# Cosmic muon flux attenuation methods for superconducting qubit experiments

E. Bertoldo, M. Martínez, B. Nedyalkov, P. Forn-Díaz

###### Abstract

We propose and demonstrate two mitigation methods to attenuate the cosmic muon
flux compatible with experiments involving superconducting qubits. Using a
specifically-built cosmic muon detector, we find that chips oriented towards
the horizon compared to chips looking at the sky overhead experience a
decrease of a factor 1.6 of muon counts at the surface. Then, we identify
shielded shallow underground sites, ubiquitous in urban environments, where
significant additional attenuation, up to a factor 35 for 100-meter depths,
can be attained. The two methods here described are the first proposed to
directly reduce the effects from cosmic rays on qubits by attenuating the
noise source, complementing existing on-chip mitigation strategies. We expect
that both on-chip and off-chip methods combined will become ubiquitous in
quantum technologies based on superconducting qubit circuits.

## Introduction

Superconducting qubit circuits are sensitive probes to many types of
fluctuations present in their environment. These fluctuations place a limit to
the qubit coherence time. The qubit quality already achieved renders qubits
sensitive to very subtle environmental events, such as random ionizing
radiation impacts coming either from background radioactivity [1] or from
cosmic muons [2]. While environmental radioactivity could be screened using
radiopure materials as well as a dedicated shielding, as is routinely
implemented in low-background experiments [3, 4, 5, 6], cosmic rays cannot be
easily screened. As such, the only method to drastically reduce the muon flux
nowadays is to move the experiment to deep underground laboratories.

Ionizing radiation events are currently not the dominant source of
superconducting qubit decoherence [7]. Nevertheless, increasing evidence
points to this radiation being responsible for qubit device stability and
performance [1, 8, 9], through its effects lingering on the qubit substrate in
the form of phonons, quasiparticles, and the scrambling of two-level defects.
Therefore, techniques to mitigate the impact of ionizing radiation events are
already necessary today. While on-chip mitigation strategies, such as phonon
[10] or quasiparticle traps [11, 12, 13], are being developed, it is also
necessary to understand the physics behind ionizing radiation, particularly
the cosmic muon events, in order to find ways to attenuate their flux. In this
work, we propose two such methods. The first method studies device-orientation
dependence with respect to the zenith angle, since muons reaching the Earth’s
surface have a strong angular dependence due to their absorption within the
atmosphere. The second method is to consider shallow underground sites of
moderate depths of up to 100 m, where significant attenuation factors of muon
flux have been characterized [3, 14, 15].

We perform muon flux characterization at different depths and for different
chip orientations using a home-built muon detector. We provide a detailed
description of the detector so that other groups can develop their own. We
believe that superconducting qubit technology developers need to start
bringing attention to ionizing radiation, and cosmic muons in particular, in
order to push even further the quality and stability of the devices in order
to develop universal quantum computers and quantum sensors for, e.g., rare
physics events.

This work is organized as follows: in Sec. 1 we give an introduction of the
physics of cosmic muons and their connection with qubits. In Sec. 2 we
describe our muon detector and present its characterization. Section 3
presents our main results of the dependence of cosmic muon flux on the zenith
angle and on the rock depth for shallow sites. Finally, we conclude in Sec. 4.

##  1 Introduction to Cosmic muon physics for qubits

###  1.1 Cosmic rays

There are multiple particles of cosmic origin that reach Earth. These
particles can be divided in two broad categories: charged and neutral. Cosmic
rays is a terminology that usually defines charged particles that originate
outside the solar system [16, 17]. The neutral particles reaching Earth are
neutrinos and photons, with neutrinos generally not interacting with the
atmosphere. This differentiates neutrinos from charged particles and photons,
which normally interact with the atmosphere and generate air showers of
secondary cosmic rays. Primary cosmic rays are generated within our galaxy and
by extragalactic sources. Primary cosmic rays which reach our atmosphere are
almost exclusively protons ($\sim$90%) and alpha particles ($\sim$9%), with
small percentages of heavier nuclei and electrons. An even tinier percentage
is composed by antimatter particles in the form of positrons and antiprotons.
The energy of this primary cosmic rays spans several orders of magnitude, with
maximum energies exceeding 1020 eV [17].

In this work, we will focus on the type of cosmic rays that have the ability
to reach the surface and the underground laboratories on Earth. The particles
reaching the surface are mostly the products of the interactions between
primary cosmic rays and the atmosphere, while the particles reaching
underground labs can also be the product of the interactions of secondary
cosmic rays with the surrounding rocks and materials.

####  1.1.1 Cosmic rays at sea level

When primary cosmic rays enter the atmosphere they start interacting with
atoms and molecules, producing a cascade of particles, often referred to as
secondary cosmic rays (which include photons, protons, alpha particles, pions,
muons, electrons, neutrinos, and neutrons). At this point, the picture starts
to become more complicated, with numerous unstable and fast-decaying particles
created, and a variety of more stable particles that keep advancing towards
Earth’s surface. In general, many photons are produced in this interaction
which produce electromagnetic showers [16]. The rest of particles might also
eventually decay into photons and electrons, feeding the electromagnetic
showers. If hadrons survive or are newly produced, the shower continues its
development. However, after a certain number of generations, the shower starts
to fade, with fewer new particles produced than the ones absorbed in the
interaction with the atmosphere [16]. Muons and neutrinos created in these
processes do not contribute to the shower development and approximately
continue freely their path towards our planet. As a result, the vast majority
of particles reaching the sea level are neutrinos and muons, while electrons,
positrons, photons, neutrons, and protons compose a small fraction of the
total flux [18].

Muons at sea level exhibit a rate of approximately 1 count per minute per cm2
of horizontal surface and a mean energy around 4 GeV [19].

####  1.1.2 Cosmic rays underground

In deep underground laboratories, the muon flux decreases almost exponentially
with the thickness of the rock overburden above the laboratory [20] (see Sec.
3.2). Normally, to compare the depth of different laboratories, since the rock
composition varies, the overburden is converted in meter water equivalent
(m.w.e.)111The conversion between m.w.e. and real meters depends on the rock
composition, with typical values in the range of 2.4-3 m.w.e./m..

At shallow depths ($\sim 10$ m.w.e.), only muons, protons, and neutrons
constitute significant sources of cosmic ray background. In particular,
interactions caused by neutrons are extremely difficult to suppress. These
neutrons are usually a product of proton and muon spallation in the rocks and
materials surrounding the experimental setup. Designing effective shields
against neutrons is particularly cumbersome, due to their ability to cross
large distances.

Thick lead shields, ordinarily used to effectively suppress environmental
radioactivity, are the perfect example of the challenges that can arise in
above-ground laboratories when trying to reduce the ionizing radiation
reaching the experimental volume. In fact, ordinary lead shields are also
contaminated with radioisotopes, the most concerning of which is 210Pb [21].
To avoid this problem, archaeological lead with a low concentration of
radionuclides can be employed [3, 4]. However, this is not a solution at all:
due to its high density, lead is a perfect medium for spallation events, which
not only create neutrons in the medium, but also cause cosmogenic activation
of the material. Cosmogenic activation is the production of radioisotopes due
to cosmic ray interactions within the material. As such, it can be immediately
understood that even employing archaeological lead at sea level is not a
complete solution, since that lead would progressively become more radioactive
due to the high cosmic ray flux. To reduce the flux of neutrons generated by
spallation in lead and other materials surrounding the cryostat, usually a
polyethylene shield is interposed between the cryostat and the lead shield
[3]. It should be noted that the polyethylene shield thickness should be
carefully selected, in order to not have a relevant production of spallation
on polyethylene itself [3].

At the surface level, almost all spallation events are due to proton
interactions, but below 15 m.w.e., due to the strong attenuation of the proton
flux, spallation induced by muons becomes dominant [14]. As such, at 30 m.w.e.
the muon flux is reduced by approximately a factor 6, but the neutron flux is
reduced by a factor of almost 100 [22, 14]. In fact, several shallow
laboratories exist around the world where muon flux screening factors of 5-10
are attained without the need of very large depths [14].

These shallow laboratories offer various advantages: they reduce the amount of
direct muon interactions with the sample, they reduce the number of fast
neutrons generated by spallation events caused by cosmic protons and muons,
and they also reduce the cosmogenic activation of materials constituting or
surrounding the experimental setup [23, 24].

At significant depths ($>1$ km.w.e.), only muons and neutrinos can cause
events in the experimental volume, both by direct hits or by inducing tertiary
fluxes of particles due to interactions with surrounding rocks and materials
[18].

The only viable solution to drastically reduce the cosmic radiation background
due to cosmic muons is to perform experiments in deep underground
laboratories. Several laboratories of this type exist around the world and
they generally host experiments focused on astroparticle physics 222 The
current list of operative deep underground laboratories in Europe includes the
Baksan Neutrino Observatory (BNO) in the Russian Federation [25], Boulby in
the United Kingdom [26], the Canfranc Laboratory (LSC) in Spain [27], the
Centre for Underground Physics in Pyhäsalmi (CUPP) in Finland [28], the Gran
Sasso Laboratory (LNGS) in Italy [29], and the Modane Laboratory (LSM) in
France [30]. In Asia, we can find the China Jinping Underground Laboratory
(CJPL) in China [31] and the Kamioka Observatory in Japan [32]. Finally, in
North America, one counts the Sanford Underground Research Facility (SURF)
[33], the Soudan Underground Laboratory [34], the Waste Isolation Pilot Plant
(WIPP) [35], and the Kimballton Underground Research Facility (KURF) [36] in
the USA, and SNOLAB in Canada [37].. The deepest operating underground lab is
CJPL in China while the biggest in size is LNGS in Italy [38].

###  1.2 Qubit decoherence due to ionizing radiation

In recent years, evidence has been mounting on the detrimental effect of
ionizing radiation on superconducting qubits [39], particularly from cosmic
rays. At least two mechanisms have been identified so far coming from hits on
the bulk of the device substrate, indirectly impacting qubit performance.

The first mechanisms corresponds to an impinging high energy particle or
photon that causes Compton scattering with an atom of the substrate bulk. The
impact produces a highly energetic electron which excites electron-hole pairs,
themselves producing phonons upon recombining. These highly energetic phonons
travel ballistically throughout the chip, some of them eventually scattering
at the boundary with the superconductor leading to the breaking of Cooper
pairs and thus creating quasiparticles which are a known source of qubit
decoherence [40].

The second mechanism has been recently identified to be a scrambling effect on
the two-level systems [9] (TLSs) believed to reside mostly at the
superconductor-insulator interface of the qubit circuit[41]. These TSLs
display a very broad frequency spectrum, often coming into resonance with
qubits they couple to, leading to either fluctuations in $T_{1}$ [9] or gaps
in the energy spectrum [42]. The scrambling effect causes shifts in the
frequency of TLSs and thus it is responsible for both frequency noise in
qubits as well as fluctuations in the relaxation time $T_{1}$ and the
decoherence $T_{2}$ time [43].

Both mechanisms are particularly harmful as they cause correlated noise [8]
which cannot be accounted for in currently explored quantum error correcting
methods such as the surface code [44]. Another possible mechanism not studied
are direct hits on the superconducting material, similar to the detection
mechanism of single photons in superconducting nanowire photon detectors [45].
The deleterious effect of cosmic rays was also demonstrated on superconducting
resonators placed in the deep underground laboratory in Gran Sasso [2]. Other
evidence of the impact of ionizing radiation has been demonstrated on multi-
qubit devices, were correlated charge fluctuating events between neighboring
qubits were observed [46].

The first direct evidence of the impact of ionizing radiation on a
superconducting qubit used a radioactive source installed directly inside of a
dilution refrigerator mixing chamber stage [1]. A transmon qubit placed in the
vicinity of the source experimented a drastic decrease of its lifetime due to
the activity of the source. Based on those measurements, an upper bound of the
transmon qubit lifetime due to background ionizing radiation of more than 3
milliseconds was established, with an estimated 40% of the external background
events corresponding to cosmic rays. Other works using fluxonium qubits have
reached millisecond-range lifetimes [47] and coherence [48], possibly limited
by QPs partially generated by ionizing radiation as well as stray microwave
fields [49].

Assuming one is able to eliminate all background radioactivity by proper
shielding inside and outside the cryostat (for instance working with a
radipopure cryostat in an underground laboratory), only cosmic rays will
impact qubit coherence. A decrease of the cosmic muon flux would lead to a
reduction of the steady-state amount of excess QPs, given the reduction of
energetic phonons in the substrate. The relation between high energy impacts
and QPs is complex and depends on many material properties. In addition, a
reduction of cosmic muon flux would also reduce the rate of TLS fluctuating
frequency, leading to more stable qubit frequency and coherence times.

Environmental radioactivity from neighboring sources inside the dilution
refrigerator are progressively being identified [50]. Attention will be placed
in future experimental setups and laboratory spaces towards more radiopure
systems, such as those employed in complex, low-background experiments [4].
Eventually, the only remaining source of ionizing radiation will be the cosmic
rays, which, as stated in the previous section, cannot be easily screened
above ground. Therefore, mitigation strategies are needed to suppress the
effects of cosmic rays. In this work, we propose two methods: the specific
device orientation (Sec. 3.1) and the screening obtained in shallow
underground sites (Sec. 3.2), as direct means to attenuate the cosmic muon
flux. Other initiatives in the field have been addressing the same problem
with direct on-chip mitigation strategies, such as lower-gap superconducting
quasiparticle traps [11], phonon traps [10], or suspending the device on a
membrane [51]. All strategies have evidenced a decrease on the impact of
ionizing events on the qubits and/or resonators employed as sensors. Very
likely, the final solution will become a combination of both external as well
as internal mitigation methodologies.

##  2 Compact muon detector

In this section, we present a compact and portable cosmic muon detector that
we built ourselves to characterize the flux of muons under different
circumstances, such as the orientation of the detector with respect to the sky
and at shallow underground sites of different depths.

###  2.1 Detection principle

In this work, we used a detector based on two superimposed plastic
scintillators coupled to photo multiplier tubes (PMTs) and readout in
coincidence for the detection of cosmic muons. The plastic scintillator used
emits fast pulses of light when radiation crosses it, with a spectrum and
speed well matched to the spectral sensitivity and time response of the PMT.

In principle, these detector setups are meant to be used to detect any type of
radiation strong enough to create a detectable signal. Therefore, in order to
discriminate muons, since they are penetrating particles, we have chosen to
use two identical detectors physically superimposed to each other and
connected to a coincidence circuit. A thick metal shielding made of copper
between both detectors helps in increasing the filtering for particles other
than muons (see Fig. 1).

Figure 1: Illustration of the detector-coincidence concept to detect muons and
filter out the signals from $\alpha$, $\beta$ and $\gamma$ radiation, that do
not produce coincidences.

###  2.2 Description of the detector

The scintillator plus PMT setup comes already pre-assembled as a commercial
detector unit (SCIONIX model R30*20B250/1.1-E3-P-X) which includes an Ej200
plastic scintillation detector of dimensions $20\times 30\times 250~{}$mm3
wrapped in reflector and surrounded by light tight vinyl coupled to 30 mm
diameter R6094 PMT equipped with a voltage divider and counting electronics.
The first 1.5 mm wrapping layer of this detector is not sensitive to
radiation.

The detector has been built in a very compact setup for easy transportation
and fits into a standard size “executive” suitcase (see Fig. 2). It is powered
by a 12 V lead battery or alternatively connected to the 220 V AC electrical
network. In this second case, the battery is recharged in parallel. From
either the 12 V from the battery or the 220 V AC power supply delivering 12 V
output, a power management circuitry delivers all the needed voltages (see
Fig. 3).

Figure 2: The muon detector inside the suitcase. The upper part is the power
circuitry including the network inlet, the power supply and low voltage
regulators, the HV generators and the lead battery for standalone operation
without power connection. Below we positioned the two scintillator detector
set stacked with a copper absorber in between. On the bottom-right, one finds
the coincidence, counting, recording, and displaying circuitry.

Using DC-DC converters, from these 12 V the setup generates the high voltages
needed to power the PMTs (700 V) and, using linear regulators, the 5 V needed
to power the coincidence circuit and the microcontroller that takes care of
the DAQ, displaying and recording.

Each scintillator plus PMT detector setup provides already a TTL digital
discriminated output, that is used as input for the coincidence electronics.
The length of the digital output is of 50 ns while the signal-overlap time
requested for detecting a coincidence (resolving time) is of about 20 $\mu$s.

Each single detector signal, as well as their coincidences, are input to an
Arduino Mega microcontroller that, using interrupts, counts them and computes
their rate and their statistical uncertainty. The microcontroller also
measures the setup temperature, humidity and atmospheric pressure and displays
all the quantities mentioned into a large display while recording all that
data into an SD card in easily readable text format for later analysis. The
suitcase can be easily carried and installed in any place, while the battery
provides an autonomy of a few days. Furthermore, it can easily incline to
check the measured rates as a function of the zenith angle, as shown in Sec.
3.1.

###  2.3 Detector characterization

In order to assess the accidental coincidence rate $N_{a}$, one can use the
single rate measurements $N_{i}$ from each detector together with the
coincidence resolving time $\tau$ in a simple probabilistic calculation using
the standard formula [52]:

$N_{a}=2\tau\times N_{1}\times N_{2}.$ (1)

In our detector, typical values measured for the single rates at ground level
after taking data for about 20 hours are $N_{i}=701.57\pm 0.78$ CPM and, as
mentioned before, $\tau=20\pm 2\,\,\mu$s, so that the calculated accidental
coincidence rate is

$N_{a}=0.33\pm 0.03\,\,\mathrm{CPM}.$ (2)

Alternatively, for a direct measurement of the accidental rate the setup also
allows a simple disassembly of the upper scintillator detector in such a way
that it can be placed sideways, in a geometrical position in which the chances
for any particle crossing both detectors are completely negligible (see Fig.
3). With this setup, the measured accidental rate at ground level after taking
data for about 20 hours is

$N_{a}=0.31\pm 0.02\,\,\mathrm{CPM}.$ (3)

Therefore, both approaches give identical results within uncertainties.

The accidental rate is almost negligible (less than 1%) when compared with the
typical coincidence rate at ground level $N_{c}$ after about 20 hours running,

$N_{c}=40.00\pm 0.18\,\,\mathrm{CPM}.$ (4)

For shallow and deep underground sites, this error becomes very relevant as it
is comparable or larger than the actual reading (see Sec. 3.2).

Figure 3: The muon detector setup for measuring the accidental coincidence
rate. The upper scintillator detector set is turned 180 degrees in such a way
that the geometrical overlap with the lower scintillator detector set is
completely negligible .

##  3 Cosmic ray mitigation strategies

By employing the cosmic muon detector presented in the previous section, we
demonstrate in this section different methods to lead to a smaller flux of
cosmic muon counts, with a direct impact on superconducting qubit-based
experiments.

###  3.1 Angular dependence of cosmic muon flux

The first mitigation strategy to reduce the total number of cosmic muon
impacts on a device is to consider the angular dependence of the muon flux due
to the absorption through the atmosphere.

In order to evaluate the flux of muons at sea level we use the aforementioned
muon detector at varying orientations, this way measuring the muon flux at
different zenith angles $\theta$, as shown in Fig. 4. The measurements are
performed from an horizontal position $\theta=0$ of the detector looking
directly at the sky overhead to a complete vertical one $\theta=\pi/2$ where
the detector looks at the horizon (from both sides). The individual
measurements are shown in Fig. 5, and they display a clear dependence on the
detector orientation.

In order to fit the observed counts, the muon flux can be estimated roughly
with a cosine law of the form $I(\theta)=I_{0}\cos^{2}(\theta)$ as shown by
Bae, et al. [53]. However, here we require a more detailed model, which
accounts for the finite thickness of the Earth’s atmosphere with the change of
the zenith angle (see the Appendix 5.1 for more details). While the
$I(\theta)$ law holds for a muon flux through an infinitesimal point in space
for a certain incoming zenith angle $\theta$, the actual geometry of the
detector needs to be accounted for.

The differential number of muons incident on a differential surface is given
by

$\mathrm{d}N=I(\theta)\,\mathrm{d}A\,\mathrm{d}\Omega\,\mathrm{d}t.$ (5)

Here, $I(\theta)$ is the muon flux intensity, incident on the differential
area $\mathrm{d}A$, over the solid angle $\mathrm{d}\Omega$, over a time
$\mathrm{d}t$. In order to obtain the total rate of muon impacts
$\Phi\equiv\Delta$N/$\Delta$t upon the detector one must integrate both over
the angular aperture of the device and over its area:

$\Phi(\theta)=\int_{\Omega}\int_{\mathrm{Area}}I(\theta)\,\mathrm{d}\Omega\,\mathrm{d}A.$
(6)

Figure 4:  Depiction of the inclination of the muon detector, being oriented
at anangle $\theta$ with respect to the zenith.

Figure 5: Muon flux as a function of the zenith angle. One the left we have
experimental data for each measured angle and a fit using our model, Eq. (6).
The fit is performed according to the corrected cosine-squared law (see
Appendix). On the right, we apply the same model to deduce the muon impacts
per minute for a typical chip of area $5\times 5$ mm2, and $0.5$ mm thickness
as a function of its inclination.

From the calculated angular flux dependence $\Phi(\theta)$, one obtains the
total amount of cosmic muon impacts depending on the orientation of the
detector. The result is plotted in Fig. 5 as the green trace. We use Eq. (6)
to fit a muon flux intensity of $I_{0}=0.420$ cm-2sr-1min-1, comparable to the
work of Rossi [54], where $I_{0}=0.498$ cm-2sr-1min-1 was obtained. The
discrepancy may come from the lack of angular resolution from our detector due
to the close proximity of the two scintillators.

Having validated our geometry-dependent muon flux model in Eq. (6), we can now
use it to predict the angular dependence impinging on a typical qubit chip
geometry. This is possible by using the exact same model as for our muon
detector, but setting the aperture angles $\alpha$ and $\beta$ (defined in
Appendix 5.3) of the detector to 180 degrees, which is equivalent to having
one solid rectangular body. Taking a 0.5 mm thick silicon substrate with an
area of $5\times 5$ mm2 yields the result shown as the blue line on Fig. 5. A
clear trend of decrease of the muon particle flux with the tilt of the device
can be observed. Between $0^{\circ}$ and $90^{\circ}$ inclination, there is a
decrease factor of $1.6$ in the muon rate. This demonstrates that the tilt of
the device can impact the total amount of energy it receives from cosmic
muons.

Based on these results, it is clear that, in order to minimize the cross-
section with cosmic muons, superconducting qubit chips must be oriented with
the surface looking towards the horizon.

###  3.2 Underground shielding

As explained in Sec. 1, experiments requiring the lowest possible noise
background level locate their sensors in deep underground laboratories around
the world, where the rock overburden screens most of the incoming cosmic
radiation. Typical depths of underground laboratories are in the 1 km range,
obtaining screening factors of cosmic muon flux of $10^{5}-10^{6}$ [55, 56].
Due to their remote location and scarcity, these underground stations do not
represent a practical solution for improving the quality of any qubit type
affected by cosmic rays. As the lifetime of superconducting qubits limited by
ionizing radiation lies in the millisecond range for transmons [1], it may not
be necessary to attain the screening factors achieved in deep underground
laboratories and, instead, much shallower sites [14] may already offer a
significant enough screening to bring qubit quality to a level desirable for
applications such as error correction and sensing.

We have identified commonly-found locations in city environments providing
significant screening factors from cosmic radiation. These are underground
stations and side tunnels in underground highways. For the particular case of
the city of Barcelona, we have conducted measurements in two such locations
using our portable cosmic muon detector described in Sec. 2. The first site is
the underground station ‘El Coll-La Teixonera’ located under a hill at a depth
of approximately 100 meters (see Fig. 6).

Figure 6:  Profile of the TMB subway station ’El Coll-La Teixonera’ where
measurements were conducted.

The second site is a side tunnel in an underground highway known as the
Vallvidrera tunnels running under the ‘Collserola’ hill, with an approximated
average depth of 120 m, with a peak height of 170 m (Fig. 7).

Figure 7:  Profile of the Vallvidrera tunnel under the Collserola hill where
measurements were conducted.

As a reference of a more realistic depth where qubit laboratories could be
easily located, we also performed measurements in a much shallower site of 6 m
in the sub-basement of the ALBA synchrotron located near the Bellaterra
campus333https://www.cells.es/.

The results are shown in Fig. 8. All measurements were taken over 48 hours to
minimize statistical uncertainty (smaller than the data markers in the plot).
Above ground, we measure on average 40 CPM. The TMB and Vallvidrera shallow
sites investigated display a muon count of 1.4 CPM and 1.1 CPM, respectively,
achieving a screening factor of 29 and 36 with respect to ground. The 6 m
shallower site achieves a rate of 20 CPM, representing a screening factor of
2. In all data, we have already subtracted the spurious coincidence counts
from the apparatus dead time, as detailed in Sec. 2.3.

Figure 8:  Measurement of muon counts as function of depth. The surface
measurement is used as a reference for the two shallow underground sites
identified: the Barcelona subway (TMB) and Vallvidrera tunnels. The ALBA
synchrotron sub-basement is used as a typical laboratory location. The
statistical uncertainty of all measurements is smaller than the marker size of
the plot after averaging over 48h for each location.

For the case of the Vallvidrera tunnels, we have collected data for single
events in our detector with the result of 1047.4 counts per minute. This
represents a factor 1.68 higher than above ground in a usual office space,
which is consistent with the presence of heavy minerals in the tunnels
generating a higher amount of radioactivity. This is also consistent with
measurements in deep underground laboratories. Therefore, it is necessary to
equip qubit measurement setups with screening against this type of
radioactivity, such as a lead shield. Note that given the low amount of cosmic
ray flux in this shallow site, the re-excitation of lead isotopes detailed in
Sec. 1.1 would take a much longer time scale than above ground.

Assuming one is able to screen environmental radioactivity using lead
shielding, as is usual in underground sites, the screening factor obtained at
100 m depths would lead to a decrease of events to a level of $\sim 3\times
10^{-4}$ s-1cm-2. Using typical device dimensions for superconducting qubits
of $\sim 1$ cm-2, a device placed in such a shallow site would experience an
average of 0.018 counts per minute, or 0.3 mHz, compared to 0.54 counts per
minute, or 9 mHz, above ground.

##  4 Conclusions

In this work, we have provided two methods to reduce the flux of cosmic muons
compatible with experiments involving superconducting qubits. The first method
is to position the qubit chip pointing towards the horizon in order to
minimize its cross section to incoming muons, this way reducing the flux a
factor 1.6 above ground. The second method consists in locating qubit
experiments in a shallow underground location, which could be as low as 10 m
below surface. This small depth already provides strong protection against
neutrons, and it reduces the cosmic muon flux by a factor 2. Deeper sites at
100 m depth would observe a reduction of cosmic muon flux on the order of 30.
We have validated both methods using a home-built, portable cosmic muon
counter consisting of two plastic scintillators integrated in a coincidence
counting electronics circuit.

The unquestionable evidence of the effect of cosmic muons on qubits requires
addressing this source of noise and instability with as much priority as the
rest of sources of noise, such as TLS surface defects. Decreasing the impact
of cosmic muons will only be possible through a combination of techniques to
reduce their flux, following methods similar to the ones proposed in this
work, and their impact with on-chip mitigation techniques, such as phonon and
quasiparticle suppression. Understanding the physics of cosmic muons
interacting with qubit devices is an important future step to further suppress
their effect and lead to higher quality qubits to build more robust quantum
computers and more stable quantum sensors.

## Acknowledgements

We would like to thank Marcos González and Adela Rubio from ‘Transport
Metropolitans de Barcelona’ (TMB), Jordi Quevedo from ‘Túnels de Barcelona i
Cadí’ and Joan Casas and Ramon Pascual from the ALBA synchrotron for granting
us access to their premises to conduct the cosmic muon analysis. We also thank
Carlos Peña-Garay at LSC for fruitful discussions. We acknowledge funding from
the Ministry of Economy and Competitiveness and Agencia Estatal de
Investigación (RYC2019-028482-I, PCI2019-111838-2, PID2021-122140NB-C31), and
the European Commission (FET-Open AVaQus GA 899561, QuantERA) and program
‘Doctorat Industrial’ of the Agency for Management of University and Research
Grants (2020 DI 41; 2020 DI 42). IFAE is partially funded by the CERCA program
of the Generalitat de Catalunya. E. B. is supported by the Grant
FJC2021-046443-I funded by MCIN/AEI/10.13039/501100011033. This study was
supported by MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya.

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##  5 Appendix

###  5.1 Correction to the $\cos^{2}\theta$ law

In order to take into account the Earth’s curvature radius, and hence the real
atmosphere thickness as a function of zenith angle in the expectation of the
angular dependence of the muon flux, Fig. 9 shows the actual geometrical
dependence. There, $r$ stands for Earth’s radius (about 6400 km), $a$ stands
for the vertical atmosphere thickness (taken to be 10 km), $\theta$ is the
zenith angle and $p$ the atmosphere thickness at that zenith angle.

After applying some geometrical considerations, we obtain an expression for
the Earth’s atmosphere crossed by muons $p(\theta)$ as a function of the
zenith angle $\theta$ as follows:

$p(\theta)=-r\cos(\theta)+\sqrt{a^{2}+2ar+[r\cos(\theta)]^{2}}.$ (7)

For small angles near $0$, we recover $p\simeq a$, as expected. For horizontal
zenith (90 degrees) and since $r\gg a$, then $p\simeq\sqrt{2ar}\gg a$.

Since the muon flux decreases due to the atmosphere’s absorption, it should be
inversely proportional to the square of the atmosphere’s thickness, namely to
the square of $p$. This correction shall therefore be taken into account to
the fit of our data, particularly when comparing the results for vertical (0
degrees zenith) and horizontal (90 degrees zenith) muon fluxes.

Thus, the following expression for the flux is obtained:

$I(\theta)=I_{0}\left(\frac{a}{-r|\cos(\theta)|+\sqrt{a^{2}+2ar+[r\cos(\theta)]^{2}}}\right)^{2}.$
(8)

Figure 9:  A model of planet Earth and its atmosphere. The radius $r$ of the
planet is taken to be 6400 km. The atmosphere thickness $a$ is taken to be 10
km. $\theta$ is the zenith angle and $p$ portrays the thickness of the
atmosphere at an angle $\theta$, which is the one crossed by incoming muons.

###  5.2 The zenith angle definition

The area of the sky our detector looks at is a spherical rectangle (a
rectangle projected on a sphere). This area takes a big part of the surface of
the unit sphere one integrates over and thus it also crosses the poles of the
sphere, where one encounters a discontinuity. In order to avoid that
discountinuity, we make a change in the reference system. Instead of the
detector rotating with respect to the sky, we have the sky rotate with respect
to the detector (by an amount $\theta_{zenith}$, which spans the
range$[0,\pi/2]$), while the detector is fixed facing in direction of the
equator (the normal vector of the detector stays on the $x-y$ plane). On Fig.
10 one can see the spherical coordinate system used for integration.

The zenith angle transformed will thus be:

$\theta_{\mathrm{mod}}(\theta_{\mathrm{zenith}},\theta,\phi)=\arcsin[\sin(\theta_{\mathrm{zenith}}+\theta)\cos(\phi/2)].$
(9)

Here, $\theta_{mod}$ is the angle of deviation from the normal of the detector
surface (which is aligned with the $x$-axis, see Fig. 10).

Figure 10:  The spherical coordinate system.

###  5.3 The area correction

The geometry of our detector also needs to be accounted for when constructing
the model. This is because for a particle to be registered as a muon it must
penetrate both scintillators. However, the angles at which this can be
accomplished are limited by the geometry of the detector itself, as shown in
Fig. 11. Also, the effective area seen by particles coming with a certain
inclination angle with respect to the detector will be a function of this
inclination angle, as illustrated in Fig. 11.

Our particular detector is a rectangular cuboid with height $h$, width $W$ and
length $L$. The area of the detector is defined by $A=L\times W$. After the
correction for the area, the effective area becomes
$A^{\prime}=L^{\prime}\times W^{\prime}$, where we define $L^{\prime}$ and
$W^{\prime}$ as:

$L^{\prime}=L-h|\tan(\theta)|\quad,\mathrm{where}\quad\theta\in[-\alpha/2,\alpha/2]$
(10)
$W^{\prime}=W-h|\tan(\phi)|\quad,\mathrm{where}\quad\phi\in[-\beta/2,\beta/2]$
(11)

Here, the $\theta$ and $\phi$ are the zenith and azimuth angles, respectively,
and they sweep values within the aperture angles for the length side $-\alpha$
and the width side $-\beta$, respectively [Fig 11 and Fig 12].

The aperture angles are calculated as follows:

$\alpha=2\arctan(L/h).$ (12) $\beta=2\arctan(W/h).$ (13)

Our scintillators have dimensions $L=24.7$ cm, $W=2.7$ cm, $h=1.7$ cm.

The aperture angles ultimately express the fact that the two scintillators
have a gap in between them. Because the detection of a muon happens when a
particle is detected to pass trough both scintillators consecutively, if a
particle comes at such as steep angle that cannot pass through both
scintillators, but only trough one or none of them, it won’t be detected. This
steep angle defines the range of detection of the scintillators and thus
defines their aperture, how much does the detector ”see” (in terms of solid
angle), as shown in Fig. 11. If one increases the distance between the
scintillators, the aperture angle decreases. If the distance between them is
zero, then the aperture angle becomes 180 degrees and it allows to simulate a
solid structure, such as a rectangular chip as used to plot the curve in Fig.
5 of the main text.

Finally, note that the angular dependence experiment was conducted at the
campus of Autonomous University of Barcelona, located in a basin surrounded by
buildings and hills. To avoid the screening effect of the surrounding
environment, we neglected the shallowest 15 degrees over the horizon when we
calculate the integral of the total flux in Fig. 5.

Figure 11: The left figure shows the variation of the effective detector area
as function of the incident angle of the incoming muons. The same
consideration applies to the other side of the rectangular cuboid. The right
figure defines an aperture angle $\alpha$. All particles which fall on the
detector at angles steeper then $\alpha/2$ will not be registered. Figure 12:
The aperture angles of the detector.

###  5.4 The final expression for the muon rate

The expression used to model our particular detector is:

$\Phi(\theta_{\mathrm{mod}})=\int_{-\alpha/2}^{\alpha/2}\int_{-\beta/2}^{\beta/2}I(\theta_{\mathrm{mod}})A^{\prime}(\theta,\phi)\sin(\theta)\,\mathrm{d}\theta\,\mathrm{d}\phi.$
(14)

This expression and can be used for any class of similar detectors. Here, we
use the expressions in Eqs. (8)–(11).