# Multi-qubit time-varying quantum channels for NISQ-era superconducting
quantum processors

Josu Etxezarreta Martinez12, Patricio Fuentes25, Antonio deMarti iOlius25,
Javier Garcia-Frías3, Javier Rodríguez Fonollosa4 and Pedro M. Crespo2

1 Corresponding author<EMAIL_ADDRESS>2 Department of Basic
Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain 3
Department of Electrical and Computer Engineering, University of Delaware,
Newark, DE 19716 USA 4 Department de Teoria del Senyal i Comunicacions,
Universitat Politecnica de Catalunya, 08034 Barcelona, Spain 5 These two
authors contributed equally to this work

###### Abstract

Recent experimental studies have shown that the relaxation time ($T_{1}$) and
the dephasing time ($T_{2}$) of superconducting qubits fluctuate considerably
over time. To appropriately consider this time-varying nature of the $T_{1}$
and $T_{2}$ parameters, a new class of quantum channels, known as Time-Varying
Quantum Channels (TVQCs), has been proposed. In previous works, realizations
of multi-qubit TVQCs have been assumed to be equal for all the qubits of an
error correction block, implying that the random variables that describe the
fluctuations of $T_{1}$ and $T_{2}$ are block-to-block uncorrelated, but
qubit-wise perfectly correlated for the same block. Physically, the
fluctuations of these decoherence parameters are explained by the incoherent
coupling of the qubits with unstable near-resonant two-level-systems (TLS),
which indicates that such variations may be local to each of the qubits of the
system. This qubit-wise uncorrelated behaviour has been proven in experiments
conducted on a two-qubit system constructed with superconducting technology.
In this article, we perform a correlation analysis of the fluctuations of the
relaxation times of multi-qubit quantum processors ibmq_quito, ibmq_belem,
ibmq_lima, ibmq_santiago and ibmq_bogota. Our results show that it is
reasonable to assume that the fluctuations of the relaxation and dephasing
times of superconducting qubits are local to each of the qubits of the system.
Based on these results, we discuss the multi-qubit TVQCs when the fluctuations
of the decoherence parameters for an error correction block are qubit-wise
uncorrelated (as well as from block-to-block), a scenario we have named the
Fast Time-Varying Quantum Channel (FTVQC). Furthermore, we lower bound the
quantum capacity of general FTVQCs based on a quantity we refer to as the
ergodic quantum capacity and we prove that they both coincide for the class of
fast time-varying amplitude damping channels. Finally, we use numerical
simulations to study the performance of quantum error correction codes (QECC)
when they operate over FTVQCs. Our simulation outcomes show that QECC
performance is worse over FTVQCs than over channels that are static in time,
but this degradation is less severe than if the fluctuations are considered to
be qubit-wise fully correlated.

##  I Introduction

Fault-tolerant quantum computers have the potential to revolutionize the
fields of computing and industry as we know them [1]. However, the societal
upheaval heralded by quantum computers will only be facilitated if these
machines are equipped with Quantum Error Correction (QEC) strategies, the
primary ingredient to make these devices fault-tolerant. The purpose of QEC
methods, commonly known as QECCs, is to identify and correct the errors that
inherently corrupt quantum information. These errors are inevitable given that
they arise naturally as a result of the interaction of quantum mechanical
systems with their surrounding environment. The different physical mechanisms
through which quantum information can be corrupted are commonly amalgamated
under the term known as decoherence [2]. Understanding the way decoherence
corrupts quantum information and mathematically modeling such effects is of
paramount importance to construct QECCs that can successfully correct the
errors that occur in real quantum hardware.

The quantum noise experienced by qubits constructed as two-level coherent
quantum mechanical systems can be described accurately by considering two
different phenomena: relaxation and pure dephasing. The effects of these
mechanisms are quantified by the relaxation time, $T_{1}$, and the dephasing
time, $T_{2}$. Thermal interactions between the quantum information system and
the environment can be neglected if the temperature of the system is low
enough, which is a valid premise for state-of-the-art superconducting quantum
processors, as these devices are cooled down to temperatures in the order of
millikelvins [3, 4, 5, 6, 7, 8]. Generally, the effects of decoherence on
quantum information are modelled by means of quantum channels, which are
completely-positive trace-preserving (CPTP) linear maps betwen spaces of
operators [2]. Most of the literature on QEC assumes that the quantum noise
level experienced by the qubits in a particular system will be identical for
each quantum information processing task, independently of when the task is
performed [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. This means that the
relaxation and pure dephasing times of the qubits are assumed to be fixed and
time-invariant. However, this behaviour has been disproven in recent
experimental studies on quantum processors [3, 4, 5, 6, 7, 20, 21]. In fact,
these works have shown that $T_{1}$ and $T_{2}$ can experience time variations
of up to $50\%$ of their mean value with coefficients of variation of
approximately $25\%$. In order to correctly account for the time-varying
nature of the decoherence defining parameters of superconducting qubits,
researchers have come up with the class of time-varying quantum channels
(TVQC), a mathematical abstraction that enables the inclusion of time
fluctuations to the models of quantum noise [22, 23].

TABLE I: Pearson correlation coefficients for the $T_{1}$ measurements
obtained for the IBM quantum processors. Relaxation time is measured for two
diferent calibration cycles for each of the machines. $r_{Q_{i}Q_{j}}$ is the
obtained Pearson correlation coefficient for the relaxation times of qubit $i$
and qubit $j$ of the processor. Bootstrapping was used to determine the $95\%$
confdence intervals, presented in parentheses.

Scenario | $r_{Q_{0}Q_{1}}$ | $r_{Q_{0}Q_{2}}$ | $r_{Q_{0}Q_{3}}$ | $r_{Q_{0}Q_{4}}$ | $r_{Q_{1}Q_{2}}$
---|---|---|---|---|---
ibmq_quito S1 | $-0.207(-0.066,-0.35)$ | $0.036(-0.094,0.17)$ | $-0.287(-0.17,-0.41)$ | $-0.076(-0.194,0.049)$ | $-0.083(-0.223,0.05)$
ibmq_quito S2 | $0.096(0.022,0.167)$ | $0.154(0.054,0.256)$ | $0.53(0.444,0.6)$ | $0.133(0.052,0.26)$ | $0.024(-0.081,0.125)$
ibmq_belem S1 | $-0.002(-0.081,0.173)$ | $0.013(-0.248,0.024)$ | $-0.008(-0.047,0.059)$ | $-0.019(-0.033,0.163)$ | $-0.028(-0.085,0.03)$
ibmq_belem S2 | $-0.141(-0.23,-0.05)$ | $-0.153(-0.25,-0.057)$ | $-0.15(-0.24,-0.06)$ | $-0.38(-0.46,-0.29)$ | $0.2(0.096,0.3)$
ibmq_lima S1 | $-0.028(-0.09,0.03)$ | $-0.09(-0.16,-0.024)$ | $0.295(0.24,0.35)$ | $0.27(0.221,0.31)$ | $-0.024(-0.086,0.037)$
ibmq_lima S2 | $0.244(0.131,0.346)$ | $0.347(0.24,0.45)$ | $-0.0049(-0.125,0.113)$ | $0.044(-0.077,0.166)$ | $-0.011(-0.138,0.13)$
ibmq_santiago S1 | $-0.035(-0.175,0.11)$ | $0.1(-0.033,0.255)$ | $0.16(0.005,0.31)$ | $-0.05(-0.17,0.06)$ | $-0.035(-0.18,0.11)$
ibmq_bogota S1 | $0.15(0.006,0.29)$ | $0.093(-0.05,0.22)$ | $-0.066(-0.19,0.055)$ | $-0.11(-0.23,0.014)$ | $-0.032(-0.17,0.11)$

Scenario | $r_{Q_{1}Q_{3}}$ | $r_{Q_{1}Q_{4}}$ | $r_{Q_{2}Q_{3}}$ | $r_{Q_{2}Q_{4}}$ | $r_{Q_{3}Q_{4}}$
---|---|---|---|---|---
ibmq_quito S1 | $0.116(-0.032,0.262)$ | $0.0031(-0.128,0.138)$ | $-0.128(-0.25,0.002)$ | $-0.161(-0.275,-0.025)$ | $-0.0159(-0.154,0.133)$
ibmq_quito S2 | $0.0028(-0.078,0.08)$ | $-0.093(-0.197,0.003)$ | $0.031(-0.051,0.115)$ | $0.049(-0.0438,0.14)$ | $0.0799(-0.00441,0.159)$
ibmq_belem S1 | $0.059(10^{-5},0.117)$ | $0.297(0.233,0.35)$ | $0.069(-0.001,0.142)$ | $0.023(-0.031,0.079)$ | $-0.004(-0.055,0.04)$
ibmq_belem S2 | $0.117(0.026,0.205)$ | $0.227(0.131,0.317)$ | $0.062(-0.03,0.152)$ | $-0.027(-0.131,0.069)$ | $0.214(0.115,0.3)$
ibmq_lima S1 | $0.057(-0.006,0.11)$ | $-0.033(-0.09,0.025)$ | $0.163(0.1,0.23)$ | $0.196(0.13,0.26)$ | $0.49(0.437,0.54)$
ibmq_lima S2 | $-0.013(-0.156,0.126)$ | $-0.085(-0.2,0.03)$ | $0.047(-0.085,0.18)$ | $0.147(0.04,0.25)$ | $-0.13(-0.25,-0.008)$
ibmq_santiago S1 | $-0.66(-0.21,0.065)$ | $-0.27(-0.39,-0.14)$ | $0.162(0.017,0.3)$ | $0.063(-0.11,0.21)$ | $0.078(-0.07,0.22)$
ibmq_bogota S1 | $0.19(0.054,0.33)$ | $0.267(0.13,0.39))$ | $-0.09(-0.22,0.051)$ | $-0.086(-0.21,0.05)$ | $0.13(-0.02,0.27)$

The multi-qubit TVQCs in [22, 23] were constructed considering that the
realizations of $T_{1}$ and $T_{2}$ would be identical for all the qubits in
each particular error correction block but would change from block to block.
This assumption implies that the random variables that describe the
fluctuations of the decoherence parameters of each of the qubits are perfectly
correlated for the same block, a scenario reminiscent of the classical
block/slow fading channel. In reality, the time fluctuations of these
decoherence parameters occur because of the incoherent coupling of an ensemble
of environmentally unstable near-resonant two-level systems (TLS) with each
particular qubit [3, 4, 5, 7]. These TLS’ actually emerge due to atomic-scale
defects that are present in the Josephson Junctions (JJ) that are used to make
up the superconducting qubits [24, 25]. In light of this, it is reasonable to
assume that the origin of the time-dependent nature of $T_{1}$ and $T_{2}$ is
local to each of the constituent qubits of the quantum machine, which
ultimately means that the realizations of the random variables that describe
the fluctuations of those parameters in the TVQC model may be considered
qubit-wise uncorrelated for each particular QEC block. This intuition has been
confirmed by experiments conducted for a $2$-qubit superconducting system (See
Supplementary Note D of [3]), where the authors simultaneously measured the
relaxation times of both qubits and studied the correlation between the
obtained data for each of them.

In this article, we further cement the observation that the local TLS defects
that are responsible for the fluctuations of the decoherence parameters of
superconducting qubits are not significantly correlated and we show that these
TLS defects will only affect each of the particular qubits within which they
arise. In order to do so, we repeatedly estimate the relaxation times of the
$5$-qubit quantum processors ibmq_quito, ibmq_belem, ibmq_lima, ibmq_santiago
and ibmq_bogota and then perform a correlation analysis on the measured
fluctuations based on the Pearson correlation coefficient. We conclude that
the obtained values of the Pearson correlation coefficients are not
sufficiently high to observe significant correlation effects on the system. In
consequence, we introduce the concept of fast time-varying quantum channels
(FTVQC) as the appropriate mathematical model to describe the decoherence
effects experienced by $n$-qubit superconducting systems. In this context,
multi-qubit time varying quantum channels are then constructed with
independent realizations of the decoherence parameters from qubit-to-qubit. In
addition, we study the quantum channel capacity for the proposed family of
quantum channels. Since the FTVQC channel resembles the classical scenario of
fast fading, we discuss the ergodic quantum capacity, whose units are
expressed in logical qubits per physical qubits, as a lower bound to the
capacity of the FTVQC channels. Moreover, we prove that for the family of fast
time-varying amplitude damping channels (FTVAD), the ergodic capacity is
indeed equal to its quantum capacity. We also show that although a capacity
loss is incurred in comparison to quantum channels that are assumed to be
static, this change is not significant. Finally, we use numerical simulations
to study the performance of planar codes and quantum turbo codes (QTC) when
they operate over the FTVQC noise model. It is concluded that the performance
of the codes worsens when compared to the static quantum channels, although
this degradation is not as significant as the one codes experience over the
previously considered multi-qubit TVQCs [22, 23]. Interestingly, we observe
that the threshold of the surface codes deteriorates by a similar amount to
the quantum capacity when the FTVQC multi-qubit model is considered.

##  II Results

###  II-A Decoherence parameter fluctuations are local to each qubit

We have conducted simultaneous measurements of the relaxation times of the IBM
quantum processors ibmq_quito, ibmq_belem, ibmq_lima, ibmq_santiago and
ibmq_bogota [26] spaced out over time (See Supplementary Note 1 for a detailed
description of the experiments). All of these quantum processors are comprised
of $5$ superconducting qubits (with different architectures and connectivity).
The aim of this experiment is to verify that the fluctuations of the
decoherence parameters of each of the constituent qubits of these $5$-qubit
superconducting processors are local to the particular qubits themselves. In
this way, we want to extend the analysis undertaken in [3] for a $2$-qubit
processor to more complex Noisy Intermediate-Scale Quantum (NISQ) devices.

Table I shows the results obtained for the Pearson correlation tests we
conducted on the measured relaxation times (See Methods for the description of
the statistical analysis). Based on these outcomes, it is clear that the
$T_{1}$ fluctuations are not significantly correlated between the qubits of
the systems in the majority of the scenarios. In fact, the values of the
correlation coefficients are all below the threshold of significant
correlation (which stands at $0.6$) that is generally considered for classical
scenarios (See Methods). This holds even when considering $95\%$ confidence
intervals. These results support the hypothesis that the fluctuations of the
decoherene parameters of superconducting qubits are caused by local effects,
making it reasonable to assume that the local TLS defects responsible for
these fluctuations are not coupled amongst themselves. We must note that there
is an outlier among our results where correlation appears to be present. This
occurs for qubits $0$ and $3$ of the ibmq_quito S2 scenario, whose correlation
coefficient has a value of $0.53$ and the upper limit of the confidence
interval is set to $0.6$. However, the $0^{th}$ qubit of such scenario shows a
step size transition at the end of the experiment (See Supplementary Note 1),
an effect that may have impacted the value of the obtained Pearson
correlation. In fact, considering just the samples before the hard transition,
the results are $r_{Q_{0}Q_{3}}=0.091(-0.026,0.217)$, which are well within
the range of uncorrelated values. In light of these results, it is safe to
assume that the fluctuations of the decoherence parameters of superconducting
qubits will be due to local effects.

###  II-B Fast time-varying quantum channels

Time-varying quantum channels (TVQC) describe the coherence loss of a qubit
when the relaxation and dephasing times that describe the rate of interaction
between the qubit and its environment fluctuate as functions of time for the
same cooldown (See Methods for the description of TVQCs) [22]. The proposal of
this theoretical framework of quantum noise was motivated by the repeated
observation of such intra-cooldown stochastic behaviour of superconducting
qubit decoherence parameters in the literature [3, 4, 5, 6, 7, 20, 21].

TVQCs successfully account for the experimentally observed time-varying nature
of the decoherence experienced by single superconducting qubits. However,
quantum information processing tasks (algorithms, error correction, memories
or communications) require sets of qubits to appropriately achieve the tasks
that they are designed for. Thus, it is necessary to consider multi-qubit
time-varying quantum channels to accurately assess the impact of $T_{1}$ and
$T_{2}$ fluctuation on practical quantum computing. In [22, 23], the authors
proposed a multi-qubit time-varying quantum channel under the assumption that
the relaxation and dephasing times are constant for all the qubits inside a
particular error correction block, but vary from block to block. Thus, a
realization of the multi-qubit time-varying quantum channel for block
$m\in\mathbb{N}$ of duration $t_{\mathrm{algo}}$ can be described
mathematically as

$\begin{split}&\mathcal{N}^{(n)}(\rho,t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})=\mathcal{N}^{\otimes
n}(\rho,t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\\\
&=\sum_{\mathcal{E}\in(\\{E_{k}\\}_{k})^{\otimes
n}}\mathcal{E}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\rho\mathcal{E}^{\dagger}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}}),\end{split}$
(1)

where $t_{1}^{m}$ and $t_{2}^{m}$ refer to the realizations of the sequences
of independent random variables $\\{T_{1}^{m}(\omega)\\}_{m\in\mathbb{N}}$ and
$\\{T_{2}^{m}\\}(\omega)\\}_{j\in\mathbb{N}}$, respectively, and
$\mathcal{E}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})=\mathcal{E}_{1}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\otimes\cdots\otimes\mathcal{E}_{n-1}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\otimes\mathcal{E}_{n}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})$
with
$\mathcal{E}_{j}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\in\\{E_{k}(t_{1}^{m},t_{2}^{m},t=t_{\mathrm{algo}})\\}_{k}$
referring to the Kraus operators of the single qubit TVQCs associated to those
realizations of the decoherence parameters. $(\\{E_{k}\\}_{k})^{\otimes n}$
refers to the set of $n$-fold tensor products of the Kraus operators of the
single-qubit TVQCs. Note that, here, those Kraus operators are related to some
TVQC that depends on the relaxation and dephasing times (See Methods). This,
however, does not exclude the construction of similar time-varying quantum
channels depending on other parameters that may show similar behaviour to
$T_{1}$ and $T_{2}$.

This means that the multi-qubit channel considered in [22, 23] assumes that
the realizations of the random variables that describe the noise experienced
by each of the qubits of the system are identical, i.e, that these random
variables are perfectly correlated. As discussed in [22, 23], this model is
reminiscent of the classical slow fading scenario. For simplicity, we will
adopt this terminology to refer to the multi-qubit TVQCs that add perfectly
correlated noise to each qubit. Hence, we name these types of channels as slow
time-varying quantum channels (STVQC). If we recall our discussion in the
previous section, we now know that the STVQCs considered in [22, 23] are not
the most accurate type of multi-qubit time-varying quantum channels for the
superconducting NISQ devices we are considering in this paper. This has to do
with the fact that the origin of the decoherence parameter fluctuations of
superconducting qubits are local to each qubit.

Based on our discussions thus far, we know that the way to construct multi-
qubit time-varying quantum channels to accurately model the superconducting
hardware considered in this article is by considering that the individual
TVQCs that make up the multi-qubit channel for each QEC block,
$m\in\mathbb{N}$, are defined by sequences of random variables
$\\{T_{1}^{m^{j}}(\omega)\\}_{j=1}^{n}$ and
$\\{T_{2}^{m^{j}}(\omega)\\}_{j=1}^{n}$ whose elements are independent amongst
themselves. Thus, the realizations of the decoherence parameter random
variables will not only be independent from block to block, but also from
qubit to qubit inside a block. In this way, a realization of the multi-qubit
time-varying quantum channel for block $m\in\mathbb{N}$ of duration
$t_{\mathrm{algo}}$ can be described mathematically as

$\begin{split}&\mathcal{N}^{(n)}(\rho,\\{t_{1}^{m^{j}}\\}_{j=1}^{n},\\{t_{2}^{m^{j}}\\}_{j=1}^{n},t=t_{\mathrm{algo}})=\bigotimes_{j=1}^{n}\mathcal{N}(\rho,t_{1}^{m^{j}},t_{2}^{m^{j}},t=t_{\mathrm{algo}})\\\
&=\\!\\!\sum_{\mathcal{E}\in\bigotimes_{j=1}^{n}(\\{E_{k}(t_{1}^{m^{j}},t_{2}^{m^{j}})\\}_{k})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{E}(\\{t_{1}^{m^{j}}\\}_{j=1}^{n},\\{t_{2}^{m^{j}}\\}_{j=1}^{n})\rho\mathcal{E}^{\dagger}(\\{t_{1}^{m^{j}}\\}_{j=1}^{n},\\{t_{2}^{m^{j}}\\}_{j=1}^{n}),\end{split}$
(2)

where $\\{t_{1}^{m^{j}}\\}_{j=1}^{n}$ and $\\{t_{2}^{m^{j}}\\}_{j=1}^{n}$
refer to the realizations of the sequences of independent random variables
$\\{T_{1}^{m^{j}}(\omega)\\}_{j=1}^{n}$ and
$\\{T_{2}^{m^{j}}(\omega)\\}_{j=1}^{n}$, respectively, and
$\mathcal{E}(\\{t_{1}^{m^{j}}\\}_{j=1}^{n},\\{t_{2}^{m^{j}}\\}_{j=1}^{n})=\mathcal{E}_{1}(t_{1}^{m^{1}},t_{2}^{m^{1}})\otimes\cdots\otimes\mathcal{E}_{n-1}(t_{1}^{m^{n-1}},t_{2}^{m^{n-1}})\otimes\mathcal{E}_{n}(t_{1}^{m^{n}},t_{2}^{m^{n}})$
with
$\mathcal{E}_{j}(t_{1}^{m^{j}},t_{2}^{m^{j}})\in\\{E_{k}(t_{1}^{m^{j}},t_{2}^{m^{j}})\\}_{k}$
referring to the Kraus operators of the single qubit TVQCs associated to those
realizations of the decoherence parameters, where we have incurred in the
slight abuse of notation $E_{k}(\cdot,\cdot)=E_{k}(\cdot,\cdot,t=t_{algo})$.
Note that the sequence of random variables will also be independent from block
to block, i.e., the elements of $\\{T_{1}^{m^{j}}(\omega)\\}_{m,j},\forall
m,j\in\mathbb{N}$ and $\\{T_{2}^{m^{j}}(\omega)\\}_{m,j},\forall
m,j\in\mathbb{N}$ will be independently distributed, respectively.

In this way, each of the realizations of this multi-qubit TVQC will have a
different noise “intensity” (the actual noise operators will be the same, but
the noise level will change) for each of the qubits of the superconducting
quantum processor. This model resembles the classical scenario of fast fading
[27, 28] (See Methods). In such scenarios, the fading process changes so
quickly that each of the symbols of a transmitted codeword is subjected to a
different fading gain (where the fading gain for each of the symbols is
independent) and, thus, to different noise levels. Note that for the multi-
qubit TVQC that we are discussing, the values of the $T_{1}$ and $T_{2}$
parameters will vary slowly, but since their particular realizations are
independent from qubit to qubit, the channel actually resembles to the fast
fading scenario. Thus, we will once again borrow from the classical realm and
refer to these quantum channels as fast time-varying quantum channels (FTVQC).

In this context, we will refer to the most commonly considered construction of
multi-qubit channels as static quantum channels. A widespread assumption in
the QECC community is that all the qubits of a quantum processor experience
the same noise through time [2, 22]. This implies that all of the qubits of
the system have the same decoherence parameters, $T_{1}$ and $T_{2}$, and that
these will not vary with the passage of time. Thus, multi-qubit static
channels are constructed by evaluating the Kraus operators of the channels
(See Methods) with the mean values of those parameters. Consequently, the
static multi-qubit time-varying quantum channel for every block $\forall
m\in\mathbb{N}$ of duration $t_{\mathrm{algo}}$ is described mathematically as

$\begin{split}&\mathcal{N}^{(n)}(\rho,\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})=\mathcal{N}^{\otimes
n}(\rho,\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})\\\
&=\sum_{\mathcal{E}\in(\\{E_{k}\\}_{k})^{\otimes
n}}\mathcal{E}(\mu_{t_{1}},\mu_{t_{2}},t=t_{\mathrm{algo}})\rho\,\mathcal{E}^{\dagger}(\mu_{t_{1}},\mu_{t_{2}},t=t_{\mathrm{algo}}),\end{split}$
(3)

where it can be seen that the channel will be equal for all the QEC blocks,
since
$\mathcal{E}(\mu_{T_{1}},\mu_{T_{2}})=\mathcal{E}_{1}(\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})\otimes\cdots\otimes\mathcal{E}_{n-1}(\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})\otimes\mathcal{E}_{n}(\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})$
with
$\mathcal{E}_{j}(\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})\in\\{E_{k}(\mu_{T_{1}},\mu_{T_{2}},t=t_{\mathrm{algo}})\\}_{k}$
is independent of the block $m$. $(\\{E_{k}\\}_{k})^{\otimes n}$ denotes the
set of $n$-fold tensor products of the Kraus operators of the single-qubit
TVQCs.

###  II-C Capacity of FTVQCs

The quantum channel capacity, $C_{Q}$, for a static quantum channel
$\mathcal{N}$, is defined as the supremum of all achievable quantum coding
rates (the quantum coding rate is defined as $R_{\mathrm{Q}}=k/n$, where $k$
is the number of logical qubits and $n$ is the number of physical qubits). A
rate, $R_{Q}$, is said to be achievable for $\mathcal{N}$ if there exists a
sequence of $[[n,k]]$ quantum codes of rate $R_{Q}$ such that the probability
of error of the codes goes to zero as the blocklength $n$ of the code goes to
infinity, $n\rightarrow\infty$ (See Methods). Therefore, the quantum channel
capacity is a quantity of interest for quantum coding theorists, as it
represents the maximum rate at which QECCs can correct the effects of a
specific noise map.

For this reason, studying the quantum capacity of time-varying quantum
channels becomes a fundamental task if we are to correctly design QECCs to
operate over such decoherence models. In [23], the concept of the quantum
outage probability as the asymptotically achievable error rate was discussed
for slow time-varying quantum channels. The need for a new concept like the
outage probability stems from the fact that the quantum capacity of the
aforementioned channels is strictly zero, which makes it necessary to employ
other information theoretical limits to study this particular family of
channels. Given the similarity that exists between slow time-varying quantum
channels and the classical scenario of slow fading, it makes good sense to
adapt the outage probability (the theoretical quantity used to study classical
slow fading) to the quantum paradigm. This has resulted in the proposition of
the quantum outage probability as the most appropriate metric to study STVQCs
[23].

Let us now consider fast time-varying quantum channels (FTVQC). Because the
capacity of these channels is not strictly zero (generally), we will be able
to use it to determine the maximum coding rate over these channels. As stated
in the previous section, the elements of the sequences of random variables
$\\{T_{1}^{j}(\omega)\\}_{j=1}^{n}$ and $\\{T_{2}^{j}(\omega)\\}_{j=1}^{n}$ in
a particular QEC block are independent. From this point we will assume that
those elements are identically distributed too. The rationale behind this is
that one needs to know how the $T_{1}$ and $T_{2}$ of each of the qubits is
distributed (in general each qubit might have different means and standard
deviations) in order to analyze the capacity and to perform numerical
simulations. Under such assumption, the quantum capacity of the combined
amplitude and phase damping FTVQC will be lower bounded by (See Supplementary
Note 2 for details)

$\begin{split}C_{\mathrm{Q}}(\bar{\gamma},\bar{\lambda})&\geq
C_{\mathrm{Q}}^{\mathrm{erg}}(\bar{\gamma},\bar{\lambda})\\\
&=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}=\int\int
Q_{\mathrm{coh}}(\gamma,\lambda)p_{\gamma,\lambda}(\gamma,\lambda)d\gamma
d\lambda\\\ &=\int\int
Q_{\mathrm{coh}}(t_{\mathrm{algo}},t_{1},t_{2})p_{T_{1},T_{2}}(t_{1},t_{2})dt_{1}dt_{2},\end{split}$
(4)

where $Q_{\mathrm{coh}}$ refers to the channel coherent information (See
Methods), operator $\mathrm{E}\\{\cdot\\}$ is the mean, and
$\bar{\gamma},\bar{\lambda}$ refer to the damping and scattering probabilities
defined by the mean relaxation and dephasing times. In the last expression,
the quantum capacity is a function of $T_{1}$ and $T_{2}$, as well as of
$t_{\mathrm{algo}}$. This comes from the fact that $\gamma$ and $\lambda$ are
functions of those three parameters, and in the integral $t_{\mathrm{algo}}$
is fixed (its value can be obtained from $\bar{\gamma},\bar{\lambda}$). Note
that the quantum capacity of the combined amplitude and phase damping channel
is unknown at the time of writing [29]. This comes from the fact that the
channel is not degradable (See Methods for definitions of degradable and anti-
degradable channels) in the region where the channel coherent information is
positive, implying that it might be superadditive [29]. This unknown
superadditivity problem extends to its FTVQC version, implying that we can
only provide a bound. Since the FTVQC model resembles the classical scenario
of fast fading [27, 28] (See Methods), we will refer to the quantity
$C_{\mathrm{Q}}^{\mathrm{erg}}$ as the ergodic quantum channel capacity.

Figure 1: Quantum capacity of the FTVAD channel. The metric is calculated for
FTVADs with $c_{\mathrm{v}}(T_{1})=\\{1,25,50\\}\%$.

The time-varying amplitude damping channel (TVAD) is typically used to
describe the noise that manifests when working with $T_{1}$-limited qubits,
such as those of [3], since the pure dephasing effects suffered by this type
of qubit are negligible (the Ramsey limit, $T_{2}\approx 2T_{1}$ is
saturated). The following theorem shows how the quantum channel capacity
coincides with the ergodic capacity for this class of FTVQCs.

######  Theorem 1 (Quantum capacity of amplitude damping FTVQCs).

The quantum capacity of the amplitude damping FTVQC is equal to its ergodic
quantum channel capacity

$\begin{split}C_{\mathrm{Q}}(\bar{\gamma})=C_{\mathrm{Q}}^{\mathrm{erg}}(\bar{\gamma})&=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}\\\
&=\int Q_{\mathrm{coh}}(\gamma)p_{\gamma}(\gamma)d\gamma\\\ &=\int
Q_{\mathrm{coh}}(t_{\mathrm{algo}},t_{1})p_{T_{1}}(t_{1})dt_{1},\end{split}$
(5)

where $Q_{\mathrm{coh}}$ refers to the channel coherent information (See
Methods), operator $\mathrm{E}\\{\cdot\\}$ is the mean, and $\bar{\gamma}$
refers to the damping probabilities defined by the mean relaxation time.

The formal proof for this theorem is provided in the Supplementary Note 2, and
it is based on the additivity of the coherent information of amplitude damping
channels [30, 31, 32, 33]. Figure 1 shows the quantum capacity of the fast
TVAD as a function of the coefficient of variation of the relaxation time. In
this figure, it can be observed that the capacity of the channel changes as a
function of the coefficient of variation of the relaxation time. For low
coefficients of variation ($\approx 1\%$), the difference between the capacity
of the FTVAD channel and the quantum capacity of the static AD channel is
negligible. When the coefficient of variation increases to about
$c_{\mathrm{v}}\approx 20\%$ differences in the capacity of the FTVAD channel
and the quantum capacity of the static AD channel become greater. Note also
that for very noisy scenarios $\bar{\gamma}>0.35$, the quantum capacity of the
FTVAD channel is higher than the quantum capacity of the static AD channel.
This is a consequence of the fact that even if the mean value of the damping
probability is the anti-degradable region ($\bar{\gamma}\in(1/2,1]$) of the AD
channel, the fluctuations of the parameters imply that some of the actual
$\gamma$ will lay in the degradable region ($\gamma\in[0,1/2]$).

Figure 2: Ergodic hashing bounds for the twirled approximations of the AD
channel. The metric is calculated for $c_{\mathrm{v}}(T_{1})=\\{1,25,50\\}\%$.
a fast time-varying amplitude damping Pauli twirl approximation. b fast time-
varying amplitude damping Clifford twirl approximation.

We wrap up this discussion by analyzing the capacity of the twirled
approximated versions of the TVAPD and TVAD channels. Similarly to the
combined amplitude and phase damping channel, the quantum capacity of the
static versions of these channels is not known, as they belong to the family
of Pauli channels, which are not degradable. Actually, these channels have
been proven to have strictly superadditive coherent information for some very
noisy scenarios [30] which makes it impossible to reduce the calculation of
the quantum capacity to a single-key formula. For this reason, the so-called
hashing bound (the hashing bound is equivalent to bounding the capacity with
the coherent information of the channel) is used as a good lower bound on the
capacity of the aforementioned channels [30]. Thus, we can derive a lower
bound for the ergodic quantum capacity of the fast time-varying twirled
approximated channels as

$\begin{split}C_{\mathrm{Q}}(\mathbf{\bar{p}})&\geq
C_{\mathrm{H}}^{\mathrm{erg}}(\mathbf{\bar{p}})\\\
&=\mathrm{E}\\{C_{\mathrm{H}}(\omega)\\}\\\ &=\int
C_{\mathrm{H}}(\mathbf{p})p_{\mathbf{p}}(\mathbf{p})d\mathbf{p}\\\ &=\int\int
C_{\mathrm{H}}(t_{\mathrm{algo}},t_{1},t_{2})p_{T_{1},T_{2}}(t_{1},t_{2})dt_{1}dt_{2},\end{split}$
(6)

where $\mathbf{p}$ refers to the array
$(p_{\mathrm{x}},p_{\mathrm{y}},p_{\mathrm{z}})$ and $\mathbf{\bar{p}}$ refers
to a similar vector derived considering the mean relaxation and dephasing
times. Note that here
$C_{\mathrm{H}}(\mathbf{\bar{p}})=Q_{\mathrm{coh}}(\mathbf{\bar{p}})$. The
expression given in (6) is reduced to a single integral that depends only on
the relaxation time if the twirled versions of the amplitude damping channel
are considered. We will refer to this lower bound as the ergodic hashing
bound, $C_{\mathrm{H}}^{\mathrm{erg}}$, in order to be consistent with the
terminology used for static Pauli channels. Figure 2 presents the ergodic
hashing bounds of the twirled approximated channels of the amplitude damping
channel. These results show how the ergodic hashing bounds deviate from the
static hashing bound when the coefficient of variation of the relaxation time
increases and how both metrics coincide when $c_{\mathrm{v}}(T_{1})\approx
1\%$.

###  II-D Performance of QECCs over FTVQCs

The performance of QECCs has been shown to worsen significantly when operating
over slow time-varying quantum channels [22, 23]. In this section, we present
the outcomes of simulations we have conducted to study the performance of QTCs
and planar codes when their qubits are subjected to the effects of the FTVQC
(See Methods for the details of the Monte Carlo numerical simulations). To
conduct the simulations, we use slow and fast time-varying amplitude damping
Clifford twirl approximations (See Methods), STVADCTA and FTVADCTA
respectively, whose static counterpart is the depolarizing channel. The
asymptotic limits for error correction associated to these noise operations
will be the hashing bound and the ergodic hashing bound, respectively. In our
simulations we consider a coefficient of variation of
$c_{\mathrm{v}}(T_{1})=25\%$, a value that is typical in experimental
superconducting qubits [3, 22].

Figure 3 shows the simulation outcomes obtained for the rate $1/9$ quantum
turbo code when it operates over static channels, STVQCs and FTVQCs.

Figure 3: QTC operating over static, STVQC and FTVQC. The coefficient of
variation considered is $c_{\mathrm{v}}(T_{1})=25\%$, which is a value that is
typical in superconducting qubits [3, 22]. a (red) static performance (solid)
and hashing bound (dashed), $p^{*}=0.1602$ b (blue) STVQC performance (solid)
and quantum outage probability [23] (dashed) c (green) FTVQC performance
(solid) and ergodic hashing bound (dashed), $p^{*}_{\mathrm{erg}}=0.1545$.

The results in Figure 3 show how the performance of the QTC, assessed in terms
of of the Word Error Rate (WER), is worse over the FTVQC than over its static
couterparts. In fact, as seen by comparing the hashing bound to the ergodic
hashing bound, the loss in code performance is similar to the loss in quantum
capacity. It should be noted that the flattening of the performance curve that
QTCs suffered over STVQCs is not observed over FTVQCs. In fact, when comparing
the performance curves obtained for the FTVQC channels and those derived for
the static channels, the only difference is that the operation point of the
code, the point where the waterfall region starts, is worse (the curves
themselves have the same shape). Thus, even though the fluctuations of the
decoherence parameters also worsen the performance of the code, this loss is
much less significant if the fluctuations are local to each qubit of the
system (as is the case with the qubits that are considered in this paper) than
if they are completely correlated.

To provide further context, we have also studied the performance of
$d\in\\{3,5,7,9\\}$ planar codes over the fast multi-qubit TVQC proposed in
this paper. These results, along with the performance of the planar codes over
static channels, are presented in Figure 4. For the sake of clarity, in this
figure we have omitted the performance results of these codes when they
operate over STVQCs [22].

Figure 4: Planar codes operating over FTVQCs and static channels. The
coefficient of variation considered is $c_{\mathrm{v}}(T_{1})=25\%$, which is
a value that is typical in superconducting qubits [3, 22]. A zoom to the code
threshold region is also presented. a (red) Planar codes over static noise b
(green) Planar codes over FTVQCs.

Figure 4 shows how, in a similar manner to what was observed for QTCs, the
performance of the planar codes is slightly worse over the FTVQCs than when
considering time-invariant noise. Notably, this decrease in performance is
less significant than for QTCs, but this is reasonable, since the performance
of these planar codes over the static scenario is much worse than that of the
turbo codes in this same context. Against this backdrop, the performance of
planar codes can be said to deteriorate when the fluctuations of the
decoherence parameters are considered. However, if the fluctuations are local
to each of the qubits of the system, the loss in performance will not be
catastrophic (a phenomenon that does actually occur over the STVQCs [22]).

Generally, surface codes are benchmarked based on a metric known as the code
threshold, $p_{\mathrm{th}}$, which is defined as the physical error
probability below which increasing the code distance actually implies lowering
the $\mathrm{WER}$. In this sense, the surface code threshold is a capacity-
like metric (the fact that operating over noise levels above the threshold
does not make sense from the coding point of view is reminiscent of operating
at rates higher than the capacity) that serves to assess the true error
correction potential of this family of codes. The close-up image shown on the
top-right of Figure 4 presents the threshold for planar codes both for static
and fast time-varying scenarios. It can be observed that the planar code
threshold is $p_{\mathrm{th}}\approx 0.112$ when the static noise model is
considered, while it degrades to $p_{\mathrm{th}}\approx 0.105$ when the codes
operate over the FTVADCTA with $c_{\mathrm{v}}(T_{1})=25\%$. This slight
degradation of the code threshold is in line with the slight decrease in
quantum capacity discussed previously.

##  III Discussion

In this paper, we have discussed the way multi-qubit time-varying quantum
channels are constructed. In previous works, the fluctuations of the qubit
decoherence parameters that describe multi-qubit TVQCs were assumed to be
qubit-wise perfectly correlated in a block, and independent from block-to-
block [22, 23]. However, recent experimental results have shown that the
fluctuations of the $T_{1}$ and $T_{2}$ parameters of superconducting qubits
are caused by the coupling of the qubit with an ensemble of enviromental
unstable near-resonant two-level systems that arise from atomic-scale defects
in the Josephson Junctions that make up these supeconducting qubits [3, 4, 5,
7, 24, 25]. This means that the origin of the decoherence parameter
fluctuations is local to each particular superconducting qubit in the system,
granted that there is no coupling between the defects. The fact that the
individual-qubit decoherence parameter fluctuations are uncorrelated was
proven in [3] for a system made up of $2$ superconducting qubits. In this work
we extend this study to quantum hardware made up of $5$ superconducting qubits
by concurrently and repeatedly measuring the relaxation times of the qubits.
To do so, we have used the quantum processors ibmq_quito, ibmq_belem,
ibmq_lima, ibmq_santiago and ibmq_bogota, which can be accessed through the
cloud using the IBM Quantum Lab platform (See Supplementary Note 1 for details
about the experiments). We have studied the correlation that exists between
various measurements of $T_{1}$ values for each of the qubits in different
operational scenarios. Our results show that for most of these scenarios the
correlation coefficients are not significant enough to indicate any
correlation. Thus, we have provided further evidence to support the claims
that fluctuations of the decoherence parameters are local to each of the
superconducting qubits for the superconducting quantum hardware considered
here. Based on previous research and our own results, it is clear that
considering the fluctuations of the decoherence parameters of superconducting
qubits to be local to each particular qubit of the system is a reasonable
assumption.

Earlier, we also discussed different ways of constructing multi-qubit time-
varying quantum channels depending on the qubit-wise correlation of the
fluctuations of $T_{1}$ and $T_{2}$. We analyzed the mathematical formalism of
the previously considered multi-qubit TVQCs [22, 23], which we named slow
TVQCs due to their similarity with classical slow fading channels, and we
proposed fast TVQCs as the appropriate way to construct multi-qubit
decoherence models with parameter fluctuations that are local to each qubit.
We saw how this phenomenon of local fluctuations is present in superconducting
hardware. Moreover, and due to their similarity with classical fading
channels, we discussed the quantum capacity of FTVQCs, and proposed the
ergodic quantum capacity as a lower bound on the asymptotically achievable
error rate for QEC for these noise models. Moreover, we stated in a theorem
(the proof is in Supplementary Note 2) that the ergodic quantum capacity
coincides with the quantum capacity for the family of fast time-varying
amplitude damping channels. We computed the ergodic quantum capacity
numerically for fast time-varying amplitude damping channels and concluded
that the loss in capacity caused by decoherence parameter fluctuations is
small, similar to what happens in classical fast fading. Finally, we discussed
the performance of quantum error correction codes when the noise operation is
defined as a FTVQC by conducting numerical simulations of the performance of
quantum turbo codes and planar codes. Interestingly, our results indicated
that the word error rate of the codes worsens only slightly, similar to what
happens to the quantum capacity, and in stark contrast to the drastic
deterioration that the performance of these codes suffers over the previously
considered STVQCs, where their WER curves flatten substantially. In summary,
when operating over FTVQCs the shape of the performance curve of a QECC is the
same as in the conventional noise model in the literature (all qubits suffer
the same static noise), but its operating point is defined for lower physical
error probabilities than for the static case. We also studied the threshold of
the considered planar codes and we observed how it was slightly lowered over
FTVQCs, similar to what happens to the quantum capacity.

Another important matter that we discussed is that understanding the nature of
the qubit-to-qubit relationship of the fluctuations of the decoherence
parameters of superconducting qubits is vital in order to characterize the
real quantum noise that affects the quantum information that they encode. The
slow TVQCs that have been presented in the literature [22, 23] predict a
substantial QECC performance loss due to the time-varying behaviour of $T_{1}$
and $T_{2}$. However, we have seen that a silver lining exists when building
QEC codes in superconducting hardware. Although superconducting qubits present
substantial ($c_{\mathrm{v}}\gtrapprox 20\%$) parameter fluctuations, these
fluctuations are local to each of the elements of the processor and, thus, are
uncorrelated between qubits. While it is true that a slight performance loss
is inevitable, it is significantly milder than the deterioration predicted by
the STVQCs which assume that the fluctuations are fully qubit-wise correlated
for a QEC block. To provide specific examples, the QTC considered in this
article operates over FTVQCs at a physical error rate $\approx 7\%$ lower than
over static channels for a $\mathrm{WER}=10^{-2}$, and the threshold of the
planar codes will be $\approx 6\%$ lower over FTVQCs than over static
channels. These losses, despite being important, are much less restrictive
than the flattening effect that takes place when fully correlated fluctuations
are considered [22, 23]. These results imply that constructing qubits whose
qubit-to-TLS defect interactions are local and uncorrelated with other qubits
is critical to maintaining the excellent performance of QECCs when the
decoherence parameters of the superconducting qubits fluctuate over time. The
coupling between qubits, and the TLS defects themselves, are directly
dependent on the architecture of each particular quantum chip. Thus, the
results we present herein prove that experimentalists must consider these
effects when designing hardware, as they will play a pivotal role in
suppressing the amount of errors the hardware suffers. It must also be
mentioned that minimizing the decoherence parameter fluctuations of
superconducting qubits will also be important to obtain error correction codes
that perform as well as it would be expected based on the results obtained
using the static noise model prevalent in the literature. Another important
research topic is to study the time-varying behaviour of the decoherence
parameters that other qubit technologies (such as trapped ions, NV center
qubits or silicon spin qubits) experience. This would allow us to understand
if their noise dynamics are more accurately described by the time-varying
quantum channels discussed in this article or by the traditional static noise
models.

It is also critical to further study the fluctuations of the decoherence
parameters that superconducting qubits exhibit. It must be noted that interest
in these effects has recently picked up, especially in the experimental
research community [3, 4, 5, 7, 24, 25]. Nonetheless, more work on this topic
is needed to completely understand the time-fluctuating behaviour of
superconducting qubits and its causes. This will enable the creation of an
accurate theoretical model and, ultimately, to mitigate the impact of time-
dependant noise on quantum information. For example, in this article, we have
studied the correlations that exist between the fluctuations of the qubits of
some $5$ qubit quantum processors and we have concluded that there is no
correlation significant enough to warrant the application of slow multi-qubit
time-varying noise models. However, this might not be the case for other
quantum processors that might have other architectures or may be comprised of
more qubits. Note that the absorption of high energy particles can also
generate correlated errors in superconducting devices [34, 35, 36].
Additionally, we must also mention that the objective of our experiments was
not to obtain very accurate values of the correlation coefficients. Instead,
our goal was to verify that, for the typical case, assuming that fluctuations
are qubit-to-qubit uncorrelated is grounded on and backed up by experimental
results (note that although the obtained $95\%$ confidence intervals are wide,
they all contain negligible correlation values and prove that fluctuations are
qubit-to-qubit uncorrelated). In any case, more accurate experiments on this
topic should be conducted, as they will serve to better understand the
dynamics of multi-qubit superconducting quantum processors. If significant
correlation values are observed in future experiments, it may become necessary
to invoke TVQCs with finite correlation (perhaps by modelling these events
with Hidden Markov Models) to better represent multi-qubit time-varying noise.
Another phenomenon that must be further explored is the sharp transition of
the relaxation times that our results show for particular qubits (See
Supplementary Note 1). This behaviour must be better understood and possibly
included in the noise models if it is shown to be common for superconducting
qubits. Furthermore, note that we have considered that the decoherence
parameters of all the qubits have the same mean and standard deviation for the
derivation of the capacity and the numerical simulations. This assumption was
necessary to discuss the topics that comprise this paper, but the most
accurate FTVQC for a multi-qubit superconducting channel will most likely have
a set of parameters for each of the qubits of the processor (at least for the
NISQ devices that exist at the time of writing).

Even though we have only studied the impact of parameter fluctuations from the
point of view of quantum error correction, the time-varying decoherence models
discussed in this article will also have an impact on near term quantum
algorithm implementations on superconducting NISQ devices and on the error
mitigation techniques used to “clean” the outcomes of NISQ devices. When
implementing quantum algorithms in NISQ processors, the physical qubits that
execute the operations of these algorithms will be subjected to decoherence
(as well as gate and measurement errors) that will corrupt the desired
outcomes. So far we have seen how the decoherence mechanisms that affect
superconducting qubits are better described by FTVQCs than by STVQCs or static
channels. One way to obtain the “best” version of quantum algorithms in the
presence of fast varying quantum noise might be to allocate their resources
(qubits) as functions of the noise itself (some qubits might be able to
perform longer-lasting tasks than others). This might allow quantum software
developers to determine exactly what limits the algorithm (how many gates can
be applied before there is too much noise). Another possibility might be to
apply error mitigation techniques based on the calibration of the device in
order to post-process results and derive more accurate outputs than the raw
yield of the NISQ device. However, these techniques and simulations strongly
depend on the calibration data of the system. For this reason, fluctuations of
decoherence parameters must also be taken into account for all these design
and simulation tasks if they are be run in NISQ devices based on
superconducting technologies. In this way, it will become possible to build
better mitigation techniques and more noise resilient NISQ algorithms.

Regarding quantum information theory, additional work is still required to
fully understand the behaviour of the quantum capacity of the channels
proposed in this article. Although we have been able to prove that the ergodic
quantum capacity is actually the quantum capacity for the fast time-varying
amplitude damping channels, we have just lower bounded this quantity for the
more general fast time-varying combined amplitude and phase damping channel
that includes non-negligible pure dephasing channels. Since most of the
superconducting qubits that exist in the literature do not saturate the Ramsey
limit (the Ramsey limit refers to the scenario $T_{2}\approx 2T_{1}$), their
dynamics (including pure dephasing) are described by the latter channel.
However, as mentioned previously, it is not known if the coherent information
of the combined amplitude and phase damping channel is additive. Thus, it may
be that this quantity is superadditive, which implies that our understanding
of this topic should increase before the capacity of the FTVQC version of this
channel can be studied. This is especially relevant, since including the
fluctuations of the pure dephasing time in a quantum noise model will worsen
the capacity more than when only $T_{1}$ is considered. Additionally, the
time-varying quantum channel models discussed herein might also be adapted to
other channels (aside from the family of amplitude damping and dephasing
channels considered in this article) if the noise parameters that define them
also present behaviour similar to the relaxation and dephasing times of the
superconducting qubits.

All in all, we consider that the FTVQC model proposed in this article
describes the noise suffered by superconducting multi-qubit systems more
accurately than previously considered noise models, at least for the hardware
considered in this article. This claim is backed up by the experiments we have
conducted and by previous literature on qubit-to-TLS defect interactions.
Consequently, we expect that quantum error correction codes implemented in
these types of quantum processors will perform worse than what would be
expected based on results obtained for static quantum channels. Once more, it
is necessary for more research on the topic of decoherence parameter
fluctuation and its incorporation to the decoherence models to be conducted in
order to unveil the true performance of near- and long-term quantum error
correction codes as well as NISQ algorithms and error mitigation techniques.

##  IV Methods

###  IV-A Pearson correlation tests

In the results section we studied the statistical dependence between the
fluctuations of the relaxation times of the qubits of various different IBM
quantum processors. The objective of this analysis was to determine if the
aforementioned fluctuations were local to each particular qubit. To do so we
used the Pearson correlation coefficient, as it provides a measure of the
correlation that exists between measured sequences [37, 38, 39]. For a pair of
random variables $(\mathrm{X},\mathrm{Y})$, the sample Pearson correlation
coefficient, $r_{\mathrm{XY}}$, is defined as [37]

$\begin{split}r_{\mathrm{XY}}&=\frac{\mathrm{cov}(\mathrm{X},\mathrm{Y})}{\sigma_{\mathrm{X}}\sigma_{\mathrm{Y}}}\\\
&=\frac{n\sum_{i=1}^{n}x_{i}y_{i}-\left(\sum_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}y_{i}\right)}{\sqrt{n\sum_{i=1}^{n}x_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}\right)^{2}}\sqrt{n\sum_{i=1}^{n}y_{i}^{2}-\left(\sum_{i=1}^{n}y_{i}\right)^{2}}},\end{split}$
(7)

where $\mathrm{cov}(X,Y)$ denotes the covariance and $\sigma$ refers to the
standard deviation.

In the main text, we calculated the correlation coefficient of the measured
qubit relaxation times for each of the studied quantum processors. To ensure
the statistical significance of the obtained correlation coefficients, we
calculate the $95\%$ confidence intervals via bootstrapping [40].
Bootstrapping is a method that uses random resampling and replacement of
samples to mimic the original population from which the samples were
extracted. The bootstrap probability distribution can then be used to derive a
significance confidence interval. For the analysis we have conducted in this
paper, we obtain the $95\%$ confidence interval as the interval that
encompasses the $2.5^{th}$ and the $97.5^{th}$ percentiles of the resampled
Pearson correlation coefficient values. In this way, we can be $95\%$
confident that the correlation parameter that exists between those random
variables will fall within said confidence interval. Confidence intervals can
also be used to reject or retain the null hypothesis of a hypothesis test
[41]. Note that here we are actually performing a hypothesis test to determine
if the variables are fully correlated (null hypothesis) or if they show some
degree of correlation (alternate hypothesis). Thus, if the null hypothesis
lays in the derived confidence interval, we cannot exclude it as being the
population parameter at the chosen level of confidence.

The final component of the statistical dependence analysis we conduct herein
is an accurate interpretation of the obtained values. This allows us to
determine if significant correlations exist or not. The ranges of values for
which two parameters might be strongly correlated depends on the actual
problem (field). However, for physical sciences, there should be no doubt
about the dependence between two variables, implying that strong or
significant correlation values should be high ($|r_{\mathrm{XY}}|\gtrapprox
0.9$) [37, 38]. For low values of $r_{\mathrm{XY}}$, no considerable
dependence (weak) relationship can be concluded. Specifically, for classical
fading channels, whenever the spatial correlation coefficient values are
approximately lower than $0.6$, it is often assumed that the fading gains of
the Rayleigh channel are i.i.d. since both channels will be very similar,
i.e., the correlation is negligible [42, 43, 44].

###  IV-B Time-varying quantum channels

Time-varying quantum channels [22], $\mathcal{N}(\rho,\omega,t)$, are defined
as

$\mathcal{N}(\rho,\omega,t)=\sum_{k}E_{k}(\omega,t)\rho
E_{k}^{\dagger}(\omega,t),$ (8)

where the $E_{k}(\omega,t)$ linear operators are the so-called Kraus operators
of the operator-sum representation of a quantum channel, and are continuous-
time random processes.

Decoherence arises from a wide range of physical processes involved in the
interaction of qubits with their environment. In the context of supeconducting
technologies, the principal vehicles for decoherence are energy relaxation and
pure dephasing. The time-varying amplitude and phase damping quantum channel,
$\mathcal{N}_{\mathrm{APD}}(\rho,\omega,t)$, [22] is a model that includes
relaxation and pure dephasing effects, whose intensity (which is given by the
relaxation time, $T_{1}$, and dephasing time, $T_{2}$) varies as a function of
time. Note that whenever the Ramsey limit is saturated ($T_{2}\approx
2T_{1}$), the channel is reduced to a time-varying amplitude damping channel,
$\mathcal{N}_{\mathrm{AD}}(\rho,\omega,t)$ [22, 23, 3]. The noise level of
these quantum channels is characterized by the damping,
$\\{\gamma(\omega,t)\\}$, and scattering, $\\{\lambda(\omega,t)\\}$,
stochastic processes, which are functions of the qubit relaxation time
$\\{T_{1}(\omega,t)\\}$ and the qubit dephasing time $\\{T_{2}(\omega,t)\\}$
as

$\gamma(\omega,t)=1-e^{-\frac{t}{T_{1}(\omega,t)}}\text{ and}$ (9)

$\lambda(\omega,t)=1-e^{\frac{t}{T_{1}(\omega,t)}-\frac{2t}{T_{2}(\omega,t)}}.$
(10)

The experimental analysis presented in [3, 22] shows that $T_{1}(t,\omega)$
and $T_{2}(t,\omega)$ can be modelled by wide-sense stationary (WSS) random
processes with means $\mu_{T_{1}},\mu_{T_{2}}$, standard deviations
$\sigma_{T_{1}},\sigma_{T_{2}}$, and a stochastic coherence time,
$T_{\mathrm{c}}$ in the order of minutes. Since the processing times for
quantum algorithms and error correction rounds, $t_{\mathrm{algo}}$, are in
the order of microseconds (the surface code cycle time is estimated to be
$1\mu s$ for superconducting devices) [8, 22, 49, 50, 51],
$t_{\mathrm{algo}}\ll T_{\mathrm{c}}$, it is reasonable to assume that the
processes remain constant during the execution of a quantum algorithm. In
other words, $\\{T_{i}(\omega,t)\\}_{i=1}^{2}$ can be modelled as a set of
random variables ($t=0$ has been selected without loss of generality due to
the fact that the process is WSS.)
$\\{T_{i}(\omega)\\}_{i=1}^{2}=\\{T_{i}(t,\omega)\rvert_{t=0}\\}_{i=1}^{2},\forall
t\in[0,T],T<<T_{\mathrm{c}}$. Given that the random processes are assumed to
be Gaussian, the random variables will also be Gaussian with distributions
$\\{\mathcal{N}(\mu_{T_{i}},\sigma^{2}_{T_{i}})\\}_{i=1}^{2}$. However, since
any and all realizations of $\\{T_{i}(\omega)\\}_{i=1}^{2}$ should be
positive, they must be modelled as truncated Gaussian random variables in the
region $[0,\infty]$. Therefore, the probability density functions are modelled
as

$f_{T_{i}}(t_{i})=\begin{cases}\frac{1}{\sigma_{T_{i}}\sqrt{2\pi}}\frac{\mathrm{e}^{-\frac{(t_{i}-\mu_{T_{i}})^{2}}{2\sigma_{T_{i}}^{2}}}}{1-\mathrm{Q}\left(\frac{\mu_{T_{i}}}{\sigma_{T_{i}}}\right)}&\text{
if }t_{i}\geq 0\\\ 0&\text{ if }t_{i}<0\end{cases},$ (11)

where in the above expression, $i=\\{1,2\\}$ and $\mathrm{Q}(\cdot)$ is the
Q-function defined as

$\mathrm{Q}(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\mathrm{e}^{-\frac{x^{2}}{2}}dx.$
(12)

The twirled approximations of those time-varying quantum channels are also
interesting since they can be simulated in an efficient manner using classical
computers [2, 22]. The time-varying Pauli twirl approximation (TVPTA),
$\mathcal{N}_{\mathrm{PTA}}(\rho,\omega,t)$, is the Pauli channel [2, 22]
obtained by twirling a time-varying quantum channel by the $n$-fold Pauli
group $\mathcal{P}_{n}$. Twirling the TVAD channel will lead to the Pauli
channel (TVADPTA) described by the probabilities that each of the Pauli
matrices has of taking place. Note that in this context these probabilities
are realizations of the random processes:

$\begin{split}&p_{\mathrm{I}}(\omega,t)=1-p_{\mathrm{x}}(\omega,t)-p_{\mathrm{y}}(\omega,t)-p_{\mathrm{z}}(\omega,t),\\\
&p_{\mathrm{x}}(\omega,t)=p_{\mathrm{y}}(\omega,t)=\frac{1}{4}(1-e^{-\frac{t}{T_{1}(\omega,t)}})\text{
and}\\\
&p_{\mathrm{z}}(\omega,t)=\frac{1}{4}(1+e^{-\frac{t}{T_{1}(\omega,t)}}-2e^{-\frac{t}{2T_{1}(\omega,t)}}).\end{split}$
(13)

For the TVAPD channel, the TVAPDPTA approximation is described by the
realizations of the following stochastic processes for each of the Pauli
matrices

$\begin{split}&p_{\mathrm{I}}(\omega,t)=1-p_{\mathrm{x}}(\omega,t)-p_{\mathrm{y}}(\omega,t)-p_{\mathrm{z}}(\omega,t),\\\
&p_{\mathrm{x}}(\omega,t)=p_{\mathrm{y}}(\omega,t)=\frac{1}{4}(1-e^{-\frac{t}{T_{1}(\omega,t)}})\text{
and}\\\
&p_{\mathrm{z}}(\omega,t)=\frac{1}{4}(1+e^{-\frac{t}{T_{1}(\omega,t)}}-2e^{-\frac{t}{T_{2}(\omega,t)}}),\end{split}$
(14)

where, once again, $T_{1}(\omega,t)$ and $T_{2}(\omega,t)$ are stochastic
processes.

Another twirled channel of interest is the time-varying Clifford twirl
approximation (TVCTA), $\mathcal{N}_{\mathrm{CTA}}(\rho,\omega,t)$ [2, 22],
which for the TVAD channel will be a depolarizing channel with depolarizing
parameter

$p(\omega,t)=\frac{3}{4}-\frac{1}{4}e^{-\frac{t}{T_{1}(\omega,t)}}-\frac{1}{2}e^{-\frac{t}{2T_{1}(\omega,t)}},$
(15)

and for the TVAPD channel a depolarizing channel with depolarizing parameter

$p(\omega,t)=\frac{3}{4}-\frac{1}{4}e^{-\frac{t}{T_{1}(\omega,t)}}-\frac{1}{2}e^{-\frac{t}{T_{2}(\omega,t)}},$
(16)

where, once more, $T_{1}(\omega,t)$ and $T_{2}(\omega,t)$ are stochastic
processes.

###  IV-C Quantum channel capacity

The quantum capacity is the maximum rate at which quantum information can be
communicated/corrected over many independent uses of a noisy quantum channel.
In other words, the concept of the quantum capacity establishes the quantum
rate limit for which reliable (i.e., with a vanishing error rate) quantum
communication/correction is asymptotically possible. The definition of the
quantum capacity, often referred to as the Lloyd-Shor-Devetak (LSD) capacity,
is given by the following theorem [30, 45].

######  Theorem 2 (LSD capacity).

The quantum capacity, $C_{\mathrm{Q}}(\mathcal{N})$, of a quantum channel,
$\mathcal{N}$, is equal to the regularized coherent information of the channel

$C_{\mathrm{Q}}(\mathcal{N})=Q_{\mathrm{reg}}(\mathcal{N}),$ (17)

where

$Q_{\mathrm{reg}}(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}(\mathcal{N}^{\otimes
n}).$ (18)

The channel coherent information, $Q_{\mathrm{coh}}(\mathcal{N})$, is defined
as

$Q_{\mathrm{coh}}(\mathcal{N})=\max_{\rho}(S(\mathcal{N}(\rho))-S(\rho_{\mathrm{E}})),$
(19)

where $S$ is the von Neumann entropy and $S(\rho_{\mathrm{E}})$ measures how
much information the environment has.

For general channels, since the quantum coherent information is not additive
in general, its regularization is a complex task. Thus, there is not a general
single-key expression for the quantum capacity given in theorem 2. However,
the AD channel does have additive coherent information, and thus the
regularization results in the one-qubit channel coherent information.

The quantum capacity of an AD channel with damping parameter $\gamma\in[0,1]$
is equal to [30, 45]

$C_{\mathrm{Q}}(\gamma)=\max_{\xi\in[0,1]}\mathrm{H}_{2}((1-\gamma)\xi)-\mathrm{H}_{2}(\gamma\xi),$
(20)

whenever $\gamma\in[0,1/2]$, and zero for $\gamma\in[1/2,1]$.
$\mathrm{H}_{2}(x)$ is the binary entropy. This comes from the fact that the
AD channel is a degradable channel for $\gamma\in[0,1/2]$ and anti-degradable
for $\gamma\in[1/2,1]$. Those channels are defined as [30]

######  Definition 1 (Degradable and anti-degradable channels).

A channel $\mathcal{N}$ from system $A$ to system $B$ is said to be degradable
if there exists a CPTP map $\mathcal{D}$ from system $B$ to the environment
$E$ such that $\mathcal{N}^{c}=\mathcal{D}\circ\mathcal{N}$, where
$\mathcal{N}^{c}$ is named the complementary channel from system $A$ to the
environment $E$. Additionally, a channel $\mathcal{N}$ from system $A$ to
system $B$ is said to be anti-degradable if there exists a CPTP map
$\mathcal{D}$ from the environment $E$ to system $B$ such that
$\mathcal{N}=\mathcal{D}\circ\mathcal{N}^{c}$, where $\mathcal{N}^{c}$ is
named the complementary channel from system $A$ to the environment $E$.

Both degradable and anti-degradable channels have the property of having
additive channel coherent information [33]. Moreover, anti-degradable channels
have always vanishing quantum channel capacity [30].

An expression for the quantum capacity of the widely used Pauli channels
remains unknown since its coherent information is not additive [2, 30].
However, a lower bound that can be achieved by stabilizer codes, the hashing
bound (which equals the single-qubit coherent information of the channel),
$C_{\mathrm{H}}$, [30] is known. The reason why the quantum capacity of a
Pauli channel can be higher than the hashing bound (this is the same as saying
that the coherent information is superadditive), i.e. $C_{\mathrm{Q}}\geq
C_{\mathrm{H}}$, is the degenerate nature of quantum codes [46, 47, 48], a
quantum-exclusive phenomenon through which several distinct channel errors
affect encoded quantum states in an indistinguishable manner. In fact, the
depolarizing channel has been proven to be superadditive for vey noisy
depolarizing probabilities [30].

The hashing bound for a Pauli channel defined by the probability mass function
$\mathbf{p}=(p_{\mathrm{I}},p_{\mathrm{x}},p_{\mathrm{y}},p_{\mathrm{z}})$ is
given by [30]

$C_{\mathrm{H}}(\mathbf{p})=1-\mathrm{H}_{2}(\mathbf{p}).$ (21)

$\mathrm{H}_{2}(\mathbf{p})=-\sum_{j}p_{j}\log_{2}(p_{j})$ is the entropy in
bits of a discrete random variable with probability mass function given by
$\mathbf{p}$.

###  IV-D Classical fast fading channels

Consider the classical scenario where the received symbols, $y[m]$, are given
by

$y[m]=\alpha[m]x[m]+w[m],$ (22)

where $x[m]$ refers to the transmitted symbols, $\alpha[m]$ refers to the
fading gains, and $w[m]$ refers to i.i.d. complex additive white Gaussian
noise [28]. If the fading process has a stochastic coherence time that is
lower than the duration of a symbol, then the set of fading gains will be
given by the realization of a sequence of i.i.d. random variables. This
classical scenario is known as the fast fading channel, and has a well defined
capacity, known as the ergodic capacity, defined as [28]

$C_{\mathrm{erg}}=\mathrm{E}\\{C(\omega)\\}=\mathrm{E}\\{\log(1+|\alpha(\omega)|^{2}\mathrm{SNR})\\},$
(23)

where $\mathrm{SNR}$ refers to the signal-to-noise ratio. The intuition behind
this limit is that one can average over many independent fades of the channel
by coding over a large number of coherence time intervals. In this way, a
reliable rate given by the mean of the “instantaneous” capacities can indeed
be achieved.

###  IV-E QECC numerical simulation

Monte Carlo computer simulations of the $d\times d$ planar codes with
$d\in\\{3,5,7,9\\}$ [19, 52] and of the QTC of rate $1/9$ in [16] have been
carried out to estimate changes in their performance over various different
operational scenarios. Planar codes belong to the more general family of
surface codes [19, 52] and are $[[d^{2}+(d-1)^{2},1,d]]$ QECCs defined by the
grid length of the code $d$. A blocklength of $k=1000$ logical qubits has been
selected for the QTC, as in [15, 16].

Planar codes are decoded using a Minimum Weight Perfect Matching (MWPM)
decoder, which is implemented using the QECSIM tool [52]. The QTC is decoded
via the decoding algorithm presented in [13, 14], which combines two Soft-In
Soft-Out (SISO) decoders.

Each round of the numerical simulation is performed by generating an $N$-qubit
Pauli operator, calculating its associated syndrome, and finally running the
decoding algorithm using the syndrome as its input. Once the logical error is
estimated, it is compared with the channel error in order to decide if the
decoding round was succesful. The operational figure of merit we use to
evaluate the performance of these quantum error correction schemes is the Word
Error Rate ($\mathrm{WER}$). The WER represents the probability that at least
one qubit of the received block has been incorrectly decoded.

For the numerical Monte Carlo methods employed to estimate the $\mathrm{WER}$
of the Kitaev toric codes and the QTC, we have applied the following rule of
thumb to select the number of blocks to be transmitted, $N_{\mathrm{blocks}}$
[15, 16], as

$N_{\mathrm{blocks}}=\frac{100}{\mathrm{WER}}.$ (24)

As explained in [15, 16], under the assumption that the observed error events
are independent, this results in a $95\%$ confidence interval of about
$(0.8\hat{\mathrm{WER}},1.25\hat{\mathrm{WER}})$, where $\hat{\mathrm{WER}}$
refers to the empirically estimated value for the $\mathrm{WER}$.

##  V Data availability

The data that supports the findings of this study is available from the
corresponding authors upon reasonable request.

##  VI Code availability

The code that supports the findings of this study is available from the
corresponding authors upon reasonable request.

##  VII Acknowledgements

We acknowledge the use of IBM Quantum services for this work. The views
expressed are those of the authors, and do not reflect the official policy or
position of IBM or the IBM Quantum team. We also want to acknowledge Luis
Vitores and Iñigo Barasoain for useful discussions regarding hypothesis
testing and bootstrapping. We also want to acknowledge Bruno Sunsundegi for
numerically studying the $3\times 3$ planar code operating over the FTVQC as
part of his bachelor’s thesis.

This work was supported by the Spanish Ministry of Economy and Competitiveness
through the ADELE project (Grant No. PID2019-104958RB-C44), by the Spanish
Ministry of Science and Innovation through the proyect Few-qubit quantum
hardware, algorithms and codes, on photonic and solid-state systems
(PLEC2021-008251), by the Diputación Foral de Gipuzkoa through the DECALOQC
Project No. E 190 / 2021 (ES), by the Ministry of Economic Affairs and Digital
Transformation of the Spanish Government through the QUANTUM ENIA project call
- QUANTUM SPAIN project, and by the European Union through the Recovery,
Transformation and Resilience Plan - NextGenerationEU within the framework of
the Digital Spain 2025 Agenda. This work was funded in part by NSF Award No.
CCF-2007689.

##  VIII Author Contributions

J.E.M. conceived the research. J.E.M., P.F. and J.R.F. discussed and
implemented the experiments in the IBM machines. J.E.M. constructed the
models. J.E.M., P.M.C. and J.G.-F. proposed the asymptotical limits. J.E.M.
and A.dM.iO. performed the numerical simulations. J.E.M., P.F. and A.deM.iO.
analyzed the results and drew the conclusions. The manuscript was written by
J.E.M., P.F. and A.deM.iO., and revised by P.M.C., J.G.-F and J.R.F. The
project was supervised by J.E.M., P.M.C., J.G.-F. and J.R.F.

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## Supplementary Note 1: Intra-calibration decoherence parameter fluctuation
for the qubits of IBM quantum hardware

In the main article, we discussed the fact that the fluctuations of the
decoherence parameters of superconducting qubits are local, that is, the
random variables $T_{1}(\omega)$ and $T_{2}(\omega)$ are qubit-wise
uncorrelated. In [1], the authors proved this to be true for the relaxation
time fluctuations by using their two-qubit superconducting system. In this
supplementary note, we perform a similar analysis for five IBM $5$-qubit
superconducting processors that are accessible online: ibmq_quito, ibmq_belem,
ibmq_lima, ibmq_santiago and ibmq_bogota [2].

Qubit relaxation time, $T_{1}$, refers to the characteristic timescale at
which a qubit in an excited state, $|1\rangle$, decays to its ground state,
$|0\rangle$, caused by simple spontaneous emission. Consequently, the
experiment that is usually performed in order to estimate the parameter
$T_{1}$ of a qubit consists in collecting the statistics of the decay curve
for the probability of measuring the excited state over time, $P_{1}(t)$. This
is done by choosing a set of delay times $t_{1},\cdots,t_{n}$ and then
repeating the following protocol $N$ times for each of them [3]:

  * •

Prepare the qubit in the $|1\rangle$ state. This is usually done by exciting
the qubit in the ground state via a Pauli $\mathrm{X}$ operator.

  * •

Wait a delay time, $t_{j}$.

  * •

Measure the qubit in the computational basis ($|0\rangle,|1\rangle$).

Once the decay curve, $P_{1}(t)$, is obtained, a fit to exponential decay is
performed in order to estimate the value of the qubit relaxation time [3]. As
explained in the main text, we are interested in studying the locality of the
fluctuations of the relaxation times of some of the IBM $5$-qubit quantum
processors. Thus, we run the previously presented experiment over time,
simultaneously for the all the qubits of the systems in question. Figure 5
portrays the schematic of each of the experiments that we have conducted. We
run each experiment $4000$ times ($4000$ shots) for $20$ uniformly separated
delay times starting from a delay of $t_{1}=1\mu s$ to
$t_{20}=2*T_{1}^{cal(i)}$, where by $T_{1}^{cal(i)}$ we refer to the
relaxation time for the $i^{th}$ qubit provided by IBM for the specific
calibration cycle in which the experiments are done. Note that the calibration
data provided by IBM refers to measurements performed during that precise
calibration round. However, these values will actually fluctuate within the
calibration cycle itself, similarly to the superconducting qubits of [1].
These intra-calibration fluctuations of the relaxation parameter are precisely
what we are interested in observing.

@C=1em @R=0.5em @! & —0⟩ X Delay(t_j)
—0⟩ X Delay(t_j)
—0⟩ X Delay(t_j)
—0⟩ X Delay(t_j)
—0⟩ X Delay(t_j)

Figure 5: Schematic representation of each of the experiments done in the IBM
quantum processors in order to estimate the decay curve of each of the qubits.

As discussed in the article, we have conducted these intra-calibration $T_{1}$
measurements for the ibmq_quito, ibmq_belem, ibmq_lima, ibmq_santiago and
ibmq_bogota $5$ qubit quantum processors on different days. In Supplementary
Table II, we detail the information related to each of the scenarios that we
have tested. Note that the number of measurements for each of the considered
scenarios/processors, as well as the duration of the experiments themselves,
is different. The reason for this is that the IBM machines are calibrated at
different times with different frequencies. We observed that the ibmq_santiago
and ibmq_bogota, which are of the Falcon r4L type, are more frequently
(several times each day) calibrated than the ibmq_quito, ibmq_belem and
ibmq_lima processors (calibrated once a day approximately), which belong to
the Falcon r4T class. Since we are interested in the intra-calibration
fluctuations, the duration of those cycles is something that must be accounted
for, which leads to some scenarios having more $T_{1}$ measurements than
others. Another limitation that must be disclosed is the fact that these
machines can be accessed by the general public, resulting in queues and wait
times to run the experiments. Consequently, whenever a large number of tasks
are sent to a machine (high demand for the processor in question), our
experiments will be more spread out in time (implying that less measurements
will be made in these specific calibration cycles). To be more specific, we
have been able to run most of the experiments on days where the demand for the
IBM systems was low, approximately running the measurement of the $T_{1}$ of
the $5$ qubits of the system once every two minutes (with the exception of the
ibmq_santiago processor where the experiment was run once every four minutes).

TABLE II: Relaxation time measurement specifications for each of the scenarios. The table includes the timestamp of the beginning of the experiments and their duration, as well as the number of measurements and the calibration data provided by IBM. The calibration relaxation times are provided in $\mu s$ units. Scenario | Timestamp | # Meas. | $T_{1}^{cal(0)}$ | $T_{1}^{cal(1)}$ | $T_{1}^{cal(2)}$ | $T_{1}^{cal(3)}$ | $T_{1}^{cal(4)}$
---|---|---|---|---|---|---|---
ibmq_quito S1 | 15/04 00:00 (4 hours) | $213$ | $61.208$ | $92.812$ | $156.228$ | $90.232$ | $119.197$
ibmq_quito S2 | 27/04 12:06 (11 hours) | $401$ | $76.5$ | $109.8$ | $93.71$ | $143.49$ | $57.27$
ibmq_belem S1 | 08/04 13:55 (24 hours) | $999$ | $120.88$ | $103.85$ | $94.75$ | $88.54$ | $83.65$
ibmq_belem S2 | 26/04 15:39 (7 hours) | $401$ | $67.4$ | $77.56$ | $83.35$ | $86.44$ | $64.2$
ibmq_lima S1 | 15/04 14:21 (27 hours) | $1000$ | $136.87$ | $125.03$ | $108.45$ | $126.02$ | $21.45$
ibmq_lima S2 | 25/04 10:25 (6 hours) | $239$ | $122.99$ | $100.58$ | $73.9$ | $97.38$ | $22.74$
ibmq_santiago S1 | 18/04 01:26 (12 hours) | $200$ | $157.41$ | $138.6$ | $143.18$ | $118.58$ | $166.08$
ibmq_bogota S1 | 17/04 21:21 (4 hours) | $200$ | $123.06$ | $187.98$ | $175.79$ | $217.003$ | $168.35$

It is important to mention that conducting the relaxation time measurements
every few minutes is not actually a problem for this study. The stochastic
processes that typically define the fluctuations of $T_{1}$ were previously
studied in [1]. This work concluded that the stochastic process coherence
times (the time for which the stochastic process can be considered to be
approximately constant), $T_{c}$, are typically in the order of minutes [5].
Consequently, the stochastic process can be modeled as a random variable,
$T_{1}(\omega)$, that is considered to be constant for a time $T_{c}$. Since
we are interested in studying the correlation that exists between the random
variables of $T_{1}$ for each of the qubits of a processor, the measurements
must be performed sufficiently apart in time so that they do not belong to the
same stochastic coherence period. As discussed before, our measurements are
conducted several minutes apart from each others; hence, they are consistent
with this reasoning.

TABLE III: Sample mean relaxation times, $\hat{\mu}$, sample standard deviation, $\hat{\sigma}$, and the coefficients of variation, $c_{\mathrm{v}}=\sigma/\mu$, for each of the qubits of the considered scenarios. Qubit $j$ is labelled by $T_{1}^{(j)}$. Scenario | $c_{\mathrm{v}}(T_{1}^{(0)})\left(\frac{\hat{\sigma}_{T_{1}^{(0)}}}{\hat{\mu}_{T_{1}^{(0)}}}\right)$ | $c_{\mathrm{v}}(T_{1}^{(1)})\left(\frac{\hat{\sigma}_{T_{1}^{(1)}}}{\hat{\mu}_{T_{1}^{(1)}}}\right)$ | $c_{\mathrm{v}}(T_{1}^{(2)})\left(\frac{\hat{\sigma}_{T_{1}^{(2)}}}{\hat{\mu}_{T_{1}^{(2)}}}\right)$ | $c_{\mathrm{v}}(T_{1}^{(3)})\left(\frac{\hat{\sigma}_{T_{1}^{(3)}}}{\hat{\mu}_{T_{1}^{(3)}}}\right)$ | $c_{\mathrm{v}}(T_{1}^{(4)})\left(\frac{\hat{\sigma}_{T_{1}^{(4)}}}{\hat{\mu}_{T_{1}^{(4)}}}\right)$
---|---|---|---|---|---
ibmq_quito S1 | $15.5\%\left(\frac{11.23}{72.51}\right)$ | $19.96\%\left(\frac{11.21}{56.15}\right)$ | $22.44\%\left(\frac{21.02}{93.63}\right)$ | $12.76\%\left(\frac{12.42}{97.33}\right)$ | $7\%\left(\frac{1.73}{24.762}\right)$
ibmq_quito S2 | $30.2\%\left(\frac{17.63}{58.37}\right)$ | $15.62\%\left(\frac{10.02}{64.16}\right)$ | $24.43\%\left(\frac{16.23}{66.46}\right)$ | $26\%\left(\frac{15.61}{60.02}\right)$ | $9.39\%\left(\frac{5.4}{57.14}\right)$
ibmq_belem S1 | $13.76\%\left(\frac{12.86}{93.5}\right)$ | $12.32\%\left(\frac{11.2}{90.87}\right)$ | $7.88\%\left(\frac{6.73}{85.4}\right)$ | $11.71\%\left(\frac{12.21}{104.22}\right)$ | $20.14\%\left(\frac{13.54}{67.23}\right)$
ibmq_belem S2 | $14.03\%\left(\frac{9.1}{64.66}\right)$ | $14.93\%\left(\frac{9.26}{62.1}\right)$ | $8.9\%\left(\frac{6.38}{71.78}\right)$ | $24.38\%\left(\frac{19.6}{80.41}\right)$ | $47.41\%\left(\frac{22.79}{48.1}\right)$
ibmq_lima S1 | $31.66\%\left(\frac{26.26}{82.94}\right)$ | $9.94\%\left(\frac{11.11}{111.66}\right)$ | $26\%\left(\frac{27.54}{105.85}\right)$ | $12.04\%\left(\frac{12.81}{106.33}\right)$ | $11.8\%\left(\frac{2.52}{21.36}\right)$
ibmq_lima S2 | $13.4\%\left(\frac{13.9}{111.25}\right)$ | $11.97\%\left(\frac{12.93}{108.06}\right)$ | $16.27\%\left(\frac{9.31}{57.27}\right)$ | $19.9\%\left(\frac{18.7}{93.91}\right)$ | $5.14\%\left(\frac{1.1}{21.52}\right)$
ibmq_santiago S1 | $14.97\%\left(\frac{21.03}{140.48}\right)$ | $21.32\%\left(\frac{15.67}{73.47}\right)$ | $22.67\%\left(\frac{32.85}{144.93}\right)$ | $11.79\%\left(\frac{13.91}{117.98}\right)$ | $17.36\%\left(\frac{21.96}{126.56}\right)$
ibmq_bogota S1 | $9.69\%\left(\frac{9.71}{100.1}\right)$ | $12.91\%\left(\frac{17.2}{133.15}\right)$ | $12.64\%\left(\frac{21.14}{167.21}\right)$ | $11.95\%\left(\frac{21.26}{177.76}\right)$ | $10.9\%\left(\frac{15.64}{143.42}\right)$

Supplementary Figures 6-13 show the results of the relaxation time
measurements of the qubits of the IBM quantum processors we have considered.
The estimated mean relaxation times, as well as the estimated standard
deviations and coefficients of variation($c_{\mathrm{v}}=\sigma/\mu$) are
provided in Supplementary Table III. It can be seen that the fluctuations
exhibited by the relaxation times of the qubits of the systems are
considerable, ranging from coefficients of variation of approximately $5\%$ up
to even $47\%$. It is worth noting that the error bars (we actually do not
plot these for the sake of clarity) are not beyond $5\%$ of the estimated
data, indicating that the fluctuations are actually relevant and are not
related to errors that may have arisen due to the fitting of the data to the
relaxation decay curves [6]. We must also mention that the circuits we have
built are also affected by gate and SPAM (state preparation and measurement)
errors, which are unavoidable in current quantum hardware. However, our
experiments contain a small number of gates ($5$ $\mathrm{X}$ Pauli gates,
which present errors in the order of $10^{-4}$ [2]) and post-decay
measurements (which present errors in the order of $10^{-2}$), and the
contributions of these error sources are suppressed by the multiple executions
we are doing for each delay of each of the experiment ($4000$ shots for each
delay). Consequently, following the rationale in [3], we can say that even if
errors of this type are present in the data, their impact on the results will
be unimportant when enough shots are run.

As stated previously, the results obtained for the fluctuations of $T_{1}$ for
each of the considered scenarios are presented in Supplementary Figures 6-13.
As expected, it can be observed that the fluctuations are notable for most of
the studied cases. In fact, there are qubits in particular scenarios that
exhibit sharp transitions in the level where the relaxation time fluctuates
(See Supplementary Figures 7a, 8e and 9e). In these cases, the qubits manifest
a step-like transition at a given point in time, at which point the relaxation
time fluctuates around a different “mean” level. Notice how for scenario 8e,
the relaxation time exhibits this effect before going back to its original
level. The reason behind these sudden changes is unclear, but speculation
regarding this topic is possible. Sharp drops in the relaxation times of
qubits have previously been observed due to the absorption of high energy
particle impacts by superconducting qubits [7, 8, 9]. However, such events
result in correlated errors over the array of superconducting qubits (global
drop of the $T_{1}$ of those qubits), an effect that has not been observed in
our scenarios. We consider that further experimental study of these types of
events is necessary to fully understand what causes such sharp drops of the
relaxation times and their subsequent return to the original level.

We finish this discussion by analyzing the dephasing times of the
superconducting qubits. To obtain values for the Ramsey dephasing times,
$T_{2}^{*}$, we need to employ a similar procedure to the one used to measure
the relaxation time, albeit with a circuit that includes a Hadamard gate
(necessary to obtain a $|+\rangle$ state), the variable delay, and another
Hadamard gate applied prior to the measurement itself. If there were no
decoherence, then the obtained result would always be the $|0\rangle$ state
(Hadamard gates are unitary), but dephasing (recall that $T_{2}^{*}$ is a
combination of both relaxation and pure dephasing) will increase the
probability of measuring the $|1\rangle$ state. This probability of measuring
$1$ will have an exponentially decaying cosine shape [3], and then the
obtained results can be fitted to such a function obtaining the Ramsey
dephasing time. However, this is a significantly more nuanced experiment than
the one used to measure the relaxation time, and measuring Ramsey times in IBM
quantum processors has been shown to be unrealiable [3]. Consequently, we will
not conduct experiments to measure the Ramsey times. In any case, fluctuations
of the dephasing time have also been experimentally observed [1, 10] and
assuming that they are local to each of the qubits seems reasonable (part of
the dephasing phenomenon is caused by relaxation, which in itself is local to
each qubit).

Figure 6: $T_{1}$ measurements of the qubits of ibmq_quito S1. Figure 7:
$T_{1}$ measurements of the qubits of ibmq_quito S2. Figure 8: $T_{1}$
measurements of the qubits of ibmq_belem S1. Figure 9: $T_{1}$ measurements of
the qubits of ibmq_belem S2. Figure 10: $T_{1}$ measurements of the qubits of
ibmq_lima S1. Figure 11: $T_{1}$ measurements of the qubits of ibmq_lima S2.
Figure 12: $T_{1}$ measurements of the qubits of ibmq_santiago S1. Figure 13:
$T_{1}$ measurements of the qubits of ibmq_bogota S1.

## Supplementary Note 2: Quantum capacity of fast time-varying quantum
channels

As stated in the LSD capacity theorem in the main text (See Methods), the
quantum channel capacity $C_{\mathrm{Q}}(\mathcal{N})$ is equal to the
regularized coherent information of the channel

$C_{Q}(\mathcal{N})=Q_{\mathrm{reg}}(\mathcal{N}),$ (25)

where

$Q_{\mathrm{reg}}(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}(\mathcal{N}^{\otimes
n}).$ (26)

The channel coherent information, $Q_{\mathrm{coh}}(\mathcal{N})$, is defined
as

$Q_{\mathrm{coh}}(\mathcal{N})=\max_{\rho}(S(\mathcal{N}(\rho))-S(\rho_{\mathrm{E}})),$
(27)

where $S$ is the von Neumann entropy and $S(\rho_{\mathrm{E}})$ measures how
much information the environment has.

There is no general single-key formula to compute the regularization necessary
to calculate the quantum channel capacity. This is due to the fact that the
coherent information of the channel is not generally additive [11, 12].
However, for specific classes of degradable quantum channels, such as the
amplitude damping (AD) channel, the channel coherent information has been
shown to be additive, reducing the expression of the regularization shown
earlier to a single-letter formula:

$C_{Q}(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}(\mathcal{N}^{\otimes
n})=\lim_{n\rightarrow\infty}\frac{1}{n}nQ_{\mathrm{coh}}(\mathcal{N})=Q_{\mathrm{coh}}(\mathcal{N}),$
(28)

that is, the quantum channel capacity is actually the same as the channel
coherent information.

Let us now discuss the capacity of fast time-varying quantum channels. We can
actually use the quantum channel capacity to quantify the maximum coding rate
that can be achieved over fast time varying quantum channels. To clarify this,
let us look at how the capacity of a FTVQC is computed. We first obtain the
realizations $\\{t_{1}^{m^{j}}\\}_{j=1}^{n}$ and
$\\{t_{2}^{m^{j}}\\}_{j=1}^{n}$ for a block $m\in\mathbb{N}$ of the sequences
of the relaxation $\\{T_{1}^{m^{j}}(\omega)\\}_{j=1}^{n}$ and dephasing
$\\{T_{2}^{m^{j}}(\omega)\\}_{j=1}^{n}$ random variables and integrate them in
the FTVQC channel model (See equation (2) in the main text). Then, we can
bound the capacity of the realization of the fast time-varying quantum channel
(which will be fixed) for block $m\in\mathbb{N}$ as

$C_{\mathrm{Q}}\left(\mathcal{N}(\rho,\\{t_{1}^{m^{j}}\\}_{j=1}^{n},\\{t_{2}^{m^{j}}\\}_{j=1}^{n})\right)=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}\left(\bigotimes_{j=1}^{n}\mathcal{N}(\rho,t_{1}^{m^{j}},t_{2}^{m^{j}})\right)\geq\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}Q_{\mathrm{coh}}\left(\mathcal{N}(\rho,t_{1}^{m^{j}},t_{2}^{m^{j}})\right),$
(29)

where we incurred in the abuse of notation
$\mathcal{N}(\cdot,\cdot,\cdot)=\mathcal{N}(\cdot,\cdot,\cdot,t=t_{\mathrm{algo}})$
for simplicity. The inequality arises from the fact that the channel coherent
information might be superadditive in general, i.e.
$Q_{\mathrm{coh}}(\mathcal{N}\otimes\mathcal{M})\geq
Q_{\mathrm{coh}}(\mathcal{N})+Q_{\mathrm{coh}}(\mathcal{M})$. The combined
amplitude and phase damping channel has been proven not to be degradable for
the region where its coherent information is positive [13] and, thus, the
additivity of its coherent information remains an open question. For the Pauli
channels (the twirled approximations of the main text are included in this
family), the channel coherent information has been proven to be strictly
superadditive for some channel instances (very noisy depolarizing channel, for
example).

Moreover, note that the sequence of values $\\{t_{1}^{m^{j}}\\}_{j=1}^{n}$ and
$\\{t_{2}^{m^{j}}\\}_{j=1}^{n}$ specify the relaxation and dephasing
parameters for each one of the $n$ qubits in the $m^{\mathrm{th}}$ block.
Therefore, the channel capacity in (29) will depend on such realizations of
the decoherence parameters, which means that, once again, the channel capacity
will itself become a random variable, $C_{Q}(\omega)$. In fact, the bound we
derived will also become a random variable,
$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}Q_{\mathrm{coh}}\left(\mathcal{N}(\rho,T_{1}^{m^{j}}(\omega),T_{2}^{m^{j}}(\omega))\right)$.
However, because of the law of large numbers and due to the fact that the
elements of the sequences of random variables
$\\{T_{1}^{m^{j}}(\omega)\\}_{j=1}^{n}$ and
$\\{T_{2}^{m^{j}}(\omega)\\}_{j=1}^{n}$ are independent (this is the case for
FTVQCs, see main text) and identically distributed (refer to main text) the
following holds

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}Q_{\mathrm{coh}}\left(\mathcal{N}(\rho,t_{1}^{m^{j}},t_{2}^{m^{j}})\right)=\mathrm{E}\left\\{Q_{\mathrm{coh}}\left(\mathcal{N}(\rho,T_{1}^{m^{j}}(\omega),T_{2}^{m^{j}}(\omega)\right)\right\\}=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}.$
(30)

The above expression is true for almost all realizations of the sequences of
random variables of the decoherence parameters, or similarly, for almost all
blocks $m$. This is similar to what happens with the channel capacity of
classical fast fading channels [14], where the capacity is usually referred to
as the ergodic channel capacity. That is why, as we stated in the main text,
we will refer to the quantity in (30) as the ergodic quantum channel capacity,
$C_{\mathrm{Q}}^{\mathrm{erg}}$, and the bound in (29) can be written as

$C_{\mathrm{Q}}(\mathcal{N})\geq
C_{\mathrm{Q}}^{\mathrm{erg}}(\mathcal{N})=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}.$
(31)

As discussed previously, the bound in (31) is the best we can do for the fast
time-varying combined amplitude and phase damping channels, as well as for its
twirled approximations (which belong to the family of Pauli channels), because
it is not known if the channel coherent information of these channels is
additive, or because it is strictly superadditive for specific instances.
However, and as stated in Theorem 1 of the main text, we can actually prove
that the channel capacity coincides with the ergodic quantum capacity for the
class of fast time-varying amplitude damping channels.

######  Theorem 3 (Quantum capacity of amplitude damping FTVQCs).

The quantum capacity of the amplitude damping FTVQC is equal to its ergodic
quantum channel capacity

$\begin{split}C_{\mathrm{Q}}(\bar{\gamma})=C_{\mathrm{Q}}^{\mathrm{erg}}(\bar{\gamma})&=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}\\\
&=\int Q_{\mathrm{coh}}(\gamma)p_{\gamma}(\gamma)d\gamma\\\ &=\int
Q_{\mathrm{coh}}(t_{\mathrm{algo}},t_{1})p_{T_{1}}(t_{1})dt_{1},\end{split}$
(32)

where $Q_{\mathrm{coh}}$ refers to the channel coherent information, operator
$\mathrm{E}\\{\cdot\\}$ is the mean, and $\bar{\gamma}$ refers to the damping
probabilities defined by the mean relaxation time.

###### Proof.

We need to prove that the bound $C_{\mathrm{Q}}(\mathcal{N})\geq
C_{\mathrm{Q}}^{\mathrm{erg}}(\mathcal{N})$ is actually an equality. To do so,
we must prove that the channel coherent information in (29) is actually
additive for the family of FTVAD channels. The static amplitude damping
channel exhibits degradability for damping parameters $\gamma\in[0,1/2]$ and
antidegradability otherwise, i.e. $\gamma\in[1/2,1]$. In this way, the tensor
product of the limit

$\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}\left(\bigotimes_{j=1}^{n}\mathcal{N}(\rho,t_{1}^{m^{j}},t=t_{\mathrm{algo}})\right),$
(33)

will consist of some amplitude damping channels that are degradable and some
others that are anti-degradable. In this sense, additivity of the channel
coherent information is assured for such a combination of channels:

  * •

Degradable channels fulfill additivity, i.e.
$Q_{\mathrm{coh}}(\mathcal{N}\otimes\mathcal{M})=Q_{\mathrm{coh}}(\mathcal{N})+Q_{\mathrm{coh}}(\mathcal{M})$,
when both $\mathcal{N}$ and $\mathcal{M}$ are degradable [11].

  * •

Degradable and anti-degradable channels fulfill additivity, i.e.
$Q_{\mathrm{coh}}(\mathcal{N}\otimes\mathcal{M})=Q_{\mathrm{coh}}(\mathcal{N})+Q_{\mathrm{coh}}(\mathcal{M})$,
when $\mathcal{N}$ is degradable and $\mathcal{M}$ is anti-degradable [12,
15].

  * •

Anti-degradable channels fulfill additivity, i.e.
$Q_{\mathrm{coh}}(\mathcal{N}\otimes\mathcal{M})=Q_{\mathrm{coh}}(\mathcal{N})+Q_{\mathrm{coh}}(\mathcal{M})$,
when both $\mathcal{N}$ and $\mathcal{M}$ are anti-degradable [12, 16].

Consequently, the following holds for FTVAD channels

$\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}\left(\bigotimes_{j=1}^{n}\mathcal{N}(\rho,t_{1}^{m^{j}},t=t_{\mathrm{algo}})\right)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}Q_{\mathrm{coh}}\left(\mathcal{N}(\rho,t_{1}^{m^{j}},t=t_{\mathrm{algo}})\right).$
(34)

Finally, considering the discussion from before (See equation (30)), we can
conclude that the quantum capacity is actually equal to the ergodic quantum
capacity for FTVAD channels:

$\begin{split}C_{\mathrm{Q}}(\bar{\gamma})=C_{\mathrm{Q}}^{\mathrm{erg}}(\bar{\gamma})&=\mathrm{E}\\{Q_{\mathrm{coh}}(\omega)\\}\\\
&=\int Q_{\mathrm{coh}}(\gamma)p_{\gamma}(\gamma)d\gamma\\\ &=\int
Q_{\mathrm{coh}}(t_{\mathrm{algo}},t_{1})p_{T_{1}}(t_{1})dt_{1},\end{split}$
(35)

where in the last step we use the fact that the coherent information for AD
channels is a function of the relaxation time $T_{1}$, which will be the
random variable, as well as of the error correction cycle time (algorithm
time), $t_{\mathrm{algo}}$. $\bar{\gamma}$ refers to the damping probability
defined by the mean relaxation time, $\mu_{T_{1}}$ for the cycle time in
consideration.

∎

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