# A quantum moving target segmentation algorithm for grayscale video

###### Abstract

The moving target segmentation (MTS) aims to segment out moving targets in the
video, however, the classical algorithm faces the huge challenge of real-time
processing in the current video era. Some scholars have successfully
demonstrated the quantum advantages in some video processing tasks, but not
concerning moving target segmentation. In this paper, a quantum moving target
segmentation algorithm for grayscale video is proposed, which can use quantum
mechanism to simultaneously calculate the difference of all pixels in all
adjacent frames and then quickly segment out the moving target. In addition, a
feasible quantum comparator is designed to distinguish the grayscale values
with the threshold. Then several quantum circuit units, including three-frame
difference, binarization and AND operation, are designed in detail, and then
are combined together to construct the complete quantum circuits for
segmenting the moving target. For a quantum video with $2^{m}$ frames (every
frame is a $2^{n}\times 2^{n}$ image with $q$ grayscale levels), the
complexity of our algorithm can be reduced to O$(n^{2}+q)$. Compared with the
classic counterpart, it is an exponential speedup, while its complexity is
also superior to the existing quantum algorithms. 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 video processing, QVNEQR, Moving target segmentation, Three-frame
difference, IBM Q

Wenjie Liu* Lu Wang Qingshan Wu

W. Liu, Q. Wu
School of Software, Nanjing University of Information Science and Technology,
Nanjing, 210044, Jiangsu, China
Email<EMAIL_ADDRESS>
L. Wang
School of Information Science and Engineering, Southeast University, Nanjing,
211189, Jiangsu, China
Email<EMAIL_ADDRESS>
##  1 Introduction

Compared with text, image and audio, video can contain more information and
convey information more intuitively, so the video processing is an important
part in the development of computer vision in this video era. As one of the
hot topics in video processing, moving target segmentation plays an important
role in computer vision, image applications, intrusion detection, abnormal
behavior detection, etc. However, as the data of the video increases, the
real-time problem gradually emerges. Quantum computing can achieve amazing
computing speed by virtue of its quantum advantage [1, 2], so the combination
of quantum computing and video processing can effectively solve the real-time
problem.

How to store classical video into qubits is a primary problem in quantum video
processing. Because each frame of the video is an image, we first need to
represent each frame as a quantum image. The current quantum image
representation methods mainly include the flexible representation of quantum
image (FRQI) [3], the multi-channel RGB images representation of quantum
images (MCQI) [4], the normal arbitrary superposition state of quantum image
(NASS) [5], the quantum probability image encoding representation (QPIE) [6],
and the novel enhanced quantum representation (NEQR) [7].

Based on these quantum image representation models, some quantum video
representation models have been proposed, such as the quantum movie (video)
representation model based on FRQI [8], the quantum video representation model
based on MCQI model [9] and the NEQR-based quantum video representation model
QVNEQR [10]. Among them, QVNEQR stores video information (frames and pixels)
in the base state of qubits, which makes the operation of pixels in video
simpler, so it has been widely used. At present, research on quantum video
mainly focuses on quantum video representation [8, 9, 10], moving target
detection[10, 11], quantum video stabilization [12] and quantum video
encryption [13, 14, 15, 16].

In 2015, Yan et al [11]. proposed a measurements-based moving target detection
(MMTD) algorithm in a quantum video based on the measurement method, which can
collapse the quantum video to a deterministic state, and then they observe the
measurement results to judge the motion trajectory. This is also the only work
to study quantum moving target detection algorithm to date. In the same year,
they [12] proposed a quantum video stabilization strategy and they used
quantum comparators and quantum image translation operations to estimate the
motion during exposure and to compensate for motion-induced video jitter. In
2020, Song et al. [16] proposed a quantum video encryption scheme based on the
new quantum video framework, the controlled-XOR operation and the improved
logic mapping. The whole encryption process is performed by inter-frame
permutation, intra-frame pixel position geometric transformation, and high
4-intra-frame-qubit-planes scrambling, which has high efficiency with simple
calculation, low complexity and strongly filtrates. All in all, the research
of quantum video processing is now in its initial stages, which may be
attributed to the need to deal with the large amount of resources encoded in
the frames. In the past decades, the quantum image segmentation, as an
important part of computer vision, has made great progress [17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 28], but the quantum video segmentation is still
scarce. In this paper, we use the three-frame difference method to design a
quantum counterpart and design a detailed quantum circuit. Unlike classical
methods, quantum methods can take advantage of the parallelism of quantum
computing to process the quantum superposition state pixels of the input
video, which greatly speeds up the classical counterpart.

In general, the main contributions of this work are as follows:

  * •

A moving target segmentation algorithm in quantum video is proposed, which can
use quantum mechanism to simultaneously perform subtraction operation on all
pixels in a grayscale video and quickly segment out the moving target.

  * •

A quantum comparator with lower quantum cost is designed to distinguish the
grayscale values with the threshold. Then, several specific quantum circuit
units, including three-frame difference, binarization and AND operation, are
designed in detail by using fewer qubits and quantum gates. And then based on
these units, the complete quantum circuit is assembled to segment the moving
target in quantum video.

  * •

We verify the superiority and feasibility of our proposed algorithm by
analyzing the circuit complexity and performing simulation experiment on IBM Q
[29], respectively.

The rest of this paper is organized as follows. Section 2 introduces the
principle of the NEQR model, QVNEQR model and the classical moving target
segmentation algorithm. In Sect. 3, some basic quantum operation modules are
introduced, then, a series of quantum circuit units are designed and some
relevant quantum states equations are given. Section 4 analyzes the circuit
complexity of our algorithm and the experiment result. Finally, the conclusion
is drawn in Sect. 5.

##  2 Preliminaries

###  2.1 Novel enhanced quantum representation (NEQR)

In classical computers, information is stored and processed using binary
numbers, and images are encoded in matrices that contain both position and
color information [30]. The NEQR [7] quantum image representation model
entangles the grayscale value information and position information of each
pixel in the classical image, and then uses a quantum superposition state to
store a complete classical image. Assuming that the size of a classical image
is $2^{n}\times 2^{n}$, and the grayscale range is $[0,2^{q}-1]$, the
grayscale value of each pixel needs to be represented by a binary sequence
$c_{q-1}c_{q-2}\cdots c_{0}$, $c_{m}\in\\{0,1\\}(m=0,1,\cdots,q-1)$. $c_{m}$
represents each bit of a binary sequence, and $m$ represents the position of
each bit. The specific form of the NEQR model is as Eq.1.

$\displaystyle\left|I\right\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{{c_{i}}}\right\rangle\left|i\right\rangle}=\frac{1}{{{2^{n}}}}\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{c_{q-1}^{i}c_{q-2}^{i}\cdots
c_{0}^{i}}\right\rangle\left|i\right\rangle},$ (1)

where $i$ represents the position of the pixels and the position information
of each pixel can be represented in the form shown in Eq. 2.

$\left|i\right\rangle=\left|Y\right\rangle\left|X\right\rangle=\left|{{Y_{n-1}}{Y_{n-2}}\cdots{Y_{0}}}\right\rangle\left|{{X_{n-1}}{X_{n-2}}\cdots{X_{0}}}\right\rangle,$
(2)

where $Y$ and $X$ respectively represent the ordinate information and abscissa
information of each pixel. $n-1,\cdots,1,0$ represent the position of the
n-bit binary sequence.

###  2.2 Quantum video based on NEQR (QVNEQR)

The QVNEQR model [10] is based on the NEQR model, and it combines multiple
NEQR images to represent a quantum video, so each frame of the QVNEQR video is
a NEQR image. Assuming that a video has $2^{m}$ frames, the size of each frame
is $2^{n}\times 2^{n}$, and the grayscale range is $[0,2^{q}-1]$, then the
QVNEQR model can be expressed as,

$\left|V\right\rangle=\frac{1}{{{2^{m/2}}}}\sum\limits_{j=0}^{{2^{m}}-1}{\left|{{I_{j}}}\right\rangle\otimes}\left|j\right\rangle,$
(3)

$\left|{{I_{j}}}\right\rangle=\frac{1}{{{2^{n}}}}\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{{c_{j,i}}}\right\rangle\otimes\left|i\right\rangle}=\frac{1}{{{2^{n}}}}\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{c_{q-1}^{j,i}c_{q-2}^{j,i}\cdots
c_{0}^{j,i}}\right\rangle\left|i\right\rangle},$ (4)

where $c_{s}^{j,i}\in\left\\{{0,1}\right\\}$, $s=0,1,\cdots,q-1$. The quantum
circuit of a quantum video is shown in Fig. 1, where $n$ and $m$ represent the
number of required qubits. If the video is a grayscale video, then $q=8$; if
the video is a RGB video, then $q=24$.

Figure 1: Quantum circuit of QVNEQR.

###  2.3 The classical moving target segmentation algorithm

Moving target segmentation is to find out the moving objects in the video. The
most common methods are frame difference method [31, 32, 33], background
difference method [34, 35] and optical flow method [36]. Among them, the frame
difference method is widely used because of its small calculation and high
real-time performance. The video captured by the camera is a continuous
sequence of images (frames), therefore, if there is no moving target in the
video, the change from frame to frame is extremely weak; if there is a moving
target in the video, the change from frame to frame will become very obvious.
Based on this principle, the frame difference method performs a subtraction
operation on several consecutive frames in the video. That is, subtraction
operations are performed between the pixels at the same position in different
frames, and when the absolute value of the grayscale value difference exceeds
a certain threshold, the pixel is determined as a part of the moving target.
Commonly used frame difference methods include two-frame difference method and
three-frame difference method, and the three-frame difference method combines
the results of two adjacent frame difference methods, which can effectively
avoid the ghosting problem of segmentation results and is widely used. Its
mathematical expression is shown below.

${d_{1}}(x,y)=\left\\{{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{1,}&{\left|{{f_{k}}(x,y)-{f_{k-1}}(x,y)}\right|}\end{array}\geq
T}\\\
{\begin{array}[]{*{20}{c}}{0,}&{\left|{{f_{k}}(x,y)-{f_{k-1}}(x,y)}\right|<T}\end{array}}\end{array}}\right.,$
(5)

${d_{2}}(x,y)=\left\\{{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{1,}&{\left|{{f_{k}}(x,y)-{f_{k+1}}(x,y)}\right|}\end{array}\geq
T}\\\
{\begin{array}[]{*{20}{c}}{0,}&{\left|{{f_{k}}(x,y)-{f_{k+1}}(x,y)}\right|<T}\end{array}}\end{array}}\right.,$
(6)

$D(x,y)={d_{1}}(x,y)\cap{d_{2}}(x,y),$ (7)

where $(x,y)$ represents the position information of each pixel in the frames
of the video (horizontal and vertical coordinates), $f_{k-1}(x,y)$,
$f_{k}(x,y)$ and $f_{k+1}(x,y)$ represent the $(k-1)$th, $k$th, and $(k+1)$th
frames in the video $(k=2,3,4,5,\cdots)$. The corresponding $d(x,y)$
represents the image after the absolute value subtraction of adjacent frames,
and $d(x,y)=1$ indicates that the pixel’s value at $(x,y)$ in the obtained
frame is 1, otherwise the pixel value at $(x,y)$ is 0. $T$ is the threshold
value (it is artificially set according to the actual situation) and $D(x,y)$
represents the result frame in the video.

##  3 The quantum moving target segmentation algorithm for grayscale video

###  3.1 Quantum operations

  1. (1)

Quantum comparator
The quantum comparator can compare the magnitude of two sequences of binary
qubits, and it takes two qubit sequences as input, and then outputs these two
sequences $a$ and $b$ and their comparison result $y_{1}y_{0}$. Based on the
idea of qantum bit string comparator (QBSC) [27], we designed a simpler
quantum comparator (QC). In the process of comparison, we check bit by bit in
order from high to low to see which binary number appears first as 1. If one
of the bits in the same position of both numbers is 1 and the other is 0, the
comparison is over and the result is output directly. Otherwise, the
comparison will continue until the last bit. The detailed comparator quantum
circuit is shown in Fig. 2, where $a$ and $b$ denote the two numbers to be
compared, $h$ denotes the auxiliary quantum bits, and $y_{1}y_{0}$ denotes the
comparison result. If $y_{1}y_{0}$=10, then $a>b$; if $y_{1}y_{0}=01$, then
$a<b$; if $y_{1}y_{0}=00$ or 11, then $a=b$. Compared with existing quantum
comparators, our proposed comparator requires fewer quantum gates and
auxiliary qubits, and the quantum cost is also lower, as shown in Table. 1.

Table 1:  Comparison of different quantum comparators. Quantum comparators | Auxiliary qubits | Quantum cost | Distinguishable state
---|---|---|---
Reference[27] | 3$n$-1 | 30$n$-15 | 3
Reference[39] | 1 | 12$n$-8 | 2
Reference[18] | 5 | 28$n$-15 | 3
Our comparator | 3 | 7$n$+6 | 3
Figure 2: Quantum circuit realization of quantum comparator (QC).

  2. (2)

Quantum subtractor
The quantum subtractor [28] we designed enables the subtraction operation of
two binary qubit sequences $a$ and $b$, and it takes $a$ and $b$ as input,
then takes the result of $a-b$ and $a$ as output. By $a_{0}-b_{0}$,
$a_{1}-b_{1}$, $\cdots$, $a_{n-1}-b_{n-1}$, the subtraction operation is
performed bit by bit, and the borrowing information is reserved for the next
bit, so that the subtraction operation can be completed using 3 auxiliary
qubits, as shown in Fig. 3. To prevent negative numbers during the
calculation, we use a combination of a quantum comparator, a quantum
subtractor, a Tofoli gate, four reset gates and $q$ CSWAP gates, as shown in
Fig. 4, so that it is always a larger number minus a smaller number.

Figure 3: Quantum circuit realization of quantum subtractor (QS). Figure 4:
Quantum circuit realization of quantum absolute value subtractor (QAS).

  3. (3)

Quantum copy operation
In order to make a copy of the original video, we need to use the copy
operation [37] to do so. It is implemented by several CNOT gates and auxiliary
qubits. The specific quantum circuit is shown in Fig. 5, where $|x\rangle$
represents the original value and $|0\rangle$ is used to store the copied
value.

Figure 5: Quantum circuit realization of copy.

  4. (4)

Cycle shift transformation operation
The cycle shift transformation (CT) operation [8] between frames in quantum
video allows all frames to be shifted cyclically so that every frame in the
video can get its neighboring frames at the same time. For example, if we
shift the video forward by one unit (CTs-), then the k-th frame in the video
becomes the (k-1)-th frame. The unitary operation for cyclic shifting of a
quantum video containing $2^{m}$ frames is shown below and the specific
quantum circuit is shown in Fig. 6.

$CT(j\pm)\left|V\right\rangle=\frac{1}{{{2^{m/2}}}}\sum\limits_{j=0}^{{2^{m}}-1}{\left|{{I_{j}}}\right\rangle\otimes}\left|{(j\pm
1)\bmod{2^{m}}}\right\rangle.$ (8)

Figure 6: The schematic diagram of the X-axis cyclic shift transformation.

###  3.2 The proposed algorithm

In order to segment the moving targets in quantum video, our proposed
segmentation algorithm can be described as follows (also shown in Fig. 7).

  1. Step 1:

Quantum Video $\left|V_{k}\right\rangle$ is prepared based on the QVNEQR
model, ${V_{K}}\xrightarrow{{QVNEQR}}\left|{{V_{K}}}\right\rangle$.

  2. Step 2:

The copy operation is used to copy $V_{k}$ and store it in the auxiliary
qubits for backup,
$\left|{{V_{K}}}\right\rangle{\left|0\right\rangle^{\otimes
q}}\xrightarrow{{copy}}\left|{{V_{K}}}\right\rangle\left|{{V_{K}}}\right\rangle$.

  3. Step 3:

The original video is cycle shifted forward by one frame (from frame $k$ to
frame $k-1$) by using a cycle shift transformation to obtain a new video
$\left|V_{k-1}\right\rangle$,
$\left|{{V_{K}}}\right\rangle\xrightarrow{{CT}}\left|{{V_{K-1}}}\right\rangle$.

  4. Step 4:

The quantum absolute value subtraction is used on the above two videos
($\left|V_{K-1}\right\rangle$ and $\left|V_{k}\right\rangle$), i.e. absolute
value subtraction for each pixel of each frame,
$\left|{\left|{{V_{K-1}}}\right\rangle-\left|{{V_{K}}}\right\rangle}\right|=\left|{{D_{K-1}}}\right\rangle$.

  5. Step 5:

The copy operation is used to reset $\left|V_{k}\right\rangle$,
$\left|{{V_{K}}}\right\rangle{\left|0\right\rangle^{\otimes
q}}\xrightarrow{{copy}}\left|{{V_{K}}}\right\rangle\left|{{V_{K}}}\right\rangle$.

  6. Step 6:

A new video $\left|V_{k+1}\right\rangle$ is obtained by shifting the original
video backwards by one frame (from frame $k$ to frame $k+1$) using a cycle
shift transformation,
$\left|{{V_{K}}}\right\rangle\xrightarrow{{CT}}\left|{{V_{K+1}}}\right\rangle$.

  7. Step 7:

The quantum absolute value subtraction is used on the above two videos
($\left|V_{k+1}\right\rangle$ and $\left|V_{k}\right\rangle$), i.e. absolute
value subtraction for each pixel of each frame,
$\left|{\left|{{V_{K+1}}}\right\rangle-\left|{{V_{K}}}\right\rangle}\right|=\left|{{D_{K+1}}}\right\rangle$.

  8. Step 8:

The result videos after two subtractions are binarized simultaneously,
$\left|{{D_{K\pm 1}}}\right\rangle\xrightarrow{{QB}}\left|{{b_{K\pm
1}}}\right\rangle$.

  9. Step 9:

The two videos after binarization are subjected to the AND operation and the
final result quantum video is obtained, $\left|{{b_{K-1}}}\right\rangle
AND\left|{{b_{K+1}}}\right\rangle\xrightarrow{{}}\left|V\right\rangle$.

  10. Step 10:

The quantum measurement operation is performed, then the classical image is
retrieved and the algorithm ends,
$\left|V\right\rangle\xrightarrow{{Measurement}}V$.

Figure 7: The workflow of our algorithm.

###  3.3 The quantum circuit implementation of the algorithm

  1. (1)

Quantum video preparation
If a video has $2^{m}$ frames, and each frame is an image of size $2^{n}\times
2^{n}$ with grayscale level $q$, then according to the QVNEQR model, we need
$2n$ qubits to store the position information and $q$ qubits to store the
grayscale value information of the pixels in each frame, besides, $m$ qubits
are also needed to store the frame information. Since we need to perform the
frame shift operation on the quantum video, the obtained video and the
original video share the position information, so we also need additional $2q$
qubits to store the obtained video ($V_{k-1}$ and $V_{k+1}$) and to copy the
original video ($V_{k}$). The quantum state expression is shown below.

$\left|V\right\rangle\otimes{\left|0\right\rangle^{2q}}=\frac{1}{{{2^{\frac{m}{2}+n}}}}\sum\limits_{j=0}^{{2^{m}}-1}{\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{{c_{j,i}}}\right\rangle\left|i\right\rangle}}\left|j\right\rangle{\left|0\right\rangle^{q}}{\left|0\right\rangle^{q}}.$
(9)

  2. (2)

Video frame cycle shift
Since all pixels in all frames of the video are in the superposition state,
the processing of a pixel is also the processing of all pixels in the video.
So we shift the video one frame forward and one frame backward respectively,
so that we have three videos ($V_{k-1}$, $V_{k}$, $V_{k+1}$) , and the
corresponding frames $K$ are the three adjacent frames of the original video
(frame $k-1$, $k$, $k+1$). This allows us to operate on all frames at once
when we perform the frame difference operation on a particular frame. The
expression for the quantum state of the quantum video set after the cycle
shift is shown as follows.

$\frac{1}{{{2^{\frac{m}{2}+n}}}}\sum\limits_{j=0}^{{2^{m}}-1}{\sum\limits_{i=0}^{{2^{2n}}-1}{\left|{{c_{j-1,i}}}\right\rangle\left|{{c_{j,i}}}\right\rangle\left|{{c_{j+1,i}}}\right\rangle\left|i\right\rangle}}\left|j\right\rangle.$
(10)

  3. (3)

Frame difference operation
After shifting the video, we have to perform an absolute value subtraction
operation (difference operation) on these three corresponding frames of the
video. This is also known as $|V_{k-1}-V_{k}|$ and $|V_{k+1}-V_{k}|$. The
operation specific to each pixel is the absolute value subtraction of the
pixels at the same position in the k-th frame of the three videos. Since all
pixels in the video are in the superposition state, our operation on one pixel
is also an operation on all pixels. The specific quantum circuit is shown in
Fig. 8.

Figure 8: Quantum circuit realization of Frame Difference.

  4. (4)

Binarization
In order to be able to reduce the amount of computation and quickly
distinguish the pixel points that have changed, we need to binarise the two
images after the frame difference operation. First, we compare the pixels to
the threshold $T$ by using a quantum comparator. If $c\geq T$, then the result
of the comparator is $y_{1}y_{0}\neq 01$. At this point, if $c_{0}=0$, then
$c_{0}$ is set to 1 by using the CNOT and Toffoli gates. If the result of the
comparator is $y_{1}y_{0}=01$, this means that $c<T$. At this point, if
$c_{0}=1$, then $c_{0}$ is set to 0 by using the CNOT and Toffoli gates. The
specific quantum circuit is shown in Fig. 9.

Figure 9: Quantum circuit realization of quantum binarization (QB).

  5. (5)

AND operation
After the above operations, we get two binary videos which contain the moving
targets between two adjacent frames, however, when the target is moving more
slowly, the ghosting will occur. Therefore, we perform the AND operation on
the two binary videos to accurately segment the moving targets, and we use the
Toffoli gate for this operation. This means that we decide that two targets
segmented from the video are moving targets only when they overlap. The
specific quantum circuit is shown in Figure. 10, and the complete quantum
algorithm circuit is shown in Figure. 11.

Figure 10: Quantum circuit realization of AND operation.

Figure 11: Quantum circuit realization of the Complete quantum algorithm.

Figure 12: The video including a moving target.(a,b,c and d are four
consecutive frames in the video. Figure 13: The probability histogram of the
resulting video

Figure 14:  The result video processed by our proposed quantum algorithm.
(a,b,c and d are four consecutive frames in the video, which include a moving
target maked with 1.)

Figure 15: The result video processed by the classical algorithm. (a,b,c and d
are four consecutive frames in the video, which include a moving target maked
with 1.)

##  4 Circuit complexity and experiment analysis

###  4.1 Circuit complexity analysis

The proposed algorithm uses a total of $q+2n+m+q+q+3$ qubits, as shown in
Figure 11. $C_{k}$ represents the grayscale value of each pixel in the
original video. Because the pixel qubits in the video are entangled with
position qubits, only $q$ qubits are required for a video with a grayscale
value of $[0,2^{q-1}]$. $P$ represents the position information of each
frame’s pixels in the original video, and the qubits are in superposition
state. Therefore, for all frames with size $2^{n}\times 2^{n}$, only $2n$
qubits are required to represent the position information of all pixels in all
frames. $S$ represents the position information of the frame, and its qubits
are also in superposition state. Therefore, for a video containing $2^{m}$
frames, only $m$ qubits are required to represent it. The remaining qubits are
used as auxiliary qubits to replicate the original video and form the
computational unit. Each replication only requires $q$ qubits to store the
grayscale value information, as pixel position information and frame position
information share qubits with the original video.

The complexity of a quantum circuit is determined by the number of underlying
logic gates in the circuit, and the entire quantum circuit for processing
quantum video is made up of several quantum logic gates. Each quantum gate can
be composed of single or double qubit gates. Therefore, we consider the
complexity of a single and double qubit gate as unit 1, so a NOT gate, a CNOT
gate, and a reset gate all have a complexity of 1. A Toffoli gate can be
composed of 5 two-qubit gates, so its quantum cost is 5 [1], and a CSWAP gate
has a complexity of 3 [39]. Supposing a quantum video has $2^{m}$ frames, each
of which has a size of $2^{n}\times 2^{n}$ and a grayscale range of
$[0,2^{q-1}]$. Therefore, we discuss the complexity of the quantum circuit in
the following parts.

Since there are no devices to obtain quantum video directly nowadays, we need
to convert classical video into quantum video, which consumes a lot of quantum
resources. But in quantum video processing or quantum image processing, we
usually ignore the complexity of this part because our algorithm is for
quantum video. This is just like classical algorithms which also do not take
into account the complexity of video preparation when calculating the
algorithm complexity. Therefore, the complexity of this stage is 0.

After the quantum video has been prepared, we need to copy the video into the
auxiliary qubits by using the copy operation and recover the video after the
adjacent frame difference. The copy operation is composed of $q$ CNOT gates,
so its complexity is O($q$).

In the algorithm we need to perform two CT operations on the video, which
include the forward and backward shift of each frame in the video. The CT
operation consists of a series of Toffoli gates with multiple control qubits,
so this part has a complexity of O$(n^{2})$ [39, 40].

The quantum absolute value subtraction (QAS) between two adjacent frames is
required in our algorithm and each frame difference is performed by an
absolute value subtractor. Each QAS is composed of a quantum comparator, a
quantum subtractor, a Toffoli gate, four reset gates and $q$ CSWAP gates and
their complexity is O($q$),O($q$), 5, 4 and O($q$) respectively, so that the
complexity of the QAS operation is O($q+q+5+4+q$)=O($q$).

Since the threshold is a known value, we set it using $q$ reset gates and $q$
NOT gates. The complexity of this part is O($q+q$)=O($q$).

The binarization operation requires a quantum comparator, two CNOT gates,
three Toffoli gates and $q+4$ reset gates, which have a complexity of O($q$),
2, 15, O($q+4$) respectively; hence the complexity of this part is
O($q+2+15+q+4$)= O($q$).

The AND operation is implemented by using a Toffoli gate with a complexity of
5 (O(1)). In order to compose the complete quantum circuit, we need to use
$4q$ reset gates at different places, which has the complexity of O($q$).

In summary, the complexity of our proposed quantum video segmentation
algorithm is O($q+n^{2}+q+q+q+1+q$)=O($n^{2}+q$). The classical video
segmentation algorithms need to process each pixel of each frame individually,
so the complexity is no less than O($2^{2n+m}$). Therefore, in comparison, the
performance of our algorithm can reach exponential improvement, which can be a
good solution to the real-time problem in the classical algorithm, as can be
seen in Table 2.

In addition, the existing moving target detection algorithm MMTD in quantum
video [11] is based on measurement. It uses MCQI model to encode quantum
image, and the complexity of image preparation is O$(2^{4n})$, where $n$ is
the number of quantum bits required. In order to detect moving targets in
quantum video with $2^{m}$ frames, the complexity of the algorithm is
O$(q\cdot 2^{m+4n})$. If the video preparation process is not considered, the
complexity is O$(q\cdot 2^{m})$, which is much higher than the complexity of
our algorithm. Since the development of segmentation algorithms in quantum
video is still immature, there are fewer related works. And the video is
composed of many images, therefore, we choose some quantum image segmentation
algorithms for comparison. For convenience, we abbreviate the segmentation
algorithms in Ref. [20] and Ref. [22] as EDCS and EDFS. EDCS employs the
classical Sobel operator for edge detection thereby segmenting the target in
the image. It requires gradient computation and threshold comparison for each
pixel, thus the complexity is O($n^{2}+2^{q+4}$). While EDFS uses an improved
Sobel operator to detect pixel gradients in more directions. In addition, non-
maximum suppression technique and double threshold detection technique are
also added into the algorithm so that the edges of the target can be segmented
more accurately. So the complexity of the algorithm is O($n^{2}+q^{3}$). From
the above comparison, it can be seen that even though our algorithm deals with
video with $m$ frame images, its complexity is much lower than the image
segmentation algorithms, which is very meaningful in this NISQ era.

Table 2: Complexity comparison of different algorithms. Algorithm | Application Scenario | Complexity
---|---|---
Classic algorithm[33] | Video | O($2^{2n+m}$)
MMTD[11] | Video | O($q\cdot 2^{m}$)
EDCS[20] | Image | O($n^{2}+2^{q+4}$)
EDFS[22] | Image | O($n^{2}+q^{3}$)
Our algorithm | Video | O($n^{2}+q$)

###  4.2 Experiment

To verify the practical feasibility of the quantum algorithm, we used the
quantum cloud platform provided by IBM Q (IBM Quantum Experience) [29]. The
environment created with the Qiskit [38] extension package and anaconda
compiles the Python language into the OpenQASM language, which can then create
quantum circuits, but because of the preciousness of qubit, we can only use a
quantum computer with 5 qubits, which is far from enough. So we can only use
the quantum simulator “ibm-qasm-simulator” to run our quantum circuits, which
has 32 qubits and supports all the quantum logic gates we use. For every task,
the run time of this simulator is limited to 10,000 second, and when used, it
is mainly called by the statement
“backend=IBMQ.get_provider(‘ibm-q’).get_backend(‘ibmq_qasm_simulator’)”.

For the currently available experimental conditions, we have chosen a video
containing $2^{2}$ frames with a fixed background and each frame has a size of
$2^{2}\times 2^{2}$ and a grayscale range of $[0,2^{3}-1]$, as shown in Fig.
12. The pixels marked in the figure are the moving targets, and the grayscale
of each pixel is marked in the figure in binary form.

We use the proposed algorithm to segment the moving targets in the video, and
because of the small range of grayscale values in the video, we set the
threshold value to 001, which can be adjusted according to the segmentation
effect in practice. In addition, to save experimental time, we only measured
the key qubits (the number of measurements is 1024), i.e. the frame position
$S$ of the video, the pixel position $P$ of each frame and the grayscale value
$C$. The measurement results are shown in Fig. 13. The horizontal coordinates
of the graph indicate the specific information of each pixel in the video and
some auxiliary qubits. The vertical coordinates indicate the measured
probability of each qubit sequence. Based on the probability histogram
information, we plotted the segmented quantum video as shown in Fig. 14. The
pixels with a grayscale value of 1 are the segmented moving targets. As can be
seen from the figure, our algorithm can accurately segment the moving targets
in the video.

Figure 15 shows the video processed by the classical algorithm, from which it
can be seen that the processing effect of the classical algorithm is the same
as that of our quantum algorithm. This is because our quantum algorithm
utilizes quantum mechanisms to accelerate classical algorithms and does not
change the way classical algorithm processes the video. Therefore, our
algorithm’s processing results is the same as that of classical algorithm.
However, our algorithm has an exponential improvement in complexity, which can
solve the real-time problem that occurs with the classical algorithm without
affecting the segmentation results.

##  5 Conclusion

Although the classical moving target segmentation algorithms in video are
mature, how to improve their processing efficiency by using quantum mechanisms
is still in its infancy in this video era. In this paper, a quantum moving
target segmentation algorithm for grayscale video is proposed, which can use
quantum mechanisms to process all pixels in the video at the same time and
segment out the moving target. In addition, a quantum comparator with lower
quantum cost is designed and a complete circuit for quantum video segmentation
is given. The circuit complexity and experiment analysis demonstrate the
superiority and feasibility of our algorithm.

When segmenting the moving target in a video, our algorithm is only adapted to
the video with a constant background, such as surveillance. However, in
practical applications, there are many scenarios that are variable, so it is
our future research direction to segment the video containing changing
background.

Acknowledgements

This work is supported by the National Natural Science Foundation of China
(62071240), the Innovation Program for Quantum Science and Technology
(2021ZD0302900), and the Priority Academic Program Development of Jiangsu
Higher Education Institutions (PAPD).

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

All data generated or analysed during this study are included in this
published article [and its supplementary information files].

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# Fast Ion Isotropization by Current Sheet Scattering in Magnetic Reconnection
Jets

Louis Richard<EMAIL_ADDRESS>Swedish Institute of Space Physics,
Uppsala, Sweden  Department of Physics and Astronomy, Space and Plasma
Physics, Uppsala University, Sweden  Yuri V. Khotyaintsev  Daniel B. Graham
Swedish Institute of Space Physics, Uppsala, Sweden  Andris Vaivads  Space and
Plasma Physics, School of Electrical Engineering, KTH Royal Institute of
Technology, Stockholm, Sweden  Daniel J. Gershman  NASA Goddard Space Flight
Center, Greenbelt, Maryland 20771, USA  Christopher T. Russell  University of
California, Los Angeles, California 90095, USA

###### Abstract

We present a statistical analysis of ion distributions in magnetic
reconnection jets using data from the Magnetospheric Multiscale spacecraft.
Compared with the quiet plasma in which the jet propagates, we often find
anisotropic and non-Maxwellian ion distributions in the plasma jets. We
observe magnetic field fluctuations associated with unstable ion
distributions, but the wave amplitudes are not large enough to scatter ions
during the observed travel time of the jet. We estimate that the phase-space
diffusion due to chaotic and quasi-adiabatic ion motion in the current sheet
is sufficiently fast to be the primary process leading to isotropization.

††preprint: APS/123-QED

Fast plasma flows, referred to as jets, are ubiquitous phenomena in the
universe [1, 2, 3]. The dynamical evolution of the plasma jets is thought to
be driven by changes in the magnetic field configuration (e.g., the Blandford-
Znajek mechanism [4]). Magnetic reconnection, in particular, is a topological
reconfiguration of the magnetic field, which results in the conversion of the
magnetic field energy into particle energy through particle acceleration and
heating [5, 6]. Complex non-Maxwellian ion velocity distribution functions
(VDFs) characterize collisionless plasmas [7]. For example, cold Alfvénic
counter-streaming ion beams entering into the reconnection exhaust [8, 9]
provide a source of free energy for ion beam-driven instabilities [10].
Simulations [11, 12] have shown that as the beams cross the reconnection jet,
heating of the beams results in the formation of a single VDF with temperature
anisotropy $R_{i}=T_{i\perp}/T_{i\parallel}<1$, where $T_{i\parallel}$ and
$T_{i\perp}$ are the ion temperature parallel and perpendicular to the local
magnetic field $\mathbf{B}$. Deviations from the thermodynamic equilibrium can
drive micro-instabilities [13], contributing to ion heating in the
reconnection region.

Spacecraft observations suggest wave generation in the reconnection jet due to
ion temperature anisotropy-driven instabilities [14, 15]. As the waves grow
and saturate, pitch-angle scattering due to wave-particle interaction limits
deviations from isotropy ($T_{i\perp}=T_{i\parallel}$). Simulations indicate
that the growth rate of the electromagnetic ion temperature anisotropy-driven
instabilities is much slower than the generation of the unstable VDFs [16,
12]. Thus, wave-particle interactions can provide isotropization of the ion
VDFs in the jets only after hundreds of ion gyroperiods, i.e, after the jet
has convected hundreds $d_{i}$ downstream of the reconnection site [12], with
$d_{i}=c/\omega_{pi}$ the ion inertial length. However, observations show a
dominance of isotropic ion VDFs already at distances $\sim 50~{}d_{i}$ from
the reconnection region [15]. This suggests that other mechanisms relax the
temperature anisotropy in the reconnection jets (e.g., current sheet pitch-
angle scattering [17, 18, 19, 20, 21, 22]). Investigating the processes
limiting the ion temperature anisotropy is thus crucial to understanding the
energy budget and partition in the reconnection jets.

In this Letter, we use data from the Magnetospheric Multiscale (MMS)
spacecraft [23] in the Earth’s magnetotail to study the temperature anisotropy
and non-Maxwellianity of ion VDFs in the reconnection jets. We use 516 jets
from a database of jets observed by MMS in the central plasma sheet (CPS) of
the Earth’s magnetotail ($\beta_{i}\geq 0.5$ [24], where
$\beta_{i}=n_{i}k_{B}T_{i}/P_{mag}$, $n_{i}$ is the ion number density,
$T_{i}$ the ion temperature and $P_{mag}=|\mathbf{B}|^{2}/2\mu_{0}$ is the
magnetic pressure) [25]. CPS plasma jets are thought to result from magnetic
reconnection or kinetic ballooning/interchange instability [26]. It is
reasonable to assume that the plasma jets are primarily reconnection outflows
since we observe the jets relatively close (the median distance is $\delta
x_{c}\approx 2.8~{}R_{E}$) to the statistical location of the reconnection
X-line in the near-Earth magnetotail at $X_{GSM}\approx-25~{}R_{E}$ in
geocentric solar magnetospheric (GSM) coordinates [25, 27]. The magnetic field
is measured by the FGM instrument [28], and the ion VDFs are measured by the
FPI-DIS instrument [29] with corrections removing a background population to
account for the penetrating radiations [30]. To increase the counting
statistics and thus reduce the measurement uncertainties, we average the
measurements across the four spacecraft and over a $450~{}\textrm{ms}$ (or
three ion VDFs) running time window (see Supplemental Material [31]).

Overall, the ion VDFs are predominantly isotropic $T_{i\perp}\sim
T_{i\parallel}$ both in the reconnection jets
($T_{i\perp}/T_{i\parallel}=0.9_{-0.3}^{+0.25}$) [Fig. 1a] and in the quiet
CPS ($T_{i\perp}/T_{i\parallel}=1.0_{-0.13}^{+0.09}$) [Fig. 1b]. However,
compared to the quiet CPS, a significant fraction of the ion VDFs in the jets
show a large deviation from $T_{i\perp}=T_{i\parallel}$ (see also Supplemental
Material [31]), indicating that magnetic reconnection generates anisotropic
ion VDFs.

Here we refer to the anisotropy in the sense of the gyrotropic, double-
adiabatic or Chew-Goldberger-Low (CGL) closure [32]. Indeed, quantifying the
non-gyrotropic anisotropy [33]
$A^{ng}=(\lambda_{2}-\lambda_{3})/(\lambda_{2}+\lambda_{3})$, where
$\lambda_{i}$ are the eigenvalues of the temperature tensor such as
$\lambda_{1}>\lambda_{2}>\lambda_{3}$, we observe that the ion VDFs are
generally gyrotropic [Fig 1c]. Hence, in what follows, we use the gyrotropic,
double adiabatic anisotropy $T_{i\perp}/T_{i\parallel}$.

Figure 1: Distribution of measurements. (a) and (b) Histograms of
$R_{i}=T_{i\perp}/T_{i\parallel}$ versus $\beta_{i\parallel}$ in the
reconnection jets and quiet CPS ($\beta_{i}>0.5$,
$|V_{i}|<100~{}\textrm{km}~{}\textrm{s}^{-1}$), respectively. The black line
in panels (a)-(b) indicates the median of $R_{i}$ as a function of
$\beta_{i\parallel}$, and the red curves indicate the 10th and 90th
percentiles. (c) and (d) Conditional average of non-gyrotropic anisotropy
$A^{ng}$ and non-Maxwellianity $\varepsilon_{i}$ in the
$(\beta_{i\parallel},R_{i})$ space. The black dash line in panels (a)-(d)
indicates the CPS threshold $\beta_{i}\geq 0.5$. (e) Histogram of
$\varepsilon_{i}$ versus $|B_{xy}|/B_{ext}$. (f) Histogram of
$\varepsilon_{i}$ versus $R_{i}$. The black line in panels (e)-(f) indicates
the median of $\varepsilon_{i}$ as a function of $|B_{xy}|/B_{ext}$ and
$R_{i}$ and the red curves indicate the 10th and 90th percentiles. Grey-shaded
bins in panels (a), (b), (e), and (f) indicate counts below
$5\sigma=5\sqrt{n}$.

Observations in the solar wind [34] have shown that the threshold
$\mathcal{R}_{i}$ for the ion temperature anisotropy driven instabilities
scales with $\beta_{i\parallel}=n_{i}k_{B}T_{i\parallel}/P_{mag}$ with:

$\mathcal{R}_{i}=1+\frac{a}{\left(\beta_{i\parallel}-\beta_{0}\right)^{b}}\,,$
(1)

where $a$, $b$, and $\beta_{0}$ are constants determined from linear Vlasov
analysis of the kinetic dispersion relation in a homogeneous, magnetized,
collisionless plasma with bi-Maxwellian ion VDFs [13]. We consider thresholds
corresponding to the growth rate $\gamma=10^{-2}\omega_{ci}$ for a proton-
electron plasma with isotropic electrons given in Ref. [35]. We observe that
$70\%$ of the ion VDFs lie within the instabilities thresholds, which, in
analogy to the solar wind [36], suggests that the temperature anisotropy is
regulated in the reconnection jets by wave-particle interaction. Still, we
will show below that this is not the case.

Fig. 2 presents an example of a reconnection jet observed by MMS on August 21,
2017. The large amplitude oscillations of the magnetic field indicate the
flapping of the magnetotail current sheet (CS) [37, 38, 39]. The flapping
enables sampling of the ion VDFs across the reconnection jet. We observe
$T_{i\perp}<T_{i\parallel}$ at the edges of the CS (local $|\mathbf{B}|$
maxima) and $T_{i\perp}>T_{i\parallel}$ at the CS center (local $|\mathbf{B}|$
minima) [Fig. 2c]. Such temperature anisotropies can result from two counter-
streaming beams VDFs at the CS edges [11, 12] and Speiser-like meandering ions
at the CS center [40, 12]. To investigate these large changes in temperature
anisotropy, we compute the eigenvalues [Fig. 2d] and the eigenvectors of the
temperature tensor [33]. We find that the eigenvectors are nearly constant
throughout the interval with $\mathbf{\hat{e}}_{1}=[-0.99,0.13,0.03]$, and
$\mathbf{\hat{e}}_{3}=[-0.01,-0.28,0.96]$ in GSM coordinates, and it is the
rotation of $\mathbf{B}$ with respect to the constant eigenvectors [Fig. 2e]
causing the large spikes in $T_{i\perp}/T_{i\parallel}$. Such behavior is
consistent with the demagnetization of the ions in the CS center.

Fig. 2f shows the ion non-Maxwellianity

$\varepsilon_{i}=(2n_{i})^{-1}\int|f_{i}-f_{i\mathrm{bM}}|\textrm{d}^{3}v\,,$
(2)

which is the normalized ($\varepsilon_{i}\in[0,1]$) zeroth-order moment of the
difference between the observed ion VDF $f_{i}$ and a bi-Maxwellian model
$f_{i\mathrm{bM}}$ with the same moments [41]. The ion counting statistics are
below the instrument noise floor at speeds $v<v_{Ti}/3$, where
$v_{Ti}=\sqrt{2k_{B}T_{i}/m_{i}}$, due to the high ion bulk energy. For this
reason, we compute $\varepsilon_{i}$ using energy channels $K_{i}>T_{i}/9$.
$\varepsilon_{i}$ peaks at the CS edges [Fig. 2f]. Simulations have shown that
at the exhaust boundaries, the ion VDFs typically consist of cold Alfvénic
counter-streaming beams [11, 12]. Such VDFs are non-velocity space-filling and
thus yield a large $\varepsilon_{i}$ consistent with our observations. In
contrast, $\varepsilon_{i}$ drops at the center of the CS, consistent with
Speiser-like meandering ions filling the velocity space and resulting in a
lower $\varepsilon_{i}$.

Figure 2:  Example reconnection jet with current sheet flapping motion. (a)
Magnetic field. (b) Ion bulk velocity. (c) Ion temperature anisotropy
$T_{i\perp}/T_{i\parallel}$, (d) ratios of the eigenvalues of the temperature
tensor $\mathbf{T_{i}}$ with $\lambda_{1}>\lambda_{2}>\lambda_{3}$. (e)
$\mathbf{\hat{e}_{1}}.\mathbf{B}$ and $\mathbf{\hat{e}_{3}}.\mathbf{B}$ where
$\mathbf{\hat{e}_{1}}$ and $\mathbf{\hat{e}_{3}}$ are the eigenvectors of
$\mathbf{T_{i}}$ associated with $\lambda_{1}$ and $\lambda_{3}$. (f) Non-
Maxwellianity $\varepsilon_{i}$. (g) Adiabaticity parameter $\kappa$ from 4
spacecraft method $\kappa=\sqrt{r_{c}/\rho_{i}}$ (green) and time scales ratio
$\kappa=\tau_{b}f_{ci}$ (blue).

We observe that $\varepsilon_{i}$ increases with decreasing
$\beta_{i\parallel}$ [Fig. 1d]. We estimate the background value
$\varepsilon_{i}=0.35\pm 0.09$ in the quiet CPS for comparison (see
Supplemental Material [31]). The same analysis repeated for a longer time
averaging windows (5, 7, and 9 ion VDFs) gives similar results (see
Supplemental Material [31]). Fig. 1e presents $\varepsilon_{i}$ as a function
$|\mathbf{B}_{xy}|/B_{ext}$, which is a proxy of the distance to the CS center
[42, 43], and where $B_{ext}=\sqrt{1+\beta_{i}}|\mathbf{B}|$ is the external
field obtained from pressure balance assumption [44]. We observe that
$\varepsilon_{i}$ increases with $|\mathbf{B}_{xy}|/B_{ext}$ [Fig. 1e],
meaning that deviations from a bi-Maxwellian are statistically more pronounced
at the CS edges similar to that seen in Fig. 2f. In addition, enhancements of
$\varepsilon_{i}$ are associated with large temperature anisotropy [Fig. 1d
and 1f], indicating that the magnetic reconnection drives both the temperature
anisotropy and non-Maxwellianity.

The non-equilibrium VDFs can be unstable to a range of instabilities. To
investigate this, we compute the magnetic field fluctuations
$|\delta\mathbf{B}|/B_{0}$ normalized to the background magnetic field
$B_{0}=B_{ext}/2$ [40]. To eliminate the large-scale fluctuations, such as
flapping, we high-pass filter $\delta\mathbf{B}$ at $f\geq 0.1~{}\mathrm{Hz}$,
which corresponds to the highest expected flapping frequency [38]. Since the
energy of the magnetic field fluctuations cascades to smaller scales, the
power is dominated by the fluctuations at $f\approx 0.1~{}\mathrm{Hz}$, i.e.,
assuming Taylor’s hypothesis applies [45], $k\rho_{i0}\approx 1.4\pm 0.8$
where $\rho_{i0}=v_{Ti}/\omega_{ci0}$ with $\omega_{ci0}=eB_{0}/m_{i}$. We
observe strong magnetic wave activity in concert with the unstable ion VDFs
[Fig 3a] consistent with previous observations [15, 14]. This enhanced wave
activity associated with unstable VDFs indicates the growth of the
instabilities in the reconnection jets.

The majority of the observed ion VDFs are nearly isotropic $T_{i\perp}\sim
T_{i\parallel}$ [Fig. 1a], meaning that there must be a physical mechanism
capable of reducing the anisotropy on time scales of the order of the time it
takes for a plasma parcel to travel from the reconnection region to the
spacecraft, which is the maximum existence time of the anisotropic ion VDFs.
For this to be the ion kinetic instabilities, they must first grow to large
amplitudes and then have sufficient time to scatter the ions. Assuming that
the reconnection outflow is Alfvénic, we estimate the travel time of the jet
as $\tau_{t}=\delta x_{t}/V_{A0}$, where
$V_{A0}=B_{0}/\sqrt{\mu_{0}n_{i}m_{i}}$ is the Alfvén speed and $\delta x$ is
the jet travel distance between the location of the spacecraft and the
statistical location of the reconnection X-line at $X_{GSM}=-25~{}R_{E}$ [27].
We normalize $\tau_{t}$ to $f_{ci0}=\omega_{ci0}/2\pi$,
$\tau_{t}f_{ci0}=(2\pi)^{-1}\delta x_{t}/d_{i}$. The regions beyond the
instabilities thresholds correspond to $\tau_{t}f_{ci0}\sim 10$, and slightly
higher $\tau_{t}f_{ci0}$ values in the region bounded by the thresholds [Fig.
3b]. Thus, for the instability to grow to a sufficiently large amplitude, the
growth rate needs to be very large, $\gamma>10^{-1}\omega_{ci}$, which
requires substantial temperature anisotropy [46]. There is a small fraction of
points with anisotropies compatible with such high growth rates for
$T_{i\parallel}>T_{i\perp}$, but none for $T_{i\parallel}<T_{i\perp}$.
Therefore it is unlikely the fluctuations can grow to sufficiently large
amplitudes.

The pitch-angle scattering time $\tau_{s}$ due to ion scale, $k\rho_{i0}\sim
1$, Alfvénic fluctuations can be estimated as
$\tau_{s}f_{ci0}\sim(|\delta\mathbf{B}|/B_{0})^{-2}$ [36, 47]. We find that
$\tau_{s}$ is typically much larger than the jet travel time $\tau_{t}$ [Fig.
3c], indicating that the wave amplitudes are not sufficiently large to scatter
ions over the travel time of the jet. In particular, the time required for
waves to scatter ions is $\tau_{t}f_{ci0}\sim\tau_{s}f_{ci0}$ and during that
time, the jet would travel a distance $\delta x\sim
2\pi(|\delta\mathbf{B}|/B_{0})^{-2}~{}d_{i}\sim 2\pi\times
10^{3}~{}d_{i}\approx 400~{}R_{E}$ which is much larger than the average
distance from reconnection region to spacecraft $\delta x_{t}\approx
2.8~{}R_{E}$. This suggests that another more efficient isotropization
mechanism is at play.

Figure 3: Comparison of diffusion time scales and travel time of the jet. (a)
Magnetic field fluctuation normalized to the background field. (b) The travel
time of the jet $\tau_{t}$. (c) Pitch-angle scattering time $\tau_{s}$ due to
Alfvénic fluctuations versus $\tau_{t}$. (d) Phase-space diffusion time
$\tau_{D}$ versus $\tau_{t}$. Grey-shaded bins in panels (c) and (d) indicate
counts below $5\sigma=5\sqrt{n}$.

The isotropization can be provided by scattering in the CS. Observed ion VDFs
are close to isotropic, and $\varepsilon_{i}$ is minimum at the CS center
[Fig. 2] suggesting efficient scattering in highly curved magnetic fields
($r_{c}\ll\rho_{i}$ where $r_{c}=|\mathbf{b}.\nabla\mathbf{b}|^{-1}$ is the
magnetic field curvature radius with $\mathbf{b}=\mathbf{B}/|\mathbf{B}|$).
Such scattering results from the breakdown of the conservation of the first
adiabatic invariant $\mu=m_{i}v_{\perp}^{2}/2|\mathbf{B}|$ of the ion motion,
which transitions to a chaotic regime, in the sense of exponentially diverging
ion phase-space trajectories [48], when the curvature radius $r_{c}$
approaches $\rho_{i}$, i.e., $\kappa=\sqrt{r_{c}/\rho_{i}}\sim 1$ [18, 21].
Alternatively, the ion motion in the CS can be described in terms of
competition between bouncing motion across the CS and gyromotion in the CS
around a finite normal magnetic field $B_{n}$ [49, 50]. Hence, the curvature
parameter $\kappa$ is equivalent to the ratio of bouncing and gyromotion time
scales $\kappa=\tau_{b}f_{ci}$ [21, 11], where
$\tau_{b}=2\pi/\omega_{b}=2\pi/\sqrt{v_{0}\omega_{ci0}/L}$ with $v_{0}$ the
characteristic particle velocity and $L$ the CS thickness. We estimate
$\kappa$ as the ratio of bouncing to the gyromotion time scale, assuming
$L\approx 20\rho_{i}$ the average CS thickness from statistical studies [51].
Since we observe $T_{i\parallel}>T_{i\perp}$ at the CS edges, we assume that
the ion population entering and escaping from the CS consist of two cold
Alfvénic counter streaming beams [8] so that $v_{0}=V_{A0}$. We also estimate
$\kappa$ using the multi-spacecraft methods [52, 53]. We find excellent
agreement between the two estimates indicating that the assumptions of
$v_{0}=V_{A0}$ and $L\approx 20\rho_{i}$ are reasonable.

We use the measured $\kappa$ as a proxy for the regime of ion motion. The ion
motion is adiabatic for $\kappa\gg 1$ with the magnetic moment $\mu$ conserved
when the gyromotion is faster than the bouncing motion. For $\kappa\sim 1$,
the ion motion becomes stochastic [18, 21, 54]. For $\kappa\ll 1$, the ion
motion is quasi-adiabatic with the generalized magnetic moment
$I_{z}=(2\pi)^{-1}\int p_{z}dz$ as an integral of motion, where $(z,p_{z})$
are the variables of the fast bouncing motion [55, 56, 21]. We observe in Fig.
2g that $\kappa\leq 1$ near the CS center, which implies that the ions are in
the chaotic and quasi-adiabatic regime in a thin $l\approx\rho_{i}$ [40] layer
in the vicinity of the CS center consistent with the observed demagnetization
of the ions in the CS center [Fig. 2e]. Consistent with previous observations
[57], we find $\kappa\leq 1$ for $73\%$ of the measured ion VDFs, which
implies that the majority of the ions in the reconnection jets are in the
chaotic and quasi-adiabatic motion regime.

We expect strong pitch-angle scattering in the CS in these regimes
($\kappa\leq 1$), which reduces the anisotropy and is related to the
destruction of $I_{z}$ [17, 18, 19, 20, 21, 22]. Studies of the Hamiltonian of
the ion motion in the CS showed that the characteristic phase-space diffusion
time is $\tau_{D}f_{ci0}\sim\kappa^{-3}$ [21, 50]. As the solution of the
diffusion equation is exponentially decaying in time, the steady ($99\%$)
state isotropic VDF is reached after $t=2\log(10)\tau_{D}$. Since the
effective ion pitch-angle scattering occurs in the thin
($l\approx\rho_{i}\approx L/20$) layer where $\kappa\leq 1$ [40], the
diffusion time is increased by a factor of 20 compared to scattering in the
entire thick CS. These yield the effective diffusion time $\tau_{eff}\approx
40\log(10)\tau_{D}\sim 100\tau_{D}$. From Fig. 3d, we find that $\tau_{D}\sim
0.01\tau_{t}$, so that the effective diffusion time is similar to the travel
time of the observed jets, $\tau_{t}\approx\tau_{eff}$. In particular, we find
that the effective characteristic scattering time is $\tau_{eff}\approx
18~{}\mathrm{s}\approx 3.3f_{ci0}^{-1}$. Thus the travel distance from the
X-line necessary to scatter the ions is $\delta x\sim 2\pi\times
3.3~{}d_{i}\approx 20~{}d_{i}\approx 1.5~{}R_{E}<\delta x_{t}\approx
2.8~{}R_{E}$. This indicates that the interaction with the CS can efficiently
scatter ions during the observed travel time of the jets.

In summary, we have presented a statistical investigation of the ion VDFs in
reconnection jets. Our results show that overall the observed ion VDFs are
isotropic ($T_{i\perp}/T_{i\parallel}=0.9_{-0.3}^{+0.25}$) but are driven out
of thermal equilibrium by magnetic reconnection compared to the quiet CPS.

Our primary finding is that the phase-space diffusion due to chaotic and
quasi-adiabatic ion motion in the CS is sufficiently fast to be the primary
process leading to isotropization. In particular, the travel distance needed
to scatter ions is only $\delta x\approx 20~{}d_{i}$. Similar isotropization
scales $\sim 10~{}d_{i}$ are found in simulations [58, 59]. Studies of the
Hamiltonian of the ion motion [60, 61] have shown that in the presence of a
guide-field such as at astrophysical plasma jets, the phase-space diffusion is
even faster, suggesting that for these environments the ion isotropization may
be very efficient.

The unstable VDFs excite micro-instabilities resulting in enhanced wave
activity, which is however too weak to explain the observed isotropization. We
observe an enhancement of wave activity for plasma conditions corresponding to
$\gamma\leq 10^{-2}\omega_{ci0}$ meaning that the instabilities generated
waves can grow to large amplitudes after $\sim 10-100~{}f_{ci0}^{-1}$. Then,
wave-particle interaction can isotropize ion VDFs after $\sim
10^{3}f_{ci0}^{-1}$. This implies that during this time, the jets would need
to travel a distance $\delta x\sim 2\pi\times 10^{3}~{}d_{i}$, much larger
than the observed distance to the X-line. Thus, wave-particle scattering is
too slow to explain the predominantly isotropic ion VDFs within the observed
jets. We note that the wave-particle scattering is more efficient in denser
environments, allowing the wave-particle interaction to scatter ions over
shorter distances compared with the system size (e.g., for coronal loops
$n_{i}\sim 1\times 10^{15}~{}\mathrm{m}^{-3}$, $d_{i}\sim 7~{}\mathrm{m}$
[62], so that $\delta x\sim 2\pi\times 10^{3}~{}d_{i}\sim 45~{}\mathrm{km}\ll
L_{loop}\sim 10^{5}~{}\mathrm{km}$).

MMS data are available at the MMS Science Data Center; see Ref. 111See
https://lasp.colorado.edu/mms/sdc/public.. Data analysis was performed using
the `pyrfu` analysis package 222See https://pypi.org/project/pyrfu/..

We thank the MMS team and instrument PIs for data access and support. This
work is supported by the Swedish National Space Agency Grant 139/18.

Supplemental Material for “Fast Ion Isotropization by Current Sheet Scattering
in Magnetic Reconnection Jets”

##  I Introduction

We present here additional material to support the analysis presented in the
Letter.

##  II Data distribution in reconnection jets and quiet plasma sheet

As discussed in the Letter, we observe that: (a) the reconnection jets and the
quiet CPS are overall isotropic, and (b) a large fraction of the VDFs in the
jets show anisotropy. To better illustrate the two aspects and to better
highlight the difference in the distributions for the reconnection jets
($2.6\times 10^{5}$ ion VDFs) versus the quiet CPS ($2.6\times 10^{6}$ ion
VDFs), we provide in Figure S1 histograms of $T_{i\perp}/T_{i\parallel}$ for
different ranges of $\beta_{i}=n_{i}k_{B}T_{i}/P_{mag}$ where $n_{i}$ is the
ion number density, $T_{i}$ is the ion temperature, and
$P_{mag}=|\mathbf{B}|^{2}/2\mu_{0}$ is the magnetic pressure. The histograms
of temperature anisotropies are centered around $T_{i\perp}=T_{i\parallel}$,
both in the reconnection jets and the quiet CPS. However, as seen in Figure 1
in the Letter and stated in the text, the spread in the distribution of
$T_{i\perp}/T_{i\parallel}$ is much larger for reconnection jets than in the
quiet CPS, meaning that a large fraction of the VDFs in the jets show
anisotropy.

Figure S1: Histograms of ion temperature anisotropy in the reconnection jets
and quiet CPS for different ranges of $\beta_{i}$. The black lines correspond
to the reconnection jets ($\beta_{i}>0.5,~{}V_{i}\geq
300~{}\mathrm{km}~{}\mathrm{s}^{-1}$ as defined in [24, 25]) and the red lines
to the quiet CPS $(\beta_{i}>0.5,~{}V_{i}\leq
100~{}\mathrm{km}~{}\mathrm{s}^{-1})$.

##  III Ion non-Maxwellianity

###  III.1 Effect of time-averaging

To understand the relative changes of non-Maxwellianity $\varepsilon_{i}$
across the reconnection jet, we must ensure that our estimate of
$\varepsilon_{i}$ is reliable in the sense that measurement uncertainties do
not cause the observed relative changes. The relatively low counting
statistics in the Earth’s magnetotail due to the low number density,
$n_{i}\sim 0.1~{}\mathrm{cm}^{-3}$, results in uncertainties in the ion VDFs,
$\delta f_{i}/f_{i}=1/\sqrt{n}$, where $n$ the number of counts measured by
the instrument assuming Poisson counting statistics. This can result in an
artificially large non-Maxwellianity $\varepsilon$. To verify the reliability
of the $\varepsilon$ estimate, we apply a test based on averaging over
different windows, which have been previously applied to electron VDFs in Ref.
[41].

Figure S2: Comparison of the different averaging windows for the flapping
event presented in Figure 2 in the Letter. (a) The magnetic field in
Geocentric Solar Magnetospheric (GSM) coordinates. (b) Non-Maxwellianity
$\varepsilon_{i}$ ion VDFs computed using different averaging windows.

To improve the counting statistics, we employ time-averaging over sliding
windows of $0.45~{}\mathrm{s}$, $0.75~{}\mathrm{s}$, $1.05~{}\mathrm{s}$,
$1.35~{}\mathrm{ms}$, and $1.65~{}\mathrm{s}$. These correspond to averaging
over 3, 5, 7, 9, and 11 ion VDFs, respectively. We apply the averaging to the
phase-space density $f_{i}(K_{i},\phi,\theta)$ for each energy $K_{i}$,
azimuthal angle $\phi$, and elevation angle $\theta$ bins of the FPI-DIS
instrument [29]. Figure S2 shows $\varepsilon_{i}$ for the different averaging
windows for the flapping event presented in the Letter. We observe that
$\varepsilon_{i}$ decreases with the increasing length of the averaging
window. This behavior is expected as the VDFs receive more smoothing, which
filters out the small velocity-space features. For the window longer than
$1~{}\mathrm{s}$ corresponding to averaging over seven measured ion VDFs, the
non-Maxwellianity converges toward similar values.

Figure S3: Comparison of the conditional average of the non-Maxwellianity
$\varepsilon_{i}$ for different averaging windows. The black dashed line
indicates $\beta_{i}=0.5$.

Comparing $\varepsilon_{i}$ estimated using the $0.45~{}\mathrm{s}$ and
$1.65~{}\mathrm{s}$ window (i.e., averaging over 3 vs 11 ion VDFs), we see
that the relative changes in $\varepsilon_{i}$ persist for different
averaging, indicating that these changes are physical. In particular, the
relative changes across the reconnection jet are similar for the different
averaging windows. Furthermore, in Figure S3, we present the conditional
average of $\varepsilon_{i}$ for different averaging windows. We see that the
relative changes of $\varepsilon_{i}$ between the different regions of the
$(\beta_{i}\parallel,T_{i\perp}/T_{i\parallel})$ phase-space persist for all
averaging window sizes. This indicates that the observed changes in
$\varepsilon_{i}$ across the reconnection jet, as described in the Letter, are
physical and not caused by measurement uncertainties.

###  III.2 Comparison to the quiet CPS

To provide a reference value for the non-Maxwellianity, we estimate
$\varepsilon_{i}$ in the quiet CPS using all ion VDFs available from 2017 to
2021 in burst mode ($150~{}\mathrm{ms}$). After time-averaging (see previous
section), it yields $5.7\times 10^{4}$ ion VDFs in the quiet CPS. Here we note
that we use burst mode ($150~{}\mathrm{ms}$ time resolution) to compare the
values of non-Maxwellianity using identical procedures with similar time-
averaging windows.

Figure S4: Comparison of data distribution and non-Maxwellianity in
reconnection jets (left) and quiet CPS (right).

We present the distribution of measured ion VDFs and the non-Maxwellianity in
the reconnection jets (left column) and the quiet plasma sheet (right column)
in Figure S4. As described earlier, the temperature anisotropy in the quiet
CPS is much more narrowly distributed compared to the reconnection jets.
Furthermore, we find that $\varepsilon_{i}$ has a weak dependence on
$\beta_{i\parallel}$, meaning that the quiet CPS $\varepsilon_{i}$ can be
summarized to its mean $\langle\epsilon_{i}\rangle=0.35$ and its standard
deviation $\sigma(\epsilon_{i})=0.09$. We use
$\langle\epsilon_{i}\rangle=0.35$ as a reference value of background non-
Maxwellianity.

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