we compare the behaviour of this metric for the AD and APD channels to its
behaviour for the approximated channels. Figure 3.5 shows how the value of the
mean diamond norm distance is smaller for the approximated channels obtained
via twirling than for the original channels. This stems from the fact that
performing the twirl operation eliminates the off-diagonal elements of the
original channels (due to the symmetrization of the channel), and thus their
contribution to the diamond norm distance is also eliminated. In any case,
Figure 3.5 shows that the approximated channels also experience time-variation
in the decoherence parameters. Furthermore, observe that even though the value
of the diamond norm distance is lower for the twirled channels, the shape of
the curve that embodies the time dependance of this metric is similar to the
curves obtained for the AD and APD channels.

#####  3.4.2 Variability of the diamond norm distance

The mean diamond norm distance
$\mathrm{E}\\{||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}\\}$
has enabled us to analyze the average difference between static and TV quantum
channels. However, while studying the average distance corroborates that
static channels and TVQCs differ, it fails to portray the variability of
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$.
Thus, looking solely at the average of this metric may be shortsighted in the
sense that additional trends and results regarding the nature of TVQCs may be
overlooked. Consequently, a boxplot analysis of the realizations of the
diamond norm distance is performed with the goal of studying the variability
of
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
(see Appendix E for description of adjusted boxplots for skewed
distributions).

######  $T_{1}$-limited qubits

Figure 3.6 presents the boxplots for the AD and the ADPTA channels,
considering different coefficients of variation of the relaxation time random
variable $T_{1}(\omega)$. When $c_{\mathrm{v}}(T_{1})=1\%$ (Figure 3.6a and
3.6d), the size of the boxplots are negligible for all the range of algorithm
times. This means that when the fluctuations of $T_{1}$ around its mean are
small, the dispersion of
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
around its mean is nearly zero and, thus, a static channel approach will be a
valid assumption. Increasing the coefficient of variation to
$c_{\mathrm{v}}(T_{1})=10\%$ (Figure 3.6b and 3.6e) the box size and the
whisker length begins to widen as the algorithm time increases, meaning that a
significantly large number of realizations of
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
will be higher than its mean given by Figure 3.5. Thus, a static noise
assumption may start to fail when algorithm times are sufficiently large.
Finally, for $c_{\mathrm{v}}(T_{1})=25\%$ (Figure 3.6c and 3.6f) one obtains
the large box sizes and whiskers of the boxplots, showing a large variation of
the diamond norm distance around its mean. Therefore, adopting static noise
modelling strategies in such scenarios represents an erroneous approach.

Figure 3.6: Boxplot representation of the simulations of the diamond norm
distance
$\mathbf{||\mathcal{N}(\mu_{T_{1}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}}$
for the $\mathbf{T_{1}}$-limited scenario. a-c AD channels with
$c_{\mathrm{v}}(T_{1})=\\{1,10,25\\}\%$. d-f ADPTA/ADCTA channels with
$c_{\mathrm{v}}(T_{1})=\\{1,10,25\\}\%$. Outlier data is represented with red
crosses.

It is also important to discuss the presence of outliers on these boxplots.
These events appear to be significant in number when the coefficient of
variation $c_{\mathrm{v}}(T_{1})$ is large. Consequently, even for algorithms
that require short processing times, where the mean of the diamond norm
distance is very small, the use of a static channel when
$c_{\mathrm{v}}(T_{1})$ is large might be misleading. For example, for a
normalized algorithm time of $0.1$, the mean value,
$\mathrm{E}\\{||\mathcal{N}(\mu_{T_{1}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}\\}$,
is $0.04$ and $0.02$ for the AD and ADPTA channels, respectively (refer to
Figures 3.5a and 3.5b) considering a $c_{\mathrm{v}}(T_{1})=25\%$. Therefore,
by looking at these values it will be reasonable to assume that a static
channel approach when simulating QECCs will be valid. However, realizations of
$||\mathcal{N}(\mu_{T_{1}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}\approx
1.2$ may occur for both the AD and ADPTA channels (refer to Figure 3.6c and
3.6f), and the frequency of outlier events seems to be important. Thus, the
above assumption may fail.

From the above results, it is clear that when the coefficients of variation
are large, the diamond norm distance
$||\mathcal{N}(\mu_{T_{1}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$ exhibits
a significant amount of spread. This entails that events where the TV channel
and the static channel substantially differ will be pretty common, even if the
mean value of the diamond norm is not very high. Thus, adopting static noise
modelling strategies in such scenarios will be an erroneous approach.

######  $T_{1}\approx T_{2}$ and $T_{2}$-dominated qubits

We also provide boxplots for the $T_{1}\approx T_{2}$ and $T_{2}$-dominated
superconducting qubit scenarios.

Figure 3.7: Boxplot representation of the simulations of the diamond norm
distance
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
for the $T_{1}\approx T_{2}$ scenario. a-i APD channels with
$c_{\mathrm{v}}(T_{1})=\\{1,10,25\\}\%$ and
$c_{\mathrm{v}}(T_{\phi})=\\{1,10,25\\}\%$. Outlier data is represented with
red crosses.

Figures 3.7 and 3.8 show the resulting diamond norm distance boxplots for the
$T_{1}\approx T_{2}$ and $T_{2}$-dominated scenarios studied in section 3.3
considering the combined amplitude and phase damping channel (we omit the
results for their twirled channels since they will be similar). The trend
observed for $T_{1}$-limited superconducting qubits can also be appreciated in
these figures. For both figures, increasing the coefficient of variation of
the random variables $T_{1}$ and $T_{\phi}$ results in a higher spread of the
diamond norm distance between the static and TV channels. Consequently, for
low coefficients of variation of the decoherence parameters (see Figures 3.7a,
3.7b, 3.8a and 3.8b), considering a static channel a is valid assumption. For
milder coefficients of variation (see Figures 3.7c, 3.7d, 3.7e,3.8c, 3.8d and
3.8e) the static assumption starts to diverge from the TV channel model, and
implications for the QECCs will start to occur. Finally, for the high values
of $c_{\mathrm{v}}(T_{1})$ and $c_{\mathrm{v}}(T_{\phi})$ (see Figures 3.7f,
3.7g, 3.7h, 3.7i, 3.8f, 3.8g, 3.8h and 3.8i), both channel models differ
substantially and the non-dynamic assumption is flawed. Note that as discussed
before, the variation of the relaxation time has more impact in the channel
fluctuations than the variation of the pure dephasing (recall that relaxation
is part of the dephasing time $T_{2}$). Therefore, the scenarios with low
$c_{\mathrm{v}}(T_{1})$, but moderately large $c_{\mathrm{v}}(T_{\phi})$ can
still be considered static (as it can be seen in Figures 3.7b and 3.8b).

Figure 3.8: Boxplot representation of the simulations of the diamond norm
distance
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
for the $T_{2}$-dominated scenario. a-i APD channels with
$c_{\mathrm{v}}(T_{1})=\\{1,10,25\\}\%$ and
$c_{\mathrm{v}}(T_{\phi})=\\{1,10,25\\}\%$. Outlier data is represented with
red crosses.

The above boxplot analysis for $T_{1}\approx T_{2}$ and $T_{2}$-dominated
qubits show similar results as the one obtained for the $T_{1}$-limited
scenario. In this way, superconducting qubits that present pure dephasing
channels that require attention are also affected by the variation of the
decoherence parameters. Thus, the TVQC model is relevant for all types of
superconducting qubits that present fluctuations of $T_{1}$ and $T_{\phi}$
with sufficiently large coefficients of variations.

####  3.5 Discussion

In this chapter we have presented a time-varying (TV) quantum channel model as
a way to include the time dependant fluctuations of the parameters that define
the decoherence effects experienced by superconducting qubits. To that end, we
have used state-of-the-art experimental results to model the time-variations
of these parameters and provided an extensive numerical analysis of the
proposed TVQC models. Therefore, the proposed model serves to coalesce recent
experimental results regarding the fluctuations of the relaxation time,
$T_{1}$, and the dephasing time, $T_{2}$, into a quantum channel model that
provides a more realistic portrayal of decoherence effects to the quantum
error correction community. The proposed channel models are sufficiently
general to include any type of slow variation in the random processes
$T_{1}(\omega,t)$ and $T_{2}(\omega,t)$. Therefore, even though the present
analysis is motivated on superconducting qubits, the proposed model is also
applicable to other quantum-coherent two-level systems that present similar
time-dependant parameter oscillations, such as trapped ions or quantum dots.
In our analysis, the criterium chosen to measure the deviation between time-
variant and time invariant quantum channels has been the diamond norm distance
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$.
Based on the statistical characterization of the random processes
$T_{1}(\omega,t)$, $T_{\phi}(\omega,t)$ and $T_{2}(\omega,t)$, it is found
that their stochastic process coherence times, $T_{c}$, are much smaller than
the algorithm times. Therefore, these decoherence parameters can be modelled
as random variables taking independent values from round to round. This in
turn results in
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$
being independent random variables from round to round. Computing the diamond
norm distance average over all rounds in a quantum processing task reveals
that by not considering time fluctuations in quantum channel models, an
integral part of the decoherence effects are neglected.

In order to asses the dispersion of the realizations of the random variable
$||\mathcal{N}(\mu_{T_{1}},\mu_{T_{2}})-\mathcal{N}(\rho,\omega,t)||_{\diamond}$,
a boxplot analysis has been carried out. The primary conclusion drawn from
this boxplot study is that the diamond norm distance exhibits a high degree of
variability if the coefficients of variation of the decoherence times are
sufficiently large, say larger than $\approx 10\%$. This is due to the fact
that under this condition, the whisker length from the median and the number
of outliers in this region is considerably high. It should be mentioned that
the state-of-the-art superconducting processors in [51] have a
$c_{\mathrm{v}}(T_{1})\approx 20\%$ (refer to Tables 3.1). Therefore, the TV
quantum channel models proposed in this article are important to describe the
decoherence effects suffered by their qubits when studying QECCs requiring a
large number of error correction cycles (e.g., a quantum memory or a long
quantum algorithm such as the one in [149]). If a QECC is run just once or a
few number of times, the parameters will not fluctuate and the effects of the
proposed channel model will not be noticeable.

These results also speak towards 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 in such parameter. In other words, the results presented here point
out that 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.

Finally, note that in this chapter, we did not discuss how the proposed time-
varying channel models will affect the performance of quantum error correction
codes. This will be done in Chapter 5, where QTCs and Kitaev toric codes
operating over the TVQCs will be simulated.

###  CHAPTER 4 Quantum outage probability

The time-varying quantum channel model was presented in Chapter 3. In this
chapter we focus on channel capacity. Quantum channel capacity gives the
quantum rate limit at which a QECC can operate with asymptotically vanishing
error rate. However, in the literature, quantum capacity has been derived for
static quantum channels. Here, we study the asymptotical limits of error
correction for time-varying quantum channel models. Motivated by the
similarity between the TVQCs and classical slow or block fading scenarios,
i.e., when the channel remains constant over the duration of the coded block
[150]; we define the quantum outage probability of a TVQC as the
asymptotically achievable error rate for a QECC with quantum rate
$R_{\mathrm{Q}}$ that operates over the aforementioned noise model.
Additionally, we also introduce the concept of the quantum hashing outage
probability to provide an upper bound on the asymptotically achievable error
rate for TVQC channels whose quantum capacity (of their static counterparts)
is unknown, but for which a lower bound known as the hashing limit exists.
Based on the experimentally determined statistical distribution of $T_{1}$
discussed in Chapter 3, we provide closed-form expressions for the following
TVQCs: time-varying amplitude damping channel (TVAD), time-varying amplitude
damping Pauli twirl approximated channel (TVADPTA) and time-varying amplitude
damping Clifford twirl approximated channel (TVADCTA). We analyze the quantum
outage probability and quantum hashing outage probabilities of the
aforementioned TVQCs for different scenarios.

####  4.1 Classical outage probability for the block fading additive Gaussian
noise channel

The TVQC model proposed in this dissertation clearly resembles the paradigm of
classical block fading [150]. This occurs because the algorithm processing
times [51] are much smaller than the coherence times of the stochastic
processes that define the dynamic behaviour of the relaxation and dephasing
times. Consequently, the realization of $T_{i}(\omega)$, $i=1,2$ remains
constant for all the qubits in the block, where we have also assumed that all
qubits in the codeword are affected equally by the noise. With this we mean
that the $T_{1}(\omega)$ and $T_{2}(\omega)$ of each of the qubits of the QECC
block are completely correlated implying that the realizations of each of the
decoherence times are the same for each of the qubits111In general this is not
the only possibility, but here we study such scenario. We study this case
following the usual reasoning in the QECC literature that the error
probability for each of the qubits of the system is the same [31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 60, 111, 128, 137].. Mathematically, this can be
written as

$\rho_{T_{i}^{j},T_{i}^{k}}=\frac{\mathrm{E}\\{(T_{i}^{j}(\omega)-\mu_{T_{i}^{j}})(T_{i}^{k}(\omega)-\mu_{T_{i}^{k}})\\}}{\sigma_{T_{i}^{j}}\sigma_{T_{i}^{k}}}=1,\forall
i\in\\{1,2\\},\forall j,k\in\\{1,\cdots,n\\}$ (4.1)

where $\rho_{X,Y}$ is named the Pearson correlation coefficient and $n$ is the
number of qubits of the error correction block.

In this way, the realization of the $n$-qubit TVQC at block $l$,
$\mathcal{N}_{l}^{(n)}(\rho)$, is equal to

$\mathcal{N}_{l}^{(n)}(\rho)=\mathcal{N}^{\otimes
n}(\rho,t_{1}^{l},t_{2}^{l}),$ (4.2)

where $t_{1}^{l}$ and $t_{2}^{l}$ refer to the realizations of random
variables $T_{1}(\omega)$ and $T_{2}(\omega)$ at block $l$, realizations that
are assumed to be equal for each of the $n$ qubits of the block.

This is reminiscent of classical block fading scenarios in which a realization
of the fading random process, $\alpha(\omega,t)$ (channel gain), is considered
to be constant for the entire codeword length [150]. We use this similarity to
develop the information theoretical concepts for TVQCs. To that end, we begin
by explaining in more detail the slow block fading channel model use in
classical communications and define the concept of outage probability for
these channels.

The general input-output relation of a continuous-time, additive gaussian
noise fading channel is given by

$Y(\omega,t)=\alpha(\omega,t)X_{cd}(t)+N(\omega,t)$ (4.3)

where $X_{cd}(t)$ denotes the input codeword, and $N(\omega,t)$ and
$\alpha(\omega,t)$ are the wide-sense stationary (WSS) random processes
modeling the gaussian noise channel and the fading process, respectively.
Whenever the duration of the codewords, $X_{cd}(t)$, is much smaller than the
coherence time, $T_{c}$, of the fading gain $\alpha(\omega,t)$, (i.e., the
time where $\alpha$ can be considered constant) the channel is said to be
operating under slow fading conditions. Therefore, the value of the channel
gain during the transmission of a codeword can be considered to be
approximately constant and given by a realization of the random variable
$\alpha(\omega)$. Furthermore, when considering the transmission of a sequence
of $K$ codewords which are sufficiently separated in time, say $T$ seconds
then the output sequence will be

$Y(\omega,t)=\sum_{j=0}^{K-1}\alpha_{j}(\omega)X^{(j)}_{cd}(t-jT)+N(\omega,t),$

where the sequence of random variables $\\{\alpha_{j}(\omega)\\}_{j=0}^{K-1}$
can be considered independent and identically distributed (iid).

The operational concept of channel capacity for these channels is meaningless
and has to be replaced by what is known the outage probability [150]. Let us
now define this parameter. Note that conditional on a realization of the
channel gain $\alpha(\omega)$, channel Eq. 4.3 reduces to an additive white
Gaussian noise (AWGN) channel with received signal-to-noise ratio (SNR)
$|\alpha(\omega)|^{2}\mathrm{SNR}$. By the Shannon channel coding theorem, the
maximum rate of reliable communication supported by this channel is
$\log_{2}(1+|\alpha(\omega)|^{2}\mathrm{SNR})$ in bits per channel use. This
quantity is a function of the gain $\alpha(\omega)$ and is therefore random.
Channel capacity is defined as the maximum coding rate that is achievable (an
error correction scheme of such rate with vanishing error rate exists) for a
noisy channel. This, however, cannot be done for the slow fading scenario
considered here since there will be a non-zero probability that the channel
realization has a capacity (realization) lower than the rate considered.
Hence, the quantum capacity for such channel in the strict sense is zero. Now
suppose that the transmitter encodes data at a rate $R$ bits per channel use.
If the channel realization $\alpha$ is such that
$C(\omega)=\log(1+|\alpha|^{2}\mathrm{SNR})<R$, then whatever code that was
used by the transmitter, the decoding error probability cannot be made
arbitrarily small. The system is said to be in outage, and the outage
probability is given by [150]

$\begin{split}p_{\mathrm{out}}(R,\mathrm{SNR})&=\mathrm{P}(\\{\omega\in\Omega:C(\omega)<R\\})\\\
&=P(\\{\omega\in\Omega:\log_{2}(1+|\alpha(\omega)|^{2}\mathrm{SNR})<R\\}).\end{split}$
(4.4)

The outage probability will depend on the probability distribution of the
channel gain random variable, $\alpha(\omega)$. For the widely used Rayleigh
fading model, for which the channel gain follows a circularly symmetric
complex normal distribution, $\mathcal{CN}(0,1)$, the outage probability can
be shown to be [150]

$p_{\mathrm{out}}(R,\mathrm{SNR})=1-\mathrm{e}^{\frac{-(2^{R}-1)}{\mathrm{SNR}}}.$
(4.5)

####  4.2 Quantum outage probability for TVQCs

Let us now look for the outage probability of time-varying quantum channels.
As stated before, during the duration of a QECC codeword, the realization of
the random processes $T_{1}$ and $T_{2}$, defining the dynamics of the quantum
channel, can be considered constant for the duration of the algorithm
processing time. Therefore, we can assume that for each of the realizations of
the qubit relaxation and dephasing times, $T_{1}$ and $T_{2}$, associated to a
codeword, a static quantum channel will result. This channel will be valid for
the current codeword under consideration, and it will render a specific value
for its quantum channel capacity, $C_{\mathrm{Q}}$ qubits per channel use.
Here is where one can observe the similarities with the case slow fading in
classical communications.

Therefore, if the realization of the decoherence parameters leads to a channel
capacity lower that the quantum coding rate, $R_{\mathrm{Q}}$, then the
quantum bit error rate (QBER) will not vanish asymptotically with the
blocklength, independently of the selected QECC. For such realizations we say
the channel is in outage, and the probability of such outage event will be

$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}})=\mathrm{P}(\\{\omega\in\Omega:C_{\mathrm{Q}}(\omega)<R_{\mathrm{Q}}\\}),$
(4.6)

which we name as quantum outage probability.

In other words, with probability
$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}})$, the capacity of the channel
$C_{\mathrm{Q}}(\omega)$ will be lower than the rate of the code, and thus,
the error rate will not vanish asymptotically. Conversely, with probability
$1-p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}})$, reliable quantum correction
will be possible. Thus, the quantum outage probability will be the
asymptotically achievable error rate for quantum error correction when the
rate is $R_{\mathrm{Q}}$.

Note that the quantum outage probability is the metric that has to be taken
into account in order to benchmark the performance of QECCs when they operate
over TVQCs that fulfill the assumptions stated in this chapter.

####  4.3 Computation of the quantum outage probability for the family of
time-varying amplitude damping channels

In Chapter 2, we provided analytic expressions of the quantum capacity of the
amplitude damping (AD) channel. We also mentioned that for its Pauli/Clifford
twirl approximations a closed-form expression for its capacity remains
unknown, but it can be upper bounded by the hashing bound. Therefore, for the
family of TVAD channels, we derive a closed-form expression for the quantum
outage probability (refer to Theorem 4.3.1). On the other hand, for the
Pauli/Clifford twirl approximated channels of the TVAD we propose (refer to
Corollary 4.3.1.1) the quantum hashing outage probability,
$p_{\mathrm{out}}^{\mathrm{H}}$, as an upper bound of their quantum outage
probability. Note that since we are studying AD channels, we deal with
$T_{1}$-limited qubits, that is, qubits that do not present pure dephasing.

#####  4.3.1 Outage probability for the time-varying amplitude damping channel

Before presenting the main result of this section, Theorem 4.3.1, we begin by
stating some facts. From expression Eq. 2.66, the quantum capacity
$C_{\mathrm{Q}}$ of the AD channel is a monotonically decreasing function of
the damping parameter, $\gamma$. Therefore, there will be a unique
$\gamma^{*}(R_{\mathrm{Q}})$ that makes the value of the channel capacity
$C_{\mathrm{Q}}$ equal to $R_{\mathrm{Q}}$, i.e.,
$C_{\mathrm{Q}}(\gamma^{*}(R_{\mathrm{Q}}))=R_{\mathrm{Q}}$. That is,

$C_{\mathrm{Q}}(\gamma^{*}(R_{\mathrm{Q}}))=R_{\mathrm{Q}}\Leftrightarrow\gamma^{*}(R_{\mathrm{Q}})=C^{-1}_{\mathrm{Q}}(R_{\mathrm{Q}}).$
(4.7)

We will refer to $\gamma^{*}(R_{\mathrm{Q}})$ as the noise limit. Note that
codes of rate $R_{\mathrm{Q}}$ operating over channels with a damping
parameter $\gamma>\gamma^{*}(R_{\mathrm{Q}})$ will not be reliable since
$R_{\mathrm{Q}}>C_{\mathrm{Q}}(\gamma)$.

In addition, from Eq. 3.7 we define the critical relaxation time
$T_{1}^{*}(R_{\mathrm{Q}},t_{\mathrm{algo}})$ as

$T_{1}^{*}(R_{\mathrm{Q}},t_{\mathrm{algo}})=\frac{-t_{\mathrm{algo}}}{\ln{(1-\gamma^{*}(R_{\mathrm{Q}}))}},$
(4.8)

which is a function of the algorithm time, $t_{\mathrm{algo}}$. In order to
compare quantum channels with different mean relaxation times, $\mu_{T_{1}}$,
we would like rewrite the critical time as a function of the damping
parameter, $\gamma$, that results when assuming a static channel222This is
similar to the normalization done in Chapter 3 for the diamond norm analysis,
as the damping rate is a function of the algorithm time and the relaxation
time. with mean $\mu_{T_{1}}$. The reason for it is that if the calculations
were done as a function of the algorithm time, the comparison between qubits
with different mean relaxation times would not be fair, since for a given
$t_{algo}$, the higher the values of $\mu_{T_{1}}$ are, the lower the values
of $\gamma$ will be, so that we will be comparing channels with different
degrees of noise. It is obvious that longer mean relaxation times are more
favorable for computing applications, as they allow for longer algorithm
times. However, we are interested in calculating the quantum outage
probability versus the noise level of the channel, i.e, we want to know how
much noise a qubit is able to tolerate.

To that end, given a damping parameter $\gamma$ and a $\mu_{T_{1}}$, the
algorithm time will be set to

$t_{\mathrm{algo}}=-\mu_{T_{1}}\ln{(1-\gamma)}.$ (4.9)

That is, for quantum channels with different $\mu_{T_{1}}$’s, the above
expression gives the corresponding algorithm times that will produce the same
damping parameter $\gamma$ in all of them.

Substituting such value in Eq. 4.8, yields,

$T^{*}_{1}(R_{\mathrm{Q}},\gamma)=\frac{\mu_{T_{1}}\ln(1-{\gamma})}{\ln(1-{\gamma^{*}(R_{\mathrm{Q}})})}.$
(4.10)

In this way, the critical relaxation time in Eq. 4.8 is independent of the
algorithm time, and is only a function of the damping parameter $\gamma$
associated to the static AD channel.

We are now ready to introduce the theorem that derives the quantum outage
probability for TVAD channels for $T_{1}$-limited qubits [51].

######  Theorem 4.3.1 (TVAD quantum outage probability).

The quantum outage probability for the time-varying amplitude damping channels
associated to the damping parameter, $\gamma\in[0,1-{e}^{-1}]$, is equal to

$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}},\gamma)=1-\frac{\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\left[\frac{\ln(1-{\gamma})}{\ln(1-{\gamma^{*}(R_{\mathrm{Q}})})}-1\right]\right)}{1-\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\right)},$
(4.11)

where $\mathrm{Q}(\cdot)$ is the Q-function, $c_{\mathrm{v}}(T_{1})$ is the
coefficient of variation of $T_{1}$, $\gamma^{*}(R_{\mathrm{Q}})$ is the noise
limit, $\mu_{T_{1}}$ is the mean relaxation time and $\sigma_{T_{1}}$ is the
standard deviation of the relaxation time.

###### Proof.

In order to compute the outage probability
$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}},\gamma)$, we use the decreasing
monotonicity of $C_{\mathrm{Q}}$ and $T_{1}$ with respect to $\gamma$. This
implies that the events
$\\{\omega\in\Omega:C_{\mathrm{Q}}(\gamma(\omega))<R_{\mathrm{Q}}\\}$,
$\\{\omega\in\Omega:\gamma(\omega)<\gamma^{*}(R_{\mathrm{Q}})\\}$ and
$\\{\omega\in\Omega:T_{1}(\omega)<T_{1}^{*}(R_{\mathrm{Q}},\gamma)\\}$ are all
the same. Therefore,

$\begin{split}p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}},\gamma)&=\mathrm{P}(\\{\omega\in\Omega:C_{\mathrm{Q}}(\gamma(\omega))<R_{\mathrm{Q}}\\})\\\
&=\mathrm{P}(\\{\omega\in\Omega:T_{1}(\omega)<T_{1}^{*}(R_{\mathrm{Q}},\gamma)\\}).\end{split}$
(4.12)

Next, we compute Eq. 4.12 based on the fact that the random variable
$T_{1}(\omega)\sim\mathcal{GN}_{[0,\infty]}(\mu_{T_{1}},\sigma^{2}_{T_{1}})$
has a probability density function

$f_{T_{1}}(t_{1})\begin{cases}\frac{1}{\sigma_{T_{1}}\sqrt{2\pi}}\frac{\mathrm{e}^{-\frac{(t_{1}-\mu_{T_{1}})^{2}}{2\sigma_{T_{1}}^{2}}}}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}&\text{
if }t_{1}\geq 0\\\ 0&\text{ if }t_{1}<0\end{cases},$ (4.13)

where in the above expression, $\mathrm{Q}(\cdot)$ is the Q-function defined
as

$\mathrm{Q}(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\mathrm{e}^{-\frac{x^{2}}{2}}dx.$
(4.14)

The outage probability of the TVAD channel can be calculated as

$\begin{split}p_{\mathrm{out}}^{\mathrm{Q}}(&R_{\mathrm{Q}},\gamma)=\mathrm{P}(\\{\omega\in\Omega:T_{1}(\omega)<T_{1}^{*}(R_{\mathrm{Q}},\gamma)\\})=\int_{-\infty}^{T_{1}^{*}(R_{\mathrm{Q}},\gamma)}f_{T_{1}}(t_{1})dt_{1}\\\
&=\int_{0}^{T_{1}^{*}(R_{\mathrm{Q}},\gamma)}\frac{1}{\sigma_{T_{1}}\sqrt{2\pi}}\frac{\mathrm{e}^{-\frac{(t_{1}-\mu_{T_{1}})^{2}}{2\sigma_{T_{1}}^{2}}}}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}dt_{1}\\\
&=\frac{1}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}\left(\int_{-\infty}^{T_{1}^{*}(R_{\mathrm{Q}},\gamma)}\frac{1}{\sigma_{T_{1}}\sqrt{2\pi}}\mathrm{e}^{-\frac{(t_{1}-\mu_{T_{1}})^{2}}{2\sigma_{T_{1}}^{2}}}dt_{1}\right.\\\
&\left.-\int_{-\infty}^{0}\frac{1}{\sigma_{T_{1}}\sqrt{2\pi}}\mathrm{e}^{-\frac{(t_{1}-\mu_{T_{1}})^{2}}{2\sigma_{T_{1}}^{2}}}dt_{1}\right)\\\
&=\frac{1}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}\left(\int_{-\infty}^{\frac{T_{1}^{*}(R_{\mathrm{Q}},\gamma)-\mu_{T_{1}}}{\sigma_{T_{1}}}}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{\eta^{2}}{2}}d\eta\right.\\\
&\left.-\int_{-\infty}^{\frac{-\mu_{T_{1}}}{\sigma_{T_{1}}}}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{\eta^{2}}{2}}d\eta\right)\\\
&=\frac{1}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}\left(1-\int_{\frac{T_{1}^{*}(R_{\mathrm{Q}},\gamma)-\mu_{T_{1}}}{\sigma_{T_{1}}}}^{\infty}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{\eta^{2}}{2}}d\eta\right.\\\
&\left.-\int_{\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}}^{\infty}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{\eta^{2}}{2}}d\eta\right)\\\
&=\frac{1-\mathrm{Q}\left(\frac{T_{1}^{*}(R_{\mathrm{Q}},\gamma)-\mu_{T_{1}}}{\sigma_{T_{1}}}\right)-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}=1-\frac{\mathrm{Q}\left(\frac{T_{1}^{*}(R_{\mathrm{Q}},\gamma)-\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}\\\
&=1-\frac{\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\left[\frac{\ln(1-{\gamma})}{\ln(1-{\gamma^{*}(R_{\mathrm{Q}})})}-1\right]\right)}{1-\mathrm{Q}\left(\frac{\mu_{T_{1}}}{\sigma_{T_{1}}}\right)}=1-\frac{\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\left[\frac{\ln(1-{\gamma})}{\ln(1-{\gamma^{*}(R_{\mathrm{Q}})})}-1\right]\right)}{1-\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\right)},\end{split}$
(4.15)

as we wanted to prove.

Note that the quantum outage probability as a function of the damping
parameter $\gamma$ does not depend on the absolute value of the mean
relaxation time, but on the coefficient of variation of $T_{1}$. This way, we
decouple the time-varying effects from the fact that longer mean relaxation
times admit longer quantum algorithm processing times. Consequently, we
present a result agnostic to the impact that longer coherence times have and
we can provide conclusions for all superconducting qubits.

Finally, we need to find a valid range of $\gamma$ that will result in
$t_{\mathrm{algo}}$ that are much smaller than the coherence time,
$T_{\mathrm{c}}$, of the random process $T_{1}(t,\omega)$, (i.e., verifying
the initial hypothesis of slow time variations). To that end, note that
algorithm times longer than $\mu_{T_{1}}$ makes no sense since for such a time
frame, the qubit will be with high probability in an equilibrium state, which
is useless as a resource. Therefore, we select
$t_{\mathrm{algo}}<\mu_{T_{1}}$. Since $\mu_{T_{1}}$ is of order of
microseconds for superconducting qubits (as explained in Chapter 3), and
$T_{\mathrm{c}}$ is of order of minutes [51] the hypothesis
$t_{\mathrm{algo}}\ll T_{\mathrm{c}}$ holds. From Eq. 4.9, we conclude that
valid for the range for $\gamma$ is $\gamma\in[0,1-e^{-1}]$. ∎

#####  4.3.2 Quantum hashing outage probability for the time-varying twirl
approximated channels

Since the LSD capacity of the Pauli channels that result from applying the
Pauli/Clifford twirled approximation to the AD channel is not known, the
quantum outage probability for this family of approximated channels cannot be
calculated. Nevertheless, by means of the hashing bound,
$C_{\mathrm{H}}(\mathbf{p})$, (refer to Eq. 2.69), we define the quantum
hashing outage probability for the time-varying Pauli channels defined by the
probability mass function
$\mathbf{p}(\omega)=(p_{\mathrm{I}},p_{\mathrm{x}},p_{\mathrm{y}},p_{\mathrm{z}})$
random vector as

$p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}})=\mathrm{P}(\\{\omega\in\Omega:C_{\mathrm{H}}(\mathbf{p}(\omega))<R_{\mathrm{Q}}\\}).$
(4.16)

Note that since the hashing limit is a lower bound of the LSD capacity
$C_{Q}$, we have that

$\\{\omega\in\Omega:C_{Q}(\mathbf{p}(\omega))<R_{\mathrm{Q}}\\}\subseteq\\{\omega\in\Omega:C_{\mathrm{H}}(\mathbf{p}(\omega))<R_{\mathrm{Q}}\\}$

Consequently, $p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}})$ gives an upper
bound of the actual quantum outage probability
$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}})$, i.e.,
$p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}})\geq
p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}})$.

It is important to realise that the Hashing bound in Eq. 2.69

$C_{\mathrm{H}}(\mathbf{p}(\gamma))=1+\sum_{k\in\\{\mathrm{I,x,y,z}\\}}p_{k}(\gamma)\log_{2}p_{k}(\gamma)=1-\mathrm{H}_{2}(\mathbf{p}(\gamma)),$
(4.17)

for the twirled approximated channels of the AD channel with the probability
distributions given in Eq. 2.40 and Eq. 2.52333Note that expressions Eq. 2.40
and Eq. 2.52 are for the twirl approximations of the complete amplitude and
phase damping channel. However, it is easy to obtain the expressions for the
ADPTA and ADCTA channels by considering $\lambda=0$, i.e., the absence of pure
dephasing channels., is a monotonic decreasing function of the damping
probability, $\gamma$. This is justified by the fact that, as $\gamma\in[0,1]$
increases, the values of $p_{\mathrm{x}},p_{\mathrm{y}},p_{\mathrm{z}}$ from
either Eq. 2.40 or Eq. 2.52 also increase. This results in the uncertainty of
the discrete random variables associated to each of these distributions, and
consequently their corresponding entropy values, becoming higher. Therefore,
as we did for the AD channel, we define the noise limit for these Pauli
channels as the unique value of the damping parameter,
$\gamma_{\mathrm{T}}^{*}(R_{\mathrm{Q}})$, such that:

$1-C_{\mathrm{H}}(\mathbf{p}(\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}}))=R_{\mathrm{Q}}\Leftrightarrow\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}})=C^{-1}_{\mathrm{H}}(1-R_{\mathrm{Q}}).$
(4.18)

From Eq. 3.7, the critical relaxation time (note that we have added the
subindex $\mathrm{T}$ to indicate we are twirling the AD channel) is

$T^{*}_{1,\mathrm{T}}(R_{\mathrm{Q}},t_{\mathrm{algo}})=\frac{-t_{\mathrm{algo}}}{\ln(1-\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}}))},$
(4.19)

where the probability mass function $\mathbf{p}$ in Eq. 4.18 should be taken
as $\mathbf{p}_{\mbox{\tiny{ADPTA}}}$ or $\mathbf{p}_{\mbox{\tiny{ADCTA}}}$
when considering the twirled ADPTA or ADCTA channels, respectively. Similarly
to the TVAD channel, we can write the critical relaxation time as a function
of the damping parameter

$T^{*}_{1,\mathrm{T}}(R_{\mathrm{Q}},\gamma)=\frac{\mu_{T_{1}}\ln(1-{\gamma})}{\ln(1-{\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}})})}.$
(4.20)

The following corollary yields the hashing outage probability for the twirled
TVADPTA and TVADCTA Pauli channels that consider $T_{1}$-limited qubits as the
ones of [51].

######  Corollary 4.3.1.1 (Quantum hashing outage probability).

The quantum hashing outage probability for the time-varying twirled
approximated channels associated to the damping parameter,
$\gamma\in[0,1-e^{-1}]$, is equal to

$p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}},\gamma)=1-\frac{\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\left[\frac{\ln(1-{\gamma})}{\ln(1-{\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}})})}-1\right]\right)}{1-\mathrm{Q}\left(\frac{1}{c_{\mathrm{v}}(T_{1})}\right)},$
(4.21)

where $\mathrm{Q}(\cdot)$ is the Q-function, $c_{\mathrm{v}}(T_{1})$ is the
coefficient of variation of $T_{1}$, $\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}})$
is the noise limit that depends on the considered twirled approximation,
$\mu_{T_{1}}$ is the mean relaxation time, and $\sigma_{T_{1}}$ is the
standard deviation of the relaxation time.

###### Proof.

In order to compute the hashing outage probability
$p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}},\gamma)$, we use the fact that
$C_{\mathrm{H}}$ and $T_{1}$ are monotonically decreasing functions of
$\gamma$. This implies that the events
$\\{\omega\in\Omega:C_{\mathrm{H}}(\omega)<R_{\mathrm{Q}}\\}$,
$\\{\omega\in\Omega:\gamma(\omega)<\gamma^{*}_{\mathrm{T}}(R_{\mathrm{Q}})\\}$
and
$\\{\omega\in\Omega:T_{1}(\omega)<T_{1,\mathrm{T}}^{*}(R_{\mathrm{Q}},\gamma)\\}$
are all same. Therefore,

$\begin{split}p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}},\gamma)&=P(\\{\omega\in\Omega:C_{\mathrm{H}}(\omega)<R_{\mathrm{Q}}\\})=\\\
&\mathrm{P}(\\{\omega\in\Omega:T_{1}(\omega)<T_{1,\mathrm{T}}^{*}(R_{\mathrm{Q}},\gamma)\\}.\end{split}$
(4.22)

Thus, the hashing outage corresponds to events where the realization of the
relaxation time is lower than the critical relaxation time.

The calculation of the quantum hashing outage probability now follows the same
steps as in the proof of Theorem 4.3.1, since equation Eq. 4.22 is the same as
Eq. 4.12. ∎

Even though the final expression for the quantum hashing outage probability
looks the same as the one of Theorem 4.3.1, what makes their values different
is that their corresponding noise limit values are computed in a different
way.

#####  4.3.3 Numerical simulations

We now discuss the behaviour of the quantum outage probability and the quantum
hashing outage probability. Following the reasoning of Chapter 3, we will
compare scenarios set by different coefficients of variation of the random
variable $T_{1}$. In particular, we consider the following values of the
coefficient of variation; $c_{\mathrm{v}}(T_{1})=\\{1,10,15,20,25\\}\%$.

######  4.3.3.1 Quantum outage probability of the TVAD channel

Figure 4.1 plots the quantum outage probability versus the damping parameter,
$10^{-3}\leq\gamma\leq 0.6$, for a quantum rate $R_{\mathrm{Q}}=1/9$, and for
all the coefficients of variation of the relaxation time random variable.

Figure 4.1: Quantum outage probability of the TVAD channel. The metric is
calculated for TVADs with $c_{\mathrm{v}}(T_{1})=\\{1,10,15,20,25\\}\%$ and
for a quantum rate of $R_{\mathrm{Q}}=1/9$.

Figure 4.1 further cements the conclusions derived in Chapter 3, that is, the
impact of the fluctuations of the decoherence parameters can be accurately
quantified by the coefficient of variation of $T_{1}$. Notice that when the
coefficient of variation is very low, i.e. $c_{\mathrm{v}}(T_{1})=1\%$, the
quantum outage probability of the TVAD channel almost coincides with the
quantum capacity (represented herein by the noise limit, $\gamma^{*}$).
Consequently, QECCs operating over TVQCs that present low coefficients of
variation will behave asymptotically in a similar manner to static channels.
However, by increasing the variability of the relaxation time around its mean
causes the outage probability to diverge from the static capacity. In this
case, the asymptotical bounds flatten and the achievable error rate of QECCs
operating over TVQCs does not vanish. Therefore, the higher
$c_{\mathrm{v}}(T_{1})$ is, the worse the achievable error rate will be.

The previous discussion indicates that the coefficient of variability of the
random variable $T_{1}(\omega)$ can be used to describe the effect that
decoherence parameter time fluctuations will produce on the aymptotical limits
of QECCs. These results show the importance of qubit construction and
cooldown: if optimized correctly, the fluctuations relative to the mean will
be mild, and the outage events will be significantly less likely. Obviously,
it is desirable for qubits to exhibit large $\mu_{T_{1}}$, so that algorithms
with longer lifespans can be handled appropriately. However, as important as
that is the minimization of the dispersion of $T_{1}$ around its
$\mu_{T_{1}}$, so the outage probability can be reduced.

Figure 4.2: Quantum outage probability of the TVAD channel for different
quantum rates. The considered quantum rates are
$R_{\mathrm{Q}}\in\\{1/49,1/9,1/4,1/3,1/2\\}$. We plot the quantum outage
probability for $c_{\mathrm{v}}(T_{1})=\\{10,25\\}\%$. The quantum capacities
(noise limits, $\gamma^{*}$) for the static quantum channels are also
represented.

Let us now discuss how the quantum rate affects the quantum outage
probability. Figure 4.2 shows the quantum outage probability versus $\gamma$
for $c_{\mathrm{v}}(T_{1})=\\{10,25\\}\%$, and for the set of quantum rates
$R_{\mathrm{Q}}\in\\{1/49,1/9,1/4,1/3,1/2\\}$. The figure also shows the
corresponding noise limits $\gamma^{*}(R_{Q})$ for the different rates. As
expected, the results portrayed in this figure show that increasing the rate
leads to an increase of
$p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}},\gamma)$. Note also that for the
same coefficient of variation, the shape of outage probability versus $\gamma$
curves remains fairly the same. The main conclusion is that, by increasing the
rate of an error correction code reduces the overall resource consumption,
however, this occurs at the expense of a degradation in the asymptotical error
correction performance. Furthermore, given a $R_{Q}$, the larger the
coefficient of variability $c_{\mathrm{v}}(T_{1})$ is, the larger the gap
between the outage probability curve and the noise limits $\gamma^{*}(R_{Q})$
curve will be. It is important to mention that this degradation does not occur
because there is higher sensitivity to time fluctuations at higher rates.
Further inspection of Figure 4.2 reveals that the noise limits for each rate
change as expected, and that the outage probabilities behave similarly
according to those noise limits. Thus, similarly to classical coding, the
quantum rate does indeed impact the quantum outage probability, but not due to
a higher sensitivity to time-variance. Furthermore, similar to what happens in
static channels, there is trade-off between resource consumption and how
demanding (in terms of noise) the quantum channel is.

######  4.3.3.2 Quantum hashing outage probability of the time-varying twirl
approximated channels

We proceed by comparing the outage of the TVAD channel and its twirled
approximated channels. Figure 4.3 plots the hashing outage probability results
of the TVADPTA and TVADCTA channels for a coding rate $R_{Q}=1/9$. Note that
the x-axis is still $\gamma$, despite the fact that the defining parameter for
the TVADPTA and TVADCTA channels is $\mathbf{p}$. However, the $\gamma$
associated to a given $\mathbf{p}$ can be easily obtained from Eq. 2.40 and
Eq. 2.52. The quantum outage probability of the TVAD channel from Figure 4.1
are also shown. Note that the hashing outage probabilities for the ADPTA and
ADCTA channels are slightly higher than the quantum outage probability of the
TVAD channel. This is due to the fact the noise limits for those channels are
smaller than the one for the TVAD.

Also note that even though the hashing outage probabilities for these
approximated channels are higher than the quantum outage probability for the
TVAD, one can not conclude that the actual quantum outage probability for
these twirled channels will be worse than the one for the AD channel (recall
that the hashing outage probability provides an upper bound on the actual
outage probability).

Figure 4.3: Quantum outage and hashing outage probabilities for TVAD, TVADPTA
and TVADCTA when $\mathbf{R_{\text{{Q}}}=1/9}$. The noise limits are:
$\gamma^{*}_{\mathrm{AD}}=0.432$, $\gamma^{*}_{\mathrm{ADPTA}}=0.3354$ and
$\gamma^{*}_{\mathrm{ADCTA}}=0.3065$.

####  4.4 Discussion

In this chapter, we have introduced the concept of quantum outage probability
as the asymptotically achievable error rate for quantum error correction when
slow time-varying quantum channels are considered. This probability is based
on the fact that the realizations of the $T_{1}$ and $T_{2}$ remain constant
and the hypothesis that they are the same for all the qubits in a quantum
codeword (they are fully correlated). We have also introduced the quantum
hashing outage probability as an upper bound on the quantum outage probability
useful when Pauli channels are considered, since the actual quantum capacity
of these channels is not known. We have provided closed-form expressions of
these probabilities for the TVAD, TVADPTA and the TVADCTA channels. The
information theoretical analysis presented in this chapter is essential to
benchmark the behaviour of quantum error correction codes in TV scenarios.
This comes from the fact that for slow time-varying quantum channels the
operational meaning of quantum capacity does not give information about the
channel and must be replaced by the quantum outage probability.

Results show that given a damping parameter, $\gamma$, the quantum outage
probability is a monotonic increasing function of the coefficient of
variation, $c_{\mathrm{v}}(T_{1})$. Therefore, it is important to
experimentally look for qubits that, not only have a large mean relaxation
time, $\mu_{T_{1}}$, but also exhibit a low standard deviation relaxation
time, $\sigma_{T_{1}}$. By doing so, the asymptotically achievable error
correction performance of QECCs under time-varying conditions will approach
the performance achieved by static channels, since the largest rate that would
make $p_{\mathrm{out}}^{\mathrm{Q}}(R_{\mathrm{Q}},\gamma)\approx 0$ (or
$p_{\mathrm{out}}^{\mathrm{H}}(R_{\mathrm{Q}},\gamma)\approx 0$) will be near
to the quantum capacity of the corresponding static channel (or to the hashing
bound).

In summary, we have shown that the time-varying nature of the decoherence
parameters may have a significant impact on the performance of future QECCs
that will be used to protect quantum information. To further corroborate this
fact, in the next chapter, numerical simulations on the performance of QECCs
over TVQCs will be done.

###  CHAPTER 5 Time-varying quantum channels and quantum error correction
codes

In Chapter 3, we showed that the diamond norm distance between TVQCs and
static channel models deviates significantly. As discussed later, the effect
that this norm divergence has on error correction is not straightforward. To
that end, we will numerically study the performance of QECCs when they operate
over the proposed time-varying quantum channels. The error correction codes
used for the numerical simulations are quantum turbo codes (QTCs) [36, 37, 58,
60] and Kitaev toric codes [38]. We will asses the performance of these codes
by means of the quantum outage probability of Chapter 4. Note that this is
equivalent to what is done in the literature when benchmarking the performance
of QECC codes operating over static channels respect to the quantum capacity.

We begin by briefly introducing the quantum error correction codes considered
in the simulations. We follow by presenting the numerical results obtained
from these codes when operating over the proposed time-varying channel models,
and discuss how the time-variations of these channels degrade their
performance. We close the chapter by using the quantum outage probability to
benchmark their performance.

####  5.1 Considered Quantum Error Correction Codes

The next section introduces the Kitaev toric codes and the unassisted QTCs
that will be numerically simulated. The decoding of these codes is discussed
in Appendix B and the Monte Carlo numerical simulation methods in Appendix C.
As it was explained in Chapter 2, the classical simulation of QECCs on
classical computers is based on the twirled approximations of the combined
amplitude and phase damping channel. Therefore, the channels considered in the
current simulations are the static and time-varying Pauli channels.

##### Kitaev Toric codes

The Kitaev toric code is a quantum error correction code that belongs to the
family of surface codes [38, 151]. It is defined as a 2D lattice of qubits
with periodic boundaries and, thus, it has the shape of a torus. Figure 5.1
shows a graphical representation of the toric code.

Figure 5.1: Graphical representation of the toric code. The blue and red lines
represent logical $\mathrm{X}$ and $\mathrm{Z}$ operators acting on the
encoded logical qubit.

In the toric code, the stabilizers are implemented in the vertices and in the
plaquettes (vertices of the dual lattice) of the lattice that constitutes the
code. By using those vertex and plaquette qubits, the syndrome of the errors
that occur in the toric code can be obtained. The errors of each of the grid
qubits occur with the probabilities given by the PTA or CTA in consideration.
Finally, the syndrome is fed to the minimum-weight perfect matching (MWPM)
decoder that estimates the logical error that has occurred in an error
correction round of the toric code (see Appendix B).

A $d\times d$ Kitaev toric code is said to be a $[[2d^{2},2,d]]$ quantum error
correction code [151] since it is able to protect two logical qubits by means
of $2d^{2}$ physical qubits with code distance $d$. For the numerical
simulations that will be conducted in this chapter, Kitaev toric codes with
distances $d\in\\{3,5,7,9\\}$ will be considered.

##### Quantum Turbo Codes

The Quantum Turbo Codes considered in this chapter consist of the interleaved
serial concatenation of unassisted111In this context, unassisted refers to
error correction codes that are not aided by entanglement. quantum
convolutional codes acting as outer and inner codes [60]. Figure 5.2 presents
the full schematic representation of such a quantum error correction system.
The $k$ input logical qubits that make up the information word
$\ket{\psi_{1}}$ are first fed to the outer convolutional encoder
$\mathcal{V}_{1}$, and encoded into $n^{\prime}$ physical qubits with the help
of ancilla qubits and memory qubits.

Figure 5.2: Schematic of the QTC. $\mathrm{P}_{i}^{a}(.)$ and
$\mathrm{P}_{i}^{e}(.)$ denote the a-priori and extrinsic probabilities
related to each of the SISO decoders used for turbo decoding (see Appendix B).

The codeword $\ket{\bar{\psi}_{1}}$ consists of $n^{\prime}$ physical qubits
generated by the first encoder, which are then passed through a quantum
interleaver $\Pi$, before being used as the input to the inner convolutional
encoder $\mathcal{V}_{2}$. Such an encoder is an unassisted device that
encodes the interleaved sequence of $n^{\prime}$ qubits $\ket{\psi_{2}}$ into
the codeword $\ket{\bar{\psi}_{2}}$ of length $n$, aided by ancilla qubits and
memory qubits. The codeword $\ket{\bar{\psi}_{2}}$ is then transmitted through
a quantum Pauli channel (static or time-varying) with overall error
probability $p=p_{\mathrm{x}}+p_{\mathrm{y}}+p_{\mathrm{z}}$, which inflicts
an $n$-qubit Pauli error $\mathcal{E}_{2}\in\mathcal{G}_{n}$ on the codeword.
The Pauli channel is independently applied to each of the qubits of the stream
$\ket{\bar{\psi}_{2}}$, and, consequently, each of the qubits may experience a
bit-flip ($\mathrm{X}$ operator) with probability $p_{\mathrm{x}}$, a phase-
flip ($\mathrm{Z}$ operator) with probability $p_{\mathrm{z}}$ or a
combination of both ($\mathrm{Y}$ operator) with probability $p_{\mathrm{y}}$.
Those probabilities will be defined depending on the static or TV twirl
approximation being in consideration.

At the output of the Pauli channel, the state
$\mathcal{E}_{2}\ket{\bar{\psi}_{2}}$ is fed to the inverse of the inner
encoder $\mathcal{V}_{2}^{\dagger}$, which outputs the decoded state
$\mathcal{L}_{2}\ket{\psi_{2}}$, where
$\mathcal{L}_{2}\in\mathcal{G}_{n^{\prime}}$ refers to the logical error
suffered by the decoded state due to the operation of the channel; and the
classical syndrome bits222Note that we emphasize the fact that when the QTC is
unassisted, the syndrome measured just gives information of the $\mathrm{X}$
operators applied to the ancillas. This terminology is common for QTCs [37].
$\bar{s}_{2}^{x}$ obtained from $\mathrm{Z}$ basis measurements on the ancilla
qubits. The corrupted logical qubits are then passed through a de-interleaver
$\Pi^{-1}$ resulting in the state $\mathcal{E}_{1}\ket{\bar{\psi}_{1}}$, which
is supplied to the inverse of the outer encoder $\mathcal{V}_{1}^{\dagger}$.
The resulting output is the state $\mathcal{L}_{1}\ket{\psi_{1}}$, which
corresponds to the information quantum state corrupted by a logical error
$\mathcal{L}_{1}\in\mathcal{G}_{k}$; and the classical syndrome bits
$\bar{s}_{1}^{x}$ obtained after measuring the ancilla qubits on the
$\mathrm{Z}$ basis. The classical syndromes $\bar{s}_{2}^{x}$ and
$\bar{s}_{1}^{x}$, obtained in the inverse decoders
$\mathcal{V}_{2}^{\dagger}$ and $\mathcal{V}_{1}^{\dagger}$, respectively, are
then provided to the iterative syndrome decoder made up of two serially
concatenated SISO decoders, as shown in Figure 5.2. Based on $\bar{s}_{2}^{x}$
and $\bar{s}_{1}^{x}$, the SISO decoders estimate the most probable logical
error (see Appendix B). Based on the aforementioned estimation, a recovery
operation $\mathcal{R}$ is applied to the corrupted state
$\mathcal{L}_{1}\ket{\psi_{1}}$, yielding the recovered output
$\ket{\tilde{\psi}_{1}}$.

The QCCs used for the quantum turbo code in Figure 5.2 are the “PTO1R”
unassisted convolutional encoders introduced in [60]. The seed
transformation333The seed transformation is a smaller encoding circuit that is
repeatedly executed in order to construct the whole convolutional encoding
unitary. The seed transformation $\mathcal{U}$ is represented using the
decimal representation presented in [37]. associated with the convolutional
encoders is

$\begin{split}\mathbf{\mathcal{U}}=\\{1355,2847,558,2107,3330,739,\\\
2009,286,473,1669,1979,189\\}.\end{split}$ (5.1)

Following previous work in the literature, we will refer to the serial
concatenation of two similar interleaved PTO1R QCCs as the “PTO1R-PTO1R”
configuration. Both QCC encoders have rate $1/3$ resulting in an error
correction scheme of rate $R_{Q}=R_{in}\times R_{out}=1/3\times 1/3=1/9$. Each
of the “PTO1R” encoders is aided by $3$ memory qubits in order to perform the
convolutional encoding operation. The turbo code in consideration encodes
blocks of $k=1000$ qubits.

####  5.2 QECCs operating over TVQCs

Next, we provide the numerical results obtained for the performance of QECCs
when they operate over TVQCs. Similar of what we did in Chapter 4, it is
assumed that the realizations of the decoherence times for the TVQCs are the
same for each qubits of a block of error correction. Thus, we are considering
that the processes are fully correlated.

We begin by providing an example444In what follows, we select a set of
realistic operating scenarios to asses the performance of error correction
codes when operating on real hardware that exhibits time fluctuations [51]. to
asses the impact that the deviations of the diamond norm distance may have on
quantum error correction codes when implemented in the superconducting qubits
of [51]. Consider an ADCTA channel555Note that as we are considering qubits
that are $T_{1}$-limited [51], the family of amplitude damping channels
describe the decoherence experienced by those qubits. like the one of scenario
QA_C5 ($c_{\mathrm{v}}(T_{1})\approx 25\%$) at $t/\mu_{T_{1}}=0.1$ (refer to
Chapter 3 for specific data regarding the scenarios discussed). The static
version of this channel will have a depolarizing probability of $p\approx
0.05$. Knowing that the diamond norm for depolarizing channels can be
simplified to
$||\mathcal{N}_{D}(p_{1})-\mathcal{N}_{D}(p_{2})||_{\diamond}=2|p_{1}-p_{2}|$,
it can be shown that for a channel instance of the TVADCTA, when the diamond
norm distance between the static and TV channels is
$||\mathcal{N}_{\mathrm{ADCTA}}(\mu_{T_{1}})-\mathcal{N}_{\mathrm{ADCTA}}(\rho,\omega,t)||_{\diamond}=0.29$,
the corresponding depolarizing probability for the TV channel will be
$p\approx 0.15$. Let us now consider the toric codes with $d\in\\{7,9\\}$ [38]
and the QTC from [60] designed for the depolarizing channel (the WER for those
scenarios when the channels are static are the red curves in figures 5.3 and
5.4). These error correction codes are capable of operating at a Word Error
Rate of $\mathrm{WER}\approx\\{5\cdot 10^{-3},10^{-3},10^{-6}\\}$,
respectively, for $p=0.05$, while for $p=0.15$ they fail completely, resulting
in $\mathrm{WER}\approx 1$. This shows that the selected QECCs will be
operating with a significantly worse $\mathrm{WER}$ than that which would be
expected based on a static channel model. Although there will also be channel
instances for which the particular realization of $\\{T_{1}(\omega,t)\\}$ will
result in operation within a region where the $\mathrm{WER}$ of the QECC is
actually smaller than what is assumed for the static channel, on average, the
effect of these realizations on the overall performance of the code will be
much smaller than that of the events where the real channel action is worse
than what is reflected by the static model. This winds up degrading the
performance of the QECCs, which is reflected by a flattening of the
$\mathrm{WER}$ curve of the code as a result of its waterfall region falling
at a lower rate. Given that the existence of channel parameter fluctuations
with time has been proven by recent experimental outcomes, it is likely that
current error correction schemes will actually yield worse performance than
what would be expected based on the results that have been obtained for static
quantum channels. It is worth mentioning that the diamond norm distance we
have chosen for this this example is pretty low. In fact, Figure 3.6 in
Chapter 3 portrays that values of up to
$||\mathcal{N}_{\mathrm{ADCTA}}(\mu_{T_{1}})-\mathcal{N}_{\mathrm{ADCTA}}(\rho,\omega,t)||_{\diamond}\approx
1.4$ can occur for a scenario with $c_{\mathrm{v}}(T_{1})\approx 25\%$ at
$t/\mu_{T_{1}}=0.1$. Note that events with high diamond norm will all be due
to events where the channel instance is very noisy since as
$||\mathcal{N}_{\mathrm{ADCTA}}(\mu_{T_{1}})-\mathcal{N}_{\mathrm{ADCTA}}(\rho,\omega,t)||_{\diamond}=2|p(\mu_{T_{1}})-p(\omega)|$,
$p(\mu_{T_{1}})$ is fixed and $p(\omega)\geq 0$, the realizations
$||\mathcal{N}_{\mathrm{ADCTA}}(\mu_{T_{1}})-\mathcal{N}_{\mathrm{ADCTA}}(\rho,\omega,t)||_{\diamond}\geq
2|p(\mu_{T_{1}})|$ must occur because $p(\omega)\geq 2p(\mu_{T_{1}})$. For the
example we are considering, the events
$||\mathcal{N}_{\mathrm{ADCTA}}(\mu_{T_{1}})-\mathcal{N}_{\mathrm{ADCTA}}(\rho,\omega,t)||_{\diamond}\geq
0.1$ will all belong to this kind of TV channel realizations.

Figure 5.3: Performance of the $d\times d$ Kitaev toric codes with
$d\in\\{3,5,7,9\\}$ [38]. Each panel presents performance for i) the static
channel (red), ii) the TVADCTA for scenario QA_C5
($c_{\mathrm{v}}(T_{1})\approx 25\%$) (blue) and iii) the TVADCTA for scenario
QA_C6 ($c_{\mathrm{v}}(T_{1})\approx 22\%$) (purple).

Let us now consider the toric codes with $d\in\\{3,5\\}$ [38] (the WER for
those scenarios when the channels are static are the red curves in Figure
5.3). These surface codes operate at a $\mathrm{WER}\approx\\{8\cdot
10^{-2},2\cdot 10^{-2}\\}$ for $p=0.05$ and at a $\mathrm{WER}\approx 1$ for
$p=0.15$. Considering that the error rate at the static point is not low
enough and given that the waterfall region is pretty flat, for these codes the
$\mathrm{WER}$ for a good channel realization will not be very different from
that of a bad channel realization. Thus, although these codes will experience
time-variation effects, these effects will not be as significant as for the
previosuly mentioned codes.

Figure 5.4: Performance of the QTC discussed in [60]. Each panel presents
performance for i) the static channel (red), ii) the TVADCTA for scenario
QA_C5 ($c_{\mathrm{v}}(T_{1})\approx 25\%$) (blue) and iii) the TVADCTA for
scenario QA_C6 ($c_{\mathrm{v}}(T_{1})\approx 22\%$) (purple).

We prove this intuition by simulating the discussed QECCs for the time-varying
amplitude damping Clifford twirl approximated (TVADCTA) channel obtained for
the configurations QA_C5 ($c_{\mathrm{v}}(T_{1})\approx 25\%$) and QA_C6
($c_{\mathrm{v}}(T_{1})\approx 22\%$) and comparing the results with the
corresponding static channel. Figures 5.3 and 5.4 show the results of these
simulations. The blue curves are for the QA_C5 scenario and the purple ones
for the QA_C6 scenario. A quick inspection of this figures reveals the
expected outcomes. It can be observed that the low distance toric codes
($d\in\\{3,5\\}$, figures 5.3a and 5.3b) do not exhibit a significant
degradation of their error rate performance (especially the $3\times 3$ toric
code; the $5\times 5$ toric code shows signs of performance degradation, but
at very low depolarizing probabilities). However, the higher distance toric
codes ($d\in\\{7,9\\}$, figures 5.3c and 5.3d) and the QTC (Figure 5.4) are
significantly affected by the time fluctuations. This is reflected by a
reduction in the steepness of the waterfall region (it becomes flattened) in
the case of the time-varying channel, which compromises the excellent
performance that QECCs exhibit for static channel models. The impact of the
coefficient of variation can also be observed, as the scenario with higher
$c_{\mathrm{v}}$ is flattened more.

The above discussion suggests that the manner in which QECCs are affected by
the proposed time-varying quantum channel model is a combination of two
factors: how much the decoherence parameters fluctuate (which is quantified by
the coefficients of variation) and the steepness of the $\mathrm{WER}$ curve
when the QECCs operate over the static channel. This occurs because if the
code presents a rather flat $\mathrm{WER}$ curve (as a function of the
depolarizing probability) in the static channel, the difference in
$\mathrm{WER}$ between benign and malign channel realizations of the time-
varying channel is not significant, and thus, on average, the overall
$\mathrm{WER}$ will be similar to that of the corresponding static channel.
Since, everything else being equal, shorter error correction codes present
flatter waterfall regions, they will be less susceptible to the mismatch. On
the other hand, the contrary effect is observed for error correction codes
that exhibit a steep waterfall region in the static channel. In these cases,
the difference between a good and a bad channel realization might be either
complete decoding success or utter failure, implying that on average, the
overall performance in the time-varying channel will experience substantial
degradation. Thus, the better the code performance in the static channel, the
more it will be affected in the time-varying channel model.

####  5.3 Benchmarking the performance of QECCs operating over TVQCs

We finish this chapter by proposing the quantum outage probability as the
information theoretical metric to asses the performance of QECCs when
operating over the TVQCs proposed when there is full correlation between the
realizations of the decoherence parameters for each of the qubits of a code
block. Since the ADCTA depolarizing static channel and its time-varying
version TVADCTA belong to the family of Pauli channels, the information
theoretical benchmarks that will be considered are the hashing bound (for the
static channels) and the quantum hashing outage probability (for the TV
channels).

The following simulations are based on a $9\times 9$ Kitaev toric code with
coding of rate $R_{\mathrm{Q}}=1/81$ and on the previous QTC of
$R_{\mathrm{Q}}=1/9$. We have selected these two codes since they experience a
large degradation (with respect to the static channel) under channel time
variations.

Figure 5.5: Performance of the $9\times 9$ Kitaev toric code [38]. The
$[[162,2,9]]$ quantum error correction code has rate $R_{\mathrm{Q}}=1/81$.
The hashing bound and the hashing outage probabilities are also plotted.

By considering the same previous channel scenarios, Figures 5.5 and 5.6 show
the $\mathrm{WER}$ versus $p$ of these two codes when they operate on the
static channels (red curve) and on the TVQCs (blue and purple). The figures
also plot their respective quantum hashing outage probabilities. Both figures
show that the quantum hashing outage probability presents a shape similar to
the performance of the QECCs. Therefore, the quantum outage probability curves
are a good indicator of the performance that some QECC will have when it
operates over a TVQC with certain coefficients of variation of their
decoherence parameters. Also, the closer the $\mathrm{WER}$ curves are to the
corresponding quantum outage probability, the better the performance of the
codes will be.

Figure 5.6: Performance of the quantum turbo code from [60]. The quantum error
correction code has rate $R_{\mathrm{Q}}=1/9$ and encodes blocks of $1000$
qubits. The hashing bound and the hashing outage probabilities are also
plotted.

It is also interesting to point out that at low depolarizing probabilities,
the $\mathrm{WER}$ curves approach to the quantum hashing outage probability
curves. This effect is clearly observed for the $9\times 9$ Kitaev toric code
in Figure 5.5. For the QTC in Figure 5.6 such saturation is not observed
mainly due to the range of values of $\mathrm{WER}$ and $p$ considered in the
simulation. A possible explanation for this effect is that at low depolarizing
probabilities, $p$, the errors produced by the channel are all mainly of low
weight (i.e., these errors will affect only a few qubits in the codeword), and
the codes will be capable of correcting all of them. Therefore, the code for
this scenario of low depolarizing probabilities behaves as an ideal code as
the one used to compute the quantum hashing outage probability.

To quantify the distance to the hashing outage bound, we use a metric similar
to the one proposed in [42]. It measures the gap in dBs between the
depolarizing probabilities of a code and the hashing outage at a given
$\mathrm{WER}=\chi$:

$\delta_{\mathrm{out}}(@\chi)=10\log\left(\frac{p(p_{\mathrm{out}}^{\mathrm{H}}=\chi)}{p(\mathrm{WER}_{\mathrm{code}}=\chi)}\right).$
(5.2)

Therefore, the lower $\delta_{\mathrm{out}}$ is, the better the performance of
the code will be since its performance will be more similar to the
asymptotically achievable error rate. Table 5.1 shows the
$\delta_{\mathrm{out}}(@10^{-3})$ for the two codes in consideration.

Code | Static | QA_C5 | QA_C6
---|---|---|---
$9\times 9$ Toric | $3.93$ | $2.04$ | $2.81$
QTC | $1.79$ | $1.75$ | $1.67$
Table 5.1: Distance to the quantum hashing outage probability of the codes in
consideration when WER = $10^{-3}$. $\delta_{\mathrm{out}}(@10^{-3})$ is given
in dBs.

It can be seen that the $9\times 9$ Kitaev toric codes are farther from the
quantum hashing outage probability than the QTC in consideration at
WER=$10^{-3}$. This fact is also true for the distance of the codes to their
hashing bounds when they operate over static channels so the result is not
surprising. Thus the QTC seems to be a better error correction method than the
Toric code since its performance is closer to the hashing bound and its
quantum rate is lower. However, it is important to state that this QTC
requires codewords of $9000$ physical qubits while the $9\times 9$ Toric code
just requires $162$.

Finally, note that for the toric code, the value of
$\delta_{\mathrm{out}}(@10^{-3})$ for scenario QA_C5 is smaller than for
scenario QA_C6. At a first glance, this seems counterintuitive, since the
coefficient of variation is higher for QA_C5. However, this can be explained
by observing the $\mathrm{WER}$ curve for scenario QA_C5 approaches earlier
the quantum hashing outage probability curve than for scenario QA_C6.

####  5.4 Discussion

In light of the discussion presented in this chapter, and based on the
experimental evidence that decoherence parameters vary with time, the proposed
time-varying quantum channel models will be very important when studying and
optimizing error correction codes capable of protecting qubit-based quantum
computers from the dynamics of the decoherence phenomenon. However, it should
be pointed out that the proposed time-varying quantum channel model would be
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 [149]).
If a very short algorithm or if a QECC is run just once or a few number of
times, the parameters will not fluctuate as much, and the effects of the
proposed channel model will not be as noticeable. 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 above), 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.

In addition, we have proposed the quantum outage probability as a benchmark of
the performance of the QECCs operating over TVQCs. We conclude that the
quantum hashing outage probability does represent well the behaviour of QECCs
on those dynamic noise scenarios. It has also been observed that the
performance of the codes approach the quantum hashing outage probability limit
when the depolarizing noise is very small.

##  Part II Quantum Error Correction: Optimization of Quantum Turbo Codes

In this second part of the dissertation we focus on quantum error correction
schemes. Sepecifically, the objective is to optimize the error correction
capabilities of Quantum Turbo Codes (QTC) when they operate under different
scenarios. First, we lower the error floor performance of QTCs by considering
interleavers with some construction rather than just performing a random
scrambling of the coded random information that comes out of the first
convolutional encoder of the QTC. Additionally, we study the performance of
this family of quantum error correction methods when the Channel State
Information (CSI) is unknown for the decoder. We consider this scenario to be
a more realitic one since, in the reality, the decoder of a QECC must operate
with estimates of the level of noise of the quantum channel. In particular, we
study how the error correction capabilities of QTCs are compromised when there
exists a mismatch between the true channel probability and the estimated
channel probability and discuss existing estimation methods that aid the
decoder to operate properly. Moreover, we propose an on-line estimation method
that uses the information being processed in the decoder in order to obtain
accurate estimates of the channel probability so that the decoder of QTCs is
able to operate “blind” to the true CSI without losing performance. Both the
depolarizing channel (Clifford twirl approximation) and the general Pauli
channel (Pauli twirl approximation) are considered. Before delving into the
proposed schemes and with the aim of providing some context, we begin by
including a general overview of QTCs and their development.

Classical turbo codes, introduced by Berrou et al. in [152], marked a
breakthrough in classical coding theory as the first practical codes closely
approaching Shannon’s channel capacity [43]. Motivated by their success, QTCs
were proposed by Poulin et al. as the interleaved serial concatenation of
Quantum Convolutional Codes (QCC) [36]. The use of a serial structure in [36]
is explained by the laws of quantum mechanics, in particular by the underlying
classical-to-quantum isomorphism [57] applied in the design of quantum error
correction codes (QECC): Classical turbo codes are generally systematic,
interleaved parallel concatenations of convolutional codes but in the quantum
domain such a systematic parallel structure is prohibited by the no-cloning
theorem.

The performance of the QTCs proposed in [36] is not on par with that of
classical serial turbo codes. The reason is that unassisted QCCs cannot be
both non-catastrophic and recursive at the same time, properties that are
required for the inner code of a turbo code so that the minimum distance grows
with the block length and the iterative decoding algorithm achieves
convergence. Consequently, non-catastrophic, non-recursive QCCs were used in
[36], resulting in codes with bounded minimum distance. Utilizing the
entanglement-assistance (EA) techniques666Entanglement-assistance techniques
utilize entangled qubits pre-shared between the sender and the receiver, which
simplifies the construction of stabilizer QECCs by relaxing the stringent
commutativity conditions that the stabilizers need to satisfy. for QECCs
presented in [40], this issue was addressed by Wilde et al. in [37], which
proposed QTCs operating within 1 dB of their corresponding EA hashing bounds.
Taking a step further, Babar et al. proposed in [58] an EXtrinsic Information
Transfer (EXIT) chart technique to narrow the gap to the Hashing bound,
resulting in codes with a performance as close as 0.3 dB to the hashing
bounds.

2009 $\bullet$ |  QTCs [36]. Non-catastrophic, non-recursive. $\delta\gtrapprox 2$ dB.
---|---
2014 $\bullet$ |  EAQTCs [37]. Non-catastrophic, recursive. $\delta\approx 1$ dB.
2015 $\bullet$ |  EXIT optimized [58]. $\delta\approx 0.3$ dB. High error floor.
$\bullet$ |  QIRCC [59]. Like [58] with lower error floor.
2016 $\bullet$ |  EAQIRCC [60]. Optimized for the PTA channel.
$\bullet$ |  QURC [69]. Higher rate maintaining performance.
$\bullet$ |  Fully parallel decoder for QTCs [153]. Lower complexity.
2018 $\bullet$ |  QTC operating over channels with memory [128].
2019 $\bullet$ |  Short block, high-rate QTCs [154].
Table 5.2: Timeline of the evolution of QTCs.

However, these QTC codes present error floors that are higher than in the
original distance-spectra-based codes in [37]. Based on the use of Quantum
IrRegular Convolutional Codes (QIRCC) as outer codes, the authors in [59]
showed a performance in the turbo-cliff region similar to that of the EXIT
optimized codes in [58] but with lower error floors. An entanglement-assisted
version of QIRCCs was introduced in [60] with the aim of designing efficient
concatenated QECCs for asymmetric depolarizing channels. A Quantum Unity Rate
Code (QURC) aided QTC scheme was proposed in [69] in order to improve the
performance of the outer code without experiencing code rate reduction due to
the inner code. A modified decoder for QTCs that relies on parallel computing
was developed in [153] in order to reduce the decoding computation time.
Later, QTCs were studied in [128] when they operate over Pauli channels with
memory and a modified decoding algorithm was proposed in order to improve the
performance of the codes in such scenarios. Finally, short block, high-rate
QTC schemes with multiple rates presenting performances near the hashing bound
were proposed and studied in [154].

Table 5.2 presents the evolution of QTCs presented before. In spite of their
short history, quantum turbo codes are a promising family of error correction
methods for the quantum computers of the post-NISQ-era or for quantum
communications due to their closeness to the hashing bound, degenerate
decoding and non-complex decoding algorithm.

Note that in Part I of this thesis we proposed quantum channels that vary in
time and discussed how those degrade the performance shown by quantum error
correction codes. However, it is important to point out that in this part of
the dissertation we consider that the QTCs operate over quantum channels that
are static.

###  CHAPTER 6 Optimization of the error floor performance of QTCs via
interleaver design

Classical turbo codes are constructed by the interleaved serial or parallel
concatenation of two convolutional codes [152]. The interleaver has a central
role in turbo coding due to the fact that it is an element that permutes the
symbols of the coded block so that the error locations are randomized. This
way, the bursts of errors that are generated in the decoding of the first
convolutional code can be more efficiently corrected due to their scrambling.
Initially, the concatenation of the convolutional codes was done by means of a
random interleaver, that is, an interleaver whose permutation pattern is
selected randomly. However, due to the importance of such code concatenation
element, the construction of classical interleavers with certain structure has
been extensively studied in the literature [61, 62, 63, 64, 65, 66, 67, 68].
This way, the authors seeked the optimization of the performance of classical
turbo codes by designing permutation patterns that scramble the information in
the best possible way.

Quantum turbo codes are also constructed by concatenating two quantum
convolutional codes [36]. However, the concatenation must be done in a serial
way since the no-clonning theorem prohibits the copy of information in the
quantum realm that is needed for the parallel concatenation of QCCs. The QTCs
proposed in the literature [36, 57, 40, 37, 58, 59, 60, 69, 128] use the
quantum analogue of the random interleaver. Regarding QTCs, the only
preliminary regarding practical quantum interleaver design was addressed in
[153], where the authors discussed the implementation of an odd-even
interleaver for reducing the complexity of the parallel QTC decoding algorithm
they proposed. Nevertheless, no performance improvement was observed by the
implementation of such scrambling method. Consequently, and based on the
classical-to-quantum isomorphism [57] that is typically applied in the design
of quantum error correction codes, we aim to integrate the classical
interleaving design methods into the paradigm of quantum turbo coding. In this
manner, we expect to improve the error correction capabilities of those codes,
specially in the error floor region. Simulation results show that the QTCs
designed using the proposed interleavers present the same behavior in the
turbo-cliff region as the codes with random interleavers in [58], while the
performance in the error floor region is improved by up to two orders of
magnitude. Simulations also show reduction in memory consumption, while the
performance is comparable to or better than that of QTCs with random
interleavers.

####  6.1 System Model

The Quantum Turbo Codes considered in this chapter consist of the interleaved
serial concatenation of unassisted QCCs acting as outer codes and
entanglement-assisted QCCs as inner codes111Note that this QTC is similar to
the QTC considered in Chapter 5 but with an EA inner QCC.. Figure 6.1 presents
the full schematic representation of such quantum error correction system. The
$k$ input logical qubits that compose the information word $\ket{\psi_{1}}$
are first fed to the outer $[[n_{1},k_{1},m_{1}]]$ unassisted convolutional
encoder $\mathcal{V}_{1}$ and encoded into $n^{\prime}=k\frac{n_{1}}{k_{1}}$
physical qubits with the help of $(n_{1}-k_{1})$ ancilla qubits and $m$ memory
qubits. The $n^{\prime}$ physical qubits that form the codeword
$\ket{\bar{\psi_{1}}}$ generated by the first encoder are then passed through
a quantum interleaver $\Pi$ before being input to the inner convolutional
encoder $\mathcal{V}_{2}$. Such an encoder is an $[[n_{2},k_{2},m_{2},c]]$
entanglement-assisted encoder that encodes the interleaved sequence of
$n^{\prime}$ qubits $\ket{\psi_{2}}$ into the codeword $\ket{\bar{\psi_{2}}}$
of length
$n=n^{\prime}\frac{n_{2}}{k_{2}}=k\frac{n_{2}}{k_{2}}\frac{n_{1}}{k_{1}}$,
aided by $(n_{2}-k_{2}-c)$ ancilla qubits, $c$ pre-shared EPR pairs and
$m_{2}$ memory qubits. Codeword $\ket{\bar{\psi_{2}}}$ is then transmitted
through a quantum channel that corrupts such quantum information. For this
chapter, we will consider the depolarizing channel (Clifford twirl
approximation) and, thus, with qubit-wise depolarizing probability $p$ an
$n$-qubit Pauli error $\mathcal{E}_{2}\in\mathcal{G}_{n}$ will be inflicted to
the codeword.

Figure 6.1: Schematic of the Entanglement-Assisted Quantum Turbo Code (EAQTC).
Note that the $c$ pre-shared EPR pairs $\ket{\Phi^{+}}$ are needed for the
inner encoder to be both recursive and non-catastrophic [37].
$\mathrm{P}_{i}^{a}(.)$ and $\mathrm{P}_{i}^{e}(.)$ denote the a-priori and
extrinsic probabilities related to each of the SISO decoders used for turbo
decoding.

At the output of the depolarizing channel, the state
$\mathcal{E}_{2}\ket{\bar{\psi_{2}}}$ is fed to the inverse of the inner
encoder $\mathcal{V}_{2}^{\dagger}$, which outputs the decoded state
$\mathcal{L}_{2}\ket{\psi_{2}}$, where
$\mathcal{L}_{2}\in\mathcal{G}_{n^{\prime}}$ refers to the logical error
suffered by the decoded state due to the operation of the channel; and the
classical syndrome bits $\bar{r}_{2}=(\bar{s}^{x}_{2},\bar{e}_{2}^{x,z})$
obtained from $\mathrm{Z}$ basis measurements on the ancilla qubits and Bell
measurements on the pre-shared EPR pairs222Syndrome measurement of the EAQCC
includes measurements of ancillas $\bar{s}^{x}$ and pre-shared ebits
$\bar{e}^{x,z}$.. The corrupted logical qubits are then passed through a de-
interleaver $\Pi^{-1}$ resulting in the state
$\mathcal{E}_{1}\ket{\bar{\psi_{1}}}$, which is supplied to the inverse of the
outer encoder $\mathcal{V}_{1}^{\dagger}$. The resulting output is the state
$\mathcal{L}_{1}\ket{\psi_{1}}$, which corresponds to the information quantum
state corrupted by a logical error $\mathcal{L}_{1}\in\mathcal{G}_{k}$; and
the classical syndrome bits $\bar{s}^{x}_{1}$ obtained after measuring the
ancilla qubits on the $\mathrm{Z}$ basis. The classical syndromes
$\bar{r}_{2}$ and $\bar{s}_{1}^{x}$, obtained in the inverse decoders
$\mathcal{V}_{2}^{\dagger}$ and $\mathcal{V}_{1}^{\dagger}$, respectively, are
then provided to the iterative syndrome decoder consisted of two serially
concatenated SISO decoders, as shown in Figure 6.1. Based on $\bar{r}_{2}$ and
$\bar{s}_{1}^{x}$, as well as the channel information
$\mathrm{P}_{\mathrm{ch}}(\mathcal{E}_{2})$, both SISO decoders engage in
degenerate iterative decoding [36, 37] to estimate the most likely error coset
$\tilde{\mathcal{L}}_{1}$ that has corrupted the information quantum state.
Based on such an estimation, a recovery operation $\mathcal{R}$ is applied to
the corrupted state $\mathcal{L}_{1}\ket{\psi_{1}}$, yielding the recovered
output $\ket{\tilde{\psi_{1}}}$.

As already mentioned, the aim of this chapter is to show that interleaver
design is beneficial to reduce the error floors of QTCs. To that end, the
inner and outer QCCs we consider to construct the concatenated error
correcting system presented in Figure 6.1 are the EXIT-optimized codes
introduced in [58]. The parameters of the inner and outer codes are presented
in Table 6.1. Such configuration, utilizing a random interleaver for the
serial concatenation of the constituent QCCs, was studied extensively in [58].
Therefore, it is an excellent benchmark for the interleaver constructions
proposed in this chapter.

From the configuration parameters presented in Table 6.1, it can be seen that
the QTCs used in this chapter will be rate $1/9$ turbo codes with an
entanglement consumption rate of $6/9$. The noise limit $p^{*}$ can be found
from the entanglement-assisted hashing bound, which for such parameters is
$p^{*}=0.3779$ [37].

Config. | Encoder |  $R_{\mathrm{Q}}$ | E | $m$ |  Seed Transformation $\mathbf{\mathcal{U}}$
---|---|---|---|---|---
EXIT- optimized | Outer | $1/3$ |  $0$ | $3$ |  | $\\{1048,3872,3485,2054,$
---
$983,3164,3145,1824,$
$987,3282,2505,1984\\}_{10}$
Inner | $1/3$ | $2/3$ | $3$ |  | $\\{4091,3736,2097,1336,$
---
$1601,279,3093,502,$
$1792,3020,226,1100\\}_{10}$
Table 6.1: Parameters of the QCC encoders. $R_{\mathrm{Q}}$ and $\mathrm{E}$
refer to the coding rate and the entanglement consumption rate, respectively.
$m$ refers to the memory qubits. The seed transformations $\mathcal{U}$ are
represented using the decimal representation presented in [37].

####  6.2 Classical Interleavers for QTCs

The concatenation between the outer and inner QCCs is done by the quantum
interleaver $\Pi$ of size $N$, which is an $N$ to $N$-qubit symplectic
transformation333The notation we utilize to describe the operation of a
quantum interleaver is the so-called symplectic notation as used in [36].
composed by a permutation $\pi$ of the $N$ qubit registers and a tensor
product of single-qubit symplectic transformations. It acts by multiplication
on the right on $\bar{\epsilon}\in\Upsilon^{-1}([\mathcal{G}_{N}])$ as

$(\epsilon_{1},\cdots,\epsilon_{N})\rightarrow(\epsilon_{\pi(1)}K_{1},\cdots,\epsilon_{\pi(N)}K_{N}),$
(6.1)

where $K_{1},\cdots,K_{N}$ are some fixed symplectic matrices444Recall the
$\Upsilon$ map in section 2. acting on
$\epsilon_{\pi(j)}\in\Upsilon^{-1}([\mathcal{G}_{1}])$, where
$\mathcal{G}_{N}$ refers to the $N$-fold Pauli group [36].

In order to use classical interleavers in QTCs, they must properly match the
definition of quantum interleavers in Eq. 6.1. Classical interleavers consist
only of permutations of the information arrays with no such thing as
individual symplectic transformations. Consequently, those operations will be
taken as identity operators acting on the registers $K_{i}=I,\forall i$; and
so the only non-trivial operation of such quantum interleavers will be the
scrambling of the qubits in the quantum information stream555The random
interleavers used in [36, 57, 40, 37, 58, 59, 60, 69, 128] also follow this
approach.. Following this approach, the quantum interleavers presented here
will be $N$ to $N$ qubit symplectic transformations acting as

$(\epsilon_{1},\cdots,\epsilon_{N})\rightarrow(\epsilon_{\pi(1)},\cdots,\epsilon_{\pi(N)})$
(6.2)

to the qubit data stream, and where the permutation pattern $\pi$ corresponds
to the classical interleaver under consideration.

#####  6.2.1 Considered Classical Interleavers

All the QTC schemes in [36, 57, 40, 37, 58, 59, 60, 69, 128] are based on
random interleavers, that is, the interleaving patterns $\pi$ are selected at
random. However, it is known from classical turbo codes that the usage of
interleavers with some added structure is beneficial to reduce either the
error floors or the memory requirements [61, 62, 63, 64, 65, 66, 67, 68]. In
order to show that interleaver design is also beneficial when implementing
quantum turbo codes, we consider the following three types of classical
interleavers.

The first classical interleavers considered here are the $S$-random
interleavers [61]. They are randomly generated by imposing the following
condition on the interleaving distance or spread ($S$):

$|\pi(i)-\pi(j)|>S\text{ for $i$ and $j$ with }|i-j|\leq S.$ (6.3)

If the heuristic recommendation $S<\sqrt{\frac{N}{2}}$ is satisfied, where $N$
is the blocklength, $S$-random interleavers can usually be produced in
reasonable time by repeatedly generating random integers until condition Eq.
6.3 is satisfied [61, 65]. However, as indicated in [68], reasonable values of
$S$ are sometimes lower when $N$ is large. We next consider the deterministic
Welch-Costas [67] interleaver, which is defined by two parameters: $p=N+1$, a
prime number, and $\alpha$, a primitive element modulo $p$. Once $\alpha$ is
chosen for the selected blocklength, it is proven in [67] that

$\pi(i)=(\alpha^{i}\bmod{(N+1)})-1,\quad\forall i\in\\{0,\cdots,N-1\\}$ (6.4)

forms a permutation which is selected as the interleaving rule. Although the
spreads obtained with Welch-Costas interleavers are low, their dispersion
parameter, $\eta$, is maximized. The dispersion of an interleaver $\pi$ is a
parameter that represents the randomness of the permutation, and it is
calculated as [67]

$\eta=\frac{|D(\pi)|}{\binom{N}{2}},$ (6.5)

where $|\cdot|$ indicates cardinality and $D(\pi)$ is the set of displacement
vectors of $\pi$ defined as

$D(\pi)=\\{(j-i,\pi(j)-\pi(i))|0\leq i<j<N\\}.$ (6.6)

The final classical interleaver we will consider for quantum error correction
is the JPL interleaver [67], recommended in the “CCSDS Recommendation for
Telemetry Channel Coding” standard [155] because it has good spreading and
dispersion parameters. The permutations that define this family of
interleavers are constructed by applying the following algorithm:

  * •

Factorize the length of the interleaver $N=k_{1}k_{2}$, where $k_{1}=8$
usually.

  * •

For $s=1$ to $s=N$ do

    1. 1.

$m=(s-1)\bmod{2}$

    2. 2.

$i=\lfloor\frac{s-1}{2k_{2}}\rfloor$

    3. 3.

$j=\lfloor\frac{s-1}{2}\rfloor-ik_{2}$

    4. 4.

$t=(19i+1)\bmod{\frac{k_{1}}{2}}$

    5. 5.

$q=t\bmod{8}+1$

    6. 6.

$c=(p_{q}j+21m)\bmod{k_{2}}$

    7. 7.

$\pi(s)=2(t+c\frac{k_{1}}{2}+1)-m$

where $p_{q}$ is defined as the primes
$p_{1}=31,p_{2}=37,p_{3}=43,p_{4}=47,p_{5}=53,p_{6}=59,p_{7}=61,p_{8}=67$.

####  6.3 Numerical Results

In this section we present Monte Carlo simulations (see Appendix C for
details) to asses the performance of quantum turbo codes when using the
interleavers proposed in Section 6.2.1. To that end, the EXIT chart optimized
QCCs defined by the parameters in Table 6.1 have been used. The interleaver
length is $N=3000$, so that the input quantum information consists of $1000$
qubits. As in [58], a maximum of $15$ iterations has been set for the turbo
decoder. The operational figure of merit selected in order to evaluate the
performance of these quantum error correction schemes is the Word Error Rate.

The following interleavers have been implemented:

  1. 1.

$S$-random interleaver with parameter $S=25$,

  2. 2.

Welch-Costas interleaver with parameter $\alpha=2987$ and

  3. 3.

JPL interleaver.

The spread ($S$) and dispersion ($\eta$) parameters of such designed
interleavers are indicated in Table 6.2.

Name |  Spread ($S$) |  Dispersion ($\eta$)
---|---|---
Random [67] |  $1$ | $\approx 0.81$
$S$-random |  $25$ | $0.8136$
Welch-Costas |  $1$ | $1$
JPL |  $16$ | $0.35$
Table 6.2: Spread and dispersion parameters.

Figure 6.2 shows the results of the Monte Carlo simulations for the different
QTC schemes in the depolarizing channel. It can be seen that the performance
of all QTCs is similar in the turbo-cliff region, achieving a gap666Note that
the gap is calculated in a similar fashion as in [58], that is, the distance
between the QTC convergence threshold $p=0.35$ and the EA Hashing limit is
taken as $10\log_{10}(0.3779/0.35)=0.3$ dB. to the EA Hashing limit of $0.3$
dB, which is also the gap obtained by the random interleaver QTCs of [58].
However, different behaviors can be observed in the error floor region. As
depicted in Figure 6.2, both the JPL and the $S$-random interleavers present a
much lower error floor than that of random interleaving. Specifically, the
error floor of the original ramdomly interleaved QTCs is of the order of
$10^{-2}$, while the error floor of the $S$-random interleaver is around
$10^{-4}$ and that of the JPL is of order of $10^{-3}-10^{-4}$. Notice that
the error floor of the JPL interleaver presents a larger slope than that of
the other interleavers, achieving levels close to those of the $S$-random
interleaver when the channel quality improves. This means that quantum error
correction systems using $S$-random and JPL interleavers present error floors
around two orders of magnitude better than those obtained with random
interleavers.

Figure 6.2: Word error rate (WER) performance curves for the $1/9$-QTCs in
Table 6.1 when different interleaving methods are used.

Figure 6.2 shows that the performance of the Welch-Costas interleaver is
similar to that of the baseline random interleaver. However, the benefit of
using the Welch-Costas interleaver is that its memory requirements are much
smaller, as the permutation is defined just by parameters $N$ and $\alpha$. On
the other hand, the random (and $S$-random) permutations must be stored
completely. This means that using the Welch-Costas interleaver is a good
option for systems with strict storage requirements. The memory requirements
for the JPL interleaver are also lower than for the random and $S$-random
ones, as the permutation can be defined by parameters $N$, $k_{1}$ and the
primes $p_{q}$, while its performance is better than that of random
interleavers but worse than that of $S$-random ones. Welch-Costas interleavers
do have less memory requirements than JPL interleavers, but their performance
is far worse in the error floor region. Table 6.3 summarizes the memory
requirements for each of the interleavers. Note that both random and
$S$-random interleavers require a memory consumption that increases linearly
with $N$, while the memory requirements for the JPL and Welch-Costas
interleavers are constant. Regarding the overall complexity of the decoding
algorithm we have corroborated that it is practically the same for all
interleavers.

Name | Parameters | Storage Requirements
---|---|---
Random [67] |  $\pi$ | $N$
$S$-random |  $\pi$ | $N$
Welch-Costas |  $N$,$\alpha$ | $2$
JPL |  $N$,$k_{1}$,$p_{q}$ | $10$
Table 6.3: Memory requirements.

As explained in [67], the spread parameter is related to the number of
distinct low weight error patterns that are possible to appear in the
decoding, while the dispersion is related to the multiplicities of those
existing low weight errors. Consequently, high spread is desired to avoid
those harmful error sequences, while a dispersion parameter that is close to
that of a random interleaver ($\eta\approx 0.81$) will mean that those harmful
errors will happen less regularly. This is why in classical turbo coding it is
known that to achieve good error correction performance in the error floor
region, an interleaver must have a high spread parameter and a dispersion
parameter similar to the dispersion of random permutations [67]. The results
presented in Figure 6.2 show that such rationale also applies to the QTC
scenario.

####  6.4 Discussion

We have adapted three interleaving design methods used for classical turbo
codes to the quantum domain, demonstrating the importance of interleaver
construction in quantum turbo codes. By using Monte Carlo simulations, we have
shown that these constructions do enhance the error correction capability of
quantum codes in the error floor region, leading to error rate improvements of
up to two orders of magnitude. We have also shown 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 EXIT
optimized QTCs.

###  CHAPTER 7 QTCs without channel state information

Decoding for QTCs typically assumes that perfect knowledge about the channel
is available at the decoder, i.e. such a decoder has perfect channel state
information (CSI). However, in realistic systems, such information must be
estimated, and thus, there exists a mismatch between the true channel
information and the estimated one. The effect of channel mismatch on QECCs has
been studied for quantum low density parity check (QLDPC) codes in [72, 73],
which showed that such codes are pretty insensitive to the errors introduced
by the imperfect estimation of the depolarizing probability and proposed
methods to improve the performance of such QECCs when the depolarizing
probability is estimated. For classical parallel turbo codes, several studies
showed their insensitivity to SNR mismatch [74, 75, 76, 77]. In such works, it
was found that the performance loss of parallel turbo codes is small if the
estimated SNR is above the actual value of the channel. However, QTCs can only
be constructed as serially concatenated convolutional codes, and classical
studies about channel information mismatch for such codes [78, 79] showed that
such structures are more susceptible to channel identification mismatch,
suffering a significant performance loss if the SNR is either overestimated or
underestimated. This strongly suggests that (serially concatenated) QTCs will
be more sensitive to CSI mismatch.

In this section, we first analyze the performance loss incurred by QTCs due to
the channel’s depolarizing probability mismatch. Then, we study how different
off-line estimation protocols affect the overall QTCs’ performance and provide
some heuristic guidelines to selecting the number of quantum probes required
to have an estimate of the depolarizing channel (Clifford twirl approximation)
that keeps the performance degradation of such codes low. Moreover, we propose
an on-line estimation procedure by utilizing a modified version of the turbo
decoding algorithm that allows, at each decoding iteration, estimating the
channel information to be fed to the inner SISO decoder. Finally, we propose
an extension of the on-line estimation method presented for the depolarizing
channel to cover the more general case of the Pauli channel with asymmetries
(Pauli twirl approximation).

####  7.1 Quantum Turbo Decoder Performance with Depolarizing Probability
Mismatch

In this section, we study the performance sensitivity of quantum turbo codes
when transmitting over the depolarizing channel, assuming that their turbo
decoders have a mismatched value, $\hat{p}$, of the actual channel
depolarizing probability, ${p}$. To that end, Monte Carlo computer simulations
of the entanglement-assisted QTC presented in section 6 have been carried out.

Figure 7.1 shows the WER versus the mismatched depolarizing probability,
$\hat{p}$, fed to the turbo decoder. Six curves are plotted in the same
figure, corresponding to six different depolarizing channels with
probabilities $p\in\\{0.35,0.34,0.33,0.32,0.31,0.30\\}$. Figure 7.1 shows the
strong relationship between the mismatch sensitivity and the actual channel
depolarizing probability, ${p}$. The smaller ${p}$ is, the wider the flat111We
define the flat region as the zone in the sensitivity curve where the WER
performance is not considerably degraded as a consequence of the mismatch.
region is. As expected, the ability of a QTC to correct errors improved as the
depolarizing probability of the channel, ${p}$, decreased. Comparing these
results to the mismatched sensitivity work presented in [72, 73] for QLDPCs,
we can conclude that QLDPCs were more robust against channel mismatch than
quantum turbo codes, since QLDPCs had a behavior flatter than that of QTCs
when the depolarizing probability was under-estimated. This type of behavior
can be explained by comparing QTCs with classical turbo codes. In the realm of
classical turbo codes, the same kind of222Note that in the classical realm,
mismatch means that the noise variance of the channel has to be estimated.
behavior is encountered for parallel and serial turbo codes [74, 75, 76, 77].
That is, classical parallel turbo codes present a sensitive region flatter
than serial ones [78, 79] when the variance of the AWGN channel is
underestimated, which may be explained by the fact that in serial
configurations, contrary to what happens in parallel turbo codes, the channel
state information is only used in the inner decoding stage, and thus, errors
due to mismatch only appear at the first decoding stage and propagate to the
outer decoder in the turbo decoder.

Figure 7.1: WER variation for QTCs using an EXIT-random interleaver as a
function of the mismatched depolarizing probability, $\hat{p}$. Each one of
the curves corresponds to a value of the actual channel depolarizing
probability. The QTCs considered here have an entanglement consumption rate of
$6/9$.

#####  7.1.1 Interleaver Impact

In Chapter 6, we have shown that the selection of the interleaver plays a
central role in the design of QTCs due to their ability to reduce the error
floors in such codes. This raises the question of whether different classes of
interleavers may also influence the sensitivity behavior of the QTC under
channel mismatch. To investigate this possibility, the previous simulation was
also carried out for the following two interleavers, which according to Figure
6.2 were the ones with the lowest error floors:

  * •

$S$-random interleaver with parameter $S=25$ and

  * •

JPL interleaver.

Figure 7.2: WER variation for the QTCs with three different interleavers as a
function of the mismatched depolarizing probability $\hat{p}$. The true
depolarizing probabilities selected are $p=0.33$ and $p=0.30$.

The next set of simulations was performed by setting the depolarizing channel
probability to $p=0.33$ and $p=0.30$, which by looking at the curves in Figure
6.2 were the values of $p$ where the WERs started to diverge in the waterfall
region and where the WERs lied in the error floor region, respectively. The
results are plotted in Figure 7.2. Note that in the same figure, we also
included the curves corresponding to the EXIT-random interleaver previously
plotted in Figure 7.1. Observe that the curves of WER versus $\hat{p}$ were
very similar for the all three interleavers under consideration as the flat
region for each of them had approximately the same width. It is true that for
$p=0.30$, both the JPL and the $S$-random interleavers presented a lower WER
in the flat region, but that was due to the fact that their error floor was
lower. Therefore, we conclude that even though the selection of the
interleaver had a significant impact on the resulting error floor of the QTC,
the influence that interleavers had on the mismatch sensitivity is negligible.

The explanation for this invariance of the mismatch sensitivity to the
interleaver choice is that the depolarizing probability was used only in the
inner SISO decoder. This means that the extra errors that appeared in the
decoding due to the mismatched depolarizing probability will be generated in
the inner part of the turbo decoder, and they will propagate through the
interleaver to the outer convolutional decoder. Consequently, even if using
the $S$-random or the JPL interleaver means a higher error correction
capability in the error floor region, the sensitivity of the overall decoder
to channel mismatch, defined by the inner decoder, will not be affected by
interleaver choice.

####  7.2 Estimating the Depolarizing Probability

The problem of quantum channel identification (quantum process tomography
[156]) is of fundamental relevance for several quantum information processing
tasks. In the context of quantum error correction, it is often assumed that
the decoder knows exactly the quantum channel. Unfortunately, such an
assumption may not be valid as the channel is usually unknown. Consequently,
the QTC decoder should be provided with an estimate of the channel. In this
case, since the quantum channel acting on an arbitrary quantum state $\rho$
was the depolarizing channel, $\mathcal{N}_{\mathrm{D}}$, the decoder of the
QTC should be fed with an estimate of the channel depolarizing probability,
$p$. It will be assumed that $p$ remains constant for at least the duration of
a QTC channel block.

#####  7.2.1 Off-Line Estimation Framework

In this framework, the channel depolarizing probability is estimated before
block transmission can begin. Note that if the channel remains constant during
the transmission of all blocks (i.e., the quantum channel is unknown but
constant) then the QTC rate loss will be asymptotically negligible as the
number of transmitted blocks increases. On other hand, if the channel varies
at every transmitted block (but remains constant during a block), i.e. the
TVQC model discussed in Chapters 4 and 5, this off-line estimation method
substantially reduces the overall rate of the QTC.

######  7.2.1.1 Quantum Channel Estimation

In general, quantum channel identification requires suitably prepared quantum
states to be subjected to the operation of the quantum channel, $\Gamma(p)$,
[157]. Figure 7.3 shows the general estimation scheme used for channel
identification. It used $N$ equal quantum probes, $\sigma$, that were then
passed through unitary operators, $U_{i}$, while being submitted to the action
of the target quantum channel $m$ times. The output quantum state,
$\sigma_{f}(p)$, which depends on the parameter $p$, was then measured qubit-
wise to obtain the classical information $(x_{1},\cdots,x_{N})$ that will be
used to get the estimate $\hat{p}=p_{est}(x_{1},\cdots,x_{N})$.

Figure 7.3: General channel identification method for a $p$ parameter
dependent quantum channel $\Gamma(p)$ [157]. For the scenario of this section,
$\Gamma(p)=\mathcal{N}_{\mathrm{D}}$. This general scenario uses $n$ quantum
probes $\sigma$ and $m$ channel invocations. $U_{1},\cdots,U_{m+1}$ are
parameter independent transformations that can be effectively selected to
estimate $p$. $p_{est}$ is an estimator function that uses the classical
information $(x_{1},\cdots,x_{N})$ obtained from measuring the output quantum
state $\sigma_{f}(p)$ in order to get the estimate $\hat{p}$.

Since the estimation of ${p}$ depends on the measurements obtained from the
state $\sigma_{f}(p)$ and such measurements in quantum physics are
statistically distributed, the estimated depolarization probability,
$\hat{p}$, will be a random variable. Consequently, the aim of channel
identification is to chose a procedure that yields estimations of $p$ with
minimal fluctuations around the actual parameter value. The way to quantify
these fluctuations is by computing the variance of the estimation,
$\mathrm{var}(\hat{p})=\mathrm{E}\\{(\hat{p}-p)^{2}\\}$, where it is assumed
that the estimator is unbiased, i.e., $\mathrm{E}\\{\hat{p}\\}=p$. Obviously,
the variance will depend on the actual choice of the estimator used. For the
purposes of this section, we will assume that the estimation scheme achieves
the information-theoretical optimal performance, that is the quantum Cramér-
Rao bound. In that sense, the results in this section will be bounds for off-
line channel estimation.

The quantum Cramér-Rao bound states that the variance on any estimator is
bounded below as [157]:

$\mathrm{var}(\hat{p})\geq\frac{1}{J(p)},$ (7.1)

where $J(p)$ is the quantum Fisher information, which depends only on the
output quantum state, $\sigma_{f}(p)$, but neither on the choice of
measurement333Note that the classical Fisher information $F(p)$ depends on the
choice of measurement, but as $F(p)\leq J(p)$, the Cramér-Rao bound can be
rewritten as a function of the measurement-independent quantum Fisher
information [157]. $(x_{1},\cdots,x_{N})$, nor on the applied estimation
scheme (refer to Figure 7.3). It is calculated as:

$J(p)=\mathrm{Tr}[\sigma_{f}(p)\hat{L}^{2}(p)],$ (7.2)

where $\hat{L}(p)$ is the symmetric logarithmic derivative (SLD) defined
implicitly via:

$\frac{\partial\sigma_{f}(p)}{\partial
p}=\frac{1}{2}[\hat{L}(p)\sigma_{f}(p)+\sigma_{f}(p)\hat{L}(p)].$ (7.3)

Quite generally, there exists a measurement procedure that asymptotically
attains the quantum Cramér-Rao bound [158]. From the fact that $\sigma_{f}(p)$
depends only on the resources used to test the channel, the same dependence
will hold for $J(p)$, and in turn for the quantum Cramér-Rao bound. Note that
there are many possible combinations of these resources, which will lead to
different Fisher informations [157, 159], but for the purposes of this study,
we will only consider some of them.

  1. 1.

Number of channel invocations: For our encoding-decoding system in Figure 6.1,
sequential channel invocations are not considered. The reason is that once a
quantum state goes through the operation of the depolarization channel, it
cannot be sent again through the channel. Therefore, the number of channel
invocations will be set to $m=1$.

  2. 2.

Unitary $\hat{U}_{1}$ transformation: The goal of the unitary transformation,
$\hat{U}_{1}$, applied to the $N$ input quantum probes, $\sigma$, is to
introduce correlations among the quantum probes. In the particular case where
the transformation is diagonal, it results in independent instances of the
quantum probes, i.e., in an independent channel use protocol. Figure 7.4 shows
such an estimation protocol.

Furthermore, if $N$ is set to one, the above scheme reduces to the simplest
estimation protocol, called single-qubit, single-channel (SQSC). If we denote
by $J_{1}(p)$ the Fisher information of this SQSC estimation protocol, then
for any $N$ greater than one, it can be shown [157] that the corresponding
overall Fisher information $J_{N}(p)$ is given by $J_{N}(p)=NJ_{1}(p)$.
Therefore, for $N$ channel invocations, the quantum Cramér-Rao bound is:

$\mathrm{var}(\hat{p})\geq\frac{1}{NJ_{1}(p)},$ (7.4)

so that the variance bound decreases linearly with $N$. In this section, we
will use the estimation protocol shown in Figure 7.4.

Figure 7.4: Estimation protocol used in this section. $N$ independent channel
invocations are performed in order to estimate the value of $p$.

  3. 3.

Input probe $\sigma$: Two different state probes $\sigma$ are considered444We
consider noiseless probes that are only affected by the depolarizing channel.
Research about constructing robust quantum probe states to face such adverse
noise has been addressed in [160, 161]..

     * •

Unentangled pure states: The Fisher information for unentangled pure states as
probes has been calculated in [162] to be:

$J_{1}(p)=\frac{9}{8p(3-2p)}\;\;\mbox{and}\;\;J_{N}(p)=N\frac{9}{8p(3-2p)}.$
(7.5)

It should be mentioned that for SQSC protocols with no entanglement available,
Eq. 7.5 is the largest value that the Fisher information can attain. In other
words, under pure quantum state probes, the scheme in Figure 7.4 is optimal.

     * •

Maximally entangled pure states: When entanglement is available, maximally
entangled pure states or EPR pairs,
$|\Phi^{+}\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$, can be used as probes for
the depolarizing channel. It can be shown that if just one of the qubits of
$|\Phi^{+}\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ is transformed by the
depolarizing channel (i.e., the EPR pair goes through an extended channel
$\mathcal{N}_{\mathrm{D}}\otimes\mathrm{id}$), then the corresponding Fisher
information is [159] :

$J_{1}(p)=\frac{9}{16p(1-p)}\;\;\mbox{and}\;\;J_{N}(p)=N\frac{9}{16p(1-p)}.$
(7.6)

Note that this type of protocol requires that one of the entangled qubits is
not affected by noise. This is not an issue in our scenario, since the codes
we consider are entanglement assisted (there is pre-shared entanglement
between the coder and the decoder), and thus, this protocol is suitable for
the estimation of the depolarizing probability. It can be shown that the
Fisher information value in Eq. 7.6, higher than in Eq. 7.5 due to the
entanglement, is the largest value that can be achieved by SQSC estimation
protocols for the depolarizing channel [157].

Although the Fisher information given by the maximally entangled probes Eq.
7.6 is larger than the one for pure states Eq. 7.5, entanglement consumption
is an expensive resource, and thus, a tradeoff between performance and
implementation complexity exists for this kind of estimation protocol.

######  7.2.1.2 Computation of the Average Word Error Rate

Based on the results of Section 7.1 regarding the variation of the WER as a
function of the mismatched depolarizing probability $\mathrm{WER}(\hat{p})$
(refer to Figure 7.1), one can obtain the average $\tilde{\mathrm{WER}}(p)$
versus the actual depolarization probability $p$ of the channel as:

$\tilde{\mathrm{WER}}(p)=\int\mathrm{WER}(\hat{p})\mathrm{P}(\hat{p})d\hat{p},$
(7.7)

where $\mathrm{P}(\hat{p})$ is the probability density function of the
estimator being considered555Note that both $\mathrm{WER}(\hat{p})$ and
$\mathrm{P}(\hat{p})$ are functions of the true depolarizing probability $p$..
As in [72], this probability density will be assumed to be a truncated normal
distribution between zero and one with the mean equal to the true depolarizing
probability $p$ and variance $[J_{N}(p)]^{-1}$, that is,
$\hat{p}\sim\mathcal{GN}_{[0,1]}(p,[J_{N}(p)]^{-1})$. The selection of the
inverse of the Fisher information as the variance of the distribution of the
estimator, $\mathrm{P}(\hat{p})$, will give us a bound on the best performance
that the system will asymptotically attain. We followed the reasoning of
[157], that is, as the Fisher information is asymptotically achievable, this
will quantify the best possible accuracy of the quantum estimation process,
giving us a bound on the performance.

Figure 7.5 shows the performance results of the QTCs considered in this
section versus $p$ as a function of the number of probes, $N$, used. In
particular, Figure 7.5a and 7.5b show the results for pure state probes and
for maximally entangled state probes, respectively. In both cases, the
estimation protocol converged after $N\approx 1000$. However, for $N=100$, the
protocol based on EPR pairs converged faster than the pure protocol. This
agrees with the fact that the variance of the entanglement probe estimator was
smaller than the variance of the pure probe estimator (refer to Expressions
Eq. 7.5 and Eq. 7.6). The drawback is that pre-shared entanglement was needed,
which is an expensive resource.

(a) Pure quantum probes.

(b) Maximally-entangled quantum probes (EPR).

Figure 7.5: WER degradation for the QTCs as a function of the number of
probes, $N$, used for the initial depolarizing probability estimation. Note
that the number of probes used applies in the asymptotic setting. (a)
Estimation using pure quantum probes. (b) Estimation using maximally-entangled
quantum probes (EPR pairs).

#####  7.2.2 On-Line Estimation Framework

In this section, we propose a modified version of the turbo decoding algorithm
presented in Section 6.1 (described in detail in [36]) that will allow
estimating the depolarizing probability jointly with the decoding process.
Contrary to what happened in the off-line scheme considered in Section 7.2.1,
this on-line estimator has the advantage of not requiring channel estimation
before transmission begins. Thus, latency increase and rate reduction are
avoided even when the quantum channel is block-to-block time-varying, such as
the TVQC model discussed in Chapters 4 and 5.

As mentioned in Section 6.1, at each iteration $j$ of the turbo decoder, the
inner decoder computes the probability distribution666Note that
$\mathrm{P}^{e}_{2}(\mathcal{L}_{2})$ also depends on iteration $j$, but such
dependency is usually omitted for simplicity.
$\mathrm{P}^{e}_{2}(\mathcal{L}_{2})$ of the logical errors $\mathcal{L}_{2}$
associated with the inner convolutional code and the probability distribution
$\mathrm{P}^{(j)}(\mathcal{E}_{2}^{i}|\bar{r}_{2})$ that the $i^{\text{th}}$
qubit of the transmitted block will endure a physical error conditioned on the
measured syndrome. Note that although $\mathrm{P}_{2}^{e}(\mathcal{L}_{2})$ is
sent to the outer SISO decoder through the interleaver,
$\mathrm{P}^{(j)}(\mathcal{E}_{2}^{i}|\bar{r}_{2})$ is left unused. Such
information may effectively be used to estimate the value of the depolarizing
probability of the channel at decoding iteration $j$ as:

$\hat{p}^{(j)}=1-\frac{1}{n}\sum_{i=1}^{n}\mathrm{P}^{(j)}(\mathcal{E}_{2}^{i}=\mathrm{I}|\bar{r}_{2}),$
(7.8)

where $n$ is the blocklength transmitted through the channel and $\mathrm{I}$
is the identity operator.

Once $\hat{p}^{(j)}$ is obtained, it is fed to the inner SISO decoder.

Under the low WER decoding operation, it is expected that the sequence of
estimates $\\{\hat{p}^{(j)}\\}$ will converge to the actual depolarization
probability of the channel $p$. This procedure is schematically represented in
Figure 7.6.

Figure 7.6: Modified QTC decoder to perform on-line estimation of the
depolarizing probability. The figure only presents the inner SISO part of the
decoder, as the rest of the turbo decoder remains unchanged.

A question that remains to be answered is what criterion one should follow to
provide the initial value of the depolarization probability, $\hat{p}^{(1)}$,
to the inner SISO decoder. The selection of the initial value may affect the
convergence rate to the final value of $p$, and in turn the decoder operation.
Figure 7.7 presents the results when the initial value, $\hat{p}^{(1)}$, is
equal to the hashing limit of the QTC, i.e., $\hat{p}^{(1)}=p^{*}$. The
rationale behind this selection is that no QECC operating over a channel with
a hashing limit $p^{*}$ will be able to operate at low WER whenever the
depolarization probability $p$ is larger than $p^{*}$, and thus,
initialization with $\hat{p}^{(1)}>p^{*}$ makes no sense in principle. In our
case, $p^{*}=0.3779$ (refer to Figure 6.2).

Figure 7.7 shows the QTC performance in terms of WER versus $p$ when using the
proposed on-line estimation method with initial value
$\hat{p}^{(1)}=p^{*}=0.3779$. For comparison purposes, the figure also shows
the WER achieved when perfect channel information is available to the inner
SISO decoder. Notice that there was no WER degradation when using on-line
estimation, since the turbo code with the on-line estimation of the
depolarizing probability achieved the same error correction capability as when
the channel information was perfectly available at the decoder. Importantly,
this performance was achieved without requiring extra qubits and, therefore,
without decreasing the rate of the QTC, since the information used for this
on-line protocol was based on information that the SISO decoding algorithm
already calculated.

Figure 7.7: WER performance for the QTCs when the proposed on-line estimation
procedure for the depolarizing probability is utilized with initial value
$\hat{p}^{(1)}=p^{*}=0.3779$. For comparison purposes, the figure also shows
the WER achieved when perfect channel information is available.

In order to assess the effect of the choice of the initial value of the
depolarizing probability, Figure 7.8 depicts the WER performance when
$\hat{p}^{(1)}$ is chosen as $\hat{p}$ in the $x$-axis. Each curve corresponds
to a different actual value of the depolarizing probability, $p$. For
comparison purposes, the curves in Figure 7.1 are also depicted here (in that
case, $\hat{p}$ is not estimated jointly with the decoding process). Notice
the excellent performance achieved by the on-line estimation method
independently of the initial value $\hat{p}^{(1)}$, which manifests by the
much flatter nature of the curves obtained by the on-line estimation method
when compared with those of Figure 7.1.

Figure 7.8: WER variation for the QTCs as a function of the initial value of
the depolarizing probability, $\hat{p}^{(1)}$. The continuous lines represent
the sensitivity of the original turbo decoder [36] (no estimation of the
depolarizing probability is performed, and the decoder always utilizes
$\hat{p}^{(1)}$), while the dashed lines represent the sensitivity of the
modified decoder using the proposed on-line estimation method.

####  7.3 On-line estimation method for the Pauli channel

In this section, we describe a modified on-line estimation algorithm for the
asymmetric Pauli channel. Furthermore, we present results that support the
claim that the proposed method successfully achieves the task at hand.

#####  7.3.1 On-line method for the Pauli channel

Here, we consider the PTA channel obatined by Pauli twirling the combined
amplitude and phase damping channel (see Chapter 2). Thus, the on-line
estimation method for such asymmetric Pauli channel should estimate the
channel probabilities, $\hat{p}_{g}^{(j)}$, with
$g\in\\{\mathrm{X},\mathrm{Y},\mathrm{Z}\\}$ at each iteration $j$. Following
the same reasoning as for the depolarizing channel, we propose the following
estimator:

$\hat{p}_{g}^{(j)}=\frac{1}{n}\sum_{i=1}^{n}\mathrm{P}^{j}(\mathcal{E}_{2}^{i}=g|\bar{r}_{2}),$
(7.9)

where $g\in\\{\mathrm{X},\mathrm{Y},\mathrm{Z}\\}$. Once such values are
obtained, they are used to derive the a priori channel probability
distribution $\hat{\mathrm{P}}_{ch}(\mathcal{E}_{2})$, which is fed back to
the inner SISO decoder for the next iteration of the turbo decoding sum-
product algorithm, similar to Figure 7.6.

Note that in Eq. 7.9, each of the probabilities associated with each of the
non-identity Pauli matrices is calculated independently. One might wonder why
not to use the fact that for the PTA channel being considered the
probabilities for bit-flip and bit-and-phase-flip are equal, and thus to force
the estimator to estimate
$\hat{p}_{\mathrm{x}}^{(j)}=\hat{p}_{\mathrm{y}}^{(j)}$ by calculating the
mean of both. In principle, this would lead to better performance, as
information about the channel structure would be indirectly fed to the
decoder. However, as it will be seen in the simulation results, the proposed
decoder achieves the same performance as a decoder that has knowledge of the
channel parameters. Consequently, the only benefit that might be obtained by
applying such restriction in the estimation procedure would be to lower the
decoding complexity. Reducing the complexity is desirable, but we consider
that the method proposed in Eq. 7.9 is more interesting, as it will be valid
for all types of Pauli channels and not just for the asymmetric model that the
PTA channel represents.

The final matter that must be established in order to appropriately define the
iterative on-line estimation algorithm is the selection of the values for the
first iteration,
$\hat{p}^{(1)}_{g},g\in\\{\mathrm{X},\mathrm{Y},\mathrm{Z}\\}$. Following the
reasoning done in the previous section for the depolarizing channel, it seems
logical to select $\hat{p}_{g}^{(1)}=\alpha p^{*}_{\alpha}/(\alpha+2)$ for
$g=\mathrm{Z}$ and $\hat{p}_{g}^{(1)}=p^{*}_{\alpha}/(\alpha+2)$ for
$g=\\{\mathrm{X},\mathrm{Y}\\}$, where $p^{*}_{\alpha}$ refers to the noise
limit for the code with channel asymmetry, $\alpha$. However, the parameter
$\alpha$ is unknown by the receiver, and so we cannot set such a starting
point for the algorithm. Consequently, we will select the value for the first
iteration as a constant for each $g$. Observing Figure 7.8, it can be inferred
that the selection of the initial values is not crucial in terms of code
performance if the number of iterations for the turbo decoding algorithm is
sufficient. In light of these results, and of the fact that the noise limit,
$p^{*}_{\alpha}$, increases777Since operating beyond the noise limit of a code
does not make sense, it is logical to select the smallest noise limit as the
starting point of all possible asymmetry coefficients $\alpha$. The smallest
noise limit is given by the symmetric case $\alpha=1$ [60]. with the value of
$\alpha$ [60], we will select the starting point of the algorithm as
$\hat{p}_{g}^{(1)}=p^{*}_{\alpha=1}/3,\forall g$. The estimated asymmetry
parameter for each iteration can be obtained as

$\hat{\alpha}^{(j)}=\frac{\hat{p}^{(j)}_{g=Z}}{\hat{p}^{(j)}_{g=X}}.$ (7.10)

The proposed estimation method is applicable for all possible values of
$\alpha$.

#####  7.3.2 Simulation Results

In this section we perform Monte Carlo simulations to asses the performance of
Quantum Turbo Codes using the proposed on-line estimation method for
asymmetric Pauli channels. For this purpose, the unassisted QTC scheme
presented in Chapter 5 has been used with a block length of $k=1000$ logical
qubits. Note that this scheme is not the same as the one used for the
depolarizing channel estimation in the previous section. The reason for doing
so is that previous literature on QTCs operating over asymmetric Pauli
channels was done for such scheme [60].

(a) $\mathrm{WER}$ performance of the QTC for $\alpha=1$ and $\alpha=10^{4}$.

(b) $\mathrm{WER}$ performance of the QTC for $\alpha=10^{2}$ and
$\alpha=10^{6}$.

Figure 7.9: $\mathrm{WER}$ performance for the QTC under consideration when
the proposed asymmetric on-line estimation procedure is utilized. The obtained
performance curves are compared to the ones obtained in [60] for the case
where the QTC decoders have knowledge of the parameters $p$ and $\alpha$.

Figure 7.9 shows the results obtained for the QTCs described in Chapter 5
using the generalized on-line estimation method presented in section 7.3.1.
The waterfall shaped curves depicted in said figure compare the performance of
the “PTO1R-PTO1R” configuration when it corrects errors of asymmetric nature
and its decoder has access to perfect channel information, to the performance
of the decoder that estimates those channel parameters while simultaneously
decoding the information. It is obvious that both methods exhibit similar
error correcting capabilities in terms of $\mathrm{WER}$ performance.
Consequently, the generalized on-line estimation method allows the QTCs to
attain excellent performance regardless of the nature of the channel. This
outcome is of significant relevance, since it means that the QTC decoder can
be applied to any Pauli channel. Therefore, no channel-specific QTC decoders
that need to know the exact noise model beforehand are required.

To finalize the study of the QTCs that use the on-line estimator to tackle the
lack of knowledge about the channel parameters, we discuss the number of
iterations required by the turbo decoder algorithm to succesfully estimate the
logical error coset. In [60], the authors claim that setting the maximum
number of iterations beyond $I_{max}=4$ produced no benefits in the decoding
process. However, the utilization of the on-line method introduced in our work
requires the decoder to execute more iterations so that the recovery operation
can be calculated correctly. This need arises because the decoder estimates
the channel parameters during every decoding iteration, which inadvertently
implies that subsequent estimates become increasingly accurate as more
decoding rounds take place. Thus, the decoder requires sufficient iterations
for the channel information to be accurate enough to produce satisfactory
estimates of the logical error coset through additional turbo decoder rounds.
For the codes considered in this section, we have observed that setting the
maximum number of iterations to less than $I_{max}=32$ implies a loss in the
$\mathrm{WER}$ performance, while increasing the parameter beyond this value
yields no benefit for the QTCs under consideration. As a result, the worst
case decoding scenario using our estimator takes $8$ times the number of
decoding rounds required by a decoder that has knowledge of the channel
parameters. However, the increase in complexity is justified by the
flexibility provided by the on-line estimation method, since if we assume that
the channel remains constant for the duration of the block, the QTC
performance will not be compromised888Here we mean that the code will not be
degraded due to the imperfect CSI. However, the performance of the code will
suffer from the effects of time-variation as seen in Chapter 5. if the nature
of the channel changes from block to block, regardless of whether the change
occurs in the gross error probability, $p$, or in the asymmetry level of the
Pauli channel, $\alpha$. Therefore, QTCs will achieve excellent performance
independently of the nature of the channel or its time evolution.

Figure 7.10: $\mathrm{WER}$ performance of the QTCs using the symmetric on-
line estimator against the performance of those QTCs using the on-line
asymmetric estimator proposed. The considered asymmetry coefficients are