# A quantum segmentation algorithm based on local adaptive threshold for NEQR
image

Lu Wang  School of Automation, Nanjing University of Information Science and
Technology,
Nanjing, 210044, Jiangsu, China
<EMAIL_ADDRESS>Wenjie Liu  School of Computer and Software, Nanjing
University of Information Science and Technology,
Nanjing, 210044, Jiangsu, China
Engineering Research Center of Digital Forensics, Ministry of Education,
Nanjing University of Information Science and Technology,
Nanjing, 210044, Jiangsu, China
<EMAIL_ADDRESS>
###### Abstract

The classical image segmentation algorithm based on local adaptive threshold
can effectively segment images with uneven illumination, but with the increase
of the image data, the real-time problem gradually emerges. In this paper, a
quantum segmentation algorithm based on local adaptive threshold for NEQR
image is proposed, which can use quantum mechanism to simultaneously compute
local thresholds for all pixels in a gray-scale image and quickly segment the
image into a binary image. In addition, several quantum circuit units,
including median calculation, quantum binarization, etc., are designed in
detail, and then a complete quantum circuit is designed to segment NEQR images
by using fewer qubits and quantum gates. For a $2^{n}\times 2^{n}$ image with
$q$ gray-scale levels, the complexity of our algorithm can be reduced to
O$(n^{2}+q)$, which is an exponential speedup compared to the classic
counterparts. Finally, the experiment is conducted on IBM Q to show the
feasibility of our algorithm in the noisy intermediate-scale quantum (NISQ)
era.

###### keywords:

Quantum image processing; Quantum image segmentation; Local adaptive
threshold; Quantum comparator; Quantum subtractor.

Received (Day Month Year)Revised (Day Month Year)

PACS No.: 03.67.-a

##  1 Introduction

With the increase of our requirements for image quality, the amount of image
data also increases rapidly, which requires a lot of computing power to
process image. However, the computing power of classical computers is now
approaching the limit, so the real-time problem is gradually revealed. Quantum
image processing (QIP) is an emerging interdisciplinary subject that combines
quantum computing and classical image processing, and has received extensive
attention and research by researchers in recent years. It uses the unique
superposition and parallelism characteristics of quantum computing to quickly
improve the computing speed. Compared with classical image processing, QIP
algorithms can achieve an exponential improvement [1, 2].

The main work of QIP starts with how to represent images in a quantum
computer. Some researchers have proposed some QIP representation models by
encoding image information into probability magnitudes of quantum states, such
as qubit lattice representation (QLR) [3], real ket representation (RKR) [4],
the flexible representation of quantum image (FRQI) [5], the multi-channel RGB
images representation of quantum images (MCQI) [6], a normal arbitrary
superposition state of quantum image (NASS) [7] and quantum probability image
encoding representation (QPIE) [8]. This encoding method can efficiently
encode images using fewer qubits, but it also makes it more difficult to
retrieve images. To solve this problem, a novel enhanced quantum
representation (NEQR) [9] is proposed, which is an encoding method that uses
some sequences of qubits to encode images, which encodes the position
information and the gray-scale value information into three entangled qubit
sequences, such that images can be retrieved quickly with few measurements.
Therefore, the NEQR model is widely used due to its simplicity of operation.
By improving the position information and color information, a series of
encoding methods have been proposed, such as an improved NEQR (INEQR) [10], a
generalized model of NEQR (GNEQR) [11], and a novel quantum representation of
color digital images (NCQI ) [12]. Besides image coding based on probability
magnitude and qubit sequence, there are some new techniques applied to image
coding, such as a quantum log-polar image (QUALPI) [13] for processing log-
polar sampled images, a Fourier transform qubit representation (FTQR) [14] for
encoding color information into a quantum superposition state, and a bit-plane
representation for log-polar quantum images (BRLQI) [15]. With the development
of quantum image representation models, corresponding quantum image processing
algorithms have also developed rapidly, such as geometrical transformation of
quantum image [16, 17, 18, 19],quantum image steganography based on least
significant bit (LSB) [20], feature extraction of quantum image [21], quantum
image scrambling [22], quantum image watermarking [23, 24, 25, 26, 27],
quantum image filtering [28], quantum image scaling [29], quantum image
matching [30], quantum image edge detection [31, 32, 33, 34], quantum image
segmentation [35, 36, 37, 38, 39], etc. However, most of the current QIP
algorithms are only theoretically feasible because they use too many quantum
resource, which is inappropriate in this noisy intermediate-scale
quantum(NISQ) era.

Quantum image segmentation algorithm is the basis of image processing, and in
recent years, it have gradually developed from theoretically feasible to
practically feasible. In 2013, Li et al. [35] used a quantum search algorithm
to segment the quantum image, but no specific oracle was provided. In 2014,
Caraiman et al. [36] proposed a histogram-based quantum image segmentation
algorithm, and they segmented the image according to the distribution of
pixels’ gray-scale values in the histogram and obtained exponential
acceleration, but there was no specific oracle implementation circuit. One
year later, they [37] proposed a quantum image segmentation algorithm based on
a single threshold and gave an oracle circuit, but they could not be simulated
in the noisy intermediate-scale quantum (NISQ) era due to the high quantum
cost. In 2019, Xia et al. [38] designed a quantum comparator and used a single
threshold to binarize the quantum image. They designed a detailed segmentation
circuit but the complexity is still too high. Although many researchers have
conducted research on quantum image segmentation algorithms, but none of the
previous algorithms are simulated on the quantum platform. In 2020, Yuan et
al. [39] proposed a dual-threshold quantum image segmentation algorithm based
on the quantum comparator they designed, which is feasible in this NISQ era
due to its low quantum cost. In addition, the algorithm is simulated on the
IBM Q platform [40] to demonstrate its feasibility. However, like all the
above algorithms, it can only use global fixed thresholds to segment the
image, which is not effective for complex image (images with uneven
illumination) segmentation, Therefore, in order to better segment the quantum
image and adapt to the current quantum situation, we conducted in-depth
research in this paper, and the main contributions are as follows.

  * •

A quantum segmentation algorithm based on local adaptive threshold for NEQR
image is proposed, which can accurately segment the image with uneven
illumination into a binary image (the existing quantum segmentation algorithms
cannot), and can solve the real-time problem of the classical counterpart.

  * •

Several specific quantum circuit units are individually designed in detail,
and then, based on these units, a complete quantum image segmentation circuit
is designed to segment the NEQR image by using fewer qubits and quantum gates.

  * •

By analyzing the circuit quantum cost and the MSE value of result image, we
verify the superiority of our proposed algorithm. Then, the the simulation
experiment is performed on IBM Q platform [40] to show the feasibility of our
algorithm.

The rest of this paper is organized as follows: In Sect. 2, we briefly
introduce the basic quantum gates, the NEQR model and the classical local
adaptive threshold for image segmentation. In Sect. 3, several quantum
operations is introduced firstly, and then, the quantum segmentation algorithm
is proposed. Next, several specific quantum circuit units are designed in
detail, and a complete quantum image segmentation circuit is designed. In
Sect. 4, the circuit complexity is analyzed, and the simulation experiment is
conducted on the IBM Q platform to show the feasibility of our algorithm.
Finally, the conclusion is given in Sect. 5.

##  2 Preliminaries

###  2.1 Quantum gates

The single qubit states $\lvert 0\rangle$ and $\lvert 1\rangle$ are defined as
follows,

$\lvert 0\rangle{\rm{=}}\left[{\begin{array}[]{*{20}{c}}1\\\
0\end{array}}\right],\lvert
1\rangle{\rm{=}}\left[{\begin{array}[]{*{20}{c}}0\\\ 1\end{array}}\right].$
(1)

Unitary matrices of Hadamard (H), NOT gate (X), Controlled-NOT gate (CNOT),
Toffoli gate (CCNOT) and Fredkin gate (CSWAP) can be expressed as follows.

$\begin{split}{\rm{H=}}\frac{1}{{\sqrt{2}}}\left[{\begin{array}[]{*{20}{c}}1&1\\\
1&{-1}\end{array}}\right],{\rm{X=}}\left[{\begin{array}[]{*{20}{c}}0&1\\\
1&0\end{array}}\right],{\rm{CNOT=}}\left[{\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}}\right],\\\
{\rm{Toffoli=}}\left[{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}}&{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}}\\\
{\begin{array}[]{*{20}{c}}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}}&{\begin{array}[]{*{20}{c}}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}}\end{array}}\right],{\rm{CSWAP=}}\left[{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}}&{\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}}\\\
{\begin{array}[]{*{20}{c}}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}}&{\begin{array}[]{*{20}{c}}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{array}}\end{array}}\right].\end{split}$ (2)

In quantum image processing, the complexity of the quantum circuit depends on
the number of basic logic gate used. Single-qubit and two-qubit gates are
basic quantum logic gates, and they can operate on arbitrary qubits.
Therefore, we calculate the complexity of the circuit in terms of the number
of single-qubit and two-qubit gates, which is also called quantum cost. The
quantum cost of a NOT gate, a reset gate or a CNOT gate is all 1. A Toffoli
gate can be composed of 5 two-qubit gates, so its quantum cost is 5 [41, 39].

###  2.2 NEQR model

Classical images contain position information and gray-scale value
information, and the NEQR model uses three entangled sequences of qubits to
represent these two kinds of information. Suppose a classical image’s size is
$2^{n}\times 2^{n}$, and the grayscale range is $[0,2^{q}-1]$. Therefore, two
$n$-length qubit sequences are needed to represent the position information,
and one $q$-length qubit sequences are needed to represent the gray-scale
value information. The NEQR expression can be written as Eq. 3 [9].

$\lvert{I_{YX}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{YX}}}\rangle\otimes\lvert
Y\rangle\lvert
X\rangle}}=\frac{1}{{{2^{n}}}}\sum\limits_{YX=0}^{{2^{2n}}-1}{\mathop{\otimes}\limits_{k=0}^{q-1}\lvert{C^{K}_{YX}}\rangle\mathop{\otimes}\limits\lvert{YX}\rangle}$
(3)

where
$\lvert{{C_{YX}}}\rangle=\lvert{C_{YX}^{q-1},C_{YX}^{q-2},{\cdots}C_{YX}^{1}C_{YX}^{0}}\rangle$
represents the quantum image gray-scale values,
$C_{YX}^{k}\in\left\\{{0,1}\right\\}$, $k=q-1,q-2,\cdots,0$.
$\lvert{YX}\rangle=\lvert Y\rangle\lvert
X\rangle=\lvert{{Y_{n-1}},{Y_{n-2}},\cdots{Y_{0}}}\rangle\lvert{{X_{n-1}},{X_{n-2}},\cdots{X_{0}}}\rangle$
represents the position of the pixel in a quantum image,
${Y_{t}},{X_{t}}\in\left\\{{0,1}\right\\}$.

Fig. 1 shows an example of a grayscale image of size 2×2, and the
corresponding NEQR expression of which is given as follows.

$\begin{split}\begin{array}[]{l}\lvert I\rangle=\frac{1}{2}\left({\lvert
0\rangle\lvert{00}\rangle+\lvert{100}\rangle\lvert{01}\rangle+\lvert{200}\rangle\lvert{10}\rangle+\lvert{255}\rangle\lvert{11}\rangle}\right)\end{array}\\\
\begin{array}[]{l}=\frac{1}{2}\left(\begin{array}[]{l}\lvert{00000000}\rangle\lvert{00}\rangle+\lvert{01100100}\rangle\lvert{01}\rangle\\\
+\lvert{11001000}\rangle\lvert{10}\rangle+\lvert{11111111}\rangle\lvert{11}\rangle\end{array}\right).\end{array}\end{split}$
(4)

Figure 1: An example of a 2×2 image.

###  2.3 The classical local adaptive threshold for image segmentation

Threshold-based image segmentation works by setting all pixels with
intensities above a threshold as one value and all remaining pixels as another
value. Traditional threshold operations use a global threshold for all pixels,
which is inappropriate for images with uneven illumination. The local adaptive
threshold operation can dynamically change the threshold according to the
gray-scale values of the local neighborhood pixels, which can adapt to the
illumination changes in the image. Adaptive threshold usually takes a gray-
scale or color image as input, and in the simplest implementation, outputs a
binary image. For each pixel in the image, it is set to 0 if the pixel value
is below the threshold, otherwise set to 1. Using the median value of the
neighborhood pixel intensities as a threshold is one of the most commonly used
local adaptive threshold methods. In order to better find the threshold, the
neighborhood window must be large enough to cover enough foreground and
background pixels, but this violates the principle of local similarity, so it
is necessary to subtract a constant $Z$ from the found median value as a new
threshold. Using different neighborhood windows for different situations will
achieve better results, and the commonly used windows are square, cross-
shaped, diagonal, etc., as shown in Fig. 2. A square window is suitable for
images containing objects with long outer contour lines, a cross-shaped window
is suitable for images with more spikes and angles, and a diagonal window is
suitable for images containing radial objects. Choosing the right neighborhood
window can preserve the details in the image better. In this paper, we
introduce the design of quantum circuits using a cross-shaped neighborhood
window as an example.

Figure 2:  The schematic diagram of the neighborhood window.

The gray-scale value of the original image is represented by $f(x,y)$, and $T$
is the threshold, which is obtained by subtracting the constant $Z$ from the
median of the neighborhood window. The corresponding gray-scale value of the
segmented image is represented by $b(x,y)$. The specific expression is
described as follows:

$b(x,y)=\left\\{\begin{array}[]{l}0,0\leq f(x,y)<T\\\ 1,T\leq f(x,y)\leq
255\end{array}\right.$ (5)

where $T=median-Z$, $Z$ is a constant we set.

##  3 Quantum image segmentation algorithm and its quantum circuit
implementation

In this section, several quantum operations are first introduced, and then a
quantum image segmentation algorithm based on local adaptive thresholding is
proposed. In addition, the corresponding quantum circuit units are designd
according to the algorithm steps, and finally a complete quantum segmentation
circuit is given. We only mark the output information that will continue to be
used, other information we do not need to consider and will be used as
auxiliary qubits for the next operation.

###  3.1 The quantum operations

  1. (1)

Quantum comparator

The quantum comparator (QC) can compare the numerical magnitude relationship
of two n-qubit sequences $a$ and $b$, where $a=\lvert
a_{n-1},a_{n-2},\dots,a_{0}\rangle$, and $b=\lvert
b_{n-1},b_{n-2},\dots,b_{0}\rangle$. The comparator takes two qubit sequences
to be compared as input, and through a series of quantum logic gate
operations, outputs the original two n-qubit sequences and their magnitude
relationship result $y$. According to the comparison idea of quantum bit
string comparator (QBSC) [43], the quantum comparator in this paper is also
compared bit by bit. But in this paper, we only need to know the two relations
$a\geq b$ or $a<b$, so we only need to use one auxiliary qubit to store the
comparison result of each bit. As shown in Fig. 3, we implement the quantum
comparator by using a combination of CNOT gates and Toffoli gates. We compare
$a$ and $b$ bit by bit in order from low to high. There are four cases of
$a_{0}b_{0}$, namely 00, 01, 10, and 11. Among them, 01 indicates $a<b$, and
the other three cases indicate $a\geq b$. Therefore, only one Toffoli gate is
needed to compare the numerical relationship between $a_{0}$ and $b_{0}$. If
$a_{0}b_{0}=01$, then the auxiliary qubit $h_{1}$ is set to 1. Otherwise, it
remains unchanged. $a_{1}b_{1}$ also has four cases of 00, 01, 10, and 11. We
use the same processing method as $a_{0}b_{0}$. At this time, $h_{2}$ is set
to 1 or unchanged. But when we synthesize the two comparison results,
$a_{1}b_{1}a_{0}b_{0}=1001$ needs to be considered separately, so we use 2
Toffoli gates and one CNOT gate to process the two comparison results, which
also requires an auxiliary qubit $h_{0}$ to store the second comparison
result. After the two qubits comparisons, we use two reset operations [44] to
set the auxiliary qubits $h_{2}h_{0}$ to 00, so that we can reuse the two
auxiliary qubits. Therefore, according to this comparison method, we can
compare two n-qubit sequences, and if $a\geq b$, then $y=0$; if $a<b$, then
$y=1$.

Figure 3: The implementation circuit of the quantum comparator (QC).

  2. (2)

Quantum subtractor

A quantum subtractor(QS) can perform subtraction operations on two binary
qubits sequences $a=\lvert a_{n-1},a_{n-2},\dots,a_{0}\rangle$ and $b=\lvert
b_{n-1},b_{n-2},\dots,b_{0}\rangle$. As shown in Fig. 4, the quantum
subtractor takes two binary qubit sequences $a$ and $b$ as input and the
result of $a-b$ as the output, where $a\geq b$. According to the idea of the
ripple carry adder [45], the quantum subtractor in this paper also operates
bit by bit in the order from low to high. There are four cases of
$a_{0}-b_{0}$: $0-0$, $0-1$, $1-0$, $1-1$. Among them, if $a_{0}-b_{0}$ is
$0-1$, then $a_{0}$ needs to borrow from $a_{1}$. At this time, we use the
auxiliary qubit $h_{1}$ to store the borrow information, and the output result
of $a_{0}-b_{0}$ is 0. If $a_{0}-b_{0}$ is $0-0$ or $1-1$, then the output
result is also 0. If $a_{0}-b_{0}$ is $1-0$, then the output result is 1.
Therefore, if $b_{0}=1$, then we XOR $a_{0}$ to get the result of
$a_{0}-b_{0}$. To save qubits, we use the reset operation and CNOT gate to
transfer the borrow information of $a_{0}$ from $h_{1}$ to $b_{0}$, so that
$h_{1}$ can be used in subsequent operations. When calculating $a_{1}-b_{1}$,
if $a_{1}-b_{1}=0-1$, then $a_{1}$ needs to borrow from $a_{2}$, and we store
the borrow information in $h_{1}$; if $b_{1}=1$ or $a_{1}=0$, and $a_{0}$
borrows from $a_{1}$, then $a_{1}$ also need to borrow from $a_{2}$. Finally,
we aggregate the overall borrow information into $b_{0}$. The calculation
method of $a_{1}-b_{1}$ is the same as that of $a_{0}-b_{0}$, except that we
need to consider whether $a_{0}$ has borrowed from $a_{1}$. If $a_{0}$ has
borrowed, then we need to perform an XOR operation on the result again. After
these operations, we use the reset operation to reset the auxiliary qubits so
that we can continue to use them. The operation from $a_{2}-b_{2}$ to
$a_{n-2}-b_{n-2}$ is the same as $a_{1}-b_{1}$, so we just need to repeat the
operation to get the results. When calculating $a_{n-1}-b_{n-1}$, because
$a>b$, $a_{n-1}$ will not borrow from higher qubits, we only need to consider
whether $a_{n-2}$ borrows from $a_{n-1}$. At this time, in the four cases of
$0-0$, $0-1$, $1-0$, $1-1$, if $b_{n-1}=1$, then the result of
$a_{n-1}-b_{n-1}$ is the XOR of $a_{n-1}$, if $b_{n-1}=0$, then the result of
$a_{n-1}-b_{n-1}$ is $a_{n-1}$. If $a_{n-2}$ borrows from $a_{n-1}$, then we
need to XOR the result again. At this point, the results of $a-b$ can be
completely output from the quantum subtractor.

Figure 4: The implementation circuit of the quantum subtractor (QS).

  3. (3)

Quantum image cyclic shift transformation

The quantum image cyclic shift transformation (CT) operation (X shift and Y
shift) [17] can shift the positions of all pixels in an image , so that each
pixel in the image can obtain its neighborhood pixels at the same time. For
example, if we move the image one unit down (CT${}_{y}-$), the pixels in the
image change from $S(X,Y)$ to $S(X,Y-1)$. The unitary operation for cyclic
shift ransformation of a $2^{n}\times 2^{n}$ NEQR image is shown as follows.

$\mathrm{CT}(X\pm)\lvert
I\rangle=\frac{1}{2^{n}}\sum_{Y=0}^{2^{n}-1}\sum_{X=0}^{2^{n}-1}\lvert
CT_{YX^{\prime}}\rangle\lvert Y\rangle\lvert(X\pm 1)\bmod 2^{n}\rangle$ (6)
$\mathrm{CT}(Y\pm)\lvert
I\rangle=\frac{1}{2^{n}}\sum_{Y=0}^{2^{n}-1}\sum_{X=0}^{2^{n}-1}\lvert
CT_{Y^{\prime}X}\rangle\lvert(Y\pm 1)\bmod 2^{n}\rangle\lvert X\rangle$ (7)

where $X^{\prime}=(X\mp 1)\bmod 2^{n}$, $Y^{\prime}=(Y\mp 1)\bmod 2^{n}$,
$\mathrm{CT}_{(X+)}\,\&\,\mathrm{CT}_{(Y+)}=\left[{\begin{array}[]{*{20}{c}}0&1\\\
{I_{2}^{n}-1}&0\end{array}}\right]$,
$\mathrm{CT}_{(X-)}\,\&\,\mathrm{CT}_{(Y-)}=\left[{\begin{array}[]{*{20}{c}}0&{I_{2}^{n}-1}\\\
1&0\end{array}}\right]$. Fig. 5 shows a schematic diagram of the X-axis cyclic
shift ransformation.

From the above analysis, we can know that when an image is subjected to CT
operation, the edge of the image will be cyclically shifted to the other side,
as shown in Fig. 6.

Figure 5: Quantum circuit for cyclic shift transformation (CT): a CT($X+$), b
CT($X-$).

Figure 6: The schematic diagram of cyclic shift: a An example of original
image of size $4\times 4$, b the image transformed after CT($X+$) operation, c
the image transformed after CT($X-$) operation.

###  3.2 The quantum segmentation algorithm

The image segmentation based on threshold is a simple and fast segmentation
algorithm, but the use of global threshold cannot deal with the uneven
illumination in the image. Therefore, in this paper, a quantum image
segmentation algorithm based on local adaptive threshold is proposed. First
the original image is cyclically shifted according to the neighborhood window
to prepare a quantum image set, so we can process all pixels at the same time.
Then, we use the median of the neighborhood window pixels minus the constant
$Z$ as the threshold to segment the center pixel, and the pixels greater than
or equal to the threshold are converted to 1, the remain are converted to 0.
Finally, we can get a binary image. The cross-shaped neighborhood mask is used
as an instance to show the procedure of image segmentation in detail.
According to this circuit design method, it can be extended to any
neighborhood window mask. The specific workflow is shown in Fig. 7.

Figure 7: The workflow of our proposed algorithm.

###  3.3 Quantum circuit implementation

Step 1  Because quantum image cannot be directly obtained at present, we need
to quantize the classical image into NEQR image. $(2n+q)$ qubits are needed to
store a $2^{n}\times 2^{n}$ classical image with $q$ gray-scale levels.
Furthermore, additional $4q$ qubits are needed to store the gray-scale value
of the 4-neighborhood pixel window, and another $q$ qubits are also needed to
replicate the gray-scale values of the original image. The quantum state
expression can be represented as

$\begin{array}[]{l}{\lvert 0\rangle^{\otimes
5q}}\otimes\lvert{{I_{YX}}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{{{\lvert
0\rangle}^{\otimes 5q}}\lvert{{C_{YX}}}\rangle}}\lvert Y\rangle\lvert
X\rangle\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}{}\hfil&{}\hfil&{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}{}\hfil&{}\hfil&{}\hfil&=\end{array}}\end{array}\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{{{\lvert
0\rangle}^{\otimes q}}}}{\lvert 0\rangle^{\otimes q}}{\lvert 0\rangle^{\otimes
q}}{\lvert 0\rangle^{\otimes q}}{\lvert 0\rangle^{\otimes
q}}\lvert{{C_{YX}}}\rangle\lvert Y\rangle\lvert X\rangle\end{array}$ (8)

Step 2 According to the neighborhood window, we need to perform a cyclic shift
transformation on the original image $\lvert{I_{YX}}\rangle$ to prepare a
quantum image set, so that we can process all pixels in the image
simultaneously. First, we translate the position of the original image by one
unit up, down, left, and right, respectively, as shown in Table 3.3. A cyclic
shift in each direction results in a new NEQR image, so we have a total of 5
NEQR images. Each pixel at the same location in these 5 NEQR images is exactly
the 4-neighborhood window pixel in the original image. Since the gray-scale
values of the NEQR image are superimposed, when we operate on the neighborhood
window of a pixel in the original image, we can operate on the neighborhood
windows of all pixels at the same time. In addition, since we still need to
use the original image after sorting, but the five images have been mixed
together during the sorting process, we need to use the CNOT gate to copy the
original image into an additional 3 qubits for later binarization operation.
The shifted quantum image sets can be encoded as follows.

$\begin{array}[]{l}\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{Y+1X}}}\rangle}}\otimes\lvert{{C_{YX+1}}}\rangle\otimes\lvert{{C_{Y-1X}}}\rangle\\\
\otimes\lvert{{C_{YX-1}}}\rangle\otimes\lvert{{C_{YX}}}\rangle\otimes\lvert{{C_{YX}}}\rangle\otimes\lvert
Y\rangle\lvert X\rangle\end{array}$ (9)

Computation prepared algorithm for shifting the image.  1\. Input: the
original NEQR image
${I_{YX}}$,$\lvert{{I_{YX}}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{YX}}}\rangle}}\lvert
Y\rangle\lvert X\rangle$ 2\. Shift ${I_{YX}}$one unit upward,
then$\lvert{{I_{Y+1X}}}\rangle=\frac{1}{{{2^{n}}}}\\!\sum\limits_{Y=0}^{{2^{n}}-1}\\!{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{Y+1X}}}\rangle}}\lvert
Y\rangle\lvert X\rangle$ 3\. Shift${I_{YX}}$one unit downward,
then$\lvert{{I_{Y-1X}}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{Y-1X}}}\rangle}}\lvert
Y\rangle\lvert X\rangle$ 4\. Shift${I_{YX}}$ one unit leftward, then
then$\lvert{{I_{YX+1}}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{YX+1}}}\rangle}}\lvert
Y\rangle\lvert X\rangle$ 5\. Shift${I_{YX}}$one unit rightward, then
$\lvert{{I_{YX-1}}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{C_{YX-1}}}\rangle}}\lvert
Y\rangle\lvert X\rangle$

Step 3  In order to facilitate the sorting of the neighborhood window pixels,
we construct the QCS according to the QC, as shown in Fig. 8. When the output
of the QC is $y=0$, we use the CSWAP gates to swap the positions of $a$ and
$b$ and let the larger value be the input of the next comparator, which can
find the largest number in the neighborhood window. We compare the central
pixel of the four-neighborhood window with other pixels in turn. After each
comparison, we output the larger number from the position of $b$ to the next
QCS and then compare it with another pixel, and finally find the largest
number $C_{1}$. In this way, we can find three larger numbers
$C_{1}C_{2}C_{3}$ after three rounds of comparison. At this point, $C_{3}$ is
the median we are looking for, and the specific operation circuit is shown in
Fig. 9. In addition, to adjust the threshold, we need to use QS to subtract a
constant $Z$ from the median found. $Z$ is used to adjust the threshold value,
and we need to set it manually according to the actual segmentation results.
So that we can get the final threshold $T$, as shown in Fig. 10. Because we
only need to keep $C_{3}$, the constant $Z$ can be set in other pixel
neighborhood pixel positions, and only the corresponding position needs to be
reset [44], which can save qubits. What’s more, $Z$ is a definite value, it
only needs to be initialized with the reset operation and the NOT gate.

Figure 8: The implementation circuit of QCS. Figure 9: The implementation
circuit for calculating the median. Figure 10: The implementation circuit for
calculating the threshold $T$.

Step 4  After getting the threshold $T$, we need to change the pixel greater
than or equal to $T$ to 1, and change the other pixels to 0. In this way, we
can clearly segment the object we want. In this step, we use the reset
operation to set $c_{q-1}\dots c_{1}$ to $0\dots 0$. If QC outputs $y=0$ and
$c_{0}=0$, then we use CNOT gate and Toffoli gate to set $c_{0}$ to 1. If QC
outputs $y=1$ and $c_{0}=1$, then we use CNOT gate and Toffoli gate to set
$c_{0}$ to 0. Finally, we can get a binary image $\lvert B_{YX}\rangle$, and
the specific operation is shown in Fig. 11. The result image $\lvert
B_{YX}\rangle$ can be represented as follows.

$\lvert{B_{YX}}\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{Y=0}^{{2^{n}}-1}{\sum\limits_{X=0}^{{2^{n}}-1}{\lvert{{b_{YX}}}\rangle\otimes\lvert
Y\rangle\lvert X\rangle}}$ (10)

The complete quantum circuit of our quantum image segmentation algorithm is
shown in Fig. 12.

Figure 11: The implementation circuit of quantum binarization (QB). Figure 12:
The complete quantum circuit of our quantum image segmentation algorithm.

##  4 Circuit complexity and experiment analysis

###  4.1 Circuit complexity analysis

Taking a $2^{n}\times 2^{n}$ classical image with gray-scale $[0,q]$ as an
example, we will discuss the complexity of the circuit in 4 steps.

In step 1, we need to prepare the classical image into a NEQR image, and it is
known from Ref. [9] that the computational complexity of this step does not
exceed O($qn2^{2n}$), because each pixel will be operated one after another.
However, our algorithm is to process quantum images, not classical images, so
the complexity of this step is not considered in the study of quantum image
processing algorithms [33, 39, 34], so we consider the complexity of this step
to be 0.

In step 2, we need to use CT operation to cycle shift the image, and the
complexity of CT is O$(n^{2})$ [19, 17]. In addition, we also use $q$ CNOT
gates to replicate the gray-scale value of the original image, the quantum
cost of this is $q$. So the complexity of this step is O$(n^{2}+q)$.

In step 3, we use 9 QCS and 1 QS to calculate the threshold $T$. The QCS is
composed of a QC and $q$ CSWAP gates. One QC requires $3q-2$ Toffoli gates,
$(q-1)$ CNOT gates, and $2(q-1)$ reset gates. So the quantum cost of a QC is
$18q-13$. In addition, the quantum cost of a CSWAP gate is 3 [41], and $q$
CSWAP gates are needed. So the quantum cost of a QCS is $21q-13$. A QS
requires $4q-7$ Toffoli gates, $4q-4$ CNOT gates, and $3q-4$ reset gates.
Therefore, the quantum cost of a QS is $27q-43$. As can be seen from the above
analysis, the complexity of this step is O$(q)$.

In step 4, we perform a quantum binarization (QB) operation on the image. As
shown in the Fig. 11, this circuit requires 2 Toffoli gates, 2 CNOT gates and
$q-1$ reset gates. So the quantum cost of this step is $q+11$, and the
complexity is O$(q)$.

In addition, in order to set the constant $Z$ , we also need $q$ reset gates
and $q$ NOT gates, so the quantum cost is $2q$, and the complexity is O$(q)$.
Therefore,the complexity of the complete algorithm is O$(n^{2}+q)$. On
classical computers, for images of size $2^{n}\times 2^{n}$, image
segmentation need to be processed individually for each pixel. So, the
complexity of the classical segmentation algorithm is no less than O$(2^{2n})$
. Thus, our scheme achieves an exponential acceleration relative to the
classical segmentation algorithm, so the real-time problem in classical
algorithm can be solved well.

###  4.2 Experiment analysis

In order to verify the feasibility of the algorithm we proposed, the
simulation experiment is conducted on the IBM Q. The IBM Q [40] currently
provides 24 quantum computers and 5 quantum simulators in the cloud. But due
to certain factors, the quantum computer available to us has only 5 qubits,
which is far from enough. Therefore, in this paper, we use a quantum simulator
called ’ibmq_qasm_simulator’ to conduct the experiment through the Qiskit
extension package [42]. This simulator has 32 qubits and can run all the basic
logic gates included in our circuit. So in this paper, all quantum circuits
have been simulated and verified on this simulator. But the actual image
processing requires a large number of qubits, therefore, due to the limitation
of the number of qubits, we selected a classic image with a size of $4\times
4$ and a gray-scale value of [0,7]. The schematic diagram of the image is
shown in Fig. 13. Y and X represent the coordinates of the pixel, and the
gray-scale value information is marked in the figure in binary form. In
addition, noise will reduce the accuracy of segmentation, and how to reduce
the noise by using quantum mechanisms is another interesting topic. Our
algorithm only considers how to implement the classical algorithm using the
quantum mechanism and does not consider the noise effect. Therefore, we assume
that the experiment is performed without noise.

Figure 13: The schematic diagram of the original image Figure 14: The
implementation circuit of quantum image preparation.

First, we need to prepare a NEQR image based on the gray-scale value
information and position information of the classical image. Because the NEQR
model makes full use of quantum entanglement and superposition, we only need
four qubits to store $4\times 4$ position information, and then we entangle
the gray-scale value and position. So only three qubits are needed to store
all the gray-scale values in the image. The quantum image preparation circuit
is shown in the Fig. 14, where $C_{2}C_{1}C_{0}$ represent the gray-scale
value information qubits, and $P_{3}P_{2}P_{1}P_{0}$ represent the position
information qubits. Because on the IBM Q platform, the qubits are initialized
to 0 by default, so we use H gates to make the position information qubits
into a superposition state. The gray-scale value information is entangled with
position by using basic quantum gates. Besides, in the experiment we set the
constant $Z=001$ randomly for the adjustment of the threshold to to verify the
feasibility of the algorithm. Because $Z$ is a definite value, it only needs
to be initialized with the reset operations and the NOT gates.

Figure 15: The probability histogram of the result image. Figure 16: The
schematic diagram of the result image.

Figure 17: The comparison of different segmentation results: a original
images, b the image segmented by fixed threshold ($T=3$), c the image
segmented by our algorithm.

Fig. 15 shows the probability histogram of the result quantum image, where the
ordinate represents the probability of the measured qubits sequences, and the
abscissa represents each pixel information and other auxiliary qubit
information. Each qubit sequence is in order from top to bottom, gray-scale
value information C and the position information P of the result image are
marked in the figure. The remaining qubits are used to form a complete
circuit, which has no effect on image representation, so to reduce the
complexity of the measurement, we only measured the gray-scale value and
position information qubits of the result image. A schematic diagram of the
result image is shown in Fig. 16, and the position information and gray-scale
value information have been marked in it. In addition, We randomly selected
three images as test images to compare our algorithm with the fixed-threshold
algorithm, as shown in Fig. 17. For better comparison of segmentation results,
we use the mean square error (MSE) to evaluate our algorithm. MSE can be used
to determine the similarity of two images. When the MSE value is smaller, it
means that the two images are more similar, which indicates that the
segmentation algorithm will be more effective. For two gray-scale images Q and
R with size $2^{n}\times 2^{n}$, MSE is defined as

$MSE=\frac{1}{2^{2n}}\sum_{Y=0}^{2^{n}-1}\sum_{X=0}^{2^{n}-1}[Q(Y,X)-R(Y,X)]^{2}$
(11)

where Y and X represent the position information of the images.

As shown in Tab. 4.2, the MSE values of our algorithm are less than the fixed
threshold segmentation, which indicates that our algorithm has a better
segmentation effect. Since we use local adaptive threshold and find the
threshold within the neighborhood window, this prevents the effect of uneven
illumination on image segmentation.

The MSE comparison of different segmentation results.  Input image MSE Fixed
threshold Our algorithm a 7.0625 6.5625 b 14.2500 11.8750 c 11.8750 11.5000

##  5 Conclusion

Since most of the current quantum image segmentation algorithms use global
fixed thresholds and require more quantum gates and qubits, whis is not
applicable in this NISQ era. In this paper, a quantum segmentation algorithm
based on local adaptive threshold for NEQR image is proposed, which can use
quantum mechanism to simultaneously compute local thresholds for all pixels in
a gray-scale image and quickly segment the image into a binary image.
Furthermore, we designed several quantum circuit units and a complete quantum
segmentation circuit by using fewer qubits and quantum gates. Circuit
complexity analysis and experiment on IBM Q demonstrate the superiority and
feasibility of our algorithm.

However, the local adaptive threshold segmentation algorithm only focuses on
the neighborhood window and does not consider the noise effect, which may lead
to some noise in the result image, so our future work is to investigate
quantum image noise reduction algorithms and apply them to quantum image
segmentation.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(62071240), and the Priority Academic Program Development of Jiangsu Higher
Education Institutions (PAPD).

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# Casimir effect for Dirac lattices

M. Bordag  Institut für Theoretische Physik, Universität Leipzig, Germany
<EMAIL_ADDRESS>I.G. Pirozhenko  Bogoliubov Laboratory of Theoretical
Physics, JINR, Dubna, Russia
Dubna State University, Dubna, Russia<EMAIL_ADDRESS>
###### Abstract

We consider polarizable sheets, which recently received some attention,
especially in context of the dispersion interaction of thin sheets like
graphene. These sheets are modeled by a collection of delta function
potentials and resemble ‘zero range potentials’, known in quantum mechanics.
We develop a theoretical description and apply the so-called ‘TGTG’-formula to
calculate the interaction of two such lattices. Thereby we make use of the
formulation of the scattering of waves off such sheets provided earlier. We
consider all limiting cases, providing link to earlier results. Also, we
discuss the relation to the pairwise summation method.

###### pacs:

03.65.Nk,03.65.-w

##  I Introduction

This paper is a contribution to the discussion of van der Waals and Casimir
forces between surfaces. Last years, much attention was paid to the
interaction between slabs of finite, and especially small, thickness. This
triggered by a growing interest to two dimensional structures (sheets) like 2d
electron gas, monoatomically layers and graphene, as well as to the
interaction between them. As discussed in Barton (2013), the situation with
sheets having only in-plane polarizability is relatively clear. Here one can
formulate a hydrodynamic plasma model and calculate the quantities of interest
Barton (2005). The same holds for graphene described by the Dirac equation
model for the $\pi$-electrons Bordag  _et al._ (2009a), which are responsible
for the interaction with the electromagnetic field.

The situation with perpendicular polarizability is more complicated. While in
Parashar  _et al._ (2012) no response to the electromagnetic field was found,
in Barton (2015) it was shown that such response takes place. In that paper
the sheet was modelled as a lattice of harmonic oscillators (dipoles),
vibrating in direction perpendicular to the sheet, and the limit of zero
spacing of this lattice was investigated. This was extended in Bordag (2014)
to a sheet, continuous from the very beginning, having parallel or
perpendicular polarizabilities. Later, a similar setup was considered using
point dipoles Bordag and Munoz-Castaneda (2015). These can be represented by
Dirac delta functions forming a so-called ’Dirac lattice’. In fact, there
represent point scatterers.

Such lattices, taken alone, have a well known internal dynamics, the simplest
case being the Kronig-Penney model (’Dirac comb’) Kronig and Penney (1931). In
quantum mechanics their use is known as ’method of zero-range potential’
Demkov and Ostrovskii (1988). As well known, in more than one dimension a
Hamilton operator with a delta function potential is not self adjoint Berezin
and Faddeev (1961). In electrodynamics, the self energy of a delta function
potential is singular and one needs a renormalization. In terms of quantum
field theory this setup was considered in Jackiw (1995).

Recently, a sheet of delta function potentials was used in Bordag and Munoz-
Castaneda (2015) to model a polarizable sheet. For instance, scattering off
such sheet was investigated and subsequently the transition to a continuous
sheet as well.

In the present note we consider a setup of two such sheets, hold in parallel
at some separation. This is a typical situation for the Casimir effect. We use
the well known scattering approach, also called ’TGTG’-formula, which can be
considered also as a generalization of the Lifshitz formula. Making use of the
translational invariance of the lattice, the $T$-matrices can be calculated in
momentum representation; for a one dimensional lattice even explicitly.
Further, we allow for a displacement of one lattice with respect to the other.
Also we study the limiting cases of small and large separation, resp. of large
and small lattice spacing. For large separation, the limiting case corresponds
to the interaction of two sheets carrying delta function potential, and for
small separation the limiting case corresponds to the interaction of two
points carrying delta function potential. In this paper we consider only a
scalar field. Extensions to the electromagnetic field should be quite straight
forward. It is also an intention of this paper to provide a framework for
calculating dispersion forces in realistic, experimentally relevant
situations, where the atomistic structure must be accounted for.

The paper is organized as follows. In the next section we provide the
necessary formulas to calculate the vacuum energy for two and one dimensional
lattices. In the third section we derive all limiting cases and discuss their
interrelations. In the fourth section we compare the results with pairwise
summation. Some conclusions and an appendix complete the paper.

##  II Vacuum energy for point scatterers

We consider two types of lattices. First, two dimensional lattices, A and B,
given by the lattice vectors

$\vec{a}^{\rm\,A}_{\mathbf{n}}=\left(\begin{array}[]{c}\mathbf{a}_{\mathbf{n}}+\mathbf{c}\\\
b\end{array}\right),\quad\vec{a}^{\rm\,B}_{\mathbf{n}}=\left(\begin{array}[]{c}\mathbf{a}_{\mathbf{n}}\\\
0\end{array}\right),$ (1)

For these, we use the following notations. A three dimensional vector is
denoted by an arrow, a two dimensional vector parallel to the $(x,y)$-plane is
denoted in bold, for instance $\vec{x}=\left(\begin{array}[]{c}\mathbf{x}\\\
x_{3}\end{array}\right)$. The lattice sites are given by
$\mathbf{a}_{\mathbf{n}}=a\left(\begin{array}[]{c}n_{1}\\\
n_{2}\end{array}\right)$, where $n_{1}$ and $n_{2}$ are integers, $a$ is the
lattice spacing; the lattice is rectangular. The lattice B is in the
$(x,y)$-plane and the lattice A is parallel to B on a separation $b$ and
shifted within that plane by the displacement vector $\mathbf{c}$.

Second we consider one dimensional lattices (chains), A and B, given by the
lattice vectors

$\vec{a}^{\rm\,A}_{n}=\left(\begin{array}[]{c}an+c\\\ 0\\\
b\end{array}\right),\quad\vec{a}^{\rm\,B}_{n}=\left(\begin{array}[]{c}an\\\
0\\\ 0\end{array}\right),$ (2)

where $n$ is a (single) integer. These chains are on the $x$-axis (B) resp.
parallel to it (A). Both are in the plane $y=0$. Their separation is $b$ as
above, the displacement $c$ is a number.

With these notations, we consider a scalar field $\phi(\vec{x})$, whose wave
equation after Fourier transform in time is

$\left(-\omega^{2}-\Delta+g\sum_{\mathbf{n}}\left(\delta(\vec{x}-\vec{a}^{\rm\,A}_{\mathbf{n}})+\delta(\vec{x}-\vec{a}^{\rm\,B}_{\mathbf{n}})\right)\right)\phi(\vec{x})=0,$
(3)

where $\vec{a}^{\rm\,A}_{\mathbf{n}}$ and $\vec{a}^{\rm\,B}_{\mathbf{n}}$ are
given by 1. In this equation, the delta functions are three dimensional, hence
the coupling $g$ has the dimension of length. As mentioned in the
Introduction, one should remember that the above equation is not well defined
and $g$ must be viewed as a bare coupling which needs to undergo a
renormalization.

The vacuum energy of the interaction of two lattices is given by the mentioned
’TGTG’-formula,

$E_{0}=\frac{1}{2\pi}\int_{0}^{\infty}d\xi\,{\rm Tr}\ln(1-{\cal M}),$ (4)

and $\xi$ is the imaginary frequency, $\omega=i\xi$. The kernel ${\cal M}$ is
the product of two other kernels,

${\cal M}(\vec{x},\vec{x}^{\prime})=\int d{x^{\prime\prime}}{\cal
N}_{A}(\vec{x},\vec{x}^{\prime\prime}){\cal
N}_{B}(\vec{x}^{\prime\prime},\vec{x}^{\prime}),$ (5)

each of which is given by

${\cal N}_{A,B}(\vec{x},\vec{x}^{\prime})=\int
d{x^{\prime\prime}}G_{0}(\vec{x},\vec{x}^{\prime\prime})T_{A,B}(\vec{x}^{\prime\prime},\vec{x}^{\prime}),$
(6)

in terms of the $T$-operators of the corresponding lattice. Further, the free
Green’s function,

$\displaystyle G_{0}(\vec{x}-\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\,\frac{e^{i\vec{k}(\vec{x}-\vec{x}^{\prime})}}{\omega^{2}-\mathbf{k}^{2}-k_{3}^{2}+i0},$
(7)

enters this formula. We will use below also the following two representations,

$G_{0}(\vec{x})=\frac{e^{i\omega|\vec{x}|}}{4\pi|\vec{x}|}=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}^{\prime})+i\Gamma|\mathbf{x}_{3}-x_{3}^{\prime}|}}{2i\Gamma(k)},$
(8)

with $\Gamma(k)=\sqrt{\omega^{2}-k^{2}+i0}$ and $k=|\mathbf{k}|$.

The $T$-operators can be defined from

$\displaystyle G(\vec{x},\vec{x}^{\prime})$ (9)
$\displaystyle=G_{0}(\vec{x}-\vec{x}^{\prime})-\int dz\,\int
dz^{\prime}\,G_{0}(\vec{x}-\vec{z})T(\vec{z},\vec{z}^{\prime})G_{0}(\vec{z}^{\prime}-\vec{x}^{\prime}),$

where $G(\vec{x},\vec{x}^{\prime})$ is the Green’s function of equation 3 and
can be related to the ansatz

$G(\vec{x},\vec{x}^{\prime})=G_{0}(\vec{x}-\vec{x}^{\prime})-\sum_{\mathbf{n},\mathbf{n}^{\prime}}G_{0}(\vec{x}-\vec{a}_{\mathbf{n}})\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}G_{0}(\vec{a}_{\mathbf{n}^{\prime}}-\vec{x}^{\prime}).$
(10)

Inserting this ansatz into the equation

$\displaystyle\left(-\omega^{2}-\Delta+g\sum_{\mathbf{n}}\left(\delta(\mathbf{x}-\vec{a}^{\rm\,A}_{\mathbf{n}})+\delta(\mathbf{x}-\vec{a}^{\rm\,B}_{\mathbf{n}})\right)\right)G(\vec{x},\vec{x}^{\prime})$
$\displaystyle=\delta(\vec{x}-\vec{x}^{\prime})$ (11)

and using the equation

$\left(-\omega^{2}-\Delta\right)G_{0}(\vec{x},\vec{x}^{\prime})=\delta(\vec{x}-\vec{x}^{\prime})$
(12)

for the free Green’s function, one comes to the equation

$-\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}+g\delta_{\mathbf{n},\mathbf{n}^{\prime}}-g\sum_{\mathbf{m}}G_{0}(\vec{a}_{\mathbf{n}}-\vec{a}_{\mathbf{m}})\Phi^{-1}_{\mathbf{m},\mathbf{n}^{\prime}}=0.$
(13)

With the definition

$\Phi_{\mathbf{n},\mathbf{n}^{\prime}}=\frac{1}{g}\delta_{\mathbf{n},\mathbf{n}^{\prime}}-G_{0}(\vec{a}_{\mathbf{n}}-\vec{a}_{\mathbf{n}^{\prime}})$
(14)

Eq. 13 becomes

$\sum_{\mathbf{m}}\Phi_{\mathbf{n},\mathbf{m}}\Phi^{-1}_{\mathbf{m},\mathbf{n}^{\prime}}=\delta_{\mathbf{n},\mathbf{n}^{\prime}}$
(15)

and $\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}$ is the inverse matrix to
$\Phi_{\mathbf{n},\mathbf{n}^{\prime}}$.

The diagonal elements of $\Phi_{\mathbf{n},\mathbf{n}^{\prime}}$ contain
$G_{0}(0)$ which is not well defined. Referring to Bordag and Munoz-Castaneda
(2015) for a discussion, we renormalize the coupling,

$\frac{1}{g}-G_{0}(0)\to\frac{1}{g},$ (16)

and after that we get

$\Phi_{\mathbf{n},\mathbf{n}^{\prime}}=\left\\{\begin{array}[]{lc}\frac{1}{g},&(\mathbf{n}=\mathbf{n}^{\prime})\\\\[5.0pt]
-G_{0}(\vec{a}_{\mathbf{n}}-\vec{a}_{\mathbf{n}^{\prime}}),&(\mathbf{n}\neq\mathbf{n}^{\prime})\end{array}\right.$
(17)

which is well defined.

Now, comparing Eqs. II and 10, we identify the $T$-operator to be

$T^{\rm
A}(\vec{z},\vec{z}^{\prime})=\sum_{\mathbf{n},\mathbf{n}^{\prime}}\delta(\vec{z}-\vec{a}^{\rm\,A}_{\mathbf{n}})\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}\delta(\vec{a}^{\rm\,A}_{\mathbf{n}^{\prime}}-\vec{z}^{\prime})$
(18)

for one lattice and by the same formula with $B$ in place of $A$ for the
other. Inserting this expression into 6 we get

${\cal
N}_{A}(\vec{x},\vec{x}^{\prime})=\sum_{\mathbf{n},\mathbf{n}^{\prime}}G_{0}(\vec{x}-\vec{a}^{\rm\,A}_{\mathbf{n}})\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}\delta(\vec{a}^{\rm\,A}_{\mathbf{n}^{\prime}}-\vec{x}^{\prime})$
(19)

and, again, the same with $A\to B$. Further, inserting these into 5, we get

${\cal M}(\vec{x},\vec{x}^{\prime})=\int dx^{\prime\prime}\,{\cal
N}_{A}(\vec{x},\vec{x}^{\prime\prime}){\cal
N}_{B}(\vec{x}^{\prime\prime},\vec{x}^{\prime}),$ (20)

for the kernel ${\cal M}$. Eqs. 19 and 20 represent the general expressions
for the kernel entering the ’TGTG’-formula 4 for a generic lattice of delta
functions. In the following we consider first two dimensional lattices as
given by 1, and, subsequently one dimensional lattices (chains) as given by
Eq. 2.

###  II.1 Vacuum energy for two dimensional lattices

In this subsection we consider lattices as given by Eq. 1. These have
translational invariance with respect to a lattice step. Equivalently, we have
$\mathbf{a}_{\mathbf{n}}-\mathbf{a}_{\mathbf{n}^{\prime}}=\mathbf{a}_{\mathbf{n}-\mathbf{n}^{\prime}}$,
and the quantities entering Eq. 14 and 15, depend only on differences,
$\Phi_{\mathbf{n},\mathbf{n}^{\prime}}=\Phi_{\mathbf{n}-\mathbf{n}^{\prime}}$
and the same for its inverse. As a consequence, the inversion of this matrix
can be calculated by Fourier transform,

$f_{\mathbf{n}}=a^{2}\int\frac{d\mathbf{k}}{(2\pi)^{2}}e^{i\mathbf{k}\mathbf{a}_{\mathbf{n}}}\tilde{f}(\mathbf{k}),\quad\tilde{f}(\mathbf{k})=\sum_{\mathbf{n}}e^{-i\mathbf{k}\mathbf{a}_{\mathbf{n}}}f_{\mathbf{n}}.$
(21)

This way we get

$\Phi^{-1}_{\mathbf{n},\mathbf{n}^{\prime}}=a^{2}\int\frac{d\mathbf{k}}{(2\pi)^{2}}e^{i\mathbf{k}(\mathbf{a}_{\mathbf{n}}-\mathbf{a}_{\mathbf{n}^{\prime}})}\frac{1}{\tilde{\phi}(\mathbf{k})}$
(22)

with

$\tilde{\phi}(\mathbf{k})=\sum_{\mathbf{n}}e^{-i\mathbf{k}\mathbf{a}_{\mathbf{n}}}\Phi_{\mathbf{n}}.$
(23)

Here we insert 17 and come to

$\displaystyle\tilde{\phi}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\frac{1}{g}-{\sum_{\mathbf{n}}}^{\prime}e^{-i\mathbf{k}\mathbf{a}_{\mathbf{n}}}G_{0}(\mathbf{a}_{\mathbf{n}}),$
(24) $\displaystyle=$
$\displaystyle\frac{1}{g}-\frac{1}{4\pi}J_{1}(\omega,\mathbf{k}),$

where Eq. 8 was used and we introduced

$J_{1}(\omega,\mathbf{k})={\sum_{\mathbf{n}}}^{\prime}\frac{1}{|\mathbf{a}_{\mathbf{n}}|}e^{i\omega|\mathbf{a}_{\mathbf{n}}|+i\mathbf{k}\mathbf{a}_{\mathbf{n}}}.$
(25)

As usual, in the primed sum the term with $\mathbf{n}=0$ is dropped. Finally,
we mention the relation

$\sum_{\mathbf{m}}e^{i\mathbf{k}\mathbf{a}_{\mathbf{m}}}\Phi^{-1}_{\mathbf{m},\mathbf{m}^{\prime}}=\frac{1}{\tilde{\phi}(\mathbf{k})}e^{i\mathbf{k}\mathbf{a}_{\mathbf{m}^{\prime}}},$
(26)

which follows with the Fourier transform 21.

Now we return to the kernel ${\cal N}$, 19, and insert 8 for the Green’s
function,

${\cal
N}_{A}(\vec{x},\vec{x}^{\prime})=\sum_{\mathbf{n},\mathbf{m}}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{i\mathbf{k}(\mathbf{x}-\mathbf{a}_{\mathbf{n}}-\mathbf{c})+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})}\Phi^{-1}_{\mathbf{n}-\mathbf{m}}\delta(\mathbf{a}_{\mathbf{m}}+\mathbf{c}-\mathbf{x}^{\prime})\delta(b-x_{3}^{\prime}).$
(27)

Next we use 26 and get

$\displaystyle{\cal N}_{A}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{i\mathbf{k}(\mathbf{x}-\mathbf{c})+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(\mathbf{k})}\sum_{\mathbf{m}}e^{-i\mathbf{k}\mathbf{a}_{\mathbf{m}}}\delta(\mathbf{a}_{\mathbf{m}}+\mathbf{c}-\mathbf{x}^{\prime})\delta(b-x_{3}^{\prime}).$
(28)

The corresponding expression for ${\cal N}_{B}(\vec{x},\vec{x}^{\prime})$
follows from here with $\mathbf{c}=0$ and $b=0$. Using these, we can write
down the kernel ${\cal M}(\vec{x},\vec{x}^{\prime})$, Eq. 5,

$\displaystyle{\cal M}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\int
dx{{}^{\prime\prime}}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{i\mathbf{k}(\mathbf{x}-\mathbf{c})+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(\mathbf{k})}\sum_{\mathbf{m}}e^{-i\mathbf{k}\mathbf{a}_{\mathbf{m}}}\delta(\mathbf{a}_{\mathbf{m}}+\mathbf{c}-\mathbf{x}^{\prime\prime})\delta(b-x_{3}^{\prime\prime})$
(29)
$\displaystyle\cdot\int\frac{d^{2}\mathbf{k}^{\prime}}{(2\pi)^{2}}\,\frac{e^{i\mathbf{k}^{\prime}\mathbf{x}^{\prime\prime}+i\Gamma^{\prime}|x_{3}^{\prime\prime}|}}{2i\Gamma(\mathbf{k}^{\prime})\tilde{\phi}(\mathbf{k}^{\prime})}\sum_{\mathbf{m}^{\prime}}e^{-i\mathbf{k}^{\prime}\mathbf{a}_{\mathbf{m}^{\prime}}}\delta(\mathbf{a}_{\mathbf{m}^{\prime}}-\mathbf{x}^{\prime})\delta(x_{3}^{\prime}).$

Carrying out the integration over $\vec{x}^{\prime\prime}$ using the delta
functions, we come to a sum over $\mathbf{m}$. Before doing this sum, we split
the momenta $\mathbf{k}$ and $\mathbf{k}^{\prime}$ into quasi momentum and
integer part,

$\mathbf{k}=\mathbf{q}+\frac{2\pi}{a}\mathbf{N},\quad\mathbf{k}^{\prime}=\mathbf{q}^{\prime}+\frac{2\pi}{a}\mathbf{M},$
(30)

where $\mathbf{N}$ and $\mathbf{M}$ are integer vectors like $\mathbf{n}$ in
$\mathbf{a}_{\mathbf{n}}$. The integration becomes $\int d^{2}\mathbf{k}=\int
d^{2}\mathbf{q}\sum_{\mathbf{N}}$ and the components of
$\mathbf{q}=(q_{1},q_{2})$ are restricted to
$-\frac{\pi}{a}\leq{q}_{1,2}<\frac{\pi}{a}$. Now the sum over $\mathbf{m}$
appearing in 29 gives

$\sum_{\mathbf{m}}e^{-i(\mathbf{k}-\mathbf{k}^{\prime})\mathbf{a}_{\mathbf{m}}}=\left(\frac{2\pi}{a}\right)^{2}\delta^{(2)}(\mathbf{q}-\mathbf{q}^{\prime}),$
(31)

where the dependence on $\mathbf{N}$ and $\mathbf{M}$ dropped out. Then 29
turns into

$\displaystyle{\cal M}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\,\sum_{\mathbf{N}}\frac{e^{i\mathbf{x}(\mathbf{q}+\frac{2\pi}{a}\mathbf{N})-i\frac{2\pi}{a}\mathbf{N}\mathbf{c}+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(\mathbf{k})}\sum_{\mathbf{M}}\frac{e^{i\frac{2\pi}{a}\mathbf{M}\mathbf{c}+i\Gamma^{\prime}|b|}}{2i\Gamma(\mathbf{k}^{\prime})\tilde{\phi}(\mathbf{k}^{\prime})}\sum_{\mathbf{m}^{\prime}}e^{-i\mathbf{q}\mathbf{a}_{\mathbf{m}^{\prime}}}\delta(\mathbf{a}_{\mathbf{m}^{\prime}}-\mathbf{x}^{\prime})\delta(x_{3}^{\prime}).$
(32)

This expression for ${\cal M}(\vec{x},\vec{x}^{\prime})$ must be inserted into
the generalized Lifshitz formula 4 under the sign of the trace. In doing so,
the delta function in one factor ${\cal M}(\vec{x},\vec{x}^{\prime})$ turns
the $\mathbf{x}$ in the next factor into $\mathbf{a}_{\mathbf{m}^{\prime}}$.
Again, the summation over $\mathbf{m}^{\prime}$ gives a delta function for the
quasi momenta. In this way, the products of the factors ${\cal
M}(\vec{x},\vec{x}^{\prime})$ is diagonal in $q$ and we can define

$h(\omega,\mathbf{q})=\frac{1}{a^{2}}\sum_{\mathbf{N}}\,\frac{e^{i\Gamma(\mathbf{k})b+i\frac{2\pi}{a}\mathbf{N}\mathbf{c}}}{2i\Gamma(\mathbf{k})\tilde{\phi}(\mathbf{k})}$
(33)

with $\mathbf{k}$ given by 30, and we rewrite 32 in the form

${\cal
M}(\vec{x},\vec{x}^{\prime})=a^{2}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\,e^{i\mathbf{q}\mathbf{x}}h(\omega,\mathbf{q})h(\omega,\mathbf{q})^{*}\sum_{\mathbf{m}}e^{-i\mathbf{q}\mathbf{a}_{\mathbf{m}}}\delta(\mathbf{a}_{\mathbf{m}}-\mathbf{x}^{\prime})\delta(x_{3}^{\prime}).$
(34)

In this way, the factors ${\cal M}(\vec{x},\vec{x}^{\prime})$ entering II
become diagonal. The $\mathbf{x}$-integration in the trace is to be taken over
one cell and it turns the sum into unity,

$\int
d\vec{x}\,e^{i\mathbf{q}\mathbf{x}}\sum_{\mathbf{m}}e^{-i\mathbf{q}\mathbf{a}_{\mathbf{m}}}\delta(\mathbf{a}_{\mathbf{m}}-\mathbf{x})\delta(x_{3})=1.$
(35)

As a consequence, for the vacuum energy per lattice cell we get the expression

$E_{0}=\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\,a^{2}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\ln\left(1-|h(i\xi,\mathbf{q})|^{2}\right).$
(36)

This is the final general formula for the vacuum energy of two lattices as
given by Eq. 1. The numerical result for the vacuum energy for one cell of the
lattices is presented at Fig. 1.

###  II.2 Vacuum energy for one dimensional lattices (chains)

In this subsection we consider lattices as given by Eq. 2. The calculations
are mostly in parallel to the preceeding subsection. The Fourier transform is
now one dimensional,

$f_{\mathbf{n}}=a\int\frac{dk_{1}}{2\pi}e^{ik_{1}an}\tilde{f}(k_{1}),\quad\tilde{f}(k_{1})=\sum_{\mathbf{n}}e^{-ik_{1}an}f_{n},$
(37)

where we used for the momentum the notation $k_{1}$ since the lattice is
parallel to the $x$-axis. Following Eqs. 22 and 23, we get

$\tilde{\phi}(k_{1})=\sum_{n}e^{-ik_{1}an}\Phi_{n},$ (38)

which after renormalization gives

$\tilde{\phi}(k_{1})=\frac{1}{g}-\frac{1}{4\pi}\sum_{n\neq
0}\frac{e^{i\omega|n|+ik_{1}an}}{a|n|}.$ (39)

The sum is one dimensional and can be done explicitly,

$\tilde{\phi}(k_{1})=\frac{1}{g}-\frac{1}{4\pi a}\ln\left(1+e^{2i\omega
a}-2\cos(k_{1}a)e^{i\omega a}\right).$ (40)

By the way, the zeros of this function determine the zones in the Kronig-
Penney model.

Now we consider the kernel ${\cal N}_{A}$, Eq. 6, and insert the Green’s
function from 8 and the relation

$\sum_{n}e^{ik_{1}an}\Phi^{-1}_{mm^{\prime}}=\frac{1}{\tilde{\phi}(k_{1})}e^{-k_{1}am^{\prime}}$
(41)

to get

$\displaystyle{\cal N}_{A}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{ik_{1}(x_{1}-an-c)+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(k_{1})}\sum_{m}e^{-ik_{1}am}\delta(am+c-x_{1}^{\prime})\delta(x_{2}^{\prime})\delta(b-x_{3}^{\prime}).$
(42)

In parallel to 29, but carrying out the integration over $x^{\prime\prime}$,
we get further

$\displaystyle{\cal M}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{a}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\,\frac{e^{ik_{1}(x_{1}-c)+ik_{2}x_{2}+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(k_{1})}\sum_{m}e^{-ik_{1}am}$
(43)
$\displaystyle\cdot\int\frac{d^{2}\mathbf{k}^{\prime}}{(2\pi)^{2}}\,\frac{e^{ik_{1}^{\prime}(am+c)+i\Gamma^{\prime}|x_{3}^{\prime\prime}|}}{2i\Gamma(\mathbf{k}^{\prime})\tilde{\phi}(k_{1}^{\prime})}\sum_{m^{\prime}}e^{-ik_{1}^{\prime}am^{\prime}}\delta(am-x^{\prime}_{1})\delta(x_{2}^{\prime})\delta(x_{3}^{\prime}).$

We introduce a splitting of the momenta like 30, but now only for $k_{1}$,

$k_{1}=q+\frac{2\pi}{a}N,\quad k_{1}^{\prime}=q+\frac{2\pi}{a}M,$ (44)

where $N$ and $M$ are integers, and come to

$\displaystyle{\cal M}(\vec{x},\vec{x}^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{a}\int\frac{dq}{2\pi}\int\frac{dk_{2}}{2\pi}\sum_{N}\frac{e^{iq(x_{1}-c)+i\frac{2\pi}{a}Nc+ik_{2}x_{2}+i\Gamma|x_{3}-b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(k_{1})}$
(45)
$\displaystyle\cdot\int\frac{dk^{\prime}_{2}}{2\pi}\sum_{M}\frac{e^{iqc+i\frac{2\pi}{a}Mc+i\Gamma^{\prime}|b|}}{2i\Gamma(k^{\prime})\tilde{\phi}(k_{1}^{\prime})}\sum_{m}e^{-iqam}\delta(am-x^{\prime}_{1})\delta(x_{2}^{\prime})\delta(x_{3}^{\prime}).$

Following the same discussion concerning the trace as in the preceding
subsection, we are lead to introduce the notation

$h(\omega,q)=\frac{1}{a}\int\frac{dk_{2}}{2\pi}\sum_{N}\frac{e^{i\frac{2\pi}{a}Nc+i\Gamma|b|}}{2i\Gamma(\mathbf{k})\tilde{\phi}(k_{1})}$
(46)

with

$\Gamma=\sqrt{\omega^{2}-k_{1}^{2}-k_{2}^{2}+i0},\quad
k_{1}=q+\frac{2\pi}{a}N.$ (47)

Finally, we get the vacuum energy for one cell of the chains,

$E_{0}=\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\,a\int\limits_{-\pi/a}^{\pi/a}\frac{dq}{2\pi}\ln\left(1-|h(i\xi,q)|^{2}\right).$
(48)

This is the final general formula for the vacuum energy of two lattices as
given by Eq. 2.

Figure 1: The vacuum energy per unit as a function of distance between
lattices for different values of the coupling. Here lattices are exactly
opposite to one another, $c=0$. From left to right $g/a=0.01,0.05,0.1$. Inset:
the same for two parallel Dirac chains.

It should be mentioned that the formula for $h(i\xi,q)$ can be simplified
since the integration over $k_{2}$ in 46 can be carried out. First, we perform
the Wick rotation and define

$\Gamma=i\gamma,\quad\gamma=\sqrt{\xi^{2}+k_{1}^{2}+k_{2}^{2}}.$ (49)

We get

$h(i\xi,q)=\frac{1}{a}\int\frac{dk_{2}}{2\pi}\sum_{N}\frac{e^{i\frac{2\pi}{a}Nc-\gamma
b}}{-2\gamma\tilde{\phi}(k_{1})}.$ (50)

Changing the integration from $k_{2}$ for $\gamma$ and using the integral
representation

$K_{0}(z)=\int_{0}^{\infty}e^{z\cosh(\theta)}d\theta$ (51)

one comes to

$h(i\xi,q)=\frac{1}{2\pi
a}\sum_{N}\frac{K_{0}(\sqrt{\xi^{2}+k_{1}^{2}}b)}{-2\tilde{\phi}(k_{1})}\,e^{i\frac{2\pi}{a}Nc}$
(52)

with $\tilde{\phi}(k_{1})$ given by Eq. 40. The numerical result for the
vacuum energy for one cell of the chains is presented at Fig. 1 (inset).

The influence of the displacement $c$ on the interaction between the chains is
shown in Fig. 2. Similar profiles may be found in Buhmann  _et al._ (2016) for
the Casimir-Polder potential between an atom and a one-dimensional grating.

Figure 2: The factor $\eta=E(c\neq 0)/E(c=0)$ is plotted as a function of
$c/a$ for $g/a=0.1$ and different separations $\beta=b/a$ of the chains. From
bottom to top $\beta=0.1,0.2,0.4,0.6$.

##  III Limiting cases

In this section we discuss the limiting cases of the interaction lattices of
delta functions. There are two limiting cases for each, two and one
dimensional lattices. One limiting case is for large separation, or
equivalently for dimensional reasons, for small lattice step. The other is for
small separation, or equivalently, large lattice step.

For large separation, the two dimensional lattices turn into planes, carrying
a delta function potential. For these the Casimir effect was first
investigated in Bordag  _et al._ (1992). The one dimensional lattices (chains)
turn into lines, carrying a (two dimensional) delta function potential. This
case was not considered so far. However, it turns out that that it is
equivalent to the finite size cylinders with Dirichlet boundary conditions for
turning their radius to zero, as considered in Emig  _et al._ (2006) and
Bordag (2006).

For small separation, in the limit, only one delta function for the one
lattice interacts with the closest one in the other lattice. This is
equivalent to the Casimir-Polder interaction of two points carrying a delta
function potential each. This is the scalar version of the well known
interaction of two dipoles, which was considered within the scattering
approach in Power and Thirunamachandran (1993). Also, it can be considered as
limiting case of the interaction of two spheres carrying delta function
potential for vanishing radius of the spheres. The corresponding calculation
is shown in the Appendix.

The considered cases and their interrelations are shown in Fig. 3.

Figure 3: Possible transitions from one dimensional lattices (chains) and two
dimensional lattices of delta functions for large and small separation. For
convenience, the two dimensional lattices are shown for $c=0$, i.e., without
displacement. For large separation, the lattices turn into continuous plates
and the chains into wires. For small separation both turn into the closes
pints. Also shown are the transitions from spheres and cylinders to plates and
wires.

###  III.1 Two dimensional lattices at large separation

We start from Eq. 36 for the vacuum energy and consider the limit of lattice
spacing $a\to 0$, or equivalently, separation $b\to\infty$. At the end, we
will get tow parallel continuous sheets for which we need the energy density
per unit area which is $\frac{1}{a^{2}}E_{0}$. The integration region in
$\mathbf{q}$, given by $-\frac{\pi}{a}\leq{q}_{1,2}<\frac{\pi}{a}$, turns into
a whole $\mathbb{R}^{2}$. The function $h(\omega,\mathbf{k})$, 33, reads with
imaginary frequency,

$h(i\xi,\mathbf{q})=\frac{1}{a^{2}}\sum_{\mathbf{N}}\,\frac{e^{-\gamma(\mathbf{k})b+i\frac{2\pi}{a}\mathbf{N}\mathbf{c}}}{-2\gamma(\mathbf{k})\tilde{\phi}(\mathbf{k})}$
(53)

with $\gamma=\sqrt{\xi^{2}+\mathbf{k}^{2}}$, $k_{i}=q_{i}+\frac{2\pi}{a}N_{i}$
(i=1,2). For $a\to 0$ all contributions except that for ${\cal N}=0$ vanish
and we get $k_{i}=q_{i}$. Also we need the function $\tilde{\phi}$, given by
24 and the function $J_{1}$, 25, which reads now

$J_{1}(i\xi,\mathbf{k})={\sum_{\mathbf{n}}}^{\prime}\frac{1}{|\mathbf{a}_{\mathbf{n}}|}e^{-\xi|\mathbf{a}_{\mathbf{n}}|+i\mathbf{k}\mathbf{a}_{\mathbf{n}}}.$
(54)

For $a\to 0$, the sum turns over into an integral, which can be calculated
easily,

$J_{1}(i\xi,k)\raisebox{-5.0pt}{${\to\atop a\to 0}$}\
\frac{2\pi}{a^{2}\gamma},$ (55)

see Eq. (134) in Bordag and Munoz-Castaneda (2015). This way we get

$h(i\xi,k)\raisebox{-5.0pt}{${\to\atop a\to 0}$}\ r\,e^{-\xi b},$ (56)

where

$r=\frac{1}{1-\frac{a^{2}}{g}2\gamma}.$ (57)

The vacuum energy 36 becomes

$\frac{1}{a^{2}}E_{0}\raisebox{-5.0pt}{${\to\atop a\to 0}$}\
\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\int\frac{d\mathbf{q}}{(2\pi)^{2}}\
\ln\left(1-\left(r\,e^{-\gamma b}\right)^{2}\right).$ (58)

In fact, $r$, Eq. 57, is the well known reflection coefficient for a plane
carrying a delta function potential and 58 gives the Casimir effect for two
such sheets. The ratio $g/a^{2}$, having the meaning of coupling per unit
area, is the strength of the delta function potential in the corresponding
equation, where the delta function is one dimensional and well defined such
that this strength carries over from the equation to the reflection
coefficient without change. This is in opposite to the strength $g$ in Eq. 36
which after renormalization 16 has little to do with the $g$ in equation 3.
The other way round, one can use the established above relation to the
continuous sheets as normalization of the coupling $g$ after renormalization.

###  III.2 One dimensional lattices (chains) at large separation

This case is to a large extend in parallel to the preceding subsection. We
start from 48 and consider the energy per unit length, $\frac{1}{a}E_{0}$.
Further, we substitute $\xi\to\xi/b$ and $\mathbf{q}\to q/b$. This way we get
a factor $1/b^{2}$ in front and all further dependence is on $a/b$.

Again in the function $h(i\xi,k_{1})$, Eq. 40, accounting for 44, only $N=0$
gives a non vanishing contribution. A somehow different feature appears from
$\tilde{\phi}$, LABEL:2.2.17, which has a logarithmic behaviour now,

$\displaystyle\tilde{\phi}(k_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{g}-\frac{b}{4\pi
a}\left(-\frac{a}{b}\xi+\ln\left(2\left(\cosh\left(\frac{a}{b}\xi\right)^{2}-\cos\left(\frac{a}{b}k_{1}\right)^{2}\right)\right)\right),$
(59) $\displaystyle=$ $\displaystyle\frac{1}{g}-\frac{b}{4\pi
a}\left(2\ln\left(\frac{a}{b}\right)-\frac{a}{b}\xi+\ln\left(\xi^{2}+k_{1}^{2}\right)+\dots\right),$
${\to\atop a\to 0}$ $\displaystyle-\frac{\ln(\frac{a}{b})}{2\pi a/b}.$

The last line is the leading order and one observes that the dependence on the
coupling $g$ is lost. Inserting into 52 we get

$h(\xi,k_{1})\raisebox{-5.0pt}{${\to\atop a\to 0}$}\
\frac{b}{\ln(a/b)}\,K_{0}\left(\sqrt{\xi^{2}+k_{1}^{2}}\right).$ (60)

Finally, inserting into 48, we get

$\frac{1}{a}E_{0}\raisebox{-5.0pt}{${\to\atop a\to 0}$}\
\frac{1}{2b^{2}}\int_{0}^{\infty}\frac{d\xi}{\pi}\int\frac{dk_{1}}{2\pi}\
\ln\left(1-\left(\frac{1}{\ln(a/b)}K_{0}\left(\sqrt{\xi^{2}+k_{1}^{2}}\right)\right)^{2}\right),$
(61)

and further expanding the logarithm,

$\frac{1}{a}E_{0}\raisebox{-5.0pt}{${\to\atop a\to 0}$}\ -\frac{1}{8\pi
b^{2}}\frac{1}{\ln(a/b)^{2}},$ (62)

where $\int_{0}^{\infty}dp\,p\,K_{0}(p)^{2}=\frac{1}{2}$ was used.

It is to be mentioned, that this result coincides with the large separation
limit of the interaction between two parallel cylindrical shells carrying
$\delta$-potentials (see Appendix B), considered also in Milton and Wagner
(2008a). The same limit may be reproduced from Bordag (2006), Eq. (88), for
the interaction of a conducting cylinder with a conducting plane after taking
into account the different geometries and a factor of 2 for the polarizations.

###  III.3 Two dimensional lattices at short separation

Here we consider the vacuum energy 36 for the interaction of two dimensional
lattices for small separation, or equivalently for large lattice steps. In
this case the integration region for the integration over $\mathbf{q}$ in 36
shrinks to zero and this integral gives

$a^{2}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}f(\mathbf{q})\raisebox{-5.0pt}{${\to\atop
a\to\infty}$}\ f(0),$ (63)

where $f(\mathbf{q})$ is some arbitrary function. In the function
$h(\omega,\mathbf{q})$, 24, we first consider the function
$J_{1}(\omega,\mathbf{k})$, 25. For $a\to\infty$, it simply vanishes. As a
consequence, $\tilde{\phi}(\mathbf{k})$, 24, turns into

$\tilde{\phi}(\mathbf{k})\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\
\frac{1}{g}.$ (64)

Now in the function $h(\omega,\mathbf{q})$, the sum over ${\cal N}$ turns into
an integration according to

$\frac{2\pi}{a}{\cal N}\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\\\
k,\qquad\frac{1}{a^{2}}\sum_{\cal N}\raisebox{-5.0pt}{${\to\atop
a\to\infty}$}\ \int\frac{d\mathbf{k}}{(2\pi)^{2}}.$ (65)

This way, we get

$h(i\xi,\mathbf{q})\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\
\int\frac{d\mathbf{k}}{(2\pi)^{2}}\frac{g}{-2\gamma}e^{-\xi
b+i\mathbf{k}\mathbf{c}}.$ (66)

The integration can be carried out and after some calculation we arrive at

$h(i\xi,\mathbf{q})\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\ \frac{g}{4\pi
d}e^{-\xi d},$ (67)

where $d=\sqrt{b^{2}+c^{2}}$ is the separation between the closes delta
functions from the two lattices. With these, the vacuum energy turns into

$E_{0}\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\
\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\ln\left(1-\left(\frac{g}{4\pi
d}e^{-\xi d}\right)^{2}\right),$ (68)

which is the Casimir-Polder interaction of two points at separation $d$
carrying a delta function potential each. It coincides with the interaction of
two spheres carrying delta function potential at large separation (or small
radius). The corresponding formula is displayed in the Appendix. Again, we
remark that the coupling $g$ in 68 is after renormalization and, therefore, is
not fixed.

###  III.4 One dimensional lattices (chains) at short separation

We consider the vacuum interaction of two chains as given by Eq. 48 for short
separation. Similar to the preceding subsection the integration over q shrinks
to a point,

$a\int\frac{dq}{2\pi}f(q)\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\ f(0),$
(69)

Next we consider the function $\tilde{\phi}(k_{1})$, 40, and have

$\tilde{\phi}(k_{1})\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\ \frac{1}{g}.$
(70)

in the function $h(\omega,\mathbf{q})$. The sum over ${\cal N}$ turns into an
integration according to

$\frac{2\pi}{a}N\raisebox{-5.0pt}{${\to\atop
a\to\infty}$}\mathbf{k},\qquad\frac{1}{a}\sum_{N}\raisebox{-5.0pt}{${\to\atop
a\to\infty}$}\ \int\frac{dk}{2\pi}.$ (71)

and we get from 52

$\displaystyle h(i\xi,q)$ ${\to\atop a\to\infty}$ $\displaystyle
g\int_{-\infty}^{\infty}\frac{dk}{2\pi}K_{0}(\sqrt{\xi^{2}+k^{2}}b)e^{ikc},$
(72) $\displaystyle=\frac{g}{4\pi d}e^{-\xi d},$

where again the integration over $k$ was carried out and
$d=\sqrt{b^{2}+c^{2}}$ is the separation between the closest delta functions
from the two chains.

Now, inserting 69 and 72 into 48, we get

$E_{0}\raisebox{-5.0pt}{${\to\atop a\to\infty}$}\
\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\ln\left(1-\left(\frac{g}{4\pi
d}e^{-\xi d}\right)^{2}\right),$ (73)

which is the same expression as 68 and represents the Casimir-Polder
interaction of two points carrying delta function potential as discussed at
the end of the preceding subsection.

##  IV Comparison with pairwise summation

A special feature of van der Waals and Casimir forces is their multi particle
character. While in certain cases a pairwise summation may give a good
approximation, in general it will not. In this section we compare the results
of the exact calculation for point scatterers with the result of pairwise
summation, i.e., without multi particle forces.

Figure 4: The difference between exact result 48 (dashed line) and pairwise
summation of individual Casimir-Polder interactions for chains 73, $g/a=0.1$.

To this end we perform a pairwise summation of all individual Casimir-Polder
interactions

$F(z)=\sum_{\mathbf{n}}F_{n}\cos(\varphi_{n}),\;F_{n}(z)=-\left.\frac{dE_{CP}(r)}{dr}\right|_{r=r_{n}},$
(74)

where $E_{CP}$ is given by 68. For two chains $\mathbf{n}=n$,
$r_{n}=\sqrt{z^{2}+n^{2}a^{2}}$ and $\cos(\varphi_{n})=1/\sqrt{1+(an/z)^{2}}$;
for two 2D-lattices a double summation is required
$\mathbf{n}=\\{n_{1},n_{2}\\}$,
$r_{\mathbf{n}}=\sqrt{z^{2}+n_{1}^{2}a^{2}+n_{2}^{2}a^{2}}$ and
$\cos(\varphi_{n})=1/\sqrt{1+(an_{1}/z)^{2}+(an_{2}/z)^{2}}$. And finally, the
interaction energy per one $\delta$-function is

$E_{pw}(b)=\int_{b}^{\infty}F(z)dz=\sum_{\mathbf{n}=-\infty}^{\infty}E_{CP}(r_{\mathbf{n}})$
(75)

As follows from Section III, Eqs. 68 and 73, the pairwise summation is a good
approximation at short distances, where the exact formulas tend to a two-point
Casimir-Polder interaction. At medium and large distances the pairwise
summation overestimates the vacuum energy. The corresponding curves for Dirac
chains at medium distances are presented at Fig. 4. Here the pairwise
summation of Casimir-Polder interactions is performed for $N=1000$ Dirac
$\delta$-s. The evaluation according to the exact formula, Eq. 48, was also
truncated at $N=1000$ in Eq. 40, though the result only weakly depends on $N$
at medium and large distances.

The result of pairwise summation depends considerable on the coupling $g$ at
large separations, while the asymptote of the exact result, Eq. 62, is
coupling independent.

##  V Conclusions

We considered $T$-operators for a two-dimensional and a one dimensional
lattices of $\delta$-functions. These involve lattice sums, which can be
expressed in terms Hurwitz zeta function. Further we used this $T$-operator to
formulate the kernel in the TGTG formula for the dispersion interaction of two
such lattices. This can be viewed as a kind of generalized Lifshitz formula
and a represents a finite (converging) expression for the interaction energy.
We considered the cases of the interaction of two parallel two-dimensional
lattices and, of 1-dimensional parallel lattices (chains). The generalization
to rotations is left for future work.

We consider in detail to limiting cases and show the transition from lattices
to plans for large separation and to the Casimir-Polder interaction of two
lattice sites at small separation. These limiting cases are in agreement with
earlier results. Our formulas appear to interpolate between these and
establish the link between these..

## acknowledgement

We acknowledge partial support from the Heisenberg-Landau Programme.

##  Appendix A The Casimir effect for two spheres carrying delta function
potentials

In this appendix we display basic formulas for the interaction of two spheres
carrying delta function potential (’semitransparent’ spheres) using the by now
well known scattering approach (’TGTG’-formula). Although such kind of
calculations are, in much more general form, contained in a number of papers,
for example in Emig (2008), specific formulas for one of the most simple
special cases may be of use. The first calculation involving delta function
potential is Bordag (2007), which was focused on the corrections beyond PFA.

The basic setup is given by the equation

$\left(-\omega^{2}-\Delta+\frac{g}{4\pi
R^{2}}\left(\delta(|\vec{x}|-R)+\delta(|\vec{x}-\vec{d}|-R)\right)\right)\phi(\vec{x})=0,$
(76)

where $R$ is the radius of the spheres, one at the origin, the other at
separation $d=|\vec{d}|$. These delta functions are one dimensional and the
coupling $g$ does not undergo any renormalization.

Within the chosen approach, we need for the $T$-operator for a single sphere.
In this case the equation is

$\left(-\omega^{2}-\Delta+\frac{g}{4\pi
R^{2}}\delta(|\vec{x}|-R)\right)\phi(\vec{x})=0.$ (77)

The scattering problem for a single sphere was considered, for example, in
Bordag  _et al._ (1999) and we use some notations from there. The Green’s
function for Eq. 77, in a spherical basis, is

$G(\vec{x},\vec{x}^{\prime})=\sum_{lm}Y_{lm}(\Omega)d_{l}(r,r^{\prime})Y_{lm}(\Omega^{\prime})^{*}$
(78)

and the free radial Green’s function is

$d^{(0)}(r,r^{\prime})=\frac{1}{rr^{\prime}}j_{l}(\omega
r_{<})h^{(1)}_{l}(\omega r_{>}),$ (79)

where $j_{l}(z)=\sqrt{\pi/(2z)}J_{l+1/2}(x)$ and
$h^{(1,2)}_{l}(z)=\sqrt{\pi/(2z)}H^{(1,2)}_{l+1/2}(x)$ are spherical Bessel
functions. Their Wronskian is
$j_{l}^{\prime}(z)h_{l}(z)-j_{l}(z)h_{l}^{\prime}(z)=1/iz^{2}$. The delta
function in 77 results in matching conditions on the radial function
$d_{l}(r,r^{\prime})$, which must be continuous and obey

$\partial_{r}d_{l}(r,r^{\prime})_{|}{{}_{r=R+0}}-\partial_{r}d_{l}(r,r^{\prime})_{|}{{}_{r=R-0}}=-\frac{g}{4\pi
R^{2}}d_{l}(R,r^{\prime}).$ (80)

With the ansatz

$d_{l}(r,r^{\prime})=d^{(0)}(r,r^{\prime})-d^{(0)}(r,R)\Phi^{-1}(\omega)d^{(0)}(R,r^{\prime})$
(81)

we get

$\Phi^{-1}(\omega)=\frac{i\omega g/4\pi}{1+\frac{g}{4\pi R}i\omega
j_{l}(\omega R)h_{l}(\omega R)}.$ (82)

After Wick rotation, this expression becomes

$\Phi^{-1}(i\xi)=\frac{-gR/4\pi}{1+\frac{g}{4\pi R^{2}}I_{l+\frac{1}{2}}(\xi
R)K_{l+\frac{1}{2}}(\xi R)}$ (83)

with the modified Bessel functions $I_{\nu}(z)$ and $K_{\nu}(z)$. Eq. 83 is in
agreement with Eq. (23) in Bordag  _et al._ (1999).

The ’TGTG’-formula can be written in a spherical basis,

$E_{0}=\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}{\rm Tr}\ln(1-{\cal M})$
(84)

where the kernel is

${\cal
M}_{l,l^{\prime};m}=\sum_{l^{\prime\prime}=|l-l^{\prime}|}^{l+l^{\prime}}{\cal
N}_{l,l^{\prime\prime};m}{\cal N}_{l^{\prime\prime},l^{\prime};m}$ (85)

and the trace is over the orbital momenta. All expressions are diagonal in the
magnetic quantum number $m$. The kernels in 85 are

$\displaystyle{\cal
N}_{l,l^{\prime};m}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(86) $\displaystyle=\frac{1}{R^{3}}\sqrt{\frac{\pi}{2\xi
R}}K_{l^{\prime\prime}+\frac{1}{2}}(\xi d)H^{l^{\prime\prime}}_{l,l^{\prime}}$
$\displaystyle I_{l+\frac{1}{2}}(\xi R)I_{l^{\prime}+\frac{1}{2}}(\xi
R)\Phi^{-1}(i\xi)$

with the factors $H^{l^{\prime\prime}}_{l,l^{\prime}}$ resulting from the
matrix elements for the transition between the two centers. These are given,
e.g., by Eq. (10.129) in Bordag  _et al._ (2009b) or Eq. (5) in Teo  _et al._
(2011), and correspond to the $\cal U$ in Emig (2008), Eq. (2.25). For
$g\to\infty$, this formula turns into that for Dirichlet boundary conditions
on the spheres.

For us, the important property is that for $d\to\infty$, the leading order
comes from $l=l^{\prime}=l^{\prime\prime}=0$ and with $H^{0}_{0,0}=1$ we get

${\cal N}_{00}\raisebox{-5.0pt}{${\to\atop d\to\infty}$}\ \frac{g}{4\pi
d}E^{-\xi d}.$ (87)

The dependence on the radius $R$ of the spheres drops out in this limit. By
inserting this into 84 we get

$E_{0}\raisebox{-5.0pt}{${\to\atop d\to\infty}$}\
\frac{1}{2}\int_{0}^{\infty}\frac{d\xi}{\pi}\ln\left(1-\left(\frac{g}{4\pi
d}e^{-\xi d}\right)^{2}\right)$ (88)

for the interaction of two spheres carrying delta function potential in the
limit of large separation.

##  Appendix B The Casimir effect for two cylinders carrying delta function
potentials

The derivation of the vacuum energy for two parallel cylinders carrying delta
function potentials may be found, for example in Milton and Wagner (2008a) or
Milton and Wagner (2008b),

$\frac{E}{L}=\frac{1}{4\pi}\int\limits_{0}^{\infty}d\xi\,\xi\,{\rm
Tr}\ln(1-{\cal M}),$ (89)

with ${\cal M}$ given by (85) and

$\displaystyle{\cal N}_{l,l^{\prime};m}=\frac{gRK_{l+l^{\prime}}(\xi
d)I_{l^{\prime}}^{2}(\xi R)}{1+gRI_{l^{\prime}}(\xi R)K_{l^{\prime}}(\xi R)}.$
(90)

Here we compute the large distance limit of this vacuum energy. The leading
contribution for large separation $d$ comes from the s-wave. After the
substitution $\xi\to\xi/d$ in (89), the relevant matrix elements entering the
trace may be rewritten in the form

${\cal
M}_{ll^{\prime}}=\delta_{l}^{0}\delta_{l}^{0}\frac{g^{2}R^{2}K_{0}^{2}(\xi)I_{0}^{4}(\xi
R/d)}{(1+gRI_{0}(\xi R/d)K_{0}(\xi R/d))^{2}}.$ (91)

With allowance for the behaviour of the Bessel functions at small arguments,
$I_{0}(x)=1+O(z^{2})$ and $K_{0}(x)=-\gamma-\ln(x/2)+O(z^{2})$, one arrives at
the expression

$\displaystyle{\cal M}_{ll^{\prime}}=$ (92)
$\displaystyle\frac{\delta_{l}^{0}\delta_{l^{\prime}}^{0}K_{0}^{2}(\xi)}{\ln^{2}(R/d)}\left[1+\frac{\gamma+\ln(\xi/2)-(gR)^{-1})}{\ln(R/d)}\right]^{-2}+O((R/d)^{2}),$

which can be easily expanded in powers of small $1/\ln(R/d)$. This expansion
we substitute into 89 and in the leading order arrive at

$\frac{E}{L}=-\frac{1}{4\pi
d^{2}}\int\limits_{0}^{\infty}d\xi\;\xi\frac{K_{0}^{2}(\xi)}{\ln^{2}(R/d)}=-\frac{1}{8\pi
d^{2}\left(\ln(R/d)\right)^{2}},$ (93)

which coincides with 73 which was obtained from the chains.

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