# Measuring of fissile isotopes partial antineutrino spectra in direct
experiment at nuclear reactor

V.V. Sinev  Institute for Nuclear Research RAS, Moscow

###### Abstract

The direct measuring method is considered to get nuclear reactor antineutrino
spectrum. We suppose to isolate partial spectra of the fissile isotopes by
using the method of antineutrino spectrum extraction from the inverse beta
decay positron spectrum applied at Rovno experiment. This admits to increase
the accuracy of partial antineutrino spectra forming the total nuclear reactor
spectrum. It is important for the analysis of the reactor core fuel
composition and could be applied for non-proliferation purposes.

## Introduction

Energy spectrum of antineutrinos from nuclear reactor is a fundamental
characteristic of a reactor. When outgoing from a reactor antineutrinos
penetrate through any shielding. These particles carry out the information
concerning the chain reaction in the reactor core. Their spectrum is unique
for every reactor type and depends on the reactor fuel composition. That is
why just when the difference in energy spectra of the fuel components became
clear there appeared an idea of reactor control through the neutrino emission
[1].

There are mainly four fissile isotopes which undergo the fission in the core,
235U, 239Pu, 238U and 241Pu. Others give an input less than 1% and may be
neglected. Detector at some distance can detect the total flux from all these
components. But during the reactor operational run the shape of total spectrum
changes because of the burn up effect. So, one can know the fuel composition
of the reactor by fitting the total spectrum with the sum of four partial
spectra. But what are the uncertainties of these partial spectra.

At the beginning of the era of experiments with reactor neutrinos the partial
spectra of individual isotopes were calculated [2, 3], but the accuracy of the
calculations was not so good. Mainly because of insufficient knowledge of
fission fragments antineutrino spectra. Later the situation becomes better,
while the base of fragments was growing [4, 5, 6]. It becomes much better when
the first experimental spectra appeared. They were obtained by converting
measured beta-spectra from fissile isotopes [7, 8, 9]. Electron spectra were
measured at Grenoble in ILL by using magnetic spectrometer. These spectra are
accounted as the best. Their uncertainty in main spectrum part (2$-$7 MeV) is
3.8$-$4.2% at 90% CL.

In spite of high enough accuracy this is not sufficient, reactor control
demands at least 1control problem this accuracy needs to be improved.

Just now several International experiments are under preparation. They are
Double Chooz [10] in France, Daya Bay [11] in China and RENO [12] in Korea. In
all of them the detectors of a new generation will be used. These detectors
can give a possibility of obtaining high statistics while measuring
antineutrino spectrum. And the high statistics admits to isolate individual
fissile isotopes spectra and compare them with measured by conversion
technique. The appearing estimated uncertainty could be close to the
uncertainty of ILL spectra.

We know about some projects with a goal to measure spectra ratio for beta
particles of 235U and 239Pu [13], which can also help to understand better
spectra behavior. In the article we consider the method of isotopes spectra
isolation by using the direct measurement of positron spectrum from inverse
beta decay reaction. As a result we hope to obtain 235U and 239Pu antineutrino
spectra, which produce about 90% of total antineutrino flux of nuclear
reactor.

##  I Antineutrino registration

Antineutrino can be registered through the inverse beta decay reaction on
proton which has the largest cross section

$\bar{\nu_{e}}+p\rightarrow n+e^{+}.$ (1)

The positron appeared as a result of the reaction carries out practically all
antineutrino energy [14, 15]. Its kinetic energy is linearly connected with
antineutrino energy

$T=E-\Delta-r_{n},$ (2)

where $T$ \- positron kinetic energy, $E$ \- antineutrino energy, $\Delta$ \-
the reaction threshold equals to 1.806 MeV and $r_{n}$ is neutron recoil
energy.

So, the positron spectrum is the same as antineutrino’s, but shifted on 1.8
MeV and convoluted with cross section. Recoil energy in the first
approximation can be neglected.

##  II Antineutrino spectrum

Antineutrino spectrum is being formed inside the reactor core from a number of
beta-decays of fission fragments. Fragments are produced by several fissile
isotopes, not only 235U as it was considered at the beginning of the period of
searching neutrino. We know that the most part of fissions is produced by four
isotopes, they are 235U, 239Pu, 241Pu and 238U.

Figure 1:  Antineutrino spectra 235U (1), 239Pu (2), 241Pu (3) from [7-9] and
238U (4) from [18].

Each isotope antineutrino spectrum can be calculated using data base for
fissile fragments

$\rho_{\nu}(E)=\sum_{i}{y_{i}A_{i}(E)},$ (3)

where $y_{i}$ \- yield of $i$-th fragment in fission and $A(E)$ is its
antineutrino spectrum. But the accuracy of evaluation the spectrum (3) is
limited by the number of nuclei with unknown decay schemes (about 25% of total
number). They have half-life periods less then 0.3 seconds.

One can take antineutrino spectrum also by another method, by converting
experimentally measured beta-spectrum from fissile isotope. This method is
accounted for the present moment as the most exact. It was used in experiments
in ILL (Grenoble). In [7, 8, 9] the beta-spectrum was measured with high level
of accuracy for three isotopes 235U, 239Pu and 241Pu , which undergo fission
through absorption of thermal neutrons. Fission of 238U goes on only by
capturing fast neutrons and has a small fission cross section. Its high enough
rate of fissions is explained by the great mass of this isotope in the content
of nuclear fuel. They fit the measured beta spectra by a set of 30
hypothetical beta-spectra with boarding energies uniformly distributed from 0
up to high energy of experimental spectrum. The coefficients found (similar to
$y_{i}$ in (3)) were used in formula (3) for calculating the total
antineutrino spectrum.

These antineutrino spectra are accounted as a standard for the moment for data
analysis in experiments with reactor antineutrinos. They are presented at
figure 1.

There is also the third method of taking antineutrino spectrum, it is a method
of direct measuring. When one measures spectrum of positrons from reaction (1)
and extracts antineutrino spectrum. This method was realized in experiments of
Rovno group [16, 17]. The authors have obtained the antineutrino spectrum as
an exponential function with polynomial of 10-th power while solving the
equation

$S_{e}(T)=\int{\rho_{\nu}(E){\sigma}_{{\nu}p}(E)R(E,T)dE},$ (4)

where $S_{e}(T)$ $-$ positron spectrum from reaction (1), $\rho_{\nu}(E)$ $-$
antineutrino spectrum, $\sigma_{{\nu}p}(E)$ $-$ inverse beta decay reaction
cross section and $R(E,T)$ $-$ response function of detector. As a result we
have a formula for antineutrino spectrum

$\rho_{\nu}(E)=5.09\cdot exp(-0.648E-0.0273E^{2}-1.411(E/8)^{10}),$ (5)

This spectrum corresponds to some reactor fuel compositions like the following
one

${}^{235}{\rm U}-0.586,^{239}{\rm Pu}-0.292,^{238}{\rm U}-0.075,^{241}{\rm
Pu}-0.047.$ (6)

Response function was simulated by Monte Carlo method. This function
transforms positron energy spectrum appeared in (1) in experimentally observed
one. Each detector has its individual response function depending on detector
features. At figure 2 one can see response functions for some values of
positron kinetic energy for detector RONS used at Rovno experiment.

The control for simulated function was done by comparing the spectra measured
and calculated from some gamma sources (60Co and 24Na) which were placed in
the center and the periphery of the detector. Calibration of the detector was
made by the beta-source 144Ce$-^{144}$Pr with boarding energy 2997 keV.

##  III The extraction of 235U and 239Pu antineutrino spectra from measured
positron spectrum

During the reactor operational run antineutrino spectrum changes its shape. At
the beginning of run the spectrum is formed mainly by 235U ($\sim$60-70%) and
at the end of run the largest or equal with 235U part of fissions comes from
239Pu ($\sim$40-50%). This is a foundation for proposed method of extraction
the partial spectra.

Let’s regard the positron spectrum (1), which could be measured in a detector
placed in the vicinity of some nuclear reactor. At figure 3 one can see
positron spectra produced by pure isotopes of uranium and plutonium and real
spectrum corresponding to (6), which one can observe during reactor
operational run. This real spectrum takes place between spectra of 235U and
241Pu. But at the beginning of the run it will be closer to 235U and at the
end closer to 239Pu..

Figure 2:  Simulated response function for detector RONS, which was used in
Rovno experiments.

So, one can divide all the data measured during the reactor run into two parts
- “beginning” and “ending”. And we can use them for extracting individual
spectra.

Spectrum “beginning” contains 235U in larger proportion. Let’s account the
input of other fissile isotopes as a background and remove it.

The main question will concern the value of uncertainty of the spectrum. We
can suppose that the total uncertainty may be about 1% including 0.1-0.2%
statistics, as they account to achieve in modern experiments like Double
Chooz. Expected statistics may be about 106 events. In this case the main
error will come from the background where the greatest part will appear from
the spectra of other fissile isotopes.

We can write the total uncertainty of three background spectra like

$\sigma_{b}=\sum_{i=8,9,1}{\alpha_{i}\sigma_{i}}.$ (7)

While experimental spectrum is known with high accuracy ${\sigma}_{e}\sim$1%,
for extracted 235U spectrum we will get the error

$\sigma_{5}=\frac{1}{\alpha_{5}}\sqrt{\sigma_{e}^{2}+\sigma_{b}^{2}}.$ (8)

In (7) and (8) the letter $\alpha_{i}$ is assigned for individual parts of
fission. One can find values of $\alpha_{i}$, which we have used for the
estimations of uncertainties, in table 1.

Table 1: Parts of isotope fission at the beginning and end of reactor operational run Isotope |  235U |  239Pu |  238U |  241Pu
---|---|---|---|---
“beginning” | 0.65 | 0.25 | 0.07 | 0.03
“ending” | 0.35 | 0.50 | 0.08 | 0.04

It is important to note that the experimental error is not a constant for a
whole spectrum, it varies from bin to bin. At figure 4 one can see the
standard behavior of experimental uncertainty for restored spectrum. At
minimum it is equal to 1% as we supposed to get.

In table 2 we show achievable values of uncertainties for235U. One can compare
these errors with uncertainties for 235U spectrum from ILL shown in the second
column. To obtain the standard spectrum we can have an average of the both
spectra and this is shown in the fourth column.

Table 2: Expected relative uncertainty for 235U antineutrino spectrum (fission part 65%, see table 1) $E$m MeV |  ${\delta}S_{5}$ (ILL 68% CL) |  ${\delta}S_{5}(experim)$ (68% CL) |  ${\delta}S_{5}(average)$ (68% CL)
---|---|---|---
2.0 | 0.026 | 0.923 | 0.026
2.5 | 0.024 | 0.155 | 0.024
3.0 | 0.023 | 0.035 | 0.019
3.5 | 0.021 | 0.025 | 0.016
4.0 | 0.020 | 0.027 | 0.016
4.5 | 0.020 | 0.028 | 0.017
5.0 | 0.024 | 0.028 | 0.018
5.5 | 0.026 | 0.029 | 0.019
6.0 | 0.030 | 0.029 | 0.021
6.5 | 0.035 | 0.031 | 0.023
7.0 | 0.040 | 0.055 | 0.032
7.5 | 0.047 | 0.109 | 0.040
8.0 | 0.061 | 0.138 | 0.056
8.5 | 0.134 | 0.276 | 0.120
9.0 | 0.486 | 0.843 | 0.421
9.5 | 0.608 | 1.610 | 0.569

Similar table 3 is constructed for 239Pu, where we used uncertainty from the
last column of table 2, while applying the same procedure for this isotope for
experimental spectrum marked “ending”.

Table 3: Expected relative uncertainty for 239Pu antineutrino spectrum (fission part 50%, see table 1) $E$m MeV |  ${\delta}S_{9}$ (ILL 68% CL) |  ${\delta}S_{9}(experim)$ (68% CL) |  ${\delta}S_{9}(average)$ (68% CL)
---|---|---|---
2.0 | 0.027 | 1.200 | 0.027
2.5 | 0.026 | 0.204 | 0.026
3.0 | 0.026 | 0.055 | 0.024
3.5 | 0.026 | 0.048 | 0.023
4.0 | 0.027 | 0.057 | 0.024
4.5 | 0.029 | 0.064 | 0.026
5.0 | 0.032 | 0.074 | 0.029
5.5 | 0.036 | 0.074 | 0.032
6.0 | 0.041 | 0.090 | 0.038
6.5 | 0.045 | 0.100 | 0.041
7.0 | 0.067 | 0.183 | 0.063
7.5 | 0.116 | 0.229 | 0.103
8.0 | 0.213 | 0.438 | 0.191
8.5 | 0.486 | 0.266 | 0.234
9.0 | 0.578 | 1.320 | 0.529
9.5 | 0.608 | 2.340 | 0.588

Figure 3:  Positron spectra: 1 - 238U, 2 - 235U, 3 - 241Pu, 4 - 239Pu. Dashed
line shows spectrum corresponding to the fuel composition (6).

The last table 4 demonstrates what the values can be achieved while applying
the procedure to 238U. This spectrum we can get to know only from
calculations. There may be a good chance to take it experimentally.

Table 4: Expected relative uncertainty for 238U antineutrino spectrum $E$m MeV |  ${\delta}S_{8}$ (Vogel 68% CL) |  ${\delta}S_{8}(experim)$ (68% CL) |  ${\delta}S_{8}(average)$ (68% CL)
---|---|---|---
2.0 | 0.05 | $-$ | 0.05
2.5 | 0.06 | 1.270 | 0.06
3.0 | 0.06 | 0.301 | 0.06
3.5 | 0.08 | 0.192 | 0.074
4.0 | 0.10 | 0.188 | 0.088
4.5 | 0.10 | 0.187 | 0.088
5.0 | 0.10 | 0.186 | 0.088
5.5 | 0.10 | 0.195 | 0.089
6.0 | 0.10 | 0.197 | 0.089
6.5 | 0.10 | 0.195 | 0.089
7.0 | 0.20 | 0.294 | 0.165
7.5 | 0.20 | 0.508 | 0.186
8.0 | 0.30 | 0.778 | 0.280
8.5 | 0.40 | 0.914 | 0.366
9.0 | 0.70 | $-$ | 0.696
9.5 | 1.00 | $-$ | 0.997

Figure 4:  Experimental uncertainty of antineutrino spectrum after
transforming from positron spectrum. Suppose that systematic and statistical
errors in total do not exceed 1% in the most part of spectrum.

##  IV Conclusion

The discussed method of obtaining 235U and 239Pu antineutrino spectra can
improve uncertainties in their spectra. It is important for neutrino control
method. The using of different methods of spectra obtaining increases the
reliability of standard spectra. The accuracy in 3% in not enough today for
measuring the fuel composition by neutrino method, 1% seems desirable. In that
case statistics 2000$-$3000 events/day may be enough to have uncertainty 5%
for 235U part of fission per month of measurement.

From one hand this method is additional to other methods of spectra obtaining,
from the other hand it is direct and is not affected by some procedures of
recalculating like converting method. Also it does not depend on knowledge of
decay schemes as calculating method because it accounts all decays
automatically.

In collaboration with ILL spectra there may be achieved better accuracy for
standard spectra. Also there may be directly determined the 238U spectrum in
spite of the high uncertainty and compared to the calculated one.

Of cause the direct method is additional to other methods. These calculations
demonstrate the importance of continuation of the experiments on measuring
beta-spectra of the fissile isotopes and improving the conversation procedure
because it is the most exact. Direct measurement may serve as an indirect test
for conversion spectra. Calculations are very important for searching time
evolution of nuclear reactor antineutrino spectrum. All three methods improve
the complete knowledge of reactor antineutrino spectrum and its behavior
during the operational run. Altogether, the usage of several methods increases
reliability of partial spectra while proposing the standard spectra for the
future analysis. As an example we can attract the attention to hard part of
ILL spectra (higher than 8 MeV) where all spectra become the same, while
calculated spectra demonstrate growing difference in this region.

Proposed method on base of Rovno experience shows its applicability in the
analysis of the data in future Double Chooz experiment.

## Acknowledgments

Author thanks L.A. Mikaelyan and A.Ya. Balysh for useful discussions and
friendly criticism.

## References

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2212-

# Scalable, Simulation-Guided Compliant Tactile Finger Design

Sandra Q. Liu, Yuxiang Ma, and Edward H. Adelson  Massachusetts Institute of
Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Yuxiang Ma1,∗, Arpit
Agarwal2,∗, Sandra Q. Liu1,∗, Wenzhen Yuan3, and Edward H. Adelson1 1Yuxiang
Ma, Sandra Q. Liu, and Edward H. Adelson are with the Massachusetts Institute
of Technology<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Agarwal is with the Robotics Institute, Carnegie Mellon University
<EMAIL_ADDRESS>Yuan is with University of Illinois Urbana-
Champaign<EMAIL_ADDRESS>with equal contribution

###### Abstract

Compliant grippers enable robots to work with humans in unstructured
environments. In general, these grippers can improve with tactile sensing to
estimate the state of objects around them to precisely manipulate objects.
However, co-designing compliant structures with high-resolution tactile
sensing is a challenging task. We propose a simulation framework for the end-
to-end forward design of GelSight Fin Ray sensors [1]. Our simulation
framework consists of mechanical simulation using the finite element method
(FEM) and optical simulation including physically based rendering (PBR). To
simulate the fluorescent paint used in these GelSight Fin Rays, we propose an
efficient method that can be directly integrated in PBR. Using the simulation
framework, we investigate design choices available in the compliant grippers,
namely gel pad shapes, illumination conditions, Fin Ray gripper sizes, and Fin
Ray stiffness. This infrastructure enables faster design and prototype time
frames of new Fin Ray sensors that have various sensing areas, ranging from 48
$\times$ 18 to 70 $\times$ 35\. Given the parameters we choose, we can thus
optimize different Fin Ray designs and show their utility in grasping day-to-
day objects.

##  I Introduction

Making robotic grippers more compliant and adaptive allows us to generalize
grasping and robots’ ability in unstructured environments. Robot grippers with
integrated sensing have shown the ability to perform complex tasks [2, 1].
These grippers are integrated with high-resolution tactile sensing (640x480
texels) that allows in-hand rotation of mugs and manipulation of wine glass
stems.

However, designing compliant grippers with high-resolution tactile sensing is
a nontrivial problem. Generally, high-resolution tactile sensors are made of
rigid hardware components, which are difficult to incorporate into softer
robotic bodies. Furthermore, manufacturing a single gripper can take at least
a day predominantly due to elastomer curing times. This lengthy manufacturing
process makes it difficult to perform rapid prototyping and generalize to
different sizes, structures, and shapes of Fin Rays. In particular, we want to
be able to make different tactile sensing Fin Rays and visualize their sensing
regions before manufacturing one.

The ability to manufacture different Fin Rays is also useful from a
utilization standpoint. Smaller Fin Rays are able to navigate through narrower
spaces and singulate objects, while larger Fin Rays can apply more torque on
an object and prevent larger objects such as mugs from rotating in their
grasp. Having stiffer or softer Fin Rays can also help interact with bulkier
or more fragile objects. Overall, we want to be able to streamline the design
process through simulation of both mechanical and tactile structures and use
this process to design and manufacture a variety of Fin Rays.

As such, the main contributions of the work are three-fold:

  * •

We propose an end-to-end simulation pipeline, composed of FEM mechanical
simulation and optical simulation with efficient fluorescent modeling.

  * •

We propose an efficient fluorescent material calibration setup and fit a low-
dimensional fluorescent model for paints used in the Fin Ray gripper.

  * •

We show design variation along gel pad shape, illumination conditions, Fin Ray
gripper sizes, and Fin Ray gripper stiffness.

Figure 1: Simulation-guided Fin Ray finger design: New design of GelSight Fin
Ray fingers can be made, based on the results of FEM mechanical simulation and
optical simulation. The simulation provides a good prediction about the finger
stiffness and tactile sensing performance.

##  II Related Work

###  II-A Compliant Grippers with Sensing

Many different soft robotic grippers and tactile sensors exist. Recently,
there has been a surge of integrating tactile sensors with soft grippers,
which generally rely on their mechanical structure or material properties to
grasp a large variety of objects. Doing so enhances soft grippers so that they
can perform a myriad of tasks using sensory feedback. These tactile sensors
take on various form factors and generally provide lower-resolution tactile
sensing.

One popular method of sensing is using strain sensors, as Georgopoulou et al.
do in their 3D-printed soft robotic gripper. For ease of manufacturing, they
integrate strain sensor elements using multi-material printing and are able to
determine the size of the object the gripper is gasping [3]. Similarly, Chin
et al. incorporate an elastic strain sensor on their handed shearing auxetic
cylinders to provide proprioceptive information and add pressure sensors along
the “tactile” surface for haptic sensing [4].

Other methods of sensing involve using “self-sensing,” or sensing that
measures the change of the soft robotic structure or actuator itself. An
example of this concept is a shape-memory alloy compliant gripper, developed
by Ganapathy et al., which is also able to sense its own deformation [5].
Thuruthel et al. also designed a bistable soft gripper, which is able to
rapidly and passively close when a force is applied on its body [6].

Another way of obtaining contact or tactile sensing is through microphones. Li
et al. use microphones to pick up the resonant frequency of a hollowed-out
channel inside a pneumatic sensor with contact sensing holes [7]. However, all
of these tactile sensors are only able to determine low-resolution details
such as if contact exists, or how much force is being pressed along an
arbitrary part of the sensor.

Recently, some works have begun to integrate camera-based tactile sensors into
soft fingers, which allows for high-resolution contact sensing [8].
Nonetheless, incorporation of these sensors can be unwieldy in terms of
electronic hardware and many soft fingers require additional hardware to
actuate them. The Fin Ray is simple to actuate, can passively comply to many
objects, and with the incorporation of fluorescent paints and flexible
mirrors, the electronics hardware can be relatively simpler to integrate [9].

###  II-B Fin Ray Gripper Design

Fin Ray grippers are able to conform to complex shapes because of the Fin Ray
structure, which consists of two long fins and horizontal ribs connected
between them. The Fin Ray structure is inspired by the adaptive deformation of
fish fins, known as the Fin Ray Effect [10]. Fin Ray grippers have shown great
performance in grasping delicate and soft objects. They also have a higher
payload than most soft robotic grippers, such as soft pneumatic grippers [11].

Some researchers are devoted to designing Fin Ray fingers that are able to
grasp a wide range of objects. Crooks et al. improved the Fin Ray’s payload by
combining soft and hard plastics in their Fin Ray finger [12]. Shin et al.
proposed a Fin Ray gripper design with an improved weight capacity and an
expanded range of graspable geometries by attaching friction pads and enabling
parallel and centric gripping modes [13]. Elgeneidy et al. investigated the
passive stiffening of Fin Ray fingers due to layer jamming and presented a
soft Fin Ray finger that can perform both delicate and high-force grasping.
There is also research which optimized different Fin Ray structure parameters,
including number, thicknesses, and angles of the ribs, to obtain higher
grasping/wrapping quality [14, 15].

In our prior research on the GelSight Fin Ray finger, we first incorporated
vision-based tactile sensing into the Fin Ray finger by hollowing out the Fin
Ray skeleton [1]. The GelSight Fin Ray finger retains good compliance after
the integration, but the hollowed out finger structure leads to inadvertent
twisting motions. To improve the robustness of the finger, we designed the
GelSight Baby, in which we placed the camera at the finger base and
synthesized a silicone adhesive-based fluorescent paint to give the finger
better compliance [9].

In this work, we design Fin Ray fingers with different sizes and stiffness
values based on simulation results. Fin Ray structures can be easily scaled up
or down to grasp a variety of objects with varying forces and torques. Without
changing materials, the stiffness of Fin Ray fingers can be tuned by the rib
arrangement in the Fin Ray structure. Moreover, we are able to achieve various
stiffness by changing the rib number and modifying the width and thickness of
ribs. In our previous work [9], we have established FEM models to analyze the
compliance of different Fin Ray designs. In this work, we combine FEM
simulation with optical simulation to provide realistic previews of tactile
images before building prototypes. However, defining a new FEM model takes
several or more hours because the components of the GelSight Fin Ray finger
have complex geometries and contact relationships. Therefore, we established a
parametric modeling pipeline using the Python-based ABAQUS Scripting Interface
to massively generate FEM models.

###  II-C Tactile Sensor Simulation

The most recent works on tactile simulation focus on finding an efficient
approximation of soft body simulation by a penetration depth-based linear
model [16] or FEM-based [17]. In Xu et al’s paper [18], the authors built a
differentiable tactile simulator for marker motion. However, all the above
simulators are focused on contact dynamics simulation. In Si et al. [19],
authors developed contact dynamics based on a linear elasticity model and
tracking marker motions. They also generate images based on a few datasets
collected from a real sensor. However, none of the above work focuses on
performing optical simulation of novel sensors. In Agarwal et al. [20],
authors used physically based rendering for simulating GelSight [21]. Their
method is fully general and allows the simulation of an arbitrary sensor
without collecting any sensor-specific data. Our work also uses physically
based rendering with the addition of an approximate fluorescent rendering
technique. The fluorescent rendering is essential for an accurate simulation
of GelSight Fin Ray sensors.

##  III Methods

###  III-A Hardware

We scale the GelSight Baby Fin Ray so that the length of the pad is 70 and the
width of the pad is 35, shown by Fig. 2 3). This 35 $\times$ 70 Fin Ray size
is chosen so that the Fin Ray will be more useful for grasping and exerting
torque on larger objects and will create more surface area for a stabler
grasp. The thickness of the gel pad is also scaled accordingly with respect to
the gel pad width.

Figure 2: The family of GelSight Fin Rays: 1a) is our original one, and 1b) is
the GelSight Baby Fin Ray. 2) represents a box like Fin Ray, which can be more
useful for applying torques to long tool handles, while 3) represents two
versions of a longer and medium-width Fin Ray, on which we performed
simulations. 3a) represents a stiffer Fin Ray structure, while 3b) represents
a softer structure with thinner ribs.

All other manufacturing processes were kept the same. We print the Fin Ray
structure out of TPU 95A and include a semi-rigid Onyx backing (Markforged) to
help mitigate twisting of the hollowed-out Fin Ray. Laser-cut cloth is used on
the sides to prevent environmental lighting interference. A 120∘ field of view
Raspberry Pi camera, which sits at the base of the structure, views a 0.15
PET-G flexible mirror, which in turn reflects the tactile sensing area.

Because the tactile pad is thicker than the gel pads on the two prior GelSight
Fin Rays, we choose to use a softer gel pad. Using a softer pad will allow us
to still pick up on finer details in tactile resolution even if the silicone
is thicker. The gel pads are made using a mixture of 1 part XP 565 catalyst
(Silicones Inc.) to 15 parts XP 565 to 3 parts plasticizer (LC1550 Phenyl
Trimethicone, Lotioncrafter). We utilize the same aluminum cornflake silicone
mixture (35μ aluminum cornflake, Schlenk Offset FM/6500) for the tactile
sensing surface which is airbrushed onto our molds before we add the silicone
mixture. After curing, we paint the sides of the silicone gel pads with red or
green fluorescent acrylic paint and silicone adhesive (A-564, Factor II)
mixture. The Chanzon blue LEDs at the base of the Fin Ray illuminate the gel
pad and cause the paint to fluoresce, giving us a tri-color sensing region.

The silicone gel pads are created using various 3D-printed high-resolution
molds. For our cylindrical and flat molds, we adhere a 0.1 mylar sheet to the
bottom of the molds. Doing so allows us to have a smooth tactile sensing
surface. We are unable to do the same for our ellipsoid molds, but attempted
to put a thicker layer of aluminum silicone paint to counteract the visible
striations formed by the 3D-printed surface. We also note that this effect
could be mitigated by polishing or sanding the surface of the mold.

To show the usefulness of the simulation on other GelSight Fin Ray sizes, we
also use our GelSight Baby Fin Ray, a less stiff version of the 35 $\times$ 70
one, and another Fin Ray with a tactile sensing pad that is 50 in height and
60 in width. The Baby Fin Ray, as shown by Fig 2 1b), will be more useful in
smaller manipulation applications, while the wider Fin Ray in Fig 2 2) will be
able to apply more torque on different handheld tools. On the other hand, the
less stiff Fin Ray in Fig 2 3b) will be more useful for grasping softer
objects.

###  III-B Optical Simulation

Our optical simulation pipeline is based on physically based rendering (PBR),
which simulates light transport inside the sensor with accurate models of
light, refractive surface, fluorescent paint, and coating material. We built
on tactile simulation techniques introduced in Agarwal et al. [20] based on
PBR. One of the key differences in the GelSight Fin Ray is the use of
fluorescent paint for illumination. Therefore, in this section, we introduce
the fluorescent material model and simplified fluorescent simulation technique
in PBR. In the first part, we will describe the simplified model for
fluorescent paint and its calibration process. In the second part, we will
describe the simplified simulation model of fluorescence lights in the PBR
framework that allows us to simulate the full sensor.

#### Fluorescent paint calibration

For accurate simulation, we need to calibrate the fluorescent paints used in
the sensor. Our imaging setup consists of a CM3-U3-13Y3C-CS 1/2” Chameleon
color camera, 450 Blue Alignment Laser Diode Module (Edmund Optics) and 8
color filters with central wavelengths – 405, 450, 500, 532, 560, 600, 630,
and 660. We calibrated two fluorescent paints (Liquitex BASICS Acrylic Paint
Red Fluorescent ASIN B07F48YG5F and Liquitex BASICS Acrylic Paint Green
Fluorescent ASIN B07F48WZWL) made in a flat sample. The calibration setup is
shown in Figure 3. We assumed that the fluorescent paint is diffuse in nature
– for any incident direction, the amount of outgoing light radiance remains
the same.

Figure 3: Fluorescent model and calibration setup: (A) shows a canonical
fluorescent material model [22]. It consists of absorption and reemission
spectra whose peaks are separated by Stokes’ shift. (B) shows the imaging
setup we created to capture the reflectance at the excitation wavelength,
$\lambda=450\text{nm}$ for calibrating fluorescent paints used in GelSight Fin
Ray. Figure 4: Fluorescent paint fit: This shows the comparison of measured
emission spectra and simulated emission spectra using a 4D parametric model
for red fluorescent paint (left) and green fluorescent paint (right).

A fluorescent material could be characterized by absorption and emission
spectra. The first defines which incident light wavelengths are absorbed and
lead to reemissions. The second describes the amount of reemission across all
incident wavelengths. The difference between the spectral positions of the
band maxima of absorption and reemission is called a Stokes shift [23].
According to Zheng et al. [24], if the spectra is not very spiky, absorption
and emission spectra can be modeled by a 4-parameter analytic distribution, a
variant of skew Cauchy distribution. The spectra value at wavelength $\lambda$
is given by the function

$\text{f}(\lambda|\lambda_{0},\gamma,\omega,\text{h})=\\\
\frac{\text{h}}{\left[\gamma^{2}+(\lambda-\lambda_{0})^{2}\right]}\Biggl{\\{}\frac{1}{\pi}\arctan\biggl{[}\frac{\omega(\lambda-\lambda_{0})}{\gamma}\biggr{]}+\frac{1}{2}\Biggl{\\}}$
(1)

where $\lambda_{0}$ is the peak wavelength, h is height parameter, $\gamma$ is
width and $\omega$ is the skewness parameter. We manually fit the measured
data and choose Stoke’s shift for the paint to be 100nm and 50nm for red and
green fluorescent respectively, based on reflectance data. For calibrating the
non-fluorescent reflectance, we collected images in room light and matched
them to the closest color in a traditional colorchecker. We found the non-
fluorescent reflectance of the red and green fluorescent paint is very similar
to colorchecker Red and Green colors respectively.

#### Fluorescent simulation model

We found the fluorescent effect leads to reflectance of around 2% - 5% in the
desired wavelength. Additionally, it depends on the incident excitation
wavelength. For a fast approximate model, we created a textured light source
whose intensity is proportional to the distance from the center of the blue
light source. The color of the LED is chosen based on our calibration model in
the previous section. Pictorially, the fluorescent light source looks as shown
in Figure 3. Our components modeled in the optical simulation are shown in
Figure 5. Thereafter, we use GPU path-tracing to generate all the images.

Figure 5: Fluorescent rendering: This image shows the visual of the efficient
rendering of fluorescent paint lights using our parametric reflectance model
for simulating GelSight Fin Ray sensor.

###  III-C Mechanical Simulation

TABLE I: Constitutive models and element types of all components  Component | Material | Constitutive Model | Parameters | Element Type
---|---|---|---|---
Fin Ray skeletons | SainSmart TPU 95A | Ogden model [25] |  | $\mu_{1}=6.279\text{MPa},\mu_{2}=1.639\text{MPa}$
---
$\alpha_{1}=1.6663,\alpha_{2}=−7.136$ [25]
C3D8R
Silicone pad | PDMS | Neo-hookean model | $C_{10}=0.1333\text{MPa}$ | C3D8R
Base, backing | Onyx | Linear elastic model | $E=2.1\text{GPa},\nu=0.38$ | C3D8R
Flexible mirror | PET-G | Linear elastic model | $E=2.8\text{GPa},\nu=0.4$ | CSS8
Mylar sheet | Mylar | Linear elastic model | $E=5\text{GPa},\nu=0.38$ | CSS8
Indenters | Onyx | Rigid | N/A | R3D4
Ogden model is a hyperelastic model with Helmholtz free energy defined as
$\Psi=\sum^{N}_{p=1}\mu_{p}$/$\alpha_{p}*(\lambda^{\alpha_{p}}_{1}+\lambda^{\alpha_{p}}_{2}+\lambda^{\alpha_{p}}_{3})$
, where
$\mu_{p}$ and $\alpha_{p}$ are material constants, and $\lambda_{i},i=1,2,3$
are principle stretches. $E$ and $v$ are Young’s modulus and Poisson’s ratio
in linear elastic model.

We use FEM to simulate the mechanical interaction between Fin Ray fingers and
indenters. The deformation of Fin Ray fingers are used to evaluate the
fingers’ stiffness and provide deformed sensing pads for optical simulation.
The models are created in Abaqus (Simulia, Dassault Systemes).

#### Parametric modeling

Abaqus Scripting Interface provides a way to define and modify all modules of
an Abaqus model, such as parts, materials, interactions, loads, and meshes,
known as parametric modeling. Using the interface, we are able to select and
combine parts to create FEM models and analysis jobs automatically, as shown
in Fig. 6.

In a Fin Ray finger assembly, the two most important components are Fin Ray
skeletons and silicone pads. Different stiffness Fin Ray fingers are created
by modifying the rib distribution as well as the thicknesses and widths of the
skeletons, while better sensing illumination can be acquired by changing the
shapes of the silicone gel pads. Tie constraints are used to assemble the
selected Fin Ray skeleton and silicone pad with other structural components.
After that, we apply displacement load to a cylindrical or cuboidal indenter
to deform the Fin Ray finger. We chose these indenters as they provide a wide
range of surface normals and are easy to analyze. Surface contact is defined
for all possible possible contact pairs, e.g. between the silicone pad and the
indenter, between adjacent ribs of the skeleton.

Figure 6: Parametric modeling pipeline of FEM simulation: New FEM models can
be created by selecting different components from candidates. Fin Ray design
with optimal illumination and desired stiffness can be found by simulating
different combinations of Fin Ray skeletons and silicone pads.

Table I summarizes material constitutive models and element types of different
components in the Fin Ray design. Since Onyx, PET-G, and Mylar are about three
orders of magnitude stiffer than TPU 95A and PDMS, small deformation
assumption and linear elastic model are adopted for their corresponding
components. We assume indenters are rigid to simplify their interaction with
silicone pads, because soft-right contact is computationally less expensive
than soft-soft contact. While the indenters are fabricated using Onyx, their
minimal deformation does not result in noticeable displacements such as
backbone deflection, thereby supporting the assumption of rigidity. TPU 95A
and PDMS are soft nonlinear hyperelastic materials, which may experience large
deformation during grasping. Ogden model is able to fit the strain-stress
curves of commercial TPU filaments, and [25] offers the Ogden model parameters
of Sainsmart TPU 95A, while Neo-hookean model can effectively capture the non-
linear elasticity of PDMS [26]. C3D8R solid elements are assigned to TPU,
PDMS, and Onyx, CSS8 shell elements are assigned to the PET-G mirror sheet and
Mylar sheet, and R3D4 rigid elements are assigned to the indenter. To capture
more deformation details on the silicone pad, we mesh the silicone pad and the
Mylar sheet with a small element size of 0.4, while the other components are
meshed with bigger element sizes between 0.6 to 1.5.

The models are analyzed with a nonlinear static step, so that nonlinear
elasticity and nonlinear geometric contacts can be well simulated. A finer
mesh is used to verify that the simulation converges well. Deformed node
coordinates of the flexible mirror, the Mylar sheet, and the silicone pad are
exported to an Abaqus results file (*.fil), which is used to generate surface
meshes for optical simulation.

#### Mesh conversion

Optical simulation requires triangular surface meshes of the silicone pad, the
Mylar sheet, and the flexible mirror, but hexahedral volumetric meshes are
applied to these components in FEM. We established a mesh conversion protocol
to deal with this mesh mismatch. First, we use Abaqus2Matlab [27] to load node
coordinates as well as element information from the exported Abaqus results
file. After that, we find all boundary nodes by sorting node coordinates and
then find all boundary elements that include these nodes. Then, we find all
boundary quadrilateral faces of the hexahedral elements and divide these
quadrilateral surfaces into two triangular faces. Finally, the nodes in each
triangular face are resorted to keep the normal directions consistent. In this
way, we create high-quality triangular meshes for deformed gel pads.

##  IV Experiments

###  IV-A Shapes of Silicone Gel Pad

We consider three different types of silicone gel pad shapes: cylindrical,
ellipsoid, and flat (Fig. 7). These shapes were considered for ease of
manufacturing and simple form factor. More complex pad geometries were not
considered as there is currently no easy parametric way for us to design and
optimize them.

For the cylindrical silicone gel pads, we utilize three different radii
(denoted r). We also consider three different sets of radii (denoted a, b, and
c) for the ellipsoid gel pads. Ellipsoid shapes are created in CAD software by
non-symmetrically transforming a unit sphere. We then take an appropriately-
sized section (width W, length L, and thickness t) from the surfaces of these
shapes to create our silicone gel pads.

Figure 7: Different gel pad profiles: On top are the different gel pad
profiles we simulated, which are color-coded by the type of shape they have.
On the bottom is an example of how we create the shape of our ellipsoid gel
pads with the corresponding ellipsoid and gel pad dimensions. Figure 8: Gel
Pad variation test: The top row shows the real-world prototype images captured
with 20 cylinder indentation. The bottom row shows the simulation results for
the same setup using our approach.

###  IV-B Change in the illumination in simulation

Choosing where to place the light source and what should be the pose of the
light with respect to the gel pad is a trial-and-error process. Therefore, we
perform forward design studies using our optical simulation for a given
mechanical simulation to obtain the best illumination. Specifically, we tested
two variations – adding another light source at the top and changing the angle
of the light source at the bottom to assess the effect on the perception. For
the following experiments we chose the flattest ellipsoid gelpad.

#### Adding blue at the top

In this experiment, we added another blue light at the top and applied the
fluorescent effect appropriately to red and green paint lights. The effect can
be seen in Figure 9A. Having two lights increases the illumination uniformity
across the gel pad. As the gel pad shape becomes bigger we think that these
additional light sources will be very helpful.

#### Changing the light angle

We changed the light source panel incident angle with respect to the gel pad.
We plotted the intensity at a point near the indenter as shown in Figure 10
left. We found that angle=30∘ leads to the highest illumination at the
indenter and thus is the best design. Subsequently, the intensity value
sharply goes down for angles greater than 90∘.

Figure 9: Illumination design results: Part A shows two light conditions –
case 1 when only one blue light is placed at the bottom to illumination the
sensor and case 2 when two blue lights are placed, with one at the top and one
at the bottom of the sensor to illuminate the gel pad. Part B shows the cases
with different incident light panel angles with respect to the gel pad. Figure
10: Light intensity vs illumination angle: (Left) shows the point on the
indenter which is used for selecting the intensity value in an image. (Right)
shows the intensity plot for the red spot for images with different incident
angles of the blue light panel with respect to the gel pad.

###  IV-C Grasping Tasks

To show the usefulness of the different simulation-driven Fin Ray designs, we
manufacture the different Fin Rays based on the simulation results of the
differing gel pads. We end up with a total of three different sizes of Fin Ray
tactile sensing pads including the GelSight Baby Fin Ray one.

In particular, we show that these different Fin Ray shapes can be useful for a
variety of different grasping tasks. Smaller Fin Rays, for instance, can reach
into narrower spaces and singulate smaller objects. The widest and largest Fin
Rays can have a stabler grasp on longer objects, which require higher torque
to manipulate. Finally, because all of the Fin Rays are compliant, they can
also interact with softer, more fragile objects.

One phenomenon we noticed was that although the 35 $\times$ 70 pad visually
had the best tri-color sensing region, when we applied a similar gel pad to
the widest Fin Ray pad (50 width), the fluorescent colors were not as vivid.
However, we were still able to view the tri-color tactile deformation.

Overall, we successfully performed simple grasping tasks with all of the
different GelSight Fin Rays (Fig. 11). The GelSight Baby Fin Ray was able to
grasp a potato chip without crushing it, dig through a bowl of objects to
singulate a pecan, and grab tape from a tape dispenser. The widest Fin Ray is
able to hold a stirring spoon in a measuring cup full of particulates, and the
35 $\times$ 70 Fin Ray can grasp a variety of objects such as broccoli, a
mushroom, a potato, and a soda can.

Figure 11: Task performing: The Fin Ray family performing a variety of tasks
and their corresponding tactile images.

##  V Conclusion

This paper introduced the first end-to-end simulation framework for compliant
GelSight Fin Ray grippers with integrated high-resolution tactile sensing.
Through this framework, we are able to explore design spaces much faster than
previous methods of prototyping and manufacturing each prototype. We then
manufacture our best design with a much wider gel pad that enables grasping a
variety of objects - uneven plastic objects, fragile chips, and tiny spatulas.
We hope that our work leads to faster design and better adoption of the
GelSight Fin Ray for various use-cases.

##  VI Acknowledgment

This work is financially supported by the Toyota Research Institute and Amazon
Science Hub. The authors would also like to thank Yichen Li for his advice in
lighting design.

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Quantum state transfer on the complete bipartite graph
Raqueline A. M. Santos

Previously it was shown that (almost) perfect state transfer can be achieved on the complete bipartite graph by a discrete-time coined quantum walk based algorithm when both the sender and receiver vertices are in the same partition of the graph and when the sender and receiver are in opposite partitions of the same size. By changing the coin operator, we analyze the state transfer problem and we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Moreover, it is also possible to use an active switch approach using lackadaisical quantum walks where the marked vertex is switched between the sender and receiver during the algorithm.

§ INTRODUCTION

Quantum walks have been a successful tool used in many algorithmic applications, specially in search algorithms [10]. Another related problem is the task of performing quantum state transfer [2] between two vertices of a graph, often called sender and receiver. In this problem, we want to transfer with high probability a state localized initially on the sender vertex to the receiver vertex.

Quantum state transfer has been investigated on different graphs by means of both discrete-time and continuous-time quantum walks. For example, quantum state transfer can be performed on the line [15, 12],  cycle [9, 8, 1, 15, 12], lattices [6, 17], star graphs and complete graphs with loops [7], regular graphs [12, 16] and different types of networks [5, 3].
In the case of continuous-time quantum walk, it was shown by [4] that perfect state transfer can be achieved for almost any graph.

Closely related to the work in this paper, quantum state transfer by means of discrete-time coined quantum walk on the complete bipartite graph was analyzed in [18]. They considered two marked vertices, the sender and receiver. They proved that (almost) perfect state transfer between these two vertices can be achieved when both the sender and receiver are in the same partition and when the sender and receiver belong to opposite partitions of the same size. Later, Štefaňák and Skoupý [13] analyzed the same problem  on the complete $M$-partite graph with partitions of the same size, by using lackadaisical quantum walks. They showed
that when the sender and the receiver are in different partitions, the algorithm succeeds with fidelity approaching
unity for a large graph. And that is not the case when the sender and the receiver are in the same partition. They also proposed an algorithm with an active switch
based on a single marked vertex which achieves high fidelity for large graphs independent on the location of the sender and receiver vertices.

In this work, we analyze quantum state transfer between two vertices (sender and receiver) of the complete bipartite graph by means of discrete-time coined quantum walks. Previously, the same problem was analyzed by [18]. By using a different coin operator from [18], we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Next, we consider a similar approach to [13] and we describe a quantum state transfer algorithm with active switch, which is based on one marked vertex. We show that for large graphs, independent on the location of the sender and receiver, the algorithm succeeds with fidelity approaching
unity. We simulate the algorithm and we find out that high fidelity can also be achieved for small graphs.

The paper is organized as follows. In section <ref>, we describe the coined quantum walk on the complete bipartite graph and how the state transfer algorithm works. We analyze state transfer when the sender and receiver belong to different partitions in section <ref>, and when they are in the same partition in section <ref>. The quantum state transfer algorithm with active switch by means of lackadaisical quantum walks is described in section <ref>. Summary and conclusions are presented in section <ref>.

§ COINED QUANTUM WALK ON THE COMPLETE BIPARTITE GRAPH

Let $\Gamma(V_1,V_2,E)$ be a complete bipartite graph with $N_1 = |V_1|$ and $N_2 = |V_2|$, as in figure <ref>.

every node/.style=draw,circle,
every fit/.style=ellipse,draw,inner sep=-2pt,text width=2cm,
-,shorten >= 0pt,shorten <= 0pt

[every node/.style=]
decoration = calligraphic brace,
amplitude=5pt] (-0.35,0.1) –  (0.35,0.1)
decoration = calligraphic brace,
amplitude=5pt] (1.35,-0.4) –  (2.05,-0.4)

[start chain=going below,node distance=4mm]
ıin 1,2,...,4
[fsnode,on chain,minimum size=12pt,inner sep=0pt] (fı) ;

[xshift=1.7cm,yshift=-0.5cm,start chain=going below,node distance=4mm]
ıin 1,2,...,3
[ssnode,on chain,minimum size=12pt,inner sep=0pt] (sı) ;

(f1) – (s1);
(f1) – (s2);
(f1) – (s3);
(f2) – (s1);
(f2) – (s2);
(f2) – (s3);
(f3) – (s1);
(f3) – (s2);
(f3) – (s3);
(f4) – (s1);
(f4) – (s2);
(f4) – (s3);

A complete bipartite graph with $N_1 = 4$ and $N_2=3$.

Associated to the graph, we define the Hilbert space $\mathcal{H}^{2|E|}$ spanned by the basis states $\ket{v,u}$ which represents the particle located at vertex $v$ pointing to vertex $u$.
A step of the quantum walk is obtained by applying the evolution operator

\begin{equation}\label{eq:U}
U = SC.
\end{equation}

The coin operator $C$ is defined by

\begin{equation}
C = \bigoplus_{v}C_v.
\end{equation}

Since the vertices can have different degree, $C$ is defined as a direct sum, so that $C_v$ acts locally at vertex $v$.
The operator $S$ is the flip-flop shift which acts as

\begin{equation}
S\ket{v,u} = \ket{u,v},
\label{eq:shift}
\end{equation}

that is, the particle will move from vertex $v$ to vertex $u$ after its application.

State transfer using discrete-time quantum walks can be realized by considering two marked vertices, the sender $s$ and the receiver $r$. In this paper, we use the Grover diffusion operator $G$ as the coin operator on the non-marked vertices, and we apply $-G$ to the marked ones. The Grover coin acts on a vertex as

\begin{equation}
G_v = \ket{v}\bra{v}\otimes(2\ket{\mu_v}\bra{\mu_v}-I),
\label{eq:grover}
\end{equation}

\begin{equation}
\ket{\mu_v} = \frac{1}{\sqrt{d_v}}\sum_{u\sim v}\ket{u},
\end{equation}

and $d_v$ is the degree of vertex $v$.
Without loss of generality, the sender vertex is always located in the partition $V_1$. And we will consider both the cases when $r\in V_1$ and $r\in V_2$.
For simplicity, we will use the sub-index to identify which partition the vertex belongs to, that is, $v_j \in V_j, j\in\{1,2\}$. And we consider that the sum over all vertices of a partition $V_j$ is represented as $\sum_{v_j} \equiv \sum_{v_j\in V_j}$.

The idea of the state transfer algorithm is as follows. The initial state is localized at the sender vertex,

\begin{equation}
\ket{\psi(0)} = \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{s,v_2}.
\label{eq:psi0}
\end{equation}

We apply the evolution operator $t$ times. The state of the walker after $t$ steps is given by

\begin{equation}
\ket{\psi(t)} = U^t\ket{\psi(0)}.
\end{equation}

Then, we analyze the walk towards the target state, $\ket{target}$, localized at the receiver vertex. The fidelity between the state at time $t$ and the target state is given by

\begin{equation}
\mathcal{F}(t) = |\braket{\psi(t)|target}|^2.
\end{equation}

We expect that after a certain number of steps we can achieve state transfer with high fidelity. Following, we analyze the cases where the sender and receiver belong to different and to the same partitions.

§ SENDER AND RECEIVER IN DIFFERENT PARTITIONS

Consider that the sender and the receiver belong to different partitions, that is, the sender $s \in V_1$ and the receiver $r \in V_2$. See figure <ref> for an example.

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(f3) – (s2);
(f3) – (s3);
(f4) – (s1);
(f4) – (s2);
(f4) – (s3);

A complete bipartite graph with the sender vertex $s$ and receiver vertex $r$ in different partitions. In this case, $N_1 = 4$ and $N_2=3$.

For this case, the coin operator is described as follows. We apply $-G$ to the marked vertices and $G$ to the non-marked, that is, $C_s = -G_s$, $C_{v_1} = G_{v_1} (v_1\neq s)$, $C_r = -G_r$, $C_{v_2} = G_{v_2}(v_2\neq r)$, where $G_v$ is given by (<ref>). The evolution operator is $U_{s,r} = SC$.

Due to the symmetry of the graph, it is possible to find an invariant subspace in which the quantum walk takes place. Since at the end of the algorithm we will be interested in the state of the walker being at  the receiver vertex $r$, we consider the states that contains vertices of $V_2$ pointing to $V_1$, that is,

\begin{eqnarray}
\ket{\phi_1} &=& \ket{r,s},\\
\ket{\phi_2} &=& \frac{1}{\sqrt{N_1-1}}\sum_{v_1\neq s}\ket{r, v_1},\\
\ket{\phi_3} &=& \frac{1}{\sqrt{N_2-1}}\sum_{v_2\neq r}\ket{v_2, s},\\
\ket{\phi_4} &=& \frac{1}{\sqrt{(N_1-1)(N_2-1)}}\sum_{\substack{v_1\neq s\\ v_2\neq r}}\ket{v_2,v_1}.
\end{eqnarray}

The walk is bipartite, then it suffices to analyze the action of $U_{s,r}^2$ in this subspace,

\begin{eqnarray}
U_{s,r}^2\ket{\phi_1} &=& \frac{(N_1-2)(N_2-2)}{N_1N_2}\ket{\phi_1}+\frac{2\sqrt{N_1-1}(N_2-2)}{N_1N_2}\ket{\phi_2}-\frac{2(N_1-2)\sqrt{N_2-1}}{N_1N_2}\ket{\phi_3}-\nonumber\\
U_{s,r}^2\ket{\phi_2} &=& -\frac{2\sqrt{N_1-1}(N_2-2)}{N_1N_2}\ket{\phi_1}+\frac{(N_1-2)(N_2-2)}{N_1N_2}\ket{\phi_2}+\nonumber\\
U_{s,r}^2\ket{\phi_3} &=& \frac{2(N_1-2)\sqrt{N_2-1}}{N_1N_2}\ket{\phi_1}+\frac{4\sqrt{N_1-1}\sqrt{N_2-1}}{N_1N_2}\ket{\phi_2}+\frac{(N_1-2)(N_2-2)}{N_1N_2}\ket{\phi_3}+\nonumber\\
U_{s,r}^2\ket{\phi_4} &=& -\frac{4\sqrt{N_1-1}\sqrt{N_2-1}}{N_1N_2}\ket{\phi_1}+\frac{2(N_1-2)\sqrt{N_2-1}}{N_1N_2}\ket{\phi_2}-\nonumber\\
\end{eqnarray}

Therefore, $\{\ket{\phi_i}:i=1..4\}$ is an invariant subspace of $U_{s,r}^2$. By considering $\cos\theta_j = 1-\frac{2}{N_j}$ and $\sin\theta_j = \frac{2}{N_j}\sqrt{N_j-1}$ for $j\in\{1,2\}$, we can express $U_{s,r}^2$ in the invariant basis as

\begin{equation}
U_{red}^2 = \matrx{
\cos\theta_1\cos\theta_2 & -\sin\theta_1\cos\theta_2 & \cos\theta_1\sin\theta_2 & -\sin\theta_1\sin\theta_2\\
\sin\theta_1\cos\theta_2 & \cos\theta_1\cos\theta_2 & \sin\theta_1\sin\theta_2 & \cos\theta_1\sin\theta_2\\
-\cos\theta_1\sin\theta_2 & \sin\theta_1\sin\theta_2 & \cos\theta_1\cos\theta_2 & -\sin\theta_1\cos\theta_2\\
-\sin\theta_1\sin\theta_2 & -\cos\theta_1\sin\theta_2 & \sin\theta_1\cos\theta_2 & \cos\theta_1\cos\theta_2
\end{equation}

Diagonalizing $U^2_{red}$ we obtain that its eigenvalues are $\lambda = \{e^{\pm i\alpha},e^{\pm i\beta}\}$, where $\alpha = \theta_1+\theta_2$ and $\beta = \theta_1-\theta_2$. Their associated eigenvectors are

\begin{equation}
\ket{\pm\alpha} = \frac{1}{2}\matrx{
\mp i\\
\pm i\\
}, \ket{\pm\beta} = \frac{1}{2}\matrx{
\pm i\\
\pm i\\
\end{equation}

The initial state (<ref>) of the algorithm cannot be expressed in the invariant basis. We have to apply the evolution operator one time in order to obtain the walker at $V_2$.
Then, we have

\begin{equation}
\ket{\psi(1)} = U_{s,r}\ket{\psi(0)} = -\frac{1}{\sqrt{N_2}}\left(\ket{\phi_1}+\sqrt{N_2-1}\ket{\phi_3}\right).
\end{equation}

In the eigenbasis of $U_{red}^2$, we can express

\begin{equation}
\ket{\psi(1)} = \frac{1}{2\sqrt{N_2}}\left[\left(1+ i\sqrt{N_2-1}\right)\left(\ket{-\beta}-\ket{+\alpha}\right)+\left(1- i\sqrt{N_2-1}\right)\left(\ket{+\beta}-\ket{-\alpha}\right)\right].
\end{equation}

The state after $2t+1$ steps is

\begin{eqnarray}
\ket{\psi(2t+1)} &=& U_{s,r}^{2t}\ket{\psi(1)} =\nonumber\\
&=&\frac{1}{2\sqrt{N_2}}\left[\left(1+ i\sqrt{N_2-1}\right)\left(e^{-i\beta t}\ket{-\beta}-e^{i\alpha t}\ket{+\alpha}\right)+\right.\nonumber\\
&&+\left.\left(1- i\sqrt{N_2-1}\right)\left(e^{i\beta t}\ket{+\beta}-e^{-i\alpha t}\ket{-\alpha}\right)\right].
\label{eq:psit}
\end{eqnarray}

The target state is given by

\begin{equation}
\ket{target} = \frac{1}{\sqrt{N_1}}\sum_{v_1}\ket{r,v_1} = \frac{1}{\sqrt{N_1}}\left(\ket{\phi_1}+\sqrt{N_1-1}\ket{\phi_2}\right).
\label{eq:target}
\end{equation}

Calculating the fidelity between $\ket{\psi(2t+1)}$ and the desired target state, from (<ref>) and (<ref>), we obtain

\begin{eqnarray}
\mathcal{F}_1(2t+1) &=& |\braket{\psi(2t+1)|target}|^2\nonumber\\
\label{eq:fid-diff}
\end{eqnarray}

For achieving state transfer with the highest possible fidelity we must choose the number of steps to be the closest odd integer to the time when the fidelity reaches its maximum.

The fidelity (<ref>) is depicted in figure <ref> in the cases $N_1=N_2=100$ (<ref>) and $N_1=100,  N_2=35$ (<ref>). When the partitions have the same size, the fidelity has a nice behavior with peaks reaching 1. The first maximum occurs at $t\approx17.68$, in this case. For the algorithm we set $t=17$ steps, then $\mathcal{F}(17) = 0.9907$.  When the partitions have different sizes, the fidelity has peaks of different sizes. The maximum in the shown interval is $\approx0.9952$ achieved at $t\approx48.45$. For the algorithm we choose $t=49$ steps, then $\mathcal{F}(49) = 0.9835$.

$N_1=100, N_2=100$

$N_1=100, N_2=35$

The fidelity $\mathcal{F}_1$ in (<ref>) represented by the blue curve. The black dots are the values for the fidelity at odd steps (at even steps, the fidelity is zero). In (a) $N_1=N_2=100$, the first maximum value $1.0$ occurs at $t\approx17.68$. In (b) $N_1=100, N_2=35$, the maximum in the interval is approximately 0.9952 at $t\approx48.45$.

By making $N_1 = N_2$, the fidelity simplifies to

\begin{equation}
\mathcal{F}_1(2t+1) = \frac{1}{N_1^2}\left(\sqrt{N_1-1}\sin(\theta_1t)-\cos(\theta_1t)\right)^4.
\end{equation}

For this case,  the first maximum occurs at

\begin{equation}
T_{max} = 2\left(\frac{\pi-\arctan\sqrt{N_1-1}}{\theta_1}\right)+1,
\end{equation}

and (almost) perfect state transfer is achieved by choosing the number of steps as the odd integer closest to $T_{max}$.

Štefaňák and Skoupý [18] analyzed state transfer in the same setting using  the coin operator as $G$ on the non-marked vertices and $-I$ on the marked vertices. We refer to it as the $(G,-I)$ case. We  will compare it with our $(G,-G)$ setting.

§.§ Comparison between fidelities of $(G,-G)$ and $(G,-I)$ cases

For the $(G,-I)$ [18] case with $s$ and $r$ in different partitions, the fidelity is given by

\begin{equation}
\begin{split}
\mathcal{F}_2(2t+1) =& \frac{1}{N_1N_2(N_2+N_1-1)^2}\left[N_1N_2-(N_1-1)(N_2-1)\cos(\omega t)+\right.\\
&\left.+\sqrt{(N_1-1)(N_2-1)(N_1+N_2-1)}\sin(\omega t)\right]^2,
\end{split}
\label{eq:fid2}
\end{equation}

where $\omega = \arccos\left(\frac{N_1N_2-2N_1-2N_2+2}{N_1N_2}\right)$. In this case, it is possible to achieve (almost) perfect state transfer only in the case $N_1=N_2$.

Now we compare $\mathcal{F}_1$ (<ref>) and $\mathcal{F}_2$ (<ref>). Figure <ref> shows the fidelity $\mathcal{F}_1$ in solid blue and $\mathcal{F}_2$ in dashed red. When the partitions have the same size, both functions achieve the maximum value of $1$, as mentioned before. $\mathcal{F}_1$ can achieve the maximum with a fewer number of steps than $\mathcal{F}_2$, which can be observed in <ref> for $N_1=N_2=100$. The biggest difference between the two functions happens when $N_1\gg N_2$ (or the opposite, since the functions are symmetric). State transfer with high fidelity is still possible to be achieved for the case $(G,-G)$ when $N_1 \neq N_2$. For example, in figure <ref>,  $N_1=100$ and $N_2=10$, the maximum value for $\mathcal{F}_1$ is $\approx0.9902$ at $t\approx 16.74$ and for $\mathcal{F}_2$ the maximum value of $\approx0.3180$ is obtained at $t\approx 9.33$.

$N_1=100, N_2=100$

$N_1=100, N_2=10$

The fidelity $\mathcal{F}_1$ depicted by the solid blue curve and $\mathcal{F}_2$ by the dashed red curve. The solid dots represent the values for the fidelity at odd steps (at even steps, the fidelity is zero). In (a) $N_1=N_2=100$, both functions achieve the maximum value of $1.0$ at $t\approx17.68$ and $t\approx22.19$ for $\mathcal{F}_1$ and $\mathcal{F}_2$, respectively.  In (b) $N_1=100, N_2=10$, for $\mathcal{F}_1$ the maximum value of $\approx0.9902$ is obtained at $t\approx 16.74$; for $\mathcal{F}_2$ the maximum value of $\approx0.3180$ is obtained at $t\approx 9.33$.

Next, in figure <ref>, we depict the maximum value of the fidelity and the time to achieve it by fixing $N_1=100$ and varying the value of $N_2$ from 1 to 1000. The red dashed curve is the case $(G,-I)$. The expressions for the maximum of $\mathcal{F}_2$ and the time of maximum can be obtained analytically [18]. We can observe in <ref> how the curve starts to grow from $N_2=1$, reaches 1 at $N_1=N_2$, and starts again to decline for $N_2 > N_1$. For the case $(G,-G)$ depicted by the solid blue curve, we were not able to calculate an analytical expression for the maximum. However, by knowing the values for $N_1$ and $N_2$, we can easily calculate it numerically by using a global optimization method. For this case we obtain the maximum in the time interval $(0,5\sqrt{N_1+N_2})$. We observe we can achieve high fidelity even when $N_1\neq N_2$, but for not all values of $N_2$ in the calculated interval. The number of steps required can be higher for this case.

Maximum of fidelity

Time of maximum

The (a) maximum value of fidelity $\mathcal{F}_1$ and $\mathcal{F}_2$ and (b) time to achieve it, for  fixed $N_1=100$ and $N_2=\{1,...,1000\}$. The case $(G,-G)$ is depicted by the solid blue curve. The maximum is obtained by using numeric global optimization method in the time interval $(0,5\sqrt{N_1+N_2})$. The case $(G,-I)$ is depicted by the dashed red curve, the expressions for the maximum fidelity and time can be obtained analytically [18].

§ SENDER AND RECEIVER IN THE SAME PARTITION

Now consider the sender and the receiver in the same partition, that is, $r, s \in V_1$. See an example in figure <ref>.

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(f1) – (s1);
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(f1) – (s3);
(f2) – (s1);
(f2) – (s2);
(f2) – (s3);
(f3) – (s1);
(f3) – (s2);
(f3) – (s3);
(f4) – (s1);
(f4) – (s2);
(f4) – (s3);

A complete bipartite graph with the sender vertex $s$ and receiver vertex $r$ in the same partition $V_1$. In this case, $N_1 = 4$ and $N_2=3$.

As before, we apply $-G$ to the marked vertices and $G$ to the non-marked, that is, $C_s = -G_s$, $C_r = -G_r$, $C_{v_1} = G_{v_1}(v_1\neq\{s,r\})$, $C_{v_2} = G_{v_2}$. The quantum walk is driven by the evolution operator $U_{s,r} = SC$.

Again we can find an invariant subspace in which the quantum walk takes place. Since the receiver vertex $r$ belongs to the first partition, we consider the states that contains vertices of $V_1$ pointing to $V_2$, that is,

\begin{eqnarray}
\ket{\phi_1} &=& \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{s, v_2},\\
\ket{\phi_2} &=& \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{r, v_2},\\
\ket{\phi_3} &=& \frac{1}{\sqrt{N_2(N_1-2)}}\sum_{\substack{v_1\neq \{s,r\}\\ v_2}}\ket{v_1,v_2}.
\end{eqnarray}

The initial state for this case is exactly $\ket{\psi(0)} = \ket{\phi_1}$ and the target state $\ket{target} = \ket{\phi_2}$.
In this subspace, $U_{s,r}^2$ acts as

\begin{eqnarray}
U_{s,r}^2\ket{\phi_1} &=& \left(1-\frac{2}{N_1}\right)\ket{\phi_1}-\frac{2}{N_1}\ket{\phi_2}-\frac{2\sqrt{N_1-2}}{N1}\ket{\phi_3},\\
U_{s,r}^2\ket{\phi_2} &=& -\frac{2}{N_1}\ket{\phi_1}+\left(1-\frac{2}{N_1}\right)\ket{\phi_2}-\frac{2\sqrt{N_1-2}}{N1}\ket{\phi_3},\\
U_{s,r}^2\ket{\phi_3} &=& \frac{2\sqrt{N_1-2}}{N1}\ket{\phi_1}+\frac{2\sqrt{N_1-2}}{N1}\ket{\phi_2}+\left(1-\frac{4}{N_1}\right)\ket{\phi_3}.
\end{eqnarray}

Notice that its behavior is independent of the size of $V_2$. This gives us the same result as in case of the algorithm that uses $(G,-I)$ [18]. The reduced evolution operator $U^2_{r}$ is also the same as for the algorithm on the star graph [7]. From [7], the fidelity is given by

\begin{equation}
\mathcal{F}(2t) = |\braket{\psi(2t)|target}|^2 =  \sin^4\left(\frac{\omega t}{2}\right),
\label{eq:fid-same}
\end{equation}

\begin{equation}
\omega = \arccos\left(1-\frac{4}{N_1}\right).
\end{equation}

We plot the fidelity for $N_1 = 100$ in figure <ref>. The maximum value of $1$ for the fidelity is achieved at $2\pi/\omega$. Then, the number of steps of the algorithm is the closest even integer to $2\pi/\omega$ in order to achieve state transfer with the highest fidelity possible.

The fidelity (<ref>) for $N_1=100$. The black dots are the values for the fidelity at even steps (at odd steps, the fidelity is zero). The maximum value of $1$ is achieved at $t\approx 22.14$. Then, for the algorithm we choose $t=22$ steps, which gives us $\mathcal{F}(22) \approx 0.9998$.

§ ACTIVE SWITCH APPROACH

Štefaňák and Skoupý [13] proposed a state transfer algorithm with an active switch strategy based on a single marked vertex. At the beggining, the marked vertex is set as the sender vertex and after some number of steps, the marked vertex is switched to the receiver vertex. Their algorithm is applied to large complete $M$-partite graphs with $N$ vertices in each partition by using lackadaisical quantum walks with self-loop weight equal to 1. Therefore, the complete bipartite graph with $N_1=N_2$ is included in it. Following, we show that a similar active switch strategy also works when $N_1 \neq N_2$.

§.§ LQW on the complete bipartite graph

Lackadaisical quantum walks are defined by having a weighted self-loop attached to each vertex [14]. In our case, we set different weights for different vertices. The self-loop weight of the vertices in $V_1$ is $l_1$ and the self-loop weight of the vertices in $V_2$ is $l_2$. See figure <ref>.

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[->]  (f2) edge [loop below] (f2) node[left=12pt,black] $l_1$;
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[->]  (s1) edge [loop below] (s1) node[right=12pt,black] $l_2$;
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[->]  (s3) edge [loop below] (s3) node[right=12pt,black] $l_2$;

A complete bipartite graph with self-loops. The self-loop weights are $l_1$ and $l_2$ for the vertices in set $V_1$ and $V_2$, respectively.

The quantum walk is a discrete-time coined quantum walk (similar as in section <ref>), where the evolution operator is given by $U = SC$. The shift operator $S$ is the flip-flop shift (<ref>). The coin operator is the generalized Grover diffusion [14] given by

\begin{equation}
\hat{G}_v = \ket{v}\bra{v}\otimes(2\ket{\hat{\mu}_v}\bra{\hat{\mu}_v}-I),
\end{equation}

\begin{equation}
\ket{\hat{\mu}_v} = \frac{1}{d_v}\left(\sum_{u\sim v}\ket{u}+\sqrt{l_v}\ket{v}\right).
\end{equation}

Notice that the degree of each vertex is the number of unweighted edges plus the value of its self-loop weight, that is, $d_{v_1} = N_1+l_1$ and $d_{v_2} = N_2+l_2$.

Let us consider the case of one marked vertex $m$. The coin operator is $C_v = \hat{G}_v (v\neq m)$ and $C_m = -\hat{G}_m$.  Without loss of generality, let the marked vertex $m \in V_1$. As we did in the previous sections, we can find an invariant subspace

\begin{eqnarray}
\ket{\phi_1} &=& \ket{m,m},\label{eq:phi1}\\
\ket{\phi_2} &=& \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{m, v_2},\\
\ket{\phi_3} &=& \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{v_2,m},\\
\ket{\phi_4} &=& \frac{1}{\sqrt{N_2}}\sum_{v_2}\ket{v_2,v_2},\\
\ket{\phi_5} &=& \frac{1}{\sqrt{N_2(N_1-1)}}\sum_{v_2,v_1\neq m}\ket{v_2,v_1},\\
\ket{\phi_6} &=&  \frac{1}{\sqrt{N_2(N_1-1)}}\sum_{v_1\neq m, v2}\ket{v_1,v_2},\\
\ket{\phi_7} &=& \frac{1}{\sqrt{(N_1-1)}}\sum_{v_1\neq m}\ket{v_1,v_1},\label{eq:phi7}
\end{eqnarray}

and obtain a reduced matrix by expressing $U_m$ in this basis. Although
it is possible to diagonalize the reduced matrix analytically in this case, it  results in quite lengthy expressions. We will use the results obtained by
Rhodes and Wong [11], who analyzed the problem of search for marked vertices by using lackadaisical quantum walks on the complete bipartite graph. They proved that for large $N_1$ and $N_2$, the success probability of the search algorithm for the case of one marked vertex is improved when $l_1 = N_2/2N_1$ (independent of $l_2$) and the
initial state is stationary under the quantum walk.

The stationary state is given by

\begin{eqnarray}
\ket{\sigma} &=& \frac{1}{\sqrt{2N_1N_2+l_1N_1+l_2N_2}}\left[\sum_{v_1}\left(\sqrt{l_1}\ket{v_1,v_1}+\sum_{v_2}\ket{v_1,v_2}\right)+\right.\\
&&\left.+\sum_{v_2}\left(\sqrt{l_2}\ket{v_2,v_2}+\sum_{v_1}\ket{v_2,v_1}\right)  \right]
\end{eqnarray}

which satisfies $U\ket{\sigma} = \ket{\sigma}$. Notice that $U$ is the evolution operator without marked vertices.

In [11], they obtained the success probability of finding a marked vertex. In our algorithm, we will be interested in the fidelity with the target vertex, which in this case is the loop on the marked vertex, $\ket{target} = \ket{m,m}$. Therefore, we need to calculate

\begin{equation}
\mathcal{F}(t) = |\braket{target|U_m^t|\sigma}|^2 = |\braket{m,m|U_m^t|\sigma}|^2.
\end{equation}

For large $N_1$ and $N_2$, we can express $\ket{\sigma}$ as

\begin{equation}
\ket{\sigma} = \frac{1}{\sqrt{2}}\left(\ket{\phi_4} + \ket{\phi_5}\right).
\end{equation}

In the asymptotic eigenbasis of $U_m$ (which was obtained by [11] and can be found in Appendix <ref> for this case), we have

\begin{equation}
\ket{\sigma} = -\frac{1}{\sqrt{2}}\ket{\psi_1}+\frac{1}{2}\ket{\psi_2}+\frac{1}{2}\ket{\psi_3}.
\end{equation}

Then, we obtain

\begin{equation}
U_m^t\ket{\sigma} = -\frac{1}{\sqrt{2}}\ket{\psi_1}+\frac{1}{2}e^{-i\theta t}\ket{\psi_2}+e^{i\theta t}\frac{1}{2}\ket{\psi_3},
\label{eq:umt}
\end{equation}

\begin{equation}
\theta = \arcsin\sqrt{\frac{2}{N_1}}.
\end{equation}

From (<ref>) we can easily calculate

\begin{equation}
\mathcal{F}(t) = |\braket{target|U_m^t|\sigma}|^2 = |\braket{m,m|U_m^t|\sigma}|^2 = \frac{1}{4}(\cos\theta t -1)^2
\label{eq:fid-lqw}
\end{equation}

Then the fidelity reaches 1 at $t=\pi/\theta$. This is exactly the same amount of time obtained by [11] so that the success probability reaches one. This means that in the search algorithm, the probability on the marked vertex is contained in the self-loop.

§.§ The state transfer algorithm with active switch

We have seen that we can start at the stationary state and reach the target state after a certain amount of time. Or we can think in the opposite way, since the walk is periodic. That is, we can start in the target state and reach the stationary state after some time. Our quantum state transfer algorithm with active switch is described as follows.

The self-loop weights are set to $l_1 = N_2/2N_1$ and $l_2=N_1/2N_2$.
The initial state of the algorithm is at the sender loop, that is,

\begin{equation}
\ket{\psi(0)} = \ket{s,s}.
\end{equation}

Then, we apply the quantum walk $U_s$ (the sender vertex is the marked vertex) for $T_1 =
round\left(\frac{\pi}{\theta_s}\right)$ steps, where

\begin{equation}
\theta_{v_j} = \arcsin\sqrt{\frac{2}{N_j}}.
\end{equation}

At this point the current state

\begin{equation}
\ket{\psi(T_1)} = U_s^{T_1}\ket{\psi(0)}
\end{equation}

has high fidelity with the stationary state $\ket{\sigma}$ (from (<ref>)). Now we switch the marked vertex to the receiver vertex, obtaining the operator $U_r$. The quantum walk is evolved for another $T_2 = round\left(\frac{\pi}{\theta_r}\right)$ steps. At the end, we have the state

\begin{equation}
\ket{\psi(T_1+T_2)} = U_r^{T_2}\ket{\psi(T_1)} = U_r^{T_2}U_s^{T_1}\ket{\psi(0)},
\end{equation}

which has high fidelity with the target state $\ket{r,r}$.

In this case, differently from what happens in the active switch algorithm of [13], we have to know in which partition the sender and receiver vertex belong to in order to correctly set the number of steps $T_1$ and $T_2$. Also notice that since we have to round the time so the number of steps is an integer,  the  algorithm  succeeds  with  fidelity
approaching unity for large $N_1$ and $N_2$.

$s\in V_1$, $r\in V_2$

$s\in V_1$, $r\in V_1$

The fidelity with the target state $\ket{r,r}$ at the end of the state transfer algorithm with active switch.

High fidelity state transfer can also be achieved for small graphs. In figure <ref>, we simulate the algorithm for a certain range of $N_1$ and $N_2$ and we plot the fidelity with the target state $\ket{r,r}$ at the end of the algorithm. In <ref>, we have the sender and receiver in different partitions. In <ref>, the sender and receiver are in the same partition. We can observe it is possible to achieve state transfer with high fidelity even for small values of $N_1$ and $N_2$. For example, the value of the fidelity is bigger than 0.9 for $N_1, N_2 > 15$ in both cases, and it will become closer to 1 as $N_1$ and $N_2$ increases. We can observe this fact in figure <ref>, where we fix $N_1 = 100$ and vary $N_2$ in the range $(2,1000)$. In this case, the sender and receiver vertices are in different partitions.

The fidelity with the target state $\ket{r,r}$ at the end of the state transfer algorithm with active switch. In this case, $s\in V_1$ and $r\in V_2$. We set $N_1=100$ and vary $N_2$ in the range $(2,1000)$.

§ SUMMARY AND CONCLUSIONS

We have analyzed state transfer by using a discrete-time coined quantum walk on the complete bipartite graph. We chose the coin operator as $(G,-G)$, that is, we apply the Grover diffusion coin $G$ to the non-marked vertices and $-G$ to the marked. We were able to analytically calculate the expression for the fidelity with the target state by expressing the evolution operator in a smaller invariant subspace. We have obtained that it is possible to achieve state transfer with high fidelity even in the case where the sender and receiver vertex belong to opposite partitions of different sizes.

We have compared our result with the case $(G,-I)$ [18]. When the sender and receiver belong to the same partition, both cases have exactly the same value for the fidelity and they achieve (almost) perfect state transfer. When the sender and receiver are in different partitions, if $N_1 = N_2$, both also achieve (almost) perfect state transfer, with our case requiring less amount of steps. When $N_1 \neq N_2$, our case can achieve high fidelity value for some instances of $N_1$ and $N_2$, specially when $N_1 \gg N_2$ (or $N_2 \gg N_1$), as it was shown numerically. For $(G,-I)$, the value of the fidelity can be quite low in such cases.

We have presented an alternative solution, which is an active switch approach (similar to the one in [13]) that uses lackadaisical quantum walks. By using previous results from [11], we have shown that for large $N_1$ and $N_2$ with proper self-loop weights and considering one marked vertex, if the initial state is the stationary state of the quantum walk, after a certain amount of time the state of the quantum walk will be contained at the self-loop on the marked vertex. With this fact, we have described a quantum state transfer algorithm based on one marked vertex. We start the algorithm by selecting the sender as the marked vertex. The initial state is the sender vertex self-loop. After we evolve the walk for some time, the marked vertex is switched to the receiver vertex. At the end of the algorithm we obtain the walker at the receiver vertex self-loop. Therefore, it is possible to achieve state transfer with  fidelity
approaching unity for a large graph, independent of the location of the sender and receiver vertices. However, in this case, we need to know the location of the sender and receiver vertices in order to correctly set up the number of steps of the algorithm. In addition, through some simulations of the algorithm, we have shown we can achieve state transfer with high fidelity even for small graphs.

§ ACKNOWLEDGEMENTS

This work was supported by ERDF project number 1.1.1.2/VIAA/1/16/002.

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§ EIGENBASIS OF $U_M$ FOR THE LQW SEARCH ON THE COMPLETE BIPARTITE GRAPH

From [11], considering the invariant subspace given by the states in (<ref>)-(<ref>), the evolution operator when searching for one marked vertex $m$ can be written as

\begin{equation}
U_m = \matrx{
\frac{N_2-l_1}{N_2+l_1} & \frac{-2\sqrt{N_2l_1}}{N_2+l_1} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{2-N_1-l_2}{N_1+l_2} & \frac{2\sqrt{l_2}}{N_1+l_2} & \frac{2\sqrt{N_1-1}}{N_1+l_2} & 0 & 0 \\
\frac{-2\sqrt{N_2l_1}}{N_2+l_1} & \frac{l_1-N_2}{N_2+l_1} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{2\sqrt{l_2}}{N_1+l_2} & \frac{l_2-N_1}{N_1+l_2} & \frac{2\sqrt{l_2(N_1-1)}}{N_1+l_2} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{N_2-l_1}{N_2+l_1} & \frac{2\sqrt{N_2l_1}}{N_2+l_1} \\
0 & 0 & \frac{2\sqrt{N_1-1}}{N_1+l_2} & \frac{2\sqrt{l_2(N_1-1)}}{N_1+l_2} & \frac{N_1-2-l_2}{N_1+l_2} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{2\sqrt{N_2l_1}}{N_2+l_1} & \frac{l_1-N_2}{N_2+l_1}}.
\end{equation}

The spectrum of $U_m$ can be approximated for large $N_1$ and $N_2$. The asymptotic eigenvectors and eigenvalues are

\begin{align}
\ket{\psi_1}=&\frac{1}{\sqrt{1+\frac{2l_1 N_1}{ N_2}}}\left[1,0,0,0,-\sqrt{\frac{l_1 N_1}{N_2}},-\sqrt{\frac{l_1 N_1}{N_2}},0\right]^\intercal, \> \lambda_1=1, \\
\ket{\psi_2}=&\frac{1}{\sqrt{2+\frac{N_2}{l_1 N_1}}}\left[1,i\sqrt{\frac{2l_1 N_1+N_2}{4l_1 N_1}},-i\sqrt{\frac{2l_1 N_1+N_2}{4l_1 N_1}},0,\sqrt{\frac{N_2}{4l_1 N_1}},\sqrt{\frac{N_2}{4l_1 N_1}},0\right]^\intercal, \>\lambda_2=e^{-i\theta}, \\
\ket{\psi_3}=&\frac{1}{\sqrt{2+\frac{N_2}{l_1 N_1}}}\left[1,-i\sqrt{\frac{2l_1 N_1+N_2}{4l_1 N_1}},i\sqrt{\frac{2l_1 N_1+N_2}{4l_1 N_1}},0,\sqrt{\frac{N_2}{4l_1 N_1}},\sqrt{\frac{N_2}{4l_1 N_1}},0\right]^\intercal, \> \lambda_3=e^{i\theta},\\
\ket{\psi_4}=&\frac{1}{\sqrt{2+\frac{N_2}{l_1 N_1}}}\left[0,1,1,0,0,0,\sqrt{\frac{N_2}{l_1N_1}}\right]^\intercal, \> \lambda_4=-1, \\
\ket{\psi_5}=&\frac{1}{\sqrt{1+\frac{l_2N_2}{l_1N_1}}}\left[0,0,0,1,0,0,\sqrt{\frac{l_2N_2}{l_1N_1}}\right]^\intercal, \> \lambda_5=-1, \\
\ket{\psi_6}=&\frac{1}{\sqrt{2+\frac{4(l_1N_1+l_2N_2)}{N_2}}}\left[0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\sqrt{2l_2},-i\sqrt{\frac{2l_1N_1+(1+2l_2)N_2}{2N_2}},\right.\nonumber\\
& \left.i\sqrt{\frac{2l_1N_1+(1+2l_2)N_2}{2N_2}},-\sqrt{\frac{2l_1 N_1}{N_2}}\right]^\intercal, \> \lambda_6=-e^{i\phi},  \\
\ket{\psi_7}=&\frac{1}{\sqrt{2+\frac{4(l_1N_1+l_2N_2)}{N_2}}}\left[0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\sqrt{2l_2},i\sqrt{\frac{2l_1N_1+(1+2l_2)N_2}{2N_2}},\right.\nonumber\\
&\left. -i\sqrt{\frac{2l_1N_1+(1+2l_2)N_2}{2N_2}},-\sqrt{\frac{2l_1 N_1}{N_2}}\right]^\intercal, \> \lambda_7=-e^{-i\phi},
\end{align}

\begin{gather*}
\sin \theta=\sqrt{\frac{2l_1 N_1+N_2}{N_1N_2}}, \\
\sin\phi=\sqrt{\frac{2l_1N_1+N_2+2l_2N_2}{N_1N_2}}.
\end{gather*}