# Microscopic parametrizations for gate set tomography under coloured noise

P. Viñas  Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de
Madrid, Cantoblanco, 28049, Madrid, Spain  A. Bermudez  Instituto de Física
Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid,
Spain

###### Abstract

Gate set tomography (GST) allows for a self-consistent characterization of
noisy quantum information processors (QIPs). The standard device-agnostic
approach treats the QIPs as black boxes that are only constrained by the laws
of physics, attaining full generality at a considerable resource cost:
numerous circuits built from the gate set must be run in order to amplify each
of the gate set parameters. In this work, we show that a microscopic
parametrization of quantum gates under time-correlated noise on the driving
phase, motivated by recent experiments with trapped-ion gates, reduces the
required resources enabling a more efficient version of GST. By making use of
the formalism of filter functions over the noise spectral densities, we
discuss the minimal parametrizations of the gate set that include the effect
of finite correlation times and non-Markovian quantum evolutions during the
individual gates. We compare the estimated gate sets obtained by our method
and the standard long-sequence GST, discussing their accuracies in terms of
established metrics, as well as showcasing the advantages of the parametrized
approach in terms of the sampling complexity for specific examples.

###### Contents

  1. I Introduction
  2. II Gate set tomography for coloured phase noise
    1. II.1 Filtered-noise parametrization of the gate set
    2. II.2 Microscopic parameters and gauge redundancy
    3. II.3  Long-sequence maximum-likelihood estimation
    4. II.4 Selection of parametrized base circuits
  3. III Numerical benchmarks with stochastic noise
    1. III.1 Parametrized GST for Ornstein-Uhlenbeck noise
    2. III.2 Comparison with fully-general GST
    3. III.3 Inclusion of amplitude noise fluctuations
  4. IV Conclusions and outlook
  5. A  Quantum tomography and GST
  6. B Coloured phase noise for single-qubit gates

##  I Introduction

Unlocking the full potential of quantum information processors (QIPs) is a
multifaceted challenge that requires key advances along various directions,
both technological and fundamental [1]. Among these, the development of
efficient techniques for the characterization of undesired interactions – or
noise – in QIPs, is of particular relevance, for it allows to develop improved
strategies to counter its undesired effects. This is the case of some schemes
[2] of quantum error mitigation [3, 4, 5], or some strategies [6, 7] for
quantum error correction [8, 8, 9, 10] that may allow to find shortcuts
towards the fault-tolerant era. In pursuit of this goal, the community working
in quantum characterization, verification, and validation (QCVV) has devised a
wide variety of methodologies [11, 12, 13]. Some of these approaches seek to
provide a unified metric to estimate device performance without explicitly
detailing the form of the errors, as exemplified by randomized benchmarking
[14, 15, 16, 17, 18]. Others strive towards a precise prediction of the actual
faulty operations in the noisy QIPs, belonging to the family of quantum
tomography techniques [19, 20].

The simplest protocol that falls in this category is quantum state tomography
(QST) [21, 22, 23, 24, 25, 26], which aims to estimate an unknown quantum
state described by a unit-trace positive-semidefinite linear operator in a
Hilbert space $\mathcal{H}$ of dimension $d={\rm dim}(\mathcal{H})$, namely
$\rho\in\mathsf{D}(\mathcal{H})\subset\mathsf{Pos}(\mathcal{H})$. For
discrete-variable systems, QST requires implementing measurements drawn from
an informationally-complete (IC) set of positive operator-valued measure
(POVM) elements $\\{M_{\mu}:\,\mu\in\mathbb{M}\\}$, where
$M_{\mu}\in\mathsf{Pos}(\mathcal{H})$ and $\sum_{\mu}M_{\mu}=\mathbb{1}_{2}$
[27] (see Fig. 1a). These allow for an univocal estimate $\hat{\rho}$ of the
state $\rho$ from the observed probabilities $p_{\mu}={\rm
Tr}\\{M_{\mu}\rho\\}$, where we will use hats to differentiate the statistical
estimates from the actual objects (see App. A). In quantum process tomography
(QPT) [28, 29, 30, 31, 32], the focus lies on the estimation of the quantum
channel $\rho\mapsto\mathcal{E}(\rho)$, a completely-positive and trace-
preserving linear operator
$\mathcal{E}\in\mathsf{C}(\mathcal{H})\subset\mathsf{L}(\mathcal{H})$ that
describes a physically-admissible operation on a quantum system transforming
input to output states [1, 27, 33, 34]. In addition to the above IC
measurements, an IC set of initial states
$\\{\rho_{s}:\,s\in\mathbb{S},\,|\mathbb{S}|=d^{2}\\}$ is also required to
perform QPT. See Fig. 1b for a schematic representation of QPT. Similarly to
QST, one can find an univocal estimate of the channel $\hat{\mathcal{E}}$ from
the measured probabilities $p_{\mu s}={\rm
Tr}\\{M_{\mu}\mathcal{E}(\rho_{s})\\}$ (see App. A).

In practice, however, any real experiment will only provide an approximation
to above POVM probabilities given a finite number of measurement shots. We
consider from now on that each POVM is described by a measurement basis $b$
and a possible outcome $m_{b}$, such that
$\mu=(b,m_{b})\in\mathbb{M}=\mathbb{M}_{b}\times\mathbb{M}_{m_{b}}$. This is
the case of the Pauli measurements in $N$-qubit systems with $d=2^{N}$, where
the POVM elements can be defined in terms of the orthonormal Pauli basis
$E^{\phantom{\dagger}}_{\alpha}\in\\!\big{\\{}\mathbb{1}_{2},\sigma_{x},\sigma_{y},\sigma_{z}\big{\\}}^{\\!\\!\bigotimes^{N}}\\!\\!\\!\\!/\sqrt{d}.$
Each Pauli measurement outcome is binary $m_{b}=\pm 1$, and the specific
number of outcomes $N_{b,m_{b},s}$ that result from $N_{b,s}$ measurement
shots per initial state and measurement basis lead to a binomial distribution
with relative frequencies $f_{\mu,s}=N_{b,m_{b},s}/N_{b,s}$. In the asymptotic
limit $N_{b,s}\gg 1$, this tends to a normal distribution $f_{\mu
s}\sim{N}(\bar{\mu},\bar{\sigma}^{2})$ centered around the aforementioned
probabilities $\bar{\mu}=p_{\mu,s}$ with standard deviation
$\bar{\sigma}=\sqrt{p_{\mu,s}(1-p_{\mu,s})/{N_{b,s}}}$. Hence, there are
intrinsic finite-sample errors that describe the deviations from the
probability distributions $p_{\mu s}$, which are typically referred to in the
literature as quantum projection or shot noise [35].

Figure 1: Schemes of quantum tomography schemes. a State tomography. An
informationally complete set of $d^{2}$ predetermined POVM elements
$\\{M_{\mu}\\}$ needs to be specified. b For process tomography, in addition
to the POVM elements , an IC set of $d^{2}$ initial states $\\{\rho_{s}\\}$
must also be specified. c Gate set tomography. The QPT scheme must be executed
for each of the $N_{G}$ gates at the set. These gates, or combinations thereof
(denoted by $H_{s}$ and $H_{b}$), act on a single experimentally-accessible
initial state $\rho_{0}$ and POVM $M_{0}$ to obtain the aforementioned IC
sets.

In analogy to QST, one can obtain a linear relation between measured relative
frequencies and the quantum channel acting as a super-operator [36]. For
instance, using the so-called Pauli Transfer Matrix (PTM) representation (see
Appendix A), the linear equations can be straightforwardly inverted. However,
this linear inversion can lead to non-physical estimates of the channel
$\hat{\mathcal{E}}\notin\mathsf{C}(\mathcal{H})$ when finite-sample errors are
significant. To avoid these problems, maximum likelihood estimation is
typically preferred [37], as the numerical optimization of a cost function can
be supplemented with constraints to ensure that all estimated channels are
indeed physically acceptable [30, 31]. One of the advantages of this strategy
is that the constrained optimization problem is convex and, thus, a unique
estimate $\hat{\mathcal{E}}$ is guaranteed.

Gate set tomography (GST) (Fig. 1c) was introduced to overcome a fundamental
shortcoming of QPT [38, 39, 40, 41, 42]. QPT, although used in several QIP
platforms [43, 44, 45, 46, 47, 48, 49, 50, 51], suffers from a self-
consistency issue: it assumes that the IC set of initial states and
measurements are error free. In reality, however, these states are usually
obtained by acting with a set of gates $\\{G_{i}\\}$ on a single fiducial
state $\rho_{0}$, which reduces to $\ket{0}\\!\bra{0}$ in the ideal $N=1$
case. Likewise, the measurements are obtained by acting with these gates on a
single binary projective measurement
$M_{z,+}=M_{0},\,M_{z,-}=\mathbb{1}-M_{0}$, which reduce to
$\ket{0}\\!\bra{0},\ket{1}\\!\bra{1}$ in the ideal $N=1$ case. Clearly,
$\\{G_{i}\\}$ are implemented similarly to the noisy gate $\hat{\mathcal{E}}$
that we aim at characterizing, and will thus contribute to a non-zero state
preparation and measurement (SPAM) error. In the case of single-qubit gates,
this error will be of the same of order as the one that afflicts the gate we
want to estimate. It is thus not justified to assume that the SPAM is ideal
and error free. GST addresses this self-consistency issue by targeting the
full set of noisy elements, including the gates used to attain IC and as well
as the fiducial operators in a full “gate set”

$\mathcal{G}=\Bigl{\\{}\rho_{0},\,\,\big{\\{}G_{i}:\,i\in\mathbb{G}\,\big{\\}},\,\,M_{0}\Bigr{\\}}.$
(1)

As emphasised in [52], QPT is not device-independent in its entirety, as it
requires a prior independent calibration of the measurement basis and initial
states. GST eliminates this requirement and provides a complete self-
consistent characterization that treats the whole QIP device as a black box.
However, to attain this full generality, it requires at least
$N_{GST}=N_{G}\times 4^{N}(4^{N}-1)+2^{3N}-1$ parameters for an $N$-qubit
system, where $N_{G}=|\mathbb{G}|$ is the number of gates in the set (1).
Estimating these parameters thus requires an exponentially-large number of
imperfect fiducial state preparation, followed by a combination of gates from
the gate set, and a final fiducial imperfect measurement. In addition to this
exponential scaling, the number of samples required to achieve a target
precision $\epsilon$ increases as $N_{b,s}\propto 1/\epsilon^{2}$. Altogether,
the resource scaling limits the applicability of a fully-general GST to
systems with small qubit numbers. The development of long-sequence GST is a
significant advancement in this regard [53], which estimates the gate set
parameters using data from specific circuits that are built from combining
elements of the gate set. This protocol linearly amplifies gate parameters as
the circuit depth $p$ increases, allowing for a Heisenberg-like scaling of the
precision estimates. In particular, in the asymptotic limit, the error falls
with the inverse of the circuit depth and not with its square root, as would
occur if one only used the resources to increase the number of repeated
measurements in circuits that only involve a single gate in addition to the
SPAM ones.

The full generality of GST is convenient to validate and verify the
functioning of a certain QIP device, regardless of having any specific
microscopic knowledge, making it useful for external users. On the other hand,
it can also help developers to calibrate their devices and guide hardware
progress in the current noisy intermediate scale quantum (NISQ) era [54]. From
that perspective, one can benefit from a microscopic modelling of these
devices to partially alleviate the stringent resource requirements of GST. In
this work, instead of using a device-agnostic approach, we advocate for a
microscopically-motivated parametrization of the gate set (1), having a
specific setup in mind: gates mainly limited by coloured phase noise. As
discussed in [55], the effect of frequency and phase stochastic fluctuations
of current trapped-ion QIPs can be accurately described with the filter-
function formalism [56, 57, 58, 59, 60, 61, 62, 63]. We use this formalism to
demonstrate that the imperfections of the gate set can be related to various
filtered integrals of the noise spectral density, reducing considerably the
number of parameters that is required in GST. In particular, for the gate set
under study in this work, the total number of parameters for fully-general
approach amounts to $N_{GST}=67$. This contrasts with that required to
characterize the gate set under the coloured phase noise, where $N_{GST}=11$
is required for a Markovian estimation or $N_{GST}=16$ for a non-Markovian
one. We do also show that it is straightforward to account for amplitude noise
fluctuations as well, at the cost of just two additional parameters in order
to completely characterize the gate set. Finally, we refer to the GitHub
repository ColouredGST [64] as a guide for the numerical implementation of the
results in this work.

This manuscript is organized as follows: Sec. II presents the theoretical
description of our parametrized GST. We delve into some peculiarities derived
from this microscopically-motivated approach, such as the simplification of
the circuit selection process and the absence of the so-called gauge
redundancy, which is a characteristic of fully-general approach to GST [38,
39, 40, 41]. Sec. III constitutes a numerical validation of our parametrized
GST. By considering specific numerical simulations of the coloured phase
noise, we can compare the GST estimates $\hat{\mathcal{G}}$ to the exact
microscopic operations $\bar{\mathcal{G}}$ that stem from the stochastic
averages over the phase fluctuations. In this way, we can analyse the accuracy
of our estimation and discuss its performance for different noise regimes,
including the effect of time correlations and non Markovian evolution as well
as the amplitude noise extension of the model. Moreover, we compare the
results of our parametrized GST approach with those obtained from fully
general GST applied to the same numerically-simulated data. We close this
manuscript in Sec. IV with our conclusions, also mentioning some future
research lines motivated by this work.

##  II Gate set tomography for coloured phase noise

The goal of this section is to give a comprehensive account of our customized
GST, comparing the parameterization for the phase-noise model on single-qubit
gates to the standard protocol of long-sequence GST. For a review of important
aspects of GST, see Appendix A, where we focus on the concepts used throughout
this work, which are discussed in more depth in the review by Nielsen et al.
[38] for fully-general GST. As outlined in the introduction, the main
advantage of employing a microscopically-motivated model lies in the reduction
of the parameters required for a complete self-consistent characterization.

Although approached from a different angle, using a simplified model to
efficiently learn the noise of a device is successfully exemplified by [65],
where one restricts the estimation to a Lindblad-Pauli channel [66, 67]. In
fact, for a Lindbladian evolution, one can estimate the generators of the
channel rather than the channel itself, which allows to reconstruct the full
dynamical quantum map [43, 68, 69, 70, 71, 72, 73, 74, 75], and obtain
advantages from compressed sensing when the microscopic noise generators have
a clear structure with leading components [76]. In some situations, however,
time correlations in the noise and non-Markovian effects forbid this approach.
Moreover, the Pauli-type channel may also be a crude approximation to the
actual native gate set. As shown in this work, even in this situations, one
can still benefit from a detailed microscopic modelling of the non-Lindblad
and non-Pauli channel through a reduced parametrization of GST. This reduction
not only leads to significant savings in computational resources, but also
results in a proportional decrease in the number of experiments required.
Moreover, as we describe now in detail, our approach circumvents certain
challenges encountered in a fully-general GST.

###  II.1 Filtered-noise parametrization of the gate set

As discussed in the introduction and in Appendix A, GST considers a noisy gate
set (1), which can be described using the process matrix representation for
the quantum channel associated to each gate

$\mathcal{E}_{i}(\rho)=\sum_{\alpha,\beta}\chi^{(i)}_{\alpha,\beta}E^{\phantom{\dagger}}_{\alpha}\rho
E_{\beta}^{\dagger},\hskip 8.61108pti\in\\{1,\cdots,N_{G}\\}.$ (2)

Here, $\\{E_{\alpha}:\alpha\in\\{1,\cdots,d^{2}\\}\\}$ can be any basis of the
space of linear operators $\mathsf{L}(\mathcal{H})$, such as the
aforementioned Pauli basis [1]. The above set of $\\{\chi^{(i)}\\}$ matrices
must be semidefinite positive and fulfill a so-called trace constraint (26).
In full generality, each of them must be parametrized via $d^{2}(d^{2}-1)$
real parameters. The fiducial operators $\rho_{0},M_{0}\in\mathcal{G}$ require
$N_{\rm SPAM}=d^{3}-1$ real parameters, leading to a total of
$N_{GST}=N_{G}\times d^{2}(d^{2}-1)+d^{3}-1$ parameters to be estimated in
GST.

Rather than treating the QIP as a black box and aiming at GST with full
generality, we advocate for a specific parametrization that employs a
physically-motivated microscopic modelling of the gates. To define a gate set,
let us start by considering the case of perfect unitary gates

$G^{\rm id}_{i}(\theta_{i},\phi_{i})={\rm exp}\left\\{-{\rm
i}\frac{\theta_{i}}{2}\big{(}\cos\phi_{i}\sigma_{x}-\sin\phi_{i}\sigma_{y}\big{)}\right\\},$
(3)

where $\theta_{i}(\phi_{i})$ is the pulse area (phase). In this work, we will
slightly abuse the notation by using the same symbol $G_{i}$ for the unitary
gate, its Pauli transfer matrix representation, and the gate label. IC can be
achieved by using a single $\theta=\pi$ pulse with $\phi=0$ that maps
$\ket{0}\\!\bra{0}\mapsto\ket{1}\\!\bra{1}$, and a pair of $\theta=\pi/2$
pulses with $\phi=0$ and $\phi=3\pi/2$, which map
$\ket{0}\\!\bra{0}\mapsto\ket{+}\\!\bra{+}$ and
$\ket{0}\\!\bra{0}\mapsto\ket{+{\rm i}}\\!\bra{+{\rm i}}$, respectively, where
$\ket{\pm w}=(\ket{0}\pm w\ket{1})/\sqrt{2}$. These $\pi/2$ pulses also
suffice to rotate the measurement basis from $b=z$ to $b=y,x$, achieving the
required IC. In the context of long-sequence GST, it will be convenient to
enlarge this gate set to $N_{G}=5$ by also including the inverse of the
$\pi/2$-pulses, as this does not require additional parameters when
considering the microscopic noisy gates, and can lead to a higher sensitivity
of the circuits to the parameters that long-sequence GST aims at estimating.
We thus define our gate set

$\begin{split}\mathcal{G}&=\Bigl{\\{}\rho_{0},\hskip
4.30554pt\big{\\{}G_{i}(\theta_{i},\phi_{i}),\,i={1,\cdots,5}\big{\\}},\hskip
4.30554ptM_{0}\Bigr{\\}},\end{split}$ (4)

where we have introduced the following angles

$(\theta_{i},\phi_{i})\in\left\\{(\pi,0),\left(\frac{\pi}{2},0\right),\left(\frac{\pi}{2},\frac{3\pi}{2}\right),\left(\frac{\pi}{2},\frac{\pi}{2}\right),\left(\frac{\pi}{2},\pi\right)\right\\}.$
(5)

In an experiment, the gates $G_{i}$ will deviate from the ideal unitaries (3).
One common source of noise that appears in several platforms and limits the
gate fidelities is that of phase noise, which can be modelled by a particular
stochastic process and characetrised by its experimentally-measurable power
spectral density $S(\omega)$. As detailed in Appendix B, using a dressed-state
master equation for the description of the noisy gates, one can find a
parametrization of the process matrices (2) that can be specified by only four
real parameters
$\chi^{(i)}\\!\left(\Gamma_{1}(t_{i}),\Gamma_{2}(t_{i}),\Delta_{1}(t_{i}),\Delta_{2}(t_{i})\right)$.
In turn, these parameters incorporate information of the noise and the single-
qubit driving leading to the gate, which are encapsulated in filtered
integrals of the power spectral density (PSD)

$\begin{split}\Gamma_{n}(t_{i})=&\int_{-\infty}^{\infty}\\!\\!{\text{d}\omega}\,S(\omega)\,F_{\Gamma_{n}}(\omega,\Omega_{i},t_{i}),\\\
\Delta_{n}(t_{i})=&\int_{-\infty}^{\infty}\\!\\!{\text{d}\omega}\,S(\omega)\,F_{\Delta_{n}}(\omega,\Omega_{i},t_{i}).\end{split}$
(6)

Here, the filter functions
$F_{\Gamma_{n}}(\omega,\Omega_{i},t_{i}),F_{\Delta_{n}}(\omega,\Omega_{i},t_{i})$
are explicitly given in Eqs. (40)-(LABEL:eq:noN_tarkov_filter_functions),
where we note that $t_{i}(\Omega_{i})$ stands for the pulse duration (Rabi
frequency) of each gate. It is interesting to observe that the noisy gates
$G^{\rm id}_{i}(\theta_{i},\phi_{i})\mapsto G_{i}(\theta_{i},\phi_{i})$ in Eq.
(4) with shared pulse duration $t_{i}$ and Rabi frequency $\Omega_{i}$, such
that the pulse area $\theta_{i}=\Omega_{i}\,t_{i}$ is the same, are actually
described by the same filtered noise parameters under our noise modelling and,
furthermore, these parameters do not depend on the driving phases $\phi_{i}$
in Eq. (5). The only change for different gates is that the microscopic
parameters enter in different places of the corresponding process matrices, as
detailed in Eqs. (48)-(52). Hence, no additional effort will be necessary to
characterize $G_{3}(\pi/2,3\pi/2)$, $G_{4}(\pi/2,\pi/2)$, $G_{5}(\pi/2,\pi)$
via GST under the assumption that $G_{1}(\pi,0),G_{2}(\pi/2,0)$ can be
estimated efficiently. In fact, including these operations in the gate set can
lead to an improved parameter sensitivity in long-sequence GST.

Let us note that the standard approach of GST emphasises the requirement of a
Markovian evolution during the noisy gates, and a failure in the gate set
reconstruction can sometimes be associated to a model violation due to the
breakdown of the the Markovianity assumption [38]. On the other hand, when
considering the full PSD of the phase noise (6), we are indeed incorporating
finite time correlations (32), and thus deviating from a purely Markovian
prediction base on a Lindbladian approach. Hence, the discussion about
Markovianity and GST is slightly more succinct. As detailed in [55], one can
rigorously connect the values of the filtered noise parameters
$\\{\Gamma_{n}(t_{i}),\Delta_{n}(t_{i})\\}$ to a non-Markovianity measure [77,
78, 79, 80, 81] that is based on the lack of a completely-positive division of
the resulting quantum dynamical map [82, 83]. If we include
$\Gamma_{2}(t_{i}),\Delta_{2}(t_{i})$ in the noisy gate set parametrization,
and the GST estimates that these parameters are indeed non-zero, the
reconstructed gate set corresponds strictly to a non-Markovian quantum
evolution. Hence, GST can indeed estimate non-Markovian gates, albeit assuming
that there are no time correlations in between the SPAM operations and the
different gates in the GST sequence, i.e. we assume that the noisy
parametrization of each gate is independent of the previous history of gates
applied within a given sequence. Considering that the coloured noise will have
a characteristic correlation time $\tau_{\rm c}$, we are thus assuming that
$t_{i}/\tau_{\rm c}$ can be sufficiently large so that non-Markovian effects
take place during each gate, but the time in between consecutive gates $\Delta
t_{i,j}\gg\tau_{\rm c}$, such that one can neglect correlations between
different gates.

In principle, one could exploit the formalism of the noise filtered integrals
to extend the modelling of fluctuations also in between gates, accounting for
non-Markovianity in the full gate sequence. However, the resource cost of GST
would increase considerably, as each circuit of the long-sequence GST would
then depend on the specific gate history, and thus require even more
parameters to be estimated. Some proposals of non-Markovian tomography have
already been made employing tensor networks [84, 85], always at the expense of
an even greater computational cost. We leave a detailed study of these
questions for future work.

In most of the discussion below, we will show that the leading-order effect of
the noisy gate set can be characterized by restricting to a non-zero
$\Gamma_{1}(t_{i})$ and $\Delta_{1}(t_{i})$, whereas
$\Gamma_{2}(t)=\Delta_{2}(t)=0$. It is worth noting that, under this Markovian
assumption, the total number of parameters – and minimal number of experiments
– required to completely characterize this gate set decreases from
$N_{GST}=67$ when working with the fully general description of the channels
to $N_{GST}=11$ with our parametrized approach. In the last part of Sec.
III.1, we present a generalization of our parametrized GST to allow for non-
zero $\\{\Gamma_{2}(t_{i}),\Delta_{2}(t_{i})\\}$, where we quantify how
certain noise regimes require GST to account for these non-Markovian effects
in order to achieve higher accuracies.

###  II.2 Microscopic parameters and gauge redundancy

Once the gate set and its parametrization have been described, we can discuss
the different estimation strategies for GST. As reviewed in Appendix A, the
first schemes of GST [40] considered a linear-inversion scheme (28) that only
requires measuring the probabilities of simple circuits composed of the SPAM
operations and the action of the individual gates $G_{i}\in\mathcal{G}$ (see
Fig. 1c). This approach leads to a certain redundancy in the estimated gate
set $\hat{\mathcal{G}}$ found by linear GST. As discussed in that Appendix, it
is easy to see in the super-operator formalism in which states and
measurements are vectorized
$|\hat{\rho}_{0}\rangle\\!\rangle,\langle\\!\langle\hat{M}_{0}|$, and quantum
channels have a matrix representation $\hat{G}_{i}$, that one can apply a
similarity transformation $T$ to obtain a new gate set
$\hat{\mathcal{G}}^{\prime}$ that is equally valid when one transforms
simultaneously the estimated noisy gates and the fiducial state and
measurement operators

$\begin{split}|\hat{\rho}_{0}\rangle\\!\rangle\rightarrow
T|\hat{\rho}_{0}\rangle\\!\rangle,\hskip 4.30554pt\hat{G}_{i}\rightarrow
T\hat{G}_{i}T^{-1},\hskip
4.30554pt\langle\\!\langle\hat{M}_{0}|\rightarrow\langle\\!\langle\hat{M}_{0}|T^{-1}.\end{split}$
(7)

This redundancy in linear GST is referred to as gauge freedom, and arises due
to the absence of a privileged reference frame, which is the price to pay when
transitioning from QPT to GST. Moreover, it is actually not exclusive to GST
but also appears in other characterization techniques such as randomised
benchmarking [86].

When using a parametrized gate model, this redundancy may be partially or even
completely eliminated, as the microscopic modelling can actually define a
privileged reference frame. This is precisely the situation for our Markovian
parametrization, which only depends on the filtered-noise parameters that do
not change under any similarity transformation. Under this transformation, the
matrix structure of our parametrization of the noisy gates must be preserved,
as it describes the whole family of imperfect gates subject to the coloured
phase noise, albeit with a different set of parameters
$\\{\Gamma_{1}(t_{i}),\Delta_{1}(t_{1})\\}\mapsto\\{\Gamma^{\prime}_{1}(t_{i}),\Delta^{\prime}_{1}(t_{1})\\}$.
Considering that under a similarity transformation
$\hat{G}_{i}^{\prime}=T^{-1}\hat{G}_{i}T$, both $\hat{G}_{i}$ and
$\hat{G}_{i}^{\prime}$ must possess the same spectrum, one can readily see
that our parametrization does not allow for gauge redundancy. By calculating
the eigenvalues of the corresponding noisy gates, we find that
$\sigma(G_{i}(\Gamma_{1},\Delta_{1}))\neq\sigma(G_{i}(\Gamma_{1}^{\prime},\Delta_{1}^{\prime}))$
for any physically-admissible $(\Gamma_{1}^{\prime},\Delta_{1}^{\prime})$,
where $\sigma(O)$ symbolizes the spectrum of the operator $O$. This indicates
that there is only one pair of parameters consistent with the GST
probabilities.

The absence of gauge redundancy renders unnecessary the need for gauge
optimization techniques used in fully-general GST [38]. Typically, these
techniques are required to compute distance metrics, which depend on the gauge
choice, adding further complexity to the already demanding estimation process
of GST. Typically, the gauge-fixed gate set is defined as the one that is
closest in a certain notion distance to the ideal unitary gate set (3), while
being compatible with the measured frequencies. Due to its simplicity, we will
opt for trace distance
$T(\rho,\rho^{\prime})=\textstyle\frac{1}{2}\sqrt{(\rho-\rho^{\prime})^{\dagger}(\rho-\rho^{\prime})}$
to get a measure of ‘closeness’ between a reconstructed gate set element and a
target one [1]. Note that the (in-)fidelity [87] is not a wise choice in this
context, as it is only well-behaved for normalized density matrices, and not
every element in our gate set satisfies this condition. For our
parametrization, the trace distance between two gates is

$\begin{split}T_{\hat{G},\bar{G}}\\!=\absolutevalue{\frac{{\rm
e}^{-\hat{\Gamma}_{\\!1}}-{\rm
e}^{-\bar{\Gamma}_{\\!1}}}{4}\\!}+\textstyle\frac{1}{2}\\!\biggl{(}\\!\\!{\rm
e}^{-\hat{\Gamma}_{\\!1}}+{\rm e}^{-\bar{\Gamma}_{\\!1}}-\biggr{.}\left.2{\rm
e}^{-\frac{\hat{\Gamma}_{\\!1}+\bar{\Gamma}_{\\!1}}{2}}\\!\\!\cos\\!\left(\\!\frac{\hat{\Delta}_{\\!1}-\bar{\Delta}_{\\!1}}{2}\\!\right)\\!\\!\\!\right)^{\\!\\!\\!\\!\frac{1}{2}}\\!\\!\\!,\end{split}$
(8)

which will be useful to estimate the error in our parametrised GST, where we
will compute an average trace distance for all elements in the gate set (trace
distances for the fiducial elements are introduced at the end of the next
subsection). We do not need to delve here into whether gauge fixing by
minimizing the trace distance to the unitary gates is a fully-consistent
technique or not, as one could imagine sufficiently-noisy instances in which,
by minimizing the distance to the unitary gate set along a gauge orbit, one is
actually moving further apart from the actual noisy channels that represent
the imperfect gate set. This problem is entirely absent in our case, as there
is no gauge redundancy left in our parametrization.

###  II.3  Long-sequence maximum-likelihood estimation

Let us now discuss the posterior GST developments that superseded linear GST
[40], showing that one can improve the accuracy and precision of linear GST
estimates by allowing for sequences of gates forming more complex circuits.
This leads to the concept of long-sequence GST (see Fig. 2) briefly discussed
in Appendix B. Let us introduce $\gamma_{s}$, $\gamma_{c}$, and $\gamma_{b}$
as labels of the different state preparation, base circuit and measurement
basis components of a particular circuit
$\bm{\gamma}=(\gamma_{s},\gamma_{c},\gamma_{b})$, respectively. The
probability characterizing the measurement outcomes of each of the $N_{C}$
circuits in long-sequence GST can be expressed in terms of the corresponding
product of parametrized process matrices

$p_{\bm{\gamma}}(m_{b})=\sum_{\bm{\omega}}\,\chi^{\gamma_{b}}_{\alpha,\beta}\chi^{\gamma_{c}}_{\tau,\zeta}\chi^{\gamma_{s}}_{\epsilon,\delta}\mathrm{Tr}\left\\{M_{0,m_{b}}E_{\alpha}^{\phantom{\dagger}}E_{\tau}^{\phantom{\dagger}}E_{\epsilon}^{\phantom{\dagger}}\rho_{0}E_{\delta}^{\dagger}E_{\zeta}^{\dagger}E_{\beta}^{\dagger}\right\\},$
(9)

where
$\bm{\omega}=(\alpha,\beta,\tau,\zeta,\epsilon,\delta)\in\\{1,\cdots,d^{2}\\}$
is a super-index that labels the components of the process matrices in the
Pauli basis for each part of the GST circuit. We recall that these matrices
are expressed as products of the parametrized $\chi^{(i)}$ matrices of the
individual noisy gates in Eqs. (48)-(52), each of which has a non-linear
dependence on the noise-filtered integrals (6)
$\chi^{(i)}\\!\left(\\{\Gamma_{n}(t_{i}),\Delta_{n}(t_{i})\\}_{n=1,2}\right)$.
We group all of these parameters, together with those of the fiducial
operations, in a single vector $\bm{\theta}$, which will allow us to formalise
the GST as a particular problem of multi-parameter point estimation [37].

Figure 2: Circuit design for long-sequence GST: The conventional long-sequence
GST scheme uses $N_{c}$ different possible combinations of circuits, here
labeled by $\bm{\gamma}$, that contains the state preparation $\gamma_{s}$,
base circuits $\gamma_{c}$, and measurement $\gamma_{b}$ sublabels. When
preparing IC sets of initial states $\rho_{\gamma_{s}}$ and POVM elements
$M_{\gamma_{b}}$, we rely on the action of single gates. For our
parametrization, the base circuit germs germs $G_{\gamma_{c}}$ are also single
gates, as these are a valid amplificationally complete set (see Sec II.4).
Each circuit results in $d$ possible outcomes resulting from the binary
Z-basis measurement for each qubit.

To avoid non-physical estimations in GST, one defines a cost function that
depends on the differences between these parametrized circuit probabilities
and the corresponding measured relative frequencies $f_{\bm{\gamma}}(m_{b})$,
which is then minimised under specific constraints to restrict to physically-
admissible gate sets. Following some of the ideas used in fully-general GST
[38], we integrate a negative log-likelihood cost function $\mathcal{C}_{\rm
ML}(\bm{\theta})$ with a weighted least-squares cost function
$\mathcal{C}_{\rm LS}(\bm{\theta})$ throughout the iterative process that
minimizes

$\begin{split}&\hat{\bm{\theta}}=\texttt{argmin}\hskip
4.30554pt\begin{cases}\mathcal{C}_{\rm
ML}(\bm{\theta})=-\sum\limits_{\bm{\gamma}}\sum\limits_{m_{b}}f_{\bm{\gamma}}(m_{b})\log
p_{\bm{\gamma}}(m_{b}),\\\ \mathcal{C}_{\rm
LS}(\bm{\theta})=\sum\limits_{\bm{\gamma}}\sum\limits_{m_{b}}\frac{1}{p_{\bm{\gamma}}(m_{b})}\big{(}f_{\bm{\gamma}}(m_{b})-p_{\bm{\gamma}}(m_{b})\big{)}^{2}\\!.\end{cases}\\\
\end{split}$ (10)

In our optimization routine, we choose circuits with a maximum depth that is
logarithmically spaced, and run an optimization iteration for each of them,
incorporating data from lower depths into the larger-depth subsequent steps.
Additionally, the solution obtained from each iteration serves as the initial
guess for the subsequent one, using the ideal unitary gate set as the initial
guess for the first iteration. This workflow, originally developed for fully-
general GST, prevents us primarily from getting stuck at an undesirable local
minima of the cost function, but also from increasing the potential ‘branch
ambiguity’ with depth. This problem occurs when any of the parameters under
estimation is multi-valued, such that $\theta\rightarrow\theta\hskip
4.30554pt\rm{mod}$$K$. Then, the Heisenberg-like increase of accuracy stemming
from considering deeper versions of the circuits also increases the multi-
value ambiguity, so that one has $\theta/p\rightarrow\theta/p\hskip
4.30554pt\rm{mod}$$K/p$. Luckily, the inclusion of the previously mentioned
logarithmically spaced maximum depth circuits is sufficient to discern the
‘true’ branch. Regarding the cost function, we opt for $\mathcal{C}_{\rm LS}$
during the optimization involving the initial low-depth stages, while
$\mathcal{C}_{\rm ML}$ is employed at later stages when a sufficiently
accurate estimation has already been reached at previous iterations that can
be used for the initial guess of the subsequent ones, as this can correct for
the potential biased behaviour of $\mathcal{C}_{\rm LS}$ for rare events [38].
We remark that the choice among the cost functions is rather heuristic, and
may vary depending on the structure of the problem.

So far, the definition of the optimization in Eq. (10) is not complete, as we
have not detailed yet the constraints. Working with the process matrix
representation (2), we need to impose the positive semi-definite and trace
preservation constraints in Eq. (26). These conditions manifest neatly with
our approach, as our parametrized process matrices are always trace
preserving, while the only condition required to ensure complete positivity in
the Markovian regime reduces to $\Gamma_{1}(t_{i})\geq 0$. Additionally, we
restrict $\rho_{0}$ and $M_{0}$ to be positive semi-definite, and set the
trace of the fiducial state to be unity. Consequently, the minimization (10)
must be constrained as

$\begin{split}\texttt{subject to}\hskip
4.30554pt\begin{cases}\Gamma_{1}(t_{i})\geq 0,\hskip
17.22217pti=1,...,N_{G},\\\ \sigma(\rho_{0})\in\mathbb{R}^{+},\hskip
8.61108pt\mathrm{Tr}\\{\rho_{0}\\}=1,\\\
\sigma(M_{0})\in\mathbb{R}^{+}.\end{cases}\end{split}$ (11)

From this point onward, we adopt the orthonormal Pauli basis with $N=1$ for
single-qubit systems. With this basis, the native state and measurements can
be expressed as
$\rho_{0}=\frac{1}{\sqrt{2}}\left(E_{0}+\sum_{{\alpha}=1}^{3}r_{\alpha}E_{\alpha}\right)$
and $\hskip
8.61108ptM_{0}=\frac{1}{\sqrt{2}}\left(\sum_{{\alpha}=0}^{3}e_{\alpha}E_{\alpha}\right)$
respectively, where $r_{\alpha}$, $e_{\alpha}$ are real coefficients. Then,
constraints on $\rho_{0}$ and $M_{0}$ can be further simplified to

$\begin{split}\sum\limits_{\alpha=1}^{3}r_{\alpha}^{2}&\leq 1,\\\
e_{0}\in[0,2],\hskip
8.61108pt\sum\limits_{\alpha=1}^{3}(e_{\alpha})^{2}&\leq\mathord{\rm
min}\\{(e_{0})^{2},(2-e_{0})^{2}\\}.\end{split}$ (12)

In contrast to fully-general GST [38], in which the CPTP conditions are not
imposed when working with the Pauli transfer matrix representation of a
quantum channel (see Appendix B), the former physical constraints are notably
simpler and can be readily imposed during the optimization stage independently
of the channel representation. Note that, in parallel with the expression in
Eq. (8), these parametrizations for the fiducial elements lead to the simple
trace distances
$T_{\hat{\rho}_{0},\bar{\rho}_{0}}=\sqrt{\sum_{\alpha=1}^{3}(\hat{r}_{\alpha}-\bar{r}_{\alpha})^{2}}$
and
$T_{\hat{M}_{0},\bar{M}_{0}}=\sqrt{\sum_{\alpha=1}^{3}(\hat{e}_{\alpha}-\bar{e}_{\alpha})^{2}}$.

###  II.4 Selection of parametrized base circuits

Having covered the main aspects of our parametrized approach to GST, we can
now discuss how to select a set of base circuits that permit achieving an
increased sensitivity in the estimation of the filtered noise parameters. The
efficacy of long-sequence GST hinges on the capability of amplifying each
model parameter in $\bm{\theta}$ by the repetition of base circuits $p\gg 1$
times. This enables a reduction of the estimation uncertainty
$\Delta\theta_{p}={\Delta\theta}/{p}$, where $\Delta\theta$ denotes the
uncertainty for a single repetition $p=1$. This contrasts the reduction of the
shot-noise uncertainty mentioned in the introduction, which scales with the
number of shots used to infer the relative frequencies as $1/\sqrt{N_{\rm
shots}}$. This idea, first explored in [40], has evolved into its current form
described in [38] for fully-general GST, including steps of so-called
circuit/germ selection and fiducial pair reduction that we now consider for
our parametrized GST.

In long-sequence GST, one has to carefully determine a set of germs – circuits
composed of gates from the gate set – that suffice to amplify each gate-set
parameter when repeated $p$ times. These sets, coined amplificationally
complete (AC), rarely result in a one-to-one correspondence between circuits
and parameters, as there are many circuits with specific combinations of gates
that end up amplifying common gate-set parameters. In addition, evaluating the
outcome of every of these germ circuits for all SPAM operations on the
fiducial pair of initial state and measurement $\rho_{0},M_{0}$ increases the
total number of circuits considerably. This can actually waste resources in
terms of experimental shots for certain measurements that do not give more
information about the parameters. In essence, there are redundant combinations
of base and SPAM circuits that increase the number of experiments beyond a
minimal GST design, and it is thus important to develop techniques for circuit
selection (CS) and fiducial pair reduction (FPR) to reduce this redundancies
[38]. In this way, the measurement resources can be optimally used for those
AC circuits that provide more information gain.

Interestingly, we can actually avoid this two-step CS and FPR process by
demonstrating that our parametrized gates already constitute an AC set, and
there is no need to combine them in more complicated circuits. This enables us
to effortlessly identify minimal GST designs that are optimal for single-qubit
gates. Finally, as explained before, our restricted model does not suffer from
gauge redundancy, obviating the need for additional gauge optimization
techniques to establish a common reference frame to compute distance metrics.

In fully-general GST, one identifies an AC set of germs $g_{i}=G_{i_{1}}\circ
G_{i_{2}}\circ...\circ G_{i_{N}}$ by finding non-vanishing quantities for

$\bm{\nabla}_{i}^{(\infty)}=\lim_{p\to\infty}\frac{1}{p}{\bm{\nabla}_{\bm{\theta}}{\tau^{p}(g_{i})}},\hskip
12.91663pti=1,...,N_{germs}$ (13)

where $\tau(g_{i})$ stands for the Pauli transfer matrix for the germ $g_{i}$.
The general approach to calculate this amplification gradient uses Schur’s
lemma to establish a relationship between the expression above and the
derivative when $p=1$, under the assumption that $\tau(g_{i})$ is
approximately unitary [38, 88]. This assumption remains valid for the high-
fidelity QIPs until the number of repetitions of the germs is too large, and
the accumulated error starts to dominate leading to a deviation from the
unitary behaviour. This transition typically occurs at depths that are
considerably larger than ${1}/{\epsilon}$, where $\epsilon$ is defined as the
decoherence rate per gate. In combination with experimental limitations, this
error accumulation prevents one from using arbitrarily long circuits in GST to
enhance the estimation accuracy and precision.

Fortunately, due to the simplicity of our parametrized expressions for the
noisy gates, we can evaluate the expression (13) analytically. Furthermore,
considering single-gate germs with a Markovian parametrization of the phase
noise, we find
$[G_{i}(\Gamma_{1}(t_{i}),\Delta_{1}(t_{i}))]^{p}=G_{i}(p\Gamma_{1}(t_{i}),p\Delta_{1}(t_{i}))$,
which is indeed the amplification behaviour one looks for when executing long
sequence GST. Coming back to Eq. (13), we note that each of the non-vanishing
matrix elements in this limit, denoted here by the indices $(j,k)$, satisfies

$\big{(}\bm{\nabla}_{i}^{(\infty)}\big{)}_{j,k}\approx{\rm
e}^{-p\Gamma_{1}(t_{i})},$ (14)

leading directly to two conclusions. Firstly, as anticipated by the linear
amplification of parameters with $p$, the individual gates indeed serve
already as valid germs for long-sequence GST. This becomes evident by
evaluating Eq. (14) in the high-fidelity regime ($p\Gamma_{1}(t_{i})\ll 1$).
Typically, short germs are prioritized to maximize precision when constrained
by a maximum circuit depth, so this is significant. In addition, as we are not
combining different gates to form germs, the sets of parameters that different
germs amplify are disjoint. Secondly, as we will discuss latter, we can use
this limit to predict the region in which the error dominates, corresponding
to $p\gtrsim{1}/{\Gamma_{1}(t_{i})}$, where long-sequence GST will not lead to
further improvements on the accuracy and precision of the estimation. It is
significant to mention that individual gates are no longer sufficient to
perform long sequence GST when the Markovian approximation is relaxed
($\Gamma_{2}(t_{i}),\Delta_{2}(t_{i})\neq 0$), as those parameters need more
complex combinations of gates to exhibit amplification with depth.
Nevertheless, as mentioned at the end of Subsec. II.1, we recall that there
are situations, as those explored in this paper, for which the contributions
introduced by these parameters can be neglected.

Once germs have been identified, one only needs to incorporate fiducial pairs
to determine the complete circuits that will be used in our routine.
Traditionally, in fully-general GST, all germs would be used – which might
probably be redundant – and only a reduced number of fiducial pairs would be
applied to mitigate further redundancies. Alternatively, per-germ fiducial
pair reduction (FPR) [88] is a process that selects germs to minimize
redundancies beforehand, resulting in near-minimal GST designs. However, it
requires solving an NP-complete problem, a two-step optimization involving
column subset selection problem [89], which becomes already challenging for
moderately large values of $N_{p}$. Our parametrization presents a structure
similar to that found in per-germ FPR, but avoids the column subset selection
problem. First, we associate each gate $G_{i}$ with the reduced set of
parameters it amplifies which, as we anticipated, forms a set that is disjoint
with respect to all the other gate-amplified parameter sets. Subsequently, we
perform a small variation from the unitary values of each of these parameters.
Then, for each of them, we select the fiducial pair that is more sensitive to
the specific variation given the base circuit built from $G_{i}^{p}$. We are
justified in the computation of this at each $p$ due to the simple analytic
form of $G_{i}^{p}$. This process can be repeated for every parameter to find
a one-to-one correspondence between parameters and circuits. In other words,
we find that $N_{p}$ circuits suffice to determine and amplify each of the
$N_{p}$ gate-set parameters from $\bm{\theta}$, constituting a minimal GST
design.

##  III Numerical benchmarks with stochastic noise

After discussing all the central aspects of our parametrized GST, we now
present a numerical benchmark, testing it with simulations and comparing its
performance against fully-general GST. To run our parametrized GST, we rely on
the practical open-source python software pyGSTi [90, 91], using stochastic
numerical simulations to obtain the noisy data. When adapting GST to our
parametrized formulation, we integrate a low-level implementation of GST with
our model’s specific requirements, all of which were detailed in the previous
section. We have independently programmed the GST maximum-likelihood
estimation from the start, which has served to test that the low-level
integration of the parametrization of pyGSTi is working seamlessly.
Ultimately, we rely on this open-source approach, as it offers extensive and
useful capabilities. This is specially convenient when comparing with the
fully-general GST, as the package is designed to offer high-level
functionalities to work easily within this ‘agnostic’ framework. When adapting
GST to our microscopic noise model, the benefits are partially reduced in
comparison to our original GST implementation due to the lower-level
implementation required, although these are still appreciable. The pyGSTi
implementation of the model in this work, used to obtain the main results of
it, can be accessed through the GitHub repository ColouredGST [64].

###  III.1 Parametrized GST for Ornstein-Uhlenbeck noise

To validate the accuracy of our results, we simulate the dynamics of the set
of quantum channels (4) under Ornstein-Uhlenbeck phase noise [92, 93]. This
choice is motivated by the facility with which we can tune the correlation
time $\tau_{\rm c}$ and the diffusion constant $c$ of this process, which
fully define the Langevin stochastic differential equation

$\frac{{\rm d}}{{\rm d}t}\tilde{\delta}(t)=-\frac{1}{\tau_{\rm
c}}\tilde{\delta}(t)+\sqrt{c}\tilde{\xi}(t),$ (15)

Here, the random process $\tilde{\delta}(t)$ models a coloured phase noise of
the driving (30) leading to single-qubit gates, and $\tilde{\xi}(t)$ stands
for a white noise seed of the fluctuations that ultimately leads to
differences with respect to the ideal unitaries (3). A useful feature of the
Ornstein-Uhlenbeck process is that there is a closed analytical expression for
its stochastic trajectories which, discretizing time as $t_{n}=t_{0}+n\Delta
t$, reads as follows

$\tilde{\delta}(t_{n+1})=\tilde{\delta}(t_{n}){\rm e}^{-\frac{\Delta
t}{\tau_{\rm c}}}+\sqrt{\frac{c\tau_{\rm c}}{2}\left(1-{\rm e}^{-\frac{2\Delta
t}{\tau_{\rm c}}}\right)}\tilde{u}_{n}.$ (16)

Here, we have introduced a unit normal random variable per time step
$\tilde{u}_{n}\in N(0,1)$, which must be statistically independent
$\mathbb{E}(\tilde{u}_{n},\tilde{u}_{m})=\delta_{n,m}$. One can thus simulate
the stochastic quantum dynamics of a noisy gate very efficiently.
Additionally, the stochastic process is Gaussian and can be fully described by
a simple Lorentzian PSD

$S(\omega)=\frac{c\tau_{\rm c}^{2}}{1+(\omega\tau_{\rm c})^{2}}.$ (17)

We can thus calculate the exact noise-filtered parameters
$\Gamma_{1}(t_{i}),\Delta_{1}(t_{i})$ in Eq. (6) for a specific Rabi frequency
and pulse duration, and easily calculate the trace distance between the
estimated and real noisy gates using either (8) or computing $T$ numerically.
Alternatively, we can use the random trajectories generated via Eq. (16) to
numerically integrate the stochastic Schröinger equation for the qubit under
each imperfect gate in $\mathcal{G}$, and calculate the trace distance with
respect to the GST estimates using the numerically averaged density matrices.
This method would not be limited by the inherent approximations used to
describe the noisy gates using a truncated time-convolutionless master
equation (see Appendix A). In the following, we will report on these two types
of benchmarks. For figures shown in this section, we will use empty markers
when benchmarking the first approach, while coloured markers are used for the
second approach. Let us note that in all these benchmarks, we always simulate
the GST measurements considering a finite number of shots, such that we
account for the finite-sample errors due to shot noise.

Let us first present our numerical results for the accuracy of our
parametrised GST as a function of the number of measurement shots. As already
noted above, we expect a $1/\sqrt{N_{\rm shots}}$ scaling in the precision of
the estimated parameters, which will lead to a similar scaling in the trace
distance of the estimated gate set with respect to the real microscopic one.
We have verified this expected behavior in Fig. 3, where we plot the average
trace distance versus the number of shots per circuit $N_{b,s}$. When using a
log-log scale, this relationship manifests as a linear function with slope
near $-{1}/{2}$, which is very close to the numerical fits obtained from the
numerical GST.

Figure 3: ${1}/{\sqrt{N_{\rm shots}}}$ scaling with number of measurements:
the average trace distance of the gate set is plotted as a function of the
number of shots per circuit for various maximum depths of long-sequence
circuits. Estimates for stochastically-generated data are displayed with
filled markers, whereas those obtained from the closed filtered-noise
expressions for the noisy channel based on the Lorentzian PSD (17) are
displayed with empty markers. Linear fits are carried out for each set of
data. For the shake of clearness, we do not plot the linear fits, but the
corresponding slopes are shown in the legend, confirming the expected values
around $m\simeq-0.5$. For the Ornstein-Uhlenbeck simulations, the noise
parameters used are $\tau_{\rm c}=5\cdot 10^{-6}{\rm s}$, $c=2\cdot 10^{3}{\rm
s}^{-3}$. To enhance accuracy, we average the process over $100$ GST estimates
for each data-point. In lighter colors, we plot confidence regions with
confidence level $\gamma=99\%$.

In a similar manner, we have also confirmed the expected ${1}/{p}$ scaling for
our parametrised long-sequence GST when computing the average trace distance
as a function of the maximum number of repetitions of germs utilized in the
resulting circuits. This time, the expected slopes on a log-log scale should
be around $-1$, which also agrees with our numerical results depicted in Fig.
4. Moreover, these results also enable us to confirm the
$p\gtrsim{1}/{\Gamma_{1}}$ regime in which errors accumulate considerably, and
the accuracy of the GST estimation indeed saturates. The deviation from the
${1}/{p}$ behaviour is not observed in the range of $p$ depths for the green
data, as the choice of parameters in this case yields $\Gamma_{1}^{-1}\simeq
10^{5}$. In contrasts, the saturation becomes noticeable for data in magenta
($\Gamma_{1}^{-1}\simeq 10^{4}$) and data in blue ($\Gamma_{1}^{-1}\simeq
10^{3}$), where one can see how the initial linear slope bends towards a
horizontal line when the accumulation of errors forbids getting more
information from going to even longer-depth circuits.

Figure 4: ${1}/{p}$ scaling with circuit depth: the average trace distance of
the gate set is plotted as a function of the maximum number of repetitions of
germs utilized in long-sequence circuits for various sets of noise parameters
and $N_{b,s}=10^{3}$ shots per circuit. This process is repeated for $100$ GST
estimates at each data-point and then the average values are plotted. Regions
in which errors begin to dominate are appreciated. This is in concordance with
the $p\gtrsim\frac{1}{\Gamma_{1}}$ condition anticipated in Sec. II. These
regions are reflected in the graph by the deviations that magenta and blue
data exhibit when compared to the linear function with slope $m=-1$ (dashed
grey), whereas green plot matches the tendency for the whole interval
displayed. We do not include values within these regions in the linear fits.
For the shake of clearness, we do not plot the linear fits, but the
corresponding slopes are shown in the legend. These results confirm the
expected slope values $m\simeq-1$. Parameters for the Ornstein-Uhlenbeck
simulations are $\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip
8.61108ptc=2\times 10^{4}{\rm s}^{-3}\\}$, $\\{\tau_{\rm c}=5\times
10^{-4}{\rm s},\hskip 8.61108ptc=2\times 10^{5}{\rm s}^{-3}\\}$ and
$\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip 8.61108ptc=2\times 10^{6}{\rm
s}^{-3}\\}$ for the green, magenta and blue sets of data, respectively. In
lighter colors, we plot confidence regions with confidence level
$\gamma=99\%$.

As we previously mentioned, we have focused mostly on the Markovian
parametrization of the noisy gates in which
$\Gamma_{2}(t_{i})=\Delta_{2}(t_{i})=0$, which is justified when the effects
brought up by the non-zero correlation time $\tau_{\rm c}$ are not too big. As
already noted above, our parametrized GST can account for non-Markovian
evolutions within each of the gates by simply allowing for
$\Gamma_{2}(t_{i}),\Delta_{2}(t_{i})\neq 0$, but not for possible correlations
between different gates. Qualitatively, this requires a separation of time
scales such that gates are fast enough for the finite $\tau_{\rm c}$ to play a
role, whereas the time lapse between switching off a driving for one gate and
switching it on again for the next gate is slow in comparison to $\tau_{\rm
c}$. This captures noise correlations present in a given gate, assuming that
correlations are ‘washed-out’ from one gate to the next.

We can select values of $\tau_{\rm c}$ for which these non-Markovian effects
become visible. We account for this effect by comparing the GST estimates
obtained without imposing the approximation $\Gamma_{2}=\Delta_{2}=0$. As can
be observed in Fig. 5, forcing $\Gamma_{2}=\Delta_{2}=0$ leads to an estimate
of a Markovian gate set that becomes less accurate when one reaches a regime
in which $\tau_{c}$ is large enough and non-Markovian effects start to kick
in. By allowing for $\Gamma_{2},\Delta_{2}\neq 0$, the parametrized GST can
incorporate such non-Markovian effects, leading to gate set estimates that are
more accurate. As pointed out in [55], as one keeps on increasing the
correlation time, none of the two parametrizations will be sufficient. In this
case, the condition $\tau_{c}^{3}c\ll 1$ will start to be violated, requiring
higher-order terms in the cumulant expansion that underlies the dressed-state
master equation and the filtered-noise effective channels. This explains the
worsening of both GTS accuracies observed in Fig. 5 as correlation time
increases more and more. In any case, we see that the non-Markovian estimate
is always better than the Markovian one. We also use a secondary Y-axis to
plot the non-Markovianity measure $\mathcal{N}_{\rm CP}(t=\pi)$ Eq. (47) [82,
83].

Figure 5: Parametrized GST and non-Markovianity: we simulate the non-Markovian
GST estimation by relaxing the approximation $\Gamma_{2}=\Delta_{2}=0$. We
then run GST on the numerically-simulated data with both a purely Markovian
and a non-Markovian parametrizations. The results from the non-Markovian
reconstructions are depicted in magenta, while those from the Markovian
reconstructions are shown in blue. The non-Markovian reconstructions
effectively capture the short-time correlated effects through the parameters
$\Gamma_{2}$ and $\Delta_{2}$, thus yielding more accurate tomographic
estimates in the regime where $\tau_{c}$ is larger. In contrast, the Markovian
reconstructions fail to capture these effects and exhibit a lower accuracy. In
both cases, as $\tau_{c}$ increases, the condition $\tau_{c}^{3}c\ll 1$ is
relaxed and a worsening of accuracies is observed. We use a secondary Y-axis
(green) to plot the non-Markovianity measure $\mathcal{N}_{\rm CP}$ Eq. (47)
in the same correlation times interval. For this graph, all the
reconstructions take circuits of a maximum long-sequence length $p=16$ and
$N_{b,s}=10^{5}$. The parameters for the Ornstein-Uhlenbeck simulations
employed are $\\{\tau_{\rm c}\in(5\times 10^{-7}s,\hskip 4.30554pt5\times
10^{-4}s),\hskip 8.61108ptc=1.6\times 10^{11}s^{-3}\\}$. For each data-point,
we average over $100$ GST estimations. Also, in lighter colors, we plot
confidence regions with confidence level $\gamma=99\%$.

###  III.2 Comparison with fully-general GST

Figure 6: Accuracy of GST estimates as a function of the number of circuits
included: we plot the average trace distance of the gate set as a function of
the number of circuits included for both the microscopically-parametrized
version of GST (left) and the fully general one (right). In both cases, an
improvement in accuracy is observed when the number of non-redundant circuits
considered reaches the total number of free parameters in the gate set, which
is 67 parameters for the general model, and 11 parameters for the
microscopically-parametrized one. The parameters used in these plots are
$\tau_{\rm c}=5\times 10^{-4}{\rm s}$ and $c=2\times 10^{9}{\rm s}^{-3}$.We
average the process over $100$ GST estimates for each data-point. In lighter
colors, we plot confidence regions with confidence level $\gamma=99\%$.

Once we have checked that our parametrised GST can reproduce the expected
scalings in accuracy, and that it can also account for a certain amount of
non-Markovian effects, we can finally compare our estimates with those
obtained utilizing fully-general GST. As we mentioned in Sec. II, constructing
a minimal GST design requires significantly fewer experiments when using our
GST based on a reduced set of noise-filtered parameters. This motivates the
study presented in Fig. 6, where we plot the average trace distance between
the estimated and microscopic gate set as a function of the number of circuits
$N_{\rm c}$ used in the GST estimations. In this case, we assume $p=1$ for
simplicity, and include the SPAM elements when computing the average trace
distance to account for every parameter in the gate set. This allows us to
compare the behaviour of parametrized and fully-general GST and, particularly,
how the accuracy of the estimation changes as we increase the number of base
circuits used in GST.

When the number of circuits is not sufficient to reach IC, Fig. 6 shows that
the GST estimation does not work, as the trace distance remains close to that
of the initial unitary guess and the actual noisy channels. Since the GST
sensitivity to the parameters is very low in this regime of a low-number of
circuits, the estimation cannot learn the noisy parameters correctly. As we
increase the number of circuits and IC is attained, an abrupt drop in the
average trace distance to the actual gate set is observed, signalling that the
GST is now accurate. This can be attributed to the fact that every parameter
in the gate set is now being optimized when performing the gradient descent of
the maximum-likelihood estimation.

We note that the number of circuits at which this drop occurs coincides, for
our parametrized model, with the number of free parameters of the noisy gate
set $N_{GST}=11$ (left panel of Fig. 6). This close agreement is a result of
the circuit selection procedure described in subsection II.4, which enables us
to order circuits according to their sensitivity, and place the first 11 non-
redundant elements at the beginning. In contrast, for fully-general GST (right
panel of Fig. 6), we observe that the drop in trace distance is only reached
further along the circuit axis, requiring thus a larger number of circuits.
This should be unsurprising, as the fully-general GST requires estimating
$N_{GST}=67$ parameters. In contrast to our parametrized GST, where the
circuit importance sampling is direct, we rely on the pyGSTi algorithms for
fully-general GST to find the minimal set of circuits sufficient to achieve
information completeness, according to which $72$ different experiments are
needed. This number coincides exactly with the point in which trace distance
drops in the right panel of Fig. 6.

To give estimates of the total measurement resources required for GST, and
compare our parametrized approach to the fully-general one, we finally analyze
the average trace distance of the gate set as a function of the total number
of shots $N_{\rm shots}$. As expected, our parametrized GST provides a
significant reduction in the required number of experiments to achieve a
target accuracy in comparison to fully-general GST (see Fig. 7). We observe
that the accuracy of our parametrized GST outperforms that of fully-general
GST by more than an order of magnitude when considering the same total number
of shots $N_{\rm shots}$. This can be seen when comparing the green data
(parametrized GST) with the magenta data (fully-general GST) in Fig. 7. Note
that in this figure we are fixing the same maximum depth for the circuits in
the long sequence scheme with both models. Thus, the total number of shots
$N_{\rm shots}$ grows by increasing the number of shots per circuit $N_{\rm
b,s}$ employed to compute each frequency $f_{\mu,s}$.

Figure 7: Comparison with fully general GST: we present the average trace
distance of the gate set as a function of the total number of shots used
$N_{\rm shots}$. In the plot, our microscopically-motivated implementation is
depicted in green with diamond markers, while fully general GST
implementations is depicted in magenta. It is notable that our parametrized
GST achieves a better estimation accuracy even with a significantly reduced
total number of circuits required. As expected, accuracy scales as
$1/\sqrt{N_{\rm shots}}$ in both cases, as evident from the linear fit slopes.
The estimate obtained for $N_{\rm shots}\simeq 10^{3}$ is only displayed for
our parametrized approach, as the number of shots is insufficient to perform
GST with the fully-general one . Each plot is generated using $p=16$ and the
simulation parameters $\\{\tau_{\rm c}=5\times 10^{-6}{\rm s},\hskip
8.61108ptc=2\times 10^{3}{\rm s}^{-3}\\}$. We also average the results over
$100$ iterations to limit the variance and we plot confidence regions with
confidence level $\gamma=99\%$ in lighter colors.

Let us now discuss how the accuracy of the estimation presented in Fig. 7 can
be further enhanced. Similar to that figure, Fig. 8 illustrates the average
trace distance as a function of $N_{\rm shots}$. However, in contrast to Fig.
7, we now more wisely choose to increase $N_{\rm shots}$ by gradually
incorporating deeper circuits in the GST scheme while maintaining the same
number of shots $N_{b,s}$. This accelerates the improvement in accuracy with
the number of shots, as already discussed and evident from Figs. 3 and 4. This
aligns with the logic behind the utility of FPR. The fact that one is able to
reduce the number of circuits needed to achieve an AC set enables employing
the excess of resources not to increase $N_{b,s}$ (which would generally yield
similar results to those without FPR), but to extend circuits further in
depth. Consequently, with this expanded range of depths, FPR provides superior
results at a constant number of shots. This can be observed when comparing the
magenta (no FPR) and the yellow (FPR) curves in Fig. 8. Likewise, our
microscopic parametrization does also reduce the number of circuits needed to
achieve an AC set. Therefore, akin to FPR, we can allocate the resources more
efficiently among deeper circuits. The corresponding plot in Fig. 8 is
displayed in green. As can be concluded from this figure, the accuracy
enhancement obtained with our parametrized approach is great when proceeding
in this way.

Figure 8: Comparison including fiducial pair reduction in fully general GST:
we present the average trace distance of the gate set as a function of the
total number of shots used $N_{\rm shots}$. In the plot, our microscopically-
parametrized GST is depicted in green with diamond markers, while fully
general GST implementations are shown in other colors. Specifically, the
yellow and magenta plots represent general GST estimations with and without
FPR, respectively. Contrary to Fig. 7, we fix the number of shots per circuit,
in this case is $N_{b,s}=10^{3}$. The total number of shots $N_{\rm Shots}$
varies by considering increasingly deeper circuits, ranging in the interval
$p_{g}\in\\{2^{i}\\}_{g=0}^{9}$. It is notable that our parametrized approach
achieves much better estimation accuracies at a fixed $N_{\rm Shots}$.
Additionally, it is worth mentioning that the improved resource allocation
stemming from FPR does also improve the estimations, as evident from the
yellow plot. Each plot is generated using the simulation parameters
$\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip 8.61108ptc=2\times 10^{4}{\rm
s}^{-3}\\}$. We also average the results over $100$ iterations to limit the
variance and plot confidence regions with confidence level $\gamma=99\%$ in
lighter colors.

###  III.3 Inclusion of amplitude noise fluctuations

As extensively detailed in previous sections of this manuscript, our work
primarily aims at describing processors subjected to phase noise using the
filter-function formalism. Nonetheless, this model can be further extended to
additionally account for amplitude or ‘intensity’ fluctuations of the drive.
As pointed out in Appendix B, these additional fluctuations are similarly
modelled by an stochastic process $\delta\tilde{\Omega}(t)$, which describes
the variations of the Rabi frequency induced by intensity noise [94].
Equivalently to the dressed-state master equation description of the pure
phase noise evolution, amplitude noise leads to a supplementary filtered
integral analogous to those in Eq. (6)

$\Delta\Gamma_{1}(t)=\int_{-\infty}^{\infty}\\!\\!\\!{\rm
d}\omega{S}_{\Omega}(\omega)F_{\Omega}(\omega,t),$ (18)

where we have introduced an additional PSD, ${S}_{\Omega}$, which exclusively
describes this process. The new filter function $F_{\Omega}(\omega,t)$ is
explicitly given in Eq.( 55).

As can be anticipated from Eq. (18), the parameter $\Delta\Gamma_{1}(t)$
neither depends on the driving phases $\phi_{i}$. Thus, the fact that gates
with shared pulse duration are described by the same set of parameters
prevails. Consistently with our previous analysis, we employ Fig. 9 as a
confirmation that depth-one germs still define an AC set. For this figure, as
well as in Fig. 10, we simulate data using two independent stochastic
processes under OU niose, i.e. they do not exhibit cross correlations (see
Apendix B). As usual, we take an average over a sufficiently large number of
random trajectories. It is noteworthy that we do not observe an emerging gauge
degree of freedom induced by the parameter $\Delta\Gamma_{1}(t)$, therefore,
the need of gauge optimization remains absent with this extended model. Using
this knowledge and taking advantage of pyGSTi adaptability, we can
effortlessly obtain a new figure equivalent to Fig. 8 which does also include
amplitude noise fluctuations. As can be expected, the inclusion of two
additional parameters to fully characterize the gate set under phase and
amplitude noise does not significantly alter the behaviour of the results (see
Fig. 10). As a consequence, the discussion concerning Fig. 8 remains accurate.

Figure 9: ${1}/{p}$ scaling with circuit depth for phase and amplitude
fluctuations: the average trace distance of the gate set is plotted as a
function of the maximum number of repetitions of germs utilized in long-
sequence circuits for various sets of noise parameters and $N_{b,s}=10^{3}$
shots per circuit. This process is repeated for $100$ GST estimates at each
data-point and then the average values are plotted. Regions in which errors
begin to dominate are appreciated. This is in concordance with the
$p\gtrsim\frac{1}{\Gamma_{1}}$ condition anticipated in Sec. II under the
substitution $\Gamma_{1}\to\Gamma_{1}+\Delta\Gamma_{1}$. These regions are
reflected in the graph by the deviations that magenta and blue data exhibit
when compared to the linear function with slope $m=-1$ (dashed grey), whereas
green plot matches the tendency for the whole interval displayed. We do not
include values within these regions in the linear fits, neither the first
data-point, which regularly encounters a local minima. For the shake of
clearness, we do not plot the linear fits, but the corresponding slopes are
shown in the legend. These results confirm the expected slope values
$m\simeq-1$. Parameters for the Ornstein-Uhlenbeck simulations are
$\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip 8.61108ptc=2\times 10^{4}{\rm
s}^{-3}\\}$, $\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip 8.61108ptc=2\times
10^{5}{\rm s}^{-3}\\}$ and $\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip
8.61108ptc=2\times 10^{7}{\rm s}^{-3}\\}$ for the amplitude fluctuations in
the green, magenta and blue sets of data, respectively. For every simulation,
we employ $\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip 8.61108ptc=2\times
10^{4}{\rm s}^{-3}\\}$ in the phase noise stochastic processes. We recall that
phase and amplitude stochastic fluctuations are not cross-correlated. In
lighter colors, we plot confidence regions with confidence level
$\gamma=99\%$. Figure 10: Comparison including fiducial pair reduction in
fully general GST and amplitude fluctuations: we present the average trace
distance of the gate set as a function of the total number of shots used
$N_{\rm shots}$. In the plot, our microscopically-parametrized GST is depicted
in green with diamond markers, while fully general GST implementations are
shown in other colors. Specifically, the yellow and magenta plots represent
general GST estimations with and without FPR, respectively. Contrary to Fig.
7, we fix the number of shots per circuit, in this case is $N_{b,s}=10^{3}$.
The total number of shots $N_{\rm Shots}$ varies by considering increasingly
deeper circuits, ranging in the interval $p_{g}\in\\{2^{i}\\}_{g=0}^{9}$. It
is notable that our parametrized approach achieves much better estimation
accuracies at a fixed $N_{\rm Shots}$. Additionally, it is worth mentioning
that the improved resource allocation stemming from FPR does also improve the
estimations, as evident from the yellow plot. Each plot is generated using the
simulation parameters $\\{\tau_{\rm c}=5\times 10^{-4}{\rm s},\hskip
8.61108ptc=2\times 10^{4}{\rm s}^{-3}\\}$ for both phase and amplitude
fluctuations. We recall that these two stochastic processes are not cross-
correlated. We also average the results over $100$ iterations to limit the
variance and plot confidence regions with confidence level $\gamma=99\%$ in
lighter colors.

##  IV Conclusions and outlook

In this work, we have presented a microscopically-motivated GST, which builds
on the quantum dynamical map of single-qubit gates subject to phase noise [55]
to find a compact parametrization of the full gate set in terms of filtered
integrals of the noise spectral density. We have generalised this work to
parametrise the full gate set, and discussed central differences with respect
to the standard fully-general GST, such as the possibility of using single
gates as base circuits to achieve amplification completeness. In addition, we
showed that germ selection, fiducial pair reduction, and thus the achievement
of minimal GST designs can be highly simplified when using this microscopic
parametrization. We also highlighted how our approach does not suffer from
gauge redundancy, saving computational resources as no gauge-fixing is
required, and avoiding reference frame issues when benchmarking the results.
We have presented a thorough numerical validation of these arguments
considering a specific stochastic model of coloured phase noise, and
numerically obtaining maximum-likelihood estimates based on long-sequence GST
with numerically-generated data that also includes the effects of shot noise.
We have shown that the accuracy of our parametrized long-sequence GST follows
the expected scalings with the number of shots and the circuit depth, and
demonstrated that our parametrization can account for non-Markovian dynamics
during the gates. We have finally showcased the advantages of our parametrized
GST with respect to fully-general GST, presenting examples in which a tenfold
reduction in the average trace distance can be achieved considering the same
number of total measurement shots and circuit depth. We also showed how this
reduction can be increased if one allocates the remaining resources in
experiments with deeper circuits. Additionally, we showed that these results
remain accurate under an extension of the model to include amplitude noise
fluctuations.

As an outlook, it would be interesting if future studies assessed if these
benefits remain valid when extending to two-qubit gates and crosstalk in an
enlarged qubit register, as required for realizing the full potential of
quantum computation. On the one hand, the fact that single gates themselves
form an amplificationally complete set is something dependent on the specific
matrix structure of our parametrized model. An equivalent treatment for
entangling gates should be developed to check whether more complex germs are
needed for amplification completeness in that case. In order to do that, an
extension of the effective quantum dynamical map to model noisy two-qubit
gates will be required. Once this is made, we believe that fiducial pair
reduction would not longer yield minimal but near-minimal GST designs. We note
here that adding entangling gates will still generate disjoint sets of per-
germ amplified parameters if the two-qubit gates do also serve as germs. This
would facilitate finding good choices for fiducial pairs, but the $d-1$
independent outcomes of each circuit would still lead to some overhead if each
parameter is linked to a single experimental setting, resulting in a more
complex combinatorial problem. On the other hand, it could also be the case
that some of the new parameters describing the entangling gates are gauge
parameters, while the single-qubit gate parameters would remain non-gauge
ones. Even in the worst-case scenario in which none of the previous features
persists in the two-qubit case, the considerable reduction in the total number
of parameters still makes it worth to explore this research avenue.
Considering entangling gates in a two-qubit system and also cross-talk in the
previous single-qubit gates, the number of parameters needed to characterize
this gate set with fully-general GST grows to $N_{GST}=5103$. On the contrary,
even with no further modelling of the entangling gate, using the present
physically-motivated parametrization yields $N_{GST}=423$ or $N_{GST}=503$ for
the Markovian and the non-Markovian estimations respectively, which will be
further reduced by a detailed modelling of the noisy entangling gate.

Another interesting question for future research will involve validating our
approach with real experimental data, assessing the extent to which our
analytical model accurately describes physical quantum devices. The use of
this model for trapped-ion QIPs is justified in [55] by the fidelity agreement
when comparing tomographic results with analytical estimations, but the model
can be ultimately extended to include other sources of error, such as
intensity noise and spontaneous decay. Once a description of the faulty
processes of a device is obtained, it can be further tested using techniques
like probabilistic error cancellation (PEC) [95], which allows to obtain ideal
expectation values in terms of the accessible faulty ones.

Finally, we also note that it would be interesting to extend the description
of non-Markovian effects briefly introduced in the final part of subsection
III.1 beyond the individual gates, such that non-Markovian correlations among
the different gates that form the base circuits can be accounted for, which
would lead to a fully non-Markovian generalization of GST.

###### Acknowledgements.

The authors warmly thank J.M. Sánchez Velázquez for his enlightening
discussions and assistance on countless occasions. Additionally, both P.V. and
A.B. thank Ch. D. Marciniak and Claire L. Edmunts for discussions during the
development of this work. Finally, P.V. would like to thank Stefan Seritan and
the team behind pyGSTi for their technical support on the use of the GST
python package. The project leading to this publication has received funding
from the US Army Research Office through Grant No. W911NF-21-1-0007. A.B
acknowledges support from PID2021-127726NB- I00 (MCIU/AEI/FEDER, UE), from the
Grant IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S, funded by
MCIN/AEI/10.13039/501100011033, from the CSIC Research Platform on Quantum
Technologies PTI-001, and from the European Union’s Horizon Europe research
and innovation programme under grant agreement No 101114305 (“MILLENION-SGA1”
EU Project). Views and opinions expressed are, however, those of the author(s)
only and do not necessarily reflect those of the European Union or the
European Commission. Neither the European Union nor the granting authority can
be held responsible for them.

##  Appendix A  Quantum tomography and GST

Quantum Tomography techniques are grounded in the ability to infer a
representation of a quantum state, a quantum channel, or a set of them from
experimental data. In this Appendix, for the sake of completeness, we review
standard tomographic approaches, which serves to set our notation.

When the aim is to estimate a quantum state $\rho\in\mathsf{D}(\mathcal{H})$,
a set of predetermined positive operator valued measure (POVM) elements
$\\{M_{\mu}:\,\mu\in\mathbb{M}\\}$ must be specified. This set must satisfy
$M_{\mu}\in\mathsf{Pos}(\mathcal{H})$ and $\sum_{\mu}M_{\mu}=\mathbb{1}_{2}$,
and should also form a basis of the $d$-dimensional Hilbert space sufficient
to uniquely identify the state of interest. The POVM meeting this criteria is
said to define an informationally complete (IC) set, and should comprise at
least $d^{2}$ elements, being overcomplete when more than $d^{2}$ elements are
included in the set. In this case, the state can be estimated from the
measurement probabilities

$p_{\mu}=\mathrm{Tr}\left\\{M_{\mu}\rho\right\\}.$ (19)

For the ongoing analysis, it will prove convenient to define the super-
operator formalism [36], in which a density matrix and a POVM element
$\rho,M_{\mu}$ are written as vectors
$|\rho\rangle\\!\rangle,|M_{\mu}\rangle\\!\rangle$ whose elements are the
coefficients associated to its decomposition in a Hilbert space basis. Quantum
channels appear as matrices acting on these vectors, such that
$\mathrm{Tr}\left\\{M_{\mu}\mathcal{E}(\rho)\right\\}=\langle\\!\langle
M_{\mu}|G_{\mathcal{E}}|\rho\rangle\\!\rangle$, where $G_{\mathcal{E}}$
denotes the super-operator corresponding to $\mathcal{E}$. When the Pauli
basis is employed, this is commonly referred to as the Pauli transfer matrix
(PTM) representation [39], in which the elements of $G_{\mathcal{E}}$ are
expressed as

$[G_{\mathcal{E}}]_{\alpha,\beta}=\mathrm{Tr}\left\\{E_{\alpha}\mathcal{E}(E_{\beta})\right\\}.$
(20)

Returning to quantum state tomography (QST), using the experimental
frequencies as an estimate of the probabilities $f_{\mu}\simeq p_{\mu}$, Eq.
(19) trivially leads to the expression

$|f\rangle\\!\rangle=\begin{pmatrix}f_{1},f_{2},\cdots,f_{d^{2}}\end{pmatrix}^{\\!T}=A|\rho\rangle\\!\rangle,$
(21)

where the matrix $A$ is built using the vectorized POVM elements as follows

$A=\begin{pmatrix}|M_{1}\rangle\\!\rangle,|M_{2}\rangle\\!\rangle,\cdots,|M_{d^{2}}\rangle\\!\rangle\end{pmatrix}^{\\!T}.$
(22)

It is then straightforward to invert Eq. (21) to obtain the estimated state
$\hat{\rho}$, namely

$|\hat{\rho}\rangle\\!\rangle=(A^{T}A)^{-1}A^{T}|f\rangle\\!\rangle,$ (23)

where we have assumed that $A$ is not generally a square matrix. A schematic
representation of QST is shown in Fig. 1a at the introduction.

When the objective is to reconstruct a quantum process $\mathcal{E}$ rather
than a state, an IC set of predetermined initial states
$\\{\rho_{s},s\in\mathbb{S}\\}$ has to be specified in addition to
$\\{M_{\mu},\mu\in\mathbb{M}\\}$. This tomographic technique is known as
quantum process tomography (QPT), and the measurements required are now
obtained from the action of $\mathcal{E}$ upon each initial state. These
‘sandwiches’ $p_{\mu s}=\langle\\!\langle
M_{\mu}|G_{\mathcal{E}}|\rho_{s}\rangle\\!\rangle$ can be visualized in
Fig.1b. Once the experimental frequencies $f_{\mu s}$ have been obtained, one
can use linear inversion to get the super-operator estimate

$\hat{G}_{\mathcal{E}}=(A^{T}A)^{-1}A^{T}FB^{T}(BB^{T})^{-1},$ (24)

where $(F)_{\mu,s}=f_{\mu s}$ and
$B=(|\rho_{1}\rangle\\!\rangle,|\rho_{2}\rangle\\!\rangle,\cdots,|\rho_{d^{2}}\rangle\\!\rangle)$.

In practice, owing to finite-sample shot noise and other experimental
imperfections, linear inversion can lead to estimated processes that are not
physical
$\hat{G}_{\mathcal{E}}\mapsto\hat{\mathcal{E}}\notin\mathsf{C}(\mathcal{H})$,
in the sense that they do not meet the complete-positivity and trace
preservation (CPTP) constraints. This motivates the use of maximum likelihood
estimation (MLE), which allows for further constraints on the process. This
method relies on the observation that, with proper normalization of the
probabilities, the multinomial distribution

$\Lambda(\mathcal{E})=\prod\limits_{\mu,s}\left(p_{\mu
s}(\mathcal{E})\right)^{f_{\mu s}},$ (25)

maximizes precisely when $p_{\mu s}(\mathcal{E})=f_{\mu s}$, where $p_{\mu
s}(\mathcal{E})$ depend on the parametrized channel. Hence, by solving for the
maximum likelihood, one obtains an estimate $\hat{\mathcal{E}}$ that is both
physical and shows the highest likelihood to reproduce the observations.
Alternatively, one can define a cost function via the negative log-likelihood
$\mathcal{L}=-\log\Lambda(\mathcal{E})$, and minimize it
$\hat{\mathcal{E}}=\texttt{argmin}\\{\mathcal{L}\\}$ subject to the
physically-admissible constrains through which CPTP is enforced. It is also
common to use a least-squares cost function $\mathcal{C}$ as a proxy of
$\mathcal{L}$, albeit behaving as a biased estimator for rare events [38]. In
the PTM representation $\hat{\mathcal{E}}\mapsto\hat{G}_{\mathcal{E}}$, TP
translates into $[\hat{G}_{\mathcal{E}}]_{1,j}=\delta_{1,j}$, such that only
the diagonal element of the first row differs from zero. Unfortunately, CP
does not have a direct effect in this PTM representation. On the other hand,
as noted in Sec. II of the main text, both CP and TP are readily imposed using
the process matrix representation. If $\chi$ denotes the process matrix of the
quantum channel of interest (2), CPTP constraints can be expressed as a
positive semidefinite condition and a so-called [96] trace constraint

$\chi\in\mathsf{Pos}(\mathbb{C}^{d^{2}}),\hskip
4.30554pt\sum_{\alpha,\beta}\chi_{\alpha,\beta}E_{\beta}^{\dagger}E_{\alpha}^{\phantom{\dagger}}=\mathbb{1}_{d^{2}}.$
(26)

As outlined during the introduction, QPT encounters self-consistency issues,
particularly when specifying the IC sets. These sets are typically obtained by
applying faulty operations to a fiducial state and measurement which, although
can be prepared reliably in the experiment, are also subject to imperfections.
It is often the case that the gate one aims at characterizing is also needed
to prepare the IC set, leading to an inherent inconsistency. Addressing this
challenge was the catalyst for gate set tomography (GST), which we now
outline.

GST aims to go beyond the characterization of an individual quantum channel,
targeting a full gate set $\mathcal{G}$ that is composed of different
elements. These elements are generally referred to as gates, and should enable
for the construction of IC sets at SPAM stages, achieved by acting on fiducial
operators with either a set of imperfect individual gates
$\\{G_{i}:\,i\in\mathbb{G}\\}$ or combinations thereof, here denoted by
$\\{H_{s},H_{b}:\,s\in\mathbb{S},b\in\mathbb{M}_{b}\\}$. In addition to these
imperfect SPAM gates, GST also includes the fiducial initial state(s) and
projective measurement(s) in the gate set $\mathcal{G}$. To keep notation
simple, we assume there is only one of each of these native fiducial elements
$\rho_{0},M_{0}$, which holds for most experimental setups such as $N=1$ where
$\rho_{0}\approx\ket{0}\\!\bra{0}$, and $M_{0}\approx(1+\sigma_{z})/2$. We
note that there are direct extensions of this situation that include more
fiducial elements [38]. We also remark that assuming that $\rho_{0}$ and
$M_{0}$ can be repeatedly accessed in an experiment is by no means equivalent
to having prior information about their representation, which could actually
be any $\rho_{0}\in\mathsf{D}(\mathcal{H}),M_{0}\in\mathsf{C}(\mathcal{H})$.
In fact, GST provides a representation not only for each gate but also for the
native elements $\rho_{0}$ and $M_{0}$, which leads to the gate set
$\mathcal{G}$ in Eq. (1).

Rather than focusing on a single process as in QPT (25), the measured
probabilities in GST account for the action of the three consecutive circuit
parts of Fig. 1c, such that

$p_{b,m_{b},i,s}=\mathrm{Tr}\left\\{M_{z,m_{b}}\mathcal{E}_{b}\circ\mathcal{E}_{i}\circ\mathcal{E}_{s}(\rho_{0})\\!\right\\}=\langle\\!\langle
M_{z,m_{b}}|H_{b}G_{i}H_{s}|\rho_{0}\rangle\\!\rangle,$ (27)

where we have defined the imperfect binary fiducial POVM elements as
$M_{z,+}=M_{0}$ and $M_{z,-}=\mathbb{1}_{2}-M_{0}$. When using the process
matrix representation, we keep the symbol $\mathcal{E}_{i}$ for the quantum
channel denoting that gate, and introduce $\mathcal{E}_{s}$ and
$\mathcal{E}_{b}$ for the imperfect operations that would transform the
fiducial operators into the IC set in the ideal error-free situation.

Paralleling the discussion of QST and QPT, GST can proceed by either inversion
or by constrained optimization. When opting for the former, one obtains the
linear GST estimates

$\begin{split}|\hat{\rho}_{0}\rangle\\!\rangle=Tg|R_{0}\rangle\\!\rangle,\hskip
6.45831pt\hat{G}_{i}=Tg^{-1}F_{i}T^{-1},\hskip
6.45831pt\langle\\!\langle\hat{M}_{0}|=\langle\\!\langle
Q_{0}|T^{-1}.\end{split}$ (28)

Here, we have introduced the so-called Gram matrix
$[g]_{\mu,s}=\langle\\!\langle
M_{\mu}|\rho_{s}\rangle\\!\rangle=\langle\\!\langle
M_{0}|G_{\mu}G_{s}|\rho_{0}\rangle\\!\rangle$ that contains information about
the imperfect SPAM. Similarly, $[R_{0}]_{\mu}=\langle\\!\langle
M_{\mu}|\rho_{0}\rangle\\!\rangle$ and $[Q_{0}]_{s}=\langle\\!\langle
M_{0}|\rho_{s}\rangle\\!\rangle$ contain information about the imperfect
measurement and imperfect state preparation, respectively. In the above
expressions, the matrices $[F_{i}]_{\mu,s}=f_{\mu is}$ contain the measured
relative frequencies for each element of the gate set. A characteristic
feature of GST is the appearance of the matrix $T$, which can actually
correspond to any invertible matrix, and cannot be fixed by looking solely at
the measured relative frequencies. This means that GST estimates a gate set up
to a similarity transformation $T$, which acts simultaneously on the initial
state, gate sets, and POVM elements according to the equation above [38]. This
ambiguity, also referred to as gauge redundancy in the literature, makes
experimental data compatible with (infinitely) many descriptions of the gate
set, and is not exclusive of GST [86]. Just as occurs for gauge theories
describing the most fundamental interactions of nature, there is a redundancy
in our description of the GST experiments involving the sequential stages of
faulty initialization, application of imperfect circuits, and faulty
measurements, which will always lead to observed quantities that are
independent of the “gauge”, i.e. the choice of the $T$ matrix.

Let us note that the parallelism is not perfect in the sense that the above
similarity transformation cannot happen ‘locally’ at a specific location of
the circuit, but must instead always occur globally. Additionally, there is no
notion of gauge fields as the fundamental degrees of redundancy and, more
importantly, this redundancy is not built on what is believed to be a
fundamental symmetry of nature, but rests instead in the type of GST
experiments that one performs. There can be other completely different
experiments where this redundancy does not appear, which ultimately rest on
our microscopic theories of light-matter interaction that underlie the
implementation of the gates in the QIP device, but actually go much beyond the
computational description of this device as a collection of qubits that can be
subject to unitaries, or faulty versions thereof. The fact that we do not have
access to the ‘true’ set of gates that is physically implemented in the
experiment, but rather to any gauge-equivalent set that results from the GST
estimation is not problematic in general. However, it makes the computation of
distance metrics to a target gate set uncertain, as those metrics typically
depend on the gauge choice. To circumvent this issue, gauge optimization
techniques are typically applied [38], establishing a privileged reference
frame which allows to fix the similarity transformation $T$.

Similarly to our discussion of linear-inversion QPT (24), non-physical
estimates $\hat{\mathcal{G}}$ of the full gate set $\mathcal{G}$ can also
result from the linear GST (28). Consequently, it is again advantageous to
employ MLE techniques, albeit bearing in mind that one will have to face
additional complications. First of all, the non-linearity of the probabilities
on the estimation parameters (27) makes the optimization problem no longer
convex. Coupled with the above gauge redundancy, this conducts to the
appearance of local and degenerate minima in the landscape of the cost
function, which complicates considerably the task of finding accurate
estimates, and underscores the importance of selecting a sufficiently-accurate
initial guess of the gate set. Secondly, and perhaps more importantly in the
long run, GST scales exponentially in the number of qubits. It is important to
recall that a quantum channel possesses $N_{p}=2^{2N}(2^{2N}-1)$ real
parameters, being $N$ the number of qubits of the processor. This parameter
count is in addition to those used to describe the native SPAM elements
$N_{SPAM}$ and is further scaled by the number of gates that one aims to
characterize, denoted as $N_{G}$. The total number of parameters is thus
$N_{GST}=N_{SPAM}+N_{G}\times 4^{N}(4^{N}-1)$, which grows very fast and
becomes intractable even for small qubit numbers.

Finally, it is noteworthy that the precision with which a gate set is
estimated can be enhanced even with a limited number of experiments [52].
Long-sequence gate set tomography (GST) addresses this by distributing shots
across deeper circuits, which are constructed by repeating certain gates
sandwiched between the SPAM elements $p$ times

$p_{b,m_{b},i,s}=\langle\\!\langle
M_{z,m_{b}}|H_{b}(G_{i})^{p}H_{s}|\rho_{0}\rangle\\!\rangle.$ (29)

This can result in an amplification of the sensitivity of the measurements to
a certain subset of parameters of the gate set, which underlies the reduction
of the estimate error by a factor of $1/p$. In practice, individual gates are
usually insufficient to amplify errors and achieving this effect [52, 88], so
combinations of them should be employed instead, which are referred to as
germs in the GST parlance. The expression of the GST probabilities (27) in
terms of the corresponding process matrices is shown in Equation (9) of the
main text, where we allow for circuits composed of several gates drawn from
the gate set for long-sequence GST. We refer to Sec. II for a detailed
discussion of the main long-sequence GST aspects, as well as to reference
[38]. Interesting problems in this regard include germ selection and fiducial
pair reduction.

##  Appendix B Coloured phase noise for single-qubit gates

In this Appendix, we describe the central aspects of the phase-noise
parametrization of noisy gates employed in the main text, which builds on the
recent results for the quantum dynamical map of trapped-ion gates presented in
[55]. In order to parametrize the full gate set in Eq. (4), we need to
generalise this quantum dynamical map to other driving phases, allowing for
the SPAM operations required for IC.

Following [55], we start by modelling phase noise by a stochastic process
$\tilde{\delta}(t)$. For single-qubit gates, the microscopic rotating-frame
Hamiltonian for a generic phase reads

$\begin{split}\tilde{H}(t)&=\frac{\Omega}{2}\big{(}\cos\phi\sigma_{x}-\sin\phi\sigma_{y}\big{)}+\frac{\tilde{\delta}(t)}{2}\sigma_{z},\end{split}$
(30)

where $\Omega$ $(\phi)$ is the Rabi frequency (phase) of the drive, and

$\tilde{\delta}(t)=\delta\tilde{\omega}(t)+\frac{{\rm d}}{{\rm
d}t}\delta\tilde{\phi}(t)$ (31)

includes the frequency (phase) fluctuations $\delta\tilde{\omega}(t)$
$(\delta\tilde{\phi}(t))$ described by a specific stochastic process. As a
result, the quantum states become stochastic
$\tilde{\rho}(t)=\ket{\tilde{\psi}(t)}\\!\\!\bra{\tilde{\psi}(t)}$, and its
Liouville–von Neumann equation which corresponds to a system of stochastic
differential equations with multiplicative noise. One can formally derive the
equations of motion for the statistical average
${\rho}(t)=\mathbb{E}[\tilde{\rho}(t)]$ by using the Nakajima-Zwanzig approach
[97, 98, 99] of projection operators. As discussed in [55], to get a tractable
and useful expansion for high-fidelity gates, one utilizes the so-called time-
convolutionless methods [100, 101] in the instantaneous or dressed-state basis
[102, 62, 103]. By truncating at second order, and setting $\phi=0$, one
arrives at a simple time-local master equation that can be expressed in terms
of integrals of the covariance

$C(t,t^{\prime})=\mathbb{E}[\tilde{\delta}(t)\tilde{\delta}(t^{\prime})],$
(32)

where we assume that $\mathbb{E}[\tilde{\delta}(t)]=0$. In the dressed-state
basis $\ket{\pm}=(\ket{0}\pm\ket{1})/\sqrt{2}$, the evolution for the
populations reads

$\rho_{\pm\pm}(t)=\textstyle\frac{1}{2}\pm\textstyle\frac{1}{2}\big{(}2\rho_{++}(0)-1\big{)}{\rm
e}^{-\Gamma_{1}(t)}\\!,$ (33)

where we have introduced the following decay function

$\Gamma_{1}(t)=\\!\\!\int_{0}^{t}\\!\\!\text{d}t^{\prime}\,\gamma_{1}(t^{\prime}),\hskip
4.30554pt\gamma_{1}(t^{\prime})=-\\!\\!\int_{0}^{t^{\prime}}\\!\\!\\!\text{d}t^{\prime\prime}C(t^{\prime}-t^{\prime\prime})\cos(\Omega(t^{\prime}-t^{\prime\prime})).$
(34)

This is precisely one of the four parameters which we aim to estimate under
GST (see Eq. 6 of the main text). The remaining parameters stem from the
evolution of the dressed-state coherences of the density matrix [55]. This
evolution is approximated by a Magnus expansion [104] to lowest order, which
yields the following integrals

$\Gamma_{n}(t)=\int_{0}^{t}\text{d}t^{\prime}\gamma_{n}(t^{\prime}),\quad\Delta_{n}(t)=\int_{0}^{t}\text{d}t^{\prime}\delta_{n}(t^{\prime}),$
(35)

where we have introduced the remaining integrals

$\begin{split}\gamma_{2}(t^{\prime})&=-\int_{0}^{t^{\prime}}\\!\\!\text{d}t^{\prime\prime}\,C(t^{\prime}-t^{\prime\prime})\cos(\Omega(t^{\prime}+t^{\prime\prime})),\\\
\delta_{1}(t^{\prime})&=-\int_{0}^{t^{\prime}}\\!\\!\text{d}t^{\prime\prime}\,C(t^{\prime}-t^{\prime\prime})\sin(\Omega(t^{\prime}-t^{\prime\prime})),\\\
\delta_{2}(t^{\prime})&=+\int_{0}^{t^{\prime}}\\!\\!\text{d}t^{\prime\prime}\,C(t^{\prime}-t^{\prime\prime})\sin(\Omega(t^{\prime}+t^{\prime\prime})).\end{split}$
(36)

These parameters contain all the information that is required in our
parametrized GST (see Sec. II.1), where the following combination shall be
useful for the parametrization

$\Theta(t)=\sqrt{\Delta_{1}^{2}(t)-\Delta_{2}^{2}(t)-\Gamma_{2}^{2}(t)}.$ (37)

We can rewrite these parameters in terms of the power spectral density (PSD)
of the noise which, when restricting to wide-sense stationary processes
$C(t,t^{\prime})=C(t-t^{\prime})$ [105], is defined by the Fourier transform

$C(t)=\int_{-\infty}^{\infty}\frac{{\rm d}\omega}{2\pi}S(\omega){\rm e}^{{\rm
i}\omega t}.$ (38)

This enables one to rewrite the integrals Eq. (35) as

$\begin{split}\Gamma_{n}(t)=&\int_{-\infty}^{\infty}\\!\\!{\text{d}\omega}\,S(\omega)\,F_{\Gamma_{n}}(\omega,\Omega,t),\\\
\Delta_{n}(t)=&\int_{-\infty}^{\infty}\\!\\!{\text{d}\omega}\,S(\omega)\,F_{\Delta_{n}}(\omega,\Omega,t),\end{split}$
(39)

where $F_{\Gamma_{i}}$ and $F_{\Delta_{i}}$ correspond to the filter functions
mentioned during the introduction. In particular, one finds

$F_{\,\Gamma_{1}}(\omega,\Omega,t)=\frac{t}{4}\\!\left(\eta_{\frac{2}{t}\\!}(\Omega-\omega)+\eta_{\frac{2}{t}\\!}(\Omega+\omega)\right),\\\
$ (40)

where we have used the nascent Dirac delta

$\eta_{\epsilon}(x)=\frac{\epsilon}{\pi
x^{2}}\sin^{2}\left(\frac{x}{\epsilon}\right)$ (41)

fulfilling $\eta_{\epsilon}(x)\to\delta(x)$ as $\epsilon\to 0^{+}$, and
$\int_{-\infty}^{\infty}{\rm d}x\,\eta_{\epsilon}(x)=1$, which clearly acts as
a Dirac-type delta filter in the long-time limit. The other filters are
defined as follows

$F_{\,\Delta_{1}}(\omega,\Omega,t)=\frac{\Omega
t}{2\pi(\Omega^{2}-\omega^{2})}+\delta F_{\,\Delta_{1}}(\omega,\Omega,t),$
(42)

where we have introduced

$\delta
F_{\,\Delta_{1}}(\omega,\Omega,t)=\frac{1}{4\pi}\left(\frac{\sin\big((\omega-\Omega)t\big{missing})}{(\omega-\Omega)^{2}}-\frac{\sin\big((\omega+\Omega)t\big{missing})}{(\omega+\Omega)^{2}}\right).$
(43)

In addition, one also finds

$\begin{split}&F_{\,\Gamma_{2}}(\omega,\Omega,t)=\frac{2\cos\Omega
t}{\pi(\omega^{2}-\Omega^{2})}\sin(\textstyle\frac{1}{2}(\omega-\Omega)t)\sin(\textstyle\frac{1}{2}(\omega+\Omega)t),\\\
&F_{\,\Delta_{2}}(\omega,\Omega,t)=\frac{2\sin\Omega
t}{\pi(\omega^{2}-\Omega^{2})}\sin(\textstyle\frac{1}{2}(\omega-\Omega)t)\sin(\textstyle\frac{1}{2}(\omega+\Omega)t).\end{split}$
(44)

These expression are of particular relevance because the phase-noise PSD can
often be accessed experimentally independently of the tomographic
characterization, e.g. by a self-heterodyne setup with a reference ultra-
stable oscillator in trapped-ion devices [106]. This would enable to adapt our
GST to a specific QIP. For instance, as mentioned in Sec. II, the validity of
long-sequence GST is limited to the region in which the sequence depth
fulfills $p\ll 1/\Gamma_{1}$. Thus, knowing the experimental PSD allows us to
give a prior estimation of this region by means of Eq. (6).

The former expressions also allow us to explicitly compute the non-
Markovianity measure [82, 83] employed in Sec. III. As a notion of quantum
non-Markovianity, we use the so-called CP-divisibility

$\mathcal{E}_{t,t_{0}}=\mathcal{E}_{t,t^{\prime}}\circ\mathcal{E}_{t^{\prime},t_{0}},\forall
t^{\prime}\in[t_{0},t];\,\,\mathcal{E}_{t,t^{\prime}}\in\mathsf{C}(\mathcal{H}_{\rm
S}).$ (45)

Within this framework, a map is said to be Markovian if one can divide it by
the composition of two CPTP maps at each intermediate time. Therefore, acting
on the tensor product space $\mathcal{H}_{\rm S}\otimes\mathcal{H}_{\rm S}$,
one can build a non-Markovianity measure on the violation of the CPTP
condition for each interval of evolution

$g(t^{\prime})=\lim_{\Delta t\to 0}\frac{1}{\Delta
t}\bigg{(}\big{\lVert}\mathcal{E}_{t^{\prime}+\Delta
t,t^{\prime}}\otimes\mathbb{1}(\ket{\Phi_{+}}\\!\bra{\Phi_{+}})\big{\rVert}_{1}-1\bigg{)},$
(46)

where the trace norm is used,
$\ket{\Phi_{+}}=\textstyle{\frac{1}{\sqrt{2}}}(\ket{00}+\ket{11})$ and the
non-Markovianity measure is defined as $\mathcal{N}_{\rm CP}=\int_{0}^{t}{\rm
d}t^{\prime}g(t^{\prime})$. Explicitly, using the integrals in (34) and (36),
one gets [55]

$\mathcal{N}_{\rm CP}(t)=\frac{1}{2}\\!\int_{0}^{t}\\!\\!{\rm
d}t^{\prime}\big{(}|\bar{\gamma}_{-}(t^{\prime})|-\bar{\gamma}_{-}(t^{\prime})\big{)},$
(47)

where
$\bar{\gamma}_{-}(t)=\textstyle\frac{1}{2}\gamma_{1}(t)-\textstyle\frac{1}{2}\sqrt{\gamma_{2}^{2}(t)+\delta_{2}^{2}(t)}$.

Finally, one can use the analytic expressions for the time evolution of the
dressed-state density matrix to derive closed expressions for the
$\chi_{\phi}(t)$ matrix of dynamical quantum map. In the Pauli basis, this
reads

$\chi_{0}(t)=\frac{1}{2}\left(\begin{matrix}\chi_{A}^{(1,1)}(t)&\chi_{A}^{(1,2)}(t)&0&0\\\
\chi_{A}^{(2,1)}(t)&\chi_{A}^{(2,2)}(t)&0&0\\\
0&0&\chi_{B}^{(1,1)}(t)&\chi_{B}^{(1,2)}(t)\\\
0&0&\chi_{B}^{(2,1)}(t)&\chi_{B}^{(2,2)}(t)\\\ \end{matrix}\right)$ (48)

where the upper and lower blocks can be expressed in terms of the parameters
and the Pauli matrices as follows

$\begin{split}\chi_{A}(t)&=\\!\\!\left(1+{\rm
e}^{-\Gamma_{1}(t)}\\!\right)\\!\\!\mathbb{1}_{2}+2{\rm
e}^{-\frac{1}{2}\Gamma_{1}(t)}\left[\\!\bigl{(}\cos\Omega
t\cos\textstyle\frac{1}{2}\Theta(t)-\frac{\Delta_{1}(t)}{\Theta(t)}\sin\Omega
t\sin\textstyle\frac{1}{2}\Theta(t)\bigr{)}\\!\sigma_{z}-\\!\\!\bigl{(}\sin\Omega
t\cos\textstyle\frac{1}{2}\Theta(t)+\frac{\Delta_{1}(t)}{\Theta(t)}\cos\Omega
t\sin\textstyle\frac{1}{2}\Theta(t)\bigr{)}\\!\sigma_{y}\right],\\\
\chi_{B}(t)&=\\!\\!\left(1-{\rm
e}^{-\Gamma_{1}(t)}\\!\right)\\!\\!\mathbb{1}_{2}-\frac{2}{\Theta(t)}{\rm
e}^{-\frac{1}{2}\Gamma_{1}(t)}\sin\textstyle\frac{1}{2}\Theta(t)\bigl{[}\bigl{(}\Gamma_{2}(t)\cos\Omega
t+\Delta_{2}(t)\sin\Omega t\bigr{)}\sigma_{z}+\bigl{(}\Gamma_{2}(t)\sin\Omega
t-\Delta_{2}(t)\cos\Omega t\bigr{)}\sigma_{x}\bigr{]}.\end{split}$ (49)

Note that nn contrast to [55], we here present the full $\chi_{\phi}(t)$
matrix and not only that of the error channel. We also note that, in the
absence of noise $\Gamma_{i}(t)=\Delta_{i}(t)=\Theta(t)=0$, the process block
matrices are simply

$\chi_{A}(t)=2(\mathbb{1}_{2}+\cos\Omega t\sigma_{z})-2{\rm i}\sin\Omega
t\sigma_{y},\hskip 4.30554pt\chi_{B}(t)=0,$ (50)

and one recovers the perfect gate $\mathcal{E}(\rho)={\rm e}^{-{\rm
i}t\frac{\Omega}{2}\sigma_{x}}\rho{\rm e}^{+{\rm
i}t\frac{\Omega}{2}\sigma_{x}}$.

For the present work, we need to generalise this derivation for different
phase angles $\phi$ in order to attain IC. We remark that the process matrices
accounting for the same pulse duration are paramterized by the same filter
integrals $\Gamma_{i}(t)$ and $\Delta_{i}(t)$, irrespectively of their phase.
In fact, for the gate set used in this work, the $G_{\pi}$ matrix is obtained
just by adding a minus sign to the non-diagonal elements of Eq. (48). For
$G_{\pi/2}$ we find

$\chi_{\pi/2}(t)=\frac{1}{2}\left(\begin{matrix}\chi_{A}^{(1,1)}(t)&0&-\chi_{A}^{(1,2)}(t)&0\\\
0&\chi_{B}^{(1,1)}(t)&0&\chi_{B}^{(1,2)}(t)\\\
-\chi_{A}^{(2,1)}(t)&0&\chi_{A}^{(2,2)}(t)&0\\\
0&\chi_{B}^{(2,1)}(t)&0&\chi_{B}^{(2,2)}(t)\end{matrix}\right),$ (51)

and finally for $G_{3\pi/2}$

$\chi_{3\pi/2}(t)=\frac{1}{2}\left(\begin{matrix}\chi_{A}^{(1,1)}(t)&0&\chi_{A}^{(1,2)}(t)&0\\\
0&\chi_{B}^{(1,1)}(t)&0&-\chi_{B}^{(1,2)}(t)\\\
\chi_{A}^{(2,1)}(t)&0&\chi_{A}^{(2,2)}(t)&0\\\
0&-\chi_{B}^{(2,1)}(t)&0&\chi_{B}^{(2,2)}(t)\end{matrix}\right).$ (52)

Let us finally consider additional amplitude noise fluctuations, as mentioned
in Subsec. III.3. As exposed in [55], the drive amplitude fluctuations will be
modeled by stochastic noise in the Rabi frequency [94] roughly as

$\Omega\to\Omega+\delta\tilde{\Omega}(t),\hskip
4.30554pt\delta\tilde{\Omega}(t)\simeq\frac{\Omega}{2}\frac{\delta\tilde{I}_{L}(t)}{I_{L,0}},$
(53)

where $\delta\tilde{\Omega}(t)$ and $\delta\tilde{I}_{L}(t)$ denote the
stochastic fluctuations of the Rabi rate and the laser intensity,
respectively, and $I_{L,0}$ denotes the ideal intensity of the driving field.

Note that the Hamiltonian Eq. (30) should now include stochastic noise in the
Rabi frequency ($\Omega\to\Omega+\delta\tilde{\Omega}(t)$). For the ongoing
analysis, we assume that the stochastic processes describing phase and
amplitude fluctuations are not cross-correlated. That is,
$\mathbb{E}\\{\tilde{\delta}(t)\delta\tilde{\Omega}(t^{\prime})\\}=0$. Under
an equivalent procedure to the that exposed for pure phase noise gates in this
appendix, one arrives to expressions for the evolution of the states which are
described by the filtered integrals Eq. (39) supplemented with a correction to
the decay $\Gamma_{1}$

$\Delta\Gamma_{1}(t)=\int_{-\infty}^{\infty}\\!\\!\\!{\rm
d}\omega{S}_{\Omega}(\omega)F_{\Omega}(\omega,t),$ (54)

where ${S}_{\Omega}$ stands for the PSD of the amplitude noise and we have
introduced an additional decay filter function

$F_{\Omega}(\omega,t)=\frac{1}{\pi}\left(\frac{1-\cos(\omega
t)}{\omega^{2}}\right)=t\eta_{\frac{2}{t}}(\omega).$ (55)

We note here that the inclusion of this new parameter does not influence the
structure of the matrices Eqs. (48), (51) and (52), which is preserved.
However, the upper and lower blocks Eq. (49) must be updated to account for
this correction

$\begin{split}\chi_{A}(t)&=\\!\\!\left(1+{\rm
e}^{-\Gamma_{1}(t)}\\!\right)\\!\\!\mathbb{1}_{2}+2{\rm
e}^{-\frac{1}{2}(\Gamma_{1}(t)+\Delta\Gamma_{1})}\left[\\!\bigl{(}\cos\Omega
t\cos\textstyle\frac{1}{2}\Theta(t)-\frac{\Delta_{1}(t)}{\Theta(t)}\sin\Omega
t\sin\textstyle\frac{1}{2}\Theta(t)\bigr{)}\\!\sigma_{z}\right.-\\\ &\hskip
258.33252pt-\left.\\!\\!\bigl{(}\sin\Omega
t\cos\textstyle\frac{1}{2}\Theta(t)+\frac{\Delta_{1}(t)}{\Theta(t)}\cos\Omega
t\sin\textstyle\frac{1}{2}\Theta(t)\bigr{)}\\!\sigma_{y}\right],\\\
\chi_{B}(t)&=\\!\\!\left(1-{\rm
e}^{-\Gamma_{1}(t)}\\!\right)\\!\\!\mathbb{1}_{2}-\frac{2}{\Theta(t)}{\rm
e}^{-\frac{1}{2}(\Gamma_{1}(t)+\Delta\Gamma_{1})}\sin\textstyle\frac{1}{2}\Theta(t)\bigl{[}\bigl{(}\Gamma_{2}(t)\cos\Omega
t+\Delta_{2}(t)\sin\Omega t\bigr{)}\sigma_{z}+\bigl{(}\Gamma_{2}(t)\sin\Omega
t-\Delta_{2}(t)\cos\Omega t\bigr{)}\sigma_{x}\bigr{]}.\end{split}$ (56)

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