# Redshift constrain of BL Lac PKS 1424+240

Sarira Sahu1, D. I. Páez-Sánchez2, B. Medina-Carrillo3, R. de J. Pacheco-Aké3,
G. Sánchez-Colón3, Subhash Rajpoot4
1Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Circuito Exterior S/N, C.U., A.P. 70-543, CDMX 04510, México.
2Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito
Exterior, C.U.,
A.P. 70-543, 04510 CDMX, México.
3Departamento de Física Aplicada, Centro de Investigación y de Estudios
Avanzados del IPN, Unidad Mérida.
A.P. 73, Cordemex, Mérida, Yucatán 97310, México.
4Department of Physics and Astronomy, California State University, 1250
Bellflower Boulevard, Long Beach,
CA 90840, USA Contact e-mail<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>
###### Abstract

In the period between 2009 and 2015, several very high-energy (VHE $>100$ GeV)
gamma-ray flaring events from the BL Lac object PKS 1424+240 were observed by
the Cerenkov telescopes VERITAS and MAGIC. It had uncertain redshift ($z$) and
using spectroscopical measurement, Paiano et al. (2017) found it to be
$z=0.604$. Using four different extragalactic background light (EBL) models
and the photohadronic model, nine independently observed VHE gamma-ray spectra
of PKS 1424+240 are analyzed and a global $\chi^{2}$ fit is performed on all
observations to estimate the best-fit value for the redshift for each EBL
model. Confidence levels (CL) intervals for the redshift are also estimated
using all the EBL models. This method is tested by comparing our analysis with
the observed value. It is shown that the photohadronic scenario provides an
excellent description of all the observed spectra. It is found that the EBL
model of Domínguez et al. (2011) is the one that provides the most restrictive
limits on the redshift of PKS 1424+240, but in our analysis, $z=0.604$ lies
within the $3\sigma$ CL interval of the EBL model of Saldana-Lopez et al.
(2021).

###### keywords:

BL Lacertae objects: general, galaxies: jets, gamma rays: galaxies

††pubyear: 2024††pagerange: Redshift constrain of BL Lac PKS 1424+240–Redshift
constrain of BL Lac PKS 1424+240

##  1 INTRODUCTION

Blazars are a subclass of active galactic nuclei (AGN), characterized by their
non-thermal spectra, originating from a relativistic jet, that is closely
aligned with the observer’s line of sight (Urry & Padovani, 1995; Acciari et
al., 2011). Blazars exhibit rapid variability throughout their entire
electromagnetic spectrum. Their spectral energy distributions (SEDs) show a
distinctive double-peak structure (Abdo et al., 2010), the first peak arises
as synchrotron photons from the propagation of low-energy electrons in the jet
magnetic field. However, the second peak can either be from the Compton
scattering of high-energy electrons with their self-produced synchrotron
photons, a phenomenon known as synchrotron self-Compton (SSC) scattering
(Maraschi et al., 1992; Dermer & Schlickeiser, 1993; Sikora et al., 1994;
Blazejowski et al., 2000; Murase et al., 2012; Gao et al., 2013) or from the
external Compton scattering with the photons from the accretion disk, broad-
line regions or the dusty torus(Dermer & Schlickeiser, 1993; Sikora et al.,
1994; Blazejowski et al., 2000).

PKS 1424+240 was discovered in the 1970s as a radio source (Fanti et al.,
1974) and was later identified as a blazar by Impey & Tapia (1988). Later, the
source was detected in $\gamma$-rays by the Fermi Large Area Telescope (LAT;
Atwood et al. (2009)). In the spring of 2009, VERITAS reported the observation
of VHE gamma-rays from this source (Ong, 2009) and, subsequently, it was
confirmed by MAGIC (Teshima, 2009). Afterwards, several VHE flaring events
from PKS 1424+240 were observed between 2009 and 2015. Nonetheless, the
frequency of the synchrotron peak $\nu^{peak}_{sync}$ for PKS 1424+240 was not
directly determined. However, it could be constrained from the optical and the
X-ray data and was found to be in the range
$10^{15}\,\mathrm{Hz}\lesssim\nu^{peak}_{sync}\lesssim 10^{17}$ Hz (Acciari et
al., 2010). From the classification of the BL Lac objects, we know that this
range of $\nu^{peak}_{sync}$ classifies the object as high-frequency peaked BL
Lac (HBL) (Padovani & Giommi, 1996; Abdo et al., 2010; Boettcher et al.,
2013).

The non-thermal emission in a BL Lac object dominates over the stellar
emission of its host galaxy, making it difficult to correctly estimate the
redshift. As a consequence, ambiguities arise in tracing the cosmic evolution,
the nature and the understanding of the intrinsic VHE spectrum of the source.
Given the fact that VHE photons are dampened by the extragalactic background
light (EBL), information of the redshift is essential to investigating the
role of the EBL.

EBL models are constructed with the convergence of different approaches as
direct measurements, semi analytical models of galaxy formation, analysis of
cosmic star formation history, and galaxy luminosity densities with SEDs. The
models used in this work belong to the last category and their main difference
is the number of galaxy samples employed, which is correlated with the
redshift value range of the data output from the model (Saldana-Lopez et al.,
2021).

The VHE gamma-ray flux of a blazar observed on Earth is given by (Hauser &
Dwek, 2001)

$F_{\gamma}(E_{\gamma})=F_{int}(E_{\gamma})\,e^{-\tau_{\gamma\gamma}(E_{\gamma},z)},$
(1)

where $E_{\gamma}$, $F_{\gamma}$, and $F_{int}$ are the observed VHE photon
energy, the observed flux, and the intrinsic flux, respectively. Here, the
optical depth $\tau_{\gamma\gamma}$ for the process $\gamma\gamma\rightarrow
e^{+}e^{-}$ depends on $E_{\gamma}$ and the redshift $z$ of the source. The
exponential factor in Eq. (1) accounts for the attenuation in the VHE flux due
to $e^{+}e^{-}$ production (Stecker et al., 1992; Ackermann et al., 2012;
Padovani et al., 2017). Thus, knowledge on $z$ is crucial to estimating the
intrinsic flux from the observed flux. Well-known and reliable EBL models by
Franceschini et al. (2008); Finke et al. (2010); Dominguez et al. (2011);
Gilmore et al. (2012); Saldana-Lopez et al. (2021) are employed by Imagining
Atmospheric Cerenkov Telescope (IACT) collaborations to analyze the observed
VHE $\gamma$-ray spectra from objects at different redshifts.

As stated previously, the redshift of HBL PKS 1424+240 is measured
spectroscopically with a value $z=0.604$ (Paiano et al., 2017). However,
alternative methods do not provide a definitive result. By making a “minimum
luminosity assumption" (host galaxy absolute magnitude to be $M=-21.9$),
Scarpa & Falomo (1995) have derived a lower limit of $z>0.06$, while
Sbarufatti et al. (2005) set a limit of $z>0.67$ by using the average value
$M=-22.9$. Acciari et al. (2010) combined the photon index (measured with
Fermi Large Area Telescope) with EBL models, to estimate the redshift
$z=0.5\pm 0.1$, with an upper limit of $z<0.66$ at $95\%$ CL. Comparing the
measured and intrinsic VHE spectra due to EBL absorption, an upper limit of
1.19 on the redshift was derived in Yang & Wang (2010). Employing a derived
empirical law, describing the relation between the upper limits and the true
redshifts, the redshift is estimated to be $0.24\pm 0.05$ in Prandini et al.
(2010, 2011). The photometric redshift, obtained through spectral fitting of
optical/UV data, has an upper limit of $z<1.11$ (Rau, A. et al., 2012).
Furniss et al. (2013) have given a lower limit, $z\geq 0.6035$, set by the
detection of Ly$\beta$ and Ly$\gamma$ lines from intervening hydrogen clouds.
By multiwavelength spectral characterization and modeling of the blazar, in
Ref. Aleksić et al. (2014), a value $z=0.61\pm 0.10$ with an upper limit of
0.81 at $2\sigma$ is reported. In Ref. Rovero, A. C. et al. (2016), the
probability of PKS 1424+240 being a member of a group of galaxies found at
$z=0.6010\pm 0.003$ was found to be $98\%$. The observed correlation between
the VHE spectral index of blazars and redshift is interpreted as a result of
EBL induced absorption effects and the absence of such correlation led to
develop a technique to estimate the redshift of distant blazars whose optical
spectrum is featureless. For PKS 1424+240, this technique was employed by
Zahoor et al. (2022) and have estimated $z=0.28\pm 0.13$ and $z=0.24\pm 0.11$.

The multi-zone leptonic models are also used to explain the VHE flaring
events. However, one needs to almost double the parameters in these models.
Models involving hadrons suffer from low efficiency. For example, in the
proton synchrotron scenario, the emission of synchrotron photons by protons in
the jet environment is suppressed by a factor of $m^{-4}_{p}$, where $m_{p}$
is the proton mass. Thus, this process requires ultra-high energy proton flux.
Also a strong magnetic field is needed, which is unusual in a blazar jet
(Mannheim et al., 1991; Mücke & Protheroe, 2001; Mücke et al., 2003). There
are several other alternative models, such as spine-layer structured jet model
and the lepton-hadron hybrid models to explain these spectra (Ghisellini et
al., 2005; Tavecchio & Ghisellini, 2008; MAGIC Collaboration et al., 2019;
Ahnen et al., 2018).

Previously, the photohadronic model (Sahu, 2019), has been successfully used
to explain the VHE $\gamma$-ray spectra of various HBLs and extreme HBLs
(EHBLs) (Sahu et al., 2020, 2021, 2022). A new classification scheme was
proposed (Sahu et al., 2019) by analyzing the spectral index of several
observed VHE $\gamma$-ray spectra from HBLs. In the foregoing, the
photohadronic model (Sahu, 2019) along with four well-known EBL models
Saldana-Lopez et al. (2021); Franceschini et al. (2008); Dominguez et al.
(2011); Gilmore et al. (2012) are used to analyze nine VHE flaring events of
PKS 1424+240 between 2009 and 2015, observed by VERITAS and MAGIC
collaborations. By performing a $\chi^{2}$ global fit to the full set of
experimental data points of these VHE spectra, the central values and CL
intervals for the redshift of PKS 1424+240 are determined for each EBL model
considered.

The plan of this paper is as follows. In Section 2 a brief review of the
photohadronic model and its features relevant to the present work is
presented. In Section 3 the analysis and results are presented. Finally,
Section 4 includes a short summary and discussion.

##  2 THE PHOTOHADRONIC MODEL

In the photohadronic model, a double jet configuration is assumed during the
VHE flaring process (Sahu, 2019; Sahu et al., 2019). A double jet
configuration has been proposed in earlier works (Ghisellini & Tavecchio,
2008; Giannios et al., 2010). In the photohadronic scenario, a compact and
constrained narrower jet, with a size $R^{\prime}_{f}$, is formed within the
wider jet of size $R^{\prime}_{b}$, $R^{\prime}_{f}<R^{\prime}_{b}$ (the
quantities in the comoving frame are indicated by primes). In the inner region
of the jet, the photon density, $n^{\prime}_{\gamma,f}$, is significantly
greater than the photon density in the outer jet region, $n^{\prime}_{\gamma}$
($n^{\prime}_{\gamma,f}\gg n^{\prime}_{\gamma}$). The photohadronic model is
based on the conventional interpretation of the first two peaks in the SED,
namely, that the first peak results from the synchrotron radiation emitted by
relativistic electrons within the jet environment and the second peak is a
consequence of the SSC process. The inner jet moves at a (slightly) higher
velocity (with bulk Lorentz factor $\Gamma_{in}$) compared to the outer jet
(with bulk Lorentz factor $\Gamma_{ext}$). For simplicity, it is assumed
$\Gamma_{in}\simeq\Gamma_{ext}\equiv\Gamma$ and a common Doppler factor
$\mathcal{D}$ (Ghisellini et al., 1998; Krawczynski et al., 2004). For HBLs,
we have $\Gamma\simeq{\cal D}$.

In the photohadronic scenario, protons are accelerated to very high-energies
in the inner jet region, and their differential spectrum follows a power-law
form, $dN_{p}/dE_{p}\propto E^{-\alpha}_{p}$ (Dermer & Schlickeiser, 1993),
where $E_{p}$ is the proton energy and $\alpha\geq 2$ is the proton spectral
index. The specific value of $\alpha$ differs, depending on the type of shock
encountered. It can be different for non-relativistic shocks, highly
relativistic shocks, and oblique relativistic shocks (Keshet & Waxman, 2005;
Summerlin & Baring, 2012). The interaction of these high-energy protons with
the SSC background seed photons in the inner jet region results in the
production of the $\Delta$-resonance through the process
$p+\gamma\rightarrow\Delta^{+}$. The $\Delta$-resonance decays to $\pi^{0}$
and $\pi^{+}$ with different probabilities. Although the direct single pion
production and the multi-pion production processes contribute, they are less
efficient in the energy range under consideration (Mücke et al., 1999; Owen et
al., 2018). Such contributions are neglected in the present work. Finally, the
neutral pion decays to $\gamma$-rays and the $\pi^{+}$ to neutrinos (Sahu et
al., 2012). In the photohadronic scenario, the $\gamma$-rays generated from
$\pi^{0}$ decay are blue-shifted to VHE $\gamma$-rays and are detected on
Earth. The positrons produced from $\pi^{+}$ decay will emit synchrotron
photons.

The observed VHE $\gamma$-ray energy $E_{\gamma}$ and the seed photon energy
$\epsilon_{\gamma}$ satisfy the condition (Sahu, 2019; Sahu et al., 2019)

$E_{\gamma}\epsilon_{\gamma}\simeq 0.032\frac{{\cal
D}^{2}}{(1+z)^{2}}\,\mathrm{GeV}^{2}.$ (2)

In the above process, the VHE photon carries approximately $10\%$ of the
proton energy, $E_{p}=10E_{\gamma}$. Given the inaccessible nature of the
inner jet region and the lack of direct means to estimate the photon density
within it, a scaling relationship between the inner and the outer jet regions
is assumed, which can be expressed as (Sahu et al., 2016)

$\frac{n^{\prime}_{\gamma,f}(\epsilon_{\gamma,1})}{n^{\prime}_{\gamma,f}(\epsilon_{\gamma,2})}\simeq\frac{n^{\prime}_{\gamma}(\epsilon_{\gamma,1})}{n^{\prime}_{\gamma}(\epsilon_{\gamma,2})}.$
(3)

In the above equation, the left-hand side represents the unknown, while the
right-hand side is known. This equation is employed to express the unknown
photon density in the inner region in terms of the known photon density in the
outer region.

The intrinsic $\gamma$-ray flux from the $\pi^{0}$ decay is deduced to be

$F_{\gamma}(E_{\gamma})\equiv
E^{2}_{\gamma}\frac{dN(E_{\gamma})}{dE_{\gamma}}\propto
E^{2}_{p}\frac{dN(E_{p})}{dE_{p}}n^{\prime}_{\gamma,f}.$ (4)

By taking into account the seed photon density and the proton flux, the
intrinsic flux can be expressed as

$F_{int}(E_{\gamma})=F_{0}\left(\frac{E_{\gamma}}{\rm
TeV}\right)^{-\delta+3}.$ (5)

By putting this in Eq. (1), the observed flux is determined to be

$F_{\gamma}(E_{\gamma})=F_{0}\left(\frac{E_{\gamma}}{\rm
TeV}\right)^{-\delta+3}e^{-\tau_{\gamma\gamma}(E_{\gamma},z)},$ (6)

where the spectral index $\delta=\alpha+\beta$. The component $\beta$
represents the power spectrum of the seed photons in the low energy tail
region of the SSC spectrum. The normalization factor $F_{0}$ can be determined
from the observed data. The value of $\delta$ is considered as the only free
parameter in the photohadronic model (Sahu et al., 2019). As can be seen from
Eq.(6), in the photohadronic model the curvature in the spectrum, if any, can
be taken care of by the exponential term (Acciari et al., 2019).

It is important to note that the photohadronic process works well for
$E_{\gamma}\gtrsim 100$ GeV. Below this energy, the leptonic processes such as
the electron synchrotron mechanism and the SSC process have the dominant
contribution to the multiwavelength SED.

##  3 ANALYSIS AND RESULTS

In the analysis of the VHE gamma-ray spectra of PKS 1424+240, the EBL
corrections are taken care of by the use of four well-known EBL models. For
convenience, the EBL models are referred to as Saldana (Saldana-Lopez et al.,
2021), Franceschini (Franceschini et al., 2008), Domínguez (Dominguez et al.,
2011), and Gilmore (Gilmore et al., 2012). Also for convenience, each
observation is named according to the telescope and the year of observation.
The observations by VERITAS in 2009 (from MJD 54881 to MJD 55003) are named as
VO09-I for the time averaged spectrum analysis and as VO09-II for the analysis
of the VHE $\gamma$-ray spectrum. In Table 1 we have listed details about all
the observations.

In accordance with the classification scheme given by Sahu et al. (2019), the
VHE emission states of a HBL are defined according to the $\delta$ value. The
emission state is considered very high when $2.5\leq\delta\leq 2.6$. The high
emission state corresponds to $2.6<\delta<3.0$ and $\delta=3.0$ implies the
low emission state. Since PKS 1424+240 is a HBL, $\delta$ must be constrained
to lie in the range $2.5\leq\delta\leq 3.0$.

The duration of time observed during each observation period by VERITAS and
MAGIC are between two to five months. Thus, the observed spectra are the
average spectra. It is very important to note that these observation periods
are too long for a spectrum to be in a very high emission state or high
emission state. Also, the average spectrum of a long observation is always in
a low emission state which corresponds to $\delta=3.0$. Thus, for further
analysis to all the nine spectra, we fixed the spectral index to $\delta=3.0$.

For our analysis and to find the best fits to the observed VHE-spectra for a
given EBL model, the EBL model is implemented in the photohadronic model and a
global $\chi^{2}$ fit is performed on all the data points by simultaneously
varying the redshift $z$ (common to all observations) and the corresponding
normalization constants $F_{0}$ of each of the nine independent observations.
The global fitting procedure is repeated for the other three EBL models too to
find the best-fit value of the redshift $z$ of the HBL PKS 1424+240 and the
normalization $F_{0}$ of the photohadronic model. The redshift CL intervals at
$1\sigma$, $2\sigma$, and $3\sigma$, are then calculated for each of the four
EBL models considered.

The best fit values for the normalization constants $F_{0}$, corresponding to
each observation and for the four EBL models, are given in Table 2. One can
observe that for a given observation, the $F_{0}$ value is very similar in all
the EBL models. This clearly shows that all the EBL models are also similar to
each other. Analogously, the best-fit values for the redshift $z$ and their CL
intervals at $1\sigma$, $2\sigma$, and $3\sigma$, are calculated which are
shown in Table 3. With 42 degrees of freedom (52 experimental data points
considered and 10 free parameters), values of minimum $\chi^{2}$ obtained for
the global fit are 61.8, 64.74, 66.47, and 63.51 for the EBL models of
Saldana, Franceschini, Domínguez, and Gilmore, respectively. The use of a
different EBL model predicts different CL intervals of $z$ for each observed
VHE spectrum. The photohadronic model with the EBL model of Domínguez provides
the most restrictive intervals on the redshift, $z=0.689$ the central value
with $0.650\leq z\leq 0.724$ at $1\sigma$, $0.622\leq z\leq 0.748$ at
$2\sigma$, and $0.610\leq z\leq 0.758$, at $3\sigma$. Although, the least
restrictive intervals on $z$ are obtained with the EBL model of Saldana, the
measured value of $z=0.604$ lies within the $3\sigma$ CL interval of this
model.

Fits to the nine observed VHE spectra of PKS 1424+240 are displayed in Fig. 1,
where, the best-fit parameters for the EBL model of Saldana (given in Tables 2
and 3) are used. The fits to the VHE spectra using the Domínguez,
Franceschini, and Gilmore EBL models are omitted because, visually, there is
no discernible difference from one EBL model to another. For comparison, all
the four EBL models are used to best fit the spectrum of VO13 which is shown
in Fig. 2. All the four EBL models fit very well to the data. However, for
$E_{\gamma}\gtrsim 0.3$ TeV we observe a small discrepancy. The Saldana EBL
model is implemented in the photohadronic model to determine the butterfly
$3\sigma$-CL regions for the nine observed spectra, which are also shown in
Fig. 1. Plots of CL regions with the other three EBL models considered are
omitted. For comparison, the redshift of PKS 1424+240 estimated by different
authors and the spectroscopic measurement are summarized in Table 4.

Table 1: Information of the PKS 1424+240 observations employed in this work. Name and instrument of the observation is presented in the first and second column, respectively. Start date, end date, period span (in MJD), and duration of the observation are found in the third, fourth, fifth, and sixth columns, respectively. The work in which each observation period data is featured is presented in the seventh column. Observation | Instrument | Start date | End date | Period (MJD) | Duration (h) | Reference
---|---|---|---|---|---|---
VO09-I | VERITAS | 19/02/2009 | 21/06/2009 | 54881 - 55003 | 37.3 | Acciari et al. (2010)
VO09-II |  |  |  |  |  |
MO09 | MAGIC | 17/04/2009 | 23/06/2009 | 54938 - 55005 | 12.5 | Aleksić et al. (2014)
MO10 | MAGIC | 14/03/2010 | 19/04/2009 | 55269 - 55305 | 11.6 | Aleksić et al. (2014)
VO11 | VERITAS | 06/02/2011 | 30/05/2011 | 55598 - 55711 | 14 | Archambault et al. (2014)
MO11 | MAGIC | 24/04/2011 | 08/05/2011 | 55675 - 55689 | 9.5 | Aleksić et al. (2014)
VO13 | VERITAS | 11/02/2013 | 04/05/2013 | 56334 - 56447 | 67 | Archambault et al. (2014)
MO14 | MAGIC | 23/03/2014 | 18/06/2014 | 56739 - 56826 | 28.19 | Acciari et al. (2019)
MO15 | MAGIC | 22/01/2015 | 13/06/2015 | 57044 - 57186 | 20.9 | Acciari et al. (2019)
Table 2: Estimated values for the normalization constant $F_{0}$ (in units of $10^{-11}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$) of the photohadronic model obtained from the best fits to the VHE spectra of PKS 14+240 for four EBL models are presented in the third column. Refer to the main text for the different observations given in the first column and for the different EBL models mentioned in the second column. The spectral index $\delta=3.0$ (low emission state) is used. Observation | EBL Model | $F_{0}$
---|---|---
VO09-I | Saldana | 1.420
| Franceschini | 1.271
| Domínguez | 1.337
| Gilmore | 1.421
VO09-II | Saldana | 1.767
| Franceschini | 1.593
| Domínguez | 1.683
| Gilmore | 1.776
MO09 | Saldana | 3.590
| Franceschini | 3.257
| Domínguez | 3.426
| Gilmore | 3.624
MO10 | Saldana | 1.191
| Franceschini | 1.074
| Domínguez | 1.126
| Gilmore | 1.194
VO11 | Saldana | 1.840
| Franceschini | 1.649
| Domínguez | 1.736
| Gilmore | 1.840
MO11 | Saldana | 3.162
| Franceschini | 2.855
| Domínguez | 2.996
| Gilmore | 3.175
VO13 | Saldana | 1.034
| Franceschini | 0.931
| Domínguez | 0.981
| Gilmore | 1.037
MO14 | Saldana | 1.907
| Franceschini | 1.709
| Domínguez | 1.797
| Gilmore | 1.913
MO15 | Saldana | 1.462
| Franceschini | 1.312
| Domínguez | 1.381
| Gilmore | 1.460
Table 3: Estimated values for the redshift in the photohadronic model obtained from the global $\chi^{2}$ fit to the VHE spectra of the HBL PKS 1424+240. Refer to the main text for the different EBL models mentioned in the first column. The redshift $z$ is given in the second column. The redshift CL intervals at $1\sigma$, $2\sigma$, and $3\sigma$ are shown in the third, the fourth, and the fifth columns for each EBL model employed, respectively. EBL Model | Redshift | Redshift CL intervals
---|---|---
| $z$ | $1\sigma$ | $2\sigma$ | $3\sigma$
Saldana | 0.660 | (0.626, 0.724) | (0.607, 0.756) | (0.597, 0.770)
Franceschini | 0.679 | (0.639, 0.722) | (0.616, 0.749) | (0.605, 0.762)
Domínguez | 0.689 | (0.650, 0.724) | (0.622, 0.748) | (0.610, 0.758)
Gilmore | 0.727 | (0.685, 0.770) | (0.661, 0.798) | (0.650, 0.812)
Table 4: Summary of the spectroscopic measurement and estimations by different authors for the redshift value $z$ of PKS 1424+240. The redshift measurement and estimated interval limits, or central values with uncertainties, are given in the second column, the third column indicates the method used to determine the redshift. Corresponding references are given in the fourth column.  n | Redshift | Method | Reference
---|---|---|---
1 | $z=0.604$ | Spectroscopic measurement | Paiano et al. (2017)
2 |  0.06 < $z$ | Minimum luminosity assumption | Scarpa & Falomo (1995)
3 |  0.67 < $z$ | Average luminosity value | Sbarufatti et al. (2005)
4 |  $z$ = 0.5 ± 0.1 | Photon index – EBL models combination | Acciari et al. (2010)
5 |  $z$ < 0.66 at 95% CL | Photon index – EBL models combination | Acciari et al. (2010)
6 |  $z$ < 1.19 | Measured – intrinsic VHE spectra comparison | Yang & Wang (2010)
7 |  $z$ = 0.24 ± 0.05 | Upper limits – true redshifts relationship | Prandini et al. (2010, 2011)
8 |  $z$ < 1.11 | Spectral fitting of optical/UV data | Rau, A. et al. (2012)
9 |  0.6035 < $z$ |  Ly$\beta$ and Ly$\gamma$ lines detection | Furniss et al. (2013)
10 |  $z$ = 0.61 ± 0.10 | Multiwavelength spectral characterization | Aleksić et al. (2014)
11 |  $z$ < 0.81 at $2\sigma$ | Multiwavelength spectral characterization | Aleksić et al. (2014)
12 |  $z$ = 0.6010 ± 0.003 | Member of a group of galaxies probability | Rovero, A. C. et al. (2016)
13 |  $z$ = 0.28 ± 0.13 | VHE spectral index – redshift correlation | Zahoor et al. (2022)
14 |  $z$ = 0.24 ± 0.11 | VHE spectral index – redshift correlation | Zahoor et al. (2022)
15 |  0.597 < $z$ < 0.770 at 95% CL | Global fit to independent observations | Present work (EBL Saldana)
Figure 1: Fits to the nine observed VHE $\gamma$-ray spectra of PKS 1424+240
using the photohadronic model with the EBL model of Saldana (solid black
curves). All these flaring events are in low emission state corresponding to
$\delta=3.0$. Here, $z=0.660$, and the corresponding $F_{0}$’s are given in
Table 2. Plots of the $3\sigma$-CL regions to all the nine observed VHE
spectra are also shown. Figure 2: Photohadronic model fits to the observed VHE
spectrum VO13 of PKS 1424+240 to illustrate the comparison of different EBL
models: Saldana (black), Franceschini (blue), Domínguez (green), and Gilmore
(orange). Corresponding values of the normalization constants $F_{0}$ and the
redshift $z$ are provided in Tables 1 and 2, respectively. As mentioned, for
VO13, the VHE flaring is in low emission state, $\delta=3.0$.

##  4 SUMMARY AND DISCUSSION

In this work, the photohadronic model is used in conjunction with the EBL
models of Saldana, Franceschini, Domínguez, and Gilmore, to impose stringent
constraints on the redshift of PKS 1424+240. This is done by analyzing nine
different VHE spectra, independently observed by VERITAS and MAGIC telescopes
in the period between 2009 and 2015. Since the photohadronic model (Sahu,
2019; Sahu et al., 2019) has demonstrated its effectiveness in constraining
the redshift of various HBLs with unknown $z$ values (Sahu et al., 2023), it
is employed again to analyze the nine VHE spectra of PKS 1424+240 to constrain
its redshift.

The observation period for each spectrum varies between two to five months,
too long for a spectrum to be in a very high emission state or in a high
emission state. It is observed that all the nine VHE emissions of the HBL PKS
1424+240 were in low emission states ($\delta=3.0$). With this information,
for the analysis, a global $\chi^{2}$ fit is performed by simultaneously
varying the redshift $z$ of the object and the normalization constants $F_{0}$
of each one of the nine independent observations, to find the best fit to the
observed VHE spectra for a given EBL model. This procedure is repeated for all
the other three EBL models to find the corresponding best fit values of $z$
and the parameters $F_{0}$. Using the best fit values of the redshift $z$ and
the $F_{0}$’s, the redshift CL intervals at $1\sigma$, $2\sigma$, and
$3\sigma$, for each one of the four EBL models considered are calculated.

Our analysis shows that the EBL model of Dominguez et al. (2011) provides the
most restrictive limits on the redshift and the least restrictive intervals on
$z$ are obtained with the EBL model of Saldana-Lopez et al. (2021). However,
the measured value of $z=0.604$ (Paiano et al., 2017) lies within the
$3\sigma$ CL interval of the EBL model of Saldana-Lopez et al. (2021) only.

## Acknowledgements

We thank the reviewer for her/his constructive remarks which helped us to
improved the manuscript substantially. The work of S.S. is partially supported
by DGAPA-UNAM (México) Project No. IN103522. B. M-C, R. de J. P-A, and G. S-C,
would like to thank CONAHCYT (México) for partial support. Partial support
from CSU-Long Beach is gratefully acknowledged.

## Data Availability

No new data were generated or analysed in support of this research.

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Stanford University Stanford University Anna Zeng and Will Crichton

###### Acknowledgements.

Titus Barik, Joshua Sunshine, and Sarah Chasins 3 9th Workshop on Evaluation
and Usability of Programming Languages and Tools (PLATEAU 2018) PLATEAU 2018
PLATEAU 2018 November 5, 2018 Boston, Massachusetts, USA 67 5

# Identifying Barriers to Adoption for Rust through Online Discourse

Anna Zeng  and  Will Crichton

###### Abstract.

Rust is a low-level programming language known for its unique approach to
memory-safe systems programming and for its steep learning curve. To
understand what makes Rust difficult to adopt, we surveyed the top Reddit and
Hacker News posts and comments about Rust; from these online discussions, we
identified three hypotheses about Rust’s barriers to adoption. We found that
certain key features, idioms, and integration patterns were not easily
accessible to new users.

###### Key words and phrases:

rust, programming language usability

###### 1991 Mathematics Subject Classification:

Human-centered computing $\rightarrow$ Human computer interaction (HCI)

###### category:

##  1\. Introduction

Rust is a new programming language designed to usher low-level programming
into the modern era. Rust uses strong type systems and functional programming
to execute programs efficiently while avoiding the many safety problems that
plague C and C++. As an open-source project with support from Mozilla, the
Rust ecosystem has grown rapidly over the last decade. Hundreds of companies
deploy Rust in production, and thousands of developers regularly use Rust in
their projects. However, Rust has a notoriously steep learning curve. A
community survey in 2017 revealed that 25% of the people who tried Rust and
dropped it felt the language was “too intimidating, too hard to learn, or too
complicated” [13]. Prominent members of the Rust community have stated that
Rust is supposed to be hard to learn: “Rust has never claimed that it is
something you can learn in half a week” [6]. Even still, the learning curve is
only one aspect of many (library support, tooling, compile times, etc. [9])
that influences the adoption of a programming language.

If Rust is hard to learn or use, the question becomes: how should its
developers prioritize language features to drive adoption, addressing the key
challenges facing its current or potential user base? A developer on the Rust
compiler told us in an interview that these decisions are usually made in an
ad hoc way, based on the intuitions of the language developers and occasional
feedback from community members in a potpourri of online forums [14]. To
address this issue, we sought to understand the challenges of adopting Rust by
analyzing online discourse within the Rust community. We conducted a holistic
survey of popular posts and comment threads about Rust to identify beliefs of
developers using Rust that highlight ongoing issues in the practice of the
language.

##  2\. Methodology

Prior work on analyzing language adoption has focused on evaluating various
languages and tools through questionnaires, surveys, in-person interviews, and
automated code base analysis. For example, Meyerovich and Rabkin [9] combined
all of the above to highlight both empirical popularity trends as well as
reported beliefs of developers about a wide range of topics from types vs.
tests to features vs. libraries. Christakis and Bird [2] used surveys and
interviews to understand the adoption of program analyzers, and Ray et al.
[10] used program analyzers to compare languages based on bugs detected in a
large corpus of GitHub repositories.

This study differs from previous work in two ways: First, we focused only on a
single language, Rust. While this focus potentially limits the generality of
our insights, it allows us to produce deeper insights via thorough
consideration of the factors driving specifically Rust’s adoption. Second, we
performed a content analysis on the existing discourse on Rust rather than
creating a new survey. We observed that Rust is frequently written about on
blogs, and subsequently discussed online by the people developing Rust itself
(henceforth referred to as “Rust language developers”), by the people learning
and using Rust (“Rust community”), and by the broader tech community. In fact,
a version of this ethnomethodological approach [4] was already used by the
Rust language developers to inform the Rust 2018 roadmap [12] and also used to
analyze other online communities like Mechanical Turk [7].

We gathered a corpus of articles and corresponding comments from Hacker News
(HN), a forum for general tech-centric discussions, and the /r/rust subreddit,
a forum specifically about Rust. We selected these two forums both because
they are frequent hosts to discussions about Rust and use upvote mechanisms to
sort content. Upvotes act as a loose proxy for what readers consider to be
good contributions to the conversation or perspectives they agree with, and
have been shown to correlate with coarse notions of quality [11]. While
upvote-based filtering can reduce exposure to controversial opinions, given
the vast amount of possible content to read, we found it critical in improving
the signal-to-noise ratio. To select the final corpus, we filtered for HN
articles with “Rust” in the title, and considered all /r/rust articles. Then,
we filtered for posts with at least 250/200 upvotes on HN/Reddit respectively,
a total of 424 posts. From there, we selected posts that we felt were most
relevant towards understanding user experiences in Rust, e.g. choosing “Three
months of Rust” and “Why I’m dropping Rust’ over “Announcing Rust 1.12”. This
ad hoc filter was not applied soundly or completely (we read as much as time
permitted), so we do not claim our survey is exhaustive. Our final corpus
contains 50 posts, with corresponding comment sections on both forums where
applicable.

For each article, we performed a content analysis on both the document and at
least the top five comments by upvotes for each forum the article appeared on.
Then we categorized the articles (categories like “community”, “ergonomics”,
“tooling”, “security”, etc.) and looked for trends within each category,
forming preliminary hypotheses about barriers to adoption that were both novel
and actionable to the Rust community. From the insights, we formed hypotheses
that help explain the experiences that users encounter in our corpus.
Concurrently, we interviewed three Rust language developers to help us
contextualize our findings and to understand prevailing attitudes towards
Rust’s usability in the community.

##  3\. Hypotheses

Hypothesis 1: Rust is primarily promoted for safety and speed; while those
aspects matter to users, the tooling around Rust is equally valuable, but its
value is not as clearly communicated by the language developers.

Potential users need to understand Rust’s features and goals in order to
determine whether to use it. The authoritative source of information on Rust
is its official website, https://rust-lang.org. On the front page and the FAQ,
Rust is promoted as “a systems programming language that runs blazingly fast,
prevents segfaults, and guarantees thread safety.” Laundry lists of language
features including “zero-cost abstractions,” “pattern matching,” and “type
inference” are also provided. The standard library, the tooling, and the
ecosystem are all absent from this messaging [1].

To evaluate whether Rust users found the language useful for its claimed
benefits, we analyzed 12 experience reports (e.g. “Trying out Rust for
Graphics Programming”) and 6 language comparisons (e.g. “Comparing Rust and
Java”). Across the 18 articles, we counted the reported pros/cons of Rust. The
first and third most reported benefits of Rust were elimination of runtime
errors (7 articles) and data races (5 articles). Runtime errors include both
avoiding memory errors through Rust’s memory safety analyzer as well as
avoiding unhandled failures through sum types, e.g. `Option<T>`. This finding
is consistent with Rust’s messaging—users empirically self-report the utility
of Rust’s safety guarantees.

By contrast, the second most reported benefit of Rust was Cargo, Rust’s build
system and dependency manager (6 articles). One article summarized the
collective sentiment [8]:

> Instead of having to invoke `pkg-config` by hand or with Autotools macros,
> wrangling include paths for header files and library files, and basically
> depending on the user to ensure that the correct versions of libraries are
> installed, you write a `Cargo.toml` file which lists the names and versions
> of your dependencies. […] It just works when you `cargo build`.

The Rust language developers likely understand the importance of Rust’s
tooling, being a major part of the Rust 2017/18 roadmaps. However, because
Rust’s promotional messaging doesn’t clearly emphasize these features, this
absence suggests a possible disconnect between what the language developers
and potential users consider the most important features of Rust.

Hypothesis 2: Complex pointer aliasing patterns are primarily implemented
through existing libraries built on unsafe code, but Rust users have a hard
time discovering these solutions.

To guarantee memory safety for low-level programs with direct access to
memory, Rust employs the “borrow checker”, a static analysis tool that
prevents memory-unsafe operations, e.g. returning a dangling pointer to a
stack-allocated value. The borrow checker does not permit mutable aliases, or
holding two mutable pointers to the same piece of memory. Aliasing patterns
that violate this rule, like reference-counted pointers, can be implemented
carefully through Rust’s `unsafe` construct, often provided as libraries.
However, this restriction is often daunting to Rust novices because it
disallows patterns that are easy to express in other low-level languages, and
it makes solutions difficult to find. Out of the 18 experience reports and
language comparisons, the complexity of the borrow checker was the second most
frequently mentioned complaint (only behind compiler version issues of stable
vs. nightly). For example, one user implementing a video codec found [5]:

> Video codecs usually operate on planes and there you’d like to operate with
> different chunks of the frame buffer (or plane) at the same time. Rust does
> not allow you to mutably borrow parts of the same array even when it should
> be completely safe like `let mut a = &mut arr[0..pivot]; let mut b = &mut
> arr[pivot..];`.

Another user implementing a GUI framework found [3]:

> For `nanogui`… each widget has a pointer to a parent and a vector of
> pointers to its children. How does this concept map to Rust? There are
> several answers: 1\. Use a naive `Vec<T>` implementation. 2\. Use `Vec<*mut
> T>`. 3\. Use `Vec<Rc<RefCell<T>>>`. 4\. Use C bindings. … I have tried
> options 1 through 3 with several drawbacks, each making them not fit for
> use. I’m currently looking at point 4 as my only remaining option to use.

In both cases, Rust users encountered a particular memory access pattern
(disjoint mutable pointers to an array, widget trees with back pointers) that
Rust disallowed. As with other complex aliasing patterns like reference
counting, the solution to these types of problems is sufficiently complex that
Rust users aren’t expected to implement it themselves, but instead defer to
external code. In both cases above, commenters pointed out standard library
functions (`Vec::split_at_mut`) and third-party libraries (`petgraph`) for
solving these issues respectively; however, the authors were not able to
independently discover these solutions. These experiences suggest that Rust
needs better resources to help users identify common aliasing patterns and
understand what tools exist to solve those problems.

Hypothesis 3: Although incremental migration of existing codebases into Rust
seems promising, Rust users aren’t pursuing this path because the cost of
integrating Rust into a different language ecosystem or toolchain is too
great.

One important path to adoption of Rust is incremental migration, or gradually
rewriting components of a large software system from a host language (like
C/C++) into Rust. Mozilla’s initial motivation for Rust was to replace
performance-critical parts of Firefox (Project Servo), and others have begun
exploring integrating with databases (Postgres) and operating systems (Linux).
However, the idea of incremental migration is almost entirely absent from the
discourse we studied. Only one of our 50 articles directly described
experiences integrating Rust into an existing system; most experience reports
detailed new projects in Rust or entire rewrites of existing projects. While
lack of discourse doesn’t necessarily imply lack of adoption, it still does
not bode well for the path of incremental migration—issues out of the public
eye are unlikely to receive attention either from the Rust compiler developers
in prioritizing work, or from the Rust community in generating documentation
and tooling.

A possible explanation is that the challenges of incremental migration of Rust
may be surmountable for larger teams, like those supporting Firefox, but are
still too challenging for most Rust users. Discussion of incremental migration
in our corpus largely occurred in scattered comments focused on describing
challenges, not hailing great successes. For example, a Servo developer
described in an interview the challenges of bridging Rust, C++, and Javascript
in Firefox: the ease of accidentally invalidating a C++ reference when in
Rust, the challenge of managing macros across the language barriers, and the
complexity of tracking when each variable would be deallocated by which
runtime [15].

Another example of a difficult challenge for most Rust users is working on
Rust without Cargo, a tool which reduces the overhead of starting (and
maintaining) a new project and reusing third-party Rust code. In low-level
programming, build systems and package management are traditionally relegated
to a hodgepodge of tools like Make, Autotools, CMake, and Apt; in contrast,
the Rust community uses Cargo for both build process and dependency
management. While Cargo is not strictly required, all major libraries must be
built with Cargo. Developing Rust without Cargo means losing easy access to
these libraries. For example, Facebook’s Mononoke project, a rewrite of
Mercurial in Rust, initially could not use any packages outside the standard
library due to integration requirements with Facebook’s custom build system,
which significantly slowed adoption of external libraries. The lack of
discourse around these issues suggest more resources should potentially be
dedicated to reducing these barriers to adoption for incremental migration
into Rust.

##  4\. Discussion

By applying an ethnomethodological approach to Rust’s online discourse, we
identified supporting evidence for three hypotheses about factors which
meaningfully influence the adoption of Rust. This methodology is neither fully
precise nor conclusive, but it provides a useful signal to direct further
evaluation of these hypotheses. While online commenters are not subject to the
same level of rigor as peer-reviewed research, our experience suggests that
blog posts and forum threads still contain an enormous amount of collective
wisdom that is perhaps under-appreciated in academic literature. Much of the
prior work has trended towards precision through controlled experiments or
breadth through surveys and code base analyses. However, we believe that
understanding these online communities can provide valuable guidance to PL/HCI
researchers seeking to address the key problems facing today’s programmers.

To that end, one key question is this study’s replicability: could other
researchers recreate our results for Rust, or perform the same study on other
programming languages? Due to the time-consuming nature of manual content
analysis, consistently filtering articles from the larger corpus is important.
Creating better filters would be simplified by looking at a smaller domain of
documents with more quantifiable filters, e.g. analyzing just experience
reports, or articles about security, or articles about language design
decisions. Conversely, studies that attempt to just categorize the discourse
by identifying common topics can reduce work done in later studies on any
given category. Lastly, applying a consistent content analysis methodology is
challenging given the open-ended, free-form nature of synthesizing connections
in document surveys. Perhaps here the PL/HCI community could provide more
guidance on standard methodologies for ethnomethodological analyses.

The second question is the applicability of this study’s results: what further
experiments do our results suggest? We believe the logical next step is to
develop surveys targeted towards understanding the extent of the issues
identified above. While usability surveys often resort to generic questions
like “where do you think Rust can improve?” (e.g. as in [13]), the hypotheses
above suggest more specific questions like:

  * •

How has your perception of Cargo’s value changed between starting with Rust
and today?

  * •

What parts of Rust do you find most useful that you weren’t told about
initially?

  * •

What are examples of times when you couldn’t solve an issue with the borrow
checker after several hours? If you eventually solved it, how did you solve
it?

  * •

What factors influence your likelihood to adopt Rust into an existing non-Rust
project?

Given feedback on these questions from the community, future work could
explore controlled experiments for more replicable identification of usability
issues, as well as analyze how insights gleaned from Rust can generalize to
other programming language communities.

## References

  * [1] The Rust Programming Language.  URL: https://www.rust-lang.org/en-US/.
  * [2] Maria Christakis and Christian Bird.  What developers want and need from program analysis: an empirical study.  In Automated Software Engineering (ASE), 2016 31st IEEE/ACM International Conference on, pages 332–343. IEEE, 2016.
  * [3] Michael de Lang.  Why I’m dropping Rust, Sep 2016.  URL: https://hackernoon.com/why-im-dropping-rust-fd1c32986c88.
  * [4] Harold Garfinkel.  Studies in ethnomethodology.  1967\.
  * [5] Kotsya.  Rust: Not so great for codec implementing, Jul 2017.  URL: https://codecs.multimedia.cx/2017/07/rust-not-so-great-for-codec-implementing/.
  * [6] Manishearth.  Rust severely disappoints me.  Reddit, Jan 2017.  URL: https://www.reddit.com/r/rust/comments/5nl3fk/rust_severely_disappoints_me/.
  * [7] David Martin, Benjamin V Hanrahan, Jacki O’Neill, and Neha Gupta.  Being a turker.  In Proceedings of the 17th ACM conference on Computer supported cooperative work & social computing, pages 224–235. ACM, 2014.
  * [8] Federico Mena-Quintero.  Rust things I miss in C, Feb 2018.  URL: https://people.gnome.org/~federico/blog/rust-things-i-miss-in-c.html.
  * [9] Leo A Meyerovich and Ariel S Rabkin.  Empirical analysis of programming language adoption.  ACM SIGPLAN Notices, 48(10):1–18, 2013.
  * [10] Baishakhi Ray, Daryl Posnett, Vladimir Filkov, and Premkumar Devanbu.  A large scale study of programming languages and code quality in github.  In Proceedings of the 22nd ACM SIGSOFT International Symposium on Foundations of Software Engineering, pages 155–165. ACM, 2014.
  * [11] Greg Stoddard.  Popularity and quality in social news aggregators: A study of reddit and hacker news.  In Proceedings of the 24th international conference on world wide web, pages 815–818. ACM, 2015.
  * [12] The Rust Core Team.  Rust’s 2018 roadmap, Mar 2018.  URL: https://blog.rust-lang.org/2018/03/12/roadmap.html.
  * [13] Jonathan Turner.  Rust 2017 survey results, Sep 2017.  URL: https://blog.rust-lang.org/2017/09/05/Rust-2017-Survey-Results.html.
  * [14] Anna Zeng, Will Crichton, and Niko Matsakis.  Interview with Niko Matsakis, May 2018.
  * [15] Anna Zeng and Josh Matthews.  Interview with Josh Matthews, May 2018.


# Superoscillating Quantum Control Induced By Sequential Selections

Yongcheng Ding  Department of Physical Chemistry, University of the Basque
Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  International Center of
Quantum Artificial Intelligence for Science and Technology (QuArtist)
and Department of Physics, Shanghai University, 200444 Shanghai, China  Yiming
Pan  School of Physical Science and Technology and Center for Transformative
Science, ShanghaiTech University, Shanghai 200031, China  Xi Chen  Department
of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644,
48080 Bilbao, Spain  EHU Quantum Center, University of the Basque Country
UPV/EHU, 48940 Leioa, Spain

###### Abstract

Superoscillation is a counterintuitive phenomenon for its mathematical feature
of ”faster-than-Fourier”, which has allowed novel optical imaging beyond the
diffraction limit. Here, we provide a superoscillating quantum control
protocol realized by sequential selections in the framework of weak
measurement, which drives the apparatus (target) by repeatedly applying
optimal pre- and post-selections to the system (controller). Our protocol
accelerates the adiabatic transport of trapped ions and adiabatic quantum
search algorithm at a finite energy cost. We demonstrate the accuracy and
robustness of the protocol in the presence of decoherence and fluctuating
noise and elucidate the trade-off between fidelity and rounds of selections.
Our findings provide avenues for quantum state control and wave-packet
manipulation using superoscillation in quantum platforms such as trapped ions.

The concept of superoscillation (SO) was originally proposed as a footnote in
a celebrated study on quantum measurement by Y. Aharonov et al. AAV  in the
late 1980s. The phenomenon occurs when a band-limited wave function varies
arbitrarily faster than its fastest Fourier components, as allowed by its
spectral content. In other words, a superoscillatory wave is a local feature
that oscillates at a much higher frequency than the overall frequency of the
global band-limited wave. This counterintuitive but physically allowed
property is dubbed as faster-than-Fourier by M. Berry FTF  and others roadmap
, which offers promising optical applications by breaking the diffraction
barrier nrp . It has been applied in super-resolution imaging, manipulating
nanoparticles, electrons, and atoms with spatiotemporally shaped light beams
manipulation ; space ; Atom ; time .

Meanwhile, with the advent of state-of-the-art quantum technologies, ingenious
protocols using weak measurements are now attainable for a variety of quantum
applications reviewweak , including quantum steering steering ; gefen ,
quantum tomography tomo ; tomosc , geometric information geometric ; eraser ;
discord , and transition detection w2s ; topological . Moreover, a sequential
weak measurement enables the production of SO by encoding the amplified weak
value of an operator from the repeatedly pre- and post-selected system into
the coupled quantum states of the apparatus pointer . This inspires us to
construct a general quantum control framework using SOs for the application
scenarios where quantum information encoding is essential for specific
proposals of quantum state steering and wave-packet manipulation.

In this Letter, we propose a framework called superoscillating quantum control
(SQC). We exert sequential pre- and post-selections on the system to construct
a superoscillating operator function that can uniformly shift the apparatus
after each round of selections, resembling shortcuts to adiabaticity stareview
. This SQC framework requires the design of optimal selections for efficient
quantum control while gently perturbing the ground state. We demonstrate two
preliminary results: a fast nonadiabatic transport of a single trapped ion and
a speed-up quantum search algorithm for general quantum computing. At the same
energy cost energetic , SQC delivers higher fidelity to conventional adiabatic
control and still outperforms when the probabilistic cost is included.
Furthermore, by investigating noise in open systems, we reveal the trade-off
between selection rounds and fidelity, showing two types of mechanisms and
their consequences for the speed and robustness of SQC protocols.

High-fidelity fast quantum transport.— Let us start by considering the use of
SQC to perform the fast transport of a single trapped ion without final
motional excitation. This is motivated by the need for coherent manipulation
of trapped ions for quantum information processing, simulations, and
metrology. In the Lamb-Dicke limit, where
$\eta\sqrt{\langle(a+a^{\dagger})^{2}\rangle}\ll 1$, as discussed Ref.
reviewtrappedion , the two-level interaction of a trapped ion with a
monochromatic photon mode of the light field is described by the Hamiltonian

$H_{\text{LD}}=\frac{\hbar}{2}\Omega{\hat{\sigma}_{+}}\left[1+i\eta\left(\hat{a}e^{-i\nu
t}+\hat{a}^{\dagger}e^{i\nu t}\right)\right]e^{i(\phi-\delta t)}+\text{H.c.},$
(1)

where $\Omega$ is the Rabi frequency, $\hat{\sigma}_{+}$ is the spin raising
operator of the two-level system, $\eta=kx_{0}$ is the Lamb-Dicke parameter
with $x_{0}=\sqrt{\hbar/(2M\nu)}$ being the characteristic length, $\hat{a}$
and $\hat{a}^{\dagger}$ the motional annihilation and creation operator, $\nu$
the trap frequency, $\phi$ and $k$ the laser phase and wave vector, $\delta$
the detuning, $M$ the ion’s mass. A spin-motion coupling can be implemented by
employing red sideband $(\delta=-\nu)$ and blue sideband $(\delta=\nu)$
resonances with laser phases $(\phi_{r},~{}\phi_{b})=(-\pi/2,~{}\pi/2)$,
resulting in a Hamiltonian with spin-orbit coupling
$H=g\hat{\sigma_{x}}\otimes\hat{p}$ Lucas . In the weak coupling regime when
$g=\eta\Omega x_{0}$, this enables high-fidelity fast transport of trapped ion
using SQC.

Figure 1:  (a) Schematic diagram of SQC in trapped ion quantum platform using
sequential selections (2), where two lasers are tuned to the first red and
blue sideband to couple the internal spin states to the motional mode.
Iterative pre-selection of the two-level system for $|i\rangle$ and post-
selection for $|f\rangle$ (or vice versa) accumulates weak value on the
position of the Gaussian-type apparatus, realizing SQC with the probability of
$P=|\langle i|f\rangle|^{2N}$ after $N$ rounds without an ensemble of $N$
particles. (b) Final apparatus states for transporting a distance of $d=x_{0}$
with interaction strength $g=1$, where SQC achieves a fidelity of
$F=|\langle\Psi(x,T)|\Psi_{G}\rangle|^{2}=0.995$ with $g\delta T=0.341$ and
$N=20$. Selections (Superoscillating Quantum Control Induced By Sequential
Selections) on the two-level system shift the initial ground state (grey) to
final states by single shot selection (red) and $N=20$ rounds(blue) after
$T=3.536$ and $6.830$, respectively. As a calibration, single-round selections
show two non-overlapping red wave packets that cannot result in the
unidirectional ion transport. The relevant dimensionless mass and trap
frequency are $M=\nu=1$, and $P=0.25$. (c) Fidelity dependence on $N$ via SQC
for transport distances of $d=7.5x_{0}$ (dot), $10x_{0}$ (triangle), and
$15x_{0}$ (cross) with $g=0.75,~{}1,~{}1.5$, and $P=0.9$. (d) Fidelity
dependence on $T$ via AQC for the same transport distances as in (c). We move
the center of the trap by defining
$x_{\text{trap}}=3dt^{2}/T^{2}-2dt^{3}/T^{3}$.

To realize SQC with the setup shown in Fig. 1(a), we apply $N$ rounds of
sequential quantum-state pre- and post-selections by alternatively projecting
the system into the initial state $|i\rangle$ and $|f\rangle$, and coupling
the apparatus weakly to the system between two selections. The sequential
selections on the system can exert a longstanding influence on the quantum
state of apparatus, leading to the final state given by

$\displaystyle|\Psi^{F}_{A}\rangle=|\langle
f|i\rangle|^{N}\underbrace{\left[\cos\left(\frac{gT}{\hbar}\frac{\hat{p}}{N}\right)-i\sigma_{xw}\sin\left(\frac{gT}{\hbar}\frac{\hat{p}}{N}\right)\right]^{N}}_{=:f(p)}|\Psi(x)\rangle,~{}~{}$
(2)

where $\sigma_{xw}=\langle f|\hat{\sigma_{x}}|i\rangle/\langle f|i\rangle$ is
the weak value of $\hat{\sigma_{x}}$, $T$ is the total operation time, and
$|\Psi(x)\rangle$ is the initial motional wave function of the apparatus. The
sequential selections evolves the wave function by a superoscillating operator
function $f(\hat{p})$ in Eq. (2). Mathematically, it allows the parameter
$\sigma_{xw}>1$ and $\sigma_{xw}\in\mathcal{R}$, even though $\sigma_{xw}$ as
a weak value can be complex in practice jozsa . The SO occurs with a low
probability of $|\langle f|i\rangle|^{2N}$ since we discard the wave function
and initialize the system once any selection on internal states fails. It
accumulates a weak value on the apparatus by kicking its position $\delta
x=gT\sigma_{xw}/N$ in each round, amplifying it by $N$ times without an
ensemble of $N$ ions.

In addition to compensating force and inverse engineering Erik ; An ;
transportreview , the SO described in Eq. (2) provides an alternative
shortcut-to-adiabaticity transport by periodically kicking the particle
without excitation. We can treat the motional wave function as the target
since there is a system-apparatus duality, making the selections on the
internal states the effective controller. To determine the cost of using SO,
we need to analyze the relation between probabilities and transport distance
and optimize the selections accordingly. A successful post-selection that
projects an arbitrary state from $|i\rangle$ to $|f\rangle$, has a probability
of $p=|\langle f|i\rangle|^{2}$. For our concern, to obtain a weak value of
both real-valued and larger-than-one, we construct the two selected states

$\displaystyle|i\rangle=\sqrt{\frac{1-\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1+\sqrt{1-p}}{2}}|1\rangle,$
$\displaystyle|f\rangle=\sqrt{\frac{1+\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1-\sqrt{1-p}}{2}}|1\rangle,$
(3)

where $p$ is the probability of successful projection, which can result in an
optimal weak value of $\sigma_{xw}=1/\sqrt{p}$. Therefore, the SQC protocol
can shift the trapped ion by $d=gT/\sqrt{p}$, at the cost of $P=p^{N}$. At
first glance, it may seem problematic because the displacement is independent
of the number of selection $N$, and the total probability falls exponentially.
That is, one can couple the internal states and the motional mode for the time
interval of $T$ in a single shot of measurement. However, the coupling
strength $gT$ maybe too large so that the coupling no longer evolves as a
translation operator on the motional wave function. From an intuitive
perspective, the post-selection on the system after evolving the system-
apparatus Hamiltonian determined by $\langle
f|\exp(-igT\hat{\sigma_{x}}\otimes\hat{p})|\Psi(x)\rangle|i\rangle$, splitting
the apparatus to the cat state (or kitty-state when two packets are not well
separated):

$\displaystyle|\text{cat}\rangle=\frac{1+\sqrt{p}}{2\mathcal{N}}|\Psi(x-gT)\rangle+\frac{-1+\sqrt{p}}{2\mathcal{N}}|\Psi(x+gT)\rangle,$
(4)

where $\mathcal{N}$ is the coefficient for normalization. Noteably, we present
the exact final quantum state of the apparatus, which is applicable for
measurements with arbitrary strengths, ranging from weak to strong coupling.
The overlap between two wave packets is crucial to define the measurement
transition, given by $\langle\Psi(x-gT)|\Psi(x+gT)\rangle=\exp(-\Gamma^{2})$,
where $\Gamma=gT/(\sqrt{2}x_{0})$ is an interference factor. Using this, we
derive the expected net shift of the apparatus:

$\langle\delta
x\rangle=\langle\text{cat}|x|\text{cat}\rangle=\frac{2\sqrt{p}gT}{1+p+\exp(-\Gamma^{2})(-1+p)},$
(5)

which corresponds to the weak-valued readout $\sigma_{xw}gT=gT/\sqrt{p}$
asymptotically as $\Gamma\rightarrow 0$, while in the opposite limit of strong
measurement $\Gamma\rightarrow\infty$, it results in a bizarre readout of
$2\sqrt{p}gT/(1+p)$ that does not match the expectation value of
$\hat{\sigma_{x}}$ in the initial or final state. More importantly, the net
shift at weak measurement limit can be amplified when $p\rightarrow 1$. To
preserve the unidirectional shift, we construct the SQC by periodically
decoupling the system and the apparatus after every $\delta T=T/N$, ensuring
each measurement is in the weak measurement regime. Fig. 1(b) plots the
difference between single-shot measurement and SQC for a target distance of
$d=10x_{0}$ with a low total probability of $P=0.25$ to increase result
contrast.

In Fig. 1(c), we set the probability to $P=0.9$ and vary the number of
selections from $N=1$ to $16$ to test the ideal performance of our protocol
under different measurement strengths. We fix the operation time corresponding
to $N$ in each set of settings by letting the coupling strength be
proportional to the transport distance as (i): $g=0.75,~{}d=7.5x_{0}$, (ii):
$g=1,~{}d=10x_{0}$, (iii): $g=1.5,~{}d=15x_{0}$. The results support our
theory that the fidelity should increase with $N$ and decrease with transport
distance.

We notice that the extra time cost (compared with $T=6.708$ for $N=1$) for SQC
is not remarkable since the nonoverlapping wave packet on the left is almost
negligible with near-parallel selection for $P\approx 1$. Furthermore, we
highlight that the SQC converges to the lossless expectation value
amplification (also the eigenvalue $\langle\hat{\sigma_{x}}\rangle=1$) with
near parallel selection of the system states. This convergence bounds the
quantum speed limit of lossless transport as $\dot{x}=g$. In this case, the
apparatus (4) reduces to a single wave packet $|\Psi(x-gT)\rangle$, and
equalizes the weak value $1/\sqrt{p}$ or the the previous bizarre readout
$2\sqrt{p}/(1+p)$ in either limit. Alsom in Fig. 1(d), we benchmark our
protocol by moving the center of the harmonic trap to the target with the
following trajectory $x_{\text{trap}}=3dt^{2}/T^{2}-2dt^{3}/T^{3}$. $T=N\delta
T$ is the operation time for the SQC of $N$ rounds. Our protocol significantly
outperforms such adiabatic quantum control (AQC) in fidelity
$F=|\langle\Psi(x,T)|\Psi_{G}\rangle|^{2}$, where $|\Psi_{G}\rangle$ is a
Gaussian ground state at the target site, and prevails even when takingt the
probability of the selections into account.

Speeding up adiabatic quantum search algorithm.— Now we generalize the
superoscillating quantum control to include the angular parameters that
characterize the wave function and enable the processing of quantum
information. Similar to the weak value accumulation on the spatial parameter,
the accumulation on the angular parameter amplifies the corresponding
coefficient of the target basis. This is analogous to the adiabatic version of
Grover’s algorithm haystack , searched for the target $|t\rangle$ from the
database $|\Psi\rangle$ with a complexity of $\mathcal{O}(\sqrt{N_{G}})$ in
the context of digitized quantum computing, where $N_{G}$ is the number of
entries. This algorithm iteratively evolves the Grover’s operator
$\hat{G}=U_{\Phi}U_{t}$, where the oracle operator $U_{t}=I-2|t\rangle\langle
t|$ and the diffusion operator $U_{\Psi}=2|\Psi\rangle\langle\Psi|-I$ flip the
phase of the wave function in the target subspace and the phase of the initial
database, respectively. By decomposing $\hat{G}$ into a gate sequence, the
algorithm rotates the wave function from the database to the target:

$\hat{G}^{N}|\Psi\rangle=\sin\left(\frac{2N+1}{2}\theta\right)|t\rangle+\cos\left(\frac{2N+1}{2}\theta\right)|\tilde{t}\rangle.$
(6)

We aim to implement an interaction Hamiltonian $H=g\hat{\sigma}\otimes\hat{J}$
that couples the system and the apparatus, where $\hat{\sigma}$ is the Pauli
operator and $\hat{J}$ is the angular momentum. By sequentially selecting the
system state, we can manipulate the quantum information encoded in angular
parameters, resulting in the following SO on the apparatus

$\displaystyle|\Psi_{A}^{F}\rangle=|\langle
f|i\rangle|^{N}|\Psi(\theta/2+N\sigma_{w}g\delta T)\rangle,$ (7)

where the interaction lasts for a duration of $\delta T=T/N$ in each sequence.
If we define $g\delta T\sigma_{w}=\theta$, the SO is equivalent to Eq. (6).
This oscillating quantum search requires both artificial design of selections,
the angular momentum operator, and the implementation of the interaction
Hamiltonian, which avoid decomposing the oracles in digitized quantum
computing. In Fig. 2(a), we demonstrate the algorithm by using a two-qubit
system, as a minimal quantum model that searches for the target state
$|t\rangle=|0\rangle$ out of $N_{G}=2$ entries. Assuming that the initial
database is
$|\Psi\rangle=|+\rangle=\sin(\pi/4)|0\rangle+\cos(\pi/4)|1\rangle$, we can
achieve a perfect query by setting $g\delta T\sigma_{yw}=\pi/4N$ instead of
using the standard Grover’s algorithm. Accordingly, we use the spin-spin
interaction for implementing the SQC, which is indeed the Heisenberg type
$H_{\text{SQC}}=-g\hat{\sigma_{y}}\otimes\hat{\sigma_{y}}$ review2021 , when
an auxiliary qubit is introduced as the controller (system). We add the
relative phase term $\exp(-i\pi/2)$ on the coefficients of Eq.
(Superoscillating Quantum Control Induced By Sequential Selections), and
trade-off between the probability and weak value accumulation remains.

We hereby present a quantitative study comparing the energy cost of SQC with
AQC. In the AQC approach, we evolve the time-dependent Hamiltonian
$H(t)=[1-\lambda(t)]H_{i}+\lambda(t)H_{f}$, with $\lambda(t)$ ranging from
zero to a certain value within the operation time. We use the type-I
Hamiltonian as a benchmark for the specific quantum search stagrover , given
by

$\displaystyle H_{\textup{I}}(t)$ $\displaystyle=$
$\displaystyle(1-t/T)\Omega\hat{\sigma_{x}}+(t/T)\Delta\hat{\sigma_{z}},$ (8)

with Rabi frequency $\Omega$, and the detuning $\Delta$. Additionally, we use
the more general type-II Hamiltonian aqcgrover

$\displaystyle H_{\textup{II}}(t)$ $\displaystyle=$
$\displaystyle(1-t/T)K(I-|\Psi\rangle\langle\Psi|)+(t/T)K(1-|t\rangle\langle
t|),$ (9)

as the baseline, where $K$ is a scaling coefficient. We evaluate the
performance of SQC by measuring fidelities $F=|\langle\Psi(T)|t\rangle|^{2}$
on different $N$ with the same probability $P=0.9$ and coupling strength $g=1$
in Fig. 2(b). It is evident that the fidelity can be improved without
increasing the operation time $T$ by using a larger Rabi frequency and
detuning in the type-I Hamiltonian, or equivalently scaling up $K$ in the
type-II Hamiltonian. Therefore, it is critical to define the energey cost
energetic  that bounds the input energy to the system for a fair comparison.
We use the Frobenius norm of the total Hamiltonian to define the instantaneous
cost of the evolution $\partial_{t}C=||H(t)||$, integrate and average it over
the operation time as

$C=\frac{1}{T}\int_{0}^{T}||H(t)||dt,$ (10)

yielding $C_{\text{SQC}}=2g$ for the superoscillating quantum search. Assuming
$\Omega=\Delta$, we derive the cost of type-I and type-II Hamiltonian as
$C_{\textup{I}}=\Omega[2\sqrt{2}-\log(-1+\sqrt{2})+\log(1+\sqrt{2})]/4$ and
$C_{\textup{II}}=K[4+3(-\log 1+\log 3)]/8$, respectively. Fig. 2(c) shows the
energy cost of adiabatic algorithms to SQC, along with the fidelities of both
types of Hamiltonian for adiabatic quantum search within $T$, corresponding to
selections of $N$ rounds. Numerical results prove that SQC dramatically
accelerates the adiabatic Grover’s algorithm, even if both probabilistic and
energetic cost are considered. We analyze its extension to arbitrary databases
by applying the multi-qubits in the Supplementary Material supplementary . The
model is physically feasible in the trapped ion platform, where Mølmer-
Sørensen gates serve naturally as analog simulators for the interaction.

Figure 2:  (a) Schematic diagram of quantum search algorithm using SQC (7).
$N=8$ rounds of selections on the controller qubit, driven by the spin-spin
interaction Hamiltonian
$H_{\text{SQC}}=-g\hat{\sigma_{y}}\otimes\hat{\sigma_{y}}$, rotate the target
qubit from the initial database $|\Psi\rangle=|+\rangle$ to the target
$|t\rangle=|0\rangle$. (b) Fidelity dependence on $N$ via SQC. The probability
of SQC and coupling strength are set to $P=0.9$ and $g=1$, respectively. (c)
Fidelity dependence on $T$ via AQC. The energetic cost is bounded, and the AQC
Hamiltonians (8) (red triangle), (9) (red cross) are evolved for $T$ that
depends on the rounds of selections in SQC algorithm.

Possible experimental implementation.— Both examples in the ideal simulations
approach the lossless SO in the limit of large $N$. To evaluate the protocol
in a laboratory environment, we generalize it to open quantum systems governed
by the Lindblad master equation. We account for imperfect selections that can
introduce atomic loss and quantum noises, reducing fidelity by a fixed
proportion $P_{\text{error}}$. These mechanisms can create a trade-off between
the fidelity and the number of selections, where the fidelity reaches a
maximum and then decreases as $N$ increases beyond a critical value. Note that
the trade-off of the first type, induced by quantum noises, does not
necessarily exist in all cases. As $N$ increases, the protocol preserves the
better property of SQC, which improves fidelity with extra operation time,
while reducing fidelity as quantum noises affect the system more. Whether this
trade-off exists depends on which factor dominates, extra fidelity gain or
loss.

To characterize the dephased two-level system (internal state) and damped
quantum harmonic oscillator (motional mode) involved in the ion transport
task, we use the collapse operators
$C_{\text{TLS}}=\sqrt{\gamma}\hat{\sigma_{z}}$,
$C_{\text{HO}}=\sqrt{\gamma}\hat{a}~{},\sqrt{\gamma}\hat{a}^{\dagger}$,
respectively. Solving the Lindblad master equation enables us to obtain the
fidelity in the formulation of density matrix approach. As a result of the
damping, the expected shift of the damped apparatus is smaller than that of
the ideal apparatus. As shown in Fig. 3(a), we set the same dephasing rate and
damping rate to $\gamma=0.01$, and the additional fidelity loss per selection
to $P_{\text{error}}=5\times 10^{-3}$. Although the only trade-off in this
parameter setting is induced by imperfect projection, the trade-off of the
first type can be observed at a small critical $N$ by tuning down the
probability of SQC. In the quantum search algorithm, we impose local dephasing
on the control qubit (system) and target qubit (apparatus) by
$C_{\text{TLS}}=\sqrt{\gamma}\hat{\sigma_{z}}$, resulting in a decrease in
fidelity due to purity loss. As calibrations, the evaluation of the algorithm
with the same parameters is shown in Fig. 3(b,c), which confirms the existence
of a trade-off mechanisms of two types.

Figure 3:  (a) Fidelity vs. $N$ in SQC-based quantum transport, with quantum
noises characterized by dephasing and damping rates of $\gamma=0.01$.
Simulation results for the open quantum system are multiplied by
$1-P_{\text{error}}=0.995$ after each selection (red markers) to mimic atomic
loss induced by imperfect projections. Parameters and markers are the same as
in Fig. 1(c). (b) Fidelity vs. $N$ in SQC-based quantum search, with local
dephasing modeled by $C_{\text{TLS}}=\sqrt{\gamma}\hat{\sigma_{z}}$ on the
controller and target qubits. Results are shown for perfect (blue) and
imperfect (red) projections with the same settings as in (a) and in Fig. 2(b).

Conclusion and outlook.— In summary, we have introduced a general SQC
framework and demonstrated its efficiency in two applications with trapped
ions. Our approach utilizes pre- and post-selections of the system to achieve
optical projection design, and SQC for atomic non-adiabatic transport and the
quantum search problem. We have shown that SQC can speed up conventional
adiabatic control, offering a promising alternative toward shortcuts to
adiabaticity. Numerical simulations demonstrate that SQC has advantages in
terms of energy cost, even when considering the probability of occurrence. We
have also extended the SQC protocol to open quantum systems, where noise
affects its performance, resulting in two types of trade-offs between
selections and fidelity.

While our focus is on the trapped ion system, insights can be gained from the
cold atoms in spin-dependent optical potentials Martin , and electrons in
semiconductor quantum dots Sherman . Additionaly, our work can be extended to
coherent matter-wave splitting, as we have presented the analytical
formulation of the apparatus after post-selection. Furthermore, SQC for
collective behaviour has applications in the emergence or suppression of
superradiant phase transition in the Dicke model dicke . We believe that SQC
can provide a deeper understanding of quantum foundations and offer potential
for various quantum technologies.

###### Acknowledgements.

Acknowledgements.— This work has been financially supported by NSFC
(12075145), EU FET Open Grant EPIQUS (899368), QUANTEK project
(KK-2021/00070), the Basque Government through Grant No. IT1470-22, and the
project grant PID2021-126273NB-I00 funded by MCIN/AEI/10.13039/501100011033
and by ”ERDF A way of making Europe” and ”ERDF Invest in your Future”. X.C.
acknowledges the Ramón y Cajal program (RYC-2017-22482).

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Supplemental Material: Superoscillating Quantum Control Induced By Sequential
Selections

##  .1 Bichromatically excited single trapped ion for atomic transport

In the Lamb-Dicke regime, the criteria
$\eta\sqrt{\langle(\hat{a}+\hat{a}^{\dagger})^{2}\rangle}\ll 1$ should hold
during the operation time. the Lamb-Dicke Hamiltonian contains the red
sideband and blue sideband as resonances. The first red sideband $\delta=-\nu$
reads

$H_{\text{rsb}}=\frac{\hbar}{2}\Omega\eta\left[\hat{a}\hat{\sigma}_{+}\exp\left(i\phi_{r}\right)+\hat{a}^{\dagger}\hat{\sigma}_{-}\exp\left(-i\phi_{r}\right)\right],$
(S1)

where $\phi_{r}$ is the phase of the laser, giving the transition
$|n\rangle|g\rangle\leftrightarrow|n-1\rangle|e\rangle$ with the effective
Rabi frequency $\Omega\sqrt{n}\eta$. It couples the internal states and the
motional mode of the single trapped ion. Similarly, we write down the first
blue sideband $\delta=-\nu$ as

$H_{\text{bsb}}=\frac{\hbar}{2}\Omega\eta\left[\hat{a}^{\dagger}\hat{\sigma}_{+}\exp\left(i\phi_{b}\right)+\hat{a}\hat{\sigma}_{-}\exp\left(-i\phi_{b}\right)\right],$
(S2)

giving another transition
$|n\rangle|g\rangle\leftrightarrow|n+1\rangle|e\rangle$ with the effective
Rabi frequency $\Omega\sqrt{n+1}\eta$. These resonances lead to the well-known
Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonian, respectively. We
implement the light-matter interaction by turning on the bichromatic laser,
and reduce the Lamb-Dicke Hamiltonian to

$H_{\text{bic}}=H_{\text{rsb}}+H_{\text{bsb}}=-i\frac{\hbar\Omega}{2}\eta\hat{\sigma_{x}}\left(\hat{a}-\hat{a}^{\dagger}\right),$
(S3)

with $\phi_{r}=-\pi/2$ and $\phi_{r}=\pi/2$. By substituting
$\hat{a}=\sqrt{M\nu/2\hbar}(\hat{x}+i\hat{p}/M\nu)$ and
$\hat{a}^{\dagger}=\sqrt{M\nu/2\hbar}(\hat{x}-i\hat{p}/M\nu)$ into (S3), we
derive the interaction Hamiltonian in the main text as
$H=g\hat{\sigma_{x}}\otimes\hat{p}$, where $g=\eta\Omega x_{0}$ and
$x_{0}=\sqrt{\hbar/2M\nu}$.

##  .2 Weak-to-strong quantum measurement, weak value, and expectation value

We present a detailed calculation of the weak-to-strong quantum measurement,
in order to verify the necessity of using SO instead of a single shot of
strong measurement. One chooses the states for pre-selection and post-
selection, and express them in the eigenbasis of the operator for measurement,
e.g.,

$\displaystyle|i\rangle=a_{0}|+\rangle+a_{1}|-\rangle,$
$\displaystyle|f\rangle=\tilde{a_{0}}|+\rangle+\tilde{a_{1}}|-\rangle,$ (S4)

if we measure the weak value of $\hat{\sigma_{x}}$. Such quantum measurement
process with arbitrary measurement strength exacly reads

$\displaystyle\langle
f|\exp(-igT\hat{\sigma_{x}}\otimes\hat{p})|i\rangle\otimes|\Psi(x)\rangle$
(S5) $\displaystyle=$ $\displaystyle
c_{0}|\Psi(x-gT)\rangle+c_{1}|\Psi(x+gT)\rangle,$

where $c_{0}=a_{0}\tilde{a_{0}}^{*}$ and $c_{1}=a_{1}\tilde{a_{1}}^{*}$. We
can renormalize these two wave packets into a cat state as

$\displaystyle|\text{cat}\rangle$ $\displaystyle=$
$\displaystyle\frac{c_{0}|\Psi(x-gT)\rangle+c_{1}|\Psi(x+gT)\rangle}{\sqrt{[c_{0}^{*}\langle\Psi(x-gT)|+c_{1}^{*}\langle\Psi(x+gT)][c_{0}|\Psi(x-gT)\rangle+c_{1}|\Psi(x+gT)\rangle]}},$
(S6) $\displaystyle=$
$\displaystyle\frac{c_{0}|\Psi(x-gT)\rangle+c_{1}|\Psi(x+gT)\rangle}{\sqrt{|c_{0}|^{2}+|c_{1}|^{2}+(c_{0}^{*}c_{1}+c_{0}c_{1}^{*})\langle\Psi(x-gT)|\Psi(x+gT)\rangle}}.$

We quantify the quantum interference in the measurement by
$\langle\Psi(x-gT)|\Psi(x+gT)\rangle=\exp(-\Gamma^{2})$, and define the
interference factor $\Gamma=gT/(\sqrt{2}x_{0})$. We obtain the measurement
outcome by calculating the central position shift from the initial Gaussian
ground state to the cat state

$\displaystyle\delta
x=\langle\text{cat}|\hat{x}|\text{cat}\rangle=\frac{(|c_{0}|^{2}-|c_{1}|^{2})gt}{|c_{0}|^{2}+|c_{1}|^{2}+(c_{0}^{*}c_{1}+c_{0}c_{1}^{*})\exp(-\Gamma^{2})}.$
(S7)

We have $\langle\delta x\rangle|_{\Gamma\rightarrow
0}=\frac{(|c_{0}|^{2}-|c_{1}|^{2})gt}{|c_{0}|^{2}+|c_{1}|^{2}+c_{0}^{*}c_{1}+c_{0}c_{1}^{*}}=\sigma_{xw}gt$
in the weak-coupling regime. However, $\langle\delta
x\rangle|_{\Gamma\rightarrow\infty}=\frac{(|c_{0}|^{2}-|c_{1}|^{2})gt}{|c_{0}|^{2}+|c_{1}|^{2}}$
does not always amplifies the expectation value $\langle
f|\hat{\sigma_{x}}|f\rangle$ in the strong-coupling regime. In our main text,
we choose the optimal selection for atomic transport

$\displaystyle|i\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1-\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1+\sqrt{1-p}}{2}}|1\rangle,$
$\displaystyle|f\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1+\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1-\sqrt{1-p}}{2}}|1\rangle,$
(S8)

which the weak value is $1/\sqrt{p}$ and the expectation value $\langle
f|\hat{\sigma_{x}}|f\rangle=\sqrt{p}$. However, in the strong measurement
limit we observe a readout of $\frac{2\sqrt{p}}{1+p}$ instead. To equalize the
measurement readout and the expectation value, we artificially design a pre-
selection and post-selection, e.g.,

$\displaystyle|i\rangle=|0\rangle,|f\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle,$
(S9)

which its weak value and expectation value are $\frac{\sqrt{1-p}}{\sqrt{p}}$
and $2\sqrt{-(-1+p)p}$, that the latter one agrees with the measurement
readout in the strong-coupling limit. We show the weak-to-strong transition
under both settings in Fig. S1.

Figure S1:  The relative central position shift from the initial Gaussian
state to the cat state $\langle\delta x\rangle/gT$ as a function of the
selection probability $p$ with different measurement strength characterized by
the interference factor $\Gamma=gt/(\sqrt{2}x_{0})$. (a) We set the states for
pre-selection and post-selection be
$|i\rangle=\sqrt{\frac{1-\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1+\sqrt{1-p}}{2}}|1\rangle$
and
$|f\rangle=\sqrt{\frac{1+\sqrt{1-p}}{2}}|0\rangle+\sqrt{\frac{1-\sqrt{1-p}}{2}}|1\rangle$,
respectively. The thick black solid curve denotes the weak value readout
$\sigma_{xw}=1/\sqrt{p}$ in the weak measurement limit $\Gamma\rightarrow 0$.
The thick black dashed curve denotes the bizzare readout $2\sqrt{p}/(1+p)$ in
the strong measurement limit $\Gamma\rightarrow\infty$, which is not in
accordance with the expectation value $\langle
f|\hat{\sigma_{x}}|f\rangle=\sqrt{p}$. Other possible readout curves lie
between them in the color according to the value of $\Gamma$. We circle the
point for eignenstate projection of $|+\rangle$, where the weak and strong
measurement meet each other losslessly. (b) We set the states for pre-
selection and post-selection be $|i\rangle=|0\rangle$ and
$|f\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, respectively, for
equalizing the expectation value and the readout in the strong measurement
limit as $2\sqrt{-(-1+p)p}$. The weak measurement output is
$\frac{\sqrt{1-p}}{\sqrt{p}}$. Here the eigenstate projection has a
probability of $0.5$. Other configurations definitions are the same as those
in the previous subfigure.

##  .3 Analog simulation of quantum search with Mølmer-Sørensen gate

In the maintext, we propose the YY interaction for recovering Grover’s search
with sequential selections. As we mentioned, we can use two trapped ion to
perform an analog simulation. We excite them bichromatically, which the
Hamiltonian reads

$H_{\text{bic}}(t)=\frac{\hbar}{2}\Omega\left(\hat{\sigma}_{+}^{(1)}+\hat{\sigma}_{+}^{(2)}\right)\left\\{1+i\eta\left[\hat{a}\exp(-i\nu
t)+\hat{a}^{\dagger}\exp(i\nu t)\right]\right\\}\left[\exp(-i\delta
t)+\exp(i\delta t)\right]+\text{H.c.},$ (S10)

exactly yielding the propagator

$U_{\text{bic}}(t)=\hat{D}\left\\{\frac{\eta\Omega}{\epsilon}\left[\exp(i\epsilon
t)-1\right]\hat{J_{y}}\right\\}\exp\left\\{i\left[\frac{\eta^{2}\Omega^{2}}{\epsilon}t-\frac{\eta^{2}\Omega^{2}}{\epsilon^{2}}\sin\left(\epsilon
t\right)\right]\hat{J_{y}^{2}}\right\\},$ (S11)

where the laser detuning from the motional sidebands is $\epsilon=\nu-\delta$,
$\hat{D}(\alpha)$ is the displacement operator, and the collective spin
operator is $\hat{J_{y}}=\hat{s_{y}}^{(1)}+\hat{s_{y}}^{(2)}$. The
displacement operator vanishes when $t_{\text{MS}}=2\pi/\epsilon$. Thus, the
propagator is the action of an effective Hamiltonian
$H=-\hbar(\eta^{2}\Omega^{2}/\epsilon)\hat{s_{y}}^{(1)}\otimes\hat{s_{y}}^{(2)}$,
which effectively entangling the controller qubit $\hat{s_{y}}^{(1)}$ and the
target qubit $\hat{s_{y}}^{(2)}$. For other initial databases, one can always
find an axis for rotation, requiring
$\hat{\sigma}_{\phi}^{1}\otimes\hat{\sigma}_{\phi}^{2}$ interaction, where
$\hat{\sigma}_{\phi}=\cos\phi\hat{\sigma_{x}}+\sin\phi\hat{\sigma_{y}}$, which
is also feasible with different phases of the laser. Note that the analog
simulation does not satisfies the energetic bound in the main text since it is
not a direct realization of YY interaction.

##  .4 Construction of general angular momentum operator

For entries $N_{G}>2$, it takes multiple qubits as the database to encode the
quantum information. Thus, the physical picture of Bloch sphere in the minimal
model is no longer valid, requiring a generalization of the rotation in the
Hilbert space of higher dimension. Let us assume that the initial database for
quantum search of target state $|t\rangle$ reads

$|\Psi\rangle=\sin\frac{\theta}{2}|t\rangle+\cos\frac{\theta}{2}|\tilde{t}\rangle,$
(S12)

where $|\tilde{t}\rangle=\sum_{N_{G}-1}c_{i}|i\rangle$ is complementary basis
in the superposition of all other entires. To realize the SO for quantum
search, we need to construct a general angular momentum $\hat{J}$ that rotates
the parameterized wave function by
$\exp(-i\phi\hat{J})|\Psi(\theta/2)\rangle=|\Psi(\theta/2+\phi)\rangle$. We
can still construct such angular momentum by mimicking the rotation in the
minimal model. For example, we can rotate the wave function toward the target
about an effective Y-axis if the amplitudes of all bases are real. We denote
the effective plus state and minus state by

$\displaystyle|\tilde{+}\rangle$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}|t\rangle+\frac{\sqrt{2}}{2}|\tilde{t}\rangle,$
$\displaystyle|\tilde{-}\rangle$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2}|t\rangle-\frac{\sqrt{2}}{2}|\tilde{t}\rangle,$
(S13)

leading to the general angular momentum operator

$\hat{J_{y}}=-\frac{i}{2}(|\tilde{-}\rangle\langle\tilde{+}|-|\tilde{+}\rangle\langle\tilde{-}|),$
(S14)

which rotates the parameterized wave function as we desire.