhua Peng, and Jiangfeng Du.  Quantum simulation of resonant transitions for solving the eigenproblem of an effective water hamiltonian.  _Physical Review Letters_ , 122:090504, Mar 2019.  URL https://link.aps.org/doi/10.1103/PhysRevLett.122.090504.
  * Li et al. [2021] Zhaokai Li, Zihua Chai, Yuhang Guo, Wentao Ji, Mengqi Wang, Fazhan Shi, Ya Wang, Seth Lloyd, and Jiangfeng Du.  Resonant quantum principal component analysis.  _Science Advances_ , 7(34):eabg2589, 2021.  URL https://www.science.org/doi/abs/10.1126/sciadv.abg2589.
  * Giovannetti et al. [2008] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.  Quantum random access memory.  _Physical Review Letters_ , 100:160501, Apr 2008.  URL https://link.aps.org/doi/10.1103/PhysRevLett.100.160501.
  * Wang [2016] Hefeng Wang.  Quantum algorithm for obtaining the eigenstates of a physical system.  _Physical Review A_ , 93:052334, May 2016.  URL https://link.aps.org/doi/10.1103/PhysRevA.93.052334.
  * Yang et al. [2023] Fan Yang, Xinyu Chen, Dafa Zhao, Shijie Wei, Jingwei Wen, Hefeng Wang, Tao Xin, and Guilu Long.  Quantum multi-round resonant transition algorithm.  _Entropy_ , 25(1), 2023.  ISSN 1099-4300.  URL https://www.mdpi.com/1099-4300/25/1/61.
  * Low and Chuang [2017] Guang Hao Low and Isaac L. Chuang.  Optimal hamiltonian simulation by quantum signal processing.  _Physical Review Letters_ , 118:010501, Jan 2017.  URL https://link.aps.org/doi/10.1103/PhysRevLett.118.010501.
  * Yuan et al. [2019] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin.  Theory of variational quantum simulation.  _Quantum_ , 3:191, 2019.  URL https://quantum-journal.org/papers/q-2019-10-07-191/.
  * Cirstoiu et al. [2020] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger.  Variational fast forwarding for quantum simulation beyond the coherence time.  _Npj Quantum Information_ , 6(1):1–10, 2020.  URL https://www.nature.com/articles/s41534-020-00302-0.
  * Li and Benjamin [2017] Ying Li and Simon C. Benjamin.  Efficient variational quantum simulator incorporating active error minimization.  _Physical Review X_ , 7:021050, Jun 2017.  URL https://link.aps.org/doi/10.1103/PhysRevX.7.021050.
  * Harrow et al. [2009] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd.  Quantum algorithm for linear systems of equations.  _Physical Review Letters_ , 103:150502, Oct 2009.  URL https://link.aps.org/doi/10.1103/PhysRevLett.103.150502.
  * Patil et al. [2022] Hrushikesh Patil, Yulun Wang, and Predrag S. Krstić.  Variational quantum linear solver with a dynamic ansatz.  _Physical Review A_ , 105:012423, Jan 2022.  URL https://link.aps.org/doi/10.1103/PhysRevA.105.012423.
  * Huang et al. [2019] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost.  Near-term quantum algorithms for linear systems of equations, 2019.  URL https://arxiv.org/abs/1909.07344.
  * Yang et al. [2024] Fan Yang, Dafa Zhao, Chao Wei, Xinyu Chen, Shijie Wei, Hefeng Wang, Guilu Long, and Tao Xin.  A parallel quantum eigensolver for quantum machine learning.  _New Journal of Physics_ , 26(4):043011, apr 2024\.  URL https://dx.doi.org/10.1088/1367-2630/ad15b4.
  * Xu et al. [2024] Feng Xu, Fan Yang, Chao Wei, Xinyu Chen, Shijie Wei, Hefeng Wang, Jun Li, and Tao Xin.  Quantum simulation of water-molecule bond angles using an nmr quantum computer.  _Physical Review A_ , 109:042618, Apr 2024.  URL https://link.aps.org/doi/10.1103/PhysRevA.109.042618.
  * Li et al. [2015] Zhaokai Li, Xiaomei Liu, Nanyang Xu, and Jiangfeng Du.  Experimental realization of a quantum support vector machine.  _Physical Review Letters_ , 114:140504, Apr 2015.  URL https://link.aps.org/doi/10.1103/PhysRevLett.114.140504.
  * Wei et al. [2022] ShiJie Wei, YanHu Chen, ZengRong Zhou, and GuiLu Long.  A quantum convolutional neural network on nisq devices.  _AAPPS Bulletin_ , 32:1–11, 2022.  URL https://doi.org/10.1007/s43673-021-00030-3.
  * Sim et al. [2019] Sukin Sim, Peter D Johnson, and Alán Aspuru-Guzik.  Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms.  _Advanced Quantum Technologies_ , 2(12):1900070, 2019.  URL https://doi.org/10.1002/qute.201900070.
  * Developer [2021] MindQuantum Developer.  Mindquantum, version 0.6.0, March 2021.  URL https://gitee.com/mindspore/mindquantum/tree/research/paper_with_code/quantum_resonant_dimensionality_reduction_and_application_in_quantum_machine_learning.


# Analytical method for calculation of the potential profiles of nitride-based
resonance tunneling structures

I.V. Boyko

(Received May 8, 2018, in final form July 2, 2018)

###### Abstract

Using the effective mass model for an electron and the dielectric continuum
model, analytical solutions of the self-consistent Schrödinger-Poisson system
of equations are obtained. Quantum mechanical theory of electronic stationary
states, the oscillator strengths of quantum transitions and a method of
potential profile calculation is developed for the experimentally constructed
three-well resonance-tunneling structure — a separate cascade of quantum
cascade detector. For the proposed method, a comparison with the results of
other methods and with the results of the experiment was carried out. A good
agreement between the calculated value of the detected energy and its
experimental value has been obtained, the difference being no more than
$2.5\%$.

Key words: quantum cascade detector, piezoelectric polarization, spontaneous
polarization, resonance tunneling structure, oscillator strength

PACS: 73.21.Ac, 78.20.hb, 78.67.-n

###### Abstract

Âèêîðèñòîâóþè ìîäåëü åôåêòèâíî¿ ìàñè äëÿ åëåêòðîíà òà ìîäåë äåëåêòðèíîãî
êîíòèíóóìó, îòðèìàí àíàëòèí ðîçâ’ÿçêè ñàìîóçãîäæåíî¿ ñèñòåìè ðâíÿíü Øðåäíãåðà-
Ïóàññîíà. Äëÿ åêñïåðèìåíòàëüíî ñòâîðåíî¿ òðèÿìíî¿ ðåçîíàíñíî-òóíåëüíî¿
ñòðóêòóðè — êàñêàäó êâàíòîâîãî êàñêàäíîãî äåòåêòîðà ðîçðîáëåíà êâàíòîâî-
ìåõàííà òåîðÿ åëåêòðîííèõ ñòàöîíàðíèõ ñòàíâ, ñèë îñöèëÿòîðâ êâàíòîâèõ ïåðåõîäâ
ìåòîä ðîçðàõóíêó ¿¿ ïîòåíöàëüíîãî ïðîôëþ. Äëÿ çàïðîïîíîâàíîãî ìåòîäó ïðîâåäåíî
ïîðâíÿííÿ ç ðåçóëüòàòàìè íøèõ ìåòîäâ òà ðåçóëüòàòàìè åêñïåðèìåíòó. Îòðèìàíî
äîáðå óçãîäæåííÿ ìæ ðîçðàõîâàíîþ âåëèèíîþ äåòåêòîâàíî¿ åíåðã¿ òà ¿¿
åêñïåðèìåíòàëüíîþ âåëèèíîþ, ÿê âäðçíÿþòüñÿ íå áëüø, íæ íà $2.5$%.

Ключов слова: êâàíòîâèé êàñêàäíèé äåòåêòîð, ï’çîåëåêòðèíà ïîëÿðèçàöÿ,
ñïîíòàííà ïîëÿðèçàöÿ, ðåçîíàíñíî-òóíåëüíà ñòðóêòóðà, ñèëà îñöèëÿòîðà

##  1 Introduction

Quantum cascade lasers (QCL) [1, 2] and detectors (QCD) [3, 4, 5, 6, 7, 8]
created experimentally on the basis of binary and ternary nitride alloys of
${\rm InN,}\,\,{\rm GaN,}\,\,{\rm AlN}$, etc., are now of considerable
practical and theoretical interest. The physical properties of Group III-
nitrides, their high temperature stability, large bandgap, and significant
optical activity in particular, allow nanodevices created on their basis to
operate efficiently in the actual infrared range of electromagnetic waves.
Besides, an effective work of nitride QCL and QCD is possible within the range
from cryogenic to room temperatures, which is a significant advantage in
comparison with the nanodevices created on the basis of arsenide semiconductor
compounds of GaAs, InAs, AlAs, which, in fact, can work only at cryogenic
temperatures.

The anisotropy of physical properties of nitride semiconductor materials is
caused by strong interatomic bonds and by the fact that their crystal lattice
is of the wurtzite type hexagonal structure. Spontaneous and piezoelectric
polarization arising in the layers of nitride semiconductor nanosystems is the
result of uncompensated total dipole moment of crystal lattices and
inconsistency of constant lattices at medium interfaces. As a result, internal
electric fields appear in the nitride multilayer nanostructures causing a
significant deformation of their potential profile.

The development of the consistent theory of electronic states and a general
method for calculating the potential profiles of plane nitride-based resonance
tunneling structures (RTS) is still an unsolved theoretical problem. The
theory, which makes it possible to calculate the internal fields arising in
the nitride semiconductor nanosystems, was developed in the papers [9, 10, 11,
12, 13, 14, 15, 16, 17]. The application of this theory to the plane
multilayer RTS is not a difficult problem, though it is not sufficient for
constructing a method for calculating the RTS potential profiles, since it,
for example, does not make it possible to take into account the contribution
of charge carriers to the value of the effective potential. Available methods
for the calculation of RTS potential profiles have significant shortcomings
both in terms of the completeness of the description of the effective
potential components and in terms of the theoretical approaches used. One of
the first methods was the one proposed in the paper [18]. Since it takes into
account only the contribution of internal electric fields in the effective
potential for an electron, as it will be shown below, it can be used only for
the qualitative estimations. In addition, widely used is the method based on
the numerical solution of the Schrödinger and Poisson equations [19, 20, 21],
as well as on the software [22] which is often employed by experimenters for
calculations [4, 6, 23, 24] based on the $k\cdot p$ method. Generally, the
disadvantages of most numerical methods for RTS potential profiles calculation
include the linearization of the Schrödinger and Poisson equations at the
initial stage of their solution, as well as the neglect of the influence of
the boundary conditions for the wave function and the fluxes of its
probability. Besides, the application of such methods considerably complicates
the possibility of a further investigation of electronic transitions and
electronic tunnel transport, electron-phonon interaction in such RTS and other
important theoretical problems.

In the presented paper, analytical solutions of the self-consistent
Schrödinger-Poisson system of equations are obtained. Using these solutions,
the theory of stationary electronic states and oscillator strengths of quantum
electronic transitions is developed for the three-well cascade of an
experimentally realized QCD of near-infrared spectral region [5]. A method for
calculating the potential RTS profile is proposed. For the proposed method,
the results are compared with the experimental results and with the results
obtained by other methods. By calculating the energy spectrum of the electron
and the oscillator strengths of quantum transitions, depending on the relative
position of the active band and the extractor, geometric configurations of the
cascade ensuring its effective operation as an active element of QCD were
established.

##  2 Statement of the problem. Components of the effective potential of
nitride-based resonance tunneling structure

In the Cartesian coordinate system, the three-well RTS–the cascade of QCD
which consists of an active band and an extractor is considered (figure 1).
The coordinate system is chosen in such a way that its axis $OZ$ is
perpendicular to the separation boundaries of the nanosystem layers. In
accordance with the experimental work [5], it is assumed that the media (0)
and (8), to the left and right of the RTS, correspond to the unstressed $\rm
AlN$ medium, medium (2) corresponds to the semiconductor material $\rm GaN$,
media (4), (6) — $\rm Al_{0.25}Ga_{0.75}N$ semiconductor material, medium (1),
(3), (5), (7) — $\rm AlN$ semiconductor material.

Figure 1: Geometrical scheme of the nitride-based RTS.
$d_{1},d_{2},\ldots,d_{7}$ are RTS layers thickness,
$z_{0},z_{1},\ldots,z_{7}$ are the boundary coordinates between these layers.

Using the effective mass model and the dielectric continuum model, the
effective electron mass and dielectric permeability of the RTS can be written
as:

$\displaystyle m(z)$
$\displaystyle=m_{0}[\theta(z-z_{1})+\theta(z-z_{2})]+m_{1}\left\\{\theta(-z)+\theta(z-z_{7})+\sum_{p=0}^{3}\left[\theta(z-z_{2p})-\theta(z-z_{2p+1})\right]\right\\}$
$\displaystyle+m_{2}\sum_{p=2}^{3}\left[\theta(z-z_{2p-1})-\theta(z-z_{2p})\right],$
(2.1) $\displaystyle\varepsilon(z)$
$\displaystyle=\varepsilon_{(0)}\left\\{\theta(-z)+\theta(z-z_{7})+\sum_{p=0}^{3}\left[\theta(z-z_{2p})-\theta(z-z_{2p+1})\right]\right\\}+\varepsilon_{(1)}[\theta(z-z_{1})+\theta(z-z_{2})]$
$\displaystyle+\varepsilon_{(2)}\sum_{p=2}^{3}\left[\theta(z-z_{2p-1})-\theta(z-z_{2p})\right],$
(2.2)

where $\theta(z)$ is the Heaviside step function, $m_{0}=m^{(2)}$;
$m_{1}=m^{(0)}=m^{(1)}=m^{(3)}=m^{(5)}=m^{(7)}=m^{(8)}$;
$m_{2}=m^{(4)}=m^{(6)}$ is the effective electron mass in the potential
barriers and wells of the RTS,
$\varepsilon_{(0)}=\varepsilon^{(0)}=\varepsilon^{(1)}=\varepsilon^{(3)}=\varepsilon^{(5)}=\varepsilon^{(7)}=\varepsilon^{(8)}$;
$\varepsilon_{(1)}=\varepsilon^{(2)}$;
$\varepsilon_{(2)}=\varepsilon^{(4)}=\varepsilon^{(6)}$ is dielectric
permeability of nanostructure material layers, respectively.

The magnitude of the macroscopic polarization $P^{(p)}$ that arises in an
arbitrary $p$-th RTS layer is the sum of spontaneous $P_{\text{Sp}}^{(p)}$ and
piezoelectric $P_{\text{Pz}}^{(p)}$ polarizations calculated according to the
general theory [9, 10, 11]:

$\displaystyle P^{(p)}=P_{\text{Sp}}^{(p)}+P_{\text{Pz}}^{(p)},$ (2.3)

where the corresponding polarization $P_{\text{Pz(Sp)}}^{(p)}$ for a ternary
semiconductor AxB1-xN, depending on the concentration $x$ of the component A,
is determined in the linear approximation:

$\displaystyle
P_{\text{Pz(Sp)}}^{(p)}(x)=P_{\text{Pz(Sp)}}^{\text{AN}\,(p)}(x)+(1-x)P_{\text{Pz(Sp)}}^{\text{BN}\,(p)}(x).$
(2.4)

The internal electric fields ($F_{p}$, $p=1,\ldots,7$) are determined from the
continuity condition of the electrical displacement vector
${D}_{p}=\varepsilon^{(p)}{F}_{p}+{P}^{(p)}$ on all boundaries of the RTS
layers [18, 25]:

$\displaystyle D_{p}=D_{p+1}\,,$ (2.5)

and also from the condition that is satisfied for the total value of the
voltage applied to the RTS [11]:

$\displaystyle\sum_{p=1}^{7}F_{p}d_{p}=0.$ (2.6)

Having solved the system of equations (2.5) and (2.6), we obtain the
expression for the electric field in an arbitrary layer of the RTS [18]:

$\displaystyle F_{p}=\mathop{\sum_{k=1}^{7}}\limits_{(k\neq
p)}\left[P^{(k)}-P^{(p)}\right]\frac{d_{k}}{\varepsilon^{(k)}}\Big{/}\varepsilon^{(p)}\sum_{k=1}^{7}\frac{d_{k}}{\varepsilon^{(k)}}\,,$
(2.7)

$d_{k}$ is the thickness of the corresponding RTS layer.

According to the papers [19, 21, 25, 26], the effective potential of the RTS
for an electron will be determined as the sum of the components:

$\displaystyle V(z)=\Delta
E_{\text{C}}(z)+V_{\text{E}}(z)+V_{\text{H}}(z)+V_{\text{HL}}(z).$ (2.8)

In the expression (2.8):

$\displaystyle\Delta
E_{\text{C}}(z)=\left\\{\begin{array}[]{l}{0.765[E_{g}({\rm AlN})-E_{g}({\rm
GaN})],\hskip 46.94704ptz<0,\,\,0\leqslant z<z_{1}\,,\,\,z_{2}\leqslant
z<z_{3}\,,}\\\ {\hskip 165.59502ptz_{4}\leqslant z<z_{5}\,,\ \ z_{6}\leqslant
z<z_{7}\,,\,\,z>z_{7}\,,}\\\ {0,\hskip 157.05919ptz_{1}\leqslant
z<z_{2}\,,}\\\ {0.765[E_{g}({\rm Al}_{0.25}{\rm Ga}_{0.75}{\rm N})-E_{g}({\rm
GaN})],\,\,\,\,z_{1}\leqslant z<z_{2}}\end{array}\right.$ (2.13)

is the potential RTS profile for an electron, the calculation of which was
performed without taking into account the electric field of the piezoelectric
and spontaneous polarizations. The dependence of the bandgap of the ${\rm
Al}_{x}{\rm Ga}_{1-x}{\rm N}$ semiconductor on temperature $T$ in expression
(2.13) is calculated from the Varshni linear-quadratic relation [27]:

$\displaystyle E_{g}(x,T)=E_{g}(x,0)-\frac{a(x)T^{2}}{b(x)+T}\,,$ (2.14)

where the bandgap at zero temperature, depending on the magnitude $x$, can be
represented as:

$\displaystyle E_{g}(x,0)=xE_{g}({\rm AlN})+(1-x)E_{g}({\rm GaN})+\alpha
x(1-x).$ (2.15)

In the expression (2.15): $E_{g}({\rm AlN})=6.25$ eV, $E_{g}({\rm GaN})=3.51$
eV is the bandgap of $\rm AlN$ and $\rm GaN$ semiconductor correspondingly,
$\alpha=0.7$ eV is a bowing parameter for Group III-nitrides [27],
$a(x)=[1.799x+0.909(1-x)]\cdot 10^{-3}$ (eV/K), $b(x)=1462x+830(1-x)$ (K) are
the Varshni parameters [27].

The component $V_{\text{E}}(z)$ of the effective potential — the potential
energy characterizing the contribution of the electron interaction with the
internal fields ($F_{p}$, $p=1,\ldots,7$) of spontaneous and piezoelectric
polarizations arising in the RTS, is determined by the expression:

$\displaystyle
V_{\text{E}}(z)=e\sum_{p=1}^{7}(-1)^{p-1}(F_{p}z-F_{p-1}z_{p-1})\left[\theta(z-z_{p-1})-\theta(z-z_{p})\right],\qquad
F_{0}=0.$ (2.16)

The potential $V_{\text{H}}(z)$ is determined by the contribution of charge
carriers within the RTS, and its immediate calculation will be carried out
further.

Component:

$\displaystyle
V_{\text{HL}}(z)=-\frac{1}{4\piup}\left(\frac{9}{4\piup^{2}}\right)^{{1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}}\left[1+\frac{0.6213r_{s}}{21}\ln\left(1+\frac{21}{r_{s}(z)}\right)\right]\frac{e^{2}}{\varepsilon_{0}r_{s}(z)\varepsilon(z)a_{\text{B}}^{*}(z)}$
(2.17)

is the Hedin-Lundquist exchange-correlation potential [26, 28], where
$r_{s}(z)=[4\piup a_{\text{B}}^{*3}n(z)/3]^{-1/3}$ is the dimensionless
function, $a_{\text{B}}^{*}(z)=a_{\text{B}}\varepsilon(z)/m(z)$,
$a_{\text{B}}$ is the Bohr radius, $n(z)$ is the concentration of carriers
forming a static space charge.

##  3 Method for determination of the solutions of the self-consistent
Schrödinger and Poisson equations system

The stationary spectrum of the electron and its wave functions $\Psi(z)$ are
determined by solutions of the self-consistent Schrödinger-Poisson system of
equations:

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{-\frac{\hbar^{2}}{2}\frac{\mathrm{d}}{\mathrm{d}z}\left[\frac{1}{m(z)}\frac{\mathrm{d}\Psi(z)}{\mathrm{d}z}\right]+V(z)\Psi(z)=E\Psi(z),\vspace{2mm}}\\\
\displaystyle{\frac{\mathrm{d}}{\mathrm{d}z}\left[\varepsilon(z)\frac{\mathrm{d}V_{\text{H}}(z)}{\mathrm{d}z}\right]=-e\rho(z)},\end{array}\right.$
(3.3)

$\rho(z)$ is the total charge density at an arbitrary point $z$ inside the
RTS.

The method of the self-consistent system solutions determination (3.3) is as
follows. At first, keeping in the effective potential the most significant
terms $\Delta E_{\text{C}}(z)$ and $V_{\text{E}}(z)$, the solutions of the
Schrödinger equation must be found. The solutions of the Schrödinger equation
are determinated in every RTS region. Therefore, the wave function that takes
into account its finiteness, can be presented as follows:

$\displaystyle\Psi_{0}(z)$
$\displaystyle=\Psi_{0}^{(0)}(z)\theta(-z)+\Psi_{0}^{(8)}(z)\theta(z-z_{7})+\sum_{p=1}^{7}\Psi_{0}^{(p)}(z)\left[\theta(z-z_{p-1})-\theta(z-z_{p})\right]$
$\displaystyle=A_{0}^{(0)}\mathrm{e}^{\,\chi^{(0)}z}\theta(-z)+\sum_{p=1}^{7}\left[A_{0}^{(p)}\operatorname{Ai}(\zeta^{(p)}(z))+B_{0}^{(p)}\operatorname{Bi}(\zeta^{(p)}(z))\right]\left[\theta(z-z_{p-1})-\theta(z-z_{p})\right]$
$\displaystyle+B_{0}^{(8)}\mathrm{e}^{-\chi^{(8)}z}\theta(z-z_{7}),$ (3.4)

where $A_{0}^{(0)}$, $B_{0}^{(8)}$ are the coefficients in the solutions of
the Schrödinger equation to the left and to the right of the RTS,
respectively, and $A_{0}^{(p)}$, $B_{0}^{(p)}$ are inside the RTS,
$\operatorname{Ai}(\zeta)$, $\operatorname{Bi}(\zeta)$ are the Airy functions,

$\displaystyle\zeta^{(p)}(z)=\left[2m^{(p)}eF_{p}/\hbar^{2}\right]^{1/3}$
$\displaystyle\left\\{[\Delta
E_{\text{C}}(z)-E]/eF_{p}-z\right\\},\quad\chi^{(0)}=\chi^{(8)}=\hbar^{-1}\sqrt{2m_{0}(U-E)},$
(3.5) $\displaystyle U$ $\displaystyle=0.765[E_{g}({\rm AlN})-E_{g}({\rm
GaN})].$ (3.6)

The lower index in the coefficients $A_{0}^{(0)}$, $B_{0}^{(8)}$,
$A_{0}^{(p)}$, $B_{0}^{(p)}$ corresponds to the order of approximation.

From the continuity condition of the wave function and the fluxes of its
probability density on all boundaries of the nanosystem:

$\displaystyle\Psi^{(p)}(z_{p})=\Psi^{(p+1)}(z_{p});\qquad\left.\frac{\mathrm{d}\Psi^{(p)}(z)}{m(z)\mathrm{d}z}\right|_{z=z_{p}-\varepsilon}=\left.\frac{\mathrm{d}\Psi^{(p+1)}(z)}{m(z)\mathrm{d}z}\right|_{z=z_{p}+\varepsilon},$
(3.7)

a dispersion equation from which the energy spectrum of the electrons
($E_{n}$) is obtained as well as all coefficients $A_{0}^{(0)}$,
$B_{0}^{(8)}$, $A_{0}^{(p)}$, $B_{0}^{(p)}$ expressed through one of them are
found. The last coefficient is determined from the condition of the wave
function normalization

$\displaystyle\int_{-\infty}^{+\infty}\Psi_{n}^{*}(E_{n},z)\Psi_{n}(E_{n},z)\mathrm{d}z=1,$
(3.8)

which uniquely determines the wave functions of all the stationary states of
the electron. The charge density within the RTS is found as follows:

$\displaystyle\rho(z)=e[N_{\text{D}}^{+}-n(z)]+\sum_{p=1}^{7}\sigma_{p}\delta(z-z_{p}),$
(3.9)

where the concentration of ionized donor impurities is given by

$\displaystyle
N_{\text{D}}^{+}=\frac{N_{\text{D}}}{1+g\exp\left(\frac{E_{\text{F}}-E_{n}}{k_{\text{B}}T}\right)}\,,$
(3.10)

where $N_{\text{D}}$ is the concentration of donor impurities, $g=2$ is the
degeneracy factor,

$\displaystyle
n(z)=n_{0}(z)\sum_{n}\left|\Psi(E_{n},z)\right|^{2}\ln\left|1+\exp\left(\frac{E_{\text{F}}-E_{n}}{k_{\text{B}}T}\right)\right|,\qquad
n_{0}(z)=\frac{\,m(z)k_{\text{B}}T}{\piup\hbar^{2}}$ (3.11)

is the electron concentration in the nanostructure,

$\displaystyle\sigma_{p}=P^{(p+1)}-P^{(p)}$ (3.12)

is the surface density of the charge carriers arising at the boundaries of the
RTS layers due to different values of the total polarization. Now, in the
region of an arbitrary layer of the RTS, the Poisson equation can be written
as follows:

$\displaystyle\frac{\mathrm{d}^{2}V_{\text{H}}^{(p)}(z)}{\mathrm{d}z^{2}}=-\frac{e}{\varepsilon^{(p)}}\left\\{e\left[N_{\text{D}}^{+}-\frac{m^{(p)}k_{\text{B}}T}{\piup\hbar^{2}}\sum_{n}\left|\Psi(E_{n},z)\right|^{2}\ln\left|1+\exp\left(\frac{E_{\text{F}}-E_{n}}{k_{\text{B}}T}\right)\right|\right]+\sigma_{p}\delta(z-z_{p})\right\\}$
(3.13)

and its exact analytic solution that takes into account the fact that for
stationary electronic states $\Psi(E_{n},z)=\Psi^{*}(E_{n},z)$ is given by

$\displaystyle V_{\text{H}}^{(p)}(z)$
$\displaystyle=-\frac{e}{\varepsilon^{(p)}}\int_{0}^{z}\int_{0}^{x_{1}}\left\\{e\left[N_{\text{D}}^{+}-\frac{k_{\text{B}}T}{\piup\hbar^{2}}\sum_{n}\left|\Psi(E_{n},x_{2})\right|^{2}\ln\left|1+\exp\left(\frac{E_{\text{F}}-E_{n}}{k_{\text{B}}T}\right)\right|\right]+\sigma_{p}\delta(x_{2}-z_{p})\right\\}\mathrm{d}x_{1}\mathrm{d}x_{2}$
$\displaystyle=-\frac{e}{\varepsilon^{(p)}}\left[\frac{eN_{\text{D}}^{+}}{2}(z-z_{p-1})^{2}+(z-z_{p})\sigma_{p}\theta(z-z_{p})\right]$
$\displaystyle+\frac{1}{3\piup}\left[\frac{em^{(p)}}{4\hbar^{4}F_{p}^{2}}\right]^{{1/3}}\frac{k_{\text{B}}T}{\varepsilon^{(p)}}\sum_{n}\ln\left|1+\exp\left(\frac{E_{\text{F}}-E_{n}}{k_{\text{B}}T}\right)\right|\bigg{(}\left\\{2[A_{0}^{(p)}]^{2}\right.[\zeta^{(p)}(z)]^{2}\operatorname{Ai}^{2}(\zeta^{(p)}(z))$
$\displaystyle\left.-\operatorname{Ai}(\zeta^{(p)}(z))\operatorname{Ai}^{\prime}(\zeta^{(p)}(z))-2\zeta^{(p)}(z)\operatorname{Ai}^{\prime
2}(\zeta^{(p)}(z))\right\\}+\left\\{2[B_{0}^{(p)}\right.]^{2}[\zeta^{(p)}(z)]^{2}\operatorname{Bi}^{2}(\zeta^{(p)}(z))$
$\displaystyle\left.-\operatorname{Bi}(\zeta^{(p)}(z))\operatorname{Bi}^{\prime}(\zeta^{(p)}(z))-2\zeta^{(p)}(z)\operatorname{Bi}^{\prime
2}(\zeta^{(p)}(z))\right\\}+\left\\{A_{0}^{(p)}B_{0}^{(p)}\operatorname{Ai}(\zeta^{(p)}(z))\right.$
$\displaystyle\times\big{\\{}4[\zeta^{(p)}(z)]^{2}\operatorname{Bi}(\zeta^{(p)}(z))\left.-\operatorname{Bi}^{\prime}(\zeta^{(p)}(z))\big{\\}}+\operatorname{Ai}^{\prime}(\zeta^{(p)}(z))\big{[}\operatorname{Bi}(\zeta^{(p)}(z))+4\zeta^{(p)}(z)\operatorname{Bi}^{\prime}(\zeta^{(p)}(z))\big{]}\right\\}\bigg{)}$
$\displaystyle+C_{1}^{(p)}(z-z_{p-1})+C_{2}^{(p)}.$ (3.14)

The coefficients $C_{1}^{(p)}$, $C_{2}^{(p)}$ are uniquely determined from the
boundary conditions of the continuity of the potential $V_{\text{H}}(z)$ and
from the corresponding electric displacement vector at all the RTS boundaries:

$\displaystyle V_{\text{H}}^{(p)}(z_{p})=V_{\text{H}}^{(p+1)}(z_{p});\qquad$
$\displaystyle\left.\varepsilon^{(p)}\frac{\mathrm{d}V_{\text{H}}^{(p)}(z)}{\mathrm{d}z}\right|_{z=z_{p}-\varepsilon}-\left.\varepsilon^{(p+1)}\frac{\mathrm{d}V_{\text{H}}^{(p+1)}(z)}{\mathrm{d}z}\right|_{z=z_{p}+\varepsilon}=-\sigma(z_{p});$
$\displaystyle\varepsilon\to+0;\qquad p=0,\ldots,7$ (3.15)

as well as from the conditions for the disappearance of potential
$V_{\text{H}}(z)$ outside the RTS:

$\displaystyle\left.V_{\text{H}}(z)\right|_{z\to 0}\to
0;\qquad\left.V_{\text{H}}(z)\right|_{z\to z_{7}}\to 0.$ (3.16)

Then, the potential $V_{\text{H}}(z)$ can be presented as:

$\displaystyle
V_{\text{H}}(z)=\sum_{p=1}^{7}V_{\text{H}}^{(p)}(z)\left[\theta(z-z_{p-1})-\theta(z-z_{p})\right].$
(3.17)

Further, for the total effective RTS potential $U_{{\rm eff}}(z)=V(z)$,
calculated according to relations (2.8), (2.13), (2.16), (2.17), (3.14), its
linearization is performed. Then, the approximated effective potential of the
RTS for an electron looks as follows:

$\displaystyle\tilde{U}_{{\rm
eff}}(z)=\sum_{p=1}^{7}\sum_{l=0}^{N}eF(z_{p_{l}})z\left[\theta(z-z_{p_{l}})-\theta(z-z_{p_{l+1}})\right],$
(3.18)

where

$\displaystyle
F(z_{p_{l}})=\frac{V(z_{p_{l+1}})-V(z_{p_{l}})}{e(z_{p_{l+1}}-z_{p_{l}})}$
(3.19)

is the effective value of the internal electric field magnitude in the RTS,

$\displaystyle z_{p_{l}}=\frac{l}{2N}(z_{p}-z_{p-1}),\qquad
p=1,\ldots,7,\qquad z_{1_{0}}=0,$ (3.20)

$N$ is the number of partitions in the $p$-th nanosystem layer.

Having substituted the effective potential of the RTS in the form (3.18) into
the Schrödinger equation in the system (3.3), taking into account (3.19) and
(3.20), we obtain its solution

$\displaystyle\Psi_{0}(z)$
$\displaystyle=A_{1}^{(0)}\mathrm{e}^{\,\chi^{(0)}z}\theta(-z)+B_{1}^{(8)}\mathrm{e}^{-\chi^{(8)}z}\theta(z-z_{7})$
$\displaystyle+\sum_{p=1}^{7}\sum_{l=0}^{N}\left[A_{1}^{(p_{l})}\operatorname{Ai}(\zeta^{(p_{l})}(z))+B_{1}^{(p_{l})}\operatorname{Bi}(\zeta^{(p_{l})}(z))\right]\left[\theta(z-z_{p_{l}})-\theta(z-z_{p_{l+1}})\right],$
(3.21)

where

$\displaystyle\zeta^{(p_{l})}(z)=\left[2m^{(p_{l})}eF(z_{p_{l}})/\hbar^{2}\right]^{{1/3}}\left\\{[\Delta
E_{\text{C}}(z)-E]/eF(z_{p_{l}})-z\right\\},$ (3.22) $\displaystyle
m^{(p_{l})}=\left\\{\begin{array}[]{l}m_{0}\,,\ \ z_{1}\leqslant z<z_{2};\\\
m_{1}\,,\,\,z<0,\ \ 0\leqslant z<z_{1}\,,\ \ z_{2}\leqslant
z<z_{3}\,,\,\,z_{4}\leqslant z<z_{5}\,,\,\,z_{6}\leqslant
z<z_{7}\,,\,\,z>z_{7};\\\ m_{2}\,,\ \ z_{3}\leqslant z<z_{4}\,,\ \
z_{5}\leqslant z<z_{6}.\end{array}\right.$ (3.26)

By substituting the relation (3.21), an expression for the component of
effective potential, calculated in the first iteration order, is given by

$\displaystyle\begin{array}[]{l}\displaystyle{V_{\text{H}}(z)=\sum_{p=1}^{7}\sum_{l=0}^{N}V_{\text{H}}^{(p_{l})}(z)\left[\theta(z-z_{p_{l}})-\theta(z-z_{p_{l}+1})\right]},\end{array}$
(3.28)

where the expression for $V_{\text{H}}^{(p_{l})}(z)$ is obtained from the
formula (3.14) by replacing the index ($p\rightarrow p_{l}$) and the
coefficients ($A_{0}^{(p)}\rightarrow A_{1}^{(p_{l})};\,B_{0}^{(p)}\rightarrow
B_{1}^{(p_{l})}$),

$\sigma_{p_{l}}=\left\\{\begin{array}[]{l}{\sigma_{p}\,,\quad
z_{p_{l}}=z_{p}\,,}\\\ {0,\qquad z_{p_{l}}\neq z_{p}.}\end{array}\right.$

The coefficients $A_{1}^{(1)}$, $B_{1}^{(8)}$, $A_{1}^{(p_{l})}$,
$B_{1}^{(p_{l})}$ are determined from boundary conditions similar to the
conditions (3.7) and the normalization condition (3.8), and the coefficients
$C_{1}^{(p_{l})}$, $C_{2}^{(p_{l})}$ — from conditions similar to the boundary
conditions (3.15).

The iterative procedure described makes it possible to establish the self-
consistent solutions of the Schrödinger and Poisson system of equations, as
well as all the components of the effective potential of the RTS for an
electron with the required accuracy, which is presented by the obvious
relation:

$\displaystyle\delta=\frac{\left|n_{\nu+1}(z)-n_{\nu}(z)\right|}{n_{\nu}(z)}\,,$
(3.29)

$\nu$ is number of approximation (iteration).

The wave function defined in an arbitrary order of iterations
[$\Psi(E,z)=\Psi^{(\nu)}(E,z)$] makes it possible to calculate the oscillator
strengths of quantum transitions within the formula

$\displaystyle
f_{n,n^{\prime}}=\frac{2(E_{n}-E_{n^{\prime}})}{\hbar^{2}}\sum_{p=1}^{N}m_{p}\left|\,\int_{z_{p-1}}^{z_{p}}z\Psi_{n}^{*(p)}(E_{n},z)\Psi_{n^{\prime}}^{(p)}(E_{n^{\prime}},z)\mathrm{d}z\right|^{2}.$
(3.30)

###  3.1 Discussion of the results

The calculation of the wave functions $\Psi_{n}(E_{n},z)$ and the energy
spectrum $E_{n}$ of the stationary electronic states localized in the RTS, the
oscillator strengths of quantum transitions $f_{n,n^{\prime}}$, and the
effective potential of the RTS $V(z)$ were performed using the theory
mentioned above.

Figure 2: Dependence of calculation accuracy $\delta$ on the number of
iterations $\nu$.

Immediate numerical calculations were performed for the three-well RTS, i.e.,
the cascade of the experimentally realized QCD [5]. The following geometric
parameters of nanosystem were chosen: thickness of potential barriers
$\Delta_{1}=\Delta_{2}=\Delta_{3}=\Delta_{4}=1.04$ nm, width of potential
wells $d_{1}=1.56$ nm, $d_{2}=1.04$ nm, $d_{3}=1.04$ nm. The physical
parameters of the RTS were taken from the papers [27, 29]. All calculations
have been performed at temperature 300 K.

The convergence of the iterative procedure for finding solutions of the system
of the Schrödinger and Poisson equations, whose accuracy of computation is
described by the expression (3.29), is presented in figure 2, which shows the
dependence of the logarithms of the accuracy of calculations $\delta$ on the
number of iterations $\nu$. As it can be seen from the figure, the accuracy of
calculations assumed in the presented paper is $\delta=10^{-6}$ is achieved by
23–25 iterations. Further iterations result in the output of the accuracy of
the calculations on the saturation, which testifies the convergence of the
iterative procedure of the proposed method.

Figure 3: (Colour online) Potential profile of RTS and square moduli of the
stationary states wave functions localized in the nanostructure.
$n=1,2,\ldots,5$ are numbers of the corresponding energy levels.

Figure 3 shows the energy scheme of the separate cascade of QCD, the
calculation of the potential profile of which, depending on the magnitude, was
performed according to the theory developed in the previous section according
to the relations (2.13)–(2.17), (3.28). The square moduli of wave functions
$\left|\Psi_{n}(E_{n},z)\right|^{2}$ for stationary electronic states
localized in the RTS are also shown in the figure. It can be seen from the
figure that the detector electronic transition is realized between the first
and the third electronic states localized in the quantum well of the cascade
active band.

In order to testify the effectiveness of the proposed method and to compare it
with the results obtained by other methods, calculations of the potential RTS
profile (figure 4) were performed using the proposed method (I — continuous
line), using the method developed in [25] (II — short dashed line), and on the
basis of the theory developed in [18] (III — dashed line). It can be seen from
figure 4 that the dependences $V=V(z)$ obtained by all three methods are
qualitatively identical. The dependences obtained by the methods (I) and (II)
are very similar, and in dependence (III), the depths of potential wells and
the height of potential barriers are larger than in dependences (I) and (II).
This is due to the fact that in the theory developed in [18], only the
contribution of internal fields arising in the nanosystem to the value of the
effective potential is taken into account, which is determined from the
expression (2.7). Calculated in all three approaches, the values of the
energies of electronic stationary states localized in the active zone of the
cascade between which the detector transition occurs are as follows:
$E_{1}^{({\rm I})}={\rm 43.9}$ meV, $E_{3}^{({\rm I})}={\rm 826.4}$ meV,
$E_{1}^{({\rm II})}=41.4$ meV, $E_{3}^{({\rm II})}={\rm 825.1}$ meV,
$E_{1}^{({\rm III})}={\rm-104.8}$ meV, $E_{3}^{({\rm III})}=802.2$ meV.

Figure 4: (Colour online) The dependence $V=V(z)$ of the effective potential
RTS calculated in three models (a) and detailing this dependence for the
potential well of the active band (b).

The value of the detected energy in the transition $1\rightarrow 3$ calculated
on the basis of the developed theory $\Omega_{{\rm 13}}^{{\rm(I)}}={\rm
782.5}$ meV, correlates well with the result obtained on the basis of the
method (II) [25] $\Omega_{{\rm 13}}^{{\rm(II)}}={\rm 783.7}\,\,{\rm meV}$ and
the experimental result $\Omega^{({\rm exp})}={\rm 8}00{\rm.}0\,{\rm meV}$,
the difference being $2.5$%. The result on the basis of the theory of paper
[18] gives the value of the transition energy $\Omega_{{\rm
13}}^{{\rm(III)}}={\rm 9}07{\rm.0}\,\,{\rm meV}$. Besides, the values of the
energies of the stationary electronic states are much smaller. Therefore, the
application of such a theory is possible only for rough estimations.

An effective operation of the QCD is possible if the intensity of quantum
transitions between the electronic states localized in the active zone of the
cascade is the greatest. In this case, it is a transition $1\rightarrow 3$
with energy $\Omega_{13}=E_{3}-E_{1}$. This means that, for a given geometric
configuration of the studied RTS, the calculated oscillator strength of the
quantum transitions between these states $f_{13}$ should be greater by an
order of magnitude than the oscillator strengths for transitions from the
first to the rest of the electronic states [30, 31], that is:

$\displaystyle f_{13}>f_{1n^{\prime}}\,,\quad
n^{\prime}=2,\ldots,5,\,\,n^{\prime}\neq 3$ (3.31)

as well as should exceed their sum:

$\displaystyle f_{13}>\sum_{n^{\prime}=1}^{5}f_{1n^{\prime}}\,,\quad
n^{\prime}\neq 3,$ (3.32)

where the condition (3.32) provides the detector reliable operation, at which
the detection of other frequencies is excluded, which could result in QCD
operating state failure.

Figure 5: (Colour online) Dependences of the energies of the stationary
electronic states ($E_{n}$, $n=1,2,\ldots,5$) (a) and the oscillator strengths
of quantum transitions from the first stationary state to the remaining
electronic states ($f_{nn^{\prime}}$) (b), calculated depending on the
magnitude of $d=d_{1}+d_{3}$.

In order to identify optimal configurations of the RTS [for which condition
(3.31) is satisfied], calculations of the stationary electron spectrum $E_{n}$
and the oscillator strengths of quantum transitions $f_{nn^{\prime}}$ were
performed as a function of the position $d$ $(0\leqslant d\leqslant
d_{1}+d_{3})$ of the inner two-barrier structure [regions (3)–(5) in figure 1]
relatively to the input [region (1)] and to the output [region (7)] of the
cascade barriers with the unchanged remaining geometric parameters of the
nanostructure. The results of these calculations are shown in figure 5 (a),
(b).

It is seen from figure 5 (a) that the dependence $E_{n}=E_{n}(d)$ depends
directly on $n$, and hence on the localization of the corresponding stationary
electronic state. For example, the dependencies $E_{1}(d)$ and $E_{3}(d)$
whose states are localized in the input potential well (figure 3), form one
and two maxima, respectively. Similarly, in dependencies $E_{4}(d)$ and
$E_{5}(d)$, whose states are localized in the output potential well, three and
four maxima are formed, respectively. It should be noted that $E_{2}(d)$
weakly depends on $d$, since this state is localized in the internal potential
well of the RTS.

As it can be seen from figure 5 (b), in the dependence
$f_{nn^{\prime}}=f_{nn^{\prime}}(d)$, two intervals are formed, for which the
magnitude of the oscillator strength $f_{13}$ is dominated by the remaining
values of the calculated oscillator strengths. Condition (3.31) is satisfied
for the first interval (I) ($0.65\,{\rm nm}\leqslant d\leqslant 0.91\,{\rm
nm}$), but condition (3.32) is not fulfilled, since
$f_{13}^{\text{(I)}}\approx
f_{12}^{\text{(I)}}+f_{14}^{\text{(I)}}+f_{15}^{\text{(I)}}$. Therefore, for
the considered geometric configuration of the cascade, the operation of the
QCD will not be effective. For the second interval (II) ($1.38\,{\rm
nm}\leqslant d\leqslant 1.69\,{\rm nm}$), which contains an experimentally
realized configuration $d_{\exp}=1.56$ nm, both conditions (3.31) and (3.32)
are satisfied, and condition (3.31) is fulfilled more strictly:
$f_{13}^{\text{(II)}}\gg
f_{12}^{\text{(II)}},\,f_{14}^{\text{(II)}},\,f_{15}^{\text{(II)}}$. That is,
for the geometric configuration $d_{\exp}$, we have: $f_{13}^{(\exp)}={\rm
0.782}$; $f_{12}^{(\exp)}={\rm 0.098}$; $f_{14}^{(\exp)}={\rm 0.024}$;
$f_{15}^{(\exp)}={\rm 0.096}$. As it can be seen from figure 5 (a), for the
geometrical configuration found, the possibility of effective quantum
detecting transitions with energies close to the experimentally obtained
result 800 meV is realised.

Having summarised the obtained results, it can be concluded that the developed
theory makes it possible to efficiently calculate the potential profiles of
the nitride-based resonance-tunnel structures and to establish, using the
proposed criteria, their geometric configurations providing an effective
detection of the electromagnetic waves during operation of these nanosystems
as active elements of the QCD.

###  3.2 Conclusions

1\. The theory of stationary electronic states and the oscillator strengths of
quantum transitions is developed on the basis of the found analytical
solutions of the self-consistent Schrödinger-Poisson system of equations for
three-well nitride RTS.

2\. Using the developed theory, an analytical method for calculation of the
nitride RTS potential profiles is proposed.

3\. Direct calculations of the electron energy spectrum, of the oscillator
strengths of quantum transitions and of the potential profile for the
experimentally realized RTS, which functioned as an active element of QCD have
been performed. Good reliability of the proposed theory in comparison with
both numerical simulation results and experimental results is established.

## Acknowledgements

The author is sincerely grateful and extends appreciation to the Head of the
Chair of Theoretical Physics and Computer Modelling of Yu. Fedkovych National
University of Chernivtsi, Dr. Sci., Professor M.V. Tkach for his detailed
comments on the obtained results and the content of the presented paper.

## References

  * [1] Ishida A., Inoue Y., Nagasawa H., Sone N., Ishino K., Kim J.J., Makino H., Yao T., Kan H., Fujiyasu H., Physica E, 2004, 21, 765–769, doi:10.1016/j.physe.2003.11.131.
  * [2] Mirzaei B., Rostami A., Baghban H., Opt. Laser Technol., 2012, 44, 378–383,
doi:10.1016/j.optlastec.2011.07.020.

  * [3] Chang H.M., Lai W.C., Chen W.S., Chang S.J., Opt. Express, 2015, 23, No. 7, A337–A345,
doi:10.1364/OE.23.00A337.

  * [4] Hofstetter D., Baumann E., Giorgetta F.R., Théron R., Wu H., Schaff W.J., Dawlaty J., George P.A., Eastman L.F., Rana F., Kandaswamy P.K., Leconte S., Monroy E., J. Phys.: Condens. Matter, 2009, 21, 174208, doi:10.1088/0953-8984/21/17/174208.
  * [5] Vardi A., Sakr S., Mangeney J., Kandaswamy P.K., Monroy E., Tchernycheva M., Schacham S.E., Julien F.H., Bahir G., Appl. Phys. Lett., 2011, 99, 202111, doi:10.1063/1.3660583.
  * [6] Sakr S., Giraud E., Dussaigne A., Tchernycheva M., Grandjean N., Julien F.H., Appl. Phys. Lett., 2012, 100, 181103, doi:10.1063/1.4707904.
  * [7] Monteagudo-Lerma L., Valdueza-Felip S., Naranjo F.B., Corredera P., Rapenne L., Sarigiannidou E., Strasser G., Monroy E., González-Herráez M., Opt. Express, 2013, 21, 27578–27586, doi:10.1364/OE.21.027578.
  * [8] Rong X., Wang X.Q., Chen G., Zheng X.T., Wang P., Xu F.J., Qin Z.X., Tang N., Chen Y.H., Sang L.W., Sumiya M., Ge W.K., Shen B., Sci. Rep., 2015, 5, 14386, doi:10.1038/srep14386.
  * [9] Bernardini F., Fiorentini V., Vanderbilt D., Phys. Rev. B, 1997, 56, No. 16, R10024–R10027,
doi:10.1103/PhysRevB.56.R10024.

  * [10] Bernardini F., Fiorentini V., Phys. Rev. B, 1998, 57, No. 16, R9427–R9430,
doi:10.1103/PhysRevB.57.R9427.

  * [11] Bernardini F., Fiorentini V., Phys. Status Solidi B, 1999, 216, No. 1, 391–398,
doi:10.1002/(SICI)1521-3951(199911)216:1<391::AID-PSSB391>3.0.CO;2-K.

  * [12] Foutz B.E., Ambacher O., Murphy M.J., Tilak V., Eastman L.F., Phys. Status Solidi B, 1999, 216, No. 1, 415–418, doi:10.1002/(SICI)1521-3951(199911)216:1<415::AID-PSSB415>3.0.CO;2-W .
  * [13] Ambacher O., Smart J., Shealy J.R., Weimann N.G., Chu K., Murphy M., Schaff W.J., Eastman L.F., Dimitrov R., Wittmer L., Stutzmann M., Rieger W., Hilsenbeck J., J. Appl. Phys., 1999, 85, No. 6, 3222–3233, doi:10.1063/1.369664.
  * [14] Bernardini F., Fiorentini V., Vanderbilt D., Phys. Rev. B, 2001, 63, No. 19, 193201,
doi:10.1103/PhysRevB.63.193201.

  * [15] Zoroddu A., Bernardini F., Ruggerone P., Fiorentini V., Phys. Rev. B, 2001, 64, No. 4, 045208, doi:10.1103/PhysRevB.64.045208.
  * [16] Ambacher O., Majewski J., Miskys C., Link A., Hermann M., Eickhoff M., Stutzmann M., Bernardini F., Fiorentini V., Tilak V., Schaff B., Eastman L.F., J. Phys.: Condens. Matter, 2002, 14, No. 13, 3399–3434, doi:10.1088/0953-8984/14/13/302.
  * [17] Bernardini F., Fiorentini V., Phys. Rev. B, 2001, 64, No. 15, 085207, doi:10.1103/PhysRevB.64.085207.
  * [18] Ridley B.K., Schaff W.J., Eastman L.F., J. Appl. Phys., 2003, 94, No. 6, 3972–3978, doi:10.1063/1.1601686.
  * [19] Li J.M., Lü Y.W., Li D.B., Han X.X., Zhu Q.S., Liu X.L., Wang Z.G., J. Vac. Sci. Technol., B, 2004, 22, 2568–2573, doi:10.1116/1.1805543.
  * [20] Gryshchenko S.V., Klymenko M.V., Shulika O.V., Sukhoivanov I.A., Lysak V.V., Superlattices Microstruct., 2012, 52, No. 4, 894–900, doi:10.1016/j.spmi.2012.06.013.
  * [21] Saha S., Kumar J., J. Comput. Electron., 2016, 15, No. 4, 1531–1540, doi:10.1007/s10825-016-0911-5.
  * [22] Nextnano software, URL http://www.nextnano.de.
  * [23] Terashima W., Hirayama H., Phys. Status Solidi C, 2009, 6, No. S2, S615–S618, doi:10.1002/pssc.200880772.
  * [24] Pesach A., Sakr S., Giraud E., Sorias O., Gal L., Tchernycheva M., Orenstein M., Grandjean N., Julien F.H., Bahir G., Opt. Express, 2014, 22, No. 17, 21069–21078, doi:10.1364/OE.22.021069.
  * [25] Boyko I.V., J. Phys. Stud., 2018, 22, No. 1, 1701, doi:10.30970/jps.22.1701.
  * [26] Piprek J., Nitride Semiconductor Devices: Principles and Simulation, Wiley-VCH, Weinheim, 2007, doi:10.1002/9783527610723.
  * [27] Yao T., Hong S-K., Oxide and Nitride Semiconductors, Springer, Berlin, Heidelberg, 2009,
doi:10.1007/978-3-540-88847-5.

  * [28] Hedin L., Lundqvist B.I., J. Phys. C: Solid State Phys., 1971, 4, No. 14, 2064–2083,
doi:10.1088/0022-3719/4/14/022.

  * [29] Kandaswamy P.K., Guillot F., Bellet-Amalric E., Monroy E., Nevou L., Tchernycheva M., Michon A., Julien F.H., Baumann E., Giorgetta F.R., Hofstetter D., Remmele T., Albrecht M., Birner S., Dang L.S., J. Appl. Phys., 2008, 104, No. 9, 093501, doi:10.1063/1.3003507.
  * [30] Tkach M.V., Seti Ju.O., Boyko I.V., Voitsekhivska O.M., Condens. Matter Phys., 2013, 16, No. 3, 33701, doi:10.5488/CMP.16.33701.
  * [31] Tkach M.V., Seti Ju.O., Boyko I.V., Voitsekhivska O.M., Rom. Rep. Phys., 2013, 65, No. 4, 1443–1453.

Ukrainian 0

Àíàëòèíèé ìåòîä ðîçðàõóíêó ïîòåíöàëüíèõ ïðîôëâ íòðèäíèõ ðåçîíàíñíî-òóíåëüíèõ
ñòðóêòóð

.Â. Áîéêî

Òåðíîïëüñüêèé íàöîíàëüíèé òåõííèé óíâåðñèòåò ìåí âàíà Ïóëþÿ,
âóë. Ðóñüêà, 56, 46001 Òåðíîïëü, Óêðà¿íà