$\alpha=10^{2}$ and $\alpha=10^{6}$.

Another question that comes up when analyzing the performance of the proposed
on-line asymmetry-considering estimator is how it measures up against the on-
line estimator proposed in the previous section for the depolarizing channel.
The answer to this question will give us information regarding the importance
of asymmetry to decode QTCs. Figure 7.10 shows the results obtained by running
the turbo decoder with the estimator from section 7.2.2, as well as the
results obtained for the extended on-line estimator proposed in this section,
when we consider the Pauli channels with asymmetry coefficients
$\alpha=10^{2}$ and $\alpha=10^{6}$. Notice that neglecting the possibility
that the channel might be asymmetric and treating it like a depolarizing
channel leads to huge performance losses. Upon further examination of Figure
7.10, it can be seen that using the estimator from section 7.2.2 for the Pauli
channel always results, independently of the nature of $\alpha$, in the QTC
performance corresponding to the symmetric case $\alpha=1$. As a consequence,
if asymmetry is not considered in the error correction system, the performance
loss experienced by the QTCs will correspond to the gap between the waterfall
region presented by the decoder that has perfect knowledge of the channel
parameters and the waterfall region of the decoder for the depolarizing
channel, which increases with the growth of the asymmetry coefficient,
$\alpha$. In practical situations, since the quantum channel might be
asymmetric, using the extended on-line estimator proposed here is paramount if
one wants to maintain excellent performance.

####  7.4 Discussion

We have studied the performance sensitivity of QTCs applied over depolarizing
channels when there exists a mismatch between the actual depolarizing
probability and the probability fed to the decoders. To that end, we fed the
turbo decoder with values different from the actual value of the depolarizing
probability and observed the degradation in the WER due to such mismatch.
Simulation results showed that the decoder is sensitive both to under- and
over-estimation of the depolarizing probability, presenting a flat region
around the actual value of the depolarizing probability.

In order to overcome the lack of knowledge regarding the depolarization
probability, we have proposed a novel on-line estimation procedure based on
the information generated in the iterative turbo decoder. Such an on-line
estimator outperformed existing off-line estimators in terms of overall coding
rate and latency, while maintaining excellent WER performance. Simulation
results have shown that the on-line estimation method is insensitive to the
initial value of the depolarizing probability, and thus, the resulting
performance experiences very little degradation in the presence of channel
mismatch.

Moreover, we have presented an extension of the on-line estimation method for
the depolarizing channel, making it capable of considering the asymmetries
that are present in the decoherence models of realistic quantum devices. The
noise in these devices is modeled using the well known Pauli twirl
approximation channel, which has been shown to be biased towards $\mathrm{Z}$
type errors for some of the qubit construction technologies that are being
used in this day and age. Monte Carlo simulations corroborate that the
proposed on-line estimation method is successful in aiding the QTC to achieve
the same performance as when it has access to the channel parameters. This
positive outcome comes at the cost of a slight increase in the complexity of
the decoding algorithm. However, the flexibility that the proposed method
provides, allowing the QTCs to be able to operate close to their hashing
bounds without requiring any knowledge of the Pauli channel parameters or of
how they evolve in time, far outweighs this minor drawback.

###  CHAPTER 8 Quantum Low-Density-Generator-Matrix Codes and Degeneracy

This chapter serves as a summary of the main results of other QEC-related
research that the author has been involved in during this Ph.D. thesis. Given
the varied nature of this work, the chapter is divided into five different
sections, each one dedicated to a specific topic:

  * •

Section 8.1 studies the design of a class of non-Calderbank-Shor-Steane (CSS)
Quantum Low-Density-Generator-Matrix (QLDGM) codes that outperforms other
QLDPC schemes of the literature. It reviews the contents of [11].

  * •

Section 8.2 discusses the scenario of non-CSS QLDGM codes when they operate
without channel state information as it was done in Chapter 7 for QTCs. The
depolarizing channel is considered. It summarizes the research presented in
[10].

  * •

Section 8.3 proposes the design of CSS QLDGM codes for the asymmetric Pauli
channel. Our scheme outerperforms other QLDPC codes in the literature for some
of the scenarios considered. This section presents the methods and results of
[9].

  * •

Section 8.4 presents a group theoretical approach to discuss the issue of
degeneracy as it relates to sparse quantum codes. It summarizes the review
article [8].

  * •

Section 8.5 discusses previously existing methods to compute the logical error
rate and presents an efficient coset-based method inspired by classical coding
strategies to estimate degenerate errors. It reviews the results and
discussions of [7].

Before moving onward, it must be mentioned that the contents shown herein are
meant only as a cursory overview. Readers should refer to the Ph.D.
dissertation of Patricio Fuentes Ugartemendia (first-author of this research)
or the journal articles themselves for a complete discussion regarding these
findings.

####  8.1 Approach for the construction of non-CSS low-density-generator-
matrix-based quantum codes

Quantum LDPC codes are built by casting classical LDPC codes in the framework
of stabilizer codes [163], which enables the design of quantum codes from any
arbitrary classical binary and quaternary codes. In [33], the authors document
the design of QLDPC codes based on their classical counterparts, detailing
numerous construction and decoding techniques along with their flaws and
merits. Among the discussed methods, the construction of QLDPC codes based on
LDGM codes is shown to yield performance and code construction improvements,
albeit at an increase in decoding complexity. This method was originally
proposed in [164, 165], where Calderbank-Steane-Shor (CSS) quantum codes based
on regular LDGM classical codes were shown to surpass the best quantum coding
schemes of the time, and performance was significantly improved in [165, 166]
by utilizing a parallel concatenation of two regular LDGM codes.

Quantum LDGM (QLDGM)-based quantum code implementations, as well as most QLDPC
designs, are based on CSS constructions. CSS codes, simultaneously proposed by
Calderbank, Shor, and Steane in [167, 168], are a particular subset of the
family of stabilizer codes. They provide a straightforward method to design
quantum codes via existing classical codes. In general, decoding of quantum
codes based on CSS designs is performed separately for bit- and phase-flip
errors, which negatively impacts their performance. In fact, CSS constructions
decoded separately are limited by an unsurpassable bound, referred to as the
CSS lower bound [169]. Joint decoding of bit- and phase-flip errors using
modified CSS decoders capable of exploiting the correlation between the
aforementioned errors has been considered in the literature [170, 171, 172,
173, 174, 175, 176], and performance improvements associated with these
modified CSS decoders have been shown in these articles. Nonetheless, the
improvements provided by these modified CSS decoders come at the expense of an
increased decoding complexity, which, along with the performance limitations
of conventional CSS decoding, inspires the search for non-CSS constructions,
as they should theoretically be able to outperform CSS codes and avoid complex
decoding strategies. Non-CSS QLDPC-based codes were proposed in [177].
However, despite showing promise, they fail to outperform existing CSS QLDPC
codes for comparable block lengths.

In such context, we introduced a technique to design non-CSS quantum codes
based on the use of the generator and parity check matrices of LDGM codes. The
proposed methods are based on modifying the upper layer of the decoding graph
in CSS QLDGM constructions. An example can be seen in Figure 8.1. The
feasibility of such modification of the upper layer of the decoding graph is
based on the othogonality properties shown by the generator and parity-check-
matrices of the classical LDGM codes used for constructing the code. The
simplicity of the proposed scheme ensures that the high degree of flexibility
in the choice of the quantum rate and the block length for the CSS code
utilized as a starting point is translated to the non-CSS design.

Figure 8.1: Schematic representation of the combination of $\mathrm{x}$ and
$\mathrm{z}$ nodes of the upper layer of the decoding graph.

Compared with quantum CSS codes based on the use of LDGM codes, the proposed
non-CSS scheme is 0.2 dB closer to the hashing bound in the depolarizing
channel, and outperforms all other existing quantum codes of comparable
complexity. The results can be seen in Figure 8.2.

Figure 8.2: Word error rate for various quantum codes when they are applied
over the depolarizing channel. The hashing bound is also included. $p$ is the
depolarizing probability.

####  8.2 Performance of non-CSS LDGM-based quantum codes over the
misidentified depolarizing channel

Most of the research related to the performance of QLDPC codes has been
conducted under the tacit premise that perfect knowledge of the quantum
channel in question is available. In reality, such a scenario is highly
unlikely, meaning that analyzing how the behaviour of these codes changes in
terms of the existing information about the quantum channel is of significant
relevance. Such a study was conducted for the quantum depolarizing channel in
[72]. In [73], the same authors designed an improved decoding strategy for
QLDPC codes when only an estimate of the channel depolarizing probability is
available. The aforementioned method makes use of quantum channel
identification, which requires the introduction of a probe (a known quantum
state) into the quantum channel and the subsequent measurement of the channel
output state to produce an accurate estimate of the depolarizing probability.
This procedure typically makes use of additional qubits and results in a
latency increase. Thus, the design of methodologies capable of minimizing this
overhead while yielding performance similar to the perfect channel knowledge
scenario is germane to this field of research.

Figure 8.3: Simulated QBER as a function of the initial estimate of the
depolarizing probability of the channel $p^{(1)}$.

In Chapter 7 of this dissertation, published in [5], we have derived an on-
line depolarizing probability estimation technique for Quantum Turbo Codes
(QTCs). This method yields similar performance to that obtained when using the
same QTCs with perfect channel information but without the need for additional
resources. In light of this outcome, we propose a similar on-line estimation
procedure for QLDPC codes.

Our analysis revealed an increasing impact of channel mismatch on decoder
performance as the depolarizing probability of the channel grows. The mismatch
effect is especially noticeable in the waterfall region of the code. As in the
case of QTCs, the on-line estimation scheme outperforms off-line channel
identification techniques in terms of overall coding rate, while maintaining
excellent performance. The simulation results may be seen in Figure 8.3. In
contrast to what happens with QTCs, simulation results showed that the on-line
estimation method is slightly dependent on the initial estimate of the
depolarizing probability when operating in the waterfall region. Selecting the
initial estimate of the depolarizing probability as the hashing limit of the
code in question yielded good performance regardless of the actual value of
the depolarizing probability.

####  8.3 Design of low-density-generator-matrix-based quantum codes for
asymmetric quantum channels

Most of the research related to the performance of quantum error correcting
codes considers the symmetric Pauli channel, generally referred to as the
depolarizing channel, which incurs bit flips, phase flips, and bit-and-phase
flips with the same probability. However, realistic quantum devices, given the
nature of the materials used to construct them, often exhibit asymmetric
behavior, where the probability of a phase flip taking place is orders of
magnitude higher than the probability of a bit flip. As seen in Chapter 2, the
behavior of these quantum devices is governed by the single-qubit relaxation
time and the dephasing time of the device itself, the former sometimes being
orders of magnitude larger than the latter. Generally, relaxation causes both
bit-flips and phase-flips, while pure dephasing only leads to phase-flip
errors. This difference in relaxation and dephasing times gives rise to the
aforementioned asymmetric behavior, where bit-flip errors are much less likely
to occur than phase flips, and it can be accurately modelled by the Pauli
twirl approximation channel. Naturally, it stands to reason that the best QEC
schemes for this asymmetric channel must somehow be able to exploit its
asymmetry. In [60], the authors introduce an EXIT-chart based methodology to
design quantum turbo codes (QTCs) specifically for the Pauli twirl
approximation channel. We have extended such work in Chapter 7, published in
[3], where an on-line estimation protocol to decode QTCs over Pauli twirl
approximation channels has been proposed. These results speak to the merit of
constructing a coding scheme tailored to the asymmetric characteristics of the
quantum channel in question, since performance of the QTCs varies depending on
the degree of asymmetry of the channel.

Consequently, we studied the performance of quantum CSS LDGM codes when they
are applied over a Pauli channel. We showed how although they are not the best
known codes for the depolarizing channel, their simplicity allows for them to
be almost seamlessly adapted to the Pauli twirl approximation channel. Based
on this result, we introduce a simple yet effective method to derive CSS QLDGM
codes that perform well over channels with varying degrees of asymmetry. Such
a strategy is necessary because CSS codes designed for the depolarizing
channel perform poorly over its asymmetric counterpart. To the extent of our
knowledge and at the time of writing, the research on designing quantum codes
specifically for asymmetric quantum channels was quite limited [111, 178],
especially when compared to results regarding the depolarizing channel. Thus,
this work represents one of the first attempts at designing QLDPC codes
specifically for asymmetric quantum channels.

Figure 8.4: Achievable coding rate at a $\mathrm{WER}$ of $10^{-3}$ for
various QLDPC codes over Pauli channels with $\alpha=10$ and $10^{2}$.

The proposed methods were based on simple modifications to the upper layer of
the decoding graph of a symmetric CSS QLDGM code designed for the depolarizing
channel. Specifically, the factor graph is adapted to an asymmetric channel by
increasing the number of syndrome nodes used to decode the $\mathrm{Z}$
operators and decreasing the number of syndrome nodes used to decode the
$\mathrm{X}$ operators. This comes from the fact that $\mathrm{Z}$ errors are
more likely to occur when the Pauli twirl approximation channel is considered
(and $T_{2}<<T_{1}$), as it can be seen in section 2 of this dissertation.
Additionally, we showed how for larger block lengths, the proposed asymmetric
CSS codes can be further optimized based on the asymmetry coefficient $\alpha$
by increasing the block length of the code and the number of $\mathrm{Z}$
syndrome nodes. Over Pauli channels with $\alpha=10$ and $10^{2}$, the schemes
proposed are closer to the theoretical limit than other existing asymmetric
codes and the best codes designed for the depolarizing channel. The results
may be seen in Figure 8.4.

####  8.4 Degeneracy and its impact on the decoding of sparse quantum codes

The quantum phenomenon known as degeneracy should theoretically improve the
performance of quantum codes if exploited appropriately. Unfortunately,
degeneracy may end up sabotaging the performance of sparse quantum codes
because its existence is neglected in the decoding process. In this work we
provided a broad overview of the role this phenomenon plays in the realm of
quantum error correction and, more specifically, in the field of QLDPC codes.
We started by introducing a group theoretical explanation of the most relevant
concepts in the field of quantum error correction. This approach is helpful to
lay out the necessary groundwork to later study the effects of degeneracy.
Following this, we examined the differences between the classical decoding
problem and its quantum counterpart. Despite the intricate similarities
between classical and quantum decoders, the coset partitioning of the $N$-fold
Pauli space (see Figure 8.5) reveals the higher complexity of the quantum
decoding task. Based on this discussion, we were able to show in a concise
manner why applying the classical decoding algorithm for sparse codes to the
quantum problem is suboptimal. Then, we provided a detailed explanation
regarding the origin of degeneracy and how it may have detrimental effects on
traditional SPA-based decoders. Finally, we provided a simple example in order
to facilitate the comprehension of the topics discussed in the work.

Figure 8.5: Schematic representation of the coset partition of the effective
Pauli group by the stabilizer.

####  8.5 On the logical error rate of sparse quantum codes

The manifestation of degeneracy in the design of sparse quantum codes and its
effects on the decoding process has been studied extensively [30, 139, 172,
175, 179, 180, 181, 182]. Unfortunately, although degeneracy should
theoretically improve performance, limited research exists on how to quantify
the true impact that this phenomenon has on Quantum Low Density Parity Check
(QLDPC) codes. This has resulted in the performance of sparse quantum codes
being assessed differently throughout the literature; while some research
considers the effects of degeneracy by computing the metric known as the
logical error rate [172, 177, 183, 184, 185, 186, 187], other works employ the
classical strategy of computing the physical error rate [33, 164, 165, 188,
189], a metric which provides an upper bound on the performance of these codes
since it ignores degeneracy. However, because these codes are highly
degenerate [39, 139, 172, 179, 190], there is a significant gap between the
upper bound provided by the physical error rate and their true performance.
Consequently, we devised a group theoretic strategy to accurately assess the
effects of degeneracy on sparse quantum codes and we explain another method
that was used in [183, 184, 185, 186] to compute the logical error rate. Then,
we used our methodology to show how sparse quantum codes should always be
assessed using the logical error rate.

Figure 8.6: Ratios of end-to-end different syndrome, identical syndrome and
degenerate errors for the QLDPC code with blocklength $N=2000$ in
consideration.

The discrepancy between the logical error rate and the physical error rate is
especially relevant to the field of sparse quantum codes because of their
degenerate nature. This was reflected by the results (see Figure 8.6) obtained
for a specific family of QLDPC codes, whose performance can be up to $20\%$
better than would be expected based on previous results in the literature. In
addition, these simulation outcomes serve to show how performance may be
improved by constructing degenerate quantum codes, and they also speak towards
the positive impact that modified decoding strategies can have on the
performance of sparse quantum codes.

###  CHAPTER 9 Conclusions and Future Work

This Ph.D. thesis set out with the following two objectives in the context of
quantum information and quantum error correction:

  * •

The analysis of decoherence as the source of the errors that corrupt quantum
information.

  * •

The optimization of the performance of quantum turbo codes when different
scenarios are considered.

With these goals in mind, we commenced the dissertation by giving the basic
tools regarding quantum computing, quantum information theory and quantum
error correction needed in order to understand the research that is presented
in the following parts of the thesis. In Chapter 2, we first introduced the
postulates of quantum mechanics, both from the state vector and density matrix
perspectives, which are the mathematical framework that describe the theory of
Quantum Mechanics. We continued by describing decoherence as the source of the
errors that make quantum computations unreliable, and provided the way it is
mathematically modelled by the use of quantum channels. Moreover, we discussed
the way in which such quantum channels can be approximated so that they can be
efficiently simulated in classical computers via the quantum information
technique named twirling. The concept of quantum capacity was also introduced.
We finished this preliminary chapter by discussing the way in which quantum
error correction codes are simulated in classical computers.

Following this, we got into Part I (consisted of Chapters 3, 4 and 5) of the
thesis which is focused on mathematically modelling decoherence and studying
the asymptotical limits of error correction of quantum channels. In Chapter 3
we proposed a time-varying quantum channel (TVQC) model. The motivation of
such proposal was the recent experimental observation that the parameters of
decoherence of superconduting qubits (relaxation time, $T_{1}$, and dephasing
time, $T_{2}$) are not fixed, but fluctuate through time. Therefore, we
studied the stochastic processes that describe the random temporal behaviour
of such parameters and integrated such randomness into the context of quantum
channels. Moreover, we compared the action of the static quantum channels with
the proposed TVQCs by making use of the metric named diamond norm distance. We
concluded that both models differ significantly in terms of diamond norm
distance indicating that the performance of QECCs will be significantly
affected as a result of the $T_{1}$ and $T_{2}$ flutctuations.

Chapter 4 is the natural continuation to Chapter 3. Quantum channel capacity
is the asymptotically achievable rate by QECCs when static quantum channels
are considered. In this chapter, we studied the asymptotical limits for
quantum error correction under the assumption that the realizations of $T_{1}$
and $T_{2}$ are the same for all the qubits of a block, which is reminiscent
of the classical scenario of slow fading channels. For classical slow fading
channels the operational concept of capacity fails to exist and has to be
replaced by the so-called outage probability. Thus, we proposed the quantum
outage probability as the asymptotically achievable error rate for QECCs when
they operate over TVQCs. We provided closed-form expressions for the family of
time-varying amplitude damping channels ($T_{1}$-limited qubits). Furthermore,
we proposed the quantum hashing outage probability as an upper bound for the
quantum outage probability when Pauli/Clifford twirl approximated channels are
considered, and provided closed-form expressions for such bound. The results
concluded that the quantum outage probability is a monotonically increasing
function with the coefficient of variation of the relaxation time. Thus, we
discussed that the asymptotically achievable performance of QECCs will be
limited not only by the mean qubit relaxation time, but also by the relative
deviation of such parameter respect to the mean.

We finished the Part I of the thesis by studying how the performance of QECCs
is affected when they operate over TVQCs. In Chapter 3 we discussed that, due
to the deviation in terms of diamond norm distance between static and time-
varying channels, the performance of QECCs may be compromised when time
variations are considered. We proved this intuition by simulating the Kitaev
toric code and quantum turbo codes operating over the TVQCs. We concluded that
the performance curves of QECCs are flattened when the channels are time-
varying. However, the flattening of the curve does not only depend on the
nature of the channel, but also on the performance of the code when it
operates over static channels. Thus, the degradation of the WER curve is a
function of the coefficients of variation of the decoherence parameters and
the steepness of WER curve of the code under static noise. This comes from the
fact that when the performance of the QECCs is flat in static scenarios, the
difference in terms of WER between “good” and “bad” channel realizations will
not be very different and, therefore, the mean performance of the code in a
TVQC will be similar to the static one. Finally, we used the quantum outage
probability in order to benchmark the performance of the codes in
consideration.

The main takeaways of Part I can be summarized as:

  * •

Decoherence parameter fluctuations are a critical issue to consider when
quantum error correction codes are constructed. Neglecting their time-varying
nature would leave out an integral part of quantum noise, making the
predictions of the performance of QECCs too optimistic in many scenarios.

  * •

The importance of superconducting qubit construction and cooldown, in the
sense that if they are optimized correctly, the fluctuations relative to the
mean will be milder. Traditionally, long mean decoherence times are seeked for
the qubits to be able to handle long time duration algorithms, however, this
should be done jointly with lowering the dispersion of such parameter.

  * •

Qubit benchmarking, at least for superconducting qubits at the moment, should
include the variation parameters of their decoherence times. However, the
current experimental research on the topic usually only focuses at the mean or
best case scenarios, which is far from being accurate when describing the
decoherence dynamics of the qubits.

  * •

Even if the current state-of-the-art experimentally implemented QECCs will not
be significantly affected by the TVQC (as they are generally short codes such
as the $3\times 3$ toric code discussed in Chapter 5), the proposed channel
model will be essential when better QECCs are implemented, such as the
$7\times 7$ and $9\times 9$ toric codes for the near term, or the more
advanced QTCs or QLDPCs for the long term.

Part II of the thesis is focused on improving the error correction performance
of QTCs. Taking this aim into acount, we attempted the optimization of the
error floor performance of QTCs by implementing practical interleavers rather
than the traditionally considered random ones (Chapter 6). Inspired by the
classical-to-quantum isomorphism, we adapted certain interleaving patterns
that proved to improve the performance of classical turbo codes to the
paradigm of quantum error correction. In particular, we implemented
$S$-random, Welch-Costas and JPL interleavers for scrambling the quantum
information that comes out of the outer quantum convolutional encoder.
Simulation results showed that the error floor performance is improved up to
two orders of magnitude. We also showed that memory requirements can be
lowered by using specific interleaver designs, while the performance in the
error floor region is comparable or even better than that of the original
QTCs. We discussed that this benefits come without any decoding complexity
costs.

Chapter 7 dealt with QTCs whose decoders operate without channel state
information. We began studying the sensitivity that the turbo decoder has to a
mismatch between the actual channel depolarizing probability and the
probability fed to the decoder. Simulation results showed that the decoder is
sensitive both to under- and over-estimation of the depolarizing probability,
presenting a flat region around the actual value of the depolarizing
probability. Therefore, we discussed that estimation techniques are necessary
for the QTCs to operate reliably over depolarizing channels. With that in
mind, we first considered existing off-line estimation techniques that
estimate the noise level of the channel previous to the operation of the QTCs.
This techniques proved to be succesful when aiding the turbo decoders to reach
the same performance as when perfect CSI is available, but at the expenses of
an increase of the qubit overhead (which in certain scenarios may lead to code
rate reduction) and system latency. To overcome such issues, we proposed an
on-line estimation technique that is able to use the information being
processed in the inner SISO decoder of the QTC for succesfully estimating the
channel depolarizing probability. Such an on-line estimator outperformed
existing off-line estimators in terms of overall coding rate and latency,
while maintaining excellent WER performance and decoding complexity.
Simulation results showed that the on-line estimation method is insensitive to
the initial value of the depolarizing probability, and thus, the resulting
performance experiences very little degradation in the presence of channel
mismatch. We finished Chapter 7 by extending the on-line estimation method to
scenarios where the channel is asymmetric, i.e. the Pauli twirled
approximation channel. Monte Carlo simulations corroborated that the extended
on-line estimation method is successful in aiding the QTC to achieve the same
performance as when it has access to the channel parameters. We discussed
that, here, this positive outcome comes at the cost of a slight increase in
the complexity of the decoding algorithm.

The main takeaways of Part II can be summarized as:

  * •

The classical-to-quantum isomorphism allows to import the knowledge of
classical coding theory to the context of quantum error correction. This way,
the performance of QECCs can be improved by utilizing classical techniques
that have been extensively developed in the literature if they are adapted
correctly.

  * •

Interleavers play a central role in the design of QTCs due to their ability to
reduce the error floors of such codes while maintaining the performance in the
waterfall region. Some interleaver constructions do also reduce the memory
requirements of QTCs.

  * •

Quantum turbo codes are susceptible to channel state information mismatch.
Feeding the turbo decoder with channel estimates that are not accurate enough
compromises the excellent performance of such family of QECCs.

  * •

Off-line estimation protocols are succesful in aiding QTCs to operate with the
same performance as when the turbo decoder has access to perfect channel
information. However, this comes at the expenses of increasing the qubit
overhead and system latency.

  * •

The on-line estimation method proposed is succesful in the task of making QTCs
that are blind to CSI to operate with a performance similar to the ones that
have access to perfect CSI. This method is succesful without needing a higher
number of qubits. The on-line estimation protocol requires lower system
latency than the discussed off-line estimation protocols. This modified
decoder is able to operate for depolarizing (CTA) and asymmetric (PTA)
channels.

In the final chapter of the dissertation, Chapter 8, we provided a brief
overview of other QEC-related works that the author has been involved in
during his time as a Ph.D. student. Such work was related to the design and
optimization of quantum low-density-parity-check-matrix codes and the study of
the quantum-unique phenomenon named degeneracy.

####  9.1 Future research lines

Many of the presented discussions and results are encouraging and suggest
further study and analysis. Here we present some future research lines that
arise from the work done in this Ph.D. thesis.

##### Asymptotically achievable limits for the combined amplitude and phase
damping channel

As discussed in Chapter 2, the quantum capacity of the combined amplitude and
phase damping channel is yet unknown. It was discussed that, in general,
quantum noise for two-level coherent systems is described by such channel that
includes both relaxation and pure dephasing effects. Therefore, the knowledge
of the asymptotically achievable limits at the time of writing reduces to the
set of $T_{1}$-limited qubits, given by the quantum capacity of the amplitude
damping channel. This, however, neglects an integral part of decoherence since
most of the run-of-the-mill experimental qubits do not saturate the Ramsey
limit ($T_{2}\approx 2T_{1}$). With this in mind, it is critical to obtain an
expression for the quantum capacity of the combined amplitude and phase
damping channel.

Moreover, in Chapter 4, we proposed the concept of quantum outage probability
as the asymptotically achievable error rate by QECCs when operating over TVQCs
that consider the realizations of the decoherence parameters to be equal for
all the qubits of a block. In such chapter, we derived closed-form expressions
of $p_{\mathrm{out}}^{Q}$ for qubits that are $T_{1}$-limited. Similar studies
for the quantum outage probability of the more general time-varying combined
amplitude and phase damping channel will be considered in future work. The
aforementioned quantum capacity for such channel under static noise is needed
for achieving this last task. This will be critical in order to have a
complete tableau of the information theoretical limits of error correction for
superconducting qubits with pure dephasing channels.

##### Fast time-varying quantum channels

In Chapters 4 and 5 we assumed that the realizations of the decoherence
parameters were the same for each of the qubits in the block of an error
correction round. Under such premise, the TVQC model resembled the classical
scenario of slow fading channels. Having such similarity in mind, we derived
the quantum outage probability as the asymptotically achievable error rate by
QECCs and simulated Kitaev toric codes and QTCs over such channels.
Nevertheless, such assumption may not be true in general, indicating that
depending on the qubit-wise correlation presented by $T_{1}$ and $T_{2}$ the
system will operate in a different way to what has been studied here. At the
time of writing, little to none experimental results regarding the qubit-wise
correlation of the fluctuations of $T_{1}$ and $T_{2}$ have been done with the
sole exception of [51]. In such study, the authors studied the correlation
between the relaxation times of two superconducting qubits and concluded that
they were completely uncorrelated. This has important implications on error
correction. First, since the realizations of the channel will be different and
independent for each of the qubits of the block, this scenario will be
reminiscent of the classical fast fading channel. That is, each of the symbols
of the block will be affected by an independent realization of the channel.
Therefore, the classical concept of ergodic capacity may be integrated in the
context of quantum error correction as the asymptotically achievable limit.
This would obviously have implications on the performance of QECCs.
Furthermore, considering a case where there is partial qubit-wise correlation
of $T_{1}$ and $T_{2}$ would generalize every possible scenario for the
proposed TVQC model.

As a consequence, this topic requires to be tackled both from the experimental
and theoretical perspectives. First of all, a more complete experimental
benchmarking of the fluctuations of the decoherence parameters of qubits needs
to be done so that they can be properly characterized. In this way, the
correlations among the qubits of the system can be correctly addressed by
theoretical models. This will probably depend on the experimental
implementation of the system (qubit technology, architecture, …) and, thus, it
is probable that each of the possible time-varying scenarios will be valid for
some of the experimental quantum hardwares. Therefore, the theoretical study
of the TVQC scenarios with uncorrelated or partially correlated fluctuations
will be very important for understanding how QECCs will operate when real
hardware is considered.

##### Switching rate QEC protocols for TVQCs

Part I of the dissertation described the theoretical modelling of the
fluctuations of the decoherence parameters of superconducting qubits and how
such fluctuations degrade the performance of QECCs. It is important to remark
that the proposed TVQC model is relevant for protocols involving a large
number of error correction cycles (e.g., a quantum memory or a long quantum
algorithm, such as the one in ref. [149]). If a very short algorithm or QECC
is run just once or a very few number of times, the parameters will not
fluctuate and the effects of the proposed channel model will not be
noticeable. Herein lies important future work on this topic, such as the
construction of optimized error correction codes that are apt to address these
dynamic scenarios. In this way, we consider relevant trying to revert the
degradation suffered by QECCs as a consequence of the decoherence parameter
fluctuations. Therefore, a possible solution might be based on a switching
rate protocol that can adapt the rate of the QECCs depending on the noise
realization. In this way, the protocol would switch to lower QECC rates when
noise level realizations are high, and to higher rates when the realizations
are low. Rate adaptive schemes have been extensively studied in the field of
classical coding theory [191, 192]. By doing this, it is possible to obtain a
protocol that assures that the WER of the code is lower than some threshold
value, say $10^{-3}$, while trying to consume as little qubit resources as
possible111Note that by designing the QECCs by considering that the
realizations are always bad will imply that the WER target would be achieved,
but the qubit consumption would be high. Qubits are an expensive resource and,
thus, designing codes that are far from being optimized is not the best
praxis.. It is important to realize that for this protocol to work
appropriately, we need to implement an estimation protocol so that, depending
on the estimated noise level, the rate is adapted accordingly. The online
estimation protocol proposed in Chapter 7 of this thesis is a perfect match
for doing so. Note that the time needed for an error correction round is much
smaller than the stochastic process coherence time and, thus, several QECC
blocks can be fitted in the same noise realization. In this way, the on-line
estimation protocol can check when the noise level has changed and, therefore,
switch the rate accordingly. Another thing to be considered is that the
selected QECC can vary its rate in a seamless manner. Protocols such as this
one may be needed for the QECCs to maintain excellent performance with low
qubit overheads when time variations are important.

##### Optimizing interleaver constructions for QTCs

In Chapter 6 of this Ph.D. dissertation we demonstrated that implementing
interleavers with some construction leads to an improvement of the performance
of QTCs in the error floor region. For doing so, we imported some well known
interleaving patterns from classical turbo coding theory to the context of
QTCs. In this context, there is still a vast amount of knowledge regarding
interleaver construction to be imported from the classical coding theory [193,
194, 195, 196, 197]. This further optimization of the scrambling patterns in
between the convolutional encoders that form the turbo code is based on the
actual structure of those two constituent codes. Therefore, in order to import
those interleaving methods to the context of quantum information, the
structure of the QCCs must be studied appropriately. Furthermore, as it has
been seen in Chapter 6, quantum interleavers have an additional degree of
freedom since $1$-qubit symplectic transformations can be included while
interleaving the quantum information stream. Nevertheless, at the time of
writing, how those would impact the performance of turbo codes remains to be
studied. Thus, we consider important to address such additional degree of
freedom when optimizing the performance of QTCs. To sum up, we consider that
the error floors of QTCs can still be improved significantly by just
constructing interleavers that are a good fit for concatenating the
constituent QCCs.

These are the main future research topics that follow the investigations
conducted throguhout this thesis. However, quantum information and computing
are rather young and, thus, a vast amount of possible paths to follow are open
at the moment. Fault-tolerant implementations of NISQ-era algorithms, hardware
specific implementations of QECCs or the construction of error correction
codes that maximize degeneracy may serve as examples of such endeavours from
the point of view of quantum error correction. Recently, claims for quantum
advantage have been made by Google and China independently [49, 50] and the
barrier of $100$ superconducting qubits has been surpassed by the new
ibm_washington processor [198]. Needless to say, quantum computing has a
bright future that will mainly be developed in this decade and QEC is
considered to be key in order to hit the jackpot. This will require an
Herculean effort by the community of quantum computing and quantum
information. Exciting times are ahead!

###  Appendix A Tensor Product

Tensor products are mathematical operations that put vector spaces together in
order to form larger vector spaces. The tensor product is a crucial part in
order to understand multiparticle systems in quantum mechanics.

Suppose that $V$ and $W$ are vector spaces of dimension $m$ and $n$
respectively. Then $V\otimes W$, or the tensor product between the vector
spaces, is an $mn$-dimensional vector space. The elements of this composite
vector space are linear combinations of tensor products
$\ket{v}\otimes\ket{w}$ of the elements $\ket{v}$ of $V$ and $\ket{w}$ of $W$.
In particular, if $\ket{i}$ and $\ket{j}$ are orthonormal bases for the spaces
$V$ and $W$ respectively, then $\ket{i}\otimes\ket{j}$ is a basis for the
composite vector space $V\otimes W$.

Tensor product in matrix vector spaces is performed by the so-called Kronecker
product.

######  Definition A.0.1 (Kronecker Product).

Suppose $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{p\times q}$ are two
arbitrary complex matrices. Then $A\otimes B\in\mathbb{C}^{mp\times nq}$ and
is defined as

$A\otimes B\equiv\begin{pmatrix}a_{11}B&a_{12}B&\cdots&a_{1n}B\\\
a_{21}B&a_{22}B&\cdots&a_{2n}B\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{m1}B&a_{m2}B&\cdots&a_{mn}B\end{pmatrix},$

where $a_{ij}$ refers to the element in row $i$ and column $j$ of matrix $A$.

The Kronecker product as defined in definition A.0.1 will be the tensor
product that will be considered in the thesis. This tensor product presents
the next properties:

  * •

Non-commutativity: $A\otimes B\neq B\otimes A$ in general.

  * •

Bilineality and associativity:

    * –

$A\otimes(B+C)=A\otimes B+A\otimes C$, with $B$ and $C$ of same dimensions.

    * –

$(A+B)\otimes C=A\otimes C+B\otimes C$, with $A$ and $B$ of same dimensions.

    * –

$(kA)\otimes B=A\otimes(kB)=k(A\otimes B)$, where $k\in\mathbb{C}$.

    * –

$(A\otimes B)\otimes C=A\otimes(B\otimes C)$.

  * •

Mixed-product: $(A\otimes B)(C\otimes D)=AC\otimes BD$, where the dimensions
must be appropriate for the ordinary matrix product. From here it is derived
that:

    * –

$(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$, from where it is clear that for
$(A\otimes B)^{-1}$ to exist, both $A^{-1}$ and $B^{-1}$ must exist.

    * –

$(A\otimes B)^{T}=A^{T}\otimes B^{T}$.

  * •

Trace: $\mathrm{Tr}(A\otimes B)=\mathrm{Tr}(A)\mathrm{Tr}(B)$.

  * •

Determinant: $\mathrm{det}(A\otimes
B)=\mathrm{det}(A)^{q}\mathrm{det}(B)^{n}$, where $A\in\mathbb{C}^{n\times n}$
and $B\in\mathbb{C}^{q\times q}$.

  * •

Rank: $\mathrm{rank}(A\otimes B)=\mathrm{rank}(A)\mathrm{rank}(B)$.

To finish with the tensor product, notation $A^{\otimes k}$ will mean
$A^{\otimes k}=A\otimes A\otimes\cdots\otimes A$, that is $A$ matrix tensored
by itself $k$ times. For example, $\ket{\psi}^{\otimes
3}=\ket{\psi}\otimes\ket{\psi}\otimes\ket{\psi}$.

###  Appendix B Decoding algorithms

In this appendix we will detail the decoding algorithms that have been used in
this dissertation. We will discuss the decoding algorithm for the Quantum
Turbo codes and the Minimum-Weight Perfect Matching (MWPM) decoder for Kitaev
toric codes. We will not go into the subtleties of these decoding methods, but
rather present their principles of operation. We provide appropriate
references for the reader that would like to go deeper in understanding the
decoding protocols for such quantum error correction codes.

####  B.1 Quantum Turbo Code decoder

Decoding of classical turbo codes rely on belief propagation methods such as
the sum-product algorithm [199] that ouput the most probable codeword
transmitted based on the syndrome of the received codeword and the information
possesed about the channel. In the quantum setting, decoding is not to find
out the most probable quantum state transmitted111Note that quantum states are
continuous, so this is impossible to do., but to find out the most probable
error coset of effective Pauli errors that have affected the codeword, and
that have been discretized after syndrome measurement. Consequently, sum-
product like algorithms in the quantum decoding world estimate the most
probable error coset and so afterwards such correction operation can be done
to the received state.

For QTCs, a Soft-Input-Soft-Output (SISO) decoding algorithm was proposed for
turbo decoding in [36] and it was later enhanced by means of the so called
extrinsic information transfer in [37]. Such decoding algorithm will be
described next. The decoding scheme for QTCs is shown in Figure B.1.

The SISO decoder for quantum convolutional codes is an adaptation of the usual
classical algorithm [36, 199]. This algorithm relies on the convolutional
structure of the convolutional encoder in order to perform a forward-backward
algorithm that will output the estimated channel error. Two of these are used
in order to decode the quantum turbo code, as the code is obtained as the
interleaved concatenation of two quantum convolutional codes. The SISO
decoding algorithm for QCCs is extensively described in [36].

Figure B.1: Circuit for SISO decoding of quantum turbo codes.

From Figure B.1 it can be seen that the receiver first decodes the inner code
based on the channel information, $\mathrm{P}_{\mathrm{ch}}(\mathcal{E}_{2})$,
and the measured syndrome, $\bar{s}_{2}$, for such inner code. The syndrome
measurement is done as for block QECCs (see Chapter 2) and the estimation of
the channel error is done by taking advantage of the structure of the inner
convolutional decoding unitary. Afterwards, the obtained probabilities of the
logical errors that may have happened during transmission conditioned to the
syndrome read, $\mathrm{P}_{2}(\mathcal{L}_{2}|\bar{s})$, are passed thorugh
the interleaver and given as a priori input information to the outer SISO
decoder, $\mathrm{P}_{1}(\mathcal{E}_{1})$. This element does the same
operation as the inner SISO decoder with the new information,
$\mathrm{P}_{1}(\mathcal{E}_{1})$, and the syndrome read for the outer code,
$\bar{s}_{1}$. The decoding for the quantum state is now done by taking
advantage of the structure of the outer convolutional decoding unitary. After
decoding, the information about the probability of the physical operator
$\mathcal{E}_{1}$ conditioned with the syndrome read,
$\mathrm{P}_{1}(\mathcal{E}_{1}|\bar{s})$, is deinterleaved and sent back to
the the inner SISO decoder, $\mathrm{P}_{2}(\mathcal{L}_{2})$. For the first
iteration, such probability is taken as equiprobable. This loop is done
several times until the estimates of the inner and outer decoders match or
some number of limit iterations is achieved (convergence is the most probable
thing to happen as stated in [37]). The output of the decoder is the
probability distribution $\mathrm{P}_{1}(\mathcal{L}_{1}|\bar{s})$ and, thus,
the most probable error coset given the syndrome read can be estimated.

It has been stated that in [37], an improved version of the turbo decoding
algorithm was presented, and we briefly present how such enchancement was
done. The algorithm presented before is based on the exchange of a posteriori
information between the constituent quantum convolutional SISO decoders of the
QTC. As the decoders pass along a posteriori information, successive
iterations of the constituent decoders are dependent on one another giving
rise to a detrimental positive feedback effect that prevents the decoding
algorithm to achieve the desired gains that iterative decoding usually
presents.

For the sake of avoiding such effect, it is necessary that the a priori
information directly related to a given information qubit is not reused in the
other decoder. Consequently, and similar to the approach employed in classical
turbo decoding, this can be achieved by making one decoder to remove the a
priori information from the a posteriori information before feeding it to the
other decoder. More explicitly, the iterative decoding procedure should
exchange only extrinsic information that is new and unknown to the other
decoder. To see how this information transfer works, consider a four-
port222Note that in the scheme presented in Figure B.1, the SISO decoders are
not exactly like that, but this is a way to see how they work. SISO decoder
that generates soft output information pertaining to a logical error
$\mathcal{L}$ and a physiscal error $\mathcal{E}$. Such circuit is presented
in Figure B.2.

Figure B.2: A SISO decoder processes a priori information and outputs a
posteriori information, from which the a priori information is removed to
obtain extrinsic information that is passed to the next decoder. Note that the
probabilities here are in logarithmic scale so that they can be just
substracted.

As it can be seen in Figure B.2, a SISO decoder exploits an A Posteriori
Probability (APP) module that accepts the a priori information
$\mathrm{P}^{a}(\mathcal{L})$ and $\mathrm{P}^{a}(\mathcal{E})$ as input and
outputs the a posteriori information $\mathrm{P}^{o}(\mathcal{L})$ and
$\mathrm{P}^{o}(\mathcal{E})$. The corresponding extrinsic probabilities
$\mathrm{P}^{e}(\mathcal{L}_{i}^{j})$ and
$\mathrm{P}^{e}(\mathcal{E}_{i}^{j})$ for the $j^{th}$ qubit at time instant
$i$ are then obtained by discarding the a priori information from the a
posteriori information as follows

$\begin{split}&\mathrm{P}^{e}(\mathcal{L}_{i}^{j})=N_{\mathcal{L}^{j}}\frac{\mathrm{P}^{o}(\mathcal{L}_{i}^{j})}{\mathrm{P}^{a}(\mathcal{L}_{i}^{j})},\\\
&\mathrm{P}^{e}(\mathcal{E}_{i}^{j})=N_{\mathcal{E}^{j}}\frac{\mathrm{P}^{o}(\mathcal{E}_{i}^{j})}{\mathrm{P}^{a}(\mathcal{E}_{i}^{j})},\end{split}$

where $N_{\mathcal{L}^{j}}$ and $N_{\mathcal{E}^{j}}$ are normalization
factors, which ensure that
$\sum_{\mathcal{L}^{j}}\mathrm{P}^{e}(\mathcal{L}_{i}^{j})=1$ and
$\sum_{\mathcal{E}^{j}}\mathrm{P}^{e}(\mathcal{E}_{i}^{j})=1$, respectively.

Furthermore, to avoid numerical instabilities and to reduce the computational
complexity, log-domain arithmetics are conventionally employed, which convert
the multiplicative operations to addition, as given below

$\begin{split}&\ln[\mathrm{P}^{e}(\mathcal{L}_{i}^{j})]=\ln[N_{\mathcal{L}^{j}}]+\ln[\mathrm{P}^{o}(\mathcal{L}_{i}^{j})]-\ln[\mathrm{P}^{a}(\mathcal{L}_{i}^{j})],\\\
&\ln[\mathrm{P}^{e}(\mathcal{E}_{i}^{j})]=\ln[N_{\mathcal{E}^{j}}]+\ln[\mathrm{P}^{o}(\mathcal{E}_{i}^{j})]-\ln[\mathrm{P}^{a}(\mathcal{E}_{i}^{j})].\end{split}$

Therefore, the inputs and outputs of a SISO decoder are the logarithmic a
priori and extrinsic probabilities, repsectively.

In [37] it was shown via simulation that this approach of extrinsic
information transfer outerperforms the original turbo decoder proposed in [36]
that relied on the a posteriori information.

The turbo decoding algorithm via SISO decoders for the convolutional codes has
been presented here. Note that the turbo decoding algorithm is a sum-product
like algorithm, similar to the ones used for decoding classical turbo codes
with the slight difference that the effective Pauli errors are estimated
instead of the codewords. Consequently, this algorithm is a classical
algorithm that will run on a classical computer, and the quantum computer will
be classically aided to solve the decoding problem and, thus, the complexity
of the algorithm will be classical. Consequently, it is very important that
the algorithm does not take more time than the decoherence time of the qubit,
as if that was the case, then the qubits may suffer from additional errors
that are not considered by the decoder. This implies that the study of quantum
computing does also involve classical algorithms, and enhancing them will
improve the quality of quantum error correction. Other approach to this
problem would be to use the inherent parallel nature of quantum computers to
develop quantum sum-product algorithms that would run in the quantum computers
with the advantage to have a potentially better complexity than the classical
ones and, therefore, improve the latency of the communications [200, 201].
Variations of those could also be useful to decode classical codes and reduce
the latency of classical communications with hybrid classical-quantum
computers [58].

####  B.2 Minimum-Weight Perfect Matching (MWPM) Decoder

The Kitaev toric code is a surface code, which are codes defined on a 2D
lattice of qubits [38]. Specifically, the toric codes have periodic boundaries
in those latices, making them to have a torus-like shape. In surface codes,
decoding a syndrome is equivalent to finding paths between the generators
whose syndrome has been triggered. There will be multiple paths associated to
the same syndrome and, consequently, the decoder must estimate the path that
describes the most likely error. In general, the optimal decoder for toric
codes has exponential complexity [151, 202], and, thus, it is impractical due
to the fact that the classical algorithm must obtain an estimation of the
error before the qubit suffers from additional errors.

A widely used decoding algorithm for the family of Kitaev toric codes is the
so called Minimum-weight Perfect Matching (MWPM) decoder [151, 202]. This
decoder is based on the MWPM problem of graph theory, where a matching333A set
of edges without common vertices. in which the sum of weights is minimized
must be found. The term perfect refers to the fact that the matching matches
all vertices of the graph. One can convert the lattice of the toric code with
errors to a complete graph, where the generators with non-trivial syndrome are
the nodes. The edges between the vetices have a weight equal to the minimum
number of qubits between them. This way, solving the MWPM problem to such
associated graph, the most likely error, i.e. the one with minimum weight, can
be estimated [151, 202]. The MWPM problem can be solved using Edmund’s Blossom
algorithm [203] with some improvements in order to have a better computational
complexity.

The toric codes of this dissertation are decoded via the MWPM decoder. We use
the implementation of this decoder in the QECSIM tool [151] for the numerical
simulations conducted.

###  Appendix C Monte Carlo numerical simulations

In this appendix we describe the numerical Monte Carlo simulations that we
have conducted in this dissertation in order to asses the performance of the
quantum error correction codes discussed. This way we are able to justify that
the numerical simulations performed are accurate and, thus, the conclusions
obtained from them valid.

The numerical simulations are based on the approximate error models that were
explained in Chapter 2. Each round of the numerical simulation is performed by
generating an $n$-qubit Pauli operator, calculating its associated syndrome
and finally running the decoding algorithm using the syndrome as its input.
The error operators are randomly generated using the distinct probability
distributions for the Pauli channel that were discussed in this thesis. Once
the logical error is estimated it is compared with the channel error111We
observe if the logical error associated to the channel error and the estimated
logical error belong to the same error coset. in order to decide if the
decoding round was succesful. The operational figure of merit selected in
order to evaluate the performance of these quantum error correction schemes is
the Word Error Rate ($\mathrm{WER}$), which is the probability that at least
one qubit of the received block is incorrectly decoded.

Since the simulations we are conducting are based on the random sampling of
the channel errors, the rules of Monte Carlo simulations are invoked in order
to obtain results that are honest. Thus, for estimating the $\mathrm{WER}$ of
the Kitaev toric codes and QTCs, the following Monte Carlo rule of thumb has
been used in order to select the number of blocks to be transmitted,
$N_{\mathrm{blocks}}$ [204]:

$N_{\mathrm{blocks}}=\frac{100}{\mathrm{WER}}.$ (C.1)

As explained in [204], under the assumption that the observed error events are
independent, this results in a $95\%$ confidence interval of about
$(0.8\hat{\mathrm{WER}},$ $1.25\hat{\mathrm{WER}})$, where
$\hat{\mathrm{WER}}$ refers to the empirically estimated value for the
$\mathrm{WER}$. This way, we can justify that the results obtained via random
sampling of the quantum channel are statistically representative and, thus,
the conclusions obtained from those throughout the dissertation are valid.

###  Appendix D Diamond norm distance

In Chapter 3, the diamond norm distance is used as the metric to compare the
action of static quantum channels versus the proposed time-varying quantum
channels. The objective of such analysis is to show that neglecting the
fluctuating of $T_{1}$ and $T_{2}$ may result in an unrealistic model for
quantum noise. Here we introduce the needed concepts regarding diamond norm
distance in order to understand such comparison. We also provide a proof that
shows that the diamond norm distances between the two ADPTA and two ADCTA
channels obtained from the same two AD channels are the same. This applies for
$T_{1}$-limited qubits.

####  D.1 Diamond norm distance

The diamond norm distance [140, 141] between two quantum channels
$\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ is defined as

$||\mathcal{N}_{1}-\mathcal{N}_{2}||_{\diamond}=\sup_{\rho}||\mathcal{N}_{1}\otimes\mathrm{I}(\rho)-\mathcal{N}_{2}\otimes\mathrm{I}(\rho)||_{1},$
(D.1)

where $||\cdot||_{1}$ is the trace norm. The diamond norm considers states
$\rho$ that might be entangled with some ancillary system that is not altered
by the action of the channels. Consider two quantum channels
$\mathcal{N}_{1},\mathcal{N}_{2}$ and a single channel use. A quantum channel
discrimination protocol aims to maximize the probability of correctly
identifying which channel has acted on the quantum state if one of them is
chosen uniformly at random. The diamond norm distance is related to the
minimum probability of error, $p_{\mathrm{e}}$, that a discrimination protocol
for $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ can achieve as

$p_{\mathrm{e}}=\frac{1}{2}-\frac{||\mathcal{N}_{1}-\mathcal{N}_{2}||_{\diamond}}{4}.$
(D.2)

Therefore, the diamond norm distance serves as a measure of how differently
two quantum channels affect input quantum states, and so it is a good metric
to assess the dissimilarity between channels.

For Pauli channels $\mathcal{N}_{\mathrm{P}}$ defined as

$\mathcal{N}_{\mathrm{P}}(\rho)=p_{\mathrm{I}}\rho+p_{\mathrm{x}}\mathrm{X}\rho\mathrm{X}+p_{\mathrm{y}}\mathrm{Y}\rho\mathrm{Y}+p_{\mathrm{z}}\mathrm{Z}\rho\mathrm{Z},$
(D.3)

with $p_{\mathrm{I}}=1-p_{\mathrm{x}}-p_{\mathrm{y}}-p_{\mathrm{z}}$, the
diamond norm distance has the closed-form expression [141]

$||\mathcal{N}_{\mathrm{P}}^{1}-\mathcal{N}_{\mathrm{P}}^{2}||_{\diamond}=\sum_{k}|p_{k}^{1}-p_{k}^{2}|,$
(D.4)

where $k\in\\{\mathrm{I,X,Y,Z}\\}$.

Amplitude damping channels $\mathcal{N}_{\mathrm{AD}}$ also have a closed-form
expression [205] for the diamond norm distance which is given by

$\begin{split}||\mathcal{N}_{\mathrm{AD}}^{1}-&\mathcal{N}_{\mathrm{AD}}^{2}||_{\diamond}=\\\
&\begin{cases}2|\gamma_{1}-\gamma_{2}|&\text{if
}\sqrt{1-\gamma_{1}}+\sqrt{1-\gamma_{2}}>1\\\ \\\
\frac{2|\sqrt{1-\gamma_{1}}-\sqrt{1-\gamma_{2}}|}{2-(\sqrt{1-\gamma_{1}}+\sqrt{1-\gamma_{2}})}&\text{otherwise}\end{cases}\end{split},$
(D.5)

where $\gamma_{1}$ and $\gamma_{2}$ are the damping probailities that define
the dynamics of each of the AD channels.

The action of phase damping or dephasing channels $\mathcal{N}_{\mathrm{PD}}$
can be easily rewritten as a function of the Pauli matrices as

$\mathcal{N}_{\mathrm{PD}}(\rho)=\frac{1+\sqrt{1-\lambda}}{2}\mathrm{I}\rho\mathrm{I}+\frac{1-\sqrt{1-\lambda}}{2}\mathrm{Z}\rho\mathrm{Z},$
(D.6)

where $\lambda$ is interpreted as the scattering probability of a photon
without loss of energy. Consequently, the PD channel is equivalent to a pure
Pauli dephasing channel, that is, a Pauli channel where the only non-trivial
errors are the phase-flip errors $\mathrm{Z}$. As a result, using Eq. D.4, the
diamond norm distance between dephasing channels has the closed-form
expression

$||\mathcal{N}_{\mathrm{PD}}^{1}-\mathcal{N}_{\mathrm{PD}}^{2}||_{\diamond}=|\sqrt{1-\lambda_{1}}-\sqrt{1-\lambda_{2}}|.$
(D.7)

Finally, for quantum channels whose diamond norm distance has no closed-form
expression, a method to efficiently compute it via semidefinite programming
(SDP) was presented in [206]. The QETLAB tool [207] for MATLAB includes the
calculation of diamond norm distances using those SDP methods. We use such
tool in order to obtain the diamond norm distances between static and time-
varying combined amplitude and phase damping channels.

####  D.2
$||\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{2})||_{\diamond}=||\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{2})||_{\diamond}$

######  Proposition D.2.1.

The diamond norm distance between the Pauli twirl approximated channels (PTA)
obtained from two amplitude damping channels (AD) of parameters $\gamma_{1}$
and $\gamma_{2}$ is the same as the diamond norm distance between the Clifford
twirl approximated channels (CTA) obtained from the same AD channels:

$||\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{2})||_{\diamond}=||\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{2})||_{\diamond}.$
(D.8)

###### Proof.

The PTA and CTA channels are Pauli channels [108, 111]. The PTA channel for an
AD channel with damping probability $\gamma$ has the following probabilities
for each of the Pauli matrices:

$p_{\mathrm{I}}=\frac{2-\gamma+2\sqrt{1-\gamma}}{4},p_{\mathrm{x}}=p_{\mathrm{y}}=\frac{\gamma}{4}\text{
and }p_{\mathrm{z}}=\frac{2-\gamma-2\sqrt{1-\gamma}}{4}.$

The CTA channel is a depolarizing channel (as a special case of the Pauli
channels), with the following probabilities:

$r_{\mathrm{I}}=\frac{2-\gamma+2\sqrt{1-\gamma}}{4}\text{ and
}r/3=r_{\mathrm{x}}=r_{\mathrm{y}}=r_{\mathrm{z}}=\frac{2+\gamma-2\sqrt{1-\gamma}}{12}.$

The diamond norm distance between Pauli channels has the closed-form
expression [141]

$||\mathcal{N}_{\mathrm{P}}^{1}-\mathcal{N}_{\mathrm{P}}^{2}||_{\diamond}=\sum_{k}|p_{k}^{1}-p_{k}^{2}|,$
(D.9)

with $k\in\\{\mathrm{I,X,Y,Z}\\}$. Knowing that the PTAs and CTAs are Pauli
channels, we can write the diamond norm distances between them obtained from
approximating two AD channels:

  1. (a)

$||\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{2})||_{\diamond}=|p_{\mathrm{I}}^{1}-p_{\mathrm{I}}^{2}|+|p_{\mathrm{x}}^{1}-p_{\mathrm{x}}^{2}|+|p_{\mathrm{y}}^{1}-p_{\mathrm{y}}^{2}|+|p_{\mathrm{z}}^{1}-p_{\mathrm{z}}^{2}|$

  2. (b)

$||\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{2})||_{\diamond}=|r_{\mathrm{I}}^{1}-r_{\mathrm{I}}^{2}|+\sum_{l=k}^{3}|r_{k}^{1}-r_{k}^{2}|=|r_{\mathrm{I}}^{1}-r_{\mathrm{I}}^{2}|+3|r^{1}/3-r^{2}/3|$

Note that $p_{\mathrm{I}}=r_{\mathrm{I}},\forall\gamma$, so the contribution
of the probabilities associated to non-identity components to the diamond norm
distances must be compared. We start by expanding (b):

$\begin{split}3|r^{1}/3-r^{2}/3|&=3\left|\frac{\gamma_{1}-\gamma_{2}+2(\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}})}{12}\right|\\\
&=\frac{1}{4}|\gamma_{1}-\gamma_{2}+2(\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}})|.\end{split}$
(D.10)

The results shown below for the terms $\gamma_{1}-\gamma_{2}$ and
$\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}$ that appear in equation Eq. D.10
will be useful in what follows:

  * •

$\gamma_{1}-\gamma_{2}>0\rightarrow\gamma_{1}>\gamma_{2}\rightarrow-\gamma_{1}<-\gamma_{2}\rightarrow
1-\gamma_{1}<1-\gamma_{2}\rightarrow\sqrt{1-\gamma_{1}}<\sqrt{1-\gamma_{2}}\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}>0$.

  * •

$\gamma_{1}-\gamma_{2}<0\rightarrow\gamma_{1}<\gamma_{2}\rightarrow-\gamma_{1}>-\gamma_{2}\rightarrow
1-\gamma_{1}>1-\gamma_{2}\rightarrow\sqrt{1-\gamma_{1}}>\sqrt{1-\gamma_{2}}\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}<0$.

  * •

$\gamma_{1}-\gamma_{2}=0\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}=0$.

From these observations, it is easy to see that $\gamma_{1}-\gamma_{2}$ and
$\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}$ share the same sign
$\forall\gamma_{1},\gamma_{2}$. Thus, we can split the sum inside the absolute
value of Eq. D.10 into a sum of absolute values as

$Eq.~{}\ref{eq:CTAStep1}=\frac{1}{4}|\gamma_{1}-\gamma_{2}|+\frac{1}{2}|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|.$
(D.11)

We continue by developing the non-identity part for the PTA as given in (a):

$\begin{split}|p_{\mathrm{x}}^{1}-p_{\mathrm{x}}^{2}|+|p_{\mathrm{y}}^{1}-p_{\mathrm{y}}^{2}|+|p_{\mathrm{z}}^{1}-p_{\mathrm{z}}^{2}|=&\frac{|\gamma_{1}-\gamma_{2}|}{4}+\frac{|\gamma_{1}-\gamma_{2}|}{4}\\\
&+\frac{|\gamma_{2}-\gamma_{1}+2(\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}})|}{4}.\end{split}$
(D.12)

We follow the same reasoning as for the CTA by making use of the following
expressions that $\gamma_{2}-\gamma_{1}$ and
$\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}$ fulfil. They are later added within
the absolute value of their corresponding terms in the expression:

  * •

$\gamma_{2}-\gamma_{1}>0\rightarrow\gamma_{2}>\gamma_{1}\rightarrow-\gamma_{2}<-\gamma_{1}\rightarrow
1-\gamma_{2}<1-\gamma_{1}\rightarrow\sqrt{1-\gamma_{2}}<\sqrt{1-\gamma_{1}}\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}<0$.

  * •

$\gamma_{2}-\gamma_{1}<0\rightarrow\gamma_{2}<\gamma_{1}\rightarrow-\gamma_{2}>-\gamma_{1}\rightarrow
1-\gamma_{2}>1-\gamma_{1}\rightarrow\sqrt{1-\gamma_{2}}>\sqrt{1-\gamma_{1}}\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}>0$.

  * •

$\gamma_{2}-\gamma_{1}=0\rightarrow\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}=0$.

From these outcomes, we can tell that $\gamma_{2}-\gamma_{1}$ and
$\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}$ have a different sign
$\forall\gamma_{1},\gamma_{2}$. In consequence we can write the sum inside the
absolute value of the third term of Eq. D.12 as

$Eq.~{}\ref{eq:PTAStep1}=\left|2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|-|\gamma_{2}-\gamma_{1}|\right|,$
(D.13)

that is to say, the absolute value of computing the substraction of each of
the original absolute values. To further expand the expression in Eq. D.13, we
use

$\begin{split}&|(\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}})(\sqrt{1-\gamma_{2}}+\sqrt{1-\gamma_{1}})|=|1-\gamma_{2}-1+\gamma_{1}|\\\
&=|\gamma_{2}-\gamma_{1}|\rightarrow
2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}||\sqrt{1-\gamma_{2}}+\sqrt{1-\gamma_{1}}|\\\
&=2|\gamma_{2}-\gamma_{1}|\rightarrow
2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|=\frac{2|\gamma_{2}-\gamma_{1}|}{|\sqrt{1-\gamma_{2}}+\sqrt{1-\gamma_{1}}|}\\\
&\rightarrow
2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|\geq|\gamma_{2}-\gamma_{1}|,\end{split}$
(D.14)

$\forall\gamma_{1},\gamma_{2}$ where the last inequality comes from the fact
that

  * •

$|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|\in(0,2]\rightarrow
0<\frac{1}{2}|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|\leq
1\rightarrow\frac{2}{|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|}\geq 1$

  * •

$|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|=0\rightarrow
2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|=|\gamma_{2}-\gamma_{1}|=0$.

Having shown this result, we can work on Eq. D.13 even further by removing the
absolute value, since we have shown that the substraction will always be $\geq
0$:

$Eq.~{}\ref{eq:PTAStep2}=2|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|-|\gamma_{2}-\gamma_{1}|.$
(D.15)

We finish by introducing Eq. D.15 into expression Eq. D.12:

$\begin{split}Eq.~{}\ref{eq:PTAStep1}=&\\\
&\frac{|\gamma_{1}-\gamma_{2}|}{4}+\frac{|\gamma_{1}-\gamma_{2}|}{4}+\frac{1}{2}|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|-\frac{|\gamma_{2}-\gamma_{1}|}{4}\\\
&=\frac{1}{4}|\gamma_{1}-\gamma_{2}|+\frac{1}{2}|\sqrt{1-\gamma_{2}}-\sqrt{1-\gamma_{1}}|\end{split}$
(D.16)

If we now compare Eq. D.11 and Eq. D.16, it is clear that both of them are the
same, which implies that

$||\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADPTA}}(\gamma_{2})||_{\diamond}=||\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{1})-\mathcal{N}_{\mathrm{ADCTA}}(\gamma_{2})||_{\diamond},$
(D.17)

and so concludes the proof for the proposition. ∎

###  Appendix E Adjusted boxplots for skewed distributions

Boxplots are a heavily employed tool in statistics to see how specific
statistical data is distributed. A boxplot is comprised of a box whose top
edge refers to the third quartile ($\mathrm{Q3}$) and whose bottom edge refers
to the first quartile ($\mathrm{Q1}$). A line inside this box represents the
median and two whiskers at both ends indicate
$\mathrm{Q3}+1.5(\mathrm{Q3}-\mathrm{Q1})$ for the top whisker and
$\mathrm{Q1}-1.5(\mathrm{Q3}-\mathrm{Q1})$ for the bottom one. Data that is
found outside whisker range are considered to be outliers in the data set and
are individually plotted. Boxplots are based on the normal distribution, and
so they are not very representative for heavily skewed distributions. Note
that the length is the same for the top and the bottom whiskers. Consequently,
the real skewness of heavily skewed data will not be represented in such
boxplots, with several data being represented as outliers as a result of the
long tails that such types of distributions have. In consequence, conventional
boxplots do not reflect a valid range of outliers, since the long tail side
requires that more data be accounted for as typical values.

Adjusted boxplots have been presented in the literature as a methodology to
take the imbalanced weight of skewed data into account [208]. The core idea of
this adjusted boxplot is to use a robust measure of the skewness of the
dataset to define the length of each of the whiskers. [208] employs a measure
known as the medcouple ($\mathrm{MC}$) [209] for this purpose. The whisker
range $\Delta$ is then defined as

$\begin{cases}[\mathrm{Q1}-1.5\mathrm{e}^{-4\mathrm{MC}}\mathrm{IQR},\mathrm{Q3}+1.5\mathrm{e}^{3\mathrm{MC}}\mathrm{IQR}]&\text{if
}\mathrm{MC}\geq 0\\\ \\\
[\mathrm{Q1}-1.5\mathrm{e}^{-3\mathrm{MC}}\mathrm{IQR},\mathrm{Q3}+1.5\mathrm{e}^{4\mathrm{MC}}\mathrm{IQR}]&\text{if
}\mathrm{MC}<0\end{cases},$ (E.1)

where $\mathrm{IQR}=\mathrm{Q3}-\mathrm{Q1}$ refers to the interquartile
range. The whiskers of the boxplot will then take the skewness of the
distribution into account, depending on whether the distribution is positively
skewed ($\mathrm{MC}>0$), negatively skewed ($\mathrm{MC}<0$) or symmetric
($\mathrm{MC}=0$). This way several data considered to be outliers in
conventinal boxplots will not be considered as such by adjusted boxplots
(skewness should be taken into account for outlier consideration).

The probability distribution of
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
discussed in Chapter 3 appears to have a heavy positive skew, and thus the
adjusted version of the boxplot presented in [208] is necessary for a correct
representation of such results.

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