X-ray Properties of Optically Variable Low-mass AGN Candidates

Alexander Messick
Washington State University, Department of Physics and Astronomy, WA 99164, USA

Vivienne Baldassare
Washington State University, Department of Physics and Astronomy, WA 99164, USA

Marla Geha
Yale University, Department of Astronomy, New Haven, CT 06511, USA

Jenny Greene
Princeton University, Department of Astrophysical Sciences, Princeton, NJ 08544, USA

We present an X-ray analysis of fourteen nearby ($z<0.044$) AGN in low mass galaxies ($M_*\lesssim5\times10^{9}M_\odot$) selected based on their optical variability [Baldassare et al., 2020].
Comparing and contrasting different AGN selection techniques in low-mass galaxies is essential for obtaining an accurate estimate of the active fraction in this regime.
We use both new and archival observations from the Chandra X-ray Observatory to search for X-ray point sources consistent with AGN.
Four objects have detected nuclear X-ray emission with luminosities ranging from $L_{0.5-7}\approx3\times10^{40}$ to $9\times10^{42} \rm{erg\,s^{-1}}$ with two more marginal detections.
All of the detected galaxies have luminosities exceeding those anticipated from X-ray binaries, and all sources are nuclear, suggesting the X-ray emission in most sources is due to an AGN.
These observations demonstrate the success of variability at identifying AGN in low-mass galaxies.
We also explore emission line diagnostics and discuss the differences in the results of these methods for AGN selection, in particular regarding low-mass and low-metallicity systems.

§ INTRODUCTION

Supermassive black holes (BHs; $M_{BH}\gtrsim 10^5 M_\odot$) occupy the centers of most massive ($M_*\gtrsim 10^{10} M_\odot$) galaxies, including our own [Kormendy, 2020].
Less is known, however, about their prevalence in dwarf galaxies ($M_*\lesssim 10^{10} M_\odot$).
This information could prove vital toward understanding the formation of massive BHs.
In particular, the occupation fraction of BHs in the centers of dwarf galaxies could place constraints on BH seed formation mechanisms by providing insight into the role of BHs in the evolution of their host galaxies [Greene et al., 2020, Volonteri et al., 2021].

Historically, detecting nucleic BHs has proven to be difficult since their gravitational spheres of influence are typically too small to be dynamically resolved in galaxies beyond $\sim5$ Mpc.
For this reason, much of the search for massive BHs in low-mass galaxies focuses on signs of accretion as evidence for the presence of an active galactic nucleus (AGN).
Large-scale spectroscopic surveys have allowed for the detection of AGN signatures using methods such as narrow-line diagnostics, which use ratios of narrow-line emission fluxes to classify galaxies <cit.>.
While these methods have been successful in identifying a large number of new AGN [Reines et al., 2013, Moran et al., 2014], there likely exists a population that is undetected by optical spectroscopy due to selection effects [Trump et al., 2015, Bradford et al., 2018].
In particular, low-mass and low-metallicity systems tend to have decreased [NII]-to-H$\alpha$ ratios, pushing them to the left and into the star-forming regime on the BPT diagram [Kauffmann et al., 2003, Cann et al., 2019].
Using alternate AGN selection techniques is necessary for obtaining an accurate estimate of the active fraction in dwarf galaxies (and thus constraining the occupation fraction).

Long-term optical variability has been a prolific tool for identifying bright AGNs [Ulrich et al., 1997, Geha et al., 2003, Schmidt et al., 2010].
More recently this has been used to identify AGN in low-mass or low-metallicity galaxies [Baldassare et al., 2018, Baldassare et al., 2020, Burke et al., 2021, Yuk et al., 2022].
Baldassare et al., 2020 used data from the Palomar Transient Factory (PTF) to search for variations in the light outputs for 35,000 low-mass galaxies, finding 238 objects in this low-mass regime.
Of this population, 75% had narrow emission lines dominated by star formation and would have been missed by performing BPT analysis alone.

X-ray observations can provide confirmation of the presence of an AGN and insight into its accretion properties.
Although some work has been done on X-rays from low-mass AGN [Desroches et al., 2009, Dong et al., 2012], most of the research on the topic has focused on galaxies more massive than dwarfs.
Additionally, X-ray emission from low-mass AGN can be difficult to disentangle from emission from X-ray binaries (XRBs).
Nevertheless, X-ray observations are thought to be one of the most reliable techniques for AGN selection, since X-ray emission from non-AGN astrophysical processes is typically comparatively weak [Hickox & Alexander, 2018].
In this work, we explore the X-ray properties of low-mass galaxies with AGN-like optical variability.
Our goals are to contrast different AGN selection techniques at the low-mass end and understand selection biases.

This paper is organized as follows: In Section <ref>, we discuss the selection of the original sample of galaxies and the origin for all the data.
In Section <ref>, we introduce the sources for all the data and the methods of analysis, displaying the results of the calculation in Section <ref>.
We then discuss the implications of the results in Section <ref>, comparing the outcomes of the different selection methods and calculating the Eddington luminosities and ratios of the objects in the sample.

c c c c c c c c c[htb]
Variability-Selected AGN with X-ray Follow-Up

NSA ID     R.A.     Decl.     Radius     Redshift     Distance     $\log{\mathrm{M_*}}$     BPT Class     FeX Emission

(hms)     (dms)     (arcsec.)     (z)     (Mpc)     ($\log{\mathrm{M_\odot}}$)

(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)

10045      08:29:12.67     +50:06:52.8     3.072      0.043     186.3     9.65     AGN       y

104881     08:00:28.55     +15:27:11.2     7.675      0.015     68.7      9.04     SF        n

124477     12:21:34.09     +04:46:46.3     25.354     0.007     31.4      9.67     —       n

124554     11:40:40.73     +59:48:50.3     8.479      0.012     50.2      9.09     —       n

15235      14:40:12.70     +02:47:43.5     7.590      0.030     126.6     9.49     AGN       y

152627     23:47:04.69     +29:28:56.2     43.790     0.017     75.1      9.48     Comp.     n

156688     08:27:23.93     +23:10:48.4     16.859     0.018     78.4      7.66     Comp.     n

51928      09:44:19.41     +09:59:05.3     8.752      0.010     45.3      9.41     SF        n

57867      16:19:02.49     +29:10:22.2     6.985      0.009     36.6      7.69     SF        n

61072      12:39:59.29     +47:38:49.7     7.086      0.031     134.9     9.11     SF        n

67333      12:26:47.95     +07:40:17.6     11.924     0.002     11.3      8.24     —       n

86652      12:42:06.47     +33:16:43.9     8.978      0.038     164.2     9.40     SF        n

88260      12:38:56.92     +38:05:24.9     9.449      0.007     33.7      8.45     SF        n

97904      15:39:50.62     +21:43:22.8     3.857      0.038     164.3     9.61     —       n

Col. (1) gives each object's ID in the NASA-Sloan Atlas. Cols. (2) and (3) give the Right Ascension and declination in units of hours:minutes:seconds and degrees:arcminutes:arcseconds, respectively. Col. (4) gives the Petrosian 90%-light radius in arcseconds, while Cols. (5), (6), and (7) give each object's redshift, distance in megaparsecs, and mass in solar masses. We assume $h=0.7$. Col. (8) gives the preliminary class found from BPT analysis [Baldassare et al., 2020], and Col. (9) gives the whether the Fe10 coronal line was found in each galaxy's spectrum.

§ DATA

§.§ Sample Selection

Our sample consists of fourteen dwarf galaxies selected for their optical variability from Baldassare et al., 2020, which used data from the Palomar Transient Factory (PTF) and NASA Sloan Atlas (NSA).
The original sample of galaxies were selected from the NSA for their mass ($M_*\lesssim 2\times10^{10} M_\odot$) and PTF coverage, then analyzed using difference imaging techniques to detect photometric variability consistent with AGNs [Ulrich et al., 1997, Padovani, 2017].
Since the focus of this paper is variable AGNs in low-mass galaxies, here we analyze observations of galaxies with stellar masses less than $\sim5\times10^9$ solar masses.
These criteria resulted in a sample of 238 dwarf galaxies with AGN-like variability.
Of these galaxies, we requested new observations for eight from the Chandra Observatory, targeting the most nearby.
We were also able to find archival data for an additional six galaxies, giving our total sample of fourteen.

The NASA Sloan Atlas is a catalog of reprocessed data from the Sloan Digital Sky Survey's <cit.> five band imaging combined with GALEX's imaging in the ultraviolet.
The SDSS images are produced using sky-subtraction, detection, and deblending techniques described in Blanton et al., 2011.
We use the catalog nsa_v0_1_2.fits[<http://sdss.physics.nyu.edu/mblanton/v0/nsa_v0_1_2.fits>], which corresponds to SDSS Data Release 8 and contains several relevant quantities. This version of the catalog extends out to redshift $z=0.055$.
These include positional information such as Right Ascension and Declination, as well as heliocentric redshift and a distance estimate using peculiar velocity, stellar mass from a K-correction fit (given in $M_\odot/h^{-2}$, so we assume $h=0.7$), the Petrosian 90%-light radius derived from the r-band with corresponding axis ratio and angle, and H$\alpha$ flux for all but 2 objects, with errors reported for the distance, mass, and flux.
Some of these values are shown in Table <ref>.

Images of the 14 objects taken from the DESI Legacy Survey [Dey et al., 2019]. Vertical bars showing 10 kiloparsecs and 10 arcseconds are shown respectively at the bottom left and right corners of each image for scale.

§.§ X-Ray Observations

NASA's Chandra X-ray Observatory is a space telescope that launched in 1999 designed to detect X-ray emission from hot regions of the universe.
Observations for these objects were made with the Chandra's Advanced CCD Imaging Spectrometer (ACIS) detector, which allows for resolution of about 1 arcsec.
Eight objects were targeted through GO 22700333 (PI: Baldassare).
These objects were chosen to be the most nearby galaxies found to have AGN-like variability in [Baldassare et al., 2020].
An additional six objects had archival Chandra data, originally observed based on their identification as as an ultra-luminous X-ray source candidate (NSA 152627), as having peculiar velocities (NSA 97904), or as AGN candidates based on optical variability (NSA 15235), transient behavior (NSA 86652), spectroscopy (NSA 61072), or merger history (NSA 10045).
Overall, total observation times range from $\sim$1.1 to $\sim$46.8 kiloseconds.

§.§ Ancillary Observations

We make use of far UV (FUV; 1350-1750 Å) observations from the Galaxy Evolution Explorer <cit.>, an ultraviolet space telescope.
We use this data for NSA 152627, for which the H$\alpha$ flux is unavailable in the NSA catalog.
Data in the infrared was taken from the Wide-field Infrared Survey Explorer <cit.>, which gives the magnitudes of each band; we used the W4 band to calculate the 25 $\mu$m flux necessary for dust corrections.
These data are available at [Wright et al., 2019].
We use single-fiber spectra from the SDSS Data Release 16 <cit.> to calculate the flux from [Fe X]$\lambda6374$ emission.

§ ANALYSIS

§.§ X-Ray Data

Data from the Chandra Observatory is processed using the Chandra Interactive Analysis of Observations software (CIAO, version 4.13).
First, we reprocess the data to create a new event list with their particle backgrounds cleaned, which is useful for very faint observations.
Next, astrometric corrections are made by running CIAO's  function on the event file, which detects X-ray point sources.
We find sources in the broad band at scales of 1, 2, 4, 6, 8, 12, 16, 24, and 32 pixels with a signal threshold of $10^{-6}$ on the X-ray images.
If more than three sources are detected, they can be  cross-matched against a known source catalog, typically the SDSS Data Release 12 [Eisenstein et al., 2011, Alam et al., 2015].
The image is energy-filtered to the correct range ($0.5-10.0$ keV) and restricted to the appropriate chip.
If an object has multiple observations, they are merged together to create a single exposure-corrected image.
also finds emission sources coinciding with four of the galaxies in our sample: NSA 10045, 104881, 15235, and 152627.
Although  did not find X-ray sources near NSA 156688 nor 51928, these galaxies have enough total X-ray photons to be included as marginal detections.

Images of the six galaxies observed in the broad X-ray band taken from the Chandra survey, with detections shown in cyan and marginal detections in yellow. Vertical bars showing 10 kiloparsecs and 10 arcseconds are shown respectively at the bottom left and right corners of each image for scale. The inner circle is the source region with a radius of 2” and the outer two rings are the inner and outer boundaries of the annular background region with radii 20” and 30”, respectively.

Aperture photometry is conducted on the Chandra data using CIAO's  software; the light signal is extracted from a $2''$ circular region around the optical galaxy center from the NSA while a source-free annulus with an inner radius of $20''$ and an outer radius of $30''$ is used for the background.
The sources are all consistent with their respective optical galaxy nuclei.
We estimated the counts and the corresponding fluxes in the broad band ($0.5-7.0$ keV) from each object assuming an absorbed power law spectrum with a photon index $\Gamma=1.7$ with a hydrogen column in the source's direction given by the National Radio Astronomy Observatory <cit.>.
Afterward, to account for contamination from X-ray binaries (XRBs), the broad band flux was converted to an unabsorbed flux in the equivalent band for the total XRB luminosity (0.5-8 keV) using the Portable Interactive Multi-Mission Simulator (PIMMS) assuming a galactic column density of $10^{21} \rm{cm^{-2}}$ and a power law source model with $\Gamma=1.8$.
The respective luminosities for all objects were then calculated from the fluxes using NSA distances.

We note that NSA 10045 had a fairly high mean count rate of 0.215 counts per second, so the effect of pileup is not negligible.
Following along the documentation provided by the [Chandra X-ray Science Center, 2010], we estimate the pileup fraction using:

\begin{equation}
\Lambda_{obs} = (1-f)\Lambda_{tot},
\end{equation}

where f is the total fraction of events lost from grade or energy migration and $\Lambda$ is the photon count rate in units of counts per second.
From these, we calculate a pileup fraction of roughly 10 to 25 percent, resulting in a total count rate of about 0.288 counts per second.
Unfortunately, these estimations are not precise enough to use in scientific calculations, but proper correction would only increase the observed luminosity from NSA 10045, so it does not affect the outcomes of this paper.

§.§ Star Formation Rates and Expected Luminosity

AGNs are not the only potential X-ray sources in these galaxies; we also anticipate a populations of XRBs that could contribute to the overall X-ray luminosity ($L_X$).
Lehmer et al., 2019 demonstrates that the X-ray luminosity from low-mass and high-mass XRBs scales with star formation rate (SFR) and nuclear stellar mass ($M_*$), respectively.
They find the total expected X-ray luminosity from XRBs to be:

\begin{equation}
\left(\frac{L_{XRB}}{erg/s}\right) =\alpha\times\left(\frac{M_*}{M_\odot}\right) + \beta\times\left(\frac{SFR}{M_\odot/yr}\right),
\end{equation}

with $\log{\alpha}=29.25$ and $\log{\beta}=39.71$.

For the stellar masses, we use the values reported in the NSA catalog.
To calculate SFR, we use the following relation given in Kennicutt & Evans, 2012 between the SFR and dust-corrected luminosity of certain “tracer” bands:

\begin{equation}
log{\left(\frac{SFR}{M_\odot/yr}\right)} = \log{\left(\frac{L_x}{erg/s}\right)} - \log{C_x},
\end{equation}

where $C_x$ is a unitless calibration constant specific to the band $x$.
We use H$\alpha$ fluxes given by the NSA catalog for all but NSA 152627, for which H$\alpha$ data is unavailable.
For this object, we obtain GALEX observations in the FUV band.
The corresponding constants are $\log{C_{H\alpha}}=41.27$ and  $\log{C_{FUV}}=43.35$.
Aperture photometry is conducted on the GALEX data; we calculate the count rate from a circle with a radius of two arcseconds.
These count rates are then converted into flux densities according to GALEX's documentation, which are then converted into fluxes by multiplying them by 1516 Å.
Just as with the Chandra data, these fluxes are also converted to luminosities using the distances from the NSA.

We implement dust corrections with the magnitudes reported in WISE's W4 band [Wright et al., 2010].
We calculate the flux density and luminosity at 25 $\mu m$ with NSA's distances, assuming the constant power law spectrum given in their documentation.
The corrections are then applied following the results of Kennicutt et al., 2009 and Hao et al., 2011 for FUV and H$\alpha$, respectively.

Once we have an estimate for the XRB luminosity, we must find the luminosity expected to come from the central 2”.
To do this, we assume that the distribution of XRBs follow that of the optical light within the galaxy.
We then multiply the total XRB luminosity by the ratio of each object's optical luminosity within a 2$''$ aperture to that within its 90%-light radius, which was found in the NSA catalog.
We note that there is scatter in scaling relations used to estimate XRB luminosity, so any individual object could have an over- or under-estimated expected XRB luminosity.
Additionally, there could be contributions to the FUV or H$\alpha$ luminosity from an AGN, leading to an over-estimate of the SFR and thus of the expected XRB luminosity.

§.§ FeX Emission

All spectra were analyzed with a pipeline using the Python package  [Astropy Collaboration et al., 2013, The Astropy Collaboration et al., 2018] and its affiliate  [Astropy-Specutils Development Team, 2019].
First, we select for objects whose [O1] flux signal-to-noise ratio is at least 3 according to the NSA file.
Then, for each spectrum we define a spectral region spanning 150 Åaround the [O1]+[Fe10] complex.
We fit the continuum within this region to a second order polynomial, masking small regions ($\mathrm{\sim5\AA}$ each) around the [O1]$\lambda6300$, [S3]$\lambda6313$ and [Fe10]$\lambda6374$ lines to ensure a good fit.
The continuum is then subtracted and we fit a single Gaussian within the small region surrounding the [O1]$\lambda6300$ line.
From this fitting, we define the [O1]$\lambda6363$ line to have the same velocity as [O1]$\lambda6300$ with a flux ratio of [O1]$\lambda6300$/[O1]$\lambda6363$=3.
We subtract the combined [O1] doublet from each of the remaining spectra, and we fit a Gaussian within the small region surrounding the [Fe10] line.
The total flux from the emission line is then the area underneath this Gaussian, which is the product of its amplitude, standard deviation, and a factor of $\sqrt{2\pi}$.
For error analysis, we resample the spectrum by adding a random error at each point drawn from a normal distribution whose scale is defined by the uncertainties reported by SDSS.
We utilize bootstrapping and repeat this process one thousand times to output a range of fluxes.
We plot a histogram to this output set, using Scott's normal reference rule to set bin widths.
For the fluxes, we set a cutoff of $10^{-20} \rm{erg\,s^{-1}\,cm^{-2}}$, above which fluxes are counted as "nonzero".
If at least 75% of the calculated fluxes from an observation were non-zero, we count this as a detection.
For these detections, we discard the zeros and fit the remaining histogram to a Gaussian, the mean and standard deviation of which represent the final [Fe10] flux and its uncertainty.

§ RESULTS

§.§ X-Ray Properties

Of the fourteen galaxies observed, four yielded detections in the broad X-ray band with luminosities ranging from $3\times10^{40}$ to $\sim9\times10^{42}\,\rm{erg\,s^{-1}}$.
For the non-detected objects, our 3-$\sigma$ upper limits range from $\sim10^{39}-2\times10^{40}\,\rm{erg\,s^{-1}}$.
The X-ray fluxes and luminosities are presented in  Table <ref>.

However, AGNs are not the only possible sources of X-rays in these galaxies.
We also expect XRBs to make a contribution to the overall X-ray luminosity of the systems.
Using the procedure described in Section <ref>, we estimate the likely X-ray binary luminosity for each object depending on SFR and stellar mass [Lehmer et al., 2019].
The estimated XRB luminosities range from $\sim 10^{37}-10^{40}\;\rm{erg\,s^{-1}}$.
All of the detected X-ray sources have luminosities exceeding the predicted XRB luminosity, while the rest are consistent.
Figure <ref> shows the observed versus expected X-ray luminosities for all objects in the sample.
Table <ref> gives the star formation rates, and observed and predicted X-ray luminosities.
We note that if there is truly an AGN in these galaxies, there will be some contribution to both the H$\alpha$ and FUV fluxes. This would likely lead to an overestimation to the SFR and therefore the expected XRB luminosity, but this is difficult to account for. We discuss this further in Section 5.1.

The logarithm of the observed X-ray luminosity versus the logarithm of the calculated expected luminosity from XRBs, with the red dashed line showing where these two values are equal. The blue points represent objects that were detected in the broad band while orange points represent marginal detections; for those that were not detected, we show their upper limits.

c c |CCC|CCC|CCC[htb]

NSA ID     Obs. Time     3cCounts     3cFlux ($10^{-15} \rm{erg\,s^{-1}\,cm^{-2}}$)     3cLuminosity ($\rm{log(erg\,s^{-1})}$)

3-5 6-8 9-11
(ks)      0.5–7 keV      2-10 keV       0.5-10 keV     0.5–7 keV      0.5-8 keV      2-10 keV     0.5–7 keV       0.5-8 keV       2-10 keV

(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)     (10)     (11)

10045$^{a}$      5.18      1015.0±95.6            200.0±42.4             1017.0±95.7            1740_-160^+160       2100_-200^+200       1030_-220^+220     42.86_-0.04^+0.04     42.94_-0.04^+0.04     42.63_-0.10^+0.08

104881           33.76     44.0_-17.3^+23.8     27.0_-13.0^+19.6     45.0_-17.5^+24.0     103_-47^+47          124_-57^+57          103_-62^+62        40.77_-0.27^+0.16     40.85_-0.26^+0.13     40.76_-0.40^+0.21

124477           7.84      2.0_-1.9^+8.9        1.0_-1.0^+7.9        2.0_-1.9^+8.9        3.5_-3.5^+15.6       4.3_-4.3^+18.8       2.1_-2.1^+23.4     38.62^+0.73             38.70^+0.73             38.40^+1.08

124554           22.13     ≤6.6                  ≤6.6                  ≤6.6                  ≤4.1                  ≤4.9                  ≤6.8                ≤39.09                 ≤39.17                 ≤39.31

15235$^{a}$      8.03      10.0_-6.9^+13.6      1.0_-1.0^+7.9        10.0_-6.9^+13.6      14.5_-9.9^+18.6      17.5_-11.9^+22.5     2.9_-2.9^+24.4     40.44_-0.50^+0.36     40.53_-0.50^+0.36     39.74^+0.98

152627$^{a}$     12.15     17.0_-9.8^+16.4      2.0_-1.9^+8.9        17.0_-9.8^+16.4      2530_-1870^+1880     3060_-2260^+2260     578_-578^+1690     42.23_-0.58^+0.24     42.31_-0.58^+0.24     41.59^+0.59

156688           46.81     5.0_-4.2^+11.0       3.0_-2.8^+9.7        5.0_-4.2^+11.0       1.3_-1.3^+3.3        1.6_-1.6^+4.0        0.6_-0.6^+4.9      38.98^+0.55             39.06^+0.55             38.65^+0.96

51928            16.27     3.0_-2.8^+9.7        4.0_-3.5^+10.4       5.0_-4.2^+11.0       2.5_-2.5^+8.2        3.0_-3.0^+9.8        5.3_-5.3^+14.9     38.79^+0.63             38.87^+0.63             39.12^+0.58

57867            11.38     ≤6.6                  ≤6.6                  ≤6.6                  ≤7.9                  ≤9.5                  ≤13.3               ≤39.10                 ≤39.19                 ≤39.33

61072$^{a}$      5.18      ≤6.6                  ≤6.6                  ≤6.6                  ≤8.6                  ≤10.4                 ≤28.8               ≤40.27                 ≤40.35                 ≤40.80

67333            1.13      ≤6.6                  ≤6.6                  ≤6.6                  ≤71.6                 ≤86.4                 ≤137                ≤39.04                 ≤39.12                 ≤39.32

86652$^{a}$      13.53      1.0_-1.0^+7.9        ≤6.6                  1.0_-1.0^+7.9        0.5_-0.5^+4.8        0.6_-0.6^+5.8        ≤13.3               39.20^+1.04             39.28^+1.04             ≤40.63

88260            8.51      ≤6.6                  1.0_-1.0^+7.9        1.0_-1.0^+7.9        ≤10.3                 ≤12.4                 1.6_-1.6^+21.6     ≤39.15                 ≤39.23                 38.32^+1.17

97904$^{a}$      36.80     ≤6.6                  ≤6.6                  ≤6.6                  ≤3.3                  ≤4.0                  ≤7.8                ≤40.03                 ≤40.12                 ≤40.40

Observed X-ray data for the objects in our sample. Col. (2) gives the total Chandra observation time for each object in kiloseconds. Cols. (3), (4), and (5) give the photon counts in the broad (0.5-7 keV) and 2-10 keV bands, as well as the total X-ray photons (0.5-10 keV). 3-$\sigma$ errors are given for each value, calculated using Poisson statistics for objects with fewer than 100 counts and Gaussian statistics otherwise. Cols. (6) and (8) respectively are the fluxes corrsponding to Cols. (3) and (4), while Col. (7) is the flux in the 0.5-8 keV band calculated from the broad band using PIMMS. Cols. (9), (10), and (11) give the logarithm of the luminosities corresponding to Cols. (6), (7), and (8), respectively. Objects marked with the superscript `a' are from the Chandra archive. Values shown with $\leq$ are upper limit values.

§.§ Emission Line Diagnostics

The BPT diagram uses optical emission line ratios to characterize the dominant source of photoionization in a galaxy  [Baldwin et al., 1981].
This diagram uses the ratios of optical narrow emission lines to determine the source of ionization in an object by placing the data points in one of three regions: star-forming (SF), AGN, or AGN-SF composite.

In particular, for a system whose primary source of ionization is an active nucleus, Groves et al., 2006 shows the line ratio [N2]$\lambda$6584 Å/H$\alpha$ to be positively correlated with both the metallicity and galaxy mass, and [O3]$\lambda$5007 Å/H$\beta$ is associated with the average ionization state and temperature of the gas and decreases with the increasing contribution of star-formation to the emission-line spectrum.

We use emission lines measurements from Baldassare et al., 2020, to place these objects on the BPT diagram when possible.
Four objects have spectra dominated by absorption for at least one of the necessary emission lines (NSA 124477, 124554, 67333, and 97904), so we do not plot these on the BPT diagram.
The remaining ten objects are plotted in Figure <ref>, which categorizes six objects in the star-forming region of the diagram, two in the composite region, and two in the AGN region.

Of the X-ray detected galaxies, one is BPT star forming (NSA 104881), one is composite (NSA 152627), and two are AGN (NSA 10045 and 15235).
The marginally detected galaxies (NSA 156688 and 51928) fall in the composite and star-forming regions.
Additionally, NSA 15235 and 10045 also have broad H$\alpha$ emission lines.
The results of this analysis are also shown in Column (8) of Table <ref>.

A BPT diagram containing the ten applicable galaxies with 3-$\sigma$ error bars, labeled by their NSA IDs. Points below the green dashed line fall within the star-forming (SF) region, objects above the red solid line are in the AGN region, and those between the two lines are in the composite (Comp.) region. The blue points represent objects that were detected in the broad X-ray band, orange points represent marginal detections, and black points were not detected.

CC | CC [htb]

NSA ID     SFR ($\rm{M_\odot\,yr^{-1}}$)     2cX-Ray Luminosity ($\rm{log(erg\,s^{-1})}$)

XRB     0.5-8 keV

(1)     (2)     (3)     (4)

10045      1.404±0.171     39.85_-0.70^+0.26     42.94_-0.04^+0.04

104881     0.139±0.040     38.65_-0.45^+0.22     40.85_-0.26^+0.13

124477     0.007±0.000     38.14_-0.17^+0.17     38.70^+0.73

124554     0.016±0.000     38.09_-0.36^+0.19     ≤39.17

15235      3.578±0.218     39.97_-0.67^+0.25     40.53_-0.50^+0.36

152627^†    1.165±0.216     37.52_-0.65^+0.25     42.31_-0.58^+0.24

156688     0.089±0.034     37.97_-1.00^+0.28     39.06^+0.55

51928      0.253±0.031     38.68_-0.46^+0.22     38.87^+0.63

57867      0.011±0.000     37.43_-0.84^+0.27     ≤39.19

61072      0.129±0.002     38.69_-0.72^+0.26     ≤40.35

67333      0.001±0.000     36.99_-0.29^+0.17     ≤39.12

86652      0.126±0.002     38.77^+0.31             39.28^+1.04

88260      0.024±0.004     37.82_-0.39^+0.20     ≤39.23

97904      0.066±0.002     38.91_-0.73^+0.26     ≤40.12

Col. (2) gives the star formation rates calculated from H$\alpha$ flux (or FUV flux for NSA 152627, which is marked with a dagger). Col. (3) gives the logarithm of the expected luminosities from X-Ray binaries. Col. (4) gives the logarithm of the observed signal in the equivalent X-ray band for comparison. All errors shown are 3-$\sigma$, and upper limits are denoted with $\leq$.

§.§ FeX Emission

From our analysis, we detect [Fe10] emission in two of our objects: NSA 10045 and 15235. We calculate fluxes of $11.8\pm3.3$ and $25.3\pm4.4\times10^{-17} erg/s/cm^2$, respectively.

§ DISCUSSION

Four of the fourteen objects in our sample were detected in the broad X-ray band, with two further marginal detections. All are consistent with the respective galaxy nucleus.
For the remaining eight galaxies, we obtain only upper limits on the X-ray luminosity ranging from $10^{39}$ to $2\times10^{40}$ erg/s.
These upper limits correspond to $\sim10^{-5}-10^{-1}\;\rm{L_{Edd}}$ for a $10^{5}\,\rm{M_{\odot}}$ black hole.

§.§ Origin of the X-ray emission

A challenge for X-ray studies of low-mass galaxies is distinguishing between emission from an AGN and X-ray binary.
We compute the expected luminosity from X-ray binaries for each object based on the SFR and stellar mass contained within the Chandra PSF.
Of the four galaxies that are detected, all have luminosities exceeding the predicted XRB luminosity.
This suggests that AGNs reside in centers of at least those four galaxies, since their X-ray luminosities likely cannot be attributed to XRBs alone.

However, we think it is unlikely that the remaining sources are solely explained by XRBs.
For all sources, there is likely some contribution to the UV luminosity from the putative AGN, leading to an overestimated SFR (and thus XRB luminosity).
Additionally, the superior angular resolution of Chandra allows us to isolate detected X-rays to the nucleus.
We find no off-nuclear X-ray sources detected in our study; all sources were consistent with their optical centers.
Since XRBs are not preferentially found in galaxy nuclei, it would be extremely unusual for these X-ray sources to be nuclear XRBs.

While we cannot rule out an XRB origin for some of these galaxies, the combination of nuclear variability and a nuclear X-ray point source leads us to conclude that most detections are due to accretion onto a central massive BH.

§.§ Comparison of AGN Detection Methods

All fourteen galaxies in our sample exhibited optical variability consistent with an AGN.
Four were detected in the broad X-ray band with two additional marginal detections, all having X-ray luminosities greater than expected from XRBs.
However, as discussed above, the fact that the X-ray sources are nuclear and/or point-like is inconsistent with most being XRBs.

Three of the four X-ray detected objects have emission lines in the AGN or composite region of a BPT diagram based on their SDSS spectroscopy.
Two galaxies, NSA 10045 and 15235, have variability, spectroscopic signatures, and X-ray luminosities significantly higher than expected from XRBs.
A visual breakdown of these results is shown in Figure <ref>.
This outcome can be characterized by discussing the concepts of "pure" versus "complete" samples.
In our study, variability selection provides a more complete sample since it finds AGN candidates that would be missed by other selection techniques, but other methods such as emission line diagnostics and Fe10 emission create a more pure sample since they more exclusive and less likely to containing false negatives.
The lack of substantial overlap in these selection techniques leaves us with some questions: If the variability is due to accretion onto a BH, why are some objects not selected by the BPT diagram or X-ray analysis?

A Venn diagram of the galaxies in our sample based on AGN detection per method: optical variability (the entire sample), X-ray detection (green), emission line diagnostics (BPT; light purple), or the Fe10 coronal line (red). Galaxies that were marginally detected in the X-ray are marked with an asterisk.

Optical spectroscopic selection has been shown to be incomplete at low galaxy masses. Low metallicity and dilution of the AGN signal by star formation can shift objects out of the AGN region of the BPT diagram [Kewley et al., 2006, Trump et al., 2015].
In Cann et al., 2019, they use modeling to show that the standard emission line diagnostics could be less reliable for lower mass systems, especially for BHs less massive than about $10^4 M_\odot$.

Our findings are consistent with these results as only four of the ten galaxies plotted on the BPT diagram were categorized as AGNs, despite their optical variability.
Of particular note is galaxy NSA 104881, which was X-ray loud with $L_X\sim 10^{41} \;\rm{erg\;s^{-1}}$, but fell in the star-forming region of the BPT.

To quantitatively explore the possibility of star formation dilution, we plot our objects on the star forming main sequence [Aird et al., 2017].
The main sequence depends only on galaxy stellar mass and redshift:

\begin{equation}
\log{\left(\frac{SFR}{M_\odot/yr}\right)} = A+B\log{\left(\frac{M_*}{M_\odot}\right)}+C\log{\left(1+z\right)}
\end{equation}

with $A=-7.6, B=0.76\pm0.06$, and $C=2.95\pm0.33$.
We use this relation to compare against the star formation rates previously calculated in Section <ref>, shown in Figure <ref>.
Many of the galaxies in our sample are consistent with being on the star forming main sequence, showing that this may play a role in some objects being missed by the BPT diagram.
However, this is likely not the only factor at play: the BPT star-forming galaxies the were detected and marginally detected in the X-ray (NSA 104881 and NSA 51928) actually fall slightly below this relation.

A comparison of the star formation rates calculated with the relation from Lehmer et al., 2019 versus stellar mass with 3-$\sigma$ errors shown. The black line shows the relation given in Aird et al., 2017 with the green region encompassing the error and variation in redshift. The blue points represent objects that were detected in the broad X-ray band, orange points represent marginal detections, and black points were not detected.

We find one object (NSA 51928) that is selected with variability and marginally detected in the X-ray, but whose observed luminosities were not significantly greater than their expected XRB luminosities.

It is possible that X-ray emission could be suppressed in low-mass AGN;
Dong et al., 2012 finds that IMBHs have suppressed X-ray luminosities relative to the UV emission.
They see a trend toward flat spectral indexes $\Gamma$, implying the presence of absorption.
Some, however, are found to have typical gamma values, but low values for $\alpha_{OX}$, the X-ray-to-optical spectral slope, raising the possibility of intrinsically weak X-ray sources.
Pu et al., 2020 also finds a distribution of $\Delta\alpha_{OX}$ that suggests the existence of X-ray weak AGNs, representing $\sim$5% of the population.
This weakness could be characterized by analyzing each object's X-ray variability, or by searching their UV or X-ray spectra for obscuration signatures.
Follow-up work with UV or X-ray spectroscopy could determine whether obscuration plays a role in this systems.

We must also consider the possibility that the optical variability is not associated with an AGN.
Recall that these galaxies were selected in Baldassare et al., 2020 for their AGN-like optical variability.
More specifically, these objects were chosen based on $\sigma_{var}$, $\sigma_{QSO}$ and $\sigma_{notQSO}$ from the software QSO_fit [Butler & Bloom, 2011].
These respectively represent the significance that the object is variable, that the object's variability fits a damped random walk, and that the object is not a quasi-stellar object (QSO), but varies in a time-independent Gaussian manner.
Objects that were selected were done so on the basis that $\sigma_{var}>2$, $\sigma_{QSO}>2$ and $\sigma_{QSO}\gtrsim \sigma_{notQSO}$.
Any objects with a burst-like light curve (a rise and fall) were also removed from the sample to eliminate possible contamination from supernovae.

There are some examples of objects displaying AGN-like variability due to non-AGN processes. The extremely metal poor blue compact emission line galaxy PHL 293B showed variability consistent with a damped random walk for almost two decades [Burke et al., 2020].
This object was suspected to be an AGN, however, it showed fading broad emission lines with a P Cygni profile and blueshifted absorption lines. [Burke et al., 2020] concluded that this was likely a long-lived, unusual Type II supernova.
While we cannot strictly rule out the possibility of contamination from non-stellar processes, PHL 293B was an extremely rare case and none of the objects in our sample show spectral features like P Cygni profiles or blueshifted absorption.
We consider it unlikely that our sample is significantly contaminated by non-AGN variability.

Finally, we consider the interesting possibility that the previously detected AGN-variability was associated with a short-lived accretion event like a tidal disruption event.
The PTF data used for constructing light curves was taken between 2009 and 2017.
Our followup Chandra X-ray observations were taken in 2020 and 2021.
It is possible that the accretion event leading to the variability has faded in the interim.
Some works have speculated that TDEs could power the entire observed population of AGN in dwarf galaxies [Zubovas, 2019].
X-ray observations that are contemporaneous with observed variability could help shed light on the nature of accretion processes occurring in dwarf galaxies.

NSA ID     $\sigma_{var}$     $\sigma_{QSO}$     $\sigma_{notQSO}$

(1)     (2)     (3)     (4)
10045         9.42        4.73          0.45

15235        12.11        7.19          0.05

88260        14.14        4.98          1.19

124477        16.48        3.75          4.31

124554         6.56        3.80          0.13

86652         3.32        3.47          0.03

51928         4.24        3.94          0.01

57867         8.13        7.39          0.00

97904         2.09        2.29          0.01

156688         3.46        3.55          0.01

104881        18.47        6.52          1.68

61072         4.11        3.72          0.02

67333         2.70        6.88          0.00

152627         4.30        2.31          0.33

Associated variances of objects within our sample.

§.§ Black Hole Mass and Eddington Luminosity

Here we consider the accretion properties of the X-ray detected AGN in our sample.
We compute or estimate BH masses and Eddington ratios.
The Eddington luminosity is a rough upper limit for energy production in an accreting system where the gravitational force of the black hole on in-falling matter is balanced by the radiation pressure from its emission.
This is related to the central BH's mass:

\begin{equation}
L_{Edd}\coloneqq\frac{4\pi Gcm_p}{\sigma_T}M_{BH}\approx1.257\times10^{38} \left(\frac{M_{BH}}{M_\odot}\right)\:\rm{erg/s}
\end{equation}

The Eddington ratio can then be thought of as a measure of efficiency for an accreting black hole, the ratio of its bolometric luminosity to the theoretical limit: $k_{Edd}=L_{bol}/L_{Edd}$.
For the sake of this calculation, we use the simple bolometric correction to hard X-rays given in Marconi et al., 2004 $L_{bol}/L_{2-10keV}=10$.

Two objects in the sample (NSA 10045 and 15235) have broad H$\alpha$ emission which we can use to estimate BH mass [Greene & Ho, 2005].
NSA 15235 was previously analyzed in [Baldassare et al., 2017], and found to have a BH mass of $1.6\times10^{5}\;M_{\odot}$ and an Eddington fraction of $0.3\%$.
NSA 10045 has a BH mass of $1\times10^{6}\;M_{\odot}$ and an Eddington fraction of $33\%$.

To estimate the mass of the remaining central BHs, we use the relation between BH mass and stellar mass found in Reines & Volonteri, 2015:

\begin{equation}
\log{(M_{BH}/M_\odot)} = \alpha + \beta \log{(M_*/10^{11}M_\odot)}
\end{equation}

where $\alpha = 7.45\pm0.08, \beta = 1.05\pm0.11$.

Assuming this sample follows the BH mass-stellar mass scaling relation, we find BH masses ranging from $10^{4} - 10^{6}\;M_{\odot}$.
For the X-ray detected objects, Eddington fractions from $\sim10^{-5} - 0.3$ (or from $0.001\%$ to $30\%$).
The median Eddington fraction is 0.003 ($0.3\%$).
This is about an order of magnitude lower than the median Eddington fraction for broad line AGN in dwarf galaxies [Baldassare et al., 2017].
This suggests that variability selection may identify lower accretion rate AGN in dwarf galaxies than searches using broad emission lines.
For the non-detected objects, the upper limits on the Eddington fractions range from $10^{-4}$ to $2\times10^{-1}$.
These results are summarized in Table <ref>.

C C C C C[htb]

NSA ID     BH Mass     $L_{Edd}$     $L_{2-10 keV}$     $k_{Edd}$

$\mathrm{log(M_\odot\:yr^{-1})}$     $\mathrm{log(erg\:s^{-1})}$     $\mathrm{log(erg\:s^{-1})}$

(1)     (2)     (3)     (4)     (5)

10045      6.00^†±0.90     44.10±0.90     42.63_-0.10^+0.08     3.39×10^-1

104881     5.39±0.69     43.49±0.69     40.76_-0.40^+0.21     1.86×10^-2

124477     6.05±0.50     44.15±0.50     38.40^+1.08             1.76×10^-5

124554     5.45±0.67     43.55±0.67     ≤39.31                 ≤2.72×10^-3

15235      5.29^†±0.90     43.39±0.90     39.74^+0.98             2.25×10^-3

152627     5.85±0.56     43.95±0.56     41.59^+0.59             4.33×10^-2

156688     3.95±1.13     42.05±1.13     38.65^+0.96             4.02×10^-3

51928      5.78±0.58     43.88±0.58     39.12^+0.58             1.72×10^-4

57867      3.97±1.12     42.07±1.12     ≤39.33                 ≤2.37×10^-1

61072      5.46±0.67     43.56±0.67     ≤40.80                 ≤8.03×10^-2

67333      4.55±0.94     42.65±0.94     ≤39.32                 ≤4.07×10^-2

86652      5.77±0.58     43.87±0.58     ≤40.63                 ≤2.20×10^-2

88260      4.77±0.88     42.87±0.88     38.32^+1.17             2.85×10^-4

97904      5.99±0.52     44.09±0.52     ≤40.40                 ≤6.72×10^-3

A table showing rough estimates for BH masses and Eddington ratios. Black hole masses marked with a dagger were calculated broad H$\alpha$ emission, while the others were calculated using the relation given in Reines & Volonteri, 2015. The Eddington ratio was estimated using the simple bolometric correction $L_{bol}/L_{2-10 keV}=10$.Upper limit values are denoted with $\leq$.

§ CONCLUSION

We analyze Chandra X-ray observations of fourteen nearby ($z\lesssim0.044$) low-mass ($\log{M_*}\lesssim9.7$) galaxies from the NASA-Sloan Atlas, selected for their optical variability in Baldassare et al., 2020.

* 4 of the 14 objects (29%) in our sample of variability-selected low-mass AGN were detected in the 0.5-7 keV X-ray band, with two further marginal detections.

* Of the 4 X-ray detected, all had luminosities exceeding that anticipated from XRBs by multiple $\sigma$, which can be interpreted as evidence for AGN activity.

* One of the galaxies with an X-ray detection and one marginally detected galaxy fall in the star forming region of the BPT diagram. This work confirms that variability can find AGN in dwarf galaxies that are missed by optical spectroscopy.

* BPT analysis, which plots the ratios of specific emission lines with empirical cuts that categorize galaxies, classified four of our objects (NSA 10045, 15235, 152627, and 156688) as AGN candidates. This method, however, is known to undercount low-mass and low-metallicity systems.

* Based on scaling relations between BH mass and stellar mass, we find Eddington fractions ranging from $10^{-5} - 0.3$ for the X-ray detected objects in our sample. These are systematically lower Eddington fraction than broad-line AGN in dwarf galaxies, implying that variability selection may find lower accretion rate objects.

With the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al., 2019) on the horizon, we can expect an abundance of new variability-selected candidate AGN in low-mass galaxies. Contamination from stellar processes may also be higher since imaging will be deeper than previous time domain surveys. In order to better understand the source of variability in these future candidates (i.e., AGN versus TDE versus stellar processes), it will be useful to combine variability observations from LSST with simultaneous X-ray observations.

§ ACKNOWLEDGEMENTS

Alexander Messick thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant #1829740, the Brinson Foundation, and the Moore Foundation; their participation in the program has benefited this work.

Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number GO1-22100X issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. The scientific results reported in this article are based to a significant degree on observations made by the Chandra X-ray Observatory and data obtained from the Chandra Data Archive. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application package CIAO.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.

This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.

Some of the data presented in this paper were obtained from the Infrared Science Archive (IRSA) at IPAC, which is operated by the California Institute of Technology under contract with the National Aeronautics and Space Administration. The specific observations analyzed can be accessed via [doi:10.26131/IRSA1]https://doi.org/10.26131/IRSA1. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via [doi:10.17909/T9H59D]https://doi.org/10.17909/T9H59D.

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# Reliable Devices Yield Stable Quantum Computations ††thanks: The manuscript
is authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the
U.S. Department of Energy. The U.S. Government retains for itself, and others
acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in
said article to reproduce, prepare derivative works, distribute copies to the
public, and perform publicly and display publicly, by or on behalf of the
Government. The Department of Energy will provide public access to these
results of federally sponsored research in accordance with the DOE Public
Access Plan: https://www.energy.gov/doe-public-access-plan.

Samudra Dasgupta${}^{1,2^{*}}$, and Travis S. Humble${}^{1,2^{\dagger}}$
1Quantum Science Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee,
USA
2Bredesen Center, University of Tennessee, Knoxville, Tennessee, USA
<EMAIL_ADDRESS>ORCID: 0000-0002-7831-745X
<EMAIL_ADDRESS>ORCID: 0000-0002-9449-0498

###### Abstract

Stable quantum computation requires noisy results to remain bounded even in
the presence of noise fluctuations. Yet non-stationary noise processes lead to
drift in the varying characteristics of a quantum device that can greatly
influence the circuit outcomes. Here we address how temporal and spatial
variations in noise relate device reliability to quantum computing stability.
First, our approach quantifies the differences in statistical distributions of
characterization metrics collected at different times and locations using
Hellinger distance. We then validate an analytical bound that relates this
distance directly to the stability of a computed expectation value. Our
demonstration uses numerical simulations with models informed by the transmon
device from IBM called washington. We find that the stability metric is
consistently bounded from above by the corresponding Hellinger distance, which
can be cast as a specified tolerance level. These results underscore the
significance of reliable quantum computing devices and the impact for stable
quantum computation.

###### Index Terms:

device reliability, program stability, spatio-temporal non-stationarity, time-
varying quantum noise

##  I Introduction

Quantum devices are subject to non-stationary noise sources, e.g. non-uniform
spontaneous decay, energy loss, cross-talk, sensitivity to imprecise control
pulses, and fluctuations in thermodynamic controls, all of which affect the
quality of the quantum register implementation. The field of quantum noise
characterization focuses on measuring and tracking noise metrics (such as CNOT
gate error) at various points in time. These characterizations inform
calibration techniques for hardware engineers as well as error mitigation
methods for programmers. However, quantum devices also exhibit temporal
variations in their noise sources, which underlies the need for frequent
calibration and adjustment of device metrics. Non-stationary noise processes
can also stymie attempts at characterization as the underlying noise models
must adapt to new and often unpredictable behaviors. How can we monitor
changes in the noise itself to better inform these efforts?

Here, we address the concern that non-stationary noise process pose to
reliable quantum computation. Device reliability is presented as a measure of
the statistical similarity of the underlying device metrics, such as gate
fidelities and coherence times. This measure captures the similarity between
device metrics considering both spatial and time-varying noise processes. We
then recall how device reliability bounds the stability of expectation values
dervied from noisy quantum computation. Moreover, we validate this bound on
stablility using numerical simulations of a circuit modeled by a multi-
dimensional correlated noise distribution.

##  II Theory

###  II-A Stability

Stability in quantum computing refers to the boundedness of the output of a
quantum circuit in the presence of noise fluctuations [1]. In this study, we
focus on the mean value of a quantum observable $O$ as a representative of
program output. Let x represent the parameter characterizing the noisy
realization of a quantum circuit $\mathcal{C}$. The mean value of $O$ obtained
from repeated executions on the noisy circuit is denoted as
$\braket{O}_{\textrm{x}}$. Considering the time-varying nature of device
noise, we introduce $f(\textrm{x};t)$ as the probability distribution function
of the quantum noise parameter x. We define $\braket{O}_{t}$ as the average
value of $\braket{O}_{\textrm{x}}$ with respect to $f(\textrm{x};t)$, the
probability distribution function for the noise parameter.

$\braket{O}_{t}=\int\limits\braket{O_{\textrm{x}}}f(\textrm{x};t)d\textrm{x}$
(1)

The stability of the quantum observable between two time points, $t_{1}$ and
$t_{2}$, is then quantified by the absolute difference in the mean values of
$\braket{O}$ obtained at those times, defined as

$s(t_{1},t_{2})=|\braket{O}_{t_{1}}-\braket{O}_{t_{2}}|$ (2)

###  II-B Reliability

We next quantify device reliability by comparing the statistical distributions
of various characterization metrics collected at different times and register
locations. When these metrics exhibit statistical similarity, the device
behavior is considered to be reliable. The statistical distance between
distributions is calculated using the Hellinger distance, which offers ease of
calculation and interpretation.

$H(f(\textrm{x};t_{1}),f(\textrm{x};t_{2}))=\sqrt{1-\int\limits_{\textrm{x}}\sqrt{f(\textrm{x};t_{1})f(\textrm{x};t_{2})}d\textrm{x}}$
(3)

The Hellinger distance above quantifies the statistical similarity of a device
at different times such that when the distance is small, the device behaves
statistically similar at both times. This is expected when the underlying
noise process is stochastic. However, larger values of the distance imply that
noise processes within the device are non-stationary processes that lead to
noticeable changes in device properties. The timescales on which such
statistically significant changes are measured represent an important metric
for evaluating the reliability of the device relative to a desired tolerance.

###  II-C Stability Bounds

We now establish an analytical and intuitive connection between output
stability and device reliability. Specifically, we show how device reliability
constrains the outcomes from a quantum program executed on a NISQ device by
examining the role of fluctuations in device metrics.

Let $s_{\textrm{tol}}$ denote a specified tolerance on the stability metric
introduced earlier. Additionally, let the reliability of the quantum device
between times $t_{1}$ and $t_{2}$ is quantified by the Hellinger distance
$H_{X}$, as discussed previously. We determine the maximum bound
$H_{X}^{\textrm{max}}(t_{1},t_{2})$ on the Hellinger distance constrained by
$s(t_{1},t_{2})<s_{\textrm{tol}}$.

We begin by noting the bound on the stability satisfies

$s^{2}(t_{1},t_{2})\leq\left(\int\left|\braket{O_{\textrm{x}}}\\{f(\textrm{x};t_{1})\textrm{dx}-f(\textrm{x};t_{2})\\}\right|\textrm{dx}\right)^{2}$
(4)

where the inequality stems from the absolute value on the integrand. Per
Holder’s inequality, if $m,n\in[1,\infty]$ and
$\nicefrac{{1}}{{m}}+\nicefrac{{1}}{{n}}=1$ then

$\int\left|f(x)g(x)\right|dx\leq\left(\int|f(x)|^{m}dx\right)^{1/m}\left(\int|g(x)|^{n}dx\right)^{1/n}$

Thus, the right-hand side of Eq. (4) becomes

$\displaystyle\left(\int\left|\braket{O_{\textrm{x}}}\\{f(\textrm{x};t_{1})\textrm{dx}-f(\textrm{x};t_{2})\\}\right|\textrm{dx}\right)^{2}$
$\displaystyle\leq\left(\int|\braket{O_{\textrm{x}}}|^{m}\textrm{dx}\right)^{2/m}\left(\int|f(\textrm{x};t_{1})-f(\textrm{x};t_{2})|^{n}\textrm{dx}\right)^{2/n}$

Choose $m=\infty,n=1$ and define
$c=\underset{\textrm{x}}{\textrm{sup}}|\braket{O_{\textrm{x}}}|$. This
circuit-specific constant satisfies

$\displaystyle\lim\limits_{m\rightarrow\infty}\left(\int|\braket{O_{\textrm{x}}}|^{m}\textrm{dx}\right)^{1/m}\leq\lim\limits_{m\rightarrow\infty}\left(\int
c^{m}\textrm{dx}\right)^{1/m}=c$

Thus, we may then reduce Eq. (4) as

$\displaystyle s(t_{1},t_{2})^{2}\leq$
$\displaystyle\lim\limits_{m\rightarrow\infty,n=1}\left(\int|\braket{O_{\textrm{x}}}|^{m}\textrm{dx}\right)^{2/m}$
$\displaystyle\left(\int|\\{f(\textrm{x};t_{1})\textrm{dx}-f(\textrm{x};t_{2})\\}|^{n}\textrm{dx}\right)^{2/n}$
$\displaystyle\leq c^{2}$
$\displaystyle\left(\int\left|\sqrt{f(\textrm{x};t_{1})}-\sqrt{f(\textrm{x};t_{2})}\right|\right.$
$\displaystyle\left.\left(\sqrt{f(\textrm{x};t_{1})}+\sqrt{f(\textrm{x};t_{2})}\right)\textrm{dx}\right)^{2}$
$\displaystyle\leq c^{2}$
$\displaystyle\int\left(\sqrt{f(\textrm{x};t_{1})}-\sqrt{f(\textrm{x};t_{2})}\right)^{2}\textrm{dx}$
$\displaystyle\int\left(\sqrt{f(\textrm{x};t_{1})}+\sqrt{f(\textrm{x};t_{2})}\right)^{2}\textrm{dx}$

Using Holder’s inequality with $m=n=2$, this yields
$s(t_{1},t_{2})=2cH\sqrt{2-H^{2}}$ which can be re-arranged to yield the
maximum

$\displaystyle H_{\textrm{max}}(t_{1},t_{2})=\sqrt{1-\sqrt{1-\phi}}$ (5)

with $\phi=s_{\textrm{tol}}^{2}/(4c^{2})$. This sets an upper limit
$H_{\textrm{max}}$ on the Hellinger distance to ensure the desired stability
criterion $s_{\textrm{tol}}$ is met.

##  III Validation

###  III-A Experimental data

We utilized data obtained from the transmon device called washington, a
127-qubit register with heavy hexagonal connectivity developed by IBM. The
publicly available characterization data for the washington device was used to
create a dataset comprising specific device metrics (refer to Table 1). This
dataset was constructed from a subset of the device characterization data
spanning a 16-month period from January 1, 2022, to April 30, 2023. The Qiskit
software library [2] was employed to access the collected characterization
data online.

These metrics correspond to the minimum requirements for the physical
implementation of quantum computing [3], which fall into one of the five
classes: SPAM (state preparation and measurement) fidelity, single-qubit gate
fidelity, two-qubit entangling gate fidelity, duty cycle (gate length to
coherence time ratio), and addressability (ability to measure a register
element without interference from other qubits). Specifically, these 16
metrics capture the noise processes of the five-qubits employed in the test
circuit illustrated in Fig. 1. Our simulation of the test circuit (described
in the next section) relies on data pertaining to these 16 metrics, which
enables us to estimate the time-varying joint distribution of circuit noise.
Utilizing this estimated distribution, Monte Carlo sampling is performed to
simulate the test circuit and validate the theory presented earlier.

Parameter |  Description |  Model
---|---|---
$\textrm{x}_{0}$ |  SPAM fidelity, register 0 |  ABC
$\textrm{x}_{1}$ |  SPAM fidelity, register 1 |  ABC
$\textrm{x}_{2}$ |  SPAM fidelity, register 2 |  ABC
$\textrm{x}_{3}$ |  SPAM fidelity, register 3 |  ABC
$\textrm{x}_{4}$ |  CNOT fidelity, control 0, target 1 |  DP$\otimes$DP
$\textrm{x}_{5}$ |  CNOT fidelity, control 2, target 1 |  DP$\otimes$DP
$\textrm{x}_{6}$ |  $T_{2}$ time, register 0 |  TR
$\textrm{x}_{7}$ |  $T_{2}$ time, register 1 |  TR
$\textrm{x}_{8}$ |  $T_{2}$ time, register 2 |  TR
$\textrm{x}_{9}$ |  $T_{2}$ time, register 3 |  TR
$\textrm{x}_{10}$ |  $T_{2}$ time, register 4 |  TR
$\textrm{x}_{11}$ |  $H$ fidelity, register 0 |  CP
$\textrm{x}_{12}$ |  $H$ fidelity, register 1 |  CP
$\textrm{x}_{13}$ |  $H$ fidelity, register 2 |  CP
$\textrm{x}_{14}$ |  $H$ fidelity, register 3 |  CP
$\textrm{x}_{15}$ |  $H$ fidelity, register 4 |  CP
TABLE I: The 16-parameter model derived from the washington data set is
composed from noise processes ‘ABC’: asymmetric binary channel, ‘CP’: coherent
phase error model, ’DP’: depolarizing noise and ‘TR’: thermal relaxation. The
two-qubit model ‘DP$\otimes$DP’ is a tensor product of depolarizing noise.

###  III-B Test circuit

We validate the stability bound using a numerical simulation of of the
Bernstein-Vazirani circuit [4] with a noise model using the characterization
data presented in the previous section. The Bernstein-Vazirani algorithm
determines a secret $n$-bit string $r$ encoded in an oracle. Our focus is on
assessing the success probability of correctly computing the secret bit string
using the fewest number of queries possible. In contrast to the classical
approach that requires $n$ queries, the Bernstein-Vazirani algorithm achieves
the same outcome with just one query. Fig. 1 illustrates the quantum circuit
corresponding to a 4-bit secret key. The observable for the problem is
$O~{}=~{}\Pi_{r}~{}=~{}\ket{r}\bra{r}$ where
$\ket{r}=\bigotimes\limits_{i=1}^{n}\ket{r_{i}}$ with $r_{i}\in\\{0,1\\}$.

###  III-C Method

We used numerical simulations to test the reliability of a model noisy quantum
device and to investigate the boundedness of the stability metric as predicted
by the theory above. We first mapped the 16 noise parameters necessary for
simulating the 5-qubit Bernstein-Vazirani circuit shown in Fig. 1 to distinct
independent noise processes, selecting them based on the criteria outlined in
[3] for the physical implementation of quantum computing. The parameters
mapped to gate and register specific noise model in Table I. For example, the
asymmetric binary channel for register $0$ flips the measured output bit
$b_{0}$ to $b_{0}\oplus 1$ with probability $x_{0}$, while the coherent phase
error channel for the Hadamard gate $H$ applied to register 0 transforms the
underlying quantum state as $CP(H\rho H)=R_{z}(\theta)H\rho
HR_{z}^{\dagger}(\theta)$. Thermal relaxation is modeled by an exponential
dephasing process that depends on the $T_{2}$ time and the duration of the
underlying gate not shown here.

While the 16 noise processes above act independently, the underlying noise
parameters are assumed to be correlated. We construct a joint distribution of
to describe these parameters using the method of Gaussian copula, cf. Fig. 2.
The copula itself is defined as

$\Theta(y)=\frac{\exp\left(-\frac{1}{2}(y-\mu)^{T}\Sigma^{-1}(y-\mu)\right)}{(2\pi)^{n/2}|\Sigma|^{1/2}}$
(6)

where the correlation elements $\Sigma_{i,j}$ are derived from Pearson
correlation coefficients computed using the daily metric values available from
the washington data set. For a Gaussian copula, the corresponding
16-dimensional noise distribution takes the form

$f_{X}(\textrm{x};t)=\Theta\left[F_{X_{1}}(\textrm{x}_{1};t),\cdots
F_{X_{d}}(\textrm{x}_{d};t)\right]\prod\limits_{j=1}^{n}f_{X_{j}}(\textrm{x}_{j};t)$
(7)

where $F_{X}(\textrm{x};t)$ is the cumulative distribution function at time
$t$ and $\Theta(\cdot)$ is the copula function. These generated distributions
are then used to calculate the Hellinger distance in Eq. 3.

Our numerical studies of the quantum circuit stability generates an ensemble
of noisy simulations by drawing samples from the multi-parameter noise
distribution represented by Eq. 7.

Figure 1: The Bernstein-Vazirani circuit for a 4-bit secret string $r=0011$.
$H$ represents the Hadamard gate and $Z$ represents a phase shift of 180
degrees to the $\ket{1}$ state. The meter symbols are measurement operation
that project into the computational basis and record results in a classical
register $c$.

We initially establish a joint distribution from the daily data gathered in
January 2022 for the washington device, utilizing copulas. Over the next 15
months, we introduce minor perturbations to this distribution, ensuring that
the Hellinger distance never exceeds $H_{\text{max}}$ between the perturbed
and original January 2022 distributions. In this perturbation scheme, the
marginal distribution of the CNOT gate error between qubits 1 and 2 is modeled
using a beta distribution, which is based on the aforementioned January 2022
daily data. Small, random perturbations to the beta distribution parameters
are incorporated over 15 months for the CNOT error, with Hellinger distance
constraint maintained. For each perturbed distribution, we generate 100,000
noise metric samples, and execute 100 qiskit simulations (each with 8192
shots). The stability metric is then computed from the obtained output, as per
Eqn. 2.

Figure 2: A two-dimensional projection of the 16-dimensional correlated
distribution modeled using the copula function. These parameter densities
present SPAM error on qubit 0 and Hadamard gate error on qubit 1 for Jan-2022.

###  III-D Results

Figure 3 presents the simulation results illustrating the relationship between
the stability metric ($s$) and the reliability of a quantum device
characterized by the Hellinger distance ($H$). The results demonstrate that
when $H\leq H_{\textrm{max}}$ the device is reliable such that the temporal
difference of the observable ($s$) remains within the specified upper bound
($s\leq s_{\textrm{max}}$).

In our simulations, we set the tolerance threshold $s_{\text{tol}}=20\%$,
which limits the maximum acceptable deviation in the expectation value over
time. According to Eqn. 5, this results in an upper limit of 7.1% for the
device reliability metric $H_{\text{max}}$. The lower panel of Fig. 3 presents
the calculated Hellinger distance between the multi-dimensional noise
processes characterizing the device. These calculations show how noise
fluctuates on a monthly basis while still respecting the $H_{\text{max}}$
constraint. While time varying, these process emulate the behavior of a
reliable device. The upper panel of Fig. 3 presents the corresponding
stability metric, which never exceeds the 20% tolerance. Moreover, we find the
stability is nearly two orders of magnitude smaller than the expected
tolerance, with an average of approximately 0.6%. By selecting a reliable
device whose temporal noise variation remains within the defined bounds, we
can ensure the stability of program output within the desired tolerance.

##  IV Conclusions

Output stability is crucial in quantum computing research as non-stationary
noise processes in quantum devices can result in unstable results that
fluctuate based on time-varying device noise characteristics, rendering them
unsuitable for meaningful interpretation and drawing scientific conclusions.
The variations in superconducting qubits, attributed to certain oxides on the
superconductors’ surface modeled as fluctuating two-level systems (TLS) [5,
6], have been extensively studied. Ongoing research focuses on addressing the
time-varying nature of quantum noise through modeling [7], characterizing
noise sources, tracking their temporal profile [8], and integrating
statistical knowledge into quantum architectures using innovative techniques
[9]. This paper explores the relationship between device reliability and
output stability by considering a user-defined upper bound on variations in
expectation values. The goal is to assess the stability of program outputs by
evaluating the reliability metric within a specified tolerance bound through
simulations.

Figure 3: Simulation demonstrating that when $H\leq H_{\textrm{max}}$ (i.e. a
reliable, slowly varying noise platform), then $s\leq s_{\textrm{max}}$ (i.e.
the temporal difference of the observable stays within the predicted upper
bound).

## ACKNOWLEDGMENTS

This work is supported by the U. S. Department of Energy (DOE), Office of
Science, Early Career Research Program. This research used computing resources
of the Oak Ridge Leadership Computing Facility, which is a DOE Office of
Science User Facility supported under Contract DE-AC05-00OR22725.

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# Ultra hypercyclicity

Martin Liu and David Walmsley

###### Abstract

Recently, new topological properties that an operator acting on a topological
vector space can have were introduced: strong hypercyclicity and hypermixing.
We introduce a new property called ultra hypercyclicity, and we compare it to
the other common topological properties, including the new ones. For the
family of weighted backward shifts on $c_{0}$ and $\ell^{p}$, we characterize
ultra hypercyclicity in terms of the weight sequence, and frame the open
question of whether or not there exists a strongly hypercyclic weighted
backward shift which is not ultra hypercyclic in terms of properties that the
weight sequence has.

##  1 Introduction

Let $T:X\to X$ be a continuous linear map (henceforth an operator) on a
separable Banach space $X$. We are interested in the dynamics, or long-term
behavior, of $T$; given some initial point $x\in X$, what can we say about the
orbit of $x$ under $T$, given by
$\operatorname{orb}(T,x)=\\{x,Tx,T^{2}x,T^{3}x,\ldots\\}$? For example, is $x$
periodic, meaning $T^{n}x=x$ for some $n\geq 1$? Or, oppositely, could
$\operatorname{orb}(T,x)$ be dense in $X$? It turns out
$\operatorname{orb}(x,T)$ being dense is equivalent to each return set
$N(x,U)\coloneqq\\{n\in\mathbb{N}:T^{n}x\in U\\}$ being nonempty for any
nomempty open set $U$ in $X$. With a more topological mindset, we might ask
how the iterates of $T$ “connect” two given nonempty open sets $U$ and $V$ in
$X$; given nonempty open subsets $U$ and $V$ of $X$, the return set $N(U,V)$
is defined as $N(U,V)=\\{n\in\mathbb{N}_{0}:T^{n}(U)\cap V\not=\emptyset\\}$.
We say $T$ is topologically transitive (resp. weakly mixing; resp. mixing) if
for all nonempty open subsets $U$ and $V$ of $X$, $N(U,V)$ is nonempty (resp.
contains arbitrarily long intervals; resp. is cofinite). In our setting of a
separable Banach space $X$, it turns out hypercyclicity is equivalent to
topological transitivity by the Birkhoff transitivity theorem [6].
Hypercyclicity has become a very active area of research in operator theory
that has many connections to other branches of mathematics. For more
background on its history, central ideas, and a detailed introduction, we
refer the interested reader to the monographs [5, 8].

Given our definitions, and remembering that hypercyclicity and topological
transitivity are equivalent, the following implications are immediate:

mixing | $\implies$ | weakly mixing | $\implies$ | hypercyclic.
---|---|---|---|---

Recently in [4, 1, 3, 2], stronger versions of hypercyclicity and mixing were
introduced and studied, namely strong hypercycliclity, supermixing, and
hypermixing. We recall these definitions, and introduce a new term as well.

######  Definition 1.1.

Let $X$ be a separable Banach space and $T$ be an operator on $X$. We say $T$
is strongly hypercyclic on $X$ if, for each nonempty open subset $U$ of $X$,

$\displaystyle X\setminus\\{0\\}\subseteq\bigcup_{n=0}^{\infty}T^{n}(U).$

We say $T$ is ultra hypercyclic if there exists an increasing sequence $n_{k}$
in $\mathbb{N}$ such that, for each nonempty open subset $U$ of $X$,

$\displaystyle X=\bigcup_{i=0}^{\infty}\bigcap_{k=i}^{\infty}T^{n_{k}}(U).$
(1)

When (1) is satisfied for a given $(n_{k})$, we say $T$ is ultra hypercyclic
for $(n_{k})$.

We say $T$ is supermixing if, for each nonempty open subset $U$ of $X$,

$\displaystyle
X=\overline{\bigcup_{i=0}^{\infty}\bigcap_{n=i}^{\infty}T^{n}(U)}.$

We say $T$ is hypermixing if, for each nonempty open subset $U$ of $X$,

$\displaystyle X=\bigcup_{i=0}^{\infty}\bigcap_{n=i}^{\infty}T^{n}(U).$

######  Remark 1.2.

Because of [2, Remark 2.5], the definition of hypermixing that appears above
coincides with the original definition of hypermixing in [2, Definition 1.1].

The following implications follow quickly from the definitions:

hypermixing | $\implies$ |  | supermixing | $\implies$ | mixing
---|---|---|---|---|---
$\Downarrow$ |  |  |  |  | $\Downarrow$
ultra hypercyclic |  |  |  |  | weakly mixing
$\Downarrow$ |  |  |  |  | $\Downarrow$
strongly hypercyclic |  |  | $\implies$ |  | hypercyclic

In [2, 7], several of the implications not yet present in the above diagram
for the family of weighted backward shift operators on $\ell^{p}$ and $c_{0}$
were explored; so far, we know:

hypermixing | $\implies$ |  | supermixing | $\iff$ | mixing
---|---|---|---|---|---
$\Downarrow$ |  |  |  |  | $\Downarrow$
ultra hypercyclic |  |  |  |  | weakly mixing
$\Downarrow$ |  |  |  |  | $\Updownarrow$
strongly hypercyclic |  |  | $\implies$ |  | hypercyclic

As we will explain later, it was shown in [7] that ultra hypercyclicity need
not imply mixing. Given this result, there is only one implication not present
in the above diagram left to explore for the family of weighted backward
shifts.

######  Question 1.3.

Does there exist a strongly hypercyclic weighted backward shift $B_{w}$ on
$\ell^{p}$ or $c_{0}$ that is not ultra hypercyclic?

At the present moment, we cannot answer this question. However, we provide a
characterization of ultra hypercyclicity in terms of the weight sequence of
the backward shift. This characterization leads us to consider the following
question whose answer would be great progress towards answering the above
question.

######  Question 1.4.

Does there exist a sequence $w_{n}$ of positive numbers with the following
properties:

  * •

$\displaystyle\inf_{n\geq 1}w_{n}>0$

  * •

$\displaystyle\sup_{n\geq 1}M_{1}^{n}=\infty$

  * •

$\displaystyle\limsup_{n\to\infty}\big{(}\inf_{i\geq 1}M_{i}^{n}\big{)}>0$

  * •

if $n_{k}$ is an increasing sequence in $\mathbb{N}$ with
$\displaystyle\lim_{k\to\infty}M_{1}^{n_{k}}=\infty$, then
$\displaystyle\limsup_{k\to\infty}\big{(}\inf_{i\geq 1}M_{i}^{n_{k}}\big{)}=0$

##  2 Weighted backward shifts

The family of weighted backward shifts on $\ell^{p}$ and $c_{0}$ are the
testing ground for any new notion in linear dynamics due to its simple yet
rich structure. Recall that $c_{0}$ is the space of all bounded sequences
$(x_{n})_{n\geq 0}$ in $\mathbb{C}$ for which $x_{n}\to 0$, and the norm in
$c_{0}$ is given by $\|x\|=\sup_{n\geq 0}|x_{n}|$. For $1\leq p<\infty$,
$\ell^{p}$ is the space of all complex sequences $(x_{n})_{n\geq 0}$ for which
$\sum_{n=0}^{\infty}|x_{n}|^{p}<\infty$, and the norm on $\ell^{p}$ is given
by $\|x\|=(\sum_{n=0}^{\infty}|x_{n}|^{p})^{1/p}$.

Let $X$ be $c_{0}$ or one of the $\ell^{p}$ spaces, $1\leq p<\infty$. We
denote by $e_{n}$, $n\geq 0$, the canonical basis vector of $X$ whose only
nonzero term is a 1 in the $n$th position. Every $x=(x_{0},x_{1},\ldots)$ in
$X$ can thus be written as $x=\sum_{i=0}^{\infty}x_{i}e_{i}$. If
$w=(w_{n})_{n\geq 0}$ is a bounded sequence of positive numbers, the weighted
backward shift $B_{w}$ is defined on the basis of $X$ by $B_{w}e_{0}=0$,
$B_{w}e_{n}=w_{n}e_{n-1}$ for $n\geq 1$. That $w_{n}$ is bounded is equivalent
to $B_{w}$ being continuous. We consider here only surjective weighted
backward shifts; it is a straightforward exercise to show that $B_{w}$ is
surjective if and only if $\inf_{n\geq 0}w_{n}>0$. Each weighted backward
shift has an associated forward shift $S$ defined on the basis of $X$ by
$Se_{n}=\frac{e_{n+1}}{w_{n+1}}$. When $B_{w}$ is surjective, $S$ is defined
on all of $X$, and the forward shift is a right inverse for $B_{w}$ on $X$,
meaning $B_{w}S=I$, where $I$ is the identity operator on $X$.

For simplicity in notations, we put $M_{i}^{j}=w_{i}w_{i+1}\cdots w_{i+j-1}$
for $i,j\geq 1$; that is to say, $M_{i}^{j}$ is the product of $j$ consecutive
weights, starting with the $i$th weight. Then for any $x\in X$, a formula for
$S^{n}x$ is given by

$\displaystyle
S^{n}x=(0,0,\ldots,0,\frac{x_{0}}{M_{1}^{n}},\frac{x_{1}}{M_{2}^{n}},\frac{x_{2}}{M_{3}^{n}},\ldots).$
(2)

Furthermore, for any $n\in\mathbb{N}_{0}$ and $i<j$, by writing out the
products in the following expressions, we have

$\displaystyle M_{i}^{n}M_{i+n}^{j-i}=M_{i}^{n+j-i}=M_{i}^{j-i}M_{j}^{n}$ (3)

The respective characterizing conditions for $B_{w}$ to be hypercyclic or
mixing are

$\displaystyle\text{hypercyclic: }\sup_{n\geq 1}M_{1}^{n}=\infty,\hskip
18.06749pt\text{mixing: }\lim_{n\to\infty}M_{1}^{n}=\infty.$

For this result and much more background on backward shifts, we refer the
interested reader to [8, Section 4.1] and the references therein.

A characterization of hypermixing weighted backward shifts was obtained by
Ansari in [2].

######  Theorem 2.1 ([2, Theorems 4.1 and 4.4]).

For a weighted backward shift $B_{w}$ on $c_{0}$ or $\ell^{p}$, the following
are equivalent.

  1. (i)

$B_{w}$ is hypermixing.

  2. (ii)

$B_{w}$ is surjective and $S^{n}x\to 0$ for every nonzero vector $x\in X$.

  3. (iii)

$\sup_{n\geq 1}M_{1}^{n}=\infty$ and $\inf_{n,k\geq 1}M_{n}^{k}>0$.

Our goal is to develop theorems analogous to Ansari’s for the ultra
hypercyclicity and strong hypercyclicity properties. Of course, Ansari has
laid much of the ground work, but we’ll need some preliminary results to
expand upon what has already been accomplished.

######  Lemma 2.2.

Suppose $B_{w}$ is surjective on $\ell^{p}$ or $c_{0}$. Then
$S^{n_{k}}e_{j}\to 0$ for all $j\in\mathbb{N}_{0}$ if and only if
$S^{n_{k}}e_{j}\to 0$ for some $j\in\mathbb{N}_{0}$. In terms of weights,
$M_{j}^{n_{k}}\to\infty$ for all $j\in\mathbb{N}_{0}$ if and only if
$M_{j}^{n_{k}}\to\infty$ for some $j\in\mathbb{N}_{0}$.

###### Proof.

Since $B_{w}$ is continuous and surjective, $\mu=\displaystyle\sup_{n\geq
1}|w_{n}|<\infty$ and $\displaystyle\delta=\inf_{n\geq 1}|w_{n}|>0$. It
suffices to show $M_{j}^{n_{k}}\to\infty$ for some $j$ implies
$M_{i}^{n_{k}}\to\infty$ for all $i$. Suppose $M_{j}^{n_{k}}\to\infty$ and let
$i\not=j$. We first consider the case when $i<j$. Rearranging equation (3), we
have
$M_{j}^{n_{k}}=\displaystyle\frac{M_{i+n_{k}}^{j-i}}{M_{i}^{j-i}}M_{i}^{n_{k}}\leq\left(\frac{\mu}{\delta}\right)^{j-i}M_{i}^{n_{k}}$,
which proves $M_{i}^{n_{k}}\to\infty$.

The case $i<j$ is similar, so we omit it. ∎

The similarities between the definition of ultra hypercyclic and hypermixing
lead immediately to the following “ultra hypercyclicity criterion.” Its proof
is entirely similar to the proof of the hypermixing criterion in [2, Theorem
2.3], so we omit it.

######  Theorem 2.3.

Suppose $X$ is a first countable topological vector space and $T:X\to X$ is a
surjective operator with right inverse map $S$. Then $T$ is ultra hypercyclic
for $(n_{k})$ if and only if, for every nonzero vector $x\in X$ and any $y\in
X$, there exists a sequence $(u_{k})$ in $X$ such that $u_{k}\in\ker
T^{n_{k}}$ and $S^{n_{k}}x+u_{k}\to y$.

Here is our main result so far.

######  Theorem 2.4.

Suppose $B_{w}$ is a weighted backward shift on $c_{0}$ or $\ell^{p}$, $1\leq
p<\infty$, and $(n_{k})$ is an increasing sequence in $\mathbb{N}$. Then the
following are equivalent.

  1. (i)

$B_{w}$ is surjective and is ultra hypercyclic for $(n_{k})$.

  2. (ii)

$B_{w}$ is surjective and for all nonzero $x\in c_{0}$, $S^{n_{k}}x\to 0$.

  3. (iii)

$\displaystyle\inf_{n\geq 1}w_{n}>0$, $\displaystyle\sup_{k\geq
1}M_{1}^{n_{k}}=\infty$, and
$\displaystyle\limsup_{k\to\infty}\big{(}\inf_{i\geq
1}M_{i}^{n_{k}}\big{)}>0$.

###### Proof.

(i)$\implies$(ii). Assume $B_{w}$ is surjective and ultra hypercyclic for
$(n_{k})$. Let $x=(x_{0},x_{1},\ldots)\in X$ be nonzero. Since the forward
shift $S$ is a right inverse for $B_{w}$ on $X$, Theorem 2.3 implies the
existence of a sequence of vectors$(u_{k})_{k}$ in $\ker B_{w}^{n_{k}}$ such
that $S^{n_{k}}x+u_{k}\to 0$. The vector $u_{k}$ must have the form
$u_{k}=(u_{k,0},u_{k,1},\ldots,u_{k,n_{k}-1},0,0,\ldots)$. Then

$\displaystyle
S^{n_{k}}x+u_{k}=(u_{k,0},u_{k,1},\ldots,u_{k,n_{k}-1},\frac{x_{0}}{M_{1}^{n_{k}}},\frac{x_{1}}{M_{2}^{n_{k}}},\ldots).$

Hence $\|S^{n_{k}}x\|\leq\|S^{n_{k}}x+u_{k}\|\to 0$.

(ii)$\implies$(i). We again use the ultra hypercyclicity criterion. Let $x\in
c_{0}$ be nonzero and let $y=(y_{0},y_{1},y_{2},\ldots)\in c_{0}$. Then
$S^{n_{k}}x\to 0$ and the sequence of vectors $(u_{k})_{k}$ given by
$u_{k}=(y_{0},y_{1},\ldots,y_{n_{k}-1},0,0,\ldots)$ belongs to $\ker
B_{w}^{n_{k}}$ and converges to $y$. Thus $S^{n_{k}}x+u_{k}\to 0+y=y$, which
shows $B_{w}$ is ultra hypercyclic for $(n_{k})$ by Theorem 2.3.

(ii) $\implies$ (iii). We prove the contrapositive. We can assume $B_{w}$ is
surjective. Furthermore, we can assume $\sup_{k\geq 1}M_{1}^{n_{k}}=\infty$.
If not, then by Lemma $B_{w}$ does not satisfy the Hypercyclicity Criterion
for $(n_{k})$, which implies $B_{w}$ is not hypercyclic for $(n_{k})$ and
hence not ultra hypercyclic for $(n_{k})$.

So suppose that $\inf_{i}M_{i}^{n_{k}}\to 0$ as $k\to\infty$. The rest of the
argument is similar to [2, Theorem 4.4]. There exists an increasing sequence
$(i_{k})$ in $\mathbb{N}$ such that $M_{i_{k}}^{n_{k}}\to 0$. Then there
exists a subsequence converging to zero fast enough such that

$\sum_{l=1}^{\infty}(M_{i_{k_{l}}}^{n_{k_{l}}})^{p}<\infty.$

Then one can construct a vector $x\in\ell^{p}$ for which $S^{n_{k_{l}}}x$
always has a 1, so $S^{n_{k}}x\not\to 0$, so $B_{w}$ is not ultra hypercyclic
for $(n_{k})$.

(iii) $\implies$ (ii). Let $X$ be $c_{0}$ or $\ell^{p}$, $p\geq 1$, and denote
by $\|\cdot\|$ the usual norm on $X$. Let $(n_{k})$ be the sequence described
in (iii). There exists some $r>0$ for which $\inf_{i}M_{i}^{n_{k}}\geq r$ for
all $k$. One can show that for any $x\in\ell^{p}$,
$\|S^{n_{k}}x\|<\frac{\|x\|}{r}$.

Let $x=(x_{0},x_{1},\ldots)\in\ell^{p}$. Let $\epsilon>0$ be given. There
exists $N\in\mathbb{N}$ such that
$\|\sum_{i=N+1}^{\infty}x_{i}\|<\frac{1}{2}r\epsilon$. Let
$x_{N}=\sum_{i=N+1}^{\infty}x_{i}$ and $x_{N}^{\prime}=\sum_{i=0}^{N}x_{i}$,
so that $x=x_{N}^{\prime}+x_{N}$. Then

$\|S^{n_{k}}x_{N}\|\leq\frac{\|x_{N}\|}{r}<\frac{\epsilon}{2}.$

We next deal with $x_{N}^{\prime}$.

Since $M_{j}^{n_{k}}\to\infty$ for each $j=1,2,\ldots,N$ by Lemma 2.2, for
large enough $n_{k}$, we can ensure

$M_{j}^{n_{k}}\geq\frac{2\|x\|}{\epsilon}.$

One can then show
$\|S^{n_{k}}x_{N}^{\prime}\|<\|x\|\frac{\epsilon}{2\|x\|}=\frac{\epsilon}{2}$.
∎

Here is the state of affairs for strong hypercyclicity, due to several results
by Ansari.

######  Theorem 2.5.

Suppose $B_{w}$ is a weighted backward shift on $c_{0}$. Then for the
following statements, the implications (i)$\iff$(ii)$\implies$(iii) hold.

  1. (i)

$B_{w}$ is surjective and is strongly hypercyclic.

  2. (ii)

$B_{w}$ is surjective and for all nonzero $x\in c_{0}$, there exists an
increasing sequence $(n_{k})$ such that $S^{n_{k}}x\to 0$.

  3. (iii)

For any increasing sequence $(i_{n})_{n}$ in $\mathbb{N}$,
$\displaystyle\sum_{n\to\infty}(M_{i_{n}}^{n})^{p}=\infty$. On $c_{0}$, the
weaker condition $\displaystyle\lim_{n\to\infty}M_{i_{n}}^{n}\not=0$ is
sufficient.

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