# Open-loop quantum control of small-size networks for high-order cumulants
and cross-correlations sensing

Antonio D’Arrigo  Dipartimento di Fisica e Astronomia "Ettore Majorana",
Università di Catania, Via Santa Sofia 64, 95123 Catania, Italy  Giulia
Piccitto  Dipartimento di Matematica e Informatica, Università di Catania,
Viale Andrea Doria, 95125 Catania, Italy  Giuseppe Falci  Dipartimento di
Fisica e Astronomia "Ettore Majorana", Università di Catania, Via Santa Sofia
64, 95123 Catania, Italy  CNR-IMM, Catania (University unit), Consiglio
Nazionale delle Ricerche, Via Santa Sofia 64, 95123 Catania, Italy  Istituto
Nazionale di Fisica Nucleare, Sezione di Catania, Via Santa Sofia 64, 95123
Catania, Italy  Elisabetta Paladino  Dipartimento di Fisica e Astronomia
"Ettore Majorana", Università di Catania, Via Santa Sofia 64, 95123 Catania,
Italy  CNR-IMM, Catania (University unit), Consiglio Nazionale delle Ricerche,
Via Santa Sofia 64, 95123 Catania, Italy  Istituto Nazionale di Fisica
Nucleare, Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy
<EMAIL_ADDRESS>
###### Abstract

Quantum control techniques represent one of the most efficient tools to attain
high-fidelity quantum operations and a convenient approach for quantum sensing
and quantum noise spectroscopy. In this work, we investigate dynamical
decoupling while processing an entangling two-qubit gate based on an
Ising-$xx$ interaction, each qubit being affected by pure dephasing classical
correlated $1/f$-noises. To evaluate the gate error, we used the Magnus
expansion introducing generalized filter functions that describe decoupling
while processing and allow us to derive an approximate analytic expression as
a hierarchy of nested integrals of noise cumulants. The error is separated in
contributions of Gaussian and non-Gaussian noise, the corresponding
generalized filter functions being calculated up to the fourth order. By
exploiting the properties of selected pulse sequences, we show that it is
possible to extract the second-order statistics (spectrum and cross-spectrum)
and to highlight non-Gaussian features contained in the fourth-order cumulant.
We discuss the applicability of these results to state-of-the-art small
networks based on solid-state platforms.

##  1 Introduction

In the current generation of solid-state devices for quantum technologies [1],
environmental noise sets the accuracy limits of quantum gates [2]. Despite the
tremendous progress in the last two decades [3, 4, 5], material-inherent noise
sources still represent a problem making unreliable even moderate-size quantum
circuits. Quantum control techniques [6] represent one of the most efficient
tools to attain high-fidelity quantum operations fulfilling given time and
power constraints. Their primary goal is to maintain noise-induced errors
below a fault-tolerance threshold required for the efficient implementation of
quantum error correction. Dynamical decoupling (DD) [7, 8, 9] is a form of
open-loop quantum control whose efficiency has been repeatedly validated in
experiments using a variety of platforms [10, 11, 12, 13, 14]. The effect of
DD can be seen as a noise filtering process [15] mathematically expressed in
terms of (generalized) filter functions (FFs) [16, 17, 18, 19].

From a different perspective, dynamical control can be turned into a tool for
quantum sensing (QS) and quantum noise spectroscopy [20] whereby properly
designed pulsed [21, 22, 23, 24, 25, 26, 27, 28, 29] or continuous-control
protocols [30, 19, 31, 32, 33, 34, 35] allow inferring microscopic
information, as noise power spectra. This provides complete statistical
information on Gaussian processes whereas the characterization of non-Gaussian
fluctuations requires estimating higher-order correlation functions, or
polyspectra in Fourier space [36, 37]. Discriminating this type of information
is of paramount relevance in state-of-the-art devices [33, 38, 39] where
evidence of microscopic two-level systems either coherently coupled to the
quantum circuit or incoherently evolving like random telegraph noise (RTN)
processes, has been demonstrated both in spectroscopy and in time-domain
measurements [40]. RTN is the archetypical non-Gaussian process and higher-
order spectral estimation using a qubit probe under pulsed control [23, 37,
41] or via a frame-based control-adapted FF formalism [42] have been recently
demonstrated. Correlated Gaussian processes [43, 44] and RTN [36] inducing
pure dephasing have been investigated via multipulse quantum noise
spectroscopy protocols. Collections of RTNs with proper distribution of
switching rates are a common model of noise with power spectrum behaving as
$1/f^{\alpha}$ where $f$ is the frequency and $\alpha\sim 1$ [45]. Inherently
microscopic noise sources with $1/f$-like spectral density and/or non-Gaussian
characteristics [40] represent one of the major problems for quantum state
processing of state-of-the-art scalable solid-state qubits[40, 5, 46, 47, 48,
49, 50, 51, 52, 53]. Preservation of entanglement in the presence of RTN or
$1/f$ noise via DD protocols has been also proposed [54, 55, 56, 57, 58, 59].

DD of local and spatiotemporal correlated noise sources with $1/f$ or RTN
spectrum in an entangling gate is a critical step to achieving high-fidelity
two-qubit gates. This issue received so far less attention, despite recent
experiments revealing spatial noise correlations [33, 60]. DD of a two-
transmon gate with a noisy tunable coupler has been recently investigated
[52], pointing out the role of $1/f$ flux noise in the coupler and observing
non-Gaussian signatures analogous to those investigated in single qubit gates
[61, 62].

In this work, we consider an entangling two-qubit gate based on an Ising-$xx$
interaction with strength $\omega_{c}$. Each qubit is affected by local pure
dephasing classical noises with power spectrum $S(f)=A/f$ in the range of
frequencies $f\in[f_{m},f_{M}]$ with some degree of correlation [40]
quantified by a non-vanishing cross-spectrum $S_{c}(f)$ between the random
forces. We consider processes characterized by $S_{c}(f)=\mu\,A/f$, where the
parameter $\mu\in[0,1]$ quantifies the strength of the correlations [63, 33,
52].

We study DD protocols implemented by sequences of instantaneous pulses acting
on each qubit locally and simultaneously designed in a way not to alter the
capability of the gate to generate entanglement at a time
$t_{e}=\pi/(2\omega_{c})$. We evaluate the gate error, both in the time and in
the frequency domain, using a Magnus expansion technique. For local
longitudinal noise, the evolution is exactly mapped to two-level problems with
transverse coupling to classical noise. By following an approach inspired by
[16, 17], we derive an approximate analytic expression for the error as a
hierarchy of nested integrals of noise cumulants and FFs. Depending on the DD
sequence and the statistical properties of the noise, the gate error is
dominated by contributions of cumulants of different order. Up to the fourth
order, we can separate the error in contributions due to Gaussian and non-
Gaussian components identifying the corresponding FFs. The different scaling
of these terms with the correlation parameter $\mu$ allows the
characterization of the noise statistics and cross-correlations. By exploiting
the filtering properties of the DD sequences considered, we show that it is
possible to extract the second-order statistical properties (spectrum and
cross-spectrum) and to highlight non-Gaussian features by the fourth-order
cumulant.

##  2 Results

###  2.1 The protocol

We start considering a system of two coupled identical qubits in the presence
of classical noise described by the Hamiltonian ${\cal
H}(t)=\mathcal{H}_{0}+\delta\mathcal{H}(t)$ where (units of $\hbar=1$ are
chosen)

$\mathcal{H}_{0}=-\frac{\Omega}{2}\,\sigma_{1z}\otimes\mathbbm{1}_{2}-\frac{\Omega}{2}\,\mathbbm{1}_{1}\otimes\sigma_{2z}+\frac{\omega_{c}}{2}\,\sigma_{1x}\otimes\sigma_{2x},\quad\delta\mathcal{H}(t)=-\frac{z_{1}(t)}{2}\,\sigma_{1z}\otimes\mathbbm{1}_{2}-\frac{z_{2}(t)}{2}\mathbbm{1}_{1}\otimes\sigma_{2z},$
(1)

where $\sigma_{\alpha x}$ and $\sigma_{\alpha z}$ are the Pauli operators
acting on the qubit $\alpha$, being the logic basis such that $\sigma_{\alpha
z}\ket{\pm}_{\alpha}=\mp\ket{\pm}_{\alpha}$. When the qubits natural
frequencies $\Omega$ are much larger than the coupling strength $\omega_{c}$,
the evolution for a time $t_{e}=\pi/2\omega_{c}$ under $\mathcal{H}_{0}$
implements a $\sqrt{i-\text{SWAP}}$ two-qubit gate which has been demonstrated
on different hardware platforms [64, 2, 65, 66]. For the sake of presentation,
we focus on the effects of local classical noise longitudinally coupled to
each qubit i.e. noise enters $\delta{\cal H}(t)$ with terms commuting with the
individual qubit Hamiltonian and discuss this choice later (see § 3). Noise is
modelled by two stochastic processes $z_{\alpha}(t)$ assumed to be of 4-th
order stationary and characterized by their power spectra
$S_{z_{\alpha}}(\omega)$ and fourth-order cumulants. Correlations of noises on
different qubits are quantified by the cross-spectrum $S_{z_{1}z_{2}}(\omega)$
(see supplemental § B).

Control is operated by applying simultaneously to both qubits a sequence made
of an even number of $\pi-$pulses around the $y-$axis of the Bloch sphere as
described by the Hamiltonian ${\cal H}_{C}(t)$ in Eq. (11). This protocol is
designed to dynamically decouple the system from the noisy environment while
executing a two-qubit gate. Indeed, in the asymptotic limit of a large number
of pulses, the sequence averages out the diagonal terms of the Hamiltonian
while keeping the coupling term $\sigma_{1x}\otimes\sigma_{2x}$. The error in
the gate operation is quantified by

$\varepsilon\,=\,1-\bra{\psi_{e}}\rho(t_{e})\ket{\psi_{e}},$ (2)

where $\ket{\psi_{e}}$ is the target state of the ideal operation and
$\rho(t_{e})$ is the actual state at $t=t_{e}$ obtained from the evolution
under the action of the controlled noisy Hamiltonian ${\cal H}(t)+{\cal
H}_{C}(t)$. The gate infidelity is the maximal error with respect to the
initial state $\ket{\psi_{0}}$.

Under the action of $\cal H$, the system evolves in two invariant subspaces
(see suppl. A). We focus on the dynamics in the single-excitation subspace
$W=\mathrm{span}\\{\ket{+-},\ket{-+}\\}$. In the basis of the Bell states
$\ket{\beta}=\big{[}\ket{+-}+(-1)^{\beta}\ket{-+}\big{]}/\sqrt{2}$ for
$\beta=1,2$ (see tab. 1), the projected Hamiltonian reads

${\cal H}_{W}(t):=P_{W}\,{\cal
H}P_{W}=-\,{\frac{\omega_{c}}{2}}\,\tau_{z}-\frac{z_{1}(t)-z_{2}(t)}{2}\,\tau_{x}\;,$
(3)

where $P_{W}$ are projection operators and $\tau$’s are Pauli matrices,
$\tau_{z}=\ket{1}\bra{1}-\ket{2}\bra{2}$ and
$\tau_{x}=\ket{1}\bra{2}+\ket{2}\bra{1}$. Therefore the effective dynamics in
the $W$ subspace is governed by a two-state Hamiltonian. The ideal gate
generated by $\sigma_{1x}\otimes\sigma_{1x}$ is projected in a two-level
unitary of the $W$-subspace which operates as a non-trivial quantum gate. The
effective noise enters via the stochastic process $\zeta(t)=z_{1}(t)-z_{2}(t)$
which couples by an operator transverse to the projected ideal Hamiltonian
$P_{W}\,{\cal H}_{0}P_{W}$.

In particular, we study the generation of a maximally entangled state obtained
from the initial factorized state $\ket{\psi_{0}}=\ket{+-}$ by evolving the
system in the absence of noise for a time $t_{e}$

$\ket{\psi}_{e}=e^{-i\mathcal{H}_{0}t_{e}}\ket{+-}=\frac{\ket{+-}-i\ket{-+}}{\sqrt{2}}.$
(4)

In the following, we focus on the error $\varepsilon$ for this operation. This
quantity will be used as the output of a QS protocol and it also provides an
indicator of the gate fidelity in the $W$-subspace since the chosen
$\ket{\psi_{0}}$ approximately maximizes the gate error for
$\zeta\ll\omega_{c}$.

###  2.2 Gate error under dynamical control

We notice that the dynamics under the $y-y$ pulse sequence preserve the
invariant subspaces of $\cal H$. Therefore, under DD control, the gate error
for the operation Eq. (4) does not contain contributions due to leakage from
the $W$ subspace. One of the key results of this work is the following formula
expressing the gate error $\varepsilon$ as an expansion in the time-
correlations of the noise truncated at the fourth-order, with the analytic
form for the FFs $F_{i}(\omega,\omega_{c},t_{e},2n)$ reported in App. D

$\displaystyle\varepsilon\,=\varepsilon^{[2]}+\varepsilon^{[4]}_{g}+\varepsilon_{ng}^{[4]}=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\,S_{\zeta}(\omega)F_{1}(\omega,\omega_{c},t_{e},2n)\,$
(5) $\displaystyle+$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,S_{\zeta}(\omega_{1})\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,S_{\zeta}(\omega_{2})\,F_{2,g}(\vec{\omega}_{2},\omega_{c},t_{e},2n)\,$
$\displaystyle+$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{3}}{2\pi}S_{\zeta
3}(\omega_{1},\omega_{2},\omega_{3})\,F_{2,ng}(\vec{\omega}_{3},\omega_{c},t_{e},2n)\,.$

The second-order $\varepsilon^{[2]}$ depends on the power spectrum
$S_{\zeta}(\omega)$ of the noise $\zeta(t)$. The latter is the sum of the
power spectra of each physical process $z_{\alpha}(t)$ and of their cross-
correlation (see supplemental § B),
$S_{\zeta}(\omega)=S_{z_{1}}(\omega)+S_{z_{2}}(\omega)-2S_{z_{1}z_{2}}(\omega)$.
The fourth-order $\varepsilon^{[4]}$ can be written in general as the sum of a
Gaussian ($\varepsilon^{[4]}_{g}$, second line) and a non-Gaussian
($\epsilon_{ng}^{[4]}$, third line) contribution. This latter depends on the
trispectrum $S_{\zeta 3}(\omega_{1},\omega_{2},\omega_{3})\,$ which is the
Fourier transform of the (stationary) fourth-order cumulant of $\zeta(t)$.

For a fixed duration $t_{e}$ of the gate operation, we analyze the dynamics
under ${\cal H}_{C}(t)$ for three different sequences of $2n$ pulses, namely
the periodic (P), the Carr-Purcell (CP) and the Uhrig (U) sequences (see § 4
for details). Information on the pulse-sequence enters Eq. (5) via the FFs
$F_{i}(\omega,\omega_{c},t_{e},2n)$. Notice that our $F_{i}$s generalize the
FFs used for standard DD and QS of longitudinal noise [10]. In our case a non-
trivial gate operation is performed during the DD sequences, thus noise
operators do not commute anymore with the Hamiltonian ${\cal H}_{0}$. The
expression of the generalized FFs, which in our case explicitly depend on the
coupling $\omega_{c}$, has been evaluated by exploiting the Magnus expansion
of the evolution operator (see § C, § D).

Figure 1: Gate error $\varepsilon$ for P (panel (a)), CP (panel (b)) and U
(panel (c)) sequences as a function of the number of pulses $2n$ for fixed
gate time $t_{e}=\pi/2\omega_{c}$, with $\omega_{c}=5\cdot
10^{9}\text{rad}/\text{sec}$ and different noise amplitudes $\Sigma_{\zeta}$.
Symbols are data from the numerical solution of the SSE, the analytical
expressions (6) for P and CP sequences, and Eq. (7) for U sequence are given
by the continuous lines. Inset of the panel (b): different contributions to
$\varepsilon$ for the CP sequence for $\Sigma_{\zeta}=4\times 10^{9}$ and
$\omega_{c}=10^{9}$ rad/s: light green filled diamonds are the numerical
solution of the SSE, large dark green diamonds are the error $\epsilon$ Eq.
(5). The second and the fourth-order contributions to the error correspond to
the dark green filled diamonds and small dark green diamonds respectively.

A first insight into the problem is gained by considering Gaussian quasi-
static noise with variance $\Sigma_{\zeta}^{2}$. In this limit, only
frequencies much lower than $\omega_{c}$ enter Eq. (5), thus the power
spectrum can be approximated by
$S_{\zeta}(\omega)=2\pi\Sigma^{2}_{\zeta}\delta(\omega)$. In Fig. 1 we show
$\varepsilon$ for the pulse sequences under study and various
$\Sigma_{\zeta}$. The symbols are the numerical solution of the stochastic
Schrödinger equation (SSE). The filled lines in panel (a) and panel (b) are
the following analytical expressions

$\varepsilon_{qs}^{(P)}\simeq\frac{\pi^{2}}{2^{6}}\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{2}\frac{1}{n^{2}}\quad,\quad\varepsilon_{qs}^{(CP)}\simeq\frac{\pi^{4}}{2^{12}}\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{2}\frac{1}{n^{4}}\;,$
(6)

derived from Eq. (5) by substituting the power spectrum. We notice that there
is an excellent agreement between numerics and analytical approximations. Both
sequences suppress noise for increasing pulse rate and decreasing ratio
${\Sigma_{\zeta}}/{\omega_{c}}$, in agreement with the analogy between DD and
the Zeno-effect [67]. Moreover, even though both errors scale quadratically
with $\Sigma_{\zeta}$, the CP ($\propto 1/n^{4}$) produces, for increasing
pulse rate, stronger error suppression than the P sequence ($\propto
1/n^{2}$). The numerical analysis for the CP sequence suggests that, for
$\Sigma_{\zeta}>10^{9}$ rad/s, the second and the fourth-order terms
contribute $\varepsilon$ with comparable magnitude. This is shown in the inset
of panel (b) where we plot the second-order and the fourth-order contribution,
$\varepsilon_{g}$ and the solution of the SSE, for $\Sigma_{\zeta}=4\times
10^{9}$ rad/s.

The approximate analytic behaviour for the U sequences (filled lines in Fig.
1(c)) is derived by fitting the SSE numerical result. For $n>4$ we found

$\varepsilon_{qs}^{(U)}\propto\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{4}\frac{1}{n^{3.6}}.$
(7)

The dependence on $\Sigma_{\zeta}^{4}$ suggests that the Uhrig filter fully
cancels the contribution of the second-order time-correlation function during
processing in the presence of transversal effective noise.

Figure 2: Gate error for the U sequence as a function of $2n$ for a Gaussian
$1/f$ noise with fixed low-frequency cutoff $\gamma_{m}=1s^{-1}$ and for
$\omega_{c}=5\cdot 10^{9}$ rad/s. Different panels refer to different upper
cut-off $\gamma_{M}$ and $\Sigma_{\zeta}$ is chosen such as to have the same
integrated power spectrum for any $\gamma_{M}$ (parameters in panel (b)
correspond to typical values of charge noise in superconducting qubits). In
each panel, open circles are the solutions of the SSE equation, grey crosses
give the error of Eq. (5), while blue and red crosses represent the 2nd and
the 4th order Gaussian contributions, respectively. The Uhrig sequence
practically cancels the 2nd order contribution for $\gamma\leq 10^{6}s^{-1}$.

Remarkably, the same results can be derived for the case of long-time
correlated noise. In Fig. 2 we report the gate error $\varepsilon$ in the
presence of noise with spectrum $S_{\zeta}(\omega)=A/\omega$ having fixed low-
frequency cut-off $\gamma_{m}=1s^{-1}$ and varying the upper cut-off
$\gamma_{M}$. Open circles are the solution of the SSE equation, grey crosses
are obtained by Eq. (5), and blue and red crosses represent respectively the
second and the fourth-order contribution in Eq. (5). The analytical
approximation Eq. (5) and the numerical SSE are in agreement also in this
case. This result assesses the ability of the DD procedure to suppress noise
at $\omega_{c}\neq 0$ and yields also in this case the Zeno effect scenario.
For low-frequencies noise ($\gamma_{M}\leq 10^{6}\ \text{s}^{-1}$) the error
is dominated by the fourth-order term, while the situation is reversed and the
second-order contribution becomes dominant, as soon as high-frequency noise
enters the game, i.e. $\gamma\geq 10^{-8}s^{-1}$. This behaviour can be used
for applications to QS.

###  2.3 Filter functions and quantum sensing

Standard FFs [68] can be designed to increase the protection of coherence from
longitudinal noise using DD techniques [10]. Moreover, modulation of properly
designed filters is a powerful tool for QS of noise [20], in particular from
the perspective of experimentally characterizing the longitudinal noise with
single-qubit [69, 22, 23, 24, 25, 26]. In this section, we investigate the
behaviour of our generalized FFs. The main result suggests that the pulse
sequences can provide distinct signatures of the local dephasing bath as non-
Gaussianity and spatial correlations for $1/f$-like noise. In Fig. 3 we show
$F_{1}(\omega,\omega_{c},t_{e},2n)$ for $t_{e}=\pi/(2\omega_{c})$ appearing in
Eq. (25), for the three pulse sequences introduced previously. Regardless of
the sequence considered, the filter has a maximum at $\omega\sim\omega_{c}$,
which becomes sharper for increasing the number of pulses $n$.

Figure 3: Plots of the second order FF $F_{1}(\omega,\omega_{c},t_{e},2n)$,
with $t_{e}=\pi/(2\omega_{c})$, as a function of $\omega$. We use
$\omega_{c}=5\cdot 10^{7}$ rad/s in the top panels and $\omega_{c}=5\cdot
10^{9}$ rad/s for the bottom ones. Different colours refer to different
sequences: P (black, panels (a) and (d)), CP (red, panels (b) and (e)) and U
(blue, panels (c) and (f)). In each figure, the curves refer to (from top to
bottom) $2n=2,\,4,\,6,\,8$. The U-filter suppresses low-frequency noise by
several orders of magnitude already with a small number of pulses.

For frequencies $\omega\ll\omega_{c}$ the filter has a plateau whose magnitude
decreases with increasing the number of applied pulses. The ratio between the
value at the peak frequency and the magnitude of the low plateau for $n\geq 3$
is moderate for the P generalized filter, but it may be very large for the U
one.

In particular, the U protocol efficiently filters out low-frequency noise in
second order. As a consequence, a relatively small number of pulses is enough
to suppress the error $\varepsilon$ by several orders of magnitude. To clarify
this property we first observe that the behaviour of the generalized second-
order FFs $F_{1}(\omega,\omega_{c},t_{e},2n)$ shown in Fig. 3 is significantly
different than that of the standard ones $F_{1}(\omega,t_{e},2n)$ for pure-
dephasing noise, the blue line in Fig. 4. We can make a connection by
rewriting the error $\varepsilon^{[2]}$ in Eq. (5) in terms of the standard
filters

$\varepsilon^{[2]}\,=\,\frac{1}{16}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\Big{[}S(\omega-\omega_{c})+S(\omega+\omega_{c})\Big{]}\,F_{1}(\omega,t_{e},2n)\;.$
(8)

The sum of the shifted spectra $S(\omega-\omega_{c})+S(\omega+\omega_{c})$,
shown in orange in Fig.(4), has a sharp peak at $\omega_{c}$, while at smaller
frequencies it exhibits a plateau whose value is orders of magnitude smaller
than the original $1/f$ spectrum $S(\omega)$. Hence, it strongly weakens the
impact of lower frequencies and behaves as a sort of narrow filter at
frequency $\omega_{c}$ for the $F_{1}(\omega,t_{e},2n)$. Thus, by using the
Uhrig sequence, we can leverage the properties of the standard Uhrig filter
$F_{1}^{U}(\omega,t_{e},2n)={\big{|}y_{n}(\omega,t_{e})\big{|}^{2}}/{\omega^{2}}$
(see sec. 4.3) to suppress frequencies up to $\sim 2\pi/t_{e}=4\omega_{c}$
very quickly as the number of pulses increases. We stress that, as can be
noticed in Fig. 7, to obtain the same result by using the P and the CP
sequences, larger $n$s are required. For larger number of pulses the rate of
change of $F_{1}^{U}(\omega,t_{e},2n)$ for $\omega\lesssim 2\pi/t_{e}$ becomes
very small and the gate error is due to the fourth-order statistical
properties of the noise in agreement with the results in Fig. 2.

Figure 4: The modified $1/f$ power spectrum
$S(\omega-\omega_{c})+S(\omega+\omega_{c})$ (orange curve) in Eq.(8) for
$\omega_{c}=5\cdot 10^{9}$ rad/s, $\gamma_{m}=10^{0}$ s-1, $\gamma_{M}=10^{6}$
s-1 and $\Sigma_{\zeta}=4\cdot 10^{8}$ s-1. It behaves as a narrow filter for
the second-order $F_{1}(\omega,t_{e},2n)$ weakening the impact of low-
frequencies. For the Uhrig filter $F_{1}^{U}(\omega,t_{e},2n)$ (blue curves)
and $t_{e}=\pi/(2\omega_{c})$, frequencies up to $\sim\leavevmode\nobreak\
2\pi/t_{e}$ are suppressed already for $2n=6$.

Non-Gaussian noise Uhrig’s dynamical control capability to suppresses more
efficiently the effect of second-order noise correlations makes this filter
valuable for providing information of non-Gaussianity. To this end, we compare
the effect of an RTN and an Ornstein–Uhlenbeck process (OU) [70] which have
the same second-order statistics (and zero average value). The error resulting
from the exact numerical solution of the SSE is reported in Fig. 5 and
compared with the analytical approximation Eq. (5). Open diamonds correspond
to OU and filled diamonds to RTN. The error due to second-order statistics
almost vanishes for $n\geq 4$, and the curves obtained by SSE are captured by
the fourth-order contributions in Eq. (5).

Figure 5: Gate error under Uhrig DD at $t_{e}$ as a function $n$, for
$\omega_{c}=5\cdot 10^{9}$ rad/s. Symbols are the solution of the SSE: for
Gaussian noise (OU, open diamonds) and non-Gaussian noise (RTN, filled
diamonds). Both processes have zero average and the same variance,
$\Sigma_{\zeta}=10^{9}$ rad/s. The Gaussian noise is obtained by an ensemble
of 256 RTNs with the same $\gamma=1$ s-1, whereas non-Gaussian noise is
produced by a single RTN with $\gamma=1$ s-1. The error given by the second-
order term in Eq. (5) (stars), the contribution to the errors given by the
Gaussian fourth-order term (second row in Eq. (5), filled circles for OU) and
Gaussian plus non-Gaussian fourth-order terms (second and third row in Eq.
(5), filled triangles for RTN), and the total error given by second and
fourth-order terms (open circles for OU, open triangles for RTN) are also
shown. Inset: the difference between gate errors due to OU and RTN processes,
normalized by the gate error due to OU noise; this highlights non-Gaussian
effects evidenced by Uhrig DD.

The error due to the OU process is entirely captured by the Gaussian fourth-
order term in Eq. (5), while in the error due to the RTN both Gaussian and
non-Gaussian terms contribute. To highlight fourth-order statistic non-
Gaussian effects in $\varepsilon$, in the inset of Figure 5 we plot the
difference between the errors due to OU (open diamonds) and RTN (filled
diamonds). We observe that non-Gaussian fourth-order effects are evident
already for $2n\geq 4$.

Spatially-correlated processes Dynamical control can also be used as a
sensitive probe of noise correlations between processes affecting the two
qubits an issue whose importance emerged in recent experiments [60]. Here we
consider spatially-correlated processes[71], and assume that $z_{\alpha}(t)$
have the same statistical properties. Under these conditions (see Suppl. B),
spatial correlations are quantified by a single correlation coefficient [70]

$\mu=\frac{S_{z_{1}z_{2}}(\omega)}{\sqrt{S_{z_{1}}(\omega)S_{z_{2}}(\omega)}},$
(9)

that can be detected by spectral analysis. It can be demonstrated along the
same lines leading to Eq. (5) that the gate error reads

$\varepsilon(\mu)=2(1-\mu)\Big{[}\varepsilon^{[2]}+(1-\mu)\big{(}2\varepsilon^{[4]}_{g}+\varepsilon_{ng}^{[4]}\big{)}\Big{]}\equiv\varepsilon^{[2]}(\mu)+\varepsilon_{g}^{[4]}(\mu)+\varepsilon_{ng}^{[4]}(\mu).$
(10)

where the $\varepsilon^{[2]}$, $\varepsilon_{g}^{[4]}$ and
$\varepsilon^{[4]}_{ng}$ are given in Eq. (5).

In Fig. 6 we show $\varepsilon(\mu)$. The symbols (squares for OU, dots for
RTN) are the analytic form Eq. (10) (reproducing the numerical solution of the
SSE, not shown). For two pulses the error is due to second-order statistics
$\varepsilon(\mu)\,\approx\,\varepsilon^{[2]}(\mu)$, therefore it does not
distinguish the OU process from RTN. The difference between the errors is
entirely due to noise correlations entering the pre-factor $2(1-\mu)$. For
larger number of pulses, the error is dominated by fourth-order statistics and
non-Gaussian effects appear (difference between the squares and dots pairs for
each colour). It is seen that the impact of correlation depends on Gaussianity
as emphasized in the inset of Fig. 5 where we plot
$[\varepsilon_{\text{OU}}(\mu)-\varepsilon_{\text{RTN}}(\mu)]/\varepsilon_{\text{OU}}(\mu)$.
For $2n=4$ the error is due to both second and fourth-order correlators which
have a different dependence on $\mu$, for a larger number of pulses the errors
are given by fourth-order correlators, resulting in a scaled difference
between errors independent on $\mu$.

Figure 6: Gate error under U in the presence of Gaussian (OU) and non-Gaussian
(RTN) correlated noise versus the number of pulses. The symbols are the
analytical result (10) (circles for OU, squares for RTN), and lines are guides
for the eye. Different colours represent different correlation coefficients
$\mu=0,0.5,0.8$ (green, blue, red). Inset: Difference between Gaussian and
non-Gaussian gate errors scaled with the Gaussian error for different
correlation coefficients.

##  3 Discussion

In this work, we have studied the protection of coherence by DD during
processing in a non-trivial quantum gate. This is an important issue for two-
qubit gates whose duration is typically much longer than single-qubit ones. To
this end, we tackled the problem of selective cancelling of non-commuting
entries of the Hamiltonian, extending to transverse noise methods introduced
for analyzing pure-dephasing longitudinal noise.

In particular, for the Ising-$xx$ coupling Hamiltonian, we studied pulsed
control in the presence of local longitudinal noise focusing on the
$W$-subspace where noise is transverse to the projected Hamiltonian. From a
complementary perspective, the two-qubit "principal system" may probe
characteristics of environmental noise. We have shown that a QS protocol based
on DD during entanglement generation may provide non-trivial information on
the noise statistics, as on spatial noise correlations and/or on the fourth-
order cumulant of the resulting stochastic process. Our result leverages
generalized FFs describing DD while processing which differ from the standard
FFs for longitudinal noise. Generalized FFs filter almost uniformly up to
frequencies $\leavevmode\nobreak\ \sim\omega_{c}$ already with a small number
of pulses yielding a very efficient protocol for QS of environmental noise.

In particular, we suggest a simple procedure to extract relevant information
on low-frequency longitudinal noise on each qubit of an entangling gate in a
fixed coupling scheme. The noise variances $\Sigma_{z}^{2}$ can be extracted
from each qubit coherence, which in the presence of quasi-static pure
dephasing noise decays with a peculiar Gaussian law
$\rho_{+-}(t)\propto\exp{(-\Sigma_{z}^{2}t^{2})}$ [72, 73], as it is observed
in Ramsey experiments. Then from an entangling gate operation, the presence of
spatial correlations of the noise can be checked from the gate error under the
P sequence. In fact, for quasi-static noise, the analysis of Section 2.2 can
be extended to correlated noise leading to
$\epsilon_{qs}^{(P)}\simeq\frac{\pi^{2}}{2^{6}}2(1-\mu)\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{2}n^{-2}$.
Due to higher order filtering properties, Uhrig dynamical control on the two-
qubit gate may be employed for distinguishing quasi-static Gaussian noise,
leading to an error scaling as $\epsilon_{qs}^{(U)}\propto
4(1-\mu)^{2}\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{4}n^{-3.6}$, from
quasi-static non-Gaussian noise. In fact, the results in Fig. 5 for a RTN and
Eq. (10) indicate that the error scales as $\epsilon\propto
2(1-\mu)^{2}\left(\frac{\Sigma_{\zeta}}{\omega_{c}}\right)^{4}n^{-3.6}$. The
quantitative distinction between the two processes requires the evaluation of
the prefactors which may depend on the specific non-Gaussian process in the
considered experimental setup.

Since our results apply directly to single-qubit devices sensitive to low-
frequency transverse noise, as the first generation of solid-state qubits
[74], properly designed single-qubit devices could work as quantum sensors of
trispectrum if biased to make the leading noise transverse. The mapping into a
single-qubit problem also suggests that there may be cases where anti-Zeno
behaviour [67] could manifest. Therefore a more complex scenario would emerge
in the two-qubit dynamics where DD may be detrimental to the accuracy of
entangling gates, possibly requiring quantum control and machine learning
methods [75, 76].

We finally comment on the gate model we have chosen and on the relevance of
different noise contributions. The $xx$-Ising interaction between qubits is a
physical description of several implementations of quantum gates with fixed
coupling, as capacitively or inductive coupled superconducting qubits [2] or
semiconducting qubits [64], as well as an effective description of cavity-
mediated interactions. In these cases, local longitudinal noise is potentially
the major semiclassical source of dephasing [77] which justifies our choice
Eq. (1), local transverse semiclassical noise, described by the Hamiltonian
$\delta\mathcal{H}(t)=-\frac{1}{2}x_{1}(t)\,\sigma_{1x}\otimes\mathbbm{1}_{2}-\frac{1}{2}\mathbbm{1}_{1}\otimes
x_{2}(t)\,\sigma_{2x}\,$ being less relevant for our work. Indeed, low-
frequency components would produce weak "transverse" dephasing between the
invariant subspaces. The main effect would be leakage from the $W$-subspace,
which should properly be described by a quantum noise model [78, 79, 80]
outside this work’s scope. In any case, the sequences of $\sigma_{\alpha y}$
pulses we consider tend to cancel also the $\sigma_{\alpha x}$ coupling with
the environment making the associated semiclassical noise less relevant, as we
checked with SSE. Finally, noise directly affecting the $xx$ qubit coupling
term is not expected to be important for fixed-coupling design or for qubits
coupled via a transmission line since it would be longitudinal in the $W$
subspace. On the contrary, it may be non-negligible when the qubit coupling is
implemented by a switchable circuit [52].

##  4 Methods

###  4.1 Open loop control

We consider a dynamical control operated by instantaneous pulses acting
locally and simultaneously on each qubit. The control sequence aims to reduce
the effect of fluctuations while performing an entangling gate operation.
These two requirements may be fulfilled by a sequence of an even number of
simultaneous $\pi$-pulses around the $y$-axis of the Bloch sphere of each
qubit which tends to average out individually all the terms of ${\cal H}(t)$
Eq. (1) but the qubit-coupling. It is described by the control

${\cal H}_{C}(t)\,=\,{\cal
V}_{1}(t)\otimes\mathbbm{1}_{2}\,+\,\mathbbm{1}_{1}\otimes{\cal
V}_{2}(t),\quad\text{with}\quad{\cal
V}_{\alpha}(t)=-i\sum_{i=1}^{2n}\delta(t-t_{i})\sigma_{\alpha y}$ (11)

where the $\delta$-function approximates the process when the duration of the
individual pulse is much smaller than the evolution time of the system under
${\cal H}$. Notice that the control Eq. (11) tends to suppress dynamically
also local transverse noise coupled to each qubit via $\sigma_{\alpha x}$,
thus we consider

$\delta\mathcal{H}(t)=-\frac{1}{2}\Big{(}x_{1}(t)\,\sigma_{1x}\,+\,z_{1}(t)\,\sigma_{1z}\Big{)}\otimes\mathbbm{1}_{2}-\frac{1}{2}\mathbbm{1}_{1}\otimes\Big{(}x_{2}(t)\,\sigma_{2x}\,+\,z_{2}(t)\,\sigma_{2z}\Big{)}\,.$
(12)

Under these conditions, we express the density matrix $\rho(t)$ of the system
as a path-integral over the realizations of the stochastic process. Denoting
by $\rho\big{(}t|\vec{\xi}(t)\big{)}$ the density matrix associated with a
single realization of the stochastic process
$\vec{\xi}(t)=\\{x_{1}(t),z_{1}(t),x_{2}(t),z_{2}(t)\\}$ we write

$\rho(t)=\int{\mathcal{D}}[\vec{\xi}(t)]\,P[\vec{\xi}(t)]\;\rho\big{(}t|\vec{\xi}(t)\big{)}\,,$
(13)

where $P[\vec{\xi}(t)]$ is the probability density for the realization
$\vec{\xi}(t)$ of the noise.

For a given realization $\vec{\xi}(t)$, the dynamics generated by a sequence
of two pulses alternated by two Hamiltonian evolutions for a time $\Delta
t_{i}=t_{i+1}-t_{i}$ is described by the propagator

${\cal U}(t_{i+1},t_{i-1}|\vec{\xi}(t))={\cal
S}\,\hat{T}e^{-i\int_{t_{i}}^{t_{i+1}}{\cal H}(t^{\prime})dt^{\prime}}\,{\cal
S}\,\hat{T}e^{-i\int_{t_{i-1}}^{t_{i}}{\cal H}(t^{\prime})dt^{\prime}}\,.$
(14)

where ${\cal S}\equiv-\sigma_{1y}\otimes\sigma_{2y}$ and
$\mathcal{H}(t)=\mathcal{H}_{0}+\delta\mathcal{H}(t)$.

For equally spaced pulses ($\Delta t_{i}\equiv\Delta t\ \ \forall i$) the gate
operation is not altered to leading order in $\Delta t$, provided that $\Delta
t\ll\min\\{{\tau}_{\xi\alpha}\\}$, being $\tau_{\xi_{\alpha}}$ the dominant
(shortest) correlation time associated with the noise $\xi_{\alpha}(t)$. Under
these conditions, the noise can be approximated, for any
$t\in[t_{i-1},t_{i+1}]$, as a static stray component[72] $\xi(t)\approx\xi$.
Consequently, the integral simply factorizes $\int_{i}^{i+1}{\cal
H}(t^{\prime})dt^{\prime}\sim{\cal H}(t_{i})\Delta t$. By expanding the
exponential in Eq. (14) the evolution operator at the first order in $\Delta
t$ reads

$\displaystyle{\cal U}(t_{i+1},t_{i-1}|\vec{\xi}(t))$
$\displaystyle\simeq\mathbbm{1}+i\,{\omega_{c}}\,\Delta
t\,\sigma_{1x}\otimes\sigma_{2x}\simeq e^{i\,\omega_{c}\Delta
t\,\sigma_{1x}\otimes\sigma_{2x}}$ (15)

Therefore the first order in $\Delta t$ ${\cal U}(t)$ implements a
$\sqrt{i-\text{SWAP}}$ at time $t_{e}=2\Delta t$, noise effects being averaged
out by the sequences of two pulses ${\cal S}$. This result extends to any
sequence of an even number $2n$ of pulses such that $\sum_{k=1}^{2n}\Delta
t_{k}=t_{e}\ll\min\\{{\tau}_{\xi_{\alpha}}\\}$.

We used the error $\varepsilon$ Eq. (5) in the entanglement generation
protocol $\ket{+-}\to\ket{\psi_{e}}$ as a tool for noise sensing. Moreover,
for $|z_{i}|<\omega_{c}$ the error $\varepsilon$ is close to the $W$-space
infidelity thus it also quantifies the performance of DD in noisy gate
processing.

Notice that while the Hamiltonian ${\cal H}_{0}$ operates in the proper limit
a $\sqrt{i-\text{SWAP}}$ gate (see supplemental A), the gate under the $\cal
S$ pulse sequence tends to preserve the ideal dynamics in the $W$-subspace and
not in the $Z$-subspace. This is not a problem since the unitary Eq. (15) when
acting on the $Z$ subspace can generate maximally entangled states. Therefore
the DD sequences we consider preserve the ability of processing a perfectly
entangling gate while decoupling. From the point of view of QS, the effective
dynamics in the $Z$-subspace under the pulse sequences provide asymptotically
information on the stochastic process $z_{1}+z_{2}$ coupled transversally to
the effective Hamiltonian.

###  4.2 Dynamical control of pure dephasing correlated noise

Here we focus on local longitudinal noise, and suppose that $z_{1}(t)$ and
$z_{2}(t)$ are distinct stochastic processes with a correlation degree
quantified by $\mu$, Eq. (9). Control pulses $\mathcal{S}$ transform to
$\pi$-rotation along the $z$-axis with propagator ${\cal S}=\tau_{z}$, such
that
$\tau_{z}\,\hat{T}e^{-\frac{i}{2}\int_{t_{k}}^{t_{k+1}}\mathcal{H}_{g}(t^{\prime})dt^{\prime}}\,\tau_{z}=\hat{T}e^{-\frac{i}{2}\int_{t_{k}}^{t_{k+1}}[-{\zeta}(t^{\prime})\tau_{x}-\omega_{c}\tau_{z}]dt^{\prime}}$.
Therefore, the effect of a control sequence can be included in the controlled-
gate Hamiltonian

$\displaystyle{\cal
H}_{cg}(t)\,=\,-\frac{\bar{\zeta}(t)}{2}\tau_{x}\,-\,\frac{\omega_{c}}{2}\tau_{z}={\cal
H}_{n}(t)\,+\,{\cal
H}_{c},\qquad\bar{\zeta}(t)=(-1)^{k+1}{\zeta}(t),\,\,\,t\in[t_{k-1},t_{k}[$
(16)

where ${\cal H}_{c}=-\,\frac{\omega_{c}}{2}\,\tau_{z}$ and ${\cal
H}_{n}(t)=-\frac{1}{2}\bar{\zeta}(t)\,\tau_{x}$. Thus, for preparation in the
single-excitation subspace, the coupled qubit evolution under local
longitudinal noise and local DD is mapped to a driven pseudo-two-state system
subject to transverse noise.

Introduced the propagator ${\cal
U}_{c}(t)=e^{\frac{i}{2}\omega_{c}t\tau_{z}}$, we can write the Hamiltonian

$\tilde{\cal H}_{n}(t)\,=\,{\cal U}_{c}^{\dagger}(t){\cal H}_{n}(t){\cal
U}_{c}(t)=-\zeta(t)\big{(}\tau_{x}\cos\omega_{c}t\,+\,\tau_{y}\sin\omega_{c}t\big{)},$
(17)

that generate the dynamics in the “toggling” frame. This dynamics is described
by
$\tilde{\mathcal{U}}(t_{e}|\zeta(t))\,=\hat{T}e^{i\int_{0}^{t_{e}}\tilde{\cal{H}}_{n}(t^{\prime})dt^{\prime}}\,$.

The overall propagator can be written as ${\cal U}(t_{e})={\cal
U}_{c}(t_{e})\tilde{\mathcal{U}}(t_{e}|\zeta(t))$ and we have
$\rho(t_{e})=\mathcal{U}(t_{e})\rho(0)\mathcal{U}^{\dagger}(t_{e})$.
Exploiting the fact that $\rho(0)=\ket{+-}\bra{+-}$ we can write the gate
error as

$\displaystyle\varepsilon=$ $\displaystyle 1-\,\bra{\psi_{e}}{\cal
U}(t_{e})\rho(0){\cal U}^{\dagger}(t_{e})\ket{\psi_{e}}$ (18) $\displaystyle=$
$\displaystyle 1-\,\bra{\psi_{e}}{\cal
U}_{c}(t_{e})\mathcal{\tilde{U}}(t_{e}|\zeta(t))\rho(0)\mathcal{\tilde{U}}^{\dagger}(t_{e}|\zeta(t)){\cal
U}_{c}^{\dagger}(t_{e})\ket{\psi_{e}}$ $\displaystyle=$ $\displaystyle
1-\,|\bra{+-}\tilde{\mathcal{U}}(t_{e}|\zeta(t))\ket{+-}|^{2}$

To find analytic expressions for the gate error, we proceed analogously to
[17] and express the time propagator in the toggling frame by its Magnus
expansion:

$\tilde{\mathcal{U}}(t_{e}|\zeta(t))\,=\,e^{\Omega_{1}(t_{e})+\Omega_{2}(t_{e})+\dots},$
(19)

where, for simplicity, we omit the dependence of $\Omega_{\alpha}(t)$ on
$\zeta(t)$. The first two terms of the expansion Eq.(19) read

$\displaystyle\Omega_{1}(t_{e})\,$ $\displaystyle=$
$\displaystyle\,i\big{[}a_{1x}(t_{e})\tau_{x}\,+\,a_{1y}(t_{e})\tau_{y}\big{]},$
(20) $\displaystyle\Omega_{2}(t_{e})\,$ $\displaystyle=$
$\displaystyle\,i\,a_{2z}(t_{e})\tau_{z},$ (21)

where

$\displaystyle
a_{1x}(t_{e})=\frac{1}{2}\int_{0}^{t_{e}}dt_{1}\,\zeta(t_{1})\cos(\omega_{c}t_{1})\,,\quad
a_{1y}(t_{e})=\int_{0}^{t_{e}}dt_{1}\,\zeta(t_{1})\sin(\omega_{c}t_{1})\,,$
(22) $\displaystyle
a_{2z}(t_{e})=\frac{1}{4}\int_{0}^{t_{e}}dt_{1}\int_{0}^{t_{1}}dt_{2}\zeta(t_{1})\zeta(t_{2})\sin(\omega_{c}(t_{1}-t_{2})),$

As discussed in App. C, by truncating the Magnus expansion to the third-order
the gate error can be approximated as follows

$\varepsilon\simeq\varepsilon^{[2]}+\varepsilon^{[4]},$ (23)

where $\varepsilon^{[2]}=\braket{a_{1y}^{2}}$ and
$\varepsilon^{[4]}=\braket{a_{2z}^{2}}$ are of second and fourth order in the
noise, respectively. The validity of these approximations is confirmed by the
results presented in Sec. 2.

###  4.3 Filter function formalism

The gate error in Eq. (23) can be expressed in terms of FFs of subsequent
noise cumulants (for details c.f. supp. D), defined from the Uhrig filter [81]

$y_{n}(\alpha,t_{e})\equiv 1+(-1)^{n+1}e^{i\alpha
t_{e}}\,+\,2\sum_{k=1}^{n}(-1)^{k}e^{i\alpha t_{k}}\,$ (24)

As an example, the second-order contribution reads

$\displaystyle\varepsilon^{[2]}$
$\displaystyle=\,\braket{\Big{(}\frac{i}{2}\int_{0}^{t_{e}}dt_{1}\,\bar{\zeta}(t_{1})\sin(\omega_{c}t_{1})\big{]}\Big{)}^{2}}$
(25)
$\displaystyle=\frac{1}{16}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S_{\zeta}(\omega)\Big{[}\frac{\big{|}y_{n}(\omega+\omega_{c},t_{e})\big{|}^{2}}{(\omega+\omega_{c})^{2}}\,+\,\frac{\big{|}y_{n}(\omega-\omega_{c},t_{e})\big{|}^{2}}{(\omega-\omega_{c})^{2}}\Big{]}$
$\displaystyle=\,\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S_{\zeta}(\omega)F_{1}(\omega,\omega_{c},t_{e},2n).$

This expression defines the FF of second order
$F_{1}(\omega,\omega_{c},t_{e},2n)=\frac{\big{|}y_{n}(\omega+\omega_{c},t_{e})\big{|}^{2}}{(\omega+\omega_{c})^{2}}+\,\frac{\big{|}y_{n}(\omega-\omega_{c},t_{e})\big{|}^{2}}{(\omega-\omega_{c})^{2}}$.

The same calculation for the forth order contribution reveals that
$\varepsilon^{[4]}$ can be decomposed in a Gaussian $\varepsilon_{g}^{[4]}$
and a non-Gaussian $\varepsilon^{[4]}_{ng}$ contributions. Analogously to what
is done for $\varepsilon^{[2]}$, we can define two additional FFs (whose
explicit expression is left to supp. D) and write

$\varepsilon^{[4]}_{g}\,=\,\,\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,S(\omega_{1})\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,S(\omega_{2})\,F_{2,g}(\vec{\omega_{2}},\omega_{c},t_{e},2n),$
(26)

and

$\displaystyle\varepsilon^{[4]}_{ng}\,=\,\,\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{3}}{2\pi}\,S_{\zeta
3}(\vec{\omega_{3}})F_{2,ng}(\vec{\omega_{3}},\omega_{c},t_{e},2n).$ (27)

and write finally Eq. (5) for the gate error.

We remark that the above expression of the gate error holds for any decoupling
sequence consisting of an even number of simultaneous $\pi_{y}$ pulses applied
at times $t_{k}=\delta_{k}t_{e}$ with $0<\delta_{k}<1$ and $k\in{1,2n}$, such
that $t_{2n}=t_{e}$. Each sequence corresponds to different filters thus
allowing either to (partly) cancel environmental effects to various orders[43]
or to filter out relevant spectral components (filtering order).

Figure 7: Plots of the second-order filters $F_{1}(\omega,t_{e},2n)$ for P
(left panel) and CP (right panel) as a function of $\omega$ and for
$\omega_{c}=5\cdot 10^{9}$ rad/s. In each figure, the different line styles
correspond to different numbers of applied pulses.

In this work, we focus on three specific decoupling sequences, the Periodic
(P), the Carr-Purcell (CP), and the Uhrig (U). They are characterized by
different $\delta_{k}$ and, consequently, different pulse intervals: For the
periodic sequence we have $\delta_{k}=k/2n$ and $\Delta t=t_{e}/2n$; for the
Carr-Purcell we have $\delta_{k}=(k-1/2)/2n$ and $\Delta t_{k}=2\Delta t_{1}$
with $\Delta t_{1}=\Delta t_{2n+1}=t_{e}/4n$; for the Uhrig case we have
$\delta_{k}=\sin^{2}[\pi k/(2n+2)]$. Notice that in the limit of a two-pulses
cycle ($n=2$) the Uhrig sequence reduces to the Carr-Purcell one.

In Fig. 7 we show the second order Urhig filters
$F_{1}(\omega,t_{e},2n)=\frac{\big{|}y_{n}(\omega,t_{e})\big{|}^{2}}{\omega^{2}}$
for the P and CP sequences, $F_{1}(\omega,t_{e},2n)$ for the Uhrig sequence is
reported in Fig. 4.

Acknowledgements This work was supported by the PNRR MUR project
PE0000023-NQSTI, by the QuantERA grant SiUCs (Grant No. 731473), by the
University of Catania, Piano Incentivi Ricerca di Ateneo 2020-22, project
Q-ICT. EP acknowledges the COST Action CA 21144 superqumap.

Data availability statement The data that support the findings of this study
are available from the corresponding author upon request.

Author Contributions A.D. and E.P. conceived the model and carried out the
analytical calculations; A.D. carried out the numerical calculations; A.D.,
G.F., G.P. and E.P. analyzed the data. All authors discussed the physics and
contributed to writing the manuscript.

Competing interests The authors declare no competing interests.

##  Appendix A Two-qubit entangling gate

We consider two qubits labelled by $\alpha=1,2$ living in the tensor product
of the two-dimensional Hilbert space $H_{1}\otimes H_{2}$. We define
$\sigma_{\alpha k}$, with $k=x,y,z$, the Pauli matrices acting on the qubit
$\alpha$. The factorized computational basis
$\\{\ket{\mu\nu}:=\ket{\mu}_{1}\otimes\ket{\nu}_{2},\;\mu,\nu=\pm\\}$ is such
that $\sigma_{\alpha z}\ket{\pm}_{\alpha}=\mp\ket{\pm}_{\alpha}$. The qubits
are coupled by an Ising $x-x$ interaction. We considered the Hamiltonian

${\cal
H}=-\frac{\Omega_{1}+z_{1}(t)}{2}\,\sigma_{1z}\otimes\mathbbm{1}_{2}-\frac{\Omega_{2}+z_{2}(t)}{2}\,\mathbbm{1}_{1}\otimes\sigma_{2z}+\frac{\omega_{c}}{2}\,\sigma_{1x}\otimes\sigma_{2x}-\frac{x_{1}(t)}{2}\,\sigma_{1x}-\frac{x_{2}(t)}{2}\sigma_{2x}.$

where terms describing both longitudinal and transverse noise affecting each
qubit appear. In the absence of transverse noise ($x_{\alpha}=0$) the
Hamiltonian $\cal H$ is block diagonal the Hilbert space being the direct sum
of two invariant subspaces denoted by $W=\mathrm{span}\\{\ket{+-},\ket{-+}\\}$
and $Z=\mathrm{span}\\{\ket{++},\ket{--}\\}$. If the qubits are identical or
if $|\Omega_{1}-\Omega_{2}|\ll\omega_{c}$ we obtain an effective ${\cal
H}_{0}$ with the structure of Eq. (1) where $\Omega=\Omega_{1}+\Omega_{2}$ and
$\omega_{c}\to\omega_{c}+|\Omega_{1}-\Omega_{2}|^{2}/(2\omega_{c})$ whose
eigenvalues and eigenvectors are reported in Tab. 1. This Hamiltonian
implements the entanglement-generation operation Eq. (4) studied in this work.

In the usual limit $\omega_{c}\ll\Omega$ the Hamiltonian in the $Z$ subspaces
presents only renormalized diagonal entries thus in this limit ${\cal H}_{0}$
can implement a gate locally equivalent to $\sqrt{{\rm i-SWAP}}$ by evolving
the system for a time $t_{e}=\pi/2\omega_{c}$.

$\beta$ | $\omega_{\beta}$ | $\ket{\beta}$
---|---|---
0 | $-\sqrt{\Omega^{2}+(\omega_{c}/2)^{2}}$ | $-(\sin\vartheta/2)|++\rangle+(\cos\vartheta/2)|--\rangle$
1 | $-\omega_{c}/2$ | $\big{[}|+-\rangle-|-+\rangle\big{]}/\sqrt{2}$
2 | $\phantom{-}\omega_{c}/2$ | $\big{[}|+-\rangle+|-+\rangle\big{]}/\sqrt{2}$
3 | $\phantom{-}\sqrt{\Omega^{2}+(\omega_{c}/2)^{2}}$ | $\cos(\vartheta/2)|++\rangle+\sin(\vartheta/2)|--\rangle$
Table 1:  Eigenvalues and eigenvectors of ${\mathcal{H}}_{0}$. Here
$\tan\vartheta=-\omega_{c}/(2\Omega)$. The two Hilbert subspaces are spanned
by $\\{|1\rangle,|2\rangle\\}$ and $\\{|0\rangle,|3\rangle\\}$

##  Appendix B Characterization of the stochastic processes

We considered noise described by two stochastic processes
$\xi_{\alpha}\equiv\\{x_{\alpha}(t),z_{\alpha}(t)\\}$ assumed to be 4-th order
stationary and with vanishing average, $\braket{\xi_{\alpha}(t)}=0$, where
$\langle\cdot\rangle$ indicates the ensemble average. The lowest order
correlation functions are
$C_{\xi_{\alpha}}(t+\tau,t)=\braket{\xi_{\alpha}(t+\tau)\xi_{\alpha}(t)}\equiv
C_{\xi_{\alpha}}(\tau)$ and the cross-covariance
$C_{\xi_{1}\xi_{2}}(\tau)\,=\,\langle\xi_{1}(t+\tau)\xi_{2}(t)\rangle-\overline{\xi}_{1}\overline{\xi}_{2}\,$
[70]. The power spectrum $S_{\xi_{\alpha}}(\omega)$ and the cross-spectrum
$S_{\xi_{1}\xi_{2}}(\omega)$ are given by

$S_{\xi_{\alpha}}(\omega)=\int_{-\infty}^{\infty}d\tau\,C_{\xi_{\alpha}}(\tau)\,e^{i\omega\tau},\qquad
S_{\xi_{1}\xi_{2}}(\omega)=\int_{-\infty}^{\infty}d\tau\,C_{\xi_{1}\xi_{2}}(\tau)\,e^{i\omega\tau}\,.$
(28)

In general, the amount of correlation between two stochastic processes
$\eta_{i}(t)$ is quantified by the correlation factor

$\displaystyle\mu=\frac{\langle[\eta_{1}(t)-\overline{\eta}_{1}][\eta_{2}(t)-\overline{\eta}_{2}]\rangle}{\sqrt{\langle[\eta_{1}(t)-\overline{\eta}_{1}]^{2}\rangle\langle[\eta_{2}(t)-\overline{\eta}_{2}]^{2}\rangle}}\,,$
(29)

where $\overline{\eta}_{i}\equiv\langle\eta_{i}(t)\rangle$. Here we assume
that each component of $\xi_{\alpha}$ is the sum of independent fluctuating
factors having the same variance, for instance,
$z_{\alpha}(t)=\sum_{k}c_{k}\delta z_{\alpha,k}(t)$ where the variance of
$\delta z_{\alpha,k}(t)$ does not depend on $k$, $\Sigma_{z,\alpha}$. This
assumption is not restrictive, for instance, it models spatially-correlated
processes and cross-talk effects in coupled transmons [71] or flux noise
correlations between two loops of a tunable flux qubit [82] or tunable
capacitively-shunted flux qubits [39], possibly due to non-local sources of
flux noise or junction critical current noise. Under these conditions, the
degree of correlations is expressed by the correlation coefficient relating
the cross-spectrum to the individual power spectra and detectable by spectral
analysis

$S_{\xi_{1}\xi_{2}}(\omega)=\mu\,\sqrt{S_{\xi_{1}}(\omega)S_{\xi_{2}}(\omega)}\,.$
(30)

To point out non-Gaussian effect we also evaluate the first non-vanishing
higher-order correlator, i.e. the fourth-order cumulant
$C_{\xi}^{(4)}(\vec{\tau}_{3})=\braket{\braket{\xi(t_{1})\xi(t_{2})\xi(t_{3})\xi(t_{4})}}$,
where $\vec{\tau}_{3}=(\tau_{1},\tau_{2},\tau_{3})$ with
$\tau_{i}=t_{i+1}-t_{1}$, and the trispectrum

$S_{\xi
3}(\vec{\omega}_{3})=\int_{-\infty}^{\infty}d\vec{\tau}_{3}e^{-i\vec{\omega}_{3}\cdot\vec{\tau}_{3}}\,C^{(4)}_{\xi}(\vec{\tau}_{3}),$
(31)

where $\vec{\omega}_{3}=(\omega_{1},\omega_{2},\omega_{3})$.

##  Appendix C Magnus expansion

In this appendix, we discuss some details of the derivation of the error in
the Eq. (23). To this purpose, we notice that the third term of the Magnus
expansion in Eq. (19) reads

$\Omega_{3}=i[a_{3x}(t_{e})\tau_{x}+a_{3y}(t_{e})\tau_{y}],$ (32)

with

$\displaystyle
a_{3x}(t_{e})=\frac{1}{12}\int_{0}^{t_{e}}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\,\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\Big{[}\cos\omega_{c}(t_{1}-t_{2}+t_{3})-\frac{1}{2}\cos\omega_{c}(t_{1}+t_{2}-t_{3})-\cos\omega_{c}(t_{1}-t_{2}-t_{3})\Big{]},$
(33) $\displaystyle
a_{3y}(t_{e})=\frac{1}{12}\int_{0}^{t_{e}}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{2}}dt_{3}\,\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\Big{[}-\sin\omega_{c}(t_{1}-t_{2}+t_{3})+\frac{1}{2}\sin\omega_{c}(t_{1}+t_{2}-t_{3})-\sin\omega_{c}(t_{1}-t_{2}-t_{3})\Big{]}\,.$

By truncating the Magnus expansion Eq. (19) to the third order, the gate error
at $t_{e}$ reads

$\varepsilon\simeq\langle a_{1y}^{2}\rangle+\,\langle
a_{2z}^{2}\rangle\,+\,\langle 2a_{1y}a_{3y}\rangle\,-\,\frac{1}{3}\langle
a_{1y}^{4}\rangle\,-\,\frac{1}{3}\langle a_{1y}^{2}a_{1x}^{2}\rangle.$ (34)

By comparing the maximum value of each contribution to the gate error, it is
possible to verify that the last three terms are negligible to the first two
justifying the approximation Eq. (23) for the gate error. This is also
confirmed by the exact numerical solution of the SSE for the considered pulse
sequences.

##  Appendix D Derivation of the filter functions

In this appendix, we report the details of the derivation of the FFs. We first
observe that the effect of dynamical control in the SWAP subspace is that of
decomposing the time evolution in Eqs. (20)-(21) as

$\int_{0}^{t_{e}}dt_{1}\,{\bar{\zeta}}(t_{1})\ldots=\sum_{k=1}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\,(-1)^{k}\,\zeta(t_{1})\ldots.$
(35)

Let us substitute this decomposition in Eq. (25).

$\displaystyle\varepsilon^{[2]}$
$\displaystyle=\,\braket{\Big{(}\frac{i}{2}\int_{0}^{t_{e}}dt_{1}\,\bar{\zeta}(t_{1})\sin(\omega_{c}t_{1})\big{]}\Big{)}^{2}}$
(36)
$\displaystyle=-\frac{1}{4}\int_{0}^{t_{e}}dt_{1}\int_{0}^{t_{e}}dt_{2}\,\sin\omega_{c}t_{1}\,\sin\omega_{c}t_{2}\,\big{\langle}\bar{\zeta}(t_{1})\,\bar{\zeta}(t_{2})\big{\rangle}$
$\displaystyle=-\frac{1}{4}\sum_{k=1}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\,(-1)^{k}\sum_{j=1}^{n+1}\int_{t_{j-1}}^{t_{j}}dt_{2}\,(-1)^{j}\,\sin\omega_{c}t_{1}\sin\omega_{c}t_{2}\,\big{\langle}\zeta(t_{1})\,\zeta(t_{2})\big{\rangle}$
$\displaystyle=-\frac{1}{4}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S_{\zeta}(\omega)\sum_{k,j=1}^{n+1}(-1)^{k+j}\int_{t_{k-1}}^{t_{k}}dt_{1}\,\int_{t_{j-1}}^{t_{j}}dt_{2}\,\sin(\omega_{c}t_{1})\sin(\omega_{c}t_{2})\,e^{i\omega(t_{1}-t_{2})}$
$\displaystyle=-\frac{1}{4}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}S_{\zeta}(\omega)\sum_{k,j=1}^{n+1}(-1)^{k+j}\int_{t_{k-1}}^{t_{k}}dt_{1}\,\int_{t_{j-1}}^{t_{j}}dt_{2}\Big{[}\cos\omega_{c}(t_{1}-t_{2})\,-\,\cos\omega_{c}(t_{1}+t_{2})\Big{]}\,\,e^{i\omega(t_{1}-t_{2})},$

where we used that by definition

$\langle\zeta(t_{1})\zeta(t_{2})\rangle\,=\,\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\,e^{i\omega(t_{1}-t_{2})}S_{\zeta}(\omega).$
(37)

We notice that

$\sum_{k=1}^{n+1}(-1)^{k}\int_{t_{k-1}}^{t_{k}}dt_{1}\,e^{i\alpha
t_{1}}\,=\,\frac{1}{i\alpha}\big{[}1+(-1)^{n+1}e^{i\alpha
t_{e}}\,+\,2\sum_{k=1}^{n}(-1)^{k}e^{i\alpha
t_{k}}\big{]}=\frac{1}{i\alpha}y_{n}(\alpha,t_{e}),$ (38)

being $y_{n}(\alpha,t_{e})$ the Uhrig filter introduced in Eq. (24), Therefore
we can compute the time integrals in the last row of Eq. (36) due to
$\cos\omega_{c}(t_{1}-t_{2})$ as follows

$\displaystyle\sum_{k,j=1}^{n+1}(-1)^{k+j}\int_{t_{k-1}}^{t_{k}}dt_{1}\,\int_{t_{j-1}}^{t_{j}}dt_{2}\,\cos\omega_{c}(t_{1}-t_{2})\,e^{i\omega(t_{1}-t_{2})}$
(39) $\displaystyle=\frac{1}{2}$
$\displaystyle\sum_{k,j=1}^{n+1}(-1)^{k+j}\Bigg{[}\int_{t_{k-1}}^{t_{k}}dt_{1}\,e^{i(\omega+\omega_{c})t_{1}}\int_{t_{j-1}}^{t_{j}}dt_{2}\,e^{-i(\omega+\omega_{c})t_{2}}\,+\,\int_{t_{k-1}}^{t_{k}}dt_{1}\,e^{i(\omega-\omega_{c})t_{1}}\int_{t_{j-1}}^{t_{j}}dt_{2}\,e^{-i(\omega-\omega_{c})t_{2}}\Bigg{]}$
$\displaystyle=\frac{1}{2}$
$\displaystyle\Big{[}\frac{1}{(\omega+\omega_{c})^{2}}\big{|}y_{n}(\omega+\omega_{c},t_{e})\big{|}^{2}\,+\,\frac{1}{(\omega-\omega_{c})^{2}}\big{|}y_{n}(\omega-\omega_{c},t_{e})\big{|}^{2}\Big{]}.$

Analogous calculations for the contribution due to
$\cos\omega_{c}(t_{1}+t_{2})$ leads to

$\sum_{k,j=1}^{n+1}(-1)^{k+j}\int_{t_{k-1}}^{t_{k}}dt_{1}\,\int_{t_{j-1}}^{t_{j}}dt_{2}\,\cos\omega_{c}(t_{1}+t_{2})\,e^{i\omega(t_{1}-t_{2})}=\frac{1}{\omega^{2}-\omega_{c}^{2}}\,\Re[y_{n}(\omega+\omega_{c},t_{e})\,y_{n}^{*}(\omega-\omega_{c},t_{e})]\,$
(40)

which is vanishing for all the considered sequences. Combining all these terms
we obtain obtain Eq. (25).

To evaluate the fourth-order contribution in Eq. (23), we observe that

$\int_{0}^{t_{e}}dt_{1}\,\int_{0}^{t_{1}}dt_{2}\,=\,\sum_{k=2}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\sum_{m=1}^{k-1}\int_{t_{m-1}}^{t_{m}}dt_{2}\,+\,\sum_{k=1}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\int_{t_{k-1}}^{t_{1}}dt_{2},$
(41)

and we introduce the integral operator

$\displaystyle\mathcal{A}=\,\frac{1}{16}$
$\displaystyle\Bigg{(}\sum_{k=2}^{n+1}(-1)^{k}\int_{t_{k-1}}^{t_{k}}dt_{1}\sum_{m=1}^{k-1}(-1)^{m}\int_{t_{m-1}}^{t_{m}}dt_{2}\,+\,\sum_{k=1}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\int_{t_{k-1}}^{t_{1}}dt_{2}\Bigg{)}\times$
(42)
$\displaystyle\Bigg{(}\sum_{j=2}^{n+1}(-1)^{j}\int_{t_{j-1}}^{t_{j}}dt_{3}\sum_{l=1}^{j-1}(-1)^{l}\int_{t_{l-1}}^{t_{l}}dt_{4}\,+\,\sum_{j=1}^{n+1}\int_{t_{j-1}}^{t_{j}}dt_{3}\int_{t_{j-1}}^{t_{3}}dt_{4}\Bigg{)}.$

Therefore we can write

$\displaystyle\varepsilon^{[4]}\,=$
$\displaystyle\Big{\langle}\Big{(}\frac{i}{4}\int_{0}^{t_{e}}dt_{1}\,\int_{0}^{t_{1}}dt_{2}\,\bar{\zeta}(t_{1})\bar{\zeta}(t_{2})\sin\omega_{c}(t_{1}-t_{2})\big{]}\Big{)}^{2}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\int_{0}^{t_{e}}dt_{1}\,\int_{0}^{t_{1}}dt_{2}\int_{0}^{t_{e}}dt_{3}\,\int_{0}^{t_{3}}dt_{4}\langle\bar{\zeta}(t_{1})\bar{\zeta}(t_{2})\bar{\zeta}(t_{3})\bar{\zeta}(t_{4})\rangle\sin\omega_{c}(t_{1}-t_{2})\sin\omega_{c}(t_{3}-t_{4})$
$\displaystyle=$ $\displaystyle{\cal
A}\Big{\\{}\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle\sin\omega_{c}(t_{1}-t_{2})\sin\omega_{c}(t_{3}-t_{4})\Big{\\}}\,,$

The average $\braket{\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})}$ can be
decomposed in a Gaussian and a non-Gaussian contributions

$\displaystyle\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle$
$\displaystyle=\,\braket{\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})}_{g}+\braket{\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})}_{ng},$
(43)

defining the Gaussian and non-Gaussian contribution to the gate error

$\displaystyle\varepsilon_{g}^{[4]}\,=\,{\cal
A}\Big{\\{}\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle_{g}\,\sin\omega_{c}(t_{1}-t_{2})\sin\omega_{c}(t_{3}-t_{4})\Big{\\}}$
(44) $\displaystyle\varepsilon^{[4]}_{ng}\,=\,{\cal
A}\Big{\\{}\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle_{ng}\,\sin\omega_{c}(t_{1}-t_{2})\sin\omega_{c}(t_{3}-t_{4})\Big{\\}}.$

Before continuing we report the explicit expression of the two contributions
of the average

$\displaystyle\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle_{g}\phantom{{}_{n}}=\,\langle\zeta(t_{1})\zeta(t_{2})\rangle\langle\zeta(t_{3})\zeta(t_{4})\rangle\,+\,\langle\zeta(t_{1})\zeta(t_{3})\rangle\langle\zeta(t_{2})\zeta(t_{4})\rangle\,+\,\langle\zeta(t_{1})\zeta(t_{3})\rangle\langle\zeta(t_{2})\zeta(t_{4})\rangle\,\,$
(45)
$\displaystyle\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle_{ng}=\,\langle\langle\zeta(t_{1})\zeta(t_{2})\rangle\langle\zeta(t_{3})\zeta(t_{4})\rangle\rangle.$

The Gaussian contributions in Eq. (43) can be expressed in terms of the noise
power spectrum Eq. (37)

$\displaystyle\langle\zeta(t_{1})\zeta(t_{2})$
$\displaystyle\zeta(t_{3})\zeta(t_{4})\rangle_{g}=\,\langle\zeta(t_{1})\zeta(t_{2})\rangle\langle\zeta(t_{3})\zeta(t_{4})\rangle\,+\,\langle\zeta(t_{1})\zeta(t_{3})\rangle\langle\zeta(t_{2})\zeta(t_{4})\rangle\,+\,\langle\zeta(t_{1})\zeta(t_{3})\rangle\langle\zeta(t_{2})\zeta(t_{4})\rangle\,\,$
(46) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,S_{\zeta}(\omega_{1})\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,S_{\zeta}(\omega_{2})\Bigg{[}e^{i\omega_{1}(t_{1}-t_{2})}e^{i\omega_{2}(t_{3}-t_{4})}\,+e^{i\omega_{1}(t_{1}-t_{3})}e^{i\omega_{2}(t_{2}-t_{4})}\,+\,e^{i\omega_{1}(t_{1}-t_{4})}e^{i\omega_{2}(t_{2}-t_{3})}\Bigg{]}$

On the other side, the non-Gaussian term can be expressed in terms of the
trispectrum (31)

$\displaystyle\langle\zeta(t_{1})\zeta(t_{2})\zeta(t_{3})\zeta(t_{4})\rangle_{ng}=\,\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,e^{i\omega_{1}(t_{2}-t_{1})}\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\text{e}^{i\omega_{2}(t_{3}-t_{1})}\int_{-\infty}^{+\infty}\frac{d\omega_{3}}{2\pi}e^{i\omega_{3}(t_{4}-t_{1})}\,S_{\zeta
3}({\omega}_{1},{\omega}_{2},{\omega}_{3}).$ (47)

Collecting these results we obtain

$\displaystyle\varepsilon^{[4]}_{g}\,=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,S(\omega_{1})\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,S(\omega_{2})\,F_{2,g}(\omega_{1},\omega_{2},\omega_{c},t_{e},2n),$
(48) $\displaystyle\varepsilon^{[4]}_{ng}\,=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega_{1}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{2}}{2\pi}\,\int_{-\infty}^{+\infty}\frac{d\omega_{3}}{2\pi}\,S_{\zeta
3}(\omega_{1},\omega_{2},\omega_{3})F_{2,ng}(\omega_{1},\omega_{2},\omega_{3},\omega_{c},t_{e},2n),$

which defines the fourth-order Gaussian and non-Gaussian FFs
$F_{2,g}(\omega_{1},\omega_{2},\omega_{c},t_{e},2n)$ and
$F_{2,ng}(\omega_{1},\omega_{2},\omega_{3},\omega_{c},t_{e},2n)$. To calculate
these filters $F_{2,(n)g}$ we first introduce the function

$\displaystyle\chi_{n}(\alpha,\beta,t_{e})$
$\displaystyle\equiv\Bigg{(}\sum_{k=2}^{n+1}(-1)^{k}\int_{t_{k-1}}^{t_{k}}dt_{1}\sum_{m=1}^{k-1}(-1)^{m}\int_{t_{m-1}}^{t_{m}}dt_{2}\,+\,\sum_{k=1}^{n+1}\int_{t_{k-1}}^{t_{k}}dt_{1}\int_{t_{k-1}}^{t_{1}}dt_{2}\Bigg{)}e^{i\alpha
t_{1}}e^{i\beta t_{2}}\,$ (49) $\displaystyle=\,(-1)^{n+2}\frac{e^{i\alpha
t_{e}}}{\alpha\beta}\,y_{n}(\beta,t_{e})\,-\,2\sum_{k=1}^{n}(-1)^{k}\frac{e^{i\alpha
t_{k}}}{\alpha\beta}\,\tilde{y}_{n,k}(\beta,t_{e})\,+\,\frac{e^{i(\alpha+\beta)t_{e}}-1}{\alpha(\alpha+\beta)},$

where $\tilde{y}_{n,k}(\beta,t_{e})\,=\,1+(-1)^{k+1}e^{i\beta
t_{k}}\,+\,2\sum_{m=1}^{k}(-1)^{m}e^{i\beta t_{m}}$, being the $t_{k}$s the
same entering in ${y}_{n}(\beta,t_{e})$. Therefore we have ${\cal
A}\Big{\\{}e^{i\alpha t_{1}}e^{i\beta t_{2}}e^{i\gamma t_{3}}e^{i\delta
t_{4}}\Big{\\}}\,=\,\chi_{n}(\alpha,\beta,t_{e})\chi_{n}(\gamma,\delta,t_{e})$.
Exploiting this result, we obtain the explicit expressions for the FFs

$\displaystyle
F_{2,g}(\vec{\omega}_{2},\omega_{c})\,=\,\frac{1}{2^{6}}\big{\\{}|$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},\omega_{2}-\omega_{c},t_{e})|^{2}\,+\,|\chi_{n}(\omega_{1}-\omega_{c},\omega_{2}+\omega_{c},t_{e})|^{2}$
(50) $\displaystyle+\,2\Re\big{[}$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},-\omega_{1}-\omega_{c},t_{e})\chi_{n}(\omega_{2}-\omega_{c},-\omega_{2}+\omega_{c},t_{e})$
$\displaystyle+$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},-\omega_{2}-\omega_{c},t_{e})\chi_{n}(-\omega_{2}-\omega_{c},-\omega_{1}+\omega_{c},t_{e})$
$\displaystyle-$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},-\omega_{1}-\omega_{c},t_{e})\chi_{n}(\omega_{2}+\omega_{c},-\omega_{2}-\omega_{c},t_{e})$
$\displaystyle-$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},\omega_{2}-\omega_{c},t_{e})\chi_{n}(-\omega_{1}+\omega_{c},-\omega_{2}-\omega_{c},t_{e})$
$\displaystyle-$
$\displaystyle\chi_{n}(\omega_{1}+\omega_{c},\omega_{2}-\omega_{c},t_{e})\chi_{n}(-\omega_{2}+\omega_{c},-\omega_{1}-\omega_{c},t_{e}),\big{]}\big{\\}},$

and

$\displaystyle
F_{2,ng}(\vec{\omega}_{3},\omega_{c},t_{e},2n)\,=\frac{1}{2^{6}}\big{\\{}-$
$\displaystyle\chi_{2n}(\omega_{c}-\omega_{1}-\omega_{2}-\omega_{3},-\omega_{c}+\omega_{1},t_{e})\,\chi_{2n}(\omega_{c}+\omega_{2},-\omega_{c}+\omega_{3},t_{e})$
(51) $\displaystyle+$
$\displaystyle\chi_{2n}(\omega_{c}-\omega_{1}-\omega_{2}-\omega_{3},-\omega_{c}+\omega_{1},t_{e})\,\chi_{2n}(-\omega_{c}+\omega_{2},\omega_{c}+\omega_{3},t_{e})$
$\displaystyle+$
$\displaystyle\chi_{2n}(\omega_{c}-\omega_{1}-\omega_{2}-\omega_{3},\omega_{c}+\omega_{1},t_{e})\,\chi_{2n}(\omega_{c}+\omega_{2},-\omega_{c}+\omega_{3},t_{e})$
$\displaystyle-$
$\displaystyle\chi_{2n}(\omega_{c}-\omega_{1}-\omega_{2}-\omega_{3},\omega_{c}+\omega_{1},t_{e})\,\chi_{2n}(-\omega_{c}+\omega_{2},\omega_{c}+\omega_{3},t_{e})\big{\\}}.$

## References

  * [1] Preskill, J.  Quantum Computing in the NISQ era and beyond.  _Quantum_ 2, 79, DOI: 10.22331/q-2018-08-06-79 (2018).
  * [2] Kjaergaard, M. _et al._ Superconducting qubits: Current state of play.  _Annual Review of Condensed Matter Physics_ 11, 369–395, DOI: 10.1146/annurev-conmatphys-031119-050605 (2020).
  * [3] Gambetta, J. M., Chow, J. M. & Steffen, M.  Building logical qubits in a superconducting quantum computing system.  _npj Quantum Information_ 3, 2, DOI: 10.1038/s41534-016-0004-0 (2017).
  * [4] Siddiqi, I.  Engineering high-coherence superconducting qubits.  _Nature Reviews Materials_ 6, 875–891, DOI: 10.1038/s41578-021-00370-4 (2021).
  * [5] Falci, G., Hakonen, P. J. & Paladino, E.  1/f noise in quantum nanoscience.  In Chakraborty, T. (ed.) _Encyclopedia of Condensed Matter Physics (Second Edition)_ , 1003–1017, DOI: 10.1016/B978-0-323-90800-9.00250-X (Academic Press, Oxford, 2024), second edition edn.
  * [6] Wiseman, H. M. & Milburn, G. J.  _Quantum Measurement and Control_ (Cambridge University Press, USA, New York, 2010).
  * [7] Viola, L. & Lloyd, S.  Dynamical suppression of decoherence in two-state quantum systems.  _Phys. Rev. A_ 58, 2733–2744, DOI: 10.1103/PhysRevA.58.2733 (1998).
  * [8] Viola, L., Knill, E. & Lloyd, S.  Dynamical decoupling of open quantum systems.  _Phys. Rev. Lett._ 82, 2417–2421, DOI: 10.1103/PhysRevLett.82.2417 (1999).
  * [9] Vitali, D., Tombesi, P. & Milburn, G. J.  Controlling the decoherence of a “meter” via stroboscopic feedback.  _Phys. Rev. Lett._ 79, 2442–2445, DOI: 10.1103/PhysRevLett.79.2442 (1997).
  * [10] Biercuk, M. J. _et al._ Experimental uhrig dynamical decoupling using trapped ions.  _Phys. Rev. A_ 79, 062324, DOI: 10.1103/PhysRevA.79.062324 (2009).
  * [11] Biercuk, M. J. _et al._ Optimized dynamical decoupling in a model quantum memory.  _Nature_ 458, 996–1000, DOI: 10.1038/nature07951 (2009).
  * [12] de Lange, G., Wang, Z. H., Ristè, D., Dobrovitski, V. V. & Hanson, R.  Universal dynamical decoupling of a single solid-state spin from a spin bath.  _Science_ 330, 60–63, DOI: 10.1126/science.1192739 (2010).
  * [13] Bluhm, H. _et al._ Dephasing time of gaas electron-spin qubits coupled to a nuclear bath exceeding 200 $\mu$s.  _Nature Physics_ 7, 109–113, DOI: 10.1038/nphys1856 (2011).
  * [14] Medford, J. _et al._ Scaling of dynamical decoupling for spin qubits.  _Phys. Rev. Lett._ 108, 086802, DOI: 10.1103/PhysRevLett.108.086802 (2012).
  * [15] Biercuk, M., Doherty, A. & Uys, H.  Dynamical decoupling sequence construction as a filter-design problem.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 44, 154002, DOI: 10.1088/0953-4075/44/15/154002 (2011).
  * [16] Green, T., Uys, H. & Biercuk, M. J.  High-order noise filtering in nontrivial quantum logic gates.  _Phys. Rev. Lett._ 109, 020501, DOI: 10.1103/PhysRevLett.109.020501 (2012).
  * [17] Green, T. J., Sastrawan, J., Uys, H. & Biercuk, M. J.  Arbitrary quantum control of qubits in the presence of universal noise.  _New Journal of Physics_ 15, 095004, DOI: 10.1088/1367-2630/15/9/095004 (2013).
  * [18] Soare, A. _et al._ Experimental noise filtering by quantum control.  _Nature Physics_ 10, 825–829, DOI: 10.1038/nphys3115 (2014).
  * [19] Frey, V. M. _et al._ Application of optimal band-limited control protocols to quantum noise sensing.  _Nature Communications_ 8, DOI: 10.1038/s41467-017-02298-2 (2017).
  * [20] Degen, C. L., Reinhard, F. & Cappellaro, P.  Quantum sensing.  _Rev. Mod. Phys._ 89, 035002, DOI: 10.1103/RevModPhys.89.035002 (2017).
  * [21] Yuge, T., Sasaki, S. & Hirayama, Y.  Measurement of the noise spectrum using a multiple-pulse sequence.  _Phys. Rev. Lett._ 107, 170504, DOI: 10.1103/PhysRevLett.107.170504 (2011).
  * [22] Álvarez, G. A. & Suter, D.  Measuring the spectrum of colored noise by dynamical decoupling.  _Phys. Rev. Lett._ 107, 230501, DOI: 10.1103/PhysRevLett.107.230501 (2011).
  * [23] Norris, L. M., Paz-Silva, G. A. & Viola, L.  Qubit noise spectroscopy for non-gaussian dephasing environments.  _Phys. Rev. Lett._ 116, 150503, DOI: 10.1103/PhysRevLett.116.150503 (2016).
  * [24] Faoro, L. & Viola, L.  Dynamical suppression of $1/f$ noise processes in qubit systems.  _Phys. Rev. Lett._ 92, 117905, DOI: 10.1103/PhysRevLett.92.117905 (2004).
  * [25] Falci, G., D’Arrigo, A., Mastellone, A. & Paladino, E.  Dynamical suppression of telegraph and $1/f$ noise due to quantum bistable fluctuators.  _Phys. Rev. A_ 70, 040101(R), DOI: 10.1103/physreva.70.040101 (2004).
  * [26] Zwick, A., Álvarez, G. A. & Kurizki, G.  Maximizing information on the environment by dynamically controlled qubit probes.  _Phys. Rev. Appl._ 5, 014007, DOI: 10.1103/PhysRevApplied.5.014007 (2016).
  * [27] Szczykulska, M., Baumgratz, T. & Datta, A.  Multi-parameter quantum metrology.  _Advances in Physics: X_ 1, 621–639, DOI: 10.1080/23746149.2016.1230476 (2016).
  * [28] Nazarov, Y. V.  _Quantum noise in mesoscopic physics_ , vol. 97 (Springer Science & Business Media, 2012).
  * [29] Ernst, R. R., Bodenhausen, G. & Wokaun, A.  _Principles of nuclear magnetic resonance in one and two dimensions_ (Oxford university press, 1990).
  * [30] Yan, F. _et al._ Rotating-frame relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution.  _Nature Communications_ 4, 2337, DOI: 10.1038/ncomms3337 (2013).
  * [31] Norris, L. M. _et al._ Optimally band-limited spectroscopy of control noise using a qubit sensor.  _Phys. Rev. A_ 98, 032315, DOI: 10.1103/PhysRevA.98.032315 (2018).
  * [32] Frey, V., Norris, L. M., Viola, L. & Biercuk, M. J.  Simultaneous spectral estimation of dephasing and amplitude noise on a qubit sensor via optimally band-limited control.  _Phys. Rev. Appl._ 14, 024021, DOI: 10.1103/PhysRevApplied.14.024021 (2020).
  * [33] von Lüpke, U. _et al._ Two-qubit spectroscopy of spatiotemporally correlated quantum noise in superconducting qubits.  _PRX Quantum_ 1, 010305, DOI: 10.1103/PRXQuantum.1.010305 (2020).
  * [34] Vepsäläinen, A., Danilin, S., Paladino, E., Falci, G. & Paraoanu, G. S.  Quantum control in qutrit systems using hybrid rabi-stirap pulses.  _Photonics_ 3, 62, DOI: 10.3390/photonics3040062 (2016).
  * [35] Falci, G., Ridolfo, A., Di Stefano, P. & Paladino, E.  Ultrastrong coupling probed by coherent population transfer.  _Sci. Rep._ 9, 9249, DOI: 10.1038/s41598-019-45187-y (2019).
  * [36] Szańkowski, P., Trippenbach, M. & Cywiński, L.  Spectroscopy of cross correlations of environmental noises with two qubits.  _Phys. Rev. A_ 94, 012109, DOI: 10.1103/PhysRevA.94.012109 (2016).
  * [37] Ramon, G.  Trispectrum reconstruction of non-gaussian noise.  _Phys. Rev. B_ 100, 161302, DOI: 10.1103/PhysRevB.100.161302 (2019).
  * [38] Rower, D. A. _et al._ Evolution of $1/f$ flux noise in superconducting qubits with weak magnetic fields.  _Phys. Rev. Lett._ 130, 220602, DOI: 10.1103/PhysRevLett.130.220602 (2023).
  * [39] Trappen, R. _et al._ Decoherence of a tunable capacitively shunted flux qubit (2023).  2307.13961.
  * [40] Paladino, E., Galperin, M., Y., Falci, G. & Altshuler, L., B.  $1/f$ noise: Implications for solid-state quantum information.  _Rev. Mod. Phys._ 86, 361, DOI: 10.1103/RevModPhys.86.361 (2014).
  * [41] Sung, Y. _et al._ Non-gaussian noise spectroscopy with a superconducting qubit sensor.  _Nature Communications_ 10, 3751, DOI: 10.1038/s41467-019-11699-4 (2019).
  * [42] Dong, W., Paz-Silva, G. A. & Viola, L.  Resource-efficient digital characterization and control of classical non-Gaussian noise.  _Applied Physics Letters_ 122, 244001, DOI: 10.1063/5.0153530 (2023).
  * [43] Paz-Silva, G. A., Norris, L. M. & Viola, L.  Multiqubit spectroscopy of gaussian quantum noise.  _Phys. Rev. A_ 95, 022121, DOI: 10.1103/PhysRevA.95.022121 (2017).
  * [44] Krzywda, J., Szańkowski, P. & Łukasz Cywiński.  The dynamical-decoupling-based spatiotemporal noise spectroscopy.  _New Journal of Physics_ 21, 043034, DOI: 10.1088/1367-2630/ab0ce7 (2019).
  * [45] Weissman, M. B.  $\frac{1}{f}$ noise and other slow, nonexponential kinetics in condensed matter.  _Rev. Mod. Phys._ 60, 537–571, DOI: 10.1103/RevModPhys.60.537 (1988).
  * [46] Yoshihara, F., Harrabi, K., Niskanen, A. O., Nakamura, Y. & Tsai, J. S.  Decoherence of flux qubits due to 1/f flux noise.  _Phys. Rev. Lett._ 97, 167001, DOI: 10.1103/PhysRevLett.97.167001 (2006).
  * [47] Kakuyanagi, K. _et al._ Dephasing of a superconducting flux qubit.  _Phys. Rev. Lett._ 98, 047004, DOI: 10.1103/PhysRevLett.98.047004 (2007).
  * [48] Bialczak, R. C. _et al._ $1/f$ flux noise in josephson phase qubits.  _Phys. Rev. Lett._ 99, 187006, DOI: 10.1103/PhysRevLett.99.187006 (2007).
  * [49] Koch, J. _et al._ Charge-insensitive qubit design derived from the cooper pair box.  _Phys. Rev. A_ 76, 042319, DOI: 10.1103/PhysRevA.76.042319 (2007).
  * [50] Bylander, J. _et al._ Noise spectroscopy through dynamical decoupling with a superconducting flux qubit.  _Nature Physics_ 7, 565–570, DOI: 10.1038/nphys1994 (2011).
  * [51] Pellegrino, M. D. F., Falci, G. & Paladino, E.  1/f critical current noise in short ballistic graphene josephson junctions.  _Comm. Phys._ 3, 6 (2020).
  * [52] McCourt, T. _et al._ Learning noise via dynamical decoupling of entangled qubits.  _Physical Review A_ 107, DOI: 10.1103/physreva.107.052610 (2023).
  * [53] Pellegrino, F., Falci, G. & Paladino, E.  Second spectrum of charge carrier density fluctuations in graphene due to trapping/detrapping processes.  _Appl. Phys. Lett._ DOI: DOI10.1063/5.0157327 (2023).
  * [54] Wang, Y. _et al._ Preservation of bipartite pseudoentanglement in solids using dynamical decoupling.  _Phys. Rev. Lett._ 106, 040501, DOI: 10.1103/PhysRevLett.106.040501 (2011).
  * [55] Franco, R. L., D’Arrigo, A., Falci, G., Compagno, G. & Paladino, E.  Spin-echo entanglement protection from random telegraph noise.  _Physica Scripta_ 2013, 014043, DOI: 10.1088/0031-8949/2013/T153/014043 (2013).
  * [56] Lo Franco, R., D’Arrigo, A., Falci, G., Compagno, G. & Paladino, E.  Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling.  _Phys. Rev. B_ 90, 054304, DOI: 10.1103/PhysRevB.90.054304 (2014).
  * [57] D’Arrigo, A., Falci, G. & Paladino, E.  Dynamical decoupling of local transverse random telegraph noise in a two-qubit gate.  _Physica Scripta_ 2015, 014037, DOI: 10.1088/0031-8949/2015/T165/014037 (2015).
  * [58] Pokharel, B., Anand, N., Fortman, B. & Lidar, D. A.  Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits.  _Phys. Rev. Lett._ 121, 220502, DOI: 10.1103/PhysRevLett.121.220502 (2018).
  * [59] Yan, H. _et al._ Entanglement purification and protection in a superconducting quantum network.  _Phys. Rev. Lett._ 128, 080504, DOI: 10.1103/PhysRevLett.128.080504 (2022).
  * [60] Yoneda, J. _et al._ Noise-correlation spectrum for a pair of spin qubits in silicon.  _Nature Physics_ 19, 1793 – 1795, DOI: 10.1038/s41567-023-02238-6 (2023).
  * [61] Paladino, E., Faoro, L., Falci, G. & Fazio, R.  Decoherence and $1/f$ noise in josephson qubits.  _Phys. Rev. Lett._ 88, 228304, DOI: 10.1103/PhysRevLett.88.228304 (2002).
  * [62] Galperin, Y. M., Altshuler, B. L., Bergli, J. & Shantsev, D. V.  Non-gaussian low-frequency noise as a source of qubit decoherence.  _Phys. Rev. Lett._ 96, 097009, DOI: 10.1103/PhysRevLett.96.097009 (2006).
  * [63] Zorin, A. B. _et al._ Background charge noise in metallic single-electron tunneling devices.  _Phys. Rev. B_ 53, 13682–13687, DOI: 10.1103/PhysRevB.53.13682 (1996).
  * [64] Chatterjee, A., Stevenson, P. & De Franceschi, S. e. a.  Semiconductor qubits in practice.  _Nat. Rev. Phys._ 3, 157–177 (2021).
  * [65] Chen, Y. _et al._ Voltage activated parametric entangling gates on gatemons.  2304.08469v1.
  * [66] Zhang, H. _et al._ Tunable inductive coupler for high fidelity gates between fluxonium qubits.  2309.05720v2.
  * [67] Facchi, P., Lidar, D. A. & Pascazio, S.  Unification of dynamical decoupling and the quantum Zeno effect.  _Phys. Rev. A_ 69, 032314, DOI: 10.1103/PhysRevA.69.032314 (2004).  Publisher: American Physical Society.
  * [68] Uys, H., Biercuk, M. J. & Bollinger, J. J.  Optimized noise filtration through dynamical decoupling.  _Phys. Rev. Lett._ 103, 040501 (2009).
  * [69] Yan, F. _et al._ Spectroscopy of low-frequency noise and its temperature dependence in a superconducting qubit.  _Phys. Rev. B_ 85, 174521, DOI: 10.1103/PhysRevB.85.174521 (2012).
  * [70] Papoulis, A. & Unnikrishna Pillai, S.  _Probability, Random Variables and Stochastic Processes_ (McGraw-Hill Inc.,US, 1984).
  * [71] D’Arrigo, A., Mastellone, A., Paladino, E. & Falci, G.  Effects of low-frequency noise cross-correlations in coupled superconducting qubits.  _New Jour. Phys._ 10, DOI: 10.1088/1367-2630/10/11/115006 (2008).
  * [72] Falci, G., D’Arrigo, A., Mastellone, A. & Paladino, E.  Initial decoherence in solid state qubits.  _Phys. Rev. lett_ 94, 167002, DOI: 10.1103/PhysRevLett.94.167002 (2005).
  * [73] Chiarello, F. _et al._ Superconducting qubit manipulated by fast pulses: experimental observation of distinct decoherence regimes.  _New Jour. Phys,_ 14, DOI: 10.1088/1367-2630/14/2/023031 (2012).
  * [74] Vion, D. _et al._ Manipulating the quantum state of an electrical circuit .  _Science_ 296, 886, DOI: 10.1126/science.1069372 (2002).
  * [75] Giannelli, L. _et al._ A tutorial on optimal control and reinforcement learning methods for quantum technologies.  _Physics Letters A_ 434, 128054, DOI: https://doi.org/10.1016/j.physleta.2022.128054 (2022).
  * [76] Brown, J. _et al._ Reinforcement learning-enhanced protocols for coherent population-transfer in three-level quantum systems.  _New Jour. Phys,_ 23, 093035, DOI: 10.1088/1367-2630/ac2393 (2021).
  * [77] Paladino, E., Mastellone, A., D’Arrigo, A. & Falci, G.  Optimal tuning of solid state quantum gates: A universal two-qubit gate.  _Phys. Rev. B_ 81, 052502, DOI: 10.1103/PhysRevB.81.052502 (2010).
  * [78] D’Arrigo, A. & Paladino, E.  Optimal operating conditions of an entangling two-transmon gate.  _New Journal of Physics_ 14, 053035, DOI: 10.1088/1367-2630/14/5/053035 (2012).
  * [79] Nesi, F., Paladino, E., Thorwart, M. & Grifoni, M.  Spin-boson dynamics beyond conventional perturbation theories.  _Phys. Rev. B_ 76, 155323, DOI: 10.1103/PhysRevB.76.155323 (2007).
  * [80] Nesi, F., Grifoni, M. & Paladino, E.  Dynamics of a qubit coupled to a broadened harmonic mode at finite detuning.  _New Journal of Physics_ 9, 316, DOI: 10.1088/1367-2630/9/9/316 (2007).
  * [81] Uhrig, G. S.  Keeping a quantum bit alive by optimized $\pi$-pulse sequences.  _Phys. Rev. Lett._ 98, 100504, DOI: 10.1103/PhysRevLett.98.100504 (2007).
  * [82] Gustavsson, S. _et al._ Dynamical decoupling and dephasing in interacting two-level systems.  _Phys. Rev. Lett._ 109, 010502, DOI: 10.1103/PhysRevLett.109.010502 (2012).