$\rho_{\textrm{test}}$ is a known density matrix used for channel
characterization.

The likelihood function is given by:

$\displaystyle\mathcal{L}=\Pr(\\{v_{\ell}\\}|\textrm{x})$
$\displaystyle=\textrm{Pr}(V_{0}=v_{0},\cdots,V_{L-1}=v_{L-1}|\textrm{x})$
(6.35) $\displaystyle=\prod_{l=0}^{L-1}\textrm{Pr}(V_{l}=v_{l}|\textrm{x})$
$\displaystyle=\prod_{l=0}^{L-1}\textrm{Categorical}_{0,\cdots,N-1}(v_{l};p_{0},\cdots,p_{N-1})$
$\displaystyle=\prod_{l=0}^{L-1}p_{0}^{\delta_{0}(v_{l})}\cdots
p_{N-1}^{\delta_{N-1}(v_{l})}$
$\displaystyle=p_{0}^{C_{0}(\textrm{data})}\cdots
p_{N-1}^{C_{N-1}(\textrm{data})}$
$\displaystyle=\prod_{i=0}^{N-1}\left[\textrm{Tr}[\Pi_{i}\tilde{\mathcal{G}}_{\textrm{x}}(\rho_{\textrm{test}})]\right]^{C_{i}(\\{v_{\ell}\\})}\;,$

where, $C_{i}(\\{v_{\ell}\\})=\sum\limits_{\ell=0}^{L-1}\delta_{i}(v_{\ell})$
(with $i\in\\{0,1,\cdots N-1\\}$) is simply a counter function that counts how
many times $i$ appeared in the experimentally observed data post-measurement
($\delta_{i}(v)$ is Kronecker delta function which is 1 if $i=v$ and zero
otherwise).

The log of the posterior is given by:

$\displaystyle f^{\textrm{posterior}}_{X}(\textrm{x}|\textrm{data})$
$\displaystyle=\sum\limits_{i=0}^{N-1}C_{i}(\textrm{data})\log\left[\textrm{Tr}[\Pi_{i}\tilde{\mathcal{G}}_{\textrm{x}}(\rho_{\textrm{test}})]\right]+\log\Gamma\left(\sum\limits_{j=0}^{N_{p}-1}\eta_{j}\right)-\sum\limits_{j=0}^{N_{p}-1}\log\Gamma(\eta_{j})$
(6.36)
$\displaystyle+\sum\limits_{j=0}^{N_{p}-1}(\eta_{j}-1)\log\textrm{x}_{j}+\textrm{terms
independent of x}\;.$

In the final step, the maximum-a-posterior (MAP) estimate is obtained as:

$\displaystyle\hat{\textrm{x}}(\tau)=$
$\displaystyle\underset{\textrm{x}}{\textrm{argmax}}\left[\sum\limits_{i=0}^{N-1}C_{i}(\textrm{data})\log\left[\textrm{Tr}[\Pi_{i}\tilde{\mathcal{G}}_{\textrm{x}}(\rho_{\textrm{test}})]\right]+\sum\limits_{j=0}^{N_{p}-1}(\eta_{j}-1)\log\textrm{x}_{j}\right]\;.$
(6.37)

$\hat{\textrm{x}}(\tau)$ is our best guess for the time-varying noise at time
$\tau$. We will use this updated estimate for error mitigation using
probabilistic error cancellation (PEC).

Probabilistic error cancellation (PEC) is a well-known error mitigation method
[72, 153, 154]. The four broad steps of the PEC workflow are as follows. We
use the convention that calligraphic symbols denote super-operators acting on
density matrices:

$\mathcal{G}(\rho)=G\rho G^{\dagger}.$ (6.38)

First, expand an ideal unitary gate $\mathcal{G}$ as a (noise-model dependent)
linear combination of implementable noisy gate set
$\\{\tilde{\mathcal{G}}_{j}\\}$ (with its ideal counterpart
$\\{\mathcal{G}_{j}\\}$), as follows:

$\mathcal{G}=\sum\limits_{j=0}^{N_{p}-1}\theta_{j}\tilde{\mathcal{G}}_{j},$
(6.39)

where $\theta_{j}$ are real coefficients, and $\mathcal{E}_{\textrm{x}}$ is an
error channel (such as Pauli noise channel), and
$\tilde{\mathcal{G}_{j}}\equiv\mathcal{E}_{\textrm{x}}\circ\mathcal{G}_{\textrm{j}}$.
The implementable noisy gate set $\\{\tilde{\mathcal{G}}_{j}\\}$ is also
called noisy basis circuit set. For example, if $G=H\otimes H$ is a two-qubit
Hadamard gate, then the noisy basis circuits are given by

$\tilde{\mathcal{G}}_{\sigma\sigma^{\prime}}=\mathcal{E}_{\textrm{x}}\circ\mathcal{P}_{\sigma\sigma^{\prime}}\circ\mathcal{G}$
(6.40)

where

$\mathcal{P}_{\sigma\sigma^{\prime}}(\cdot)\equiv(\sigma\otimes\sigma^{\prime})(\cdot)(\sigma\otimes\sigma^{\prime})\;\;\;\;,$
(6.41)

and $\sigma,\sigma^{\prime}$ are picked from the set of Pauli matrices
$\\{I,X,Y,Z\\}$.

The second step of PEC involves estimating the expectation value of the noise-
mitigated observable as:

$\sum\limits_{j=0}^{N_{p}-1}\theta_{j}\braket{\tilde{\mathcal{G}}_{j}}.$
(6.42)

The ideal gate can be approximated as:

$\mathcal{G}=\sum\limits_{w=0}^{N_{p}-1}p(w)[\gamma\textrm{sgn}(\theta_{w})\tilde{\mathcal{G}}_{w}],$
(6.43)

where

$\gamma=\sum\limits_{w=0}^{N_{p}-1}|\theta_{w}|$ (6.44)

and $p(w)=|\theta_{w}|/\gamma$ and $w$ is a random variable such that
$w\in{0,1,\cdots,N_{p}-1}$. Said differently, the super-operator
$\gamma\textrm{sgn}(\theta_{w})\tilde{\mathcal{G}}_{w}$ is an unbiased
estimator for the ideal super-operator $\mathcal{G}$ since:

$\displaystyle\mathcal{G}=$
$\displaystyle\sum\limits_{w=0}^{N_{p}-1}\textrm{sgn}(\theta_{w})|\theta_{w}|\tilde{\mathcal{G}}_{w}$
(6.45) $\displaystyle=$
$\displaystyle\sum\limits_{w=0}^{N_{p}-1}\textrm{sgn}(\theta_{w})\frac{|\theta_{w}|}{\sum|\theta_{w}|}(\sum|\theta_{w}|)\tilde{\mathcal{G}}_{w}$
(6.46) $\displaystyle=$
$\displaystyle\sum\limits_{w=0}^{N_{p}-1}p(w)\left[\gamma\textrm{sgn}(\theta_{w})\tilde{\mathcal{G}}_{w}\right]$
(6.47) $\displaystyle=$
$\displaystyle\mathds{E}_{w}\left[\gamma\textrm{sgn}(\theta_{w})\tilde{\mathcal{G}}_{w}\right].$
(6.48)

The third step of PEC involves sampling from each of the noisy implementable
circuits and computing the mean of the observable for these noisy basis
circuits.

The final step of PEC involves inferring the (hopefully) noiseless observable
using a weighted sum of the mean of the observable obtained from the noisy
basis circuits in previous step (as per Eqn. 6.45).

In our 2-qubit application ($\mathds{H}\otimes\mathds{H}$), the observables
are the projection operators $\Pi_{0}=\ket{00}\bra{00}$,
$\Pi_{1}=\ket{01}\bra{01}$, $\Pi_{2}=\ket{10}\bra{10}$, and
$\Pi_{3}=\ket{11}\bra{11}$. The complete noisy basis circuit set is shown in
Fig. 6.6.

For our noise simulation, we assume that the mean of the stochastic $T_{1}$
coherence time for the first qubit decreases uniformly in a simple step-
function-like manner over five time periods, deteriorating from 150 to 60
$\mu$s. Similarly, for the second qubit, we assume that the mean of the
$T_{1}$ time deteriorates from 200 to 10 $\mu$s in five time periods.
Additionally, we assume a simple step-function-like decrease in the mean of
the $T_{2}$ coherence time for the first qubit, deteriorating from 70 to 50
$\mu$s in five time periods. Finally, for the second qubit, we assume that the
mean of the $T_{2}$ coherence time deteriorates from 130 to 62.5 $\mu$s in
five time periods. The coefficients of the Pauli channel are then computed
using Eq. (6.21). This setup mimics intra-calibration deterioration of the
noise in a quantum circuit. We assume a typical execution time of 100 $\mu$s
for the Hadamard gate on the IBM transmon platform.

As described before, we model the distribution of the Pauli coefficients using
a time-varying Dirichlet distribution. The Hellinger distance between the
density at time $\tau$=0 and a later time is a measure of the degree of non-
stationarity, increasing from 0 to 57% as shown in Fig. 6.7. The true means of
the time-varying noise Pauli channel coefficients are shown in Table 6.1. In
period 0, the coefficient of the identity term in the Pauli noise channel is
38%, but by period 2, it degrades to 26%. This change is driven by the
deterioration in the coherence times for qubit 0 and qubit 1, respectively, as
described before.

At the outset, we assume we are in period 0, equipped with accurate knowledge
of the Pauli noise channel. With this knowledge, we obtain the super-operator
expression for the noisy basis circuits, which we linearly combine to estimate
the ideal operation ($H\otimes H$). The reconstructed operation is a weighted
average using a quasi-probability distribution that uses the true noisy basis.
Our results, shown in Fig. 6.9, indicate an accurate ideal gate implementation
in the presence of noise in period 0, with a Hellinger distance between the
expected and observed output of 0.34%. This small error stems from the shot
noise due to finite sample size.

In subsequent time periods, the noise characteristics of the Pauli channel
change as shown in Fig. 6.7 and Table 6.1. This change renders the previously
implemented PEC approach invalid as the super-operators characterizing the
noisy basis circuits are no longer accurate. Consequently, the PEC
coefficients are also invalid.

The black bars in Fig. 6.9 indicate that the Hellinger distance between the
output and ideal increases from 0.34% to 7%, and 15% in periods 1, and 2,
respectively. To examine the raw data of the obtained histograms for the non-
adaptive case, refer to the crimson colored bars in Fig. 6.8.

For the Bayesian update, we use the histogram of projective measurements
obtained from applying the PEC circuits to the input density matrix shown in
Eq. (6.49). The resulting observation stream of 2-bit strings belongs to one
of four possibilities: ${00,01,10,11}$ with probabilities $0.76,0.08,0.10,$
and $0.06$, respectively. The input density matrix must have sufficient off-
diagonal components to produce an observation rich histogram (as opposed to a
bland uniform histogram which will make it impossible to differentiate between
the Pauli coefficients).

$\rho_{\textrm{test}}=\begin{pmatrix}0.2&0.22-0.02j&0.15-0.09j&0.16-0.1j\\\
0.22+0.02j&0.24&0.16-0.08j&0.19-0.1j\\\
0.15+0.09j&0.16+0.08j&0.36&0.14+0.06j\\\
0.16+0.1j&0.19+0.1j&0.14-0.06j&0.21\\\ \end{pmatrix}\;.$ (6.49)

Table 6.2 and Table 6.3 demonstrate the MAP estimation of Pauli coefficients
and illustrates the quality of Bayesian estimation. With such updated
estimates, we re-compute the super-operators for the noisy basis circuits and
the linear combination coefficients. PEC is then implemented using the updated
super-operators, resulting in improved output quality, as shown in Fig. 6.8.
In this figure, the y-axis represents the probability of observing a
particular computational basis state. In this two-qubit case, there are four
possible states, and their probabilities sum to 1. The black bars indicate the
ideal probabilities, while the red and orange bars represent the probabilities
obtained with non-adaptive and adaptive PEC, respectively, for the fourth
time-period. The graph demonstrates that adaptive PEC, which used adaptive
estimation of the noise super-operators, improves accuracy compared to non-
adaptive PEC. Specifically, the probability of observing 00 in period 2
increases from 57% for non-adaptive PEC to 72% for adaptive PEC. The
likelihood and cost functions used in the Bayesian inference procedure are
shown in Figures 6.10 and 6.11.

Fig. 6.9 compares the performance of adaptive and non-adaptive PEC
implementations across four back-to-back time-periods in the presence of time-
varying noise. The y-axis represents the Hellinger distance, which is the
distance between two discrete probability distributions over the computational
basis states. The black bars show the experimentally observed distribution
when non-adaptive PEC is used, while the orange bars show the distribution
when adaptive PEC is used. The x-axis represents the four time-periods. When
using non-adaptive PEC, the Hellinger distance from the ideal distribution is
7%, and 15% for time-periods 1, and 2, respectively. When using adaptive PEC,
the Hellinger distance significantly improves to 1.1%, and 3.1% for the same
time-periods.

Note that we used a Pauli noise channel with all $4^{n}$ terms. However, in
practical applications, it becomes necessary to reduce the number of terms. To
achieve this, one can explore the use of a sparse Lindbladian noise model
[27], which considers noise only in nearest-neighbor connections for Pauli
terms with weight greater than 1. This reduction in terms leads to a linear
scaling instead of exponential with the number of qubits, making the model
more computationally efficient. Also note that while it has been observed that
single-qubit gate noise can be more than 10 times smaller than two-qubit gate
noise [27], it cannot be disregarded in Probabilistic Error Cancellation (PEC)
due to error propagation effects. Moreover, in the presence of time-varying
quantum noise, it becomes even more critical to account for single-qubit noise
to ensure accurate error cancellation.

Note that re-calibrating the noise model does not solve the challenge of
dynamic estimation, as experimental evidence from various studies indicates
significant fluctuations in decoherence times over time. These fluctuations,
observed in studies like [19, 37], show that decoherence times can vary by
approximately 50% within an hour due to the presence of oxides on
superconductors’ surfaces, represented as fluctuating two-level systems (TLS)
[42, 36]. These fluctuations at the scale of minutes and hours are considered
non-systematic noise, which cannot be addressed solely through re-
calibrations. This emphasizes the necessity for Bayesian algorithms that can
provide reliable error bars (as opposed to erroneous point estimates from
MLE).

To summarize, this section delved into the behavior of non-stationary noise
channels on real-world quantum computing platforms, with a focus on Pauli
noise channels in superconducting qubits. The investigation specifically
addressed the spatio-temporal non-stationarity of these noise channels,
particularly in the context of ibm_belem transmon device. Spatial correlations
within multi-qubit noise were explored by treating it as a collection of
single-qubit channels while retaining the spatial correlations between
individual qubits’ $T_{1}$ and $T_{2}$ times. The Dirichlet distribution
modeled the joint distribution of Pauli noise channel coefficients, while the
Hellinger distance gauged the reliability of error channel characterization.
The impact of time-varying Pauli noise on quantum information encoded in a
$n$-qubit register was characterized, and coefficients of a separable 2-qubit
Pauli noise channel were obtained. These coefficients directly relate to the
decoherence times of individual elements, showing strong correlations among
them. PEC is effective when noise is well-characterized, but non-stationary
noise necessitates an adaptive approach. An adaptive Bayesian inference
strategy is proposed to enhance PEC performance in the presence of time-
varying noise. This approach dynamically estimated the time-varying Dirichlet
distribution of Pauli coefficients using a Bayesian inference-based rolling
update. An application of adaptive PEC for executing a Hadamard operation on
two-qubits amidst time-varying noise was presented, revealing the need for
adaptability due to changing noise characteristics. Without adaptive
mitigation, outdated coefficients lead to inaccuracies and newly introduced
noise affects circuit execution, severely impacting output accuracy.

Figure 6.1:  Time-varying density of SPAM fidelity. Data shown for register element #26.  Figure 6.2:  SPAM fidelity time-series for qubit #26 for Dec-May 2022.  Figure 6.3: We simulated SPAM and gate error channels for a quantum circuit with 4 qubits that creates a uniform superposition across all the computational basis states using Hadamard gates. The blue bars represent the probability distribution across the measurement outcomes for the ideal, noiseless circuit while the orange bars represent the same for a realization of an execution on an unstable device. Figure 6.4: The plot shows improved accuracy achieved using Bayesian optimization in presence of time-varying noise. Mitigation using average parameters deteriorates accuracy. The number of samples used in the Bayesian estimation process is held constant at $10^{4}$. Table 6.1: True means of the time-varying noise Pauli channel coefficients Pauli term [$\textrm{qubit}0\otimes\textrm{qubit}1$] | Period 0 | Period 1 | Period 2 |
---|---|---|---|---
II | 0.379 | 0.326 | 0.26 |
IX | 0.056 | 0.064 | 0.075 |
IY | 0.056 | 0.064 | 0.075 |
IZ | 0.076 | 0.078 | 0.079 |
XI | 0.081 | 0.083 | 0.082 |
XX | 0.012 | 0.016 | 0.024 |
XY | 0.012 | 0.016 | 0.024 |
XZ | 0.016 | 0.02 | 0.025 |
YI | 0.081 | 0.083 | 0.082 |
YX | 0.012 | 0.016 | 0.024 |
YY | 0.012 | 0.016 | 0.024 |
YZ | 0.016 | 0.02 | 0.025 |
ZI | 0.127 | 0.12 | 0.108 |
ZX | 0.019 | 0.024 | 0.031 |
ZY | 0.019 | 0.024 | 0.031 |
ZZ | 0.026 | 0.029 | 0.033 |
Table 6.2: Estimated Pauli coefficients for peiod 1 Pauli term [$\textrm{qubit}0\otimes\textrm{qubit}1$] | Estimated value | True value |
---|---|---|---
II | 0.311 | 0.326 |
IX | 0.064 | 0.064 |
IY | 0.064 | 0.064 |
IZ | 0.097 | 0.078 |
XI | 0.078 | 0.083 |
XX | 0.016 | 0.016 |
XY | 0.016 | 0.016 |
XZ | 0.024 | 0.02 |
YI | 0.078 | 0.083 |
YX | 0.016 | 0.016 |
YY | 0.016 | 0.016 |
YZ | 0.024 | 0.02 |
ZI | 0.114 | 0.12 |
ZX | 0.023 | 0.024 |
ZY | 0.023 | 0.024 |
ZZ | 0.036 | 0.029 |
Table 6.3: Estimated coefficients for period 2 Pauli term [$\textrm{qubit}0\otimes\textrm{qubit}1$] | Estimated value | True value |
---|---|---|---
II | 0.243 | 0.26 |
IX | 0.073 | 0.075 |
IY | 0.073 | 0.075 |
IZ | 0.103 | 0.079 |
XI | 0.074 | 0.082 |
XX | 0.022 | 0.024 |
XY | 0.022 | 0.024 |
XZ | 0.031 | 0.025 |
YI | 0.074 | 0.082 |
YX | 0.022 | 0.024 |
YY | 0.022 | 0.024 |
YZ | 0.031 | 0.025 |
ZI | 0.103 | 0.108 |
ZX | 0.031 | 0.031 |
ZY | 0.031 | 0.031 |
ZZ | 0.044 | 0.033 |
Figure 6.5: The 4-qubit quantum circuit used for the simulation experiment.
Each qubit is assumed to have a different, independent SPAM error process.
Each Hadamard gate is similarly assumed to have a different, independent gate
error process. Figure 6.6:  The figure displays the noisy basis circuits for
the linear combination step (the first step) in PEC, where each figure
represents the noisy operations
$\mathcal{E}_{\textrm{x}}\circ\mathcal{P}_{\sigma,\sigma^{\prime}}\circ\mathcal{G}$
and $G=H\otimes H$ is the desired, ideal operation. Note that different basis
set choices are possible, depending on the hardware. Figure 6.7: We model the
degradation of a non-stationary Pauli noise channel by assuming the coherence
time steadily decreases over time. Using a non-stationary Dirichlet
distribution, we model the joint distribution of coefficients for a two-qubit
circuit, which fluctuate as coherence times deteriorate. The y-axis represents
the degree of non-stationarity, and the x-axis shows four time-periods. The
Hellinger distance between the Dirichlet distributions at time $\tau$=0 and a
later time is a measure of non-stationarity, increasing from 0 to 57%. This
model is based on transmon platforms and is used as an experimental setup for
our simulation experiments. Figure 6.8: This graph compares adaptive and non-
adaptive PEC for the $H\otimes H$ gate under time-varying noise. The y-axis
shows the probability of observing a basis state. The black bars represent the
ideal histogram for the test input, while red/orange bars are non-adaptive PEC
results. Adaptive PEC improves accuracy. For example, the $\ket{00}$
probability increases from 57% to 72%. Figure 6.9: This graph compares
adaptive and non-adaptive PEC implementations over four time-periods with
time-varying noise, using Hellinger distance to measure the difference from
the ideal distribution. The black and orange bars represent the observed
distributions with non-adaptive and adaptive PEC, respectively. Adaptive PEC
significantly outperforms non-adaptive PEC in reducing Hellinger distance to
1.1%, and 3.2% compared to 7%, and 15% for non-adaptive PEC across time-
periods. Adaptive PEC uses adaptive estimation of noise super-operators to
improve accuracy compared to non-adaptive PEC. Time-varying noise underscores
the need for adaptive PEC. Figure 6.10: The likelihood function used in the
Bayesian inference procedure for the second period.  Figure 6.11: The cost
function used in the Bayesian inference procedure for the second period.

##  Chapter 7 Adaptive probabilistic error cancellation

In this final chapter, we investigate the accuracy and stability of
probabilistic error cancellation (PEC) outcomes in the presence of non-
stationary noise, which is an obstacle to achieving accurate observable
estimates. Leveraging Bayesian methods, we design a strategy to enhance PEC
stability and accuracy.

The practical realization of quantum computing has witnessed rapid
advancements [55], with quantum devices now operating as systems with hundreds
of interacting qubits. However, these real quantum devices [30] are noisy
[34], and practical efforts to realize a quantum computer introduce various
noise processes like decay, de-coherence [155], environmental coupling, intra-
register cross-talk [40, 41], and leakage from computational space [31].
Physical operations like quantum gates and measurements rely on
electromagnetic fields susceptible to pulse distortion, attenuation, jitter,
and drift, which further increase noise [54, 156]. Imperfections in
thermodynamic controls (such as cryogenic cooling, magnetic shielding,
vibration suppression, and imperfect vacuum chambers) can disturb the
operating conditions of the quantum computer [70, 71].

These lead to computational errors that make it essential to address noise and
implement error mitigation strategies [59] to improve the accuracy of quantum
outcome [68].

Contemporary quantum computers are not only noisy but they also exhibit non-
stationarity. Non-stationary noise processes [157, 44, 46, 42, 36] in
superconducting qubits [33] are well-studied. For example, state preparation
and measurement (SPAM) fidelities have been observed to fluctuate
significantly showing more than 25% deviation from their long-term average
[110]. Similarly, the fidelity of CNOT gates have been noted to change by over
40% within similar time frames [112]. Moreover, the qubit relaxation times,
known as $T_{1}$, have experienced fluctuations of up to 400% in just 30
minutes [121]. Likewise, de-phasing times, denoted as $T_{2}$, have been
recorded to vary by over 50% within an hour [19, 37].

The non-stationarity observed in contemporary superconducting quantum
computers stems from two primary sources linked to material defects:
impurities within the material and ionization induced by cosmic rays. It is
theorized that fluctuating two-level systems, possibly stemming from certain
oxides on the superconductor’s surface, contribute to non-stationarity [42,
36]. Additionally, cosmic rays [43, 44] contribute by ionizing the substrate
upon impact, leading to the emission of high-energy phonons, which in turn
triggers a burst of quasi-particles. These quasi-particles disrupt qubit
coherence across the device. It has been shown that quantum computers can
experience catastrophic errors in multi-qubit registers approximately every 10
seconds due to cosmic rays originating from outer space [44]. Studies that
address non-stationary noise in superconducting quantum computers include
investigations on output reproducibility [45], noise modeling [46], tracking
the non-stationary profile of quantum noise [47], and quantum error mitigation
using continuous control [48].

Quantum error mitigation is a set of techniques that employs statistical tools
from estimation theory to reduce the impact of noise in quantum computations
without directly correcting the quantum state [59]. Such techniques can become
vulnerable to errors stemming from over or under estimation of noise due to
the presence of non-stationarity [158].

A quantum noise channel [11] can be described as a stochastic process,
allowing for the continuous update of its estimated characteristics in
response to varying noise conditions. Such an approach treats the channel as a
time-varying random variable. Consider Fig. 7.1 which shows the state
preparation and measurement (SPAM) fidelity for the second register element on
ibm_kolkata device on Jan 15, 2024. The probability density is clearly
changing with time, even though the variance stays consistent. In light of
such non-stationary data, a single-qubit SPAM noise channel can be described
by the model: $\mathcal{E}(\rho)=f(t)\rho+[1-f(t)]\mathds{X}\rho\mathds{X}$,
where $\rho$ represents the single-qubit density matrix,
$\mathds{X}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ denotes the Pauli-X
matrix, and $f(t)$ denotes the SPAM fidelity drawn from a time-dependent
distribution. From Fig. 7.1, non-stationarity of noise in this superconducting
device is apparent within 24 hours. The noise parameters estimated during
device re-calibration quickly becomes outdated, compromising the accuracy of
noise channel information essential for mitigation. Thus re-calibration alone
is insufficient and there is a need for adaptive error mitigation techniques
that can function in between calibration intervals in the face of changing
noise conditions.

This study focuses on probabilistic error cancellation (PEC) [72] in the
presence of non-stationary noise. PEC is a quantum error mitigation approach
that aims to construct unbiased estimates of the means of quantum observables
from noisy observations. Effective implementation requires an accurate noise
characterization [159]. For example, learning correlated noise channels in
large quantum circuits on a superconducting quantum processor has proven to be
difficult [27]. Yet, leveraging sparse noise models, PEC has been successful
in estimating the mean of observables in circuits comprising 2,880 CNOT gates,
executed on a 127-qubit noisy superconducting processor - a task that
conventional brute-force computing could not match [35].

We demonstrate that adaptive probabilistic error cancellation, which views
quantum noise channels as evolving random variables, outperforms its non-
adaptive counterpart in devices subject to non-stationary noise. To achieve
this aim, we will make use of a 5-qubit implementation of the Bernstein-
Vazirani algorithm [109].

The manuscript is organized as follows. In Sec. 7.1, we provide background for
probabilistic error cancellation (PEC). Sec. 7.2 develops a Bayesian [73, 74]
approach for adapting the method of probabilistic error cancellation to non-
stationary noise. It also sets up a performance evaluation framework for the
accuracy and stability of PEC results. Sec. 7.3 presents a numerical
validation of adaptive probabilistic error cancellation using a 5-qubit
implementation of the Bernstein-Vazirani algorithm, where we treat the noise
parameters (qubit-specific state preparation and measurement (SPAM) fidelities
and depolarizing parameters characterizing noise in CNOT gate) as non-
stationary random variables. In Sec. 7.4, we present the results of
experiments conducted on a real, noisy quantum device to test the adaptive PEC
algorithm. Concluding remarks are provided in Sec. 8.

###  7.1 Background

In this section, we provide background for the quantum error mitigation method
called probabilistic error cancellation (PEC) [72, 27, 35] which aims to
mitigate errors by approximating the noiseless mean of the observable as a
weighted sum of noisy observables.  We use calligraphic symbols to denote
super-operators acting on density matrices ($\rho$):

$\mathcal{G}\rho=G\rho G^{\dagger}.$ (7.1)

where $\mathcal{G}$ is the super-operator and $G$ is a unitary quantum
operator. For example, if $G$ denotes the CNOT operator, then $\mathcal{G}$ is
the super-operator for the CNOT operation.

Typically we do not have access to a noiseless implementation of
$\mathcal{G}$. Let $\tilde{\mathcal{G}}$ denote the super-operator
corresponding to the available, noisy implementation of $\mathcal{G}$. We have
access to other synthesized implementations of $\tilde{\mathcal{G}}$ by
subjecting $\tilde{\mathcal{G}}$ to basis operations available on a noisy
device. Originally, the noisy basis set was specified as the set of native
gates that a quantum computer could implement [72]. However, with advancements
in superconducting quantum computers, the noise associated with single-qubit
Pauli operators has become negligible [27, 28]. Consequently, Pauli operators,
which might be a composition of multiple native gates, can be employed as the
basis set [160]. We denote by $\\{\tilde{\mathcal{G}_{k}}\\}$ the set of all
noisy super-operators, with $k=0$ denoting the noisy implementation of
$\mathcal{G}$ without additional operations from the basis set.

As an example, consider a 2-qubit quantum gate $G$. The set of noisy super-
operators composed under a Pauli channel assumption are given by:
$\\{\mathcal{P}\circ\tilde{\mathcal{G}}\\}$ where
$\mathcal{P}\circ\tilde{\mathcal{G}}(\rho)=\mathcal{P}(\tilde{\mathcal{G}}(\rho))$
and
$\mathcal{P}(\cdot)\equiv[\mathds{P}_{0}\otimes\mathds{P}_{1}](\cdot)[\mathds{P}_{0}\otimes\mathds{P}_{1}]$.
Here $0$ and $1$ refer to qubit $0$ and qubit $1$ respectively and
$\mathds{P}_{0},\mathds{P}_{1}$ are picked from the set of Pauli operators
$\\{\mathds{I},\mathds{X},\mathds{Y},\mathds{Z}\\}$. Thus, by varying $P_{0}$
and $P_{1}$, we obtain the set $\\{\mathcal{P}\circ\tilde{\mathcal{G}}\\}$
which forms a basis $\\{\tilde{\mathcal{G}_{k}}\\}_{k=0}^{15}$ that spans the
super-operator space. In this 2-qubit example, the map for $k$ is derived from
the cartesian product of
$\\{\mathds{I},\mathds{X},\mathds{Y},\mathds{Z}\\}\times\\{\mathds{I},\mathds{X},\mathds{Y},\mathds{Z}\\}$,
with the sequence of this ordered set determining the value of $k$.

In general, the super-operator $\mathcal{G}$ can be expressed as a linear
combination of the basis super-operators:

$\mathcal{G}=\sum\limits_{k=0}^{N_{p}-1}\eta_{{}_{\text{k}}}\tilde{\mathcal{G}}_{k},$
(7.2)

where $N_{p}$ is the dimension of the super-operator space. The PEC
coefficients ${\eta_{{}_{\text{k}}}}$ in the linear combination are determined
either analytically, under a noise model assumption for single and two-qubit
gates, which can be extended to larger circuits, or numerically, by minimizing
the one-norm between high-dimensional matrices [160, 72]. We employ the
analytical approach.

The circuits corresponding to the super-operators $\tilde{\mathcal{G}}_{k}$ in
Eqn. 7.2 are constructed by subjecting each of the gates in the quantum
circuit for $\tilde{\mathcal{G}}$ to operations from the noisy basis set. If
we execute these noisy circuits and collect the mean of the observable, then,
from Eqn. 7.2, we recover the ideal noiseless mean of an observable
$\braket{\mathcal{O}}$ as:

$\braket{\mathcal{O}}=\textrm{Tr}\left[\mathcal{O}\mathcal{G}\rho\right]=\textrm{Tr}\left[\mathcal{O}\sum\limits_{k}\eta_{{}_{\text{k}}}\tilde{\mathcal{G}}_{k}\rho\right]$
(7.3)

where $\rho$ is the input density matrix to the circuit. Thus, an observable
$\braket{\mathcal{O}}$ may be estimated as a weighted sum of the mean of the
observables from the noisy circuits.

Eqn. 7.3 can be re-written as:

$\braket{\mathcal{O}}=\gamma\sum\limits_{k}\text{sgn}(\eta_{{}_{\text{k}}})\mathds{Q}_{{}_{\text{k}}}\textrm{Tr}\left[\mathcal{O}\tilde{\mathcal{G}_{{}_{\text{k}}}}\rho\right]$
(7.4)

with $\gamma=\sum|\eta_{{}_{\text{k}}}|$ and
$\mathds{Q}_{{}_{\text{k}}}=|\eta_{{}_{\text{k}}}|/\gamma$. The sign function,
$\text{sgn}(\cdot)$, returns +1 for positive inputs, -1 for negative inputs,
and 0 for an input of 0.

Note that the set $\\{\mathds{Q}_{k}\\}$ forms a valid probability
distribution because all its elements are positive and sum to 1. However,
$\\{\mathds{Q}_{k}\\}$ is termed a quasi-probability distribution (QPD) [72,
160]. To see this, consider a random integer $K\in\\{0,\cdots N_{p}-1\\}$
which follows the probability distribution function denoted by
$\mathds{Q}_{k}$. This random integer $K$ can be mapped one-to-one to the
random variable
$\text{sgn}(\eta_{{}_{\text{K}}})\textrm{Tr}\left[\mathcal{O}\tilde{\mathcal{G}_{{}_{\text{K}}}}\rho\right]$,
which inherits the probability distribution function $\mathds{Q}_{k}$. By
repeatedly sampling the random variable $K$, we realize a set of random
integers represented as $\\{\mathfrak{m}\\}$. For each specific $\mathfrak{m}$
that is realized, we execute the corresponding noisy quantum circuit
$\tilde{\mathcal{G}_{\mathfrak{m}}}$ multiple times and obtain a set of noisy
means of observables denoted by the set
$\left\\{\textrm{Tr}\left[\mathcal{O}\tilde{\mathcal{G}_{\mathfrak{m}}}\rho\right]\right\\}$.

When computing the average over the set of noisy means of observables
$\left\\{\textrm{Tr}\left[\mathcal{O}\tilde{\mathcal{G}_{\mathfrak{m}}}\rho\right]\right\\}$,
we need to adjust the sign of each element of the set by the sign of
$\eta_{{}_{\text{m}}}$ and scale it by $\gamma$. This average then converges
to the mean of the observable from a noiseless gate $\mathcal{G}$, in the
asymptotic limit of a large number of repeated samplings of the random
variable $K$. The need for adjustment by the sign function leads us to denote
$\mathds{Q}_{k}$ as quasi-probabilities.

Achieving an accuracy of $O(\epsilon)$ using the empirical mean of the random
variable
$\gamma\text{sgn}(\eta_{{}_{\text{K}}})\textrm{Tr}\left[\mathcal{O}\tilde{\mathcal{G}_{{}_{\text{K}}}}\rho\right]$
(which is an unbiased estimator of $\braket{\mathcal{O}}$) requires
$O(\gamma/\epsilon)^{2}$ PEC circuit samples and the result has variance of
order $O(\gamma^{2})$ [72].

###  7.2 Adaptive PEC

Estimating channel noise parameters is crucial for determining the PEC
coefficients $\\{\eta_{{}_{\text{k}}}\\}$. However, the non-stationary nature
of noise, along with drift and latency in characterization, complicates this
task. This in turn makes it difficult to accurately assess the PEC
coefficients. In this section, we demonstrate how adaptive parameter
estimation can be applied to PEC.

The parameters characterizing the noise during idle time (such as qubit
decoherence time [18]) and quantum operations (such as CNOT fidelity [110])
exhibit random non-stationary behavior in some hardware. Estimating non-
stationary stochastic processes is challenging and their predictive value is
limited because the patterns identified from historical data may not reliably
indicate future behavior, making it difficult to discern underlying trends. A
model that is effective at one time point can become inaccurate at another.

However, we can utilize intermittent incremental measurements from quantum
circuits to devise a Bayesian [73, 74, 75, 76] update for the current state of
the device noise:

$\Pr(\textbf{x}|\text{data})\propto\Pr(\text{data}|\textbf{x})\Pr(\textbf{x})$
(7.5)

where data refers to measurements obtained from circuit execution, x denotes
the multi-dimensional vector of parameters characterizing the device noise
(such as connection-specific CNOT fidelity and qubit-specific SPAM
fidelities), $\Pr(\textbf{x}|\text{data})$ is the posterior noise
distribution, $\Pr(\text{data}|\textbf{x})$ is the likelihood, and
$\Pr(\textbf{x})$ is the prior noise distribution. We use this to update
$\\{\eta_{{}_{\text{k}}}\\}$ via updates to model-specific parameters.

Obtaining the posterior distribution for x is a two-step process: first, we
estimate the posterior for uncorrelated parameters, and second, for correlated
parameters in the underlying noise model.

####  7.2.1 Uncorrelated parameters

Uncorrelated parameters refer to those model parameters for which there exist
datasets (such as the partial trace of the observable on a single qubit) where
the observed data is modeled by error due to only one noise parameter. Here,
the analysis is simpler as univariate priors can be used. For example, if we
disregard noise from single-qubit rotations, then the measurements obtained
from any qubit that was not subjected to entangling operations, can be
described by a one-parameter model under symmetric SPAM noise model.

Let $\mathcal{D}_{q}=\\{b_{q}(0),\cdots,b_{q}(L-1)\\}$ represent a dataset of
$L$ samples obtained for qubit $q$ after measurement in the computational
basis. Each $b_{q}(l)$ is a single-bit measured after the $l$-th execution of
$\tilde{\mathcal{G}}_{0}$. We can adopt a beta distribution as the prior for
the SPAM fidelity $f_{q}$ because the beta distribution is well-suited for
values restricted to the [0,1] interval and can effectively accommodate the
experimentally observed unimodal and skewed density as seen in Fig. 7.1 for
qubit 2. The beta distribution’s flexibility allows for an accurate fit to
these characteristics. The Beta distribution with parameters
$\alpha_{q},\beta_{q}$ is given by
$\text{Beta}(f_{q};\alpha_{q},\beta_{q})=f_{q}^{\alpha_{q}-1}(1-f_{q})^{\beta_{q}-1}/B(\alpha_{q},\beta_{q})$
where the normalizing denominator $B(\alpha_{q},\beta_{q})$ is the Beta
function defined as $B(m,n)=\Gamma(m)\Gamma(n)/\Gamma(m+n)$, and
$\Gamma(\cdot)$ is the gamma function given by:
$\Gamma(y)=\int\limits_{0}^{\infty}t^{y-1}e^{-t}dt$, defined for any positive
y.

The prior for the mean $\mu_{q}^{\text{prior}}$ and variance
$v_{q}^{\text{prior}}$ of the SPAM fidelity $f_{q}$ can be derived from
historical characterization data (e.g. using data post calibration). If such
data is not accessible, we can obtain the starting point from a small
perturbation to the ideal value. The prior parameters are then obtained as:
$\alpha_{q}^{\text{prior}}=\mu_{q}^{\text{prior}}\left[\mu_{q}^{\text{prior}}(1-\mu_{q}^{\text{prior}})/v_{q}^{\text{prior}}-1\right]$
and
$\beta_{q}^{\text{prior}}=(1-\mu_{q}^{\text{prior}})\left[\mu_{q}^{\text{prior}}(1-\mu_{q}^{\text{prior}})/v_{q}^{\text{prior}}-1\right]$.

The likelihood is obtained as:

$\mathcal{L}(f_{q})=f_{q}^{C_{0}[\mathcal{D}_{q}]}(1-f_{q})^{L-C_{0}[\mathcal{D}_{q}]}/B(\alpha_{q},\beta_{q})^{L}$
(7.6)

where $C_{0}[\mathcal{D}_{q}]$ counts the number of 0’s in the dataset
$\mathcal{D}_{q}$. The updated posterior density (indicated by the prime on
the updated parameters) for the SPAM fidelity ($f_{q}$) [161] :

$f_{q}|\mathcal{D}_{q}\sim\text{Beta}(\alpha_{q}^{\prime},\beta_{q}^{\prime})$
(7.7)

where $\alpha_{q}^{\prime}=\alpha_{q}+L-C_{0}[\mathcal{D}_{q}]$ and
$\beta_{q}^{\prime}=\beta_{q}+C_{0}[\mathcal{D}_{q}]$. This shows the
influence of incremental measurements on posterior noise density. The qubit-
wise updated mean [162] of the SPAM fidelity, obtained as
$\mu_{q}^{\prime}=\alpha_{q}^{\prime}/(\alpha_{q}^{\prime}+\beta_{q}^{\prime})$,
are then used in estimating the PEC coefficients.

####  7.2.2 Correlated parameters

The second task involves the adaptive estimation of correlated noise
parameters, which is more complex due to the measurements being influenced by
multiple noise processes simultaneously. These are the noise parameters for
which there exist datasets (typically the partial trace of an observable
across a subset of qubits) which reflect errors from various noise parameters
jointly. The analysis uses the probability of observing classical bit-strings
on this qubit subset as the random variables.

For a subset of $m$ qubits from a total of $n$ qubits, there are $M=2^{m}$
possible observed bit-strings, each with a probability denoted by
$p_{0},p_{1},\ldots,p_{M-1}$, where $p_{0}$ represents the probability of
observing all zeros and $p_{M-1}$ that of all ones. These probabilities are
treated as random variables which form a probability simplex, as they are all
positive and add up to 1. Hence the natural way to model the joint density is
using a Dirichlet prior given by:

$\text{Pr}(p_{0},\cdots,p_{M-1})=\frac{\Gamma(\sum
a_{i})}{\prod\limits_{0}^{M-1}\Gamma(a_{i})}\prod\limits_{0}^{M-1}p_{i}^{a_{i}-1}$
(7.8)

where the gamma function $\Gamma(\cdot)$ was already defined previously and
$\\{a_{i}:a_{i}>0,i\in\\{0,\cdots M-1\\}\\}$ are the parameters characterizing
the Dirichlet distribution that need to be estimated and updated – a multi-
variate task analogous to the uni-variate task for the beta distribution in
the previous section.

We denote the dataset derived from $L$ measurements of the $m$ qubits in the
computational basis as $\mathcal{D}=\\{w(0),\cdots,w(L-1)\\}$, where each
$w(i)$ is a binary string of length $m$. The likelihood function is given by:

$\mathcal{L}(p_{0},\cdots,p_{M-1})=\prod\limits_{i=0}^{M-1}p_{i}^{C_{v_{i}}[\mathcal{D}]}$
(7.9)

where $v_{i}\in\\{0,1\\}^{m}$, $C_{v_{i}}[\mathcal{D}]$ is the count of
occurrences of the string $v_{i}$ in the dataset $\mathcal{D}$. Note that
$w(i)$ is a specific realization post-measurement that takes one of the values
in the set $\\{v_{i}\\}$.

The posterior joint density is also a Dirichlet distribution [163]:

$\Pr(p_{0},\cdots,p_{M-1}\mid\mathcal{D})=\frac{\Gamma(\sum
a_{i}^{\prime})}{\prod\limits_{i=0}^{M-1}\Gamma(a_{i}^{\prime})}\prod\limits_{i=0}^{M-1}p_{i}^{a_{i}^{\prime}-1}$
(7.10)

with parameters $a_{i}^{\prime}=a_{i}+C_{v_{i}}(\mathcal{D})$. Upon
marginalizing this joint density, the marginals follow a Beta distribution.
For example, the random variable $p_{i}$ is Beta-distributed with parameters
$\alpha_{i}=a_{i}^{\prime}$ and
$\beta_{i}=-a_{i}^{\prime}+\sum\limits_{j=0}^{M-1}a_{j}^{\prime}$ [164].

The updated marginals from the Dirichlet distribution give us the evolving
densities for each $p_{i}$, allowing us to calculate their time-varying means
and variances. Specifically, the mean for $p_{i}$ updates to
$a_{i}^{\prime}/\sum a_{i}^{\prime}$, and its variance updates to
$a_{i}^{\prime}\left(\sum_{j}a_{j}^{\prime}-a_{i}^{\prime}\right)/\left(\sum_{j}a_{j}^{\prime}\right)^{2}/\left(1+\sum_{j}a_{j}^{\prime}\right)$.

At the final stage, the method employs the relationship between the
probabilities $p_{0},\cdots,p_{M-1}$ and the noise model parameters to derive
the time-dependent parameter means and variances of the noise parameters. The
process concludes by updating the quasi-probability distribution using the
updated noise parameters per Eqn. 7.5.

####  7.2.3 Accuracy and stability

We next evaluate the performance of PEC in presence of non-stationary noise
using the lens of accuracy and stability. A mean of a quantum observable
$\mathcal{O}$ is represented as $\braket{\mathcal{O}}_{\textrm{x}}$ (where x
labels the noise instance), while the observed mean after mitigation is
denoted as $\braket{\mathcal{O}}_{\textrm{x}}^{\text{mit}}$, which equals
$\braket{\mathcal{O}}$ only in the asymptotic limit of infinite samples and
zero noise. When we want to say the mitigated noisy mean at a specific time
$t$, we will use $\braket{\mathcal{O}}_{\textrm{x}}^{\text{mit}}(t)$.

We say that a PEC mitigated observable is $\epsilon-$accurate if the absolute
difference between the mitigated and noiseless observable is upper bounded by
$\epsilon$:

$\displaystyle|\braket{O}_{\textrm{x}}^{\text{mit}}-\braket{O}|\leq\epsilon$
(7.11)

Now, suppose the underlying noise, being non-stationary, is characterized by a
time-dependent density $f(\textrm{x};t)$. We say that a PEC mitigated
observable is $\epsilon-$stable between times $t_{1}$ and $t_{2}$ if:

$|\braket{O}_{\textrm{x}}^{\text{mit}}(t_{1})-\braket{O}_{\textrm{x}}^{\text{mit}}(t_{2})|\leq\epsilon$
(7.12)

where

$\begin{split}&\braket{O}_{\textrm{x}}^{\text{mit}}(t)=\int\limits_{\textrm{x}}\braket{O}_{\textrm{x}}^{\text{mit}}f(\textrm{x};t)d\textrm{x}\\\
&=\int\limits_{\textrm{x}}\sum\limits_{k}\eta_{{}_{\text{k}}}(\textrm{x},t)\text{Tr}\left[\mathcal{O}\tilde{\mathcal{G}}_{k}\rho\right]f(\textrm{x};t)d\textrm{x}\end{split}$
(7.13)

Note the dependence of $\eta_{{}_{\text{k}}}$ on the random variable x and
time $t$.

When measuring $\mathcal{O}_{q}$ for qubit $q$, the qubit-wise accuracy metric
is:

$\epsilon_{q}=\left|\braket{\mathcal{O}_{q}}_{\textrm{x}}-\braket{\mathcal{O}_{q}}\right|$
(7.14)

without mitigation, and

$\epsilon_{q}=\left|\braket{\mathcal{O}_{q}}_{\textrm{x}}^{\text{mit}}-\braket{\mathcal{O}_{q}}\right|$
(7.15)

with error mitigation. The register average, for a $n$-qubit register, is:

$\epsilon_{R}=\sum\limits_{q=0}^{n-1}\epsilon_{q}/n$ (7.16)

Similarly, the qubit-wise stability metric is:

$s_{q}(t)=\left|\braket{\mathcal{O}_{q}}_{\textrm{x}}(t)-\braket{\mathcal{O}_{q}}_{\textrm{x}}(0)\right|$
(7.17)

without mitigation, and

$s_{q}(t)=\left|\braket{\mathcal{O}_{q}}_{\textrm{x}}^{\text{mit}}(t)-\braket{\mathcal{O}_{q}}_{\textrm{x}}^{\text{mit}}(0)\right|$
(7.18)

with error mitigation. The register average, for a $n$-qubit register, is:

$s_{R}(t)=\sum\limits_{q=0}^{n-1}s_{q}(t)/n.$ (7.19)

We expect the accuracy ($\epsilon_{q},\epsilon_{R}$) and stability
$(s_{q},s_{R})$ metrics to be smaller when using adaptive PEC because of more
accurate estimates for the time-varying PEC coefficients
$\\{\eta_{{}_{\text{k}}}\\}$.

###  7.3 Numerical Validation

For studying the stability and accuracy of PEC in presence of non-stationary
noise, we use an implementation of the Bernstein-Vazirani algorithm [109], a
standard benchmarking circuit that requires only a modest number of gates. The
purpose of the algorithm is to recover a $n$-bit secret string $r$, encoded in
a black-box oracle function. The algorithm identifies the secret with a single
query, while the classical method needs $2^{n}$ queries (worst-case).

A quantum circuit for the Bernstein-Vazirani algorithm is shown in Fig. 7.2.
We conceptualize the circuit as having four principal layers
($\mathds{L}_{1},\mathds{L}_{2},\mathds{L}_{3}$, and $\mathds{L}_{4}$)
separated by dashed vertical lines. Note that the total number of qubits
needed by an $n-1$-bit secret string is $n$, comprising $n-1$ data qubits and
one ancilla qubit. So, a 5-qubit implementation of the Bernstein-Vazirani
algorithm has $n=5$ but the secret string length is $n-1=4$.

The circuit is initialized with all the qubits in the register in the
$\ket{0}$ state. Thus, a pure state description yields an initial state
$\ket{\psi_{0}}=\ket{0}^{\otimes n}$. In the first layer, all the qubits are
subjected to Hadamard gates. The ancilla qubit (which is the last qubit with
index n) is additionally subjected to a $\mathds{Z}$ gate. The output after
the first layer
$\mathds{L}_{1}=\mathds{H}^{\otimes{n-1}}\otimes(\mathds{Z}_{n}\mathds{H}_{n})$
is $\ket{\psi_{1}}=\ket{+}^{\otimes{n-1}}\ket{-}$.

The second layer implements the oracle function for which we can use CNOT
gates. Each CNOT’s control qubit corresponds to one of the bits in the secret
string $r$, while the target qubit remains fixed at qubit $n$. Specifically,
if the bit $r_{i}$ of the secret string $r$ is 1, we add a CNOT between qubit
$i$ (control) and qubit $n$ (target). The input to the second layer is
$\mathds{H}^{\otimes{n-1}}\ket{0}\ket{-}$ while the output is
$\ket{\psi_{2}}=\left(\mathds{H}^{\otimes{n-1}}\ket{r}\right)\ket{-}$ where
$r$ is the secret string.

To retrieve the secret string $r$, the third layer requires another layer of
Hadamard gates. A $\mathds{Z}$ gate is applied to the ancilla qubit (qubit
$n$) before the application of the Hadamard layer to make the computing
reversible. The output quantum state after the third layer
$\mathds{L}_{3}=\mathds{H}^{\otimes(n-1)}\otimes(\mathds{H}_{n}\mathds{Z}_{n})$
is $\ket{\psi_{3}}=\ket{r}\ket{0}$.

####  7.3.1 Observable

The fourth layer is for measurement in the $\mathds{Z}$-basis and is not a
unitary layer. We measure the state of each of the $n$-qubits after projection
onto the computational basis states. Post-measurement, the observation
obtained is a classical bit (either $0$ or $1$) for each of the n-qubits
measured. Thus, the final observed output is a bit-string of length $n$,
including the ancilla.

We make a distinction between the qubit-wise observables $\mathcal{O}_{q}$,
where $q\in\\{0,\cdots n\\}$ and the measurement operator for the register
$M_{Z}=\mathds{Z}_{0}\otimes\cdots\mathds{Z}_{n}$ in computational basis. The
qubit-wise observable $\mathcal{O}_{q}$ has identity across the $n$-qubit
tensor product except in the $q$-th position. For example,
$\mathcal{O}_{0}=\mathds{Z}_{0}\otimes\mathds{I}^{\otimes{n-1}}$ and
$\mathcal{O}_{n}=\mathds{I}^{\otimes{n-1}}\otimes\mathds{Z}_{n}$. The
eigenvalues of $\mathcal{O}_{q}$ are +1 (corresponding to classical bit 0) and
-1 (corresponding to classical bit 1). The theoretical mean of the qubit-wise
observable $\mathcal{O}_{q}$ is denoted by
$\braket{\mathcal{O}_{q}}=\textrm{Tr}[\mathcal{O}_{q}\rho]$ where
$q\in\\{0,\cdots,n\\}$, which reflects the measurement process being
fundamentally probabilistic.

The experimentally observed mean of the observable, denoted by
$\braket{\mathcal{O}_{q}}_{\textrm{x}}$ is calculated as the sum of
eigenvalues weighted by their empirically observed frequencies. In a noiseless
circuit, the observed mean asymptotically converges to the theoretical mean as
the sample size tends to infinity. However, in presence of shot noise and non-
zero variance of x, $\braket{\mathcal{O}_{q}}_{\textrm{x}}$ may not equal
$\braket{\mathcal{O}_{q}}$.

The experimentally observed measurements of the $M_{Z}$ operator in
computational basis belong to one of the $2^{n}$ eigenstates, each of which
can be represented by a $n$-bit binary string. These observations contain the
information necessary for computing $\mathcal{O}_{q}$ for all $q$. In the
context of the Bernstein-Vazirani algorithm, we discard the ancilla bit and
declare the search a success when the first $n$-bits of the observed value of
the $M_{Z}$ operator matches the secret string $r$.

####  7.3.2 Modeling circuit noise

Our experimental focus is on superconducting hardware in next section. The
potential sources of noise in a superconducting implementation of the circuit
depicted in Fig. 7.2 are: (i) state preparation noise, (ii) noise in the
implementation of the Hadamard gate, (iii) noise in the implementation of the
$\mathds{Z}$ gate, (iv) noise in the implementation of the CNOT gate, and (v)
measurement noise (also known as readout noise). The error resulting from the
first and last noise sources, namely state preparation noise and measurement
noise, are often measured collectively due to experimental limitations. This
combined noise is commonly referred to as SPAM (state preparation and
measurement) noise. An effective model assumes that the state preparation is
noiseless, and the noise impacts the measurement (or readout) process. The
relative magnitudes of the different types of noise are different. State
preparation and measurement (SPAM) noise typically has the largest
contribution to errors [28]. The next most significant contribution arises
from imperfect implementations of entangling gates, such as the CNOT. After
that the strength of the noise, for single-qubit rotations, decreases by two
orders of magnitude or more [27, 28]. The $\mathds{Z}$ gate is a software-
based operation [165] and error-free. After disregarding single-qubit rotation
errors, only two predominant types of noise emerge: SPAM noise and CNOT noise.

Next, we create a quantum channel based description of the circuit noise. The
circuit noise can be modeled layer-wise. The output density matrix prior to
measurement can be represented as:
$\tilde{\rho}=\mathcal{E}_{4}(\mathcal{E}_{3}(\mathds{L}_{3}\mathcal{E}_{2}(\mathds{L}_{2}\mathcal{E}_{1}(\mathds{L}_{1}\rho\mathds{L}_{1}^{\dagger})\mathds{L}_{2}^{\dagger})\mathds{L}_{3}^{\dagger})$
where $\mathcal{E}_{k}(\cdot)$ denotes the noise channel for the $k$-th layer
of the circuit in Fig. 7.2. We approximate $\mathcal{E}_{1}$ as identity
channel because we ignore single-qubit errors.

We model noise in the CNOT gate using a 2-qubit depolarizing model. Let
$\textrm{x}_{C}(t)$ and $\textrm{x}_{T}(t)$ represent the stochastic
depolarizing parameters for the control and target qubits at time t. The
depolarizing noise model for the CNOT gate is represented by:

$\begin{split}&\mathcal{E}_{\text{CNOT}}(\cdot)=[1-\textrm{x}_{T}(t)][1-\textrm{x}_{C}(t)](\cdot)+\\\
&\frac{1-\textrm{x}_{C}(t)}{3}\textrm{x}_{T}\sum\limits_{\mathds{P}_{T}^{\prime}}(\mathds{I}_{C}\otimes\mathds{P}_{T}^{\prime})(\cdot)(\mathds{I}_{C}\otimes\mathds{P}_{T}^{\prime})\\\
&+\frac{1-\textrm{x}_{T}(t)}{3}\textrm{x}_{C}\sum\limits_{\mathds{P}_{C}^{\prime}}(\mathds{P}_{C}^{\prime}\otimes
I_{T})(\cdot)(\mathds{P}_{C}^{\prime}\otimes I_{T})\\\
&+\frac{\textrm{x}_{C}(t)\textrm{x}_{T}}{9}\sum\limits_{\mathds{P}_{C}^{\prime},\mathds{P}_{T}^{\prime}}(\mathds{P}_{C}^{\prime}\otimes\mathds{P}_{T}^{\prime})(\cdot)(\mathds{P}_{C}^{\prime}\otimes\mathds{P}_{T}^{\prime})\end{split}$
(7.20)

In this expression, we use
$\mathds{P}_{C}^{\prime},\mathds{P}_{T}^{\prime}\in\\{\mathds{X},\mathds{Y},\mathds{Z}\\}$
as the single-qubit Pauli operators excluding identity $\mathds{I}$, acting on
the control (C) and target (T) qubits respectively. The sum is over all the
single-qubit Pauli operators, excluding identity. The effect of the quantum
noise channel $\mathcal{E}_{2}(\cdot)$ for layer 2, on a 5-qubit state,
combines the identity channel $\mathcal{I}$ (which acts on qubits without CNOT
connection) and $\mathcal{E}_{\text{CNOT}}$ (which acts on qubits linked by
CNOT connections).

Similar to layer 1, layer 3 also comprises single-qubit rotations only. Hence,
we treat $\mathcal{E}_{3}(\cdot)$ as identity channel. Lastly, the fourth
layer is subjected to SPAM noise which we adapt to a noise channel description
[105].  It effectively handles measurement noise by corrupting the density
matrix post execution but pre measurement, and then conducting noise-free
projective measurements on the corrupted output.  The SPAM noise channel for
qubit $q$ has two Kraus operators $M_{0}$ and $M_{1}$:

$\begin{split}M_{0}=&\sqrt{f_{q}}\ket{0}\bra{0}+\sqrt{1-f_{q}}\ket{1}\bra{1}\\\
M_{1}=&\sqrt{1-f_{q}}\ket{0}\bra{0}+\sqrt{f_{q}}\ket{1}\bra{1}\\\ \end{split}$
(7.21)

where $f_{q}$ represents the SPAM fidelity of qubit $q$. The probability of
observing $0$ is given by $\text{Tr}[M_{0}^{\dagger}M_{0}\rho]$ and the
probability of observing $1$ is given by
$\text{Tr}[M_{1}^{\dagger}M_{1}\rho]$. Note that we have assumed a symmetric
model for SPAM noise with $f_{q}$ denoting the average SPAM fidelities for the
initial states prepared as $\ket{0}$ and $\ket{1}$.

Neglecting inter-qubit cross-talk, we then have the noise channel
representation for the last layer $\mathcal{E}_{4}$ as a separable SPAM noise
channel:

$\mathcal{E}_{4}(\cdot)=\left[\bigotimes\limits_{q=0}^{n}\mathcal{E}_{q}^{\text{SPAM}}\right](\cdot)$
(7.22)

####  7.3.3 PEC coefficients under Pauli noise

We first discuss CNOT noise mitigation using PEC. Following that, we discuss
single-qubit SPAM noise mitigation using PEC. Then, we integrate the two
discussions for noise mitigation in the quantum circuit implementation of the
Bernstein-Vazirani algorithm using PEC.

#####  7.3.3.1 CNOT noise

Let $\tilde{\mathcal{G}_{0}}$ denote the super-operator for the noisy CNOT
operation. Under the Pauli channel assumption, there are 16 basis super-
operators, indexed by $k$:

$\tilde{\mathcal{G}_{k}}\rho=(\mathds{P}_{C}\otimes\mathds{P}_{T})\left(\tilde{\mathcal{G}_{0}}\rho\right)(\mathds{P}_{C}\otimes\mathds{P}_{T})$
(7.23)

where $\rho$ is a 2-qubit density matrix,
$\mathds{P}_{C},\mathds{P}_{T}\in\\{\mathds{I},\mathds{X},\mathds{Y},\mathds{Z}\\}$
are the single-qubit Paulis acting on the control (C) and target (T) qubits
respectively, and $k\in\\{0,\cdots 15\\}$. The index $k$ is determined by the
specific combination of $\mathds{P}_{C}$ and $\mathds{P}_{T}$, starting from
$\mathds{I}\otimes\mathds{I}$ for $k=0$ and ending with
$\mathds{Z}\otimes\mathds{Z}$ for $k=15$, incrementing $k$ for each subsequent
combination in the sequence $\mathds{I},\mathds{X},\mathds{Y},\mathds{Z}$
applied to $\mathds{P}_{C}$ and $\mathds{P}_{T}$.

When $\mathds{P}_{C}=\mathds{P}_{T}=\mathds{I}$, the circuit corresponds to
the noisy super-operator $\tilde{\mathcal{G}_{0}}$ denoting the noisy CNOT
gate available. An example of one of the remaining 15 PEC circuits for CNOT
noise mitigation is shown in Fig. 7.3.

Using the requirement that the linear combination
$\sum\eta_{{}_{k}}\tilde{\mathcal{G}}_{k}$ should equal the noiseless CNOT
operation, we derive the PEC coefficients as follows.

Let
$c_{0}=3/[\textrm{x}_{T}(1-\textrm{x}_{C})]^{2}+[\textrm{x}_{C}(1-\textrm{x}_{T})]^{2}-3c_{1}^{2}]$
and, $c_{1}=\textrm{x}_{C}+\textrm{x}_{T}-\textrm{x}_{C}\textrm{x}_{T}-1$.

When $\mathds{P}_{C}=\mathds{P}_{T}=\mathds{I}$,

$\eta_{{}_{\text{0}}}=c_{0}c_{1}.$

When $\mathds{P}_{C}=\mathds{I}$ and
$\mathds{P}_{T}\in\\{\mathds{X},\mathds{Y},\mathds{Z}\\}$,

$\eta_{{}_{\text{k}}}=c_{0}\frac{\textrm{x}_{T}(1-\textrm{x}_{C})}{3},\;\;\;\;k\in\\{1,2,3\\}.$

When $\mathds{P}_{T}=\mathds{I}$ and
$\mathds{P}_{C}\in\\{\mathds{X},\mathds{Y},\mathds{Z}\\}$,

$\eta_{{}_{\text{k}}}=c_{0}\frac{\textrm{x}_{C}(1-\textrm{x}_{T})}{3},\;\;\;\;k\in\\{4,8,12\\}.$

For all the remaining terms,

$\eta_{{}_{\text{k}}}=c_{0}\frac{\textrm{x}_{C}\textrm{x}_{T}}{9},\;\;\;\;k\in\\{9,10,11,13,14,15\\}.$

This yields the quasi-probability distribution as:
$\\{\mathds{Q}_{k}\\}=\left\\{|\eta_{{}_{0}}|/\gamma_{{}_{\text{CNOT}}},\cdots,|\eta_{{}_{15}}|/\gamma_{{}_{\text{CNOT}}}\right\\}$
where $\gamma_{{}_{\text{CNOT}}}=|\eta_{{}_{0}}|+\cdots+|\eta_{{}_{15}}|.$

#####  7.3.3.2 SPAM noise for qubit q

Consider a single-qubit SPAM noise channel for qubit $q$. The four noisy
super-operators that can be implemented in this case are: (i) the SPAM noise
channel:
$\tilde{\mathcal{G}}_{\mathds{I}}(\rho)=f_{q}\rho+(1-f_{q})\mathds{X}\rho\mathds{X}$,
(ii) the SPAM noise channel followed by an X error:
$\tilde{\mathcal{G}}_{\mathds{X}}(\rho)=f_{q}\mathds{X}\rho\mathds{X}+(1-f_{q})\rho$,
(iii) the SPAM noise channel followed by a Y error:
$\tilde{\mathcal{G}}_{\mathds{Y}}(\rho)=f_{q}\mathds{Y}\rho\mathds{Y}+(1-f_{q})\mathds{Z}\rho\mathds{Z}$
and, (iv) the SPAM noise channel followed by a Z error:
$\tilde{\mathcal{G}}_{\mathds{Z}}(\rho)=f_{q}\mathds{Z}\rho\mathds{Z}+(1-f_{q})\mathds{Y}\rho\mathds{Y}$.

Solving the linear equation:

$\mathcal{I}=\eta_{0}\tilde{\mathcal{G}}_{I}+\eta_{1}\tilde{\mathcal{G}}_{X}+\eta_{2}\tilde{\mathcal{G}}_{Y}+\eta_{3}\tilde{\mathcal{G}}_{Z}$
(7.24)

we get the quasi-probability distribution as:
$\left\\{|\eta_{{}_{\text{0}}}|/\gamma_{{}_{\text{SPAM}}},|\eta_{{}_{\text{1}}}|/\gamma_{{}_{\text{SPAM}}},0,0\right\\}$
with

$\begin{split}&\eta_{{}_{\text{0}}}=\frac{f_{q}}{2f_{q}-1},\;\;\;\;\eta_{{}_{\text{1}}}=-\frac{1-f_{q}}{2f_{q}-1},\;\;\;\;\eta_{{}_{\text{2}}}=\eta_{{}_{\text{3}}}=0\\\
&\text{sgn}(\eta_{{}_{\text{0}}})=+1,\;\;\;\;\text{sgn}(\eta_{{}_{\text{1}}})=-1\\\
&\gamma_{{}_{\text{SPAM}}}(q)=|\eta_{{}_{\text{0}}}|+|\eta_{{}_{\text{1}}}|\\\
\end{split}$ (7.25)

Estimating $f_{q}$ therefore provides a complete specification of the PEC
coefficients.

#####  7.3.3.3 Composite noise in 5-qubit implementation of Bernstein-Vazirani
algorithm

The two noisy basis super-operators for each of the 5 distinct SPAM noise
channels and the 16 noisy basis super-operators for the CNOT noise channel
leads to 512 ($16\times 2^{5}$) noisy basis circuits $\\{\mathcal{G}_{k}\\}$
for the 5-qubit Bernstein-Vazirani circuit, where $k$ runs from 0 to 511.

Under the SPAM noise separability assumption, $\gamma$ is obtained as:

$\gamma=\gamma_{{}_{CNOT}}\prod\limits_{q=0}^{4}\gamma_{{}_{SPAM}}(q)$ (7.26)

where $\gamma_{{}_{SPAM}}(q)$ refers to the $\gamma_{{}_{SPAM}}$ for the
$q$-th qubit. The PEC coefficients for the Bernstein-Vazirani circuit are the
elements of:

$\left\\{\eta_{{}_{\text{0}}}^{\text{CNOT}},\cdots,\eta_{{}_{\text{15}}}^{\text{CNOT}}\right\\}\times\prod\limits_{q=0}^{4}\left\\{\eta_{{}_{\text{0}}}^{\text{SPAM}}(q),\eta_{{}_{\text{1}}}^{\text{SPAM}}(q)\right\\}$
(7.27)

The quasi-probability distribution then follows as:
$\left\\{\mathds{Q}_{k}\right\\}=\left\\{|\eta_{{}_{\text{k}}}^{\text{BV}}|/\gamma\right\\}$.

####  7.3.4 Adaptive noise model

In the context of our 5-qubit Bernstein-Vazirani setup, the noise parameters
$f_{0}$, $f_{1}$, and $f_{2}$ are estimated using the method for uncorrelated
parameters as they do not have CNOT correlations. The adaptive estimation of
parameters $f_{3},f_{4},\textrm{x}_{C},$ and $\textrm{x}_{T}$ employs the
method for handling correlated parameters, detailed in Sec. 7.2. The process
is initiated within the Bayesian inference framework, which considers the
probabilities of observing outcomes 00, 01, 10, and 11 on qubits 3 and 4, as
depicted in Fig. 7.2, to be random variables. The estimation of the time-
varying means and variances of these correlated noise parameters
($f_{3},f_{4},\textrm{x}_{C},\textrm{x}_{T}$) is achieved by associating the
mean values of the estimated densities directly with the parameters of the
correlated noise model using:

$\Pr(i)=\textrm{Tr}\left[\Pi_{i}\mathcal{E}_{4}(\mathds{L}_{3}\mathcal{E}_{2}(\mathds{L}_{2}\mathds{L}_{1}\rho\mathds{L}_{1}^{\dagger}\mathds{L}_{2}^{\dagger})\mathds{L}_{3}^{\dagger})\right]$
(7.28)

where $\Pi_{i}=\ket{i}\bra{i}$ are the projection operators and
$i\in\\{00,01,10,11\\}$. The probabilities are given by:

$\begin{split}\text{Pr}(00)=&f_{3}f_{4}(-1-\textrm{x}_{C}\textrm{x}_{T}+\textrm{x}_{C}+\textrm{x}_{T})\\\
&+f_{3}(\textrm{x}_{C}\textrm{x}_{T}/2-\textrm{x}_{T}/2)\\\
&+f_{4}(1-\textrm{x}_{T}-\textrm{x}_{C}/2+\textrm{x}_{C}\textrm{x}_{T}/2)\\\
&-\textrm{x}_{C}\textrm{x}_{T}/4+\textrm{x}_{T}/2\\\ \end{split}$ (7.29)

$\begin{split}\text{Pr}(01)=&f_{3}f_{4}(1-\textrm{x}_{C}-\textrm{x}_{T}+\textrm{x}_{C}\textrm{x}_{T})\\\
&+f_{3}(-1-\textrm{x}_{C}\textrm{x}_{T}/2+\textrm{x}_{C}-\textrm{x}_{T}/2)\\\
&+f_{4}(-1-\textrm{x}_{C}\textrm{x}_{T}/2+\textrm{x}_{C}+\textrm{x}_{T})\\\
&+1+\textrm{x}_{C}\textrm{x}_{T}/4-\textrm{x}_{C}/2-\textrm{x}_{T}/2\\\
\end{split}$ (7.30)

$\begin{split}\text{Pr}(10)=&f_{3}f_{4}(1-\textrm{x}_{T}-\textrm{x}_{C}+\textrm{x}_{C}\textrm{x}_{T})\\\
&+f_{3}(\textrm{x}_{T}/2-\textrm{x}_{C}\textrm{x}_{T}/2)\\\
&+f_{4}(\textrm{x}_{C}/2-\textrm{x}_{C}\textrm{x}_{T}/2)\\\
&+\textrm{x}_{C}\textrm{x}_{T}/4\\\ \end{split}$ (7.31)

$\begin{split}\text{Pr}(11)=&f_{3}f_{4}(-1+\textrm{x}_{C}+\textrm{x}_{T}-\textrm{x}_{C}\textrm{x}_{T})\\\
&+f_{3}(1-\textrm{x}_{T}/2-\textrm{x}_{C}+\textrm{x}_{C}\textrm{x}_{T}/2)\\\
&+f_{4}(\textrm{x}_{C}\textrm{x}_{T}/2-\textrm{x}_{C}/2)\\\
&+\textrm{x}_{C}/2-\textrm{x}_{C}\textrm{x}_{T}/4\\\ \end{split}$ (7.32)

In the last step, the updated PEC coefficients are obtained using Eqn. 7.27.

####  7.3.5 Numerical simulation

For our 5-qubit circuit implementing the Bernstein-Vazirani algorithm, we used
secret string $r=``1000"$. Thus, the qubit-wise mean of the $\mathds{Z}$
observable for the noiseless case is given by:
$\braket{\mathcal{O}_{0}}=+1,\braket{\mathcal{O}_{1}}=+1,\braket{\mathcal{O}_{2}}=+1,\braket{\mathcal{O}_{3}}=-1,\braket{\mathcal{O}_{4}}=+1$,
qubit 4 being the ancilla.

To validate our method, we used a numerical experiment that conducts a density
matrix simulation of a 5-qubit noisy quantum circuit implementing the
Bernstein-Vazirani algorithm using the Qiskit [23] software. The simulation
begins with the mean of the beta distributions characterizing the SPAM
fidelities for qubits 0-4 set at $0.96,0.95,0.94,0.93,0.92$, respectively, and
the mean of the depolarizing channel parameters for the control and target
qubits in the CNOT gate both fixed at $0.017$. Over the course of ten
simulated time periods, the average SPAM fidelity for each qubit decreased by
0.01 per period, resulting in final mean SPAM fidelities of
$0.86,0.85,0.84,0.83,0.82$ for qubits 0-4, respectively. Similarly, the
average depolarizing parameter for the CNOT gate also declined by 0.01 per
time period, leading to a final mean value of $0.117$ for both control and
target qubits by the simulation’s end. The noise parameters were adaptively
estimated, per the methodology described in previous sections, using data
generated by executing each of the 512 PEC circuits using $10,000$ shots.

Fig. 7.5 demonstrates the simulated efficacy of the adaptive PEC algorithm.
The plot compares the register accuracy and stability (averaged over the 5
qubits) achieved with the adaptive approach against a non-adaptive approach.
It shows an improvement in accuracy of 59.5% in the final time period, when
device noise is at its peak, with an average accuracy improvement of 53.4%
across all ten periods. Similarly, stability improved by 58.0% in the last
period, and by an average of 51.5% over the entire span of ten periods. The
improvement in accuracy and stability of the outcomes from adaptive PEC occur
due to more accurate noise characterizations using Bayesian inference.

###  7.4 Experimental Testing

We tested the adaptive PEC method on the 27-qubit superconducting device
called ibm_kolkata. Qubits 0,1,2,3,4 in Fig. 7.2 map to physical qubits
0,1,2,3,5 on the device shown in Fig. 7.6. The CNOT gate is between the
physical qubits 3 (control) and 5 (target) on ibm_kolkata. Our dataset spans
24 hours and comprises 13 complete PEC datasets. It was collected on January
15, 2024, and have the following time-stamps: 00:01 hrs, 02:03 hrs, 03:50 hrs,
05:43 hrs, 07:23 hrs, 09:11 hrs, 10:53 hrs, 12:39 hrs, 14:22 hrs, 16:09 hrs,
17:50 hrs, 19:33 hrs, and 21:17 hrs. Each dataset is derived from measurements
made in the computational basis, with each observation being a 5-bit string.
The observations were obtained from the 512 noisy basis circuits, as described
in Sec. 7.3.3, with each circuit repeated using $L=10,000$ shots, resulting in
approximately 67 million observations in total.

####  7.4.1 Non-stationary noise estimates

Fig. 7.7 illustrates how noise in the quantum computer changed over time. The
blue line in plot (a) shows the depolarizing parameter for the target qubit of
the CNOT gate, while the black line represents the depolarizing parameter for
the control qubit. Plot (b) shows five lines, each representing the SPAM
fidelity for the register elements. The x-axis denotes intra-calibration
timestamps.

The graph shows periods where the noise levels in the depolarizing parameter
for qubit 3 (the control qubit) are steady, notably between 2:00 am and 7:30
am, contrasting with times of significant fluctuation, as observed between
9:00 am and 2:30 pm. The depolarizing parameter fluctuates between 1% and 4%
for qubit 4 (the target qubit) and between 1% and 3% for qubit 3, both peaking
sharply at 10:53 am.

Qubit 4 experiences the most significant impact from SPAM noise, with its
values fluctuating between 0.99 and 0.94, a notable range given the
sensitivity of PEC to accurate noise estimations. In contrast, Qubit 3
maintains a consistent SPAM fidelity throughout the same period, indicated by
small error margins and a steady average value of 0.99. Meanwhile, Qubit 2
demonstrates a gradual drift in its values, starting from below 0.94 and
rising to 0.96. This progressive change suggests a systematic, non-random
trend that might be rectifiable with bias shift corrections. However, such
patterns are not uniform across the entire register, implying the necessity to
consider non-stationary statistics for modeling the system. Conducting
experiments in times of significant non-stationary activity, like from 7:30 am
to 2:30 pm, lead to more unstable outcomes when using non-adaptive PEC.

####  7.4.2 Non-stationary quasi-probability distribution

Fig. 7.8 underscores the importance of considering the non-stationary nature
of the quasi-probability distribution when implementing PEC, especially given
the lengthy data collection process required for a single PEC mitigation
(approximately 2 hours in our example). The abrupt change observed at 12:39
p.m. in the quasi-probability distribution directly correlates with the sharp
change in the noise parameters characterizing the quantum circuit at the same
time, as illustrated in Fig. 7.7.

To maintain clarity in Fig. 7.8, we have not plotted all 512 bins of the
histogram in one plot. Fig. 7.8 (a) displays the time-varying weight
$\mathds{Q}_{0}$ for the Bernstein-Vazirani circuit as-is without any
additional Pauli-gate added (we also refer to this as the raw Bernstein-
Vazirani circuit). If the circuit were noiseless, then the weight for the raw
circuit will be a constant 1. We observe a decrease in weight for the raw
circuit, dipping below 78% around 10:53 am from a peak of almost 83%,
coinciding with a peak in circuit noise as seen in Fig. 7.7. This decrease in
weight is expected as the circuit noise peaks.

The values of the quasi-probability bins for the next 10 basis circuits are
shown in plot (b), with values approximately 10 times lower than the first
circuit. Ignoring seemingly small coefficients in the quasi-probability
distribution without considering the precision of final reported results can
be risky. Subsequent basis circuits, not shown here, have significantly
smaller quasi-probability weights (around $10^{-4}$). Yet their collective
impact in a sum of 500 can be substantial, contributing up to 0.05. Given our
reported accuracy and stability are around $10^{-2}$, these coefficients,
though small, can significantly influence the results. Therefore, in our
analysis, we included all basis circuits without approximation, focusing on
the effects of non-stationary noise on PEC, rather than on resource
optimization.

####  7.4.3 Non-stationary PEC outcomes

The impact of the non-stationary noise can be seen in Fig. 7.9. The first
plot, labeled No mitigation, presents the accuracy and stability metrics
defined in Eqns. 7.16 and 7.19 respectively, for the raw Bernstein Vazirani
circuit without any form of quantum error mitigation. The second plot, labeled
ROEM, which stands for readout error mitigation, displays the metrics after
performing SPAM noise mitigation. In this case, the SPAM noise parameters are
held constant after initial device characterization. It deploys the standard
matrix inversion [28] technique for mitigation. The third plot, labeled non-
adaptive PEC, exhibits the accuracy and stability metrics for the Bernstein
Vazirani circuit with non-adaptive PEC. This method incorporates SPAM noise
mitigation within the PEC framework, as detailed in Sec. 7.3.3. The fourth
plot, labeled adaptive PEC, presents the metrics for the adaptive PEC method,
as discussed in Sec. 7.2.

Fig. 7.9 (a) shows the effectiveness of the adaptive PEC in enhancing result
accuracy. It reveals approximately a 42% improvement in accuracy on average
compared to the non-adaptive method. The observed accuracy benefit ranges from
a minimum of 25% to a maximum of 78%. Fig. 7.9 (b) shows the impact on result
stabilization. It shows an approximately 60% enhancement in stability on
average compared to the non-adaptive method. The observed stability benefit
ranges from a minimum of 8% to a maximum of 200%.

Observing the plots, it is evident that adaptive PEC significantly outperforms
standard PEC. Additionally, all four methods (no mitigation, ROEM, PEC, and
adaptive PEC) exhibit a time-series trend that deteriorates notably at 10:53
am. The observation correlates with the abrupt change in the underlying quasi-
probability distribution at 10:53 am as seen in Fig. 7.8.

Both the accuracy and stability metrics at 12:39 p.m. are slightly worse for
adaptive PEC compared to non-adaptive PEC. This discrepancy stands out as the
only instance where adaptive PEC performs sub-optimally. It seems,
surprisingly, that the stale data serves as a better noise estimate for this
specific time-point. However, the noise at 12:39 p.m. is not necessarily
closer to the noise at the starting time-stamp of 00:01 a.m., as demonstrated
in Fig. 7.7. While adaptive PEC succeeds in most cases, the learning process
is not instantaneous. There exists a slight lag in learning due to the
influence of prior information on the final estimate. This phenomenon reflects
a fundamental aspect of learning methods: the presence of memory, which can
aid learning but also slows down adaptation to the fast, spiky changes. The
sharp change in noise at 10:53 a.m. leads to an overestimation of the noise
estimate compared to when using the initial value at 00:01 a.m. as a reference
point. Residual errors in the noise estimates likely arises from inaccurate
models and non-stationary processes changing at a faster rate than sampled
here.

Figure 7.1: Non-stationary distribution functions of the state preparation and
measurement (SPAM) fidelity for qubit 2 on ibm_kolkata superconducting device
collected on Jan 15, 2024. Figure 7.2: A 5-qubit implementation of the
Bernstein-Vazirani algorithm with secret bit-string $r$. Figure 7.3: Circuit
diagram for implementing one of the 16 noisy super-operators for CNOT noise
mitigation using PEC. The initial noisy CNOT gate is followed up by
$\mathds{Y}$ and $\mathds{Z}$ gates on the control and target qubits
respectively. Figure 7.4: One of the 512 noisy basis circuits for mitigation
using PEC in the 5-qubit implementation of the Bernstein-Vazirani circuit. The
initial noisy CNOT gate is followed up with $\mathds{Y}$ and $\mathds{Z}$
gates on the control and target qubits respectively. The readout lines for
qubit $0,1,2,3,4$, and $5$ are subjected to the Pauli gates
$\mathds{X},\mathds{I},\mathds{X},\mathds{X},\mathds{I}$ respetively, prior to
measurement, in this specific noisy basis circuit for PEC.

(a)

(b)

Figure 7.5:  Comparison of the adaptive PEC algorithm with a non-adaptive
approach for a density matrix simulation that implements a noisy 5-qubit
circuit for solving the Bernstein Vazirani problem. (a) Average accuracy
across the 5-qubits and (b) Average stability across the 5-qubits.  Figure
7.6: Qubit layout of the 27-qubit superconducting device ibm_kolkata. Circles
represent superconducting transmons and lines indicate possible gate
operations between sites.

(a)

(b)

Figure 7.7:  This figure depicts the non-stationary noise on the experimental
device ibm_kolkata. In plot (a), the blue line represents the depolarizing
parameter for the target qubit, while the black line denotes the depolarizing
parameter for the control qubit in the CNOT gate. Plot (b) illustrates five
lines, each indicating the SPAM fidelity for the register elements. The x-axis
corresponds to intra-calibration timestamps for January 15. The shaded regions
denote the time-varying standard deviations.

(a)

(b)

Figure 7.8:  These plots depict the non-stationary nature of the quasi-
probability distribution. To maintain clarity, we opted not to plot all 512
bins of the histogram in a single plot. In (a), we observe the time-varying
weight $\mathds{Q}_{0}$ for the raw Bernstein-Vazirani circuit which carries
the most substantial weight, accounting for almost 80% of the distribution. In
(b), we show the values of the quasi-probability bins for the subsequent 10
basis circuits, which are more than 10 times lower in magnitude compared to
the first circuit. The non-stationary nature of the quasi-probability
distribution becomes crucial given the lengthy data collection process
required for PEC mitigation because the noise estimation becomes inaccurate in
these time-frames. Our experiment took approximately 2 hours for each dataset
comprising 512 circuits.

(a)

(b)

Figure 7.9:  The figure comprises two plots, each depicting four graphs: (1)
"No mitigation" presents raw Bernstein-Vazirani metrics without error
mitigation, (2) "ROEM" shows metrics after readout error mitigation, with
constant SPAM noise parameters, (3) "Non-adaptive PEC" displays metrics for
PEC, and (4) "Adaptive PEC" exhibits metrics for PEC with adaptive noise
mitigation. The x-axis denotes intra-calibration time-stamps (UTC) for Jan 15.
Plot (a) illustrates the time-varying accuracy metric from Eqn. 7.16. It
demonstrates the adaptive PEC’s 42% accuracy improvement over non-adaptive
PEC. Plot (b) shows the time-varying stability metric from Eqn. 7.19. It
illustrates the adaptive PEC’s 60% stability enhancement compared to non-
adaptive PEC. These plots underscore the significant impact of non-stationary
noise on PEC resilience. Due to PEC’s lengthy completion time (a couple of
hours), adaptive methods are able to handle non-stationary noise conditions
better.  Figure 7.10: Non-stationary distribution function of SPAM fidelity
for qubit 0 of ibm_kolkata as observed on Jan 15, 2024. Figure 7.11: Non-
stationary distribution function of SPAM fidelity for qubit 1 of ibm_kolkata
as observed on Jan 15, 2024. Figure 7.12: Time-varying density for probability
of observing ‘00’ on qubits 3 and 4 of ibm_kolkata, for the Bernstein-Vazirani
circuit, as observed on Jan 15, 2024. Figure 7.13: Time-varying density for
probability of observing ‘01’ on qubits 3 and 4 of ibm_kolkata, for the
Bernstein-Vazirani circuit, as observed on Jan 15, 2024. Figure 7.14: Time-
varying density for probability of observing ‘10’ on qubits 3 and 4 of
ibm_kolkata, for the Bernstein-Vazirani circuit, as observed on Jan 15, 2024.
Figure 7.15: Time-varying density for probability of observing ‘11’ on qubits
3 and 4 of ibm_kolkata, for the Bernstein-Vazirani circuit, as observed on Jan
15, 2024.

##  Chapter 8 Conclusion

Quantum computing’s tremendous potential [1] is curtailed by noise.
Characterizing noise in contemporary quantum computers remains challenging due
to its non-stationary statistics. The non-stationarity manifests across time,
across different parts of the chip, and across devices. It impacts the
verification and validation of quantum computing demonstrations and adversely
impacts error mitigation strategies [59]. This hinders the production of
trustworthy results.

This dissertation focused on the non-stationarity [46, 42, 36] of noise in
superconducting processors [30]. It established a framework comprising
computational accuracy, device reliability, outcome stability, and result
reproducibility for assessing noisy outcomes. It determined upper and lower
bounds for the performance metrics, in terms of the Hellinger distance between
time-varying noise densities. It demonstrated that if the noise stays within
the theoretical bounds, outcome stability can be ensured with high confidence.
It developed a Bayesian strategy to improve the stability and accuracy of
results obtained from probabilistic error cancellation. Refinement in noise
model selection, optimization algorithms, and data collection frequencies can
yield further improvements and is a rich space to explore further.

In particular, chapter 2 discussed noise in quantum computing and
experimentally analysed the decoherence of superconducting qubits over a
24-hour period on September 12, 2023. The time-varying decoherence
characterization was connected to the coefficients of a Pauli channel model.

While the majority of quantum computing research has focused on achieving
accuracy [54, 71, 68], limited attention has been given to reproducibility,
reliability, and stability. Chapter 3 precisely defined these terms
(computational accuracy, result reproducibility, device reliability and
stability of error mitigated outcomes) for assessing the performance of noisy,
quantum computers. Chapter 4 evaluated hardware reliability by analyzing
experimental data across various time scales (monthly, daily, hourly, and
seconds). It quantified the degree of non-stationarity in SPAM fidelity, CNOT
fidelity, duty cycle, and addressability. It developed and validated a method
to examine holistic reliability using the method of copulas [124].

Chapter 5 developed bounds on a proxy parameter to encapsulate the array of
parameters characterizing a noisy device. Such a proxy parameter is valuable
for streamlining high-dimensional noise analysis. Experimental validation of
the theoretical bound was performed using a 27-qubit superconducting device.
The chapter further illustrated how to determine the minimum sample size to
achieve reproducibility with $1-\delta$ confidence. It established bounds on
device reliability necessary for achieving an $\epsilon$-stable outcome.

Chapter 6 applied adaptive techniques [73, 74, 75, 76] to enhance accuracy of
histograms, demonstrating a reduction in the Hellinger distance from 15% to
3.1%, while chapter 7 examined the impact of non-stationary noise on
probabilistic error cancellation. It introduced a Bayesian approach to improve
stability and accuracy of PEC outcomes. The algorithm was tested on the
ibm_kolkata device on January 15, 2024. The dataset covered a 24-hour period,
consisting of 13 complete PEC datasets for the Bernstein-Vazirani test
circuit, with approximately 67 million observations. Results indicated a 42%
increase in accuracy and a 60% enhancement in stability. Consistent
improvement trends across time-stamps and qubits demonstrated the
effectiveness of the algorithm.

The improvements in accuracy and stability of adaptive PEC vs non-adaptive
PEC, across time and qubits, presented in this dissertation, indicate that the
noise was well-characterized. In general, the choice of noise model influences
the noise estimation process. If the chosen noise model is incorrect or
insufficiently granular, the estimated noise parameters will reflect the
specific dataset’s patterns rather than accurately representing device noise.
Mitigation using such mis-estimated noise models will be unsuccessful because
the mitigation algorithm will introduce additional errors to the already noisy
data. However, scalability becomes an issue with the Bayesian approach in the
presence of correlations. While the method can accurately estimate numerous
noise parameters in the absence of correlations, scalability diminishes when
dealing with highly correlated parameters due to the need to estimate joint
distributions and perform Monte Carlo simulations for maximum-a-posterior
optimization.

Before concluding, I offer a perspective on the significance of this
dissertation. Firstly, this work delves into the intricate nature of non-
stationary noise in contemporary quantum computers. It sheds light on
strategies for improving the reproducibility and stability of noisy quantum
computations, by incorporating adaptive processes to manage non-stationary
noise.

Quantum error correction (QEC) methods represent a formal way to mitigate and
manage errors but in practice need to be tailored to the noise. This is
similar to 5G networks where low density parity check (LDPC) [166, 167] codes
are employed to meet the high data transfer demands. These codes use a quasi-
cyclic structure that adapt their encoding and decoding graph size depending
on prevailing error rates. The wireless channel noise is often analyzed using
methods that take into account the variability in space, time, and frequency
when using correlation functions and power spectrum densities. In parallel,
when dealing with quantum noise channels, this work adopted a strategy that
incorporates time-varying correlated distributions to reflect the non-
stationary nature of quantum noise and utilized quasi-probability
distributions (in probabilistic error cancellation) whose weights are
influenced by the strength of the prevailing noise.

This dissertation can also help draw parallels to classical cellular
communications in handling non-stationary noise. Just as non-stationary
channel conditions in cellular networks once caused frustrating service
quality issues due to inadequate noise characterization, quantum computing
faces similar issues with error mitigation fluctuating due to poor noise
characterization. Moreover, the concept of temporal decoherence, caused by
interactions with the environment, can be likened to the phenomenon of signal
fading in classical cellular systems due to atmospheric interactions. Both
provide case studies for how non-stationary processes can impact the
reliability of information transmission. Lastly, just like early cellular
communication systems suffered high error-rates from poor multiplexing
capabilities due to limited spectrum resources, similarly, contemporary
quantum computing systems are limited in their error mitigation and correction
capabilities due to constraints in qubit resources.

The unpredictable and rapidly changing nature of non-stationary noise presents
a challenge for fault-tolerant quantum computing. This type of noise can lead
to error patterns that change more quickly than contemporary error correction
codes can adjust, making it difficult to maintain fault tolerance. For
instance, cosmic rays can cause sudden and sporadic error bursts, temporarily
pushing error rates beyond the limits that quantum error correction codes are
designed to handle. This can result in uncorrected errors and, potentially,
the failure of logical qubits. Additionally, superconducting processors
exhibit varying error rates across the chip, which challenges the assumption
of a uniform threshold error rate, as often applied in techniques like the
surface code. Such issues of non-stationary temporal and spatial error rates
affecting error correction have previously been encountered in the field of
cellular communications, indicating that the challenges faced by quantum
computing are not entirely unprecedented. Note that non-stationary noise does
not prevent the application of fault-tolerant quantum error correction but
needs a tailored approach.

The results from this study can also help in noise modeling in quantum
communication scenarios involving entanglement distribution over long
distances as the quality of entanglement can be degraded by non-stationary
atmospheric conditions when using free-space optical channels. These
conditions can change rapidly due to weather phenomena, causing fluctuations
in entanglement fidelity and compromising the integrity of information
transfer.

Lastly, this dissertation underscores the need for interdisciplinary
collaboration to advance the field of quantum computing. For example, it
stresses the role of reliability engineering in improving manufacturing
processes for quantum chips to ensure high standards of quality. It urges
software developers to develop adaptive algorithms to tackle real-time
challenges posed by non-stationary noise in quantum systems. It calls for the
utilization of advanced statistical models and database management techniques
by data scientists to aid quantum physicists using models should embrace the
inherent quantum noise model uncertainties as a feature rather than a flaw.
And, it challenges information theorists to extend their work beyond simple
noise models. While foundational insights are invaluable, there is a need to
extend, validate and apply these theories using real-world data to make them
useful. By fostering collaboration across such diverse disciplines, we can
accelerate the development of quantum computing technologies that are
reliable, reproducible, and stable.

## References

  * Humble et al. [2019] Travis S Humble, Himanshu Thapliyal, Edgard Munoz-Coreas, Fahd A Mohiyaddin, and Ryan S Bennink.  Quantum computing circuits and devices.  _IEEE Design & Test_, 36(3):69–94, 2019.
  * Browaeys and Lahaye [2020] Antoine Browaeys and Thierry Lahaye.  Many-body physics with individually controlled rydberg atoms.  _Nature Physics_ , 16(2):132–142, 2020.
  * Montanaro [2016] Ashley Montanaro.  Quantum algorithms: an overview.  _npj Quantum Information_ , 2(1):1–8, 2016.
  * Shang et al. [2015] Jiangwei Shang, Yi-Lin Seah, Hui Khoon Ng, David John Nott, and Berthold-Georg Englert.  Monte carlo sampling from the quantum state space. i.  _New Journal of Physics_ , 17(4):043017, 2015\.
  * Espitia [2021] Giovanny Espitia.  On the role of quantum computing in science and cybersecurity.  _arXiv preprint arXiv:2105.09942_ , 2021.
  * How and Cheah [2023] Meng-Leong How and Sin-Mei Cheah.  Business renaissance: Opportunities and challenges at the dawn of the quantum computing era.  _Businesses_ , 3(4):585–605, 2023.
  * Feynman et al. [1965] Richard P Feynman, Robert B Leighton, Matthew Sands, and Everett M Hafner.  The feynman lectures on physics.  _American Journal of Physics_ , 1965.
  * Cohen-Tannoudji et al. [1977] C Cohen-Tannoudji, B Diu, and F Laloë.  Quantum mechanics, volume i, ii, paris-new york: Èditions hermann and john wiley & sons, 1977.
  * Sakurai and Napolitano [2020] Jun John Sakurai and Jim Napolitano.  _Modern quantum mechanics_.  Cambridge University Press, 2020.
  * Mermin [2007] N David Mermin.  _Quantum computer science: an introduction_.  Cambridge University Press, 2007.
  * Nielsen and Chuang [2002] Michael A Nielsen and Isaac Chuang.  Quantum computation and quantum information, 2002.
  * Bhatia [1997] R Bhatia.  Matrix analysis, 1997.
  * Kraus et al. [1983] Karl Kraus, Arno Böhm, John D Dollard, and WH Wootters.  _States, Effects, and Operations: Lectures in Mathematical Physics_.  Springer, 1983.
  * Zeguendry et al. [2023] Amine Zeguendry, Zahi Jarir, and Mohamed Quafafou.  Quantum machine learning: A review and case studies.  _Entropy_ , 25(2):287, 2023.
  * Schumacher [1995] Benjamin Schumacher.  Quantum coding.  _Physical Review A_ , 51(4):2738, 1995.
  * Bennett and Shor [1998] Charles H. Bennett and Peter W. Shor.  Quantum information theory.  _IEEE Transactions on Information Theory_ , 44(6):2724–2742, 1998.
  * Béjanin et al. [2021] JH Béjanin, CT Earnest, AS Sharafeldin, and M Mariantoni.  Interacting defects generate stochastic fluctuations in superconducting qubits.  _Physical Review B_ , 104(9):094106, 2021.
  * Burnett et al. [2019] Jonathan J Burnett, Andreas Bengtsson, Marco Scigliuzzo, David Niepce, Marina Kudra, Per Delsing, and Jonas Bylander.  Decoherence benchmarking of superconducting qubits.  _npj Quantum Information_ , 5(1):1–8, 2019.
  * Carroll et al. [2021] Malcolm Carroll, Sami Rosenblatt, Petar Jurcevic, Isaac Lauer, and Abhinav Kandala.  Dynamics of superconducting qubit relaxation times.  _arXiv preprint arXiv:2105.15201_ , 2021.
  * [20] Dephasing and relaxation. available online: https://wright.chem.wisc.edu/content/dephasing-and-relaxation-0/ (accessed on 2 dec 2022).
  * Cormen et al. [1990] Thomas H Cormen, CE Leiserson, Ronald L Rivest, and Cliford Stein.  Introduction to algorithms cambridge, massachusetts, 1990.
  * Cook [2023] Stephen A Cook.  The complexity of theorem-proving procedures.  In _Logic, Automata, and Computational Complexity: The Works of Stephen A. Cook_ , pages 143–152. 2023.
  * Alexander et al. [2020] Thomas Alexander, Naoki Kanazawa, Daniel J Egger, Lauren Capelluto, Christopher J Wood, Ali Javadi-Abhari, and David C McKay.  Qiskit pulse: programming quantum computers through the cloud with pulses.  _Quantum Science and Technology_ , 5(4):044006, 2020.
  * Ball et al. [2021] Harrison Ball, Michael J Biercuk, and Michael R Hush.  Quantum firmware and the quantum computing stack.  _Physics Today_ , 74(3):28–34, 2021.
  * Britt et al. [2017] Keith A. Britt, Fahd A. Mohiyaddin, and Travis S. Humble.  Quantum accelerators for high-performance computing systems.  In _2017 IEEE International Conference on Rebooting Computing (ICRC)_ , pages 1–7, 2017.  10.1109/ICRC.2017.8123664.
  * McCaskey et al. [2018a] Alexander McCaskey, Eugene Dumitrescu, Dmitry Liakh, and Travis Humble.  Hybrid programming for near-term quantum computing systems.  In _2018 IEEE International Conference on Rebooting Computing (ICRC)_ , pages 1–12, 2018a.  10.1109/ICRC.2018.8638598.
  * Van Den Berg et al. [2023] Ewout Van Den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme.  Probabilistic error cancellation with sparse pauli-lindblad models on noisy quantum processors.  _Nature Physics_ , pages 1–6, 2023.
  * Bravyi et al. [2021] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C Mckay, and Jay M Gambetta.  Mitigating measurement errors in multiqubit experiments.  _Physical Review A_ , 103(4):042605, 2021.
  * Krantz et al. [2019] Philip Krantz, Morten Kjaergaard, Fei Yan, Terry P Orlando, Simon Gustavsson, and William D Oliver.  A quantum engineer’s guide to superconducting qubits.  _Applied Physics Reviews_ , 6(2):021318, 2019\.
  * Roth et al. [2021] Thomas E Roth, Ruichao Ma, and Weng C Chew.  An introduction to the transmon qubit for electromagnetic engineers.  _arXiv preprint arXiv:2106.11352_ , 2021.
  * Vittal et al. [2023] Suhas Vittal, Poulami Das, and Moinuddin Qureshi.  Eraser: Towards adaptive leakage suppression for fault-tolerant quantum computing.  In _Proceedings of the 56th Annual IEEE/ACM International Symposium on Microarchitecture_ , pages 509–525, 2023.
  * Ben-Aroya and Ta-Shma [2011] Avraham Ben-Aroya and Amnon Ta-Shma.  Approximate quantum error correction for correlated noise.  _IEEE Transactions on Information Theory_ , 57(6):3982–3988, 2011.
  * Martinis [2015] John M Martinis.  Qubit metrology for building a fault-tolerant quantum computer.  _npj Quantum Information_ , 1(1):1–3, 2015\.
  * Preskill [2019] John Preskill.  Quantum computing in the nisq era and beyond.  _Bulletin of the American Physical Society_ , 64, 2019.
  * Kim et al. [2023] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, et al.  Evidence for the utility of quantum computing before fault tolerance.  _Nature_ , 618(7965):500–505, 2023.
  * Klimov et al. [2018] PV Klimov, Julian Kelly, Z Chen, Matthew Neeley, Anthony Megrant, Brian Burkett, Rami Barends, Kunal Arya, Ben Chiaro, Yu Chen, et al.  Fluctuations of energy-relaxation times in superconducting qubits.  _Physical Review Letters_ , 121(9):090502, 2018\.
  * McRae et al. [2021] Corey Rae H McRae, Gregory M Stiehl, Haozhi Wang, Sheng-Xiang Lin, Shane A Caldwell, David P Pappas, Josh Mutus, and Joshua Combes.  Reproducible coherence characterization of superconducting quantum devices.  _Applied Physics Letters_ , 119(10):100501, 2021\.
  * Klesse and Frank [2005] Rochus Klesse and Sandra Frank.  Quantum error correction in spatially correlated quantum noise.  _Physical Review Letters_ , 95(23):230503, 2005\.
  * Gupta et al. [2020] Riddhi Swaroop Gupta, Luke CG Govia, and Michael J Biercuk.  Integration of spectator qubits into quantum computer architectures for hardware tune-up and calibration.  _Physical Review A_ , 102(4):042611, 2020.
  * Parrado-Rodríguez et al. [2021] Pedro Parrado-Rodríguez, Ciarán Ryan-Anderson, Alejandro Bermudez, and Markus Müller.  Crosstalk suppression for fault-tolerant quantum error correction with trapped ions.  _Quantum_ , 5:487, 2021.
  * Fang et al. [2022] Chao Fang, Ye Wang, Shilin Huang, Kenneth R Brown, and Jungsang Kim.  Crosstalk suppression in individually addressed two-qubit gates in a trapped-ion quantum computer.  _Physical Review Letters_ , 129(24):240504, 2022\.
  * Müller et al. [2015] Clemens Müller, Jürgen Lisenfeld, Alexander Shnirman, and Stefano Poletto.  Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits.  _Physical Review B_ , 92(3):035442, 2015.
  * Xu et al. [2022] Qian Xu, Alireza Seif, Haoxiong Yan, Nam Mannucci, Bernard Ousmane Sane, Rodney Van Meter, Andrew N Cleland, and Liang Jiang.  Distributed quantum error correction for chip-level catastrophic errors.  _Physical Review Letters_ , 129(24):240502, 2022\.
  * McEwen et al. [2022] Matt McEwen, Lara Faoro, Kunal Arya, Andrew Dunsworth, Trent Huang, Seon Kim, Brian Burkett, Austin Fowler, Frank Arute, Joseph C Bardin, et al.  Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits.  _Nature Physics_ , 18(1):107–111, 2022.
  * Proctor et al. [2020] Timothy Proctor, Melissa Revelle, Erik Nielsen, Kenneth Rudinger, Daniel Lobser, Peter Maunz, Robin Blume-Kohout, and Kevin Young.  Detecting and tracking drift in quantum information processors.  _Nature Communications_ , 11(1):1–9, 2020.
  * Etxezarreta Martinez et al. [2021] Josu Etxezarreta Martinez, Patricio Fuentes, Pedro Crespo, and Javier Garcia-Frias.  Time-varying quantum channel models for superconducting qubits.  _npj Quantum Information_ , 7(1):1–10, 2021.
  * Danageozian et al. [2022] Arshag Danageozian, Ashe Miller, Pratik J Barge, Narayan Bhusal, and Jonathan P Dowling.  Noisy coherent population trapping: applications to noise estimation and qubit state preparation.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ , 55(15):155503, 2022.
  * Majumder et al. [2020] Swarnadeep Majumder, Leonardo Andreta de Castro, and Kenneth R Brown.  Real-time calibration with spectator qubits.  _npj Quantum Information_ , 6(1):19, 2020.
  * Martinez et al. [2020] Josu Etxezarreta Martinez, Patricio Fuentes, Pedro M Crespo, and Javier Garcia-Frias.  Approximating decoherence processes for the design and simulation of quantum error correction codes on classical computers.  _IEEE Access_ , 8:172623–172643, 2020.
  * deMarti iOlius et al. [2023] Antonio deMarti iOlius, Patricio Fuentes, Román Orús, Pedro M Crespo, and Josu Etxezarreta Martinez.  Decoding algorithms for surface codes.  _arXiv e-prints_ , pages arXiv–2307, 2023.
  * deMarti iOlius et al. [2022] Antonio deMarti iOlius, Josu Etxezarreta Martinez, Patricio Fuentes, Pedro M Crespo, and Javier Garcia-Frias.  Performance of surface codes in realistic quantum hardware.  _Physical Review A_ , 106(6):062428, 2022.
  * Vapnik [1999] Vladimir Vapnik.  _The nature of statistical learning theory_.  Springer science & business media, 1999.
  * Carrasco et al. [2021] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller.  Theoretical and experimental perspectives of quantum verification.  _PRX Quantum_ , 2(1):010102, 2021.
  * Kliesch and Roth [2021] Martin Kliesch and Ingo Roth.  Theory of quantum system certification.  _PRX Quantum_ , 2(1):010201, 2021.
  * [55] Quantum roadmap. Available online: https://www.ibm.com/roadmaps/quantum/ (accessed on 12 feb 2024).
  * Acín et al. [2018] Antonio Acín, Immanuel Bloch, Harry Buhrman, Tommaso Calarco, Christopher Eichler, Jens Eisert, Daniel Esteve, Nicolas Gisin, Steffen J Glaser, Fedor Jelezko, et al.  The quantum technologies roadmap: a european community view.  _New Journal of Physics_ , 20(8):080201, 2018\.
  * Hughes and Heinrichs [2004] Richard Hughes and Todd Heinrichs.  A quantum information science and technology roadmap.  _Rep. LA-UR-04-1778, ARDA_ , 2004.
  * Khatri et al. [2020] Sumeet Khatri, Kunal Sharma, and Mark M Wilde.  Information-theoretic aspects of the generalized amplitude-damping channel.  _Physical Review A_ , 102(1):012401, 2020.
  * Bharti et al. [2022] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S Kottmann, Tim Menke, et al.  Noisy intermediate-scale quantum algorithms.  _Reviews of Modern Physics_ , 94(1):015004, 2022\.
  * Harper et al. [2020] Robin Harper, Steven T Flammia, and Joel J Wallman.  Efficient learning of quantum noise.  _Nature Physics_ , 16(12):1184–1188, 2020.
  * Gheorghiu et al. [2019] Alexandru Gheorghiu, Theodoros Kapourniotis, and Elham Kashefi.  Verification of quantum computation: An overview of existing approaches.  _Theory of computing systems_ , 63(4):715–808, 2019.
  * Liu et al. [2020] Ye-Chao Liu, Jiangwei Shang, Xiao-Dong Yu, and Xiangdong Zhang.  Efficient verification of quantum processes.  _Phys. Rev. A_ , 101:042315, Apr 2020.  10.1103/PhysRevA.101.042315.  URL https://link.aps.org/doi/10.1103/PhysRevA.101.042315.
  * McCaskey et al. [2018b] A. McCaskey, E. Dumitrescu, M. Chen, D. Lyakh, and T. S. Humble.  Validating quantum-classical programming models with tensor network simulations.  _PLoS ONE_ , 13:e0206704, 2018b.  10.1371/journal.pone.0206704.
  * Nguyen et al. [2021] T. Nguyen, D. Lyakh, P. Lotshaw, A. McCaskey, R. Bennink, and W Jong.  Scalable programming workflows for validation of quantum computers.  _2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS)_ , 2021.  10.1109/qcs54837.2021.00013.
  * McCaskey et al. [2019] Alexander J McCaskey, Zachary P Parks, Jacek Jakowski, Shirley V Moore, Titus D Morris, Travis S Humble, and Raphael C Pooser.  Quantum chemistry as a benchmark for near-term quantum computers.  _npj Quantum Information_ , 5(1):1–8, 2019.
  * Blume-Kohout and Young [2020] Robin Blume-Kohout and Kevin C. Young.  A volumetric framework for quantum computer benchmarks.  _Quantum_ , 4:362, November 2020.  ISSN 2521-327X.  10.22331/q-2020-11-15-362.  URL https://doi.org/10.22331/q-2020-11-15-362.
  * Helsen and Wehner [2023] J. Helsen and S. Wehner.  A benchmarking procedure for quantum networks.  _npj Quantum Inf_ , 9, 2023.  10.1038/s41534-022-00628-x.
  * Ferracin et al. [2021] Samuele Ferracin, Seth T Merkel, David McKay, and Animesh Datta.  Experimental accreditation of outputs of noisy quantum computers.  _arXiv preprint arXiv:2103.06603_ , 2021.
  * DiVincenzo [2000] David P DiVincenzo.  The physical implementation of quantum computation.  _Fortschritte der Physik: Progress of Physics_ , 48(9-11):771–783, 2000.
  * Blume-Kohout [2020] Robin J Blume-Kohout.  Modeling and characterizing noise in quantum processors.  Technical report, Sandia National Lab.(SNL-NM), Albuquerque, NM (United States), 2020.
  * Blume-Kohout [2010] Robin Blume-Kohout.  Optimal, reliable estimation of quantum states.  _New Journal of Physics_ , 12(4):043034, 2010\.
  * Temme et al. [2017] Kristan Temme, Sergey Bravyi, and Jay M Gambetta.  Error mitigation for short-depth quantum circuits.  _Physical Review Letters_ , 119(18):180509, 2017\.
  * Lukens et al. [2020] Joseph M Lukens, Kody JH Law, Ajay Jasra, and Pavel Lougovski.  A practical and efficient approach for bayesian quantum state estimation.  _New Journal of Physics_ , 22(6):063038, 2020\.
  * Zheng et al. [2020] Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang.  A bayesian approach for characterizing and mitigating gate and measurement errors.  _arXiv preprint arXiv:2010.09188_ , 2020.
  * Gordon et al. [1993] Neil J Gordon, David J Salmond, and Adrian FM Smith.  Novel approach to nonlinear/non-gaussian bayesian state estimation.  In _IEE Proceedings F-radar and Signal Processing_ , volume 140, pages 107–113. IET, 1993.
  * Kotecha and Djuric [2003] Jayesh H Kotecha and Petar M Djuric.  Gaussian sum particle filtering.  _IEEE Transactions on Signal Processing_ , 51(10):2602–2612, 2003.
  * Gottesman [1998] Daniel Gottesman.  The heisenberg representation of quantum computers.  _arXiv preprint quant-ph/9807006_ , 1998.
  * Spehner et al. [2017] D Spehner, F Illuminati, M Orszag, and W Roga.  Geometric measures of quantum correlations with bures and hellinger distances.  _Lectures on general quantum correlations and their applications_ , pages 105–157, 2017.
  * Nelsen [2007] Roger B Nelsen.  _An introduction to copulas_.  Springer science & business media, 2007.
  * Wan et al. [2019] Yong Wan, Daniel Kienzler, Stephen D Erickson, Karl H Mayer, Ting Rei Tan, Jenny J Wu, Hilma M Vasconcelos, Scott Glancy, Emanuel Knill, David J Wineland, et al.  Quantum gate teleportation between separated qubits in a trapped-ion processor.  _Science_ , 364(6443):875–878, 2019.
  * Chan et al. [2018] K. W. Chan, W. Huang, C. H. Yang, J. C. C. Hwang, B. Hensen, T. Tanttu, F. E. Hudson, K. M. Itoh, A. Laucht, A. Morello, and A. S. Dzurak.  Assessment of a silicon quantum dot spin qubit environment via noise spectroscopy.  _Phys. Rev. Applied_ , 10:044017, Oct 2018.  10.1103/PhysRevApplied.10.044017.  URL https://link.aps.org/doi/10.1103/PhysRevApplied.10.044017.
  * Jin et al. [2015] XY Jin, A Kamal, AP Sears, T Gudmundsen, D Hover, J Miloshi, R Slattery, F Yan, J Yoder, TP Orlando, et al.  Thermal and residual excited-state population in a 3d transmon qubit.  _Physical Review Letters_ , 114(24):240501, 2015\.
  * Emerson et al. [2007] Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G Cory, and Raymond Laflamme.  Symmetrized characterization of noisy quantum processes.  _Science_ , 317(5846):1893–1896, 2007.
  * Silva et al. [2008] Marcus Silva, Easwar Magesan, David W Kribs, and Joseph Emerson.  Scalable protocol for identification of correctable codes.  _Physical Review A_ , 78(1):012347, 2008.
  * Horodecki et al. [1999] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki.  General teleportation channel, singlet fraction, and quasidistillation.  _Physical Review A_ , 60(3):1888, 1999.
  * Eggeling and Werner [2001] Tilo Eggeling and Reinhard F Werner.  Separability properties of tripartite states with $u\otimes u\otimes u$ symmetry.  _Physical Review A_ , 63(4):042111, 2001.
  * Dankert et al. [2009] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine.  Exact and approximate unitary 2-designs and their application to fidelity estimation.  _Physical Review A_ , 80(1):012304, 2009.
  * Magesan [2008] Easwar Magesan.  Gaining information about a quantum channel via twirling.  Master’s thesis, University of Waterloo, 2008.
  * Cai et al. [2020] Zhenyu Cai, Xiaosi Xu, and Simon C Benjamin.  Mitigating coherent noise using pauli conjugation.  _npj Quantum Information_ , 6(1):17, 2020.
  * Ferracin et al. [2019] Samuele Ferracin, Theodoros Kapourniotis, and Animesh Datta.  Accrediting outputs of noisy intermediate-scale quantum computing devices.  _New Journal of Physics_ , 21(11):113038, 2019\.
  * Beran [1977] Rudolf Beran.  Minimum hellinger distance estimates for parametric models.  _The annals of Statistics_ , pages 445–463, 1977.
  * Rahimian et al. [2019] Hamed Rahimian, Güzin Bayraksan, and Tito Homem-de Mello.  Identifying effective scenarios in distributionally robust stochastic programs with total variation distance.  _Mathematical Programming_ , 173(1-2):393–430, 2019.
  * Lilliefors [1967] Hubert W Lilliefors.  On the kolmogorov-smirnov test for normality with mean and variance unknown.  _Journal of the American statistical Association_ , 62(318):399–402, 1967.
  * Xiang et al. [2008] Shiming Xiang, Feiping Nie, and Changshui Zhang.  Learning a mahalanobis distance metric for data clustering and classification.  _Pattern recognition_ , 41(12):3600–3612, 2008\.
  * Cuesta and Matrán [1989] Juan Antonio Cuesta and Carlos Matrán.  Notes on the wasserstein metric in hilbert spaces.  _The Annals of Probability_ , pages 1264–1276, 1989.
  * Fisher and Van Ness [1969] Lloyd Fisher and John W Van Ness.  Distinguishability of probability measures.  _The Annals of Mathematical Statistics_ , pages 381–392, 1969.
  * Van Erven and Harremos [2014] Tim Van Erven and Peter Harremos.  Rényi divergence and kullback-leibler divergence.  _IEEE Transactions on Information Theory_ , 60(7):3797–3820, 2014.
  * Nielsen [2020] Frank Nielsen.  On a generalization of the jensen–shannon divergence and the jensen–shannon centroid.  _Entropy_ , 22(2):221, 2020.
  * Vigelis et al. [2019] Rui F Vigelis, Luiza HF De Andrade, and Charles C Cavalcante.  Properties of a generalized divergence related to tsallis generalized divergence.  _IEEE Transactions on Information Theory_ , 66(5):2891–2897, 2019.
  * Dajka [2011] Luczka J. Hänggi P. Dajka, J.  Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study.  _Phys. Rev. A_ , 84, 2011.  10.1103/physreva.84.032120.
  * Bennink [2019] Jasra A. Law K. J. H. Lougovski P. Bennink, R. S.  Estimation and uncertainty quantification for the output from quantum simulators.  _Foundations of Data Science_ , 1:157–176, 2019.  10.3934/fods.2019007.
  * Tóth and Apellaniz [2014] Géza Tóth and Iagoba Apellaniz.  Quantum metrology from a quantum information science perspective.  _Journal of Physics A: Mathematical and Theoretical_ , 47(42):424006, 2014.
  * Bădescu et al. [2019] Costin Bădescu, Ryan O’Donnell, and John Wright.  Quantum state certification.  In _Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing_ , pages 503–514, 2019.
  * Rudinger et al. [2019] Kenneth Rudinger, Timothy Proctor, Dylan Langharst, Mohan Sarovar, Kevin Young, and Robin Blume-Kohout.  Probing context-dependent errors in quantum processors.  _Physical Review X_ , 9(2):021045, 2019.
  * Smith et al. [2021] Alistair WR Smith, Kiran E Khosla, Chris N Self, and MS Kim.  Qubit readout error mitigation with bit-flip averaging.  _Science Advances_ , 7(47):eabi8009, 2021.
  * Maciejewski et al. [2021] Filip B Maciejewski, Flavio Baccari, Zoltán Zimborás, and Michał Oszmaniec.  Modeling and mitigation of cross-talk effects in readout noise with applications to the quantum approximate optimization algorithm.  _Quantum_ , 5:464, 2021.
  * Pérez-Delgado and Kok [2011] Carlos A Pérez-Delgado and Pieter Kok.  Quantum computers: Definition and implementations.  _Physical Review A_ , 83(1):012303, 2011.
  * Dasgupta and Humble [2022] Samudra Dasgupta and Travis S Humble.  Assessing the stability of noisy quantum computation.  In _Quantum Communications and Quantum Imaging XX_ , volume 12238, pages 44–49. SPIE, 2022.
  * Bernstein and Vazirani [1993] Ethan Bernstein and Umesh Vazirani.  Quantum complexity theory.  In _Proceedings of the twenty-fifth annual ACM symposium on theory of computing_ , pages 11–20, 1993.
  * Dasgupta and Humble [2023] Samudra Dasgupta and Travis S Humble.  Reliability of noisy quantum computing devices.  _arXiv preprint arXiv:2307.06833_ , 2023.
  * [111] Quantum computing software and programming tools. Available online: https://www.ibm.com/quantum-computing/experience/ (accessed on 21 august 2021).
  * Dasgupta and Humble [2021] Samudra Dasgupta and Travis S Humble.  Stability of noisy quantum computing devices.  _arXiv preprint arXiv:2105.09472_ , 2021.
  * [113] Processor types. Available online: https://docs.quantum.ibm.com/run/processor-types/ (accessed on 12 feb 2024).
  * Aleksandrowicz et al. [2019] Gadi Aleksandrowicz, Thomas Alexander, Panagiotis Barkoutsos, Luciano Bello, Yael Ben-Haim, D Bucher, FJ Cabrera-Hernández, J Carballo-Franquis, A Chen, CF Chen, et al.  Qiskit: An open-source framework for quantum computing.  _Accessed on: Mar_ , 16, 2019.
  * [115] Oak Ridge Leadership Computing Facility. Available online: https://www.olcf.ornl.gov/ (accessed on 12 feb 2024).
  * Knill et al. [2008] Emanuel Knill, Dietrich Leibfried, Rolf Reichle, Joe Britton, R Brad Blakestad, John D Jost, Chris Langer, Roee Ozeri, Signe Seidelin, and David J Wineland.  Randomized benchmarking of quantum gates.  _Physical Review A_ , 77(1):012307, 2008.
  * Bravyi and Kitaev [2005] Sergey Bravyi and Alexei Kitaev.  Universal quantum computation with ideal clifford gates and noisy ancillas.  _Physical Review A_ , 71(2):022316, 2005.
  * Hurant and Stancil [2020] Travis Hurant and Daniel D Stancil.  Asymmetry of cnot gate operation in superconducting transmon quantum processors using cross-resonance entangling.  _arXiv preprint arXiv:2009.01333_ , 2020.
  * Martin et al. [2007] Ivar Martin, L Bulaevskii, A Shnirman, and YM Galperin.  Stationary and non-stationary noise in superconducting quantum devices.  In _Noise and Fluctuations in Circuits, Devices, and Materials_ , volume 6600, pages 35–46. SPIE, 2007.
  * Xie et al. [2023] Tianyu Xie, Zhiyuan Zhao, Shaoyi Xu, Xi Kong, Zhiping Yang, Mengqi Wang, Ya Wang, Fazhan Shi, and Jiangfeng Du.  99.92%-fidelity cnot gates in solids by noise filtering.  _Physical Review Letters_ , 130(3):030601, 2023\.
  * Dasgupta and Humble [2020] Samudra Dasgupta and Travis S Humble.  Characterizing the stability of nisq devices.  In _2020 IEEE International Conference on Quantum Computing and Engineering (QCE)_ , pages 419–429. IEEE, 2020.
  * Novais and Baranger [2006] E. Novais and H. Baranger.  Decoherence by correlated noise and quantum error correction.  _Phys. Rev. Lett._ , 97, 2006.  10.1103/physrevlett.97.040501.
  * Biercuk et al. [2016] M. Biercuk, T. Stace, S. Flammia, and M. Biercuk.  Effect of noise correlations on randomized benchmarking.  _Phys. Rev. A_ , 93, 2016.  10.1103/physreva.93.022303.
  * Tang et al. [2013] X. Tang, D. Li, C. Zhou, and L. Zhang.  Bivariate distribution models using copulas for reliability analysis.  _Proceedings of the Institution of Mechanical Engineers, Part O:_ , 227:499–512, 2013.  10.1177/1748006x13481928.
  * Verleysen and François [2005] Michel Verleysen and Damien François.  The curse of dimensionality in data mining and time series prediction.  In _International work-conference on artificial neural networks_ , pages 758–770. Springer, 2005.
  * Emerson et al. [2005] Joseph Emerson, Robert Alicki, and Karol Życzkowski.  Scalable noise estimation with random unitary operators.  _Journal of Optics B: Quantum and Semiclassical Optics_ , 7(10):S347, 2005.
  * Oszmaniec et al. [2019] Michał Oszmaniec, Filip B Maciejewski, and Zbigniew Puchała.  Simulating all quantum measurements using only projective measurements and postselection.  _Physical Review A_ , 100(1):012351, 2019.
  * Geller [2020] Michael R Geller.  Rigorous measurement error correction.  _Quantum Science and Technology_ , 5(3):03LT01, 2020.
  * Sklar [1959] A Sklar.  Fonctions de répartition į n dimensions et leur marges.  _Publ. Inst. Stat. Paris_ , 8:131–229, 1959.
  * Klüppelberg and Kuhn [2009] C. Klüppelberg and G. Kuhn.  Copula structure analysis.  _Journal of the Royal Statistical Society Series B: Statistical Methodology_ , 71:737–753, 2009.  10.1111/j.1467-9868.2009.00707.x.
  * Dutfoy and Lebrun [2009] A. Dutfoy and R. Lebrun.  Practical approach to dependence modelling using copulas.  _Proceedings of the Institution of Mechanical Engineers, Part O:_ , 223:347–361, 2009.  10.1243/1748006xjrr226.
  * Chesneau [2023] C. Chesneau.  Extensions of two bivariate strict archimedean copulas.  _Computational Journal of Mathematical and Statistical Sciences_ , 0:159–180, 2023.  10.21608/cjmss.2023.205330.1007.
  * Zhu et al. [2022] Elton Yechao Zhu, Sonika Johri, Dave Bacon, Mert Esencan, Jungsang Kim, Mark Muir, Nikhil Murgai, Jason Nguyen, Neal Pisenti, Adam Schouela, et al.  Generative quantum learning of joint probability distribution functions.  _Physical Review Research_ , 4(4):043092, 2022\.
  * De Luca and Rivieccio [2012] Giovanni De Luca and Giorgia Rivieccio.  Multivariate tail dependence coefficients for archimedean copulae.  In _Advanced statistical methods for the analysis of large data-sets_ , pages 287–296. Springer, 2012.
  * McNeil et al. [2015] Alexander J McNeil, Rüdiger Frey, and Paul Embrechts.  _Quantitative risk management: concepts, techniques and tools-revised edition_.  Princeton university press, 2015.
  * Wilkens and Moorhouse [2023] Sascha Wilkens and Joe Moorhouse.  Quantum computing for financial risk measurement.  _Quantum Information Processing_ , 22(1):51, 2023\.
  * Genest et al. [2011] Christian Genest, Johanna Nešlehová, and Johanna Ziegel.  Inference in multivariate archimedean copula models.  _Test_ , 20:223–256, 2011.
  * Calderhead [2014] Ben Calderhead.  A general construction for parallelizing metropolis- hastings algorithms.  _Proceedings of the National Academy of Sciences_ , 111(49):17408–17413, 2014.
  * Chapeau-Blondeau [2022] F. Chapeau-Blondeau.  Modeling and simulation of a quantum thermal noise on the qubit.  _Fluct. Noise Lett._ , 21, 2022.  10.1142/s0219477522500602.
  * Wong and Santos [2017] Thomas G Wong and Raqueline AM Santos.  Exceptional quantum walk search on the cycle.  _Quantum Information Processing_ , 16:1–17, 2017.
  * Devitt [2016] Simon J Devitt.  Performing quantum computing experiments in the cloud.  _Physical Review A_ , 94(3):032329, 2016.
  * Stigler [1982] Stephen M Stigler.  Thomas bayes’s bayesian inference.  _Journal of the Royal Statistical Society: Series A (General)_ , 145(2):250–258, 1982.
  * Mezghani et al. [2008] Amine Mezghani, Michel T Ivrlac, and Josef A Nossek.  Achieving near-capacity on large discrete memoryless channels with uniform distributed selected input.  In _2008 International Symposium on Information Theory and Its Applications_ , pages 1–6. IEEE, 2008.
  * Blei et al. [2017] David M Blei, Alp Kucukelbir, and Jon D McAuliffe.  Variational inference: A review for statisticians.  _Journal of the American statistical Association_ , 112(518):859–877, 2017.
  * Bassett and Deride [2019] Robert Bassett and Julio Deride.  Maximum a posteriori estimators as a limit of bayes estimators.  _Mathematical Programming_ , 174:129–144, 2019.
  * Ziegel [2003] Eric R Ziegel.  The elements of statistical learning, 2003.
  * Kápl and Parízek [2020] Roman Kápl and Pavel Parízek.  Endicheck: Dynamic analysis for detecting endianness bugs.  In _Tools and Algorithms for the Construction and Analysis of Systems: 26th International Conference, TACAS 2020, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020, Dublin, Ireland, April 25–30, 2020, Proceedings, Part II 26_ , pages 254–270. Springer, 2020.
  * Gupta and Nadarajah [2004] Arjun K Gupta and Saralees Nadarajah.  _Handbook of beta distribution and its applications_.  CRC press, 2004.
  * Eliason [1993] Scott R Eliason.  _Maximum likelihood estimation: Logic and practice_.  Number 96. Sage, 1993.
  * Yang et al. [2022] Bo Yang, Rudy Raymond, and Shumpei Uno.  Efficient quantum readout-error mitigation for sparse measurement outcomes of near-term quantum devices.  _Physical Review A_ , 106(1):012423, 2022.
  * Krstovski et al. [2013] Kriste Krstovski, David A Smith, Hanna M Wallach, and Andrew McGregor.  Efficient nearest-neighbor search in the probability simplex.  In _Proceedings of the 2013 Conference on the Theory of Information Retrieval_ , pages 101–108, 2013.
  * Wong [1998] Tzu-Tsung Wong.  Generalized dirichlet distribution in bayesian analysis.  _Applied Mathematics and Computation_ , 97(2-3):165–181, 1998.
  * Endo et al. [2018] Suguru Endo, Simon C Benjamin, and Ying Li.  Practical quantum error mitigation for near-future applications.  _Physical Review X_ , 8(3):031027, 2018.
  * Zhang et al. [2020] Shuaining Zhang, Yao Lu, Kuan Zhang, Wentao Chen, Ying Li, Jing-Ning Zhang, and Kihwan Kim.  Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system.  _Nature Communications_ , 11(1):587, 2020.
  * Kandala et al. [2019] Abhinav Kandala, Kristan Temme, Antonio D Córcoles, Antonio Mezzacapo, Jerry M Chow, and Jay M Gambetta.  Error mitigation extends the computational reach of a noisy quantum processor.  _Nature_ , 567(7749):491–495, 2019.
  * White et al. [2022] G.A.L. White, F.A. Pollock, L.C.L. Hollenberg, K. Modi, and C.D. Hill.  Non-markovian quantum process tomography.  _PRX Quantum_ , 3:020344, May 2022.  10.1103/PRXQuantum.3.020344.  URL https://link.aps.org/doi/10.1103/PRXQuantum.3.020344.
  * Thorbeck et al. [2023] Ted Thorbeck, Andrew Eddins, Isaac Lauer, Douglas T McClure, and Malcolm Carroll.  Two-level-system dynamics in a superconducting qubit due to background ionizing radiation.  _PRX Quantum_ , 4(2):020356, 2023.
  * Henao et al. [2023] Ivan Henao, Jader P Santos, and Raam Uzdin.  Adaptive quantum error mitigation using pulse-based inverse evolutions.  _npj Quantum Information_ , 9(1):120, 2023.
  * Torlai et al. [2020] Giacomo Torlai, Christopher J Wood, Atithi Acharya, Giuseppe Carleo, Juan Carrasquilla, and Leandro Aolita.  Quantum process tomography with unsupervised learning and tensor networks.  _arXiv preprint arXiv:2006.02424_ , 2020.
  * [160] Quasi-probability representation of an ideal gate. Available online: https://mitiq.readthedocs.io/en/v0.11.1/examples/pec-tutorial.html#task-2-sampling-from-the-quasi-probability-representation-of-an-ideal-gate/ (last accessed on 25 mar 2024).
  * Gelman et al. [1995] Andrew Gelman, John B Carlin, Hal S Stern, and Donald B Rubin.  _Bayesian data analysis_.  Chapman and Hall/CRC, 1995.
  * Sivia and Skilling [2006] Devinderjit Sivia and John Skilling.  _Data analysis: a Bayesian tutorial_.  OUP Oxford, 2006.
  * Robert et al. [2010] Christian P Robert, George Casella, and George Casella.  _Introducing monte carlo methods with r_ , volume 18.  Springer, 2010.
  * Lee [1989] Peter M Lee.  _Bayesian statistics_.  Oxford University Press London:, 1989.
  * McKay et al. [2017] David C McKay, Christopher J Wood, Sarah Sheldon, Jerry M Chow, and Jay M Gambetta.  Efficient z gates for quantum computing.  _Physical Review A_ , 96(2):022330, 2017.
  * Ryan et al. [2004] William E Ryan et al.  An introduction to ldpc codes.  _CRC Handbook for Coding and Signal Processing for Recording Systems_ , 5(2):1–23, 2004.
  * Bravyi et al. [2024] Sergey Bravyi, Andrew W Cross, Jay M Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J Yoder.  High-threshold and low-overhead fault-tolerant quantum memory.  _Nature_ , 627(8005):778–782, 2024.