# Stabilizing and improving qubit coherence by engineering noise spectrum of
two-level systems

Xinyuan You  Superconducting Quantum Materials and Systems Center, Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Ziwen Huang
Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Ugur Alyanak
Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Alexander Romanenko
Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Anna Grassellino
Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Shaojiang Zhu
Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA

###### Abstract

Superconducting circuits are a leading platform for quantum computing.
However, their coherence times are still limited and exhibit temporal
fluctuations. Those phenomena are often attributed to the coupling between
qubits and material defects that can be well described as an ensemble of two-
level systems (TLSs). Among them, charge fluctuators inside amorphous oxide
layers contribute to both low-frequency $1/f$ charge noise and high-frequency
dielectric loss, causing fast qubit dephasing and relaxation. Moreover,
spectral diffusion from mutual TLS interactions varies the noise amplitude
over time, fluctuating the qubit lifetime. Here, we propose to mitigate those
harmful effects by engineering the relevant TLS noise spectral densities.
Specifically, our protocols smooth the high-frequency noise spectrum and
suppress the low-frequency noise amplitude via depolarizing and dephasing the
TLSs, respectively. As a result, we predict a drastic stabilization in qubit
lifetime and an increase in qubit pure dephasing time. Our detailed analysis
of feasible experimental implementations shows that the improvement is not
compromised by spurious coupling from the applied noise to the qubit.

##  I Introduction

Superconducting circuits hold substantial promise as quantum bits due to their
great flexibility in circuit design [1, 2, 3, 4, 5]. One main challenge of
superconducting qubits is their relatively short coherence times, caused by
the uncontrolled interaction with solid state environment [6, 7, 8, 9]. Over
the past two decades, the coherence times have been boosted more than five
orders of magnitudes, and have now reached the millisecond range [10, 11, 12,
13]. Even with these great improvements, longer coherence times are still
required to achieve the fault-tolerant computation [14, 15].

Towards understanding and reducing qubit decoherence, multiple noise channels
have been identified [6, 7, 8, 9]. One popular model capable of describing
several of the dominating noise sources is the two-level systems (TLSs) [16,
17, 18, 19, 20]. In particular, those charged fluctuators inside the amorphous
oxide layers of the circuit device couple to qubit modes via electric-dipole
interaction, causing qubit decoherence. At high frequency, they contribute to
dielectric loss, limiting the qubit lifetime $T_{1}$ [21, 22]. At low
frequency, they lead to $1/f$ charge noise, degrading the qubit pure dephasing
time $T_{\phi}$ [23, 24, 25]. To protect the qubit from those detrimental
effects, several approaches have been taken. First, the negative impacts of
TLSs can be mitigated through advancement in material science [7, 8, 20].
Recently, the lifetime of transmon qubits has been improved several times by
solely switching the host material from aluminum/niobium to tantalum [26, 27].
Second, the coupling and sensitivity of the qubit to TLSs can be greatly
suppressed by exploiting freedom in qubit design [9]. For example, with
significantly increased ratio of Josephson junction energy to charging energy,
transmon qubits demonstrate exponential insensitivity to $1/f$ charge noise,
and thus long pure dephasing time [28, 29]. Besides optimizing circuit
parameters, another useful method to suppress the qubit–TLS coupling is to
place coplanar 2D circuits into 3D cavities [30]. This strongly reduces the
electric field energy stored in the oxide layers, and therefore improves qubit
lifetime [22].

In addition to long coherence times, reliable quantum computation also
requires them to be stable over time, cooldowns, devices, etc. The latter is
attracting more attention recently, with ubiquitous experimental evidence
showing that the temporal fluctuation of qubit lifetime becomes stronger as
its average value gets longer [31, 32, 33, 34]. The origin of this temporal
fluctuation is often attributed to spectral diffusion, proposed by Müller et
al. [35]. In this model, fluctuating qubit lifetimes are caused by random
shifts in the energies of those close-resonance TLSs, which are further
induced by the interaction between TLSs and low-frequency two-level
fluctuators (TLFs). To suppress this fluctuation, Matityahu et al. [36]
propose to drive the qubit coherently, such that the TLS bath is not only
probed at the qubit frequency, but sampled across a larger range of the
spectrum. In addition, Zhao et al. [37] explore the possibility to tune the
qubit away from those high-frequency TLSs via ac Stark shift. Since both
proposals involve coherently driving the qubit, more complex single- and two-
qubit gate schemes are required [38].

Instead of driving the qubit, it is also possible and potentially advantageous
to directly address the TLSs [39, 40, 41, 42, 43, 44]. In this work, we
propose to engineer the TLS noise spectrum by implementing artificial noise.
Specifically, we apply longitudinal noise to dephase TLSs, which smooths the
high-frequency noise spectrum, and results in stabilization of qubit lifetime.
Moreover, we utilize transverse noise to depolarize TLSs, which suppress the
low-frequency noise amplitude, and leads to an increase of qubit pure
dephasing time. The protocols are summarized schematically in Fig. 1. Compared
with the methods that involve driving the qubit, our approach is compatible
with standard gate schemes of qubit controls.

Figure 1: Schematic representation of the protocols. Artificial longitudinal
noise is applied to dephase the TLS, which stabilizes the qubit lifetime
(blue). Transverse noise is introduced to depolarize the TLS, leading to
longer pure dephasing time of the qubit (red). Spurious coupling from the
applied noise to the qubit is irrelevant (gray).

The paper is structured as follows: In Sec. II, we review the standard
tunneling model of the TLS, and connect its noise spectral densities to qubit
decoherence. We then introduce our protocols to improve qubit coherence via
engineering TLS noise spectrum in Secs. III and IV, where we show that the
qubit lifetime gets stabilized by dephasing TLSs, and its pure dephasing time
is improved by depolarizing TLSs. We explore feasible experimental
realizations in Sec. V, and confirm the irrelevance of the spurious coupling
from the applied noise to the qubit. We share our conclusions in Sec. VI, and
provide additional details in the subsequent appendix.

##  II TLS as a noise source of qubit decoherence

In this section, we start with a brief review of the TLS model used in the
rest of the work. We then relate the decoherence properties of qubits to noise
spectral densities of each single TLS. This motivates our protocols developed
in later sections. We end this section by introducing two different statistics
in the presence of an ensemble of TLSs, which are prerequisites to analyze the
average and fluctuation of qubit decoherence properties.

###  II.1 Standard tunneling model

Based on the standard tunneling model [45], the dynamics of a TLS can be
characterized by a double-well potential in some configuration space, with
bias energy $\varepsilon$, tunneling amplitude $\Delta$, and thus eigenenergy
$\omega_{\text{t}}=\sqrt{\varepsilon^{2}+\Delta^{2}}$. With Pauli operators
$\\{\hat{\Sigma}_{i}\\}$ defined in this configuration basis, the Hamiltonian
of an isolated TLS is

$\hat{H}_{\textrm{TLS}}=\dfrac{1}{2}(\varepsilon\,\hat{\Sigma}_{z}+\Delta\,\hat{\Sigma}_{x}).$
(1)

The microscopic parameters of an individual TLS can be extracted
experimentally by probing the spectroscopy of a coupled qubit [46, 47, 48, 49,
50]. In practice, a TLS couples to some environment, e.g., phonon bath, which
leads to its relaxation and dephasing. To describe TLS decoherence, it is
convenient to switch to the eigenbasis $\\{\hat{\sigma}_{i}\\}$ with the
following transformations,

$\hat{\Sigma}_{z}=\cos\theta\,\hat{\sigma}_{z}-\sin\theta\,\hat{\sigma}_{x},\quad\hat{\Sigma}_{x}=\sin\theta\,\hat{\sigma}_{z}+\cos\theta\,\hat{\sigma}_{x},$

where the angle $\theta$ follows $\tan\theta=\Delta/\varepsilon$. The
decoherence dynamics can then be described with a density matrix $\rho(t)$
obeying the Lindblad master equation [51],

$\dfrac{\mathrm{d}\rho(t)}{\mathrm{d}t}=-i[\hat{H}_{\text{TLS}},\rho(t)]+\gamma_{\downarrow}\mathcal{D}[\hat{\sigma}_{-}]\rho+\gamma_{\uparrow}\mathcal{D}[\hat{\sigma}_{+}]\rho+\gamma_{\phi}\mathcal{D}[\hat{\sigma}_{z}]\rho,$

with the dissipator defined as
$\mathcal{D}[\hat{\mathcal{O}}]\rho=\hat{\mathcal{O}}\rho\hat{\mathcal{O}}^{\dagger}-\\{\hat{\mathcal{O}}^{\dagger}\hat{\mathcal{O}},\rho\\}/2$.
Taking the phonon bath as the dominant TLS decoherence channel [45], the
relaxation and excitation rates are given by

$\gamma_{\downarrow}=2\pi\dfrac{\Delta^{2}}{\omega_{\text{t}}^{2}}J(\omega_{\text{t}})[n_{\text{B}}(\omega_{\text{t}})+1],\quad\gamma_{\uparrow}=2\pi\dfrac{\Delta^{2}}{\omega_{\text{t}}^{2}}J(\omega_{\text{t}})n_{\text{B}}(\omega_{\text{t}}),$

where $J(\omega)=J_{0}\omega^{3}e^{-\omega^{2}/2\omega_{\text{D}}^{2}}$ is the
typical phonon bath spectral density with Debye cutoff frequency
$\omega_{\text{D}}$, and $n_{\text{B}}(\omega)$ is the Bose–Einstein
distribution. To directly measure the decoherence properties of a TLS,
microwave pulses are used to initialize the TLS, which time evolution is then
mapped onto the population of a coupled qubit for readout [52]. Since the
parameters $\varepsilon$ and $\Delta$ cover a broad range of frequencies, the
lifetime of a TLS varies from nanoseconds to hours [53]. This motivates the
idea to use those long-lived TLSs as quantum memory [54, 55]. In contrast, the
dephasing rate $\gamma_{\phi}$ observed in the experiment is usually on the
order of MHz [52, 56]. The above master equation thus fully characterizes the
dynamics of a single TLS coupled to an environment. Next, we study the effects
when such a noisy TLS is coupled to a qubit.

###  II.2 Noise spectral densities of a single TLS

A TLS may couple to a qubit via several ways, resulting in different noise
channels [18]. For a charged TLS with electric dipole moment, it couples to
the electric field of qubit modes, which causes charge noise dephasing [23,
24, 25] and dielectric loss relaxation [21, 22]. Similarly, a TLS with
magnetic moment may contribute to the total flux in a circuit loop, leading to
flux noise [57, 58, 59, 60, 61, 62]. Moreover, a TLS inside the junction oxide
layer may block some channels through which Cooper pairs tunnel, and generate
critical current noise [63, 64]. Here, we are in particular interested in
those charged fluctuators responsible for charge noise and dielectric loss,
which are usually the limiting noise sources of many types of superconducting
qubits. The corresponding electric-dipole interaction between a qubit and a
TLS is given by

$\hat{H}_{\text{qubit--
TLS}}=\kappa\hat{n}\hat{\Sigma}_{z}=\kappa\hat{n}(\cos\theta\,\hat{\sigma}_{z}-\sin\theta\,\hat{\sigma}_{x}),$
(2)

where $\kappa$ denotes the coupling strength, $\hat{n}$ is the qubit charge
operator proportional to its electric field, and $\hat{\Sigma}_{z}$ is related
to the TLS electric dipole moment $p\hat{\Sigma}_{z}$. From Fermi’s golden
rule [6], the relaxation rate $\Gamma_{\downarrow}$ and excitation rate
$\Gamma_{\uparrow}$ of the qubit are proportional to the relevant TLS noise
spectral density $s(\omega)$, evaluating at positive and negative qubit
frequencies, i.e.,

$\Gamma_{\downarrow}\propto s(\omega_{\text{q}}),\quad\Gamma_{\uparrow}\propto
s(-\omega_{\text{q}}).$ (3)

In thermal equilibrium, they follow the detailed balance relation
$s(-\omega_{\text{q}})=s(\omega_{\text{q}})e^{-\beta\omega_{\text{q}}}$, with
temperature $k_{\text{B}}T=1/\beta$. The qubit dephasing rate is related to
the noise spectral density close to zero frequency 111For spectral density
that is singular at zero frequency, a decoherence function has to be involved,
see Eq. (31),

$\Gamma_{\phi}\propto s(0).$ (4)

From Eqs. (3) and (4), the qubit decoherence dynamics can be fully
characterized by the TLS noise spectral density, which is derived in the
following.

The noise spectral density is defined as the Fourier transform of the
autocorrelation function of the noise operator [66], i.e., $\hat{\Sigma}_{z}$
for the electric-dipole interaction considered here,

$s(\omega)=\int_{-\infty}^{+\infty}\mathrm{d}t\,e^{i\omega
t}\big{[}\langle\hat{\Sigma}_{z}(t)\hat{\Sigma}_{z}(0)\rangle-\langle\hat{\Sigma}_{z}\rangle^{2}\big{]}.$
(5)

The notation $\langle\cdots\rangle$ refers to the quantum-mechanical
expectation in thermal equilibrium. In the eigenbasis, Eq. (5) converts to

$s(\omega)=\cos^{2}\theta\,s_{zz}(\omega)+\sin^{2}\theta\,s_{xx}(\omega),$ (6)

where $s_{\alpha\alpha}(\omega)$ denotes the spectral density of
$\hat{\sigma}_{\alpha}$. Note that the cross terms vanish in thermal
equilibrium. With the Markov approximation 222Treatment beyond the Markov
approximation can be found, e.g., in Ref. [78], we apply quantum regression
theorem [51] to derive the relevant noise spectral densities [68, 53],

$\displaystyle\begin{split}s_{xx}(\omega)&=\,\dfrac{1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}}{2}\dfrac{2\gamma_{2}}{(\omega-\omega_{\text{t}})^{2}+\gamma_{2}^{2}}\\\
&+\dfrac{1+\langle\hat{\sigma}_{z}\rangle_{\text{eq}}}{2}\dfrac{2\gamma_{2}}{(\omega+\omega_{\text{t}})^{2}+\gamma_{2}^{2}},\end{split}$
(7) $\displaystyle s_{zz}(\omega)$
$\displaystyle=\,(1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2})\dfrac{2\gamma_{1}}{\omega^{2}+\gamma_{1}^{2}}.$
(8)

Here, $\gamma_{1}=\gamma_{\downarrow}+\gamma_{\uparrow}$ is the depolarization
rate, $\gamma_{2}=\gamma_{1}/2+\gamma_{\phi}$ the dephasing rate, and
$\langle\hat{\sigma}_{z}\rangle_{\text{eq}}=(\gamma_{\uparrow}-\gamma_{\downarrow})/(\gamma_{\uparrow}+\gamma_{\downarrow})$
the equilibrium polarization. Since $s_{xx}(\omega)$ has two peaks at
$\omega=\pm\omega_{\text{t}}$, it mainly contributes to qubit depolarization.
In contrast, $s_{zz}(\omega)$ is a Lorentzian centered at $\omega=0$, and is
responsible for qubit dephasing. Following Eqs. (3) and (4), the qubit
depolarization and pure dephasing rates are

$\displaystyle\Gamma_{1}=$
$\displaystyle\,\Gamma_{\downarrow}+\Gamma_{\uparrow}\approx\kappa^{2}M_{1}^{2}\sin^{2}\theta\dfrac{2\gamma_{2}}{\omega_{\delta}^{2}+\gamma_{2}^{2}},$
(9) $\displaystyle\Gamma_{\phi}=$
$\displaystyle\kappa^{2}M_{\phi}^{2}\cos^{2}\theta\,(1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2})/\gamma_{1},$
(10)

with detuning $\omega_{\delta}=\omega_{\text{q}}-\omega_{\text{t}}$, and
matrix elements $M_{1}=|\langle 0|\hat{n}|1\rangle|$, $M_{\phi}=|\langle
0|\hat{n}|0\rangle-\langle 1|\hat{n}|1\rangle|$. Here, $|0\rangle$ and
$|1\rangle$ denote the two logical states of the qubit.

Equations (9) and (10) connect the decoherence properties of a qubit
($\Gamma_{1}$, $\Gamma_{\phi}$) to the ones of a TLS ($\gamma_{1}$,
$\gamma_{\phi}$), and thus provide intriguing insights on qubit coherence
improvement. First, this relation is asymmetric, in the sense that the
lifetime (dephasing time) of the qubit is determined by the dephasing rate
(depolarization rate) of the TLS (also illustrated schematically in Fig. 1).
Moreover, the decoherence properties of the two subsystems are negatively
correlated. Specifically, a larger $\gamma_{1}$ of the TLS results in a
smaller $\Gamma_{\phi}$ of the qubit, and a stronger $\gamma_{\phi}$ of a
close-resonance ($|\omega_{\delta}|<\gamma_{2}$) TLS leads to a weaker qubit
$\Gamma_{1}$. The above observations motivate our protocols to improve qubit
coherence times by engineering TLS noise properties, which will be discussed
in detail later in Secs. III and IV.

###  II.3 Statistics of an ensemble of TLSs

Up to this point, we only focus on a single TLS. In practice, there exists an
ensemble of TLSs in a typical qubit device. The density of avoided level
crossings in a qubit spectrum due to coupled TLSs is found to be around
$10^{2}$–$10^{3}$ $\textrm{GHz}^{-1}\mu\textrm{m}^{-3}$ [18]. Therefore, a
statistical treatment is required to describe the effects of TLSs on a qubit.
Here, it is crucial to first clarify the nature of the statistics, and to
specify the relevant random variables.

#### Statistics over TLS realizations

In the literature, TLSs are observed to have a wide range of bias energies and
tunneling amplitudes [53]. While the exact values of these parameters are
random, experimental evidence indicates that they follow certain probability
distributions. To explain both the observed $1/f$ charge noise and Ohmic noise
simultaneously [23], Shnirman et al. [68] propose the joint probability
distribution to be $P(\varepsilon,\Delta)\propto\varepsilon/\Delta$.
Physically, such probability distribution could arise from models like
Andreev-level fluctuators [69]. In comparison, Constantin et al. [53] suggest
an alternative one, $P(\varepsilon,\Delta)\propto 1/\Delta$, which leads to a
white noise behavior in the high-frequency noise spectrum. Regardless of the
forms of distributions, the specific realization of those parameters for an
ensemble of TLSs varies between different device samples and cooldown cycles.
As a result, qubit coherence times fluctuate according to Eqs. (9) and (10).

#### Statistics over time

Recently, many experiments observe that the qubit lifetime fluctuates also
within a single cooldown, with a timescale between mileseconds to days [31,
32, 33]. One popular explanation based on spectral diffusion is proposed by
Müller et al. [35]. In this model, the relevant TLSs are sorted into two
categories based on their energy scales. Those in close resonance with the
qubit are denoted as high-energy (coherent) TLSs, dominating qubit lifetime.
The role of the other low-energy (compared to temperature) TLSs, also known as
two-level fluctuators (TLFs), is to fluctuate the frequencies of those high-
energy TLSs via TLS–TLF interaction. Since the qubit lifetime depends
sensitively on its detuning with those high-energy TLSs [see Eqs. (9)], this
explains the observed instability in qubit lifetime. Here, the relevant random
variables responsible for the temporal fluctuation are the states of these
TLFs, which can be described by the associated polarization $\sigma_{z}=\pm
1$.

##  III Stabilizing qubit depolarization time by dephasing TLSs

In the previous section, we suggest that the qubit decoherence due to a single
TLS can be suppressed by making the TLS noisy. In the following two sections,
we elaborate this idea and extend it to the realistic case of an ensemble of
TLS with various statistics discussed above. We focus on qubit lifetime here
and leave the discussion on dephasing time to Sec. IV. To benchmark the
improvement, we first characterize the average and variance of qubit lifetime
without implementing our protocol.

###  III.1 Statistics of qubit lifetime over TLS realizations

We start by deriving the statistics of qubit depolarization rate over TLS
realizations. Following Refs. [68, 53], the joint probability distribution of
TLS parameters is

$P(\varepsilon,\Delta)=\mathcal{N}(\alpha)\mathcal{B}(\varepsilon,\Delta)\dfrac{\varepsilon^{\alpha}}{\Delta},$
(11)

where $\alpha\in\\{0,1\\}$ accounts for the two distributions discussed in
Sec. II.3. The boundary function $\mathcal{B}(\varepsilon,\Delta)$ equals
unity for $\varepsilon_{\text{m}}<\varepsilon<\varepsilon_{\text{M}}$ and
$\Delta_{\text{m}}<\Delta<\Delta_{\text{M}}$, and vanishes otherwise. The set
of the parameters is taken from Ref. [53], with
$\varepsilon_{\text{m}}/k_{\text{B}}=0$,
$\varepsilon_{\text{M}}/k_{\text{B}}=4\,\textrm{K}$,
$\Delta_{\text{m}}/k_{\text{B}}=2\,\mu\textrm{K}$, and
$\Delta_{\text{M}}/k_{\text{B}}=4\,\textrm{K}$. The normalization factor is
given by

$\mathcal{N}(\alpha)^{-1}=\left(\dfrac{\varepsilon_{\textrm{M}}+\varepsilon_{\textrm{m}}}{2}\right)^{\alpha}(\varepsilon_{\textrm{M}}-\varepsilon_{\textrm{m}})\log\left(\dfrac{\Delta_{\textrm{M}}}{\Delta_{\textrm{m}}}\right).$
(12)

The average qubit depolarization rate is then,

$\langle\Gamma_{1}\rangle_{\text{ens}}=\iint_{\mathbb{R}^{2}}\mathrm{d}\varepsilon\,\mathrm{d}\Delta\,P(\varepsilon,\Delta)\,\Gamma_{1},$
(13)

with $\Gamma_{1}$ taken from Eq. (9). To obtain analytical results, it is
useful to perform change of variables from $(\varepsilon,\Delta)$ to
$(\omega_{\text{t}},\gamma_{2})$, with the TLS dephasing rate,

$\gamma_{2}=\pi
J_{0}\Delta^{2}\omega_{\text{t}}\coth\left(\beta\omega_{\text{t}}/2\right)+\gamma_{\phi}.$
(14)

Here the pure dephasing rate $\gamma_{\phi}=10\,\text{MHz}$ is assumed to be a
constant for simplicity [52, 56]. With some algebra (see details in Appendix
A), Eq. (13) equals

$\langle\Gamma_{1}\rangle_{\text{ens}}\approx\begin{cases}\kappa^{2}M_{1}^{2}\mathcal{N}\pi\omega_{\text{q}},&\alpha=1\\\
\\\ 2\kappa^{2}M_{1}^{2}\mathcal{N}\pi,&\alpha=0\end{cases}.$ (15)

The results exhibit the expected Ohmic and white noise behavior for $\alpha=1$
and $\alpha=0$, respectively. We note that both expressions do not depend on
TLS dephasing rate. This seems to be inconsistent with our observation in Sec.
II.2, where the qubit depolarization rate decreases with a larger dephasing
rate of a single close-resonance TLS. The contradiction can be resolved by
noticing that Eq. (15) is derived for an ensemble average of TLSs, which also
includes those TLSs that are far-detuned from the qubit
($|\omega_{\delta}|>\gamma_{2}$). For those TLSs, increasing their dephasing
rates enhances the qubit depolarization rate. Therefore, on average, the qubit
depolarization rate is insensitive to the TLS dephasing rate. However, as we
will show in the following, this is not the case for the fluctuation of qubit
lifetime.

To derive the variance of qubit lifetime, we calculate

$\textrm{Var}_{\text{ens}}(\Gamma_{1})\approx\langle\Gamma_{1}^{2}\rangle_{\text{ens}}=\iint_{\mathbb{R}^{2}}\mathrm{d}\varepsilon\,\mathrm{d}\Delta\,P(\varepsilon,\Delta)\Gamma_{1}^{2}.$
(16)

With details in Appendix A, the above expression equals

$\textrm{Var}_{\text{ens}}(\Gamma_{1})\approx\begin{cases}\kappa^{4}M_{1}^{4}\mathcal{N}\pi\omega_{\text{q}}/2\gamma_{\phi},&\alpha=1\\\
\\\ 4\kappa^{4}M_{1}^{4}\mathcal{N}\pi/3\gamma_{\phi},&\alpha=0\end{cases}.$
(17)

Unlike the average of qubit lifetime, its variance does depend on TLS
dephasing rate, and is inversely proportional to that. Intuitively, this can
be understood as follows. The fluctuation of qubit lifetime is related to the
comb-shape TLS spectral density (each peak corresponds to a TLS), shown
schematically in Fig. 2. The noise spectral density probed by the qubit
$S(\omega_{\text{q}})$ changes significantly in the case of random shifts in
frequencies of some close-resonance TLSs. With larger TLS dephasing rate,
those peaks are flattened, which suppresses the fluctuation in qubit lifetime.

Figure 2: Schematic representation of the noise spectral density of an
ensemble of TLSs. In the presence of a random shift in a TLS frequency, the
spectral density probed by the qubit at $\omega_{\text{q}}$ fluctuates less
for flattened peaks, corresponding to larger TLS linewidth.

Note that the above results [Eqs. (15) and (17)] are the averaged contribution
from a single TLS. The total depolarization rate of the qubit
$\Gamma_{1,\text{tot}}$ sums over numerous TLSs (with number
$N_{\text{TLS}}$). Under the assumption that the TLSs are independent and
identical (i.e., they obey the same distribution), $\Gamma_{1,\text{tot}}$
follows a normal distribution by central limit theorem. Its average and
variance are given by

$\langle\Gamma_{1,\text{tot}}\rangle=N_{\text{TLS}}\langle\Gamma_{1}\rangle,\quad\text{Var}(\Gamma_{1,\text{tot}})=N_{\text{TLS}}\text{Var}(\Gamma_{1}).$

Since most experimental data is expressed in terms of $T_{1}$, it is also
useful to derive the statics of $T_{1,\text{tot}}=1/\Gamma_{1,\text{tot}}$,

$\langle
T_{1,\text{tot}}\rangle=1/\langle\Gamma_{1,\text{tot}}\rangle,\quad\text{Var}(T_{1,\text{tot}})=\text{Var}(\Gamma_{1,\text{tot}})/\langle\Gamma_{1,\text{tot}}\rangle^{4}.$

###  III.2 Statistics of qubit lifetime over time

The analysis above addresses the fluctuation of qubit lifetime over different
cooldown cycles and device samples. However, qubit $T_{1}$ also fluctuates
temporarily during a single cooldown [31, 32, 33]. Since the mechanisms of the
fluctuations are different, the relevant statistics of qubit lifetime may also
change. In the following, we derive the average and variance of qubit lifetime
regarding the temporal fluctuation.

Based on the spectral diffusion model [35], this temporal fluctuation arises
from the TLS–TLF interaction,

$\hat{\mathcal{H}}_{\text{TLS--
TLF}}=\dfrac{1}{2}\sum_{i}g_{i}\hat{\sigma}_{z}\hat{\sigma}_{z,i},$ (18)

where the subscript $i$ indexes TLFs coupled to the studied TLS (with no
subscript), and $g_{i}$ denotes their interaction strength. Since the energy
of those TLFs is much smaller than temperature, they can be treated
classically, i.e., $\hat{\sigma}_{z,i}\to\sigma_{z,i}$. The average and
variance of these random variables are

$\langle\sigma_{z,i}\rangle=-\tanh\left(\beta\omega_{\text{t},i}/2\right),\,\,\textrm{Var}(\sigma_{z,i})=1-\tanh^{2}\left(\beta\omega_{\text{t},i}/2\right).$

Due to the TLS–TLF interaction, the TLS energy depends on the states of those
coupled TLFs,

$\omega_{\text{t}}=\omega_{\text{t}}^{(0)}+\sum_{i}g_{i}\sigma_{z,i},$ (19)

with $\omega_{\text{t}}^{(0)}$ the unperturbed TLS frequency. For weak
coupling, we can Taylor expand the qubit depolarization rate in Eq. (9), and
find its first-order dependence on the coupling to TLFs,

$\Gamma_{1}=\Gamma_{1}^{(0)}+\Gamma_{1}^{(1)}\sum_{i}g_{i}\sigma_{z,i}.$ (20)

The two coefficients are defined as follows,

$\displaystyle\begin{split}\Gamma_{1}^{(0)}&=\kappa^{2}M_{1}^{2}\sin^{2}\theta\dfrac{2\gamma_{2}}{\gamma_{2}^{2}+\omega_{\delta}^{2}},\\\
\Gamma_{1}^{(1)}&=\kappa^{2}M_{1}^{2}\sin^{2}\theta\dfrac{4\gamma_{2}\omega_{\delta}}{\left(\gamma_{2}^{2}+\omega_{\delta}^{2}\right)^{2}}.\end{split}$
(21)

Then, we take the average of Eq. (20) regarding the states of those TLFs,

$\langle\Gamma_{1}\rangle_{\text{spd}}=\Gamma_{1}^{(0)}+\Gamma_{1}^{(1)}\sum_{i}g_{i}\langle\sigma_{z,i}\rangle=\Gamma_{1}^{(0)}.$
(22)

In deriving the second equality, we assume the distribution of $g_{i}$ is
symmetric about zero. As a result, Eq. (22) shows that there is no net effect
of spectral diffusion on the average qubit lifetime.

Next, we derive the variance of qubit lifetime. With the assumption that the
TLFs are uncorrelated, we obtain

$\textrm{Var}_{\text{spd}}(\Gamma_{1})=\langle\Gamma_{1}^{2}\rangle_{\text{spd}}-\langle\Gamma_{1}\rangle_{\text{spd}}^{2}=\left(\Gamma_{1}^{(1)}\right)^{2}G^{2},$
(23)

where the quantity
$G^{2}=\sum_{i}g_{i}^{2}(1-\tanh^{2}{\beta\omega_{\text{t},i}/2})$ manifests
the fluctuation of TLFs. Note that both Eqs. (22) and (23) are derived for a
specific TLS. To account for the existence of multiple TLSs, we further take
an ensemble average of Eq. (23) with respect to TLSs realizations 333Note that
we take the ensemble average of the variance instead of its standard
deviation, since $\Gamma_{1}$ of each TLS follows a normal distribution..

$\langle\textrm{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}=\iint_{\mathbb{R}^{2}}\mathrm{d}\varepsilon\,\mathrm{d}\Delta\,P(\varepsilon,\Delta)\left(\Gamma_{1}^{(1)}\right)^{2}G^{2}.$
(24)

For simplicity, we assume $G^{2}$ to be the same for all TLSs. With details in
Appendix A, we derive

$\langle\textrm{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}\approx\begin{cases}\kappa^{4}G^{2}M_{1}^{4}\mathcal{N}\pi\omega_{\text{q}}/4\gamma_{\phi}^{3},&\alpha=1\\\
\\\
2\kappa^{4}G^{2}M_{1}^{4}\mathcal{N}\pi/3\gamma_{\phi}^{3},&\alpha=0\end{cases}.$
(25)

Compared with $\textrm{Var}_{\text{ens}}(\Gamma_{1})$ in Eq. (17),
$\langle\textrm{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$ also
depends on TLS dephasing rate, but more sensitively with an inverse cubic
dependence. This indicates a significant suppression of temporal fluctuation
in qubit lifetime by increasing TLS dephasing rate. To verify the analytical
expressions in Eqs. (15), (17), and (25), we compare them with the results
obtained by numerical integration. The relative deviations as a function of
the qubit frequency are shown in Fig. 3, which shows decent agreement with
each other.

Figure 3: Relative deviations between the approximated analytical expressions
in Eq. (15), (17), and (25), and the numerically obtained results, as a
function of the qubit frequency. The solid and dashed lines correspond to the
two joint probability distributions with $\alpha=1$ and $\alpha=0$,
respectively.

###  III.3 Stabilization of qubit lifetime

The results derived above suggest that we can stabilize qubit lifetime by
increasing the TLS dephasing rate. In the following, we describe in detail our
stabilization protocol, and quantify the potential improvement. In general, a
TLS responds to external coupling through its bias energy [18],

$\varepsilon=\varepsilon_{0}+2\mathbf{d}\cdot\mathbf{F},$ (26)

with $\varepsilon_{0}$ the intrinsic TLS bias energy. The external field
$\mathbf{F}$ can be an electric field $\mathbf{E}$, or a mechanical strain
field $\mathbf{S}$ (not to be confused with the spectral density $S$), and
$\mathbf{d}$ is the corresponding electric dipole $\mathbf{d}_{\text{e}}$ or
strain tensor $\mathbf{d}_{\text{s}}$ of the TLS. For a typical TLS, the two
coupling strengths are on the orders of $|\mathbf{d}_{\text{s}}|\sim
1\text{eV}$ and $|\mathbf{d}_{\text{e}}|\sim 1\text{D}$, 4respectively [18].
Recently, static tuning of TLS frequency has been realized experimentally via
both dc electric and strain fields [71, 72, 52, 73, 74, 56, 41, 75, 76]. Here,
the technique useful for our purpose is instead to apply ac signals, or more
precisely, noisy signals. The interaction Hamiltonian is given by

$\hat{H}_{\text{int}}=\mathbf{d}\cdot\mathbf{F}\cos\theta\,\hat{\sigma}_{z}+\mathbf{d}\cdot\mathbf{F}\sin\theta\,\hat{\sigma}_{x}.$
(27)

Note that here the field $\mathbf{F}$ can either be a quantum operator or a
classical variable. From Fermi’s golden rule, $\hat{H}_{\text{int}}$ results
in additional TLS depolarization and pure dephasing,

$\displaystyle\delta\gamma_{1}=\delta\gamma_{\downarrow}+\delta\gamma_{\uparrow}=\left(d\cos{\alpha}\sin{\theta}\right)^{2}\left[S_{\text{add}}(\omega_{\text{t}})+S_{\text{add}}(-\omega_{\text{t}})\right],$
$\displaystyle\delta\gamma_{\phi}=2\left(d\cos{\alpha}\cos{\theta}\right)^{2}S_{\text{add}}(0),$
(28)

where $S_{\text{add}}(\omega)$ is the noise spectral density of the applied
field $\mathbf{F}$, with an angle $\alpha$ relative to the dipole
$\mathbf{d}$. Depending on the effective temperature of the noise,
characterized by
$k_{\text{B}}T=\omega/\log[S_{\text{add}}(\omega)/S_{\text{add}}(-\omega)]$,
the equilibrium polarization of the TLS may also change,

$\langle\hat{\sigma}_{z}\rangle^{\prime}=\dfrac{(\gamma_{\uparrow}+\delta\gamma_{\uparrow})-(\gamma_{\downarrow}+\delta\gamma_{\downarrow})}{(\gamma_{\uparrow}+\delta\gamma_{\uparrow})+(\gamma_{\downarrow}+\delta\gamma_{\downarrow})}.$
(29)

To quantify the stabilization of qubit lifetime, we reevaluate the average
[Eq. (13)] and variance [Eqs. (17) and (23)] of qubit $\Gamma_{1}$, with TLS
dephasing rate $\gamma_{2}$ modified from Eq. (14),

$\gamma_{2}^{\prime}=\pi
J_{0}\Delta^{2}\omega_{\text{t}}\coth\left(\beta\omega_{\text{t}}/2\right)+\gamma_{\phi}+\delta\gamma_{1}/2+\delta\gamma_{\phi}.$
(30)

Note that the quantity $G^{2}$ in Eq. (23) changes as well due to the modified
$\langle\hat{\sigma}_{z}\rangle^{\prime}$. For the purpose of dephasing TLSs,
it is appropriate to apply a low-frequency noise. Specifically, we take a
classical white noise $S_{\text{add}}(\omega)=S_{\text{add}}$, with a high-
frequency cutoff $\omega_{\text{c}}$ on the same order of the TLS dephasing
rate. Note that as we increase the amplitude of the noise, TLS dephasing rate
also grows. Thus, we need to adjust the cutoff frequency accordingly.

In Fig. 4, we plot the stabilization of qubit lifetime as a function of the
strength of the applied noise. The shown results are the relative magnitudes
of the variances of qubit depolarization rates between the cases when the
protocol is turned off and on. Here, the noise strength is denoted by
$d^{2}S_{\text{add}}$, which is proportional to the additional TLS dephasing
rate. Alternatively, we can express the noise strength via its standard
deviation $\sqrt{S_{\textrm{add}}\omega_{\text{c}}/\pi}$. For the largest
noise amplitude taken in Fig. 4, $d^{2}S_{\text{add}}=1$ GHz, it corresponds
to a standard deviation of strain field about $10^{-6}$ (e.g., 10-nm
deformation for a 10-mm chip) and electric field around $10$ kV/m. In addition
to the amplitude of the noise, we supplement the amplitude dependent cutoff
frequency $\omega_{\text{c}}$. Clearly, both variances
$\langle\text{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$ and
$\text{Var}_{\text{ens}}(\Gamma_{1})$ are suppressed with our protocol, with
the latter demonstrating 10 times better stabilization. While
$\text{Var}_{\text{ens}}(\Gamma_{1})$ exhibits a monotonic suppression,
$\langle\text{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$ has a minimum
regarding the noise amplitude (or maximum for the relative magnitude shown in
Fig. 4). This is because the effective temperature of TLFs are raised by the
applied classical noise, causing TLFs to switch more likely (i.e., larger
$G^{2}$) 444This will not be the case in terms of a quantum noise with
identical effective temperature with the TLS bath.. Therefore, even the change
in qubit lifetime during a TLF switch is suppressed for stronger applied
noise, the frequency of the switch is enhanced. These two competing factors
result in the local extremum of
$\langle\text{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$ observed in
Fig. 4. We also notice that the average depolarization rate is almost
independent of the applied noise: With $\alpha=1$, there is a minute 5%
increase of $\langle\Gamma_{1}\rangle$ for a 10$\times$ suppression of
$\langle\text{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$.

Figure 4: Relative magnitudes of the average and variances of qubit
depolarization rates between the cases when the protocol is turned off and on.
The lower horizontal axis denotes the strength of the noise, and the upper one
shows its cutoff frequency. The solid and dashed lines correspond to two joint
probability distributions with $\alpha=1$ and $\alpha=0$, respectively.
[Parameters: $\omega_{\text{q}}/2\pi$ = 5.7 GHz, $T$ = 20 mK, $J_{0}$ = 0.047
ps2.]

##  IV Improving qubit pure dephasing time by depolarizing TLSs

In Sec. III, we introduced our protocol to stabilize the qubit lifetime by
dephasing TLSs. Here, we instead explore ways to enhance the qubit pure
dephasing time. Early in Eq. (10), we derive the dephasing time of the qubit,
when it couples to a single TLS. The generalization to the case of an ensemble
of TLSs requires special treatment, since the relevant noise spectral density
here is singular around zero frequency (as will be shown in the following).
With the usual assumption of Gaussian noise, the qubit dephasing (free
induction) is described by [6],

$\langle
e^{i\delta\phi(t)}\rangle=\exp\left[-\dfrac{t^{2}M^{2}}{2}\int_{-\infty}^{\infty}\dfrac{\mathrm{d}\omega}{2\pi}\,S_{\text{low}}(\omega)\,\text{sinc}^{2}\dfrac{\omega
t}{2}\right],$ (31)

with $M$ the relevant matrix element, and $S_{\text{low}}(\omega)$ the low-
frequency noise spectral density. Unlike Eq. (10), where only the zero
frequency noise spectral density is probed, here in Eq. (31), the qubit
dephasing depends on a broad range of frequencies controlled by the filter
function. In the remaining of the section, we first derive the detailed
structure of $S_{\text{low}}(\omega)$ of an ensemble of TLSs. Then, we
demonstrate the protocol to suppress $S_{\text{low}}(\omega)$ by depolarizing
TLSs, which leads to longer qubit pure dephasing time.

###  IV.1 White and $1/f$ noise from an ensemble of TLSs

The low-frequency noise spectral density of an ensemble of TLSs is obtained by
taking an average of $s_{zz}(\omega)$ in Eq. (8),

$S_{\text{low}}(\omega)=N_{\text{TLS}}\iint_{\mathbb{R}^{2}}\mathrm{d}\varepsilon\,\mathrm{d}\Delta\,P(\varepsilon,\Delta)\cos^{2}\theta
s_{zz}(\omega).$ (32)

Since the qualitative behavior of $S_{\text{low}}(\omega)$ is identical with
the joint probability distribution being $\alpha=0\,\text{and}\,1$ [68, 78],
we take $P(\varepsilon,\Delta)\propto\varepsilon/\Delta$ (i.e., $\alpha=1$)
for simplicity. To evaluate Eq. (32), we apply variable transformation from
$(\varepsilon,\Delta)$ to $(\omega_{\text{t}},\gamma_{1})$, with the TLS
depolarization rate,

$\gamma_{1}=2\pi
J_{0}\Delta^{2}\omega_{\text{t}}\coth\left(\beta\omega_{\text{t}}/2\right).$
(33)

With appropriate approximations (see details in Appendix A), we derive
analytical expressions of the spectral density for different frequency ranges.
Of particular interest here is the emergence of white and $1/f$ noise at low
frequency 555The crossover from $1/f$ to a $1/f^{2}$ noise is discussed in
Ref. [78].,

$S_{\text{low}}(\omega)=\begin{cases}\dfrac{N_{\text{TLS}}\mathcal{N}k_{\text{B}}T}{2\pi
J_{0}\Delta_{\text{m}}^{2}},&\omega\ll\omega_{\text{ir}}\\\ \\\
\dfrac{2\log(2)\pi
N_{\text{TLS}}\mathcal{N}(k_{\text{B}}T)^{2}}{\omega},&\omega\gg\omega_{\text{ir}}\end{cases},$
(34)

with the infrared cutoff frequency for $1/f$ noise,
$\omega_{\text{ir}}=4\pi^{2}\log(2)J_{0}k_{\text{B}}T\Delta_{\text{m}}^{2}$.
This is further confirmed by numerical integration of Eq. (32), shown as the
gray line in Fig. 5(a). Note that we have converted $S_{\text{low}}(\omega)$
to offset charge noise following
$S_{n_{\text{g}}}(\omega)=(d_{\text{e}}/eL)^{2}S_{\text{low}}(\omega)$, with
$L$ a sample-dependent length scale. The number of TLS is chosen to match
$S_{n_{\text{g}}}(\omega)=2\pi A^{2}/\omega$, with a realistic amplitude
$A=10^{-3}$ [25]. Plugging the calculated $S_{n_{\text{g}}}(\omega)$ into Eq.
(31), we plot the qubit pure dephasing dynamics in Fig. 5(b), which shows a
typical Gaussian decay profile.

Figure 5:  (a) Low-frequency noise spectral densities of an ensemble of TLSs.
Different colors represent the cutoff frequencies of the applied noise. Solid
and dashed lines distinguish quantum and classical noise, respectively. (b)
Transmon qubit dephasing dynamics according to the noise spectral densities
shown in (a). [Parameters: $d^{2}\eta=1$, $E_{\text{C}}/h=0.3$ GHz, and
$E_{\text{J}}/h=15$ GHz.]

###  IV.2 Improvement of qubit pure dephasing time

The qubit dephasing rate derived in Eq. (10) reveals $\Gamma_{\phi}\propto
1/\gamma_{1}$, which indicates that we can improve qubit pure dephasing time
by depolarizing the TLSs. However, $\Gamma_{\phi}$ also depends on a prefactor
$1-\langle\hat{\sigma}_{z}\rangle^{2}$, which is related to the fluctuation of
TLSs. When TLSs undergo additional depolarization,
$\langle\hat{\sigma}_{z}\rangle$ may deviate from
$-\tanh{\beta\omega_{\text{t}}/2}$, depending on whether the nature of the
noise is quantum or classical [see Eq. (29)]. Therefore, it is necessary to
quantify the improvement of qubit dephasing time for both types of noise
respectively.

#### Quantum noise

In order to depolarize TLSs, we apply high-frequency noise with an Ohmic noise
spectrum
$S_{\text{add}}(\omega>0)=\theta(\omega_{\text{c}}-\omega)\,\eta\,\omega$. In
the case of quantum noise with the same effective temperature of TLSs,
$S_{n_{\text{g}}}(\omega)$ is modified due to the additional decay of those
TLSs with frequencies lower than the cutoff frequency $\omega_{\text{c}}$,

$\gamma_{1}^{\prime}=\gamma_{1}+\delta\gamma_{1}.$ (35)

When $\omega_{\text{c}}\gg k_{\text{B}}T$, most of the TLSs contributing to
qubit dephasing are affected, resulting in a suppressed white noise spectral
density (see details in Appendix A),

$S_{\text{white}}^{\prime}=\dfrac{0.58N_{\text{TLS}}\mathcal{N}(2k_{\text{B}}T)^{2}}{\Delta_{\text{m}}^{2}\sqrt{4\pi
J_{0}d^{2}\eta}}.$ (36)

Moreover, the crossover to $1/f$ noise is pushed to higher frequency. In
contrast, when $\omega_{\text{c}}\lesssim k_{\text{B}}T$, only a part of the
TLSs relevant for qubit dephasing is affected, which leads to a weaker
suppression than $S_{\text{white}}^{\prime}$ in Eq. (36). In Fig. 5(a), we
confirm the above analysis by plotting the modified noise spectral densities
when the quantum noise is applied. The cyan and blue solid curves correspond
to the same noise amplitude $d^{2}\eta$, but with different cutoff frequencies
$\omega_{\text{c}}$. Significantly, for $\omega_{\text{c}}/2\pi=$ 5 GHz, our
protocol leads to orders of magnitudes suppression of the white noise
amplitude. As a result, the qubit pure dephasing time is notably improved, as
shown by solid curves in Fig. 5(b).

Figure 6:  Schematic representation of a transmon qubit with applied strain
field. (a) An IDT is fabricated on a local piezoelectric thin-film, which
generates SAW propagating towards the transmon qubit. (b) A piezo actuator is
placed underneath the qubit chip to induce the deformation. Adapted from Ref.
[71]. (c) Geometry of a transmon qubit. Green areas denote the metal pads with
characteristic distance $d$. The yellow cross symbolize the Josephson junction
with width $l$, and the gray background represents the substrate. (d) The
qubit chip is deformed in the presence of strain field. Specifically, the chip
is elongated horizontally, and bent vertically. (e) Zoom-in of the junction
region in the presence of deformation. The width of the oxide layer is
increased by $\delta l$. Purple arrows on the interfaces represent the phonon-
induced dipoles $\mathbf{p}$.

#### Classical noise

The modified $S_{n_{\text{g}}}(\omega)$ with classical noise is shown as
dashed curves in Fig. 5(a). While the spectra are similar between the cases of
quantum and classical noise when $\omega_{\text{c}}\lesssim k_{\text{B}}T$
(cyan), they differ drastically for $\omega_{\text{c}}\gg k_{\text{B}}T$
(blue). In particular, the modified spectrum for $\omega_{\text{c}}/2\pi=5$
GHz is elevated with respect to its quantum counterpart, and is even larger
than the one without applied noise (when $\omega/2\pi>10^{-3}$ Hz). This non-
monotonic behavior of the modified spectrum regarding $\omega_{c}$ can be
understood as follows. In thermal equilibrium, only the TLSs with energy
smaller than temperature contribute significantly to qubit dephasing. When
$\omega_{\text{c}}\approx k_{\text{B}}T$, the applied noise selectively
depolarizes the same part of TLSs, suppressing the low-frequency noise
spectrum. However, for larger $\omega_{\text{c}}\gg k_{\text{B}}T$, more TLSs
start to contribute to qubit dephasing due to their reduced polarization from
the applied classical noise. This non-monotonic behavior is also reflected in
qubit dephasing time, which first increases and then decreases as a function
of the cutoff frequency, as shown by dashed curves in Fig. 5(b).

##  V Feasible experimental realizations with negligible spurious coupling to
qubits

In Secs. III and IV, we demonstrated in detail how the applied noise modifies
the noise spectral densities of an ensemble of TLSs, which results in
stabilization of qubit lifetime and improvement in qubit pure dephasing time.
In practice, the applied noise may also directly couple to the qubit, causing
unwanted decoherence. However, as will be shown in the following, this is not
a concern when the applied noise is in the form of strain field. For
concreteness, we quantitatively estimate the effects of relevant noise
channels on a transmon qubit [28, 29], one of the most popular types of
superconducting qubits. With $E_{\text{C}}$ and $E_{\text{J}}$ being its
charging and junction energies, the transmon Hamiltonian is

$\hat{H}_{\text{q}}=4E_{\text{C}}(\hat{n}-n_{\text{g}})^{2}-E_{\text{J}}\cos\hat{\varphi}.$
(37)

Here, $\hat{n}$ and $\hat{\varphi}$ are the charge and phase operators of the
qubit degree of freedom. The offset charge dependence is manifested in
$n_{\text{g}}$.

Mechanical strain field is a popular and powerful tool to study TLSs. For
example, strain field has been used to tune the frequencies of TLSs [71, 72,
56, 76], to extract their decoherence properties [52, 74], and to demonstrate
spectral hole-burning [80]. Experimentally, strain field can be generated in
various ways [81, 71, 82, 83, 84, 84, 85, 80]. In Fig. 6, we sketch two
feasible realizations that can be potentially used for the protocols developed
in Secs. III and IV. First, as shown in Fig. 6(a), an interdigital transducer
(IDT) is fabricated on a piezoelectric thin film. With applied electrical
signals, the IDT generates surface acoustic waves (SAWs) that propagate
elastically on the surface of the non-piezoelectric substrate. By placing the
transmon circuit near the IDT, strain field is induced on those TLSs coupled
to the qubit. An alternative setup is depicted in Fig. 6(b), where a stacked
piezo actuator is placed under the qubit chip. By applying voltage onto the
piezo actuator, it deforms the chip and generates the required strain field.
We may compare the two realizations by focusing on the frequency and amplitude
of the strain field that can be supported, which are two crucial parameters of
the proposed protocols. In particular, the largest frequency and amplitude
used in Secs. III and IV are on the order of $\omega/2\pi\sim 1\,\text{GHz}$
and $|\mathbf{S}|\sim 10^{-6}$, respectively. While the required field
amplitude can be achieved in both realizations [71, 80], the GHz range
frequency is routinely reached in SAWs [82, 85, 80], but challenging for piezo
actuators [83, 84].

The main advantage of using strain field to address TLS, is that it does not
produce stray electromagnetic fields that directly couple to the qubit
electric dipole. Nevertheless, it does interact with the qubit via other
mechanisms. In the following, we carefully analyze each of these coupling
mechanisms, and confirm that they barely contribute to qubit decoherence.

#### Capacitive loss

The transmon circuit [Fig. 6(c)] consists of two metal pads connected by a
Josephson junction. One consequence of the deformation is that it changes the
circuit parameters. In particular, the distance $d$ between the two metal pads
are modified [Fig. 6(d)], resulting in a modulation of the capacitance
$C\propto d^{-1}$, and thus the charging energy $E_{\text{C}}=e^{2}/2C$. For
small deformation, the perturbed qubit Hamiltonian is

$\delta\hat{H}_{\text{cap}}=\dfrac{\partial\hat{H}_{\text{q}}}{\partial
E_{\text{C}}}\dfrac{\partial E_{\text{C}}}{\partial d}\delta
d=4E_{\text{C}}(\hat{n}-n_{\text{g}})^{2}\dfrac{\delta d}{d},$ (38)

where the deformation relates to strain via $S=\delta d/d$.

#### Junction loss

Besides the distance between metal pads, the thickness of the junction tunnel
barrier $l$ also changes in the presence of strain field [Fig. 6(e)]. Since
the junction energy has an exponential dependence on the width,
$E_{\text{J}}\propto\exp(-l/l_{\text{c}})$, the modulation in $E_{\text{J}}$
is more significant compared with $E_{\text{C}}$,

$\delta\hat{H}_{\text{JJ}}=\dfrac{\partial\hat{H}_{\text{q}}}{\partial
E_{\text{J}}}\dfrac{\partial E_{\text{J}}}{\partial l}\delta
l=E_{\text{J}}\cos\hat{\varphi}\dfrac{\delta l}{l_{\text{c}}},$ (39)

with strain $S=\delta l/l$. For a typical Al/AlOx/Al Josephson junction, the
relevant parameters are $l=2$ nm and $l_{\text{c}}=0.069$ nm [86].

#### Phonon-induced loss

It is well known that thermal-excited mechanical oscillations in a
piezoelectric material produce electric fields via piezoelectric effect. The
noisy electric fields can then couple to the qubit electric dipole and cause
decoherence. To avoid this loss channel, superconducting qubits are usually
fabricated with non-piezoelectric material. However, due to surface relaxation
[87, 88], electric dipoles can still develop on the interface or surface
[purple arrows in Fig. 6(e)],

$\mathbf{p}\approx g_{\text{I}}St_{\text{I}}\delta(z)\hat{z}.$ (40)

Here, ${g}_{\text{I}}$ is the piezoelectric coefficient, and $t_{\text{I}}$ is
the thickness of the interface. The direction of the induced dipole is
perpendicular to the interface. We then apply Eq. (40) to the oxide region of
the junction, where the qubit mode has maximum electric field amplitude. For
this dielectric/metal (Al${}_{\text{2}}$O${}_{\text{3}}$/Al) interface, the
parameters are $g_{\text{I}}=0.06\,\text{C}/\text{m}^{2}$ and
$t_{\text{I}}=2.17\,$\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ [88]. The
phonon-induced dipole interaction is given by

$\delta\hat{H}_{\text{ph}}=\int\mathrm{d}\mathbf{r}\,\mathbf{p}(\mathbf{r})\cdot\hat{\mathbf{E}}_{\text{q}}(\mathbf{r})\approx\dfrac{4E_{\text{C}}g_{\text{I}}t_{\text{I}}AS}{el}\hat{n},$
(41)

where $A=100\,\textrm{nm}\times 100\,\textrm{nm}$ is the typical junction
area. In deriving the above equation, we approximate the electric field inside
the junction to be a uniform field
$|\hat{\mathbf{E}}_{\text{q}}(\mathbf{r})|\approx\hat{n}4E_{\text{C}}/(el)$.
It should be emphasized here that Eq. (41) overestimates this phonon-induced
interaction (and thus the relevant loss), with the following reason. The
electric dipoles are induced on both sides of the oxide layer, with directions
opposite with each other. Ideally, the contribution from these two sides
should cancel with each other. In practice, the cancellation is imperfect due
to factors such as surface roughness.

Having introduced the relevant noise channels, we next quantify the effects of
those spurious couplings on qubit depolarization and dephasing, respectively.
For concreteness, we take the applied noise to be white, which corresponds to
the case in Sec. III and also serves as an upper bound for Ohmic noise
considered in Sec. IV. Since the applied noise is regular at low-frequency,
the qubit pure dephasing rate for the noise channel $\lambda$ can be estimated
by

$\Gamma_{\phi}^{\lambda}=\dfrac{1}{2}\big{|}\langle
0|\partial_{S}\delta\hat{H}_{\lambda}|0\rangle-\langle
1|\partial_{S}\delta\hat{H}_{\lambda}|1\rangle\big{|}^{2}S_{\text{add}}.$ (42)

Note that we have chosen the offset charge to maximize
$\Gamma_{\phi}^{\lambda}$, such that the above expression gives an upper bound
for the dephasing rate. The predicted limits of qubit pure dephasing time
$T^{\text{lim}}_{\phi}=1/\Gamma_{\phi}^{\lambda}$ for various loss channels
are shown in Fig. 7(a), as a function of the applied noise strength. For the
largest noise amplitude considered here, all three loss mechanisms pose an
upper bound of qubit pure dephasing time above 10 ms. Therefore, the
decoherence from the spurious coupling is not a concern for the current
superconducting qubits.

The depolarization dynamics of the qubit due to additional noise requires
careful treatment. Here, the cutoff frequency $\omega_{\text{c}}$ of the
applied noise is lower than the qubit frequency. The Fermi’s golden rule would
result in zero decay rate of the qubit, i.e., $\Gamma_{1}^{\lambda}\propto
S_{\text{add}}(\omega_{\text{q}})=0$, which is incorrect. An appropriate
description of the qubit depolarization can be derived, for example, following
Ref. [89]. Assume the qubit starts in the excited state, its population
evolves as

$\rho_{\text{e}}^{\lambda}(t)=\exp\left[-S_{\text{add}}\big{|}\langle
0|\partial_{S}\delta\hat{H}_{\lambda}|1\rangle\big{|}^{2}D(t)\right],$ (43)

with the depolarization function,

$D(t)=2t^{2}\int_{-\omega_{\text{c}}}^{\omega_{\text{c}}}\dfrac{\mathrm{d}\omega}{2\pi}\,\text{sinc}^{2}\dfrac{(\omega-\omega_{\text{q}})t}{2}.$
(44)

In the limit $\omega_{\text{c}}\to\infty$, we have $D(t)\to 2t$, which
recovers the Markovian result. However, for a finite $\omega_{\text{c}}$
considered here, the depolarization function oscillates and saturates on a
timescale of $\omega_{\text{c}}^{-1}$ 666Similar saturation is discussed in
the context of qubit dephasing with a super-Ohmic noise with cutoff, for
example, in Ref. [91].,

$D(t\to\infty)\equiv
D_{\infty}=\dfrac{1}{2\pi}\dfrac{8\omega_{\text{c}}}{\omega_{\text{q}}^{2}-\omega_{\text{c}}^{2}}.$
(45)

Therefore, we choose to characterize qubit depolarization by its ground state
population at $t\to\infty$,

$\rho_{\text{g}}^{\lambda}(t\to\infty)=1-\exp\left(-S_{\text{add}}\big{|}\langle
0|\partial_{S}\hat{H}_{\lambda}|1\rangle\big{|}^{2}D_{\infty}\right).$

The results are plotted in Fig. 7(b) for $\omega_{\text{c}}=5$ GHz (solid) and
1 GHz (dashed), respectively. The largest ground state population in the
estimation is still below $10^{-6}$, which confirms the irrelevance of
spurious couplings on qubit depolarization.

Figure 7:  Estimation of qubit decoherence from spurious couplings. (a) Qubit
pure dephasing time for various loss channels, as a function of the applied
noise strength. (b) Saturated ground state population with the qubit
initialized in the excited state. Solid and dashed lines represent the cutoff
frequencies of 5 GHz and 1 GHz, respectively. [Qubit parameters:
$E_{\text{C}}/h=0.3$ GHz, $E_{\text{J}}/h=15$ GHz.]

For completeness, we briefly comment on the usage of electric field to
engineer noise spectrum of TLSs. Unlike the strain field, the electric field
directly couples to the qubit via electric-dipole interaction,

$\delta\hat{H}_{\mathbf{E}}=\mathbf{d}_{\text{q}}\cdot\mathbf{E}\,\hat{n},$
(46)

where $d_{\text{q}}=2eL_{\text{q}}$ is the electric dipole moment of the
qubit, and $L_{\text{q}}$ is its characteristic length. For a transmon with
$L_{\text{q}}\sim 10\,\mu\text{m}$ [28], its dipole moment is about
$d_{\text{q}}\sim 10^{6}$ D, six orders of magnitude larger than that of a
TLS. Even the strength of this spurious interaction could be reduced by
aligning the applied electric field to be perpendicular to the qubit chip
plane, the residual interaction is still large enough to cause strong qubit
decoherence. Therefore, while electric field is useful as a tool to dc bias
the energy of TLSs, it is not suitable as a method to dynamically engineer the
TLSs noise spectrum.

##  VI Conclusions

To summarize, we developed protocols to stabilize and improve qubit coherence
times by engineering the relevant noise spectral densities of an ensemble of
TLSs. Specifically, our calculations showed that the high-frequency noise
spectrum gets flattened in the presence of applied longitudinal noise on TLSs,
leading to an order of magnitude stabilization of qubit lifetime. Moreover, we
demonstrated that the low-frequency noise amplitude is significantly
suppressed via the implementation of transverse noise on TLSs, which results
in a longer qubit pure dephasing time. Finally, we discussed feasible
experimental realizations and confirmed that the spurious coupling from the
applied noise does not limit the qubit performance. Our results thus suggest a
complementary approach to improve qubit coherence, in addition to better qubit
design and material investigation.

###### Acknowledgements.

We thank G. Andersson, S. Chakram, P. Groszkowski, C.R.H. McRae, A. Murthy, J.
Mutus, R. Pilipenko, Y.M. Pischalnikov, O.V. Pronitchev, and J.A. Sauls for
illuminating discussions. This material is based upon work supported by the
U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers, Superconducting Quantum Materials and Systems Center
(SQMS) under contract number DE-AC02-07CH11359.

##  Appendix A Derivation of central results

###  A.1 Average qubit lifetime: $\langle\Gamma_{1}\rangle_{\text{ens}}$ in
Eq. (15)

To evaluate the integral in Eq. (13), it is convenient to perform a variable
transformation from $(\varepsilon,\Delta)$ to
$(\omega_{\text{t}},\gamma_{2})$. The joint probability distribution of the
new variables is

$P(\omega_{\text{t}},\gamma_{2})=\mathcal{N}(\alpha)\dfrac{\omega_{\text{t}}}{2(\gamma_{2}-\gamma_{\phi})}\left[\omega_{\text{t}}^{2}-\dfrac{\gamma_{2}-\gamma_{\phi}}{f(\omega_{\text{t}})}\right]^{\tfrac{\alpha-1}{2}},$

with function $f(\omega)=\pi J_{0}\omega\coth(\omega/2k_{\text{B}}T)$. In the
case of $\alpha=1$, Eq. (13) equals to

$\langle\Gamma_{1}\rangle_{\text{ens}}=\kappa^{2}M_{1}^{2}\mathcal{N}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{\mathrm{d}\omega_{\text{t}}}{\omega_{\text{t}}f(\omega_{\text{t}})}\int_{\gamma_{\text{2,m}}(\omega_{\text{t}})}^{\gamma_{\text{2,M}}(\omega_{\text{t}})}\dfrac{\gamma_{2}\,\mathrm{d}\gamma_{2}}{\gamma_{2}^{2}+\omega_{\delta}^{2}}.$

The boundaries of the integral are listed as follows:
$\omega_{\text{m}}=\Delta_{\text{m}}$,
$\omega_{\text{M}}=\sqrt{2}\Delta_{\text{M}}$,
$\gamma_{\text{2,m}}(\omega_{\text{t}})=\Delta_{\text{m}}^{2}f(\omega_{\text{t}})+\gamma_{\phi}$,
and
$\gamma_{\text{2,M}}(\omega_{\text{t}})=\omega_{\text{t}}^{2}f(\omega_{\text{t}})+\gamma_{\phi}$.
Direct integration of $\gamma_{2}$ leads to

$\langle\Gamma_{1}\rangle_{\text{ens}}=\kappa^{2}M_{1}^{2}\mathcal{N}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{\mathrm{d}\omega_{\text{t}}}{2\omega_{\text{t}}f(\omega_{\text{t}})}\log\left(\dfrac{\gamma_{\text{2,M}}^{2}+\omega_{\delta}^{2}}{\gamma_{\text{2,m}}^{2}+\omega_{\delta}^{2}}\right).$

We then approximate the logarithm with a delta function, and integrate over
$\omega_{\text{t}}$,

$\displaystyle\langle\Gamma_{1}\rangle_{\text{ens}}$
$\displaystyle=\kappa^{2}M_{1}^{2}\mathcal{N}\pi\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\mathrm{d}\omega_{\text{t}}\,\dfrac{(\gamma_{\text{2,M}}-\gamma_{\text{2,m}})\delta(\omega_{\text{t}}-\omega)}{\omega_{\text{t}}f(\omega_{\text{t}})}$
$\displaystyle=\kappa^{2}M_{1}^{2}\mathcal{N}\pi\omega_{\text{q}}.$

Taking similar steps for $\alpha=0$, Eq. (13) transforms to

$\displaystyle\langle\Gamma_{1}\rangle_{\text{ens}}$
$\displaystyle=\kappa^{2}M_{1}^{2}\mathcal{N}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{\mathrm{d}\omega_{\text{t}}}{\omega_{\text{t}}f(\omega_{\text{t}})}$
$\displaystyle\times\int_{\gamma_{\text{2,m}}(\omega_{\text{t}})}^{\gamma_{\text{2,M}}(\omega_{\text{t}})}\dfrac{\gamma_{2}\,\mathrm{d}\gamma_{2}}{\gamma_{2}^{2}+\omega_{\delta}^{2}}\left[\omega_{\text{t}}^{2}-\dfrac{\gamma_{2}-\gamma_{\phi}}{f(\omega_{\text{t}})}\right]^{-\tfrac{1}{2}},$

which results in

$\langle\Gamma_{1}\rangle_{\text{ens}}=\kappa^{2}M_{1}^{2}\mathcal{N}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{2\gamma_{\phi}\,\mathrm{d}\omega_{\text{t}}}{\gamma_{\phi}^{2}+\omega_{\delta}^{2}}=2\kappa^{2}M_{1}^{2}\mathcal{N}\pi.$

###  A.2 Variance of qubit lifetime: $\text{Var}_{\text{ens}}(\Gamma_{1})$ in
Eq. (17) and $\langle\text{Var}_{\text{spd}}(\Gamma_{1})\rangle_{\text{ens}}$
in Eq. (25)

Here, we derive the variance of qubit lifetime over TLS realizations for the
case of $\alpha=1$. The other results can be obtained similarly. After
variable transformation, Eq. (16) becomes

$\displaystyle\text{Var}_{\text{ens}}(\Gamma_{1})$
$\displaystyle=\kappa^{4}M_{1}^{4}\mathcal{N}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{2\,\mathrm{d}\omega_{\text{t}}}{\omega_{\text{t}}^{3}f^{2}(\omega_{\text{t}})}$
$\displaystyle\times\int_{\gamma_{\text{2,m}}(\omega_{\text{t}})}^{\gamma_{\text{2,M}}(\omega_{\text{t}})}\dfrac{\gamma_{2}^{2}(\gamma_{2}-\gamma_{\phi})\,\mathrm{d}\gamma_{2}}{\left(\gamma_{2}^{2}+\omega_{\delta}^{2}\right)^{2}}.$

Assuming that the TLS pure dephasing rate is much larger than its decay rate,
we can employ the approximation,

$\gamma_{2}^{2}(\gamma_{2}-\gamma_{\phi})/\left(\gamma_{2}^{2}+\omega_{\delta}^{2}\right)^{2}\approx\gamma_{\phi}^{2}(\gamma_{2}-\gamma_{\phi})/(\gamma_{\phi}^{2}+\omega_{\delta}^{2})^{2}.$

With the boundaries of $\omega_{\text{t}}$ being extended to $(0,\infty)$, we
derive the results shown in Eq. (17),

$\displaystyle\text{Var}_{\text{ens}}(\Gamma_{1})$
$\displaystyle=\kappa^{4}M_{1}^{4}\mathcal{N}^{2}\int_{\omega_{\text{m}}}^{\omega_{\text{M}}}\dfrac{2\gamma_{\phi}^{2}\omega_{\text{t}}\,\mathrm{d}\omega_{\text{t}}}{(\gamma_{\phi}^{2}+\omega_{\delta}^{2})^{2}}$
$\displaystyle=\kappa^{4}M_{1}^{4}\mathcal{N}\pi\omega_{\text{q}}/2\gamma_{\phi}.$

###  A.3 Low frequency noise spectral density of an ensemble of TLSs:
$S_{\text{low}}(\omega)$ in Eq. (34) and $S_{\text{white}}^{\prime}$ in Eq.
(36)

With variable transformation from $(\varepsilon,\Delta)$ to
$(\omega_{\text{t}},\gamma_{1})$, the low-frequency noise
$S_{\text{low}}(\omega)$ is given by

$S_{\text{low}}(\omega)=N_{\text{TLS}}\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\int_{\gamma_{\text{1,m}}}^{\gamma_{\text{1,M}}}\mathrm{d}\gamma_{1}P(\gamma_{1},\omega_{\text{t}})\cos^{2}\theta
s_{zz}(\omega).$

Here, the joint probability distribution is

$P(\gamma_{1},\omega_{\text{t}})=\mathcal{N}\omega_{\text{t}}/2\gamma_{1},$

with $\mathcal{N}$ the normalization factor. The boundaries of the integral
are
$\gamma_{\text{1,m}}(\omega_{\text{t}})=2\Delta_{\text{m}}^{2}f(\omega_{\text{t}})$,
and
$\gamma_{\text{1,M}}(\omega_{\text{t}})=2\omega_{\text{t}}^{2}f(\omega_{\text{t}})$.
With $\cos^{2}\theta=1-\gamma_{1}/\gamma_{\text{1,M}}$, we have,

$\displaystyle S_{\text{low}}(\omega)=$ $\displaystyle
N_{\text{TLS}}\mathcal{N}\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\,\omega_{\text{t}}\int_{\gamma_{\text{1,m}}}^{\gamma_{\text{1,M}}}\mathrm{d}\gamma_{1}\dfrac{1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2}}{\omega^{2}+\gamma_{1}^{2}}\left(1-\dfrac{\gamma_{1}}{\gamma_{\text{1,M}}}\right)$
$\displaystyle=$ $\displaystyle
N_{\text{TLS}}\mathcal{N}\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\,\omega_{\textrm{t}}\left\\{\dfrac{1}{\omega}\left[\tan^{-1}\left(\frac{\gamma_{\textrm{1,M}}}{\omega}\right)-\tan^{-1}\left(\dfrac{\gamma_{\textrm{1,m}}}{\omega}\right)\right]-\dfrac{1}{2\gamma_{\textrm{1,M}}}\log(\dfrac{\gamma_{\textrm{1,M}}^{2}+\omega^{2}}{\gamma_{\textrm{1,m}}^{2}+\omega^{2}})\right\\}(1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2}).$

For $\gamma_{\textrm{m}}\ll\omega\ll\gamma_{\textrm{M}}$, the spectral density
has a $1/f$ behavior,

$\displaystyle S_{\text{low}}(\omega)\approx S_{1/f}(\omega)$
$\displaystyle=\dfrac{N_{\text{TLS}}\mathcal{N}\pi}{2\omega}\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\,\omega_{\textrm{t}}(1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2})$
$\displaystyle=2\log(2)\pi
N_{\text{TLS}}\mathcal{N}(k_{\text{B}}T)^{2}/\omega.$

While for $\omega\ll\gamma_{\textrm{m}}\ll\gamma_{\textrm{M}}$, the spectral
density exhibits a white noise behavior,

$\displaystyle S_{\text{low}}(\omega)\approx S_{\text{white}}(\omega)$
$\displaystyle=N_{\text{TLS}}\mathcal{N}\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\,\omega_{\textrm{t}}\,(1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2})/\gamma_{\text{1,m}}$
$\displaystyle=\dfrac{N_{\text{TLS}}\mathcal{N}k_{\text{B}}T}{\Delta_{\text{m}}^{2}2\pi
J_{0}}.$

In the presence of applied quantum noise, the depolarization rate of the TLS
is modified according to Eq. (35). As a result, the lower boundary of the
integral becomes,

$\gamma_{\text{1,m}}^{\prime}(\omega,\alpha)=2\Delta_{\text{m}}^{2}f(\omega_{\text{t}})+2d^{2}\eta\cos{\alpha}^{2}\Delta_{\text{m}}^{2}/\omega_{\text{t}}.$

Reevaluating the white noise part results in

$\displaystyle S_{\text{white}}^{\prime}(\omega)$
$\displaystyle=\dfrac{2N_{\text{TLS}}\mathcal{N}}{\pi}\int_{0}^{\frac{\pi}{2}}\mathrm{d}\alpha\int_{\omega_{\textrm{m}}}^{\omega_{\textrm{M}}}\mathrm{d}\omega_{\textrm{t}}\,\omega_{\textrm{t}}\,\dfrac{1-\langle\hat{\sigma}_{z}\rangle_{\text{eq}}^{2}}{\gamma_{\text{1,m}}^{\prime}}$
$\displaystyle\approx
N_{\text{TLS}}\mathcal{N}\dfrac{0.58(2k_{\text{B}}T)^{2}}{\Delta_{\text{m}}^{2}\sqrt{4\pi
J_{0}d^{2}\eta}}.$

Note that an additional integration is required to average over the angle
$\alpha$ between the applied field and the corresponding TLS dipole, which is
assumed to have a uniform distribution. In the above derivation, we also make
the approximation that $\gamma_{1,\text{m}}^{\prime}$ is dominated by the
applied noise, and extend the integral over $\omega_{\text{t}}$ to
$(0,\infty)$.

## References

  * Krantz  _et al._ [2019] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019).
  * Martinis  _et al._ [2020] J. M. Martinis, M. H. Devoret, and J. Clarke, Quantum Josephson junction circuits and the dawn of artificial atoms, Nat. Phys. 16, 234 (2020).
  * Blais  _et al._ [2020] A. Blais, S. M. Girvin, and W. D. Oliver, Quantum information processing and quantum optics with circuit quantum electrodynamics, Nat. Phys. 16, 247 (2020).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Gao  _et al._ [2021] Y. Y. Gao, M. A. Rol, S. Touzard, and C. Wang, Practical Guide for Building Superconducting Quantum Devices, PRX Quantum 2, 040202 (2021).
  * Ithier  _et al._ [2005] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, and G. Schön, Decoherence in a superconducting quantum bit circuit, Phys. Rev. B 72, 134519 (2005).
  * Oliver and Welander [2013] W. D. Oliver and P. B. Welander, Materials in superconducting quantum bits, MRS Bull. 38, 816 (2013).
  * Siddiqi [2021] I. Siddiqi, Engineering high-coherence superconducting qubits, Nat. Rev. Mater. 6, 875 (2021).
  * Gyenis  _et al._ [2021a] A. Gyenis, A. Di Paolo, J. Koch, A. Blais, A. A. Houck, and D. I. Schuster, Moving beyond the Transmon: Noise-Protected Superconducting Quantum Circuits, PRX Quantum 2, 030101 (2021a).
  * Gyenis  _et al._ [2021b] A. Gyenis, P. S. Mundada, A. Di Paolo, T. M. Hazard, X. You, D. I. Schuster, J. Koch, A. Blais, and A. A. Houck, Experimental Realization of a Protected Superconducting Circuit Derived from the $0-\pi$ Qubit, PRX Quantum 2, 010339 (2021b).
  * Kalashnikov  _et al._ [2020] K. Kalashnikov, W. T. Hsieh, W. Zhang, W.-S. Lu, P. Kamenov, A. Di Paolo, A. Blais, M. E. Gershenson, and M. Bell, Bifluxon: Fluxon-Parity-Protected Superconducting Qubit, PRX Quantum 1, 010307 (2020).
  * Zhang  _et al._ [2021] H. Zhang, S. Chakram, T. Roy, N. Earnest, Y. Lu, Z. Huang, D. K. Weiss, J. Koch, and D. I. Schuster, Universal Fast-Flux Control of a Coherent, Low-Frequency Qubit, Phys. Rev. X 11, 011010 (2021).
  * Somoroff  _et al._ [2021] A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. V. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, arXiv:2103.08578 (2021).
  * DiVincenzo and Shor [1996] D. P. DiVincenzo and P. W. Shor, Fault-tolerant error correction with efficient quantum codes, Phys. Rev. Lett. 77, 3260 (1996).
  * Preskill [1998] J. Preskill, Fault-tolerant quantum computation, in  _Introduction to quantum computation and information_ (World Scientific, 1998) pp. 213–269.
  * Ashhab  _et al._ [2006a] S. Ashhab, J. Johansson, and F. Nori, Decoherence dynamics of a qubit coupled to a quantum two-level system, Physica C 444, 45 (2006a).
  * Paladino  _et al._ [2014] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, $1/f$ noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86, 361 (2014).
  * Müller  _et al._ [2019] C. Müller, J. H. Cole, and J. Lisenfeld, Towards understanding two-level-systems in amorphous solids: insights from quantum circuits, Rep. Prog. Phys. 82, 124501 (2019).
  * McRae  _et al._ [2020] C. R. H. McRae, H. Wang, J. Gao, M. R. Vissers, T. Brecht, A. Dunsworth, D. P. Pappas, and J. Mutus, Materials loss measurements using superconducting microwave resonators, Rev. Sci. Instrum. 91, 091101 (2020).
  * Murray [2021] C. E. Murray, Material matters in superconducting qubits, Mater. Sci. Eng. R Rep. 146, 100646 (2021).
  * Martinis  _et al._ [2005] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Decoherence in Josephson Qubits from Dielectric Loss, Phys. Rev. Lett. 95, 210503 (2005).
  * Wang  _et al._ [2015] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Appl. Phys. Lett. 107, 162601 (2015).
  * Astafiev  _et al._ [2004] O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto, and J. S. Tsai, Quantum noise in the josephson charge qubit, Phys. Rev. Lett. 93, 267007 (2004).
  * Schlör  _et al._ [2019] S. Schlör, J. Lisenfeld, C. Müller, A. Bilmes, A. Schneider, D. P. Pappas, A. V. Ustinov, and M. Weides, Correlating Decoherence in Transmon Qubits: Low Frequency Noise by Single Fluctuators, Phys. Rev. Lett. 123, 190502 (2019).
  * Tomonaga  _et al._ [2021] A. Tomonaga, H. Mukai, F. Yoshihara, and J. S. Tsai, Quasiparticle tunneling and $1/f$ charge noise in ultrastrongly coupled superconducting qubit and resonator, Phys. Rev. B 104, 224509 (2021).
  * Place  _et al._ [2021] A. P. M. Place, L. V. H. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman, G. Cheng, T. Madhavan, H. K. Babla, X. H. Le, Y. Gang, B. Jäck, A. Gyenis, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds, Nat. Commun. 12, 1779 (2021).
  * Wang  _et al._ [2022] C. Wang, X. Li, H. Xu, Z. Li, J. Wang, Z. Yang, Z. Mi, X. Liang, T. Su, C. Yang, G. Wang, W. Wang, Y. Li, M. Chen, C. Li, K. Linghu, J. Han, Y. Zhang, Y. Feng, Y. Song, T. Ma, J. Zhang, R. Wang, P. Zhao, W. Liu, G. Xue, Y. Jin, and H. Yu, Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds, npj Quantum Inf. 8, 3 (2022).
  * Koch  _et al._ [2007] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).
  * Houck  _et al._ [2007] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Generating single microwave photons in a circuit, Nature 449, 328 (2007).
  * Paik  _et al._ [2011] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Observation of High Coherence in Josephson Junction Qubits Measured in a Three-Dimensional Circuit QED Architecture, Phys. Rev. Lett. 107, 240501 (2011).
  * Klimov  _et al._ [2018] P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, E. Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanskiy, H. Neven, and J. M. Martinis, Fluctuations of energy-relaxation times in superconducting qubits, Phys. Rev. Lett. 121, 090502 (2018).
  * Burnett  _et al._ [2019] J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, Decoherence benchmarking of superconducting qubits, npj Quantum Inf. 5, 54 (2019).
  * Simbierowicz  _et al._ [2021] S. Simbierowicz, C. Shi, M. Collodo, M. Kirste, F. Hassani, J. Fink, J. Bylander, D. Perez Lozano, and R. E. Lake, Qubit Energy-Relaxation Statistics in the Bluefors Quantum Measurement System Measurement Bluefors and Zurich Instruments measurement (2021).
  * Carroll  _et al._ [2022] M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer, and A. Kandala, Dynamics of superconducting qubit relaxation times, arXiv:2105.15201 (2022).
  * Müller  _et al._ [2015] C. Müller, J. Lisenfeld, A. Shnirman, and S. Poletto, Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits, Phys. Rev. B 92, 035442 (2015).
  * Matityahu  _et al._ [2021] S. Matityahu, A. Shnirman, and M. Schechter, Stabilization of qubit relaxation rates by frequency modulation, Phys. Rev. Applied 16, 044036 (2021).
  * Zhao  _et al._ [2022] P. Zhao, T. Ma, Y. Jin, and H. Yu, Combating fluctuations in relaxation times of fixed-frequency transmon qubits with microwave-dressed states, Phys. Rev. A 105, 062605 (2022).
  * Huang  _et al._ [2021] Z. Huang, P. S. Mundada, A. Gyenis, D. I. Schuster, A. A. Houck, and J. Koch, Engineering dynamical sweet spots to protect qubits from $1/f$ noise, Phys. Rev. Appl. 15, 034065 (2021).
  * Burin  _et al._ [2013] A. L. Burin, M. S. Khalil, and K. D. Osborn, Universal Dielectric Loss in Glass from Simultaneous Bias and Microwave Fields, Phys. Rev. Lett. 110, 157002 (2013).
  * Khalil  _et al._ [2014] M. S. Khalil, S. Gladchenko, M. J. A. Stoutimore, F. C. Wellstood, A. L. Burin, and K. D. Osborn, Landau-zener population control and dipole measurement of a two-level-system bath, Phys. Rev. B 90, 100201(R) (2014).
  * Matityahu  _et al._ [2019] S. Matityahu, H. Schmidt, A. Bilmes, A. Shnirman, G. Weiss, A. V. Ustinov, M. Schechter, and J. Lisenfeld, Dynamical decoupling of quantum two-level systems by coherent multiple Landau–Zener transitions, npj Quantum Inf. 5, 114 (2019).
  * Yu  _et al._ [2021] L. Yu, S. Matityahu, Y. J. Rosen, C.-C. Hung, A. Maksymov, A. L. Burin, M. Schechter, and K. D. Osborn, Evidence for weakly and strongly interacting two-level systems in amorphous silicon, arXiv:2110.10747 (2021).
  * Burgelman  _et al._ [2022] M. Burgelman, P. Forni, and A. Sarlette, Quantum dynamical decoupling by shaking the close environment, arXiv:2201.09849 (2022).
  * Lisenfeld  _et al._ [2022] J. Lisenfeld, A. Bilmes, and A. V. Ustinov, Enhancing the Coherence of Superconducting Quantum Bits with Electric Fields, arXiv:2208.01570 (2022).
  * Phillips and Anderson [1981] W. A. Phillips and A. Anderson,  _Amorphous solids: low-temperature properties_ , Topics in Current Physics, Vol. 24 (Springer-Verlag Berlin Heidelberg, 1981).
  * Ashhab  _et al._ [2006b] S. Ashhab, J. R. Johansson, and F. Nori, Rabi oscillations in a qubit coupled to a quantum two-level system, New J. Phys. 8, 103 (2006b).
  * Tian and Simmonds [2007] L. Tian and R. W. Simmonds, Josephson Junction Microscope for Low-Frequency Fluctuators, Phys. Rev. Lett. 99, 137002 (2007).
  * Lupaşcu  _et al._ [2009] A. Lupaşcu, P. Bertet, E. F. C. Driessen, C. J. P. M. Harmans, and J. E. Mooij, One- and two-photon spectroscopy of a flux qubit coupled to a microscopic defect, Phys. Rev. B 80, 172506 (2009).
  * Zaretskey  _et al._ [2013] V. Zaretskey, B. Suri, S. Novikov, F. C. Wellstood, and B. S. Palmer, Spectroscopy of a Cooper-pair box coupled to a two-level system via charge and critical current, Phys. Rev. B 87, 174522 (2013).
  * Abdurakhimov  _et al._ [2021] L. V. Abdurakhimov, I. Mahboob, H. Toida, K. Kakuyanagi, Y. Matsuzaki, and S. Saito, Identification of different types of high-frequency defects in superconducting qubits, arXiv:2112.05391 (2021).
  * Breuer and Petruccione [2002] H.-P. Breuer and F. Petruccione,  _The theory of open quantum systems_ (Oxford University Press, 2002).
  * Lisenfeld  _et al._ [2016] J. Lisenfeld, A. Bilmes, S. Matityahu, S. Zanker, M. Marthaler, M. Schechter, G. Schön, A. Shnirman, G. Weiss, and A. V. Ustinov, Decoherence spectroscopy with individual two-level tunneling defects, Sci. Rep. 6, 23786 (2016).
  * Constantin  _et al._ [2009] M. Constantin, C. C. Yu, and J. M. Martinis, Saturation of two-level systems and charge noise in Josephson junction qubits, Phys. Rev. B 79, 094520 (2009).
  * Zagoskin  _et al._ [2006] A. M. Zagoskin, S. Ashhab, J. R. Johansson, and F. Nori, Quantum Two-Level Systems in Josephson Junctions as Naturally Formed Qubits, Phys. Rev. Lett. 97, 077001 (2006).
  * Neeley  _et al._ [2008] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state, Nat. Phys. 4, 523 (2008).
  * Lisenfeld  _et al._ [2019] J. Lisenfeld, A. Bilmes, A. Megrant, R. Barends, J. Kelly, P. Klimov, G. Weiss, J. M. Martinis, and A. V. Ustinov, Electric field spectroscopy of material defects in transmon qubits, npj Quantum Inf. 5, 105 (2019).
  * Yoshihara  _et al._ [2006] F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, and J. S. Tsai, Decoherence of Flux Qubits due to $1/f$ Flux Noise, Phys. Rev. Lett. 97, 167001 (2006).
  * Bialczak  _et al._ [2007] R. C. Bialczak, R. McDermott, M. Ansmann, M. Hofheinz, N. Katz, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, $1/f$ Flux Noise in Josephson Phase Qubits, Phys. Rev. Lett. 99, 187006 (2007).
  * Yan  _et al._ [2016] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, The flux qubit revisited to enhance coherence and reproducibility, Nat. Commun. 7, 12964 (2016).
  * Kumar  _et al._ [2016] P. Kumar, S. Sendelbach, M. A. Beck, J. W. Freeland, Z. Wang, H. Wang, C. C. Yu, R. Q. Wu, D. P. Pappas, and R. McDermott, Origin and reduction of $1/f$ magnetic flux noise in superconducting devices, Phys. Rev. Applied 6, 041001(R) (2016).
  * Quintana  _et al._ [2017] C. M. Quintana, Y. Chen, D. Sank, A. G. Petukhov, T. C. White, D. Kafri, B. Chiaro, A. Megrant, R. Barends, B. Campbell, Z. Chen, A. Dunsworth, A. G. Fowler, R. Graff, E. Jeffrey, J. Kelly, E. Lucero, J. Y. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, P. Roushan, A. Shabani, V. N. Smelyanskiy, A. Vainsencher, J. Wenner, H. Neven, and J. M. Martinis, Observation of Classical-Quantum Crossover of $1/f$ Flux Noise and Its Paramagnetic Temperature Dependence, Phys. Rev. Lett. 118, 057702 (2017).
  * Braumüller  _et al._ [2020] J. Braumüller, L. Ding, A. P. Vepsäläinen, Y. Sung, M. Kjaergaard, T. Menke, R. Winik, D. Kim, B. M. Niedzielski, A. Melville, J. L. Yoder, C. F. Hirjibehedin, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Characterizing and optimizing qubit coherence based on squid geometry, Phys. Rev. Applied 13, 054079 (2020).
  * Van Harlingen  _et al._ [2004] D. J. Van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. A. Crane, and J. Clarke, Decoherence in Josephson-junction qubits due to critical-current fluctuations, Phys. Rev. B 70, 064517 (2004).
  * Constantin and Yu [2007] M. Constantin and C. C. Yu, Microscopic Model of Critical Current Noise in Josephson Junctions, Phys. Rev. Lett. 99, 207001 (2007).
  * Note [1] For spectral density that is singular at zero frequency, a decoherence function has to be involved, see Eq. (31).
  * Clerk  _et al._ [2010] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  * Note [2] Treatment beyond the Markov approximation can be found, e.g., in Ref. [78].
  * Shnirman  _et al._ [2005] A. Shnirman, G. Schön, I. Martin, and Y. Makhlin, Low- and High-Frequency Noise from Coherent Two-Level Systems, Phys. Rev. Lett. 94, 127002 (2005).
  * Faoro  _et al._ [2005] L. Faoro, J. Bergli, B. L. Altshuler, and Y. M. Galperin, Models of Environment and $T_{1}$ Relaxation in Josephson Charge Qubits, Phys. Rev. Lett. 95, 046805 (2005).
  * Note [3] Note that we take the ensemble average of the variance instead of its standard deviation, since $\Gamma_{1}$ of each TLS follows a normal distribution.
  * Grabovskij  _et al._ [2012] G. J. Grabovskij, T. Peichl, J. Lisenfeld, G. Weiss, and A. V. Ustinov, Strain Tuning of Individual Atomic Tunneling Systems Detected by a Superconducting Qubit, Science 338, 232 (2012).
  * Lisenfeld  _et al._ [2015] J. Lisenfeld, G. J. Grabovskij, C. Müller, J. H. Cole, G. Weiss, and A. V. Ustinov, Observation of directly interacting coherent two-level systems in an amorphous material, Nat. Commun. 6, 6182 (2015).
  * Sarabi  _et al._ [2016] B. Sarabi, A. N. Ramanayaka, A. L. Burin, F. C. Wellstood, and K. D. Osborn, Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics, Phys. Rev. Lett. 116, 167002 (2016).
  * Brehm  _et al._ [2017] J. D. Brehm, A. Bilmes, G. Weiss, A. V. Ustinov, and J. Lisenfeld, Transmission-line resonators for the study of individual two-level tunneling systems, Appl. Phys. Lett. 111, 112601 (2017).
  * Bilmes  _et al._ [2020] A. Bilmes, A. Megrant, P. Klimov, G. Weiss, J. M. Martinis, A. V. Ustinov, and J. Lisenfeld, Resolving the positions of defects in superconducting quantum bits, Sci. Rep. 10, 3090 (2020).
  * Bilmes  _et al._ [2021] A. Bilmes, S. Volosheniuk, J. D. Brehm, A. V. Ustinov, and J. Lisenfeld, Quantum sensors for microscopic tunneling systems, npj Quantum Inf. 7, 27 (2021).
  * Note [4] This will not be the case in terms of a quantum noise with identical effective temperature with the TLS bath.
  * You  _et al._ [2021] X. You, A. A. Clerk, and J. Koch, Positive- and negative-frequency noise from an ensemble of two-level fluctuators, Phys. Rev. Research 3, 013045 (2021).
  * Note [5] The crossover from $1/f$ to a $1/f^{2}$ noise is discussed in Ref. [78].
  * Andersson  _et al._ [2021] G. Andersson, A. L. Bilobran, M. Scigliuzzo, M. M. de Lima, J. H. Cole, and P. Delsing, Acoustic spectral hole-burning in a two-level system ensemble, npj Quantum Inf. 7, 15 (2021).
  * O’Connell  _et al._ [2010] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator, Nature 464, 697 (2010).
  * Gustafsson  _et al._ [2014] M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekström, G. Johansson, and P. Delsing, Propagating phonons coupled to an artificial atom, Science 346, 207 (2014).
  * Teissier  _et al._ [2014] J. Teissier, A. Barfuss, P. Appel, E. Neu, and P. Maletinsky, Strain Coupling of a Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator, Phys. Rev. Lett. 113, 020503 (2014).
  * Barfuss  _et al._ [2015] A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, Strong mechanical driving of a single electron spin, Nat. Phys. 11, 820 (2015).
  * Satzinger  _et al._ [2018] K. J. Satzinger, Y. P. Zhong, H.-S. Chang, G. A. Peairs, A. Bienfait, M.-h. Chou, A. Y. Cleland, C. R. Conner, É. Dumur, J. Grebel, I. Gutierrez, B. H. November, R. G. Povey, S. J. Whiteley, D. D. Awschalom, D. I. Schuster, and A. N. Cleland, Quantum control of surface acoustic-wave phonons, Nature 563, 661 (2018).
  * Gloos  _et al._ [2003] K. Gloos, P. J. Koppinen, and J. P. Pekola, Properties of native ultrathin aluminium oxide tunnel barriers, J. Phys. Condens. Matter 15, 1733 (2003).
  * Ioffe  _et al._ [2004] L. B. Ioffe, V. B. Geshkenbein, C. Helm, and G. Blatter, Decoherence in Superconducting Quantum Bits by Phonon Radiation, Phys. Rev. Lett. 93, 057001 (2004).
  * Diniz and de Sousa [2020] I. Diniz and R. de Sousa, Intrinsic Photon Loss at the Interface of Superconducting Devices, Phys. Rev. Lett. 125, 147702 (2020).
  * Preskill [2015] J. Preskill,  _Lecture Notes for Ph219/CS219 : Quantum Information and Computation_ (2015) Chap. 3.
  * Note [6] Similar saturation is discussed in the context of qubit dephasing with a super-Ohmic noise with cutoff, for example, in Ref. [91].
  * Shnirman  _et al._ [2002] A. Shnirman, Y. Makhlin, and G. Schön, Noise and decoherence in quantum two-level systems, Phys. Scr. T102, 147 (2002).