# Content Provider Contributions to Capacity Expansion of a Neutral ISP:
Effect of Private Option

Pranay Agarwal  BITS Pilani Hyderabad Campus
Hyderabad, India - 500078
<EMAIL_ADDRESS>D. Manjunath  IIT Bombay
Mumbai, India - 400076
<EMAIL_ADDRESS>
###### Abstract

Increasing content consumption by users and the expectation of a better
Internet experience requires Internet service providers (ISPs) to expand the
capacity of the access network continually. The ISPs have been demanding the
participation of the content providers (CPs) in sharing the cost of upgrading
the infrastructure. From CPs’ perspective, investing in the ISP
infrastructure, termed as _public investment_ , seems rational as it will
boost their profit. However, the CPs can alternatively invest in making
content delivery more efficient, termed as _private investment_ , as it also
boosts their profit. Thus, in this work, we investigate this trade-off between
public and private investment of the CPs for a net-neutral ISP. Specifically,
we consider centralized decision and non-cooperative forms of interaction
between CPs and an ISP and determine the optimum public and private
investments of the CPs for each model. In the non-cooperative interaction, we
find that at most one CP contributes to the public infrastructure, whereas all
invest in their private infrastructure.

###### Index Terms:

Content providers (CPs), internet service providers (ISPs), network economics,
net neutrality.

##  I Introduction

Internet users today interact with two complementary service providers—content
providers (CPs) that provide the content and access providers or ISPs that
provide the connectivity. It is easy to see that content and access are indeed
complementary goods: high demand on content causes a high demand on access
bandwidth and low bandwidth availability causes a low demand on content. In
spite of the two services being complementary, the ability of the ISPs to
monetize the user surplus has been significantly lower than that of the CPs.
It can be argued that this is due to a combination of market forces (average
revenue per user has been stubbornly constant, and low, for many years in most
markets) and regulations (e.g., net neutrality and others akin to common
carrier laws).

The asymmetric economic system against the ISPs outlined above is compounded
by the march of technology and resource demand. While the content is becoming
more demanding on the network resources, the pressure to provide that last
mile capacity is on the ISPs. Thus the ISPs have to continuously invest in the
access infrastructure to both upgrade to more modern technologies and to
expand capacity. Since the CPs appear to be getting a larger share of the
consumer surplus, there is an argument, at least from some of the ISPs, that
the CPs need to contribute to investments in the access infrastructure that
they so significantly profit from. If indeed the CPs have to pay for the
public ISP infrastructure, the natural question is to understand the incentive
structures that should be created, even as the market forces and the
regulatory structure are maintained.

###  I-A Related Literature

Early work on addressing the issue of CPs paying for the access centred around
allowing ISPs to be non neutral and provide differentiated service quality to
the CPs, e.g., [1, 2, 3]. Another strain of work considered non neutrality of
ISPs where they provide differentiated pricing via schemes like sponsored data
and zero rating, e.g., [4, 5, 6, 7]. With net neutrality laws becoming either
de jure or de facto in many markets, these do not provide us any insights on
how to make the CPs pay for access infrastructure. A widely used practice that
has emerged is for CPs, including content distribution networks (CDNs), to
peer with ISPs over a direct link and cache their content deep inside the ISP.
Such peering interconnections and caching contribute to reducing cost for ISPs
while effectively favouring the traffic from the peering ISP over those that
do not peer. This has been explored in, e.g., [8]. Another scheme proposed to
contribute to ISP revenues is a profit sharing model between the ISP and the
CPs [9]. In [10], the voluntary payment by the CPs to the neutral ISP is
treated like a game to provide common good (see, e.g., [11] for a definition)
and the total contribution is analysed under different types of games. This
paper explores that idea further by including a private investment, in
addition to the public investment like in [10], that the CPs can make to
improve the experience of their users. We develop this idea further in the
following subsection.

###  I-B Problem Setup and Preview

Consider a system with a neutral ISP and several CPs that together serve a
fixed user population. The consumption of the contents of a CP depends on the
quality of service (QoS) that users experience which in turn depends on the
quality of the access network provided by the ISP. The CPs can make a
voluntary contribution to the ISP for it to invest in improving the access
network infrastructure; an improved access network will contribute to
increased content consumption and hence increased revenues to CPs. This
voluntary contribution will be called the public investment of the CP. Since
the ISP is neutral, all the CPs can benefit from this public investment of
some of the CPs and thus there is the possibility of a tragedy of commons in
which case none of the CPs will make public investment.

In addition to collectively investing in the public infrastructure, each CP
can also individually invest in its private infrastructure. This private
investment could include, better caching, encoding techniques, better
connectivity of their servers to the user population, etc. This investment
will improve the QoS to the users of only this CP and provide a consequent
increase in its revenues.

The premise for the model is similar to that in [10] but we add the
significantly more realistic framework of the CPs also investing in their own
systems. The two-part investment decision adds significantly more complexity
to the analysis and also provides for more interesting analyses. While there
are several obvious questions in this setup, we will be interested in
analysing the resulting public investment under various strategic interactions
between the CPs.

In the next section, we describe the system model and develop some preliminary
results that will be used in later sections. In Section III, we consider a
centralised decision to choose the private and public investments of each of
the CPs to maximise the sum utilities of the CPs as collective. In Section IV,
we analyse the system when the CPs decide their investments as the outcome of
a non cooperative game and characterise the public and private investments at
Nash equilibrium. In Section V, we present some numerical results and discuss
the results. Some proofs are carried in the appendix.

##  II System Model

There are $N$ CPs and one ISP that together provide Internet services to a
fixed user population. There is a basic infrastructure that is present at the
ISP and the CPs and the interest here is in new investments that need to be
made to improve the quality of service for the users. Each CP can make a
_public investment_ towards the improvement of the infrastructure of the
neutral ISP; users of all the CPs see an improvement in the service quality
from this investment. A CP can also invest in its private infrastructure to
improve the service quality to only its own users and this is the _private
investment._ Let $q_{n}$ be the public investment and $p_{n}$ be the private
investment of CP $n,$ for all $n\in\mathcal{N}$, where
$\mathcal{N}:=\\{1,2,\ldots,N\\}$. Let $Q:=\sum_{n=1}^{N}q_{n}$ be the total
public investment and the $P:=\sum_{n=1}^{N}p_{n}$ be the total private
investment. Our objective in this paper is to evaluate the relation between
$P$ and $Q$ under different assumptions of interaction between the CPs.
Specifically, we will be interested in, among others, the measure

$\gamma:=\frac{Q}{P}$ (1)

that we call the public-private trade-off.

The public and private investments lead to an improvement in the overall QoS
experience of the users, a consequent increase in traffic, and hence, a
revenue gain to the CPs. Further, since the ISP operates in the net neutrality
regime, the gain is a function of the total public investment. A total public
investment of $Q$ and a private investment of $p_{n}$ results in an increased
consumption of the content of CP $n$ by an amount $g_{n}(Q+h_{n}(p_{n})),$
where both $g_{n}(\cdot)$ and $h_{n}(\cdot)$ are concave increasing functions.
This is a reasonable assumption because the marginal benefit from increasing
investments will decrease. Further, $h_{n}(\cdot)$ is such that the inequality

$h_{n}(p_{n})\geq p_{n}$ (2)

holds for some $p_{n}$ implying

$g_{n}(Q_{-n}+q_{n}+h(p_{n}))\geq g_{n}(Q_{-n}+q_{n}+p_{n})$

for some $p_{n}$ and $q_{n}.$ Here $Q_{-n}:=\sum_{m=1,m\neq n}^{N}q_{m}.$ This
requirement only means that there exists some $(p_{n},q_{n})$ for which a CP
will make non-zero public and private investments, a reasonable technical
assumption.

######  Remark 1.

A possible view of $h_{n}(\cdot)$ is that it is the ‘equivalent public
investment’ for a private investment of $p_{n}.$

Denoting the revenue per unit traffic for CP $n$ to be $r_{n},$ the surplus,
or the utility function for CP $n,$ denoted by $U_{n},$ is

$U_{n}(q_{n},p_{n})=r_{n}g_{n}(Q+h_{n}(p_{n}))-(p_{n}+q_{n}),\forall
n\in\mathcal{N}.$ (3)

######  Remark 2.

Since $g_{n}(\cdot)$ and $h_{n}(\cdot)$ are concave increasing, it is easy to
show that $U_{n}$ is concave in both $p_{n}$ and $q_{n}.$ This is not
surprising because the marginal revenue gains from either the public or the
private investment should eventually plateau and the net gain will become
negative.

In the following sections, we will determine the $p_{n}$ and $q_{n},$
$n\in\mathcal{N}$, when the CPs make their choices under two interaction
models—a centralized allocation and a non-cooperative game. In the process, we
also obtain the $\gamma$ and the utility functions of the different players
under these interactions. In the rest of the paper, we use the following forms
of $g_{n}(\cdot)$ and $h_{n}(\cdot)$.

$g_{n}(x)=a_{n}\log(1+x)\text{ and }h_{n}(x)=b_{n}\sqrt{x}\,\,\,\,\,\,\forall\
\ x\geq 0.$ (4)

Further, we impose that $b_{n}\geq 1$ to meet the structural requirement
discussed in (2). Substituting (4) into (3), we get

$U_{n}(q_{n},p_{n})=r_{n}a_{n}\log(1+Q+b_{n}\sqrt{p_{n}})-(p_{n}+q_{n})\,.$
(5)

##  III Centralized Allocation

First, let us consider a centralised allocation where both the $Q$ and the
$p_{n}$ are determined centrally to maximize the sum utilities of all the CPs

$U_{C}(Q,\\{p_{n}\\}):=\sum_{n=1}^{N}\left(r_{n}a_{n}\log(1+Q+b_{n}\sqrt{p_{n}})-p_{n}\right)-Q.$
(6)

This is an idealisation and will be used to get the best case scenario that
will in turn be used as a standard to compare against. The optimization
problem in this _centralized_ allocation will be

$\displaystyle\max_{Q,\\{p_{n}\\}}\ \
\sum_{n=1}^{N}(r_{n}a_{n}\log(1+Q+b_{n}\sqrt{p_{n}})-p_{n})-Q,$ s. t.
$\displaystyle p_{n}\geq 0~{}\forall n\in\\{1,\ldots,N\\},$ $\displaystyle
Q\geq 0.$ (7)

Towards solving (7), we first present the following lemma that characterizes
the optimum $p_{n}$ for a given $Q.$ Proof is omitted due to space
constraints.

######  Lemma 1.

The optimum value of $p_{n}$ for a given value of $Q$, denoted by
$p_{n}^{\ast}(Q)$, is given by

$p_{n}^{\ast}(Q)=\left(\frac{\sqrt{(1+Q)^{2}+2b_{n}^{2}r_{n}a_{n}}-(1+Q)}{2b_{n}}\right)^{2}.$
(8)

######  Remark 3.

Lemma 1 says that $p_{n}^{\ast}(Q)$ is strictly positive for all $Q\geq 0$ and
is a convex decreasing function of $Q$.

Thus, given a fixed total public investment, a CP can always increase its
utility by making private investment, however small. Further, because of the
concavity of the utility function, the marginal investment that can improve
its utility decreases with increasing $Q.$

The optimum total public investment with _centralized allocation_ , i.e., the
solution to (7), denoted by $Q_{C}^{\ast}$, is characterised by the following
lemma. Once again, we skip the proof due to space constraints.

######  Lemma 2.

$Q_{C}^{\ast}$ is obtained by solving

$\sum_{n=1}^{N}\frac{\sqrt{(1+Q)^{2}+2b_{n}^{2}r_{n}a_{n}}}{b_{n}^{2}}=(1+Q)\left(\sum_{n=1}^{N}\frac{1}{b_{n}^{2}}\right)+1.$
(9)

Lemma 2 implies

$Q_{C}^{\ast}>0\iff\sum_{n=1}^{N}\frac{\sqrt{1+2b_{n}^{2}r_{n}a_{n}}}{b_{n}^{2}}>1+\sum_{n=1}^{N}\frac{1}{b_{n}^{2}}.$
(10)

Denoting the public-private trade-off under _centralized allocation_ by
$\gamma_{C},$ we have the following theorem whose proof is provided in the
appendix.

######  Theorem 1.

For centralized allocation of $\\{p_{n}\\}$ and $Q,$

$\gamma_{C}=\frac{2Q_{C}^{\ast}}{\sum_{n=1}^{N}r_{n}a_{n}-(1+Q_{C}^{\ast})}.$
(11)

##  IV Non-Cooperative Interaction

We now consider the case when the CPs choose the $p_{n}$ and $q_{n}$
strategically where they do not coordinate and choose these quantities to
selfishly maximize their utility. We begin the analysis by first using Lemma 1
and rewriting $U_{n}$ as

$\displaystyle U_{n}$ $\displaystyle=$ $\displaystyle
r_{n}a_{n}f_{n}(Q)-p_{n}(Q)-q_{n}$ (12) $\displaystyle\mbox{where }f_{n}(Q)$
$\displaystyle=$
$\displaystyle\log\left(\frac{\sqrt{(1+Q)^{2}\\!+\\!2b_{n}^{2}r_{n}a_{n}}\\!+\\!(1\\!+\\!Q)}{2}\right).$

Since $U_{n}$ is a function of $Q=\sum_{n}q_{n},$ and the latter are the
actions of the CPs, we determine the Nash equilibrium choice of $\\{q_{n}\\}$
as follows. Let $q_{n}^{\ast}$ denote the equilibrium public investment
maximizing the utility of CP $n$. Then, a pure strategy Nash equilibrium
exists if and only if

$U_{n}(q_{n}^{\ast},Q_{-n}^{\ast})\geq U_{n}(q_{n},Q_{-n}^{\ast})$ (13)

holds for $\\{q_{n}\\}_{n\in\mathcal{N}}$ where
$Q_{-n}^{\ast}=\sum_{m\in\mathcal{N},m\neq n}q_{m}^{\ast}.$ In other words,
$\\{q_{n}^{\ast}\\}_{n\in\mathcal{N}}$ is a Nash equilibrium public investment
tuple that ensures

$q_{n}^{\ast}=\operatorname*{arg\,max}_{q_{n}}\,U_{n}(q_{n},Q_{-n}^{\ast})~{}\forall
n\in\mathcal{N}.$ (14)

Now, using $Q_{N}^{\ast}:=\sum_{n\in\mathcal{N}}q_{n}^{\ast},$ we present the
following theorem whose proof is in the appendix.

######  Theorem 2.

For non-cooperative interaction, the following holds.

  * •

If
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$, $q_{n}^{\ast}=0$ for all $n\in\mathcal{N}$, and hence, $Q_{N}^{\ast}=0$.

  * •

If
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$,
let

$\mathcal{M}:=\left\\{m\in\mathcal{N}:m=\operatorname*{arg\,max}\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\right\\}.$
(15)

Then, $q_{n}^{\ast}\geq 0$ for all $n\in\mathcal{M}$ and $q_{n}^{\ast}=0$ for
all $n\in\mathcal{N}\setminus\mathcal{M}$. Further,
$1+Q_{N}^{\ast}=\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)$.

An important conclusion from Theorem 2 is that only the CP which has the
maximum value of $r_{n}a_{n}-\frac{b_{n}^{2}}{2}$ contributes to the public
investment and the remaining CPs free-ride on this. If there more than one CP
have the maximum value of $r_{n}a_{n}-\frac{b_{n}^{2}}{2}$, i.e., some
symmetry exists between the CPs in which case $q_{n}>0$ for all such CPs. All
the CPs though invest in their private infrastructure and the optimum value of
$p_{n}$ can be obtained by substituting $Q$ by $Q_{N}^{\ast}$ in (8).

Let $\eta$ denote the price of anarchy defined as

$\eta:=\frac{Q_{C}^{\ast}}{Q_{N}^{\ast}}.$ (16)

Further, let $\gamma_{N}$ denote the public-private trade-off in the case of
the non-cooperative interaction. The following theorem provides insights on
$\eta$ and $\gamma_{N}.$

######  Theorem 3.

For _Non-Cooperative_ interaction, the following holds.

  * •

If
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$, $\eta$ is unbounded and $\gamma_{N}=0$.

  * •

If
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$,
$\eta>1$. For the special case of $|\mathcal{M}|=|\mathcal{N}|$,

$\gamma_{N}=\frac{Q_{N}^{\ast}}{\sum\limits_{n\in\mathcal{N}}\frac{b_{n}^{2}}{4}}.$
(17)

From Theorem 3 it is clear that $\eta>1.$

Let $U_{N}$ denote the sum utility of all the CPs in the non-cooperative
interaction. Akin to the derivation of (6), we obtain $U_{N}$ is given by

$U_{N}=\sum_{n\in\mathcal{N}}\left(r_{n}a_{n}\log(1+Q_{N}+b_{n}\sqrt{p_{n}})-p_{n}\right)-Q_{N}.$
(18)

Defining $\Gamma$ as

$\Gamma:=\frac{U_{C}^{\ast}}{U_{N}^{\ast}},$ (19)

we see from Theorem 3 that if
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$,
$\Gamma>1$.

##  V Numerical Results

In this section, we present some numerical results for $N=2.$ We consider
$r_{n}a_{n}=2n^{-\delta}$ for all $n=\\{1,2\\}.$ Here, $\delta$ controls the
revenue a CP earns per unit advertisement. Further, the multiplicative factor
of $2$ ensures that the optimum total public investment is non-zero for all
values of $\delta$, i.e., (10) holds for all $\delta$. Note that the revenue
of the second CP per unit advertisement diminishes as $\delta$ increases. The
optimization problem formulated in (7) is solved using Pyomo.

Fig. LABEL:fig:centralized presents the variation of the optimum total public
investment, optimum total private investment, and public-private trade-off
against different values of $\delta$ and b for centralized game in Fig.
LABEL:fig:cent_pub, LABEL:fig:cent_priv, and LABEL:fig:cent_ppt, respectively.
We observe from Fig. LABEL:fig:cent_pub that the optimum total public
investment, i.e., $Q^{\ast}$, decreases as $\delta$ increases. This follows
from the fact that the revenue of CP 2 per unit advertisement is a decreasing
function of $\delta$. Since the revenue of CP 1 per unit advertisement does
not change with $\delta$, the decrease in $Q^{\ast}$ leads to an increase in
the optimum private investment of CP 1. However, since the revenue of CP 2 per
unit advertisement decreases with $\delta$, the optimum private investment of
CP 2 decreases with $\delta$. This explains the mixed trend of the optimum
total private investment, which is the sum of the optimal private investments
of both CPs, against different values of $\delta$ as observed in Fig.
LABEL:fig:cent_priv. In the case of $\textbf{b}=[2,2]$, both the CPs are
incentivized to invest more in private than public. Therefore, we observe that
the optimal total public and private investment are lowest and highest for
$\textbf{b}=[2,2]$ in Fig. LABEL:fig:cent_pub and LABEL:fig:cent_priv,
respectively. A similar explanation holds for the highest and lowest values of
optimum total public and private investments for the case of
$\textbf{b}=[1,1]$ in Fig. LABEL:fig:cent_pub and LABEL:fig:cent_priv,
respectively. This also translates into the highest public-private trade-off
for $\textbf{b}=[1,1]$ in Fig. LABEL:fig:cent_ppt. Since in
$\textbf{b}=[1,2]$, only CP 1 prefers to invest more in public than private,
we observe from Fig. LABEL:fig:cent_pub that the $Q^{\ast}$ is higher for
$\textbf{b}=[1,2]$ than $\textbf{b}=[2,1]$. The observation of lower optimum
total private investment for $\textbf{b}=[1,2]$ than $\textbf{b}=[2,1]$ in
Fig. LABEL:fig:cent_priv also follows from the same explanation. For all the
cases except $\textbf{b}=[1,1]$, CPs have invested some share of the public
investment in the private infrastructure to maximize their surplus leading to
the reduction in the public-private trade-off.

Fig. LABEL:fig:NE illustrates price of anarchy ($\eta$), $\Gamma$, and public-
private trade-off for _Non-Cooperative Interaction_ ($\gamma_{N}$) against
different values of $\psi_{n}(=r_{n}a_{n})$ for $N=2$ and $\textbf{b}=[1,1]$
in Fig. LABEL:fig:poa, LABEL:fig:NE_ppt, and LABEL:fig:gamma, respectively.
Similarly, Fig. LABEL:fig:NE_b22 illustrates price of anarchy ($\eta$),
$\Gamma$, and public-private trade-off for _Non-Cooperative Interaction_
($\gamma_{N}$) against different values of $\psi_{n}(=r_{n}a_{n})$ for $N=2$
and $\textbf{b}=[2,2]$ in Fig. LABEL:fig:poa, LABEL:fig:NE_ppt, and
LABEL:fig:gamma, respectively. We observe from Fig. LABEL:fig:poa that $\eta$
is high for the region $(\psi_{1}\leq 1.5,\psi_{2}\leq 1.5)$. This follows
from Theorem 3 as
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$ in this region. Same holds for the region $(\psi_{1}\leq 3,\psi_{2}\leq 3)$
in Fig. LABEL:fig:poa_b22 as $\textbf{b}=[2,2]$. In the remaining region, we
observe that $\eta>1$ which is consistent with Theorem 3. When
$\psi_{1}\approx\psi_{2}$, both CPs invest in public infrastructure leading to
a higher value of $Q_{C}^{\ast}$, and hence, $\eta$. However, when there is a
large asymmetry between the CPs, only a CP makes a major investment in public
infrastructure leading to a lower value of $\eta$. Similar observations can be
made from Fig. LABEL:fig:poa_b22. However, note that the optimum total public
investment is less in the case of $\textbf{b}=[2,2]$ than $\textbf{b}=[1,1]$
for both _Centralized Allocation_ as well as _Non-Cooperative Interaction_.
This is because both the CPs are incentivized to invest more in private than
public. Thus, we observe that $\eta$ is slightly higher for $\textbf{b}=[2,2]$
than $\textbf{b}=[1,1]$. In _Centralized Allocation_ , the optimum total
public investment maximizes the net utility of all the CPs. However, in _Non-
Cooperative interaction_ , CPs reduce the net public investment by playing
selfishly. This results in lower net utility in _Non-Cooperative Interaction_
than _Centralized Allocation_ as observed in Fig. LABEL:fig:gamma. Similar
holds for the case of $\textbf{b}=[2,2]$ as observed in Fig.
LABEL:fig:gamma_b22. As $\psi_{1}$ increases, $Q_{N}^{\ast}$ increases. Thus,
the optimum total private investment decreases, and $\gamma_{N}$ increases as
observed in Fig. LABEL:fig:NE_ppt. We observe from Fig. LABEL:fig:NE_ppt and
LABEL:fig:NE_ppt_b22 that $\gamma_{N}$ significantly reduces for
$\textbf{b}=[2,2]$ than $\textbf{b}=[1,1]$. This is because both the CPs are
incentivized to invest more in private infrastructure than public for
$\textbf{b}=[2,2]$ than $\textbf{b}=[1,1]$.

##  VI Conclusions and Future Works

With increasing interest by CPs to participate in improving the access network
quality, we consider the problem of how CPs can contribute under two kinds of
interactions—the idealised centralised allocation and the more realistic the
strategic model. The key finding that in the latter case at most one CP will
contribute to the public infrastructure in an interesting finding. We expect
to investigate this further. We also could consider other models for
interactions—cooperative games and a bargaining model. It must be noted that
in the bargaining model in [10], the public investment was higher than in the
cooperative setting. It would be interesting to see if this would indeed be
the case here.

## References

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###  -A Proof of Theorem 1

###### Proof.

From Lemma 2, we know that $Q_{C}^{\ast}$ is obtained by solving (9). Further,
$p_{n}^{\ast}(Q)$ is given by (8). Thus, we have

$p_{n}^{\ast}(Q)=\frac{(1+Q)}{2b_{n}^{2}}+\frac{r_{n}a_{n}}{2}-\frac{(1+Q)\sqrt{(1+Q)^{2}+2b_{n}^{2}r_{n}a_{n}}}{2b_{n}^{2}}$

Using (9), we obtain

$\displaystyle\sum_{n=1}^{N}p_{n}^{\ast}(Q_{C}^{\ast})$
$\displaystyle=\sum_{n=1}^{N}\frac{(1+Q_{C}^{\ast})}{2b_{n}^{2}}+\sum_{n=1}^{N}\frac{r_{n}a_{n}}{2}-$
$\displaystyle\frac{(1+Q_{C}^{\ast})}{2}\left(1+(1+Q_{C}^{\ast})\sum_{n=1}^{N}\frac{1}{b_{n}^{2}}\right),$
$\displaystyle=\sum_{n=1}^{N}\frac{r_{n}a_{n}}{2}-\frac{(1+Q_{C}^{\ast})}{2}.$
(20)

Substituting (20) into (1), we obtain (11). This completes the proof. ∎

###  -B Proof of Theorem 2

###### Proof.

From (12), we obtain $\frac{\partial U_{n}}{\partial q_{n}}$ as

$\frac{\partial U_{n}}{\partial
q_{n}}=\frac{2r_{n}a_{n}}{\sqrt{(1+Q)^{2}+2b_{n}^{2}r_{n}a_{n}}+(1+Q)}-1.$
(21)

From (21), we have

$\displaystyle\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{Q=0}\leq
0\implies\frac{2r_{n}a_{n}}{\sqrt{1+2b_{n}^{2}r_{n}a_{n}}+1}\leq 1,$
$\displaystyle\implies\sqrt{1+2b_{n}^{2}r_{n}a_{n}}+1\geq 2r_{n}a_{n}\implies
r_{n}a_{n}-\frac{b_{n}^{2}}{2}\leq 1.$

Therefore, if $r_{n}a_{n}-\frac{b_{n}^{2}}{2}\leq 1$, we have
$\left.\frac{\partial U_{n}}{\partial q_{n}}\right\rvert_{Q=0}\leq 0$. If
$\max\limits_{n\in\mathcal{M}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$, we have $r_{n}a_{n}-\frac{b_{n}^{2}}{2}\leq 1$ for all $n\in\mathcal{N}$.
This in turn implies $\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{Q=0}\leq 0$ for all $n\in\mathcal{N}$. From (21), we have

$\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{\\{q_{n}=0,Q_{-n}>0\\}}<\left.\frac{\partial
U_{n}}{\partial q_{n}}\right\rvert_{\\{q_{n}=Q_{-n}=0\\}}\leq 0.$ (22)

It can be concluded from (22) that $q_{n}^{\ast}=0$ for any value of $Q_{-n}$
for all $n$.

We now consider the case when
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$.
Since
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$,
$\left.\frac{\partial U_{n}}{\partial q_{n}}\right\rvert_{Q=0}>1$ holds for at
least one CP, and hence, some CPs have an incentive to invest in the public
investment. Let us define
$1+Q_{N}^{\ast}=\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)$.
Note that $Q_{N}^{\ast}>0$ follows from
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$.
Further, $\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{Q=Q_{N}^{\ast}}=0$ for all $n\in\mathcal{M}$. Let
$\\{\hat{q}_{n}\\}_{n\in\mathcal{N}}$ such that $\hat{q}_{n}\geq 0$ for all
$n\in\mathcal{N}$ be any action profile of the CPs. Further, let
$1+\mu=\sum_{n\in\mathcal{N}}\hat{q}_{N}$. We now consider the following
cases. If $1+\mu>1+Q_{N}^{\ast}$, $\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{Q=\mu}<0$ for all $n\in\mathcal{M}$. Thus, some CPs have
an incentive to decrease their public investment. This implies that $1+\mu$ is
not a Nash Equilibrium. Similarly, if $1+\mu<1+Q_{N}^{\ast}$,
$\left.\frac{\partial U_{n}}{\partial q_{n}}\right\rvert_{Q=\mu}>0$ for all
$n\in\mathcal{M}$. Therefore, some CPs have an incentive to increase their
public investment. Thus, it is not a Nash Equilibrium. Lastly, if
$1+\mu=1+Q_{N}^{\ast}$, $\left.\frac{\partial U_{n}}{\partial
q_{n}}\right\rvert_{Q=\mu}=0$ for all $n\in\mathcal{M}$. It is a Nash
Equilibrium if $\hat{q}_{n}=0$ for all $n\in\mathcal{N}\setminus\mathcal{M}$.
This completes the proof. ∎

###  -C Proof of Theorem 3

###### Proof.

We first consider the case when
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$. This implies $\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq 1$ for all
$n\in\mathcal{N}$. From Theorem 2, this in turn implies that $Q_{N}^{\ast}=0$.
It can be verified that $\frac{\sqrt{1+2b_{n}^{2}r_{n}a_{n}}-1}{b_{n}^{2}}\leq
1$ follows from $r_{n}a_{n}-\frac{b_{n}^{2}}{2}\leq 1$ for all
$n\in\mathcal{N}$. This leads to two possibilities which are as follows.

  * •

$\sum_{n=1}^{N}\frac{\sqrt{1+2b_{n}^{2}r_{n}a_{n}}-1}{b_{n}^{2}}\leq 1$: In
this case, from Lemma 2, we have $Q_{C}^{\ast}=0$, and hence, from (16), it is
clear that $\eta$ is unbounded.

  * •

$\sum_{n=1}^{N}\frac{\sqrt{1+2b_{n}^{2}r_{n}a_{n}}-1}{b_{n}^{2}}>1$: In this
case, from Lemma 2, we have $Q_{C}^{\ast}>0$. However, since $Q_{N}^{\ast}=0$,
from (16), it follows that $\eta$ is unbounded.

Since $Q_{N}^{\ast}=0$ when
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)\leq
1$, from (1), we have $\gamma_{N}=0$.

Now we consider that
$\max\limits_{n\in\mathcal{N}}\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)>1$.
From Theorem 2, $\left.\frac{\partial U_{m}}{\partial
q_{m}}\right\rvert_{q_{m}=Q_{N}^{\ast}}=0$ follows for each $m\in\mathcal{M}$,
where $\mathcal{M}$ is as defined in (15). This implies

$\frac{\sqrt{(1+Q_{N}^{\ast})^{2}+2b_{m}^{2}r_{m}a_{m}}-(1+Q_{N}^{\ast})}{b_{m}^{2}}=1~{}\forall
m\in\mathcal{M}.$ (23)

Using (23), we have

$\displaystyle\left.\frac{\partial U}{\partial
Q}\right\rvert_{Q=Q_{N}^{\ast}}$
$\displaystyle=\sum_{n\in\mathcal{N}\setminus\mathcal{M}}\frac{\sqrt{(1+Q_{N}^{\ast})^{2}+2b_{n}^{2}r_{n}a_{n}}-(1+Q_{N}^{\ast})}{b_{n}^{2}}$
$\displaystyle+\sum_{n\in\mathcal{M}}\frac{\sqrt{(1+Q_{N}^{\ast})^{2}+2b_{n}^{2}r_{n}a_{n}}-(1+Q_{N}^{\ast})}{b_{n}^{2}}-1,$

Thus, $\left.\frac{\partial U}{\partial Q}\right\rvert_{Q=Q_{N}^{\ast}}>0$.
This implies $Q_{C}^{\ast}>Q_{N}^{\ast}$ as $\left.\frac{\partial U}{\partial
Q}\right\rvert_{Q=Q_{C}^{\ast}}=0$. This follows from the concavity of the
utility function. Therefore, from (16), $\eta>1$. Using (8),
$p_{n}^{\ast}(Q_{N}^{\ast})$ for all $n\in\mathcal{N}\setminus\mathcal{M}$ is
given by

$p_{n}^{\ast}(Q_{N}^{\ast})=\left(\frac{\sqrt{(1+Q_{N}^{\ast})^{2}+2b_{n}^{2}r_{n}a_{n}}-(1+Q_{N}^{\ast})}{2b_{n}}\right)^{2}.$

For all $n\in\mathcal{M}$, $p_{n}^{\ast}(Q_{N}^{\ast})$ simplifies to

$\displaystyle p_{n}^{\ast}(Q_{N}^{\ast})$
$\displaystyle=\left(\frac{\sqrt{\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)+2b_{n}^{2}r_{n}a_{n}}-\left(r_{n}a_{n}-\frac{b_{n}^{2}}{2}\right)}{2b_{n}}\right)^{2},$
$\displaystyle=\frac{b_{n}^{2}}{4}.$

Therefore, $\gamma_{N}$ is given by

$\gamma_{N}=\frac{Q_{N}^{\ast}}{\sum\limits_{n\in\mathcal{M}}\frac{b_{n}^{2}}{4}+\sum\limits_{n\in\mathcal{N}\setminus\mathcal{M}}\left(\frac{\sqrt{(1+Q_{N}^{\ast})^{2}+2b_{n}^{2}r_{n}a_{n}}-(1+Q_{N}^{\ast})}{2b_{n}}\right)^{2}}.$
(24)

Clearly, (17) follows from (24) for the special case of
$|\mathcal{M}|=|\mathcal{N}|$. This completes the proof. ∎


††thanks: Present Address: Joint Center for Quantum Information and Computer
Science, University of Maryland-NIST, College Park, Maryland 20742, USA

# Accurate and Honest Approximation of Correlated Qubit Noise

F. Setiawan<EMAIL_ADDRESS>Riverlane Research Inc.,
Cambridge, Massachusetts 02142, USA  Alexander V. Gramolin  Riverlane Research
Inc., Cambridge, Massachusetts 02142, USA  Elisha S. Matekole  Riverlane
Research Inc., Cambridge, Massachusetts 02142, USA  Hari Krovi  Riverlane
Research Inc., Cambridge, Massachusetts 02142, USA  Jacob M. Taylor  Riverlane
Research Inc., Cambridge, Massachusetts 02142, USA

###### Abstract

Accurate modeling of noise in realistic quantum processors is critical for
constructing fault-tolerant quantum computers. While a full simulation of
actual noisy quantum circuits provides information about correlated noise
among all qubits and is therefore accurate, it is, however, computationally
expensive as it requires resources that grow exponentially with the number of
qubits. In this paper, we propose an efficient systematic construction of
approximate noise channels, where their accuracy can be enhanced by
incorporating noise components with higher qubit-qubit correlation degree. To
formulate such approximate channels, we first present a method, dubbed the
cluster expansion approach, to decompose the Lindbladian generator of an
actual Markovian noise channel into components based on interqubit correlation
degree. We then generate a $k$-th order approximate noise channel by
truncating the cluster expansion and incorporating noise components with
correlations up to the $k$-th degree. We require that the approximate noise
channels must be accurate and also “honest”, i.e., the actual errors are not
underestimated in our physical models. As an example application, we apply our
method to model noise in a three-qubit quantum processor that stabilizes a
$\llbracket 2,0,0\rrbracket$ codeword, which is one of the four Bell states.
We find that, for realistic noise strength typical for fixed-frequency
superconducting qubits coupled via always-on static interactions, correlated
noise beyond two-qubit correlation can significantly affect the code
simulation accuracy. Since our approach provides a systematic noise
characterization, it enables the potential for accurate, honest and scalable
approximation to simulate large numbers of qubits from full modeling or
experimental characterizations of small enough quantum subsystems, which are
efficient but still retain essential noise features of the entire device.

##  I Introduction

Noise presents the major stumbling block in constructing a scalable quantum
information processing device. To protect quantum information against noise
such that error rates are small enough for practical quantum advantage in
large-scale quantum processors, one needs to implement quantum error
correction (QEC) [1, 2, 3, 4], where a logical qubit is encoded using many
physical qubits. The basic idea behind QEC is that the logical error rate can
be made arbitrarily small by increasing the number of physical qubits provided
that the physical-qubit error rate is below a certain threshold [5, 6, 7, 8].
Since the assessment of QEC performance is based on the assumption of the
underlying noise model, accurate noise modelings and characterizations of
quantum processors are absolutely essential for designing reliable and
practical QEC. Moreover, faithful information on noise in realistic quantum
devices is also crucial for devising more noise-resilient qubit architectures
and hardware-efficient QEC codes with better threshold and smaller overhead.
For example, for devices with biased Pauli noise, cat qubits [9, 10, 11] and
the XZZX surface code [12] are favorable as they possess outstanding
resilience against such noise.

Knowledge about noise in quantum devices can be obtained by doing full
simulations or experimental characterizations, e.g., gate set [13, 14, 15],
quantum state and process tomography [16, 17], Lindblad tomography [18] or
randomized benchmarking [19, 20], of the actual physical device, which supply
details on the qubit state at a given moment or quantum process/channel due to
a gate application of a certain duration. Such theoretical simulations or
experimental characterizations, however, require computational or experimental
resources (such as number of measurements) that scale exponentially with the
number of qubits, which makes them impractical for large quantum circuits. The
question that naturally arises is how can one use the full characterization of
noise in small quantum processors to accurately assess the performance of
large quantum information processors?

Figure 1: Schematics of the cluster expansion approach, a perturbative series
method that decomposes the Lindbladian (noise generator) into terms based on
the interqubit correlation degrees. Shown is an example of the cluster
expansion approach applied to noise in a three-qubit quantum processor with
triangular connectivity.

One strategy used to simulate noise in large quantum systems is to assume a
Pauli noise model [21, 22, 23] and construct approximate stochastic noise
channels based on the Pauli twirling approximation [24, 25] which averages the
noise channels with respect to a set of unitaries. This oversimplified model,
however, often does not provide an accurate and honest description of noise in
real quantum systems as it does not capture coherent noise and non-Pauli
errors such as pulse miscalibration, leakage, etc, which are the dominant
noise in many realistic quantum devices. In this paper, we propose a general
systematic approach to generate approximate noise channels beyond the Pauli
noise model, which is based on actual noise data obtained from either
realistic simulations or experimental characterizations of subsystems of the
quantum device. In particular, we consider obtaining the actual noise data via
either Lindbladian dynamics simulation or experimental characterizations such
as Lindblad tomography [18], process tomography [17] which supplies data that
can be fitted to get the best-fit Lindbladian for the quantum channel [26,
27], etc.; here we focus on the former, i.e., using inputs from theoretical
simulations.

We present an approach, dubbed the cluster expansion method, to decompose the
actual noise into components according to their degree of qubit-qubit
correlations [see Fig. 1]. A $k$-th order approximate noise channel is then
generated by truncating the expansion and including noise components with
qubit-qubit correlations up to the $k$-th degree. The accuracy of the
approximate noise model can be improved (at the expense of reducing the
efficiency) by including noise components with increasingly higher interqubit
correlation degree. Further, we impose that our approximate noise channel must
be constructed in an “honest” manner, i.e., the approximate error does not
underestimate the actual error. Therefore, the aim of this paper is to provide
a systematic method to construct approximate noise models,
$\mathcal{N}_{\mathrm{approx}}$, such that

  1. 1.

The action of $\mathcal{N}_{\mathrm{approx}}$ can be made as “accurate”
(faithful) as possible in representing the action of the actual channel
$\mathcal{N}_{\mathrm{actual}}$,

  2. 2.

$\mathcal{N}_{\mathrm{approx}}$ gives an “honest” representation of the noise
in the actual system, and

  3. 3.

The construction of $\mathcal{N}_{\mathrm{approx}}$ is scalable in the number
of qubits.

This paper is organized as follows. In Sec. II, we begin by giving a short
review of the Lindblad master equation and also define the normal-form of a
noise channel. Subsequently in Sec. III, we use the cluster expansion approach
to decompose the Lindbladian into terms based on their qubit-qubit correlation
degrees. In Sec. IV, we present ways to construct approximate noise channels
using the decomposed Lindbladians and define the honesty and accuracy criteria
of the approximate noise models. We then give a detailed application of our
noise model for QEC in Sec. V, focusing on a specific example, namely the
$\llbracket 2,0,0\rrbracket$ code. We give the conclusion in Sec. VI. The
appendices contain details on the superoperator representation, example
applications of the cluster expansion method, and results for simulating the
$\llbracket 2,0,0\rrbracket$ code in a three-qubit quantum processor with
triangular connectivity.

##  II Lindblad noise model

We assume that the evolution of a quantum processor interacting with its
environment is well approximated by the Lindblad master equation [28, 29]:

$\frac{d\rho(t)}{dt}=\mathcal{L}\rho(t),$ (1)

where $\rho$ is the system density matrix and the Lindbladian $\mathcal{L}$ is
given by [28, 29]

$\mathcal{L}(\rho)=-i[\hat{H},\rho]+\sum_{j}\gamma_{j}\left(\hat{A}_{j}\rho\hat{A}_{j}^{\dagger}-\frac{1}{2}\hat{A}_{j}^{\dagger}\hat{A}_{j}\rho-\frac{1}{2}\rho\hat{A}_{j}^{\dagger}\hat{A}_{j}\right),$
(2)

with $[\cdot,\cdot]$ denoting the commutator. Here the Hamiltonian $\hat{H}$
describes the unitary dynamics of the system, which can include coherent
errors resulting from pulse miscalibration, crosstalk due to always-on qubit-
qubit interactions, etc. $\hat{A}_{j}$ is the Lindblad (jump) operator with a
jump/dissipation rate $\gamma_{j}$ representing the incoherent noise (e.g.,
dephasing and relaxation) due to the interaction between the system and the
environment. The resulting channel (propagator) that evolves the system for a
duration $t$ can be written as

$\mathcal{V}=\hat{T}\left[\exp\left(\int_{0}^{t}\mathcal{\mathcal{L}}(t^{\prime})dt^{\prime}\right)\right],$
(3)

where $\hat{T}$ denotes the time-ordering operator.

In order to describe the dynamics due to only the unwanted noise, we write the
channel (propagator) in the normal form. This normal form is defined as the
noise channel with the unitary dynamics contribution $\mathcal{U}^{\dagger}$
of the ideal Hamiltonian being factored out, i.e.,

$\mathcal{N}=\mathcal{V}\circ\mathcal{U}^{\dagger},$ (4)

where $\mathcal{U}^{\dagger}(\rho)=U^{\dagger}\rho U$ is the ideal (noiseless)
channel with $U$ being the ideal target unitary operator 111We remark that,
given the target $U$, there exists a family of such forms
$\mathcal{N}_{x}=\mathcal{U}^{-x}\circ\mathcal{V}\circ\mathcal{U}^{-1+x}$ (5)
for $x\in[0,1]$ with $\mathcal{U}^{x}(\rho)=U^{x}\rho U^{-x}$. In addition,
summation over a function $f(x)dx$ also produces appropriate forms, e.g.,
$\mathcal{N}_{f}=\int\mathcal{N}_{x}f(x)dx$ with $\int f(x)dx=1$ the only
requirement to ensure that $\mathcal{N}_{f}$ is the identity channel when
there is no noise. There can be circumstances where use of such
$\mathcal{N}_{f}$ forms may be desirable, due to averaging or simplification
for the appropriate setting, but here we fix $\mathcal{N}_{0}$ as the _normal_
form..

If the Hamiltonian or Lindblad operators are time-dependent then the
Lindbladian $\mathcal{L}$ and hence, the noise channel are also time-
dependent. We capture the effect of this time-dependent dynamics by using
$\widetilde{\mathcal{L}}$, which is an effective dimensionless time-
independent Lindbladian that serves as a generator for the system evolution
from the beginning to the end of the application of the noise channel. In the
superoperator representation (see Appendix A), it is given by

$\hat{\widetilde{\mathcal{L}}}=\mathrm{ln}(\hat{\mathcal{N}}),$ (6)

where $\hat{\widetilde{\mathcal{L}}}$ is the superoperator representation of
the Lindbladian $\widetilde{\mathcal{L}}$ and $\hat{\mathcal{N}}$ is the
superoperator representation of the normal-form noise channel $\mathcal{N}$ as
defined in Eq. (4). In general, the matrix logarithm in Eq. (6) yields an
infinite number of branch cuts. However, for small enough noise strength, the
normal-form noise channel is close to the identity channel and hence the
zeroth branch cut of the matrix logarithm taken in Eq. (6) gives a physical
Lindbladian, i.e., the Lindbladian where all its eigenvalues have non-positive
real parts.

##  III Lindbladian Decomposition using Cluster Expansion

Quantum circuits are usually transpiled into one- and two-qubit gates. Noise
associated with the gate operations are then expected to have significant
effect only in the neighborhood of the qubits where the gates are applied. As
a result, we can simplify the noise model by first decomposing the noise into
locally correlated noise and incorporating only the subset of noise that
substantially affects the gates performance.

To separate and characterize the contributions due to noise with different
degree of correlations, we present a method akin to the cluster expansion
approach [31, 32] used for interacting closed many-body systems. The cluster
expansion approach is a perturbative series expansion which sums over the
contribution from clusters with different size; see Fig. 1. Instead of doing a
power-series expansion over the partition function as in the standard cluster
expansion approach used to treat the interactions in closed many-body systems
[31, 32], our cluster expansion approach decomposes the Lindblad generator
$\hat{\widetilde{\mathcal{L}}}$ of an actual noise channel into sum of
components with different degree of qubit-qubit correlations, i.e.,

$\hat{\widetilde{\mathcal{L}}}=\hat{\widetilde{\mathcal{L}}}^{1-\mathrm{body}}+\hat{\widetilde{\mathcal{L}}}^{2-\mathrm{body}}+\cdots+\hat{\widetilde{\mathcal{L}}}^{n-\mathrm{body}},$
(7)

where $\hat{\widetilde{\mathcal{L}}}$ denotes the exact Lindbladian of the
whole system which comprises $n$ number of qubits. Here,

$\displaystyle\hat{\widetilde{\mathcal{L}}}^{m-\mathrm{body}}$
$\displaystyle=\sum_{S_{m}}\hat{\widetilde{\mathcal{L}}}^{S_{m}}$ (8)

is the component of the Lindbladian that describes noise with $m$-th
correlation degree. The sum used to evaluate the $m$-th order correlated
Lindbladian $\hat{\widetilde{\mathcal{L}}}^{m-\mathrm{body}}$ in Eq. (8) is
performed over all subsets of size $m$ in the set $S=\\{1,2,\cdots,n\\}$,
i.e., $S_{m}\subseteq S$ such that $|S_{m}|=m$. Each of the $m$-th order
correlated noise terms is given by the following recurrence relation:

$\displaystyle\hat{\widetilde{\mathcal{L}}}^{S_{m}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{\bar{S}_{m}}\left[e^{\hat{\widetilde{\mathcal{L}}}}\left(\mathds{1}_{S_{m}}\otimes\rho_{\bar{S}_{m}}\right)\right]\otimes\mathds{1}_{\bar{S}_{m}}\right)-\sum_{R\subsetneq
S_{m}}\hat{\widetilde{\mathcal{L}}}^{R}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{\bar{S}_{m}}\left[e^{\hat{\widetilde{\mathcal{L}}}}\left(\mathds{1}_{S_{m}}\otimes\frac{\mathds{1}_{\bar{S}_{m}}}{|\mathcal{H}_{\bar{S}_{m}}|}\right)\right]\otimes\mathds{1}_{\bar{S}_{m}}\right)$
$\displaystyle\hskip 14.22636pt-\sum_{R\subsetneq
S_{m}}\hat{\widetilde{\mathcal{L}}}^{R},$ (9)

where $\bar{S}_{m}=\\{x\in S:x\notin S_{m}\\}$ denotes the complement subset
of $S_{m}$. As shown in Eq. (III), $\hat{\widetilde{\mathcal{L}}}^{S_{m}}$ is
calculated by first doing a partial trace of the whole system ($n$-qubits)
noise channel over the $(n-m)$ qubits which live in the Hilbert space
$\mathcal{H}_{\bar{S}_{m}}$ with dimension
$|\mathcal{H}_{\bar{S}_{m}}|={n_{\mathrm{qlvl}}}^{(n-m)}$. We use
${n_{\mathrm{qlvl}}}$ to denote the number of the qubit states, where
${n_{\mathrm{qlvl}}}=2$ for the case considered in this paper. Here, the state
of the ($n-m$) qubits is taken to be a fully mixed state; this is a valid
assumption for QEC circuits, the focus of our paper, as noise in data qubits
makes the syndrome qubits undergoing ergodic evolutions into all possible
states. The tensor product with the identity ($\mathds{1}_{\bar{S}_{m}}$) in
the $(n-m)$-qubit subspace after the trace operation is to ensure that the
dimension of the noise channel is the same as its initial dimension.
Afterward, we take the logarithm of the resulting noise channel to get the
effective Lindbladian that describes noise in $m$-qubits subsystem where it
includes all noise components with correlation degrees up to the $m$-th order.
Subsequently, to get the $m$-th order correlated noise term
$\hat{\widetilde{\mathcal{L}}}^{S_{m}}$, we subtract from the resulting
expression all the lower ($m^{\prime}<m$)-th order correlated noise terms,
where the subtraction is over all proper subsets of $S_{m}$ ($R\subsetneq
S_{m}$), i.e., all subsets of $S_{m}$ that are not equal to $S_{m}$.

##  IV Approximate Noise Model

###  IV.1 Construction

Using the above decomposed Lindbladians, we can now ask the question: can we
truncate the total Lindblad term $\hat{\widetilde{\mathcal{L}}}$ by
considering only terms up to the $k$-th order, where $k\leq n$? As we show in
the rest of this work, we answer this in the affirmative. Specifically, we
consider the $k$-body approximation of the Lindbladian
$\hat{\widetilde{\mathcal{L}}}_{k}$ by ignoring all higher-order terms
($m>k$), defining:

$\hat{\widetilde{\mathcal{L}}}_{k}=\hat{\widetilde{\mathcal{L}}}^{1-\mathrm{body}}+\hat{\widetilde{\mathcal{L}}}^{2-\mathrm{body}}+\cdots+\hat{\widetilde{\mathcal{L}}}^{k-\mathrm{body}}.$
(10)

We remark that, in general, each term in $\hat{\widetilde{\mathcal{L}}}_{k}$
and therefore $\hat{\widetilde{\mathcal{L}}}_{k}$ is not guaranteed to be a
physical Lindbladian, as it comprises a sum of both positive and negative
Lindblad terms [see Eqs. (8) and (III)] 222While the term
$\hat{\widetilde{\mathcal{L}}}^{S_{m}}$ in Eq. (III) is not guaranteed to be a
physical Lindbladian, the first term of the right-hand side of Eq. (III),
i.e.,
$\mathrm{ln}\left(\mathrm{Tr}_{\bar{S}_{m}}\left[e^{\hat{\widetilde{\mathcal{L}}}}\left(\mathds{1}_{S_{m}}\otimes\rho_{\bar{S}_{m}}\right)\right]\otimes\mathds{1}_{\bar{S}_{m}}\right)$,
is always a physical Lindbladian as it describes noise in an $m$-qubits
subsystem.. However, all the 1-body terms
$\hat{\widetilde{\mathcal{L}}}^{S_{1}}$ and hence
$\hat{\widetilde{\mathcal{L}}}^{1-\mathrm{body}}$ are guaranteed to be
physical Lindbladians. Note that the way we calculate each term in
$\hat{\widetilde{\mathcal{L}}}_{k}$, as defined in Eqs. (8) and (III),
guarantees that the summation in Eq. (10) does not over- or under-count the
expected error, as we showcase with several examples in Appendix B.

To generate the approximate noise channel, we can in principle exponentiate
the $k$-body approximate Lindbladian $\hat{\widetilde{\mathcal{L}}}_{k}$.
However, for sufficiently small noise strength, we can further simplify the
computation for the approximate noise channel by making a Trotter
approximation. Combining the truncation of the Lindbladian cluster-expansion
and the Trotter approximation, we write the $k$-th order approximate noise
channel as

$\displaystyle\hat{\mathcal{N}}_{\mathrm{approx},k}(g)\equiv\prod_{S_{1}}\exp(g\hat{\widetilde{\mathcal{L}}}^{S_{1}})\cdots\prod_{S_{k}}\exp(g\hat{\widetilde{\mathcal{L}}}^{S_{k}}),$
(11)

where $g$ is the gain factor 333In general, one may have to use different gain
factors for different decomposed components of the Lindbladian to get a
completely-positive and trace-preserving (CPTP) approximate noise channel that
is honest and accurate. However, for our case of $\llbracket 2,0,0\rrbracket$
code in 3-qubit processors, we find that we can use the same gain factor for
all the Lindbladian components to obtain honest and accurate CPTP approximate
noise channels., a free parameter to be optimized in order to make the
approximation “honest” and as accurate as possible (see Sec. IV.2 for details
on the honesty and accuracy criteria). We use this Trotterized form in what
follows as it allows us to reason about noise in the context of more complex
quantum circuits, where each noise component can be replaced by a quantum
operation occurring over a subset $S_{m}$ with $m\leq k$.

We note that Our approximate noise model can be scaled to larger systems even
when there is crosstalk between all connected pairs of qubits as long as the
number of qubits that the gates act on are small enough. This is because the
cluster expansion can be truncated at a certain depth where the small
amplitude of higher-order terms, i.e., terms with correlation degree greater
than $k$, allows us to entirely neglect these higher-order residual errors.

###  IV.2 Honesty and accuracy

To characterize the performance of the approximate noise model, we calculate
the distances between ideal (noiseless), actual, and approximate noise
channels, which quantify the distinguishability between the channels. Based on
these distances, we can characterize two properties of the approximate noise
channels: honesty and accuracy [35, 36, 37].

An approximate noise channel is honest if it does not underestimate the
deleterious effect of the actual noise. On the other hand, the accuracy of an
approximate channel refers to how closely it can emulate the impact of the
actual noise. More explicitly, if we consider the ideal (noiseless)
$\mathcal{V}_{\mathrm{ideal}}$, actual $\mathcal{V}_{\mathrm{actual}}$, and
approximate $\mathcal{V}_{\mathrm{approx}}$ target channels map an initial
pure state $|\Psi_{\mathrm{in}}\rangle$ into the $\rho_{\mathrm{ideal}}$,
$\rho_{\mathrm{actual}}$, and $\rho_{\mathrm{approx}}$ density matrices,
respectively, we can then define the honesty and accuracy criteria of the
approximate noise channel $\mathcal{V}_{\mathrm{approx}}$ in terms of the
distance metrics $\mathcal{D}$ between these density matrices as

$\displaystyle\mathrm{honest\,if\hskip 5.69046pt}$
$\displaystyle\frac{\mathcal{D}(\rho_{\mathrm{ideal}},\rho_{\mathrm{approx}})}{\mathcal{D}(\rho_{\mathrm{ideal}},\rho_{\mathrm{actual}})}\geq
1,$ (12) $\displaystyle\mathrm{accurate\,if\hskip 5.69046pt\,}$
$\displaystyle\frac{\mathcal{D}(\rho_{\mathrm{ideal}},\rho_{\mathrm{actual}})}{\mathcal{D}(\rho_{\mathrm{actual}},\rho_{\mathrm{approx}})}\geq
1,$ (13)

for every input pure state $|\Psi_{\mathrm{in}}\rangle$.

For ease of understanding, we show in Fig. 2 the schematics of the honesty and
accuracy criteria of approximate noise models. For the approximate noise model
to be honest, its density matrix must lie outside the ball whose radius is
defined by the distance between $\rho_{\mathrm{ideal}}$ and
$\rho_{\mathrm{actual}}$ (the shaded ball shown in Fig. 2). On the other hand,
for the approximate noise model to be accurate, it must be as close as
possible to the actual model; in other words, the distance between them (the
green dashed line in Fig. 2) must be as small as possible. So, accurate and
honest approximate noise models must lie on the green arc outside the shaded
ball but close to the actual model.

Figure 2: A schematic of distances between density matrices obtained from the
ideal ($\rho_{\mathrm{ideal}}$), actual ($\rho_{\mathrm{actual}}$) and
approximate ($\rho_{\mathrm{approx}}$) models. Shaded area represents
parameter regimes where approximate noise models are not honest. Honest
approximate noise models must lie outside the ball defined by the distance
between the ideal and actual models (shown by shaded ball). For the
approximate noise models to be accurate, they must be close to the actual
model.

To obtain the best approximate noise model, we find the optimal gain factor
($g=g_{\mathrm{opt}}$) in $\hat{\mathcal{N}}_{\mathrm{approx}}$ [Eq. (11)]
that yields the highest state-averaged accuracy ratio, i.e.,

$\max_{g}\left\langle\frac{D(\rho_{\mathrm{ideal}},\rho_{\mathrm{actual}})}{D(\rho_{\mathrm{actual}},\rho_{\mathrm{approx}}(g))}\right\rangle_{|\Psi_{\mathrm{in}}\rangle},$

subject to the honesty constraint [Eq. (12)]. Here,
$\langle\cdots\rangle_{|\Psi_{\mathrm{in}}\rangle}$ denotes the averaging over
all the pure input states (for the case of QEC, which is the focus of our
paper, these input states are the stabilizer states of the QEC code).

##  V Application to QEC: $\llbracket 2,0,0\rrbracket$ code

###  V.1 Basics of $\llbracket 2,0,0\rrbracket$ code

As an example application of our noise model, we consider applying it to the
simplest QEC code, the $\llbracket 2,0,0\rrbracket$ code. As its name
suggests, this code has 2 physical qubits, zero logical qubits and zero
distance. It has zero logical qubits since the number of stabilizers ($XX$ and
$ZZ$) is the same as the number of physical data qubits. As it does not have
any logical operators, it has zero distance, which means that it cannot
correct any physical errors.

The circuit for implementing the $\llbracket 2,0,0\rrbracket$ code is shown in
Fig. 3. The input state of the circuit is taken to be any one of the four Bell
states, where each of them has a unique syndrome measurement at each round of
the implementation of an ideal (noiseless) $\llbracket 2,0,0\rrbracket$-code
circuit, as shown in Table 1.

Figure 3: Quantum circuit for implementing the $\llbracket 2,0,0\rrbracket$ code. The code consists of two data qubits (qubits 1 and 2) and one syndrome qubit (qubit 3). The input state of the data qubits $|\Psi_{\mathrm{in}}\rangle$ is one of the Bell states while the syndrome qubit is initialized in state $|0\rangle$. The syndrome qubit is reset to $|0\rangle$ at the beginning of each round of the implementation of the $\llbracket 2,0,0\rrbracket$ code. The code has 2 stabilizers: $ZZ$ and $XX$. The $ZZ$ and $XX$ syndrome measurements of each input Bell state $|\Psi_{\mathrm{in}}\rangle$ for each round of the implementation of an ideal (noiseless) $\llbracket 2,0,0\rrbracket$-code circuit are shown in Table 1. Input state $|\Psi_{\mathrm{in}}\rangle$ |  ($ZZ$, $XX$) syndrome
---|---
$\dfrac{|00\rangle+|11\rangle}{\sqrt{2}}$ |  (0,0)
$\dfrac{|00\rangle-|11\rangle}{\sqrt{2}}$ |  (0,1)
$\dfrac{|01\rangle-|10\rangle}{\sqrt{2}}$ |  (1,0)
$\dfrac{|01\rangle+|10\rangle}{\sqrt{2}}$ |  (1,1)
Table 1: Data-qubits input states together with the syndrome measurement
results of the ancilla qubit at each round of the implementation of an ideal
(noiseless) $\llbracket 2,0,0\rrbracket$-code circuit.

###  V.2 Infidelity distance measure between different quantum channels

Since noise in different quantum channels change the same initial data-qubits
states into different quantum states, we can characterize noise in different
quantum channels by first defining the fidelity between the data-qubits states
resulting from two different quantum channels $\mathcal{E}$ and $\mathcal{F}$
as [38, 39]

$F_{\mathrm{q}}(\rho_{\mathcal{E}},\rho_{\mathcal{F}})=\left[\mathrm{Tr}\left(\sqrt{\sqrt{\rho_{\mathcal{E}}}\rho_{\mathcal{F}}\sqrt{\rho_{\mathcal{E}}}}\right)\right]^{2}.$
(14)

As different data-qubits states give rise to different probabilities of the
syndrome-qubit measurement results $x$, we can similarly define the fidelity
of the syndrome qubit measurements of different quantum channels in terms of
their syndrome measurement probabilities [$p_{\mathcal{E}}(x)$ and
$p_{\mathcal{F}}(x)$] as [39]

$F_{\mathrm{c}}(p_{\mathcal{E}},p_{\mathcal{F}})=\left[\sum_{x}\sqrt{p_{\mathcal{E}}(x)p_{\mathcal{F}}(x)}\right]^{2}.$
(15)

Combining the fidelity measures characterized in terms of the data-qubits
states and syndrome qubit measurements, we can define the infidelity distance
between any two quantum channels used to implement QEC codes as

$\displaystyle
D^{\mathrm{inf}}_{|\Psi_{\mathrm{in}}\rangle}(\rho_{\mathcal{E}},\rho_{\mathcal{F}})=1-\left(\sum_{x\in\\{0,1\\}^{2l}}\bigg{[}\sqrt{p_{\mathcal{E}}(x|\Psi_{\mathrm{in}})p_{\mathcal{F}}(x|\Psi_{\mathrm{in}})}\times\mathrm{Tr}\left(\sqrt{\sqrt{\rho_{\mathcal{E}}(x|\Psi_{\mathrm{in}})}\rho_{\mathcal{F}}(x|\Psi_{\mathrm{in}})\sqrt{\rho_{\mathcal{E}}(x|\Psi_{\mathrm{in}})}}\right)\bigg{]}\right)^{2},$
(16)

where $x\in\\{0,1\\}^{2l}$ are the syndrome measurement bit strings after the
$l$-th round implementation of the $\llbracket 2,0,0\rrbracket$-code circuit
and $|\Psi_{\mathrm{in}}\rangle$ is the data-qubits input state, which is one
of the four Bell states. Here,
$\rho_{\mathcal{E},\mathcal{F}}(x|\Psi_{\mathrm{in}})$ are the data-qubits
density matrices resulting from the application of the quantum channels
$\mathcal{E}$ and $\mathcal{F}$, respectively. The density matrices
$\rho_{\mathcal{E},\mathcal{F}}(x|\Psi_{\mathrm{in}})$ are associated with the
probability distributions $p_{\mathcal{E},\mathcal{F}}(x|\Psi_{\mathrm{in}})$
of a given syndrome measurement result $x$ given a particular input state
$|\Psi_{\mathrm{in}}\rangle$. The infidelity values are given by $0\leq
D^{\mathrm{inf}}_{|\Psi_{\mathrm{in}}\rangle}(\rho_{\mathcal{E}},\rho_{\mathcal{F}})\leq
1$, where
$D^{\mathrm{inf}}_{|\Psi_{\mathrm{in}}\rangle}(\rho_{\mathcal{E}},\rho_{\mathcal{F}})=0$
means that the two density matrices are identical and
$D^{\mathrm{inf}}_{|\Psi_{\mathrm{in}}\rangle}(\rho_{\mathcal{E}},\rho_{\mathcal{F}})=1$
indicates that the two density matrices are furthest apart from each other.

In the context of QEC, the honesty and accuracy criteria [Eqs. (12) and (13)]
of our approximate noise models can then be defined in terms of the infidelity
metrics given in Eq. (16). Specifically, we require these criteria to be
satisfied for every input pure state $|\Psi_{\mathrm{in}}\rangle$ in the
logical space spanned by the stabilizer states (which are the four Bell states
for our example of the $\llbracket 2,0,0\rrbracket$ code).

###  V.3 Simulation

We now analyze the implementation of our noise model for the $\llbracket
2,0,0\rrbracket$ code in a concrete physical setup: a quantum processor
consisting of fixed-frequency qubits coupled to each other via static
transverse couplings. The system Hamiltonian is given by

$\hat{H}(t)=\hat{H}_{q}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{dr}}(t),$ (17)

where $\hat{H}_{q}$ is the bare qubit Hamiltonian, $\hat{H}_{\mathrm{int}}$ is
the qubit-qubit interaction term and $\hat{H}_{\mathrm{dr}}(t)$ is the qubit
drive term.

We consider the qubits to be a collection of two-level systems with the bare
Hamiltonian (in units $\hbar=1$)

$\hat{H}_{q}=\sum_{j}\frac{\omega_{q_{j}}}{2}\hat{\sigma}_{z,j},$ (18)

where $\hat{\sigma}_{z,j}=|0\rangle_{j}{}_{j}\langle
0|-|1\rangle_{j}{}_{j}\langle 1|$ is the $j$-th qubit Pauli-$z$ operator and
$\omega_{q_{j}}$ is the $j$-th qubit frequency. The qubits are coupled to each
other via always-on static transverse couplings with the Hamiltonian

$\hat{H}_{\mathrm{int}}=\sum_{\langle
j,k\rangle}J_{jk}(\hat{\sigma}_{+,j}\hat{\sigma}_{-,k}+\hat{\sigma}_{+,k}\hat{\sigma}_{-,j}),$
(19)

where the sum is over all pairs of interacting qubits. Here, $J_{jk}$ is the
strength of the coupling between qubits $j$ and $k$,
$\hat{\sigma}_{+}=|1\rangle\langle 0|$ and $\hat{\sigma}_{-}=|0\rangle\langle
1|$ are the qubit raising and lowering operators, respectively. We consider
two different connectivities: linear and triangle geometries (see Fig. 4),
where the linear geometry has one less qubit-qubit coupling compared to the
triangle geometry.

Figure 4: Connectivities of the 3-qubit processor used for implementing the
$\llbracket 2,0,0\rrbracket$ code: (a) Linear and (b) Triangle geometry.
Circles represent data qubits and squares represent syndrome qubits. Numbers
indicate the qubit labels as in Fig. 3.

Gates are implemented by driving the qubits with control fields where the
drive Hamiltonian is

$\displaystyle\hat{H}_{\mathrm{dr}}(t)$
$\displaystyle=\sum_{\begin{subarray}{c}j\in\mathrm{driven}\\\
\mathrm{qubits}\end{subarray}}\Omega_{\mathrm{dr},j}(t)\left(\hat{\sigma}_{+,j}e^{-i(\omega_{\mathrm{dr},j}t+\phi_{\mathrm{dr},j})}+\mathrm{H.c.}\right),$
(20)

where $\Omega_{\mathrm{dr},j}(t)$, $\omega_{\mathrm{dr},j}$ and
$\phi_{\mathrm{dr},j}$ are respectively the envelope, frequency and phase of
the control field that drives qubit $j$. The notation “H.c.” denotes the
Hermitian conjugate of the preceding term. A single-qubit gate is implemented
by driving the qubit with a control field at the qubit’s characteristic
frequency. Any single-qubit gate can be implemented in this way by choosing
appropriate amplitude, duration and phase of the driving pulse. However, here
we choose to realize single-qubit gates using sequences of $X$ and $Y$
rotations obtained using the $X$-$Y$ decomposition [39] of the gates. The two-
qubit (CNOT) gate used in our simulation is a cross-resonance [40, 41] gate
which is implemented by driving the control qubit at the frequency of the
target qubit. To cancel the unwanted effect due to crosstalk in the cross-
resonance gate, we apply an echoed version of the calibrated pulse sequence as
described in Ref. [42].

Figure 5: Amplitudes $\Omega_{\mathrm{dr},j}(t)$ and phases
$\phi_{\mathrm{dr},j}(t)$ of the control pulses for the $ZZ$ [panels (a), (b),
and (c)] and $XX$ [panels (d), (e), and (f)] stabilizers of the $\llbracket
2,0,0\rrbracket$ code implemented in a linear geometry. From top to bottom:
pulse schedules for qubit $j=1,2,3$ (the qubit labels are the same as those in
Fig. 3).

The pulse schedules of the $ZZ$ and $XX$ stabilizers, which consist of single-
and two-qubit gate pulses, are shown in the left and right panels of Fig. 5,
respectively; here we choose Gaussian and Gaussian-square pulse envelopes.
These pulse schedules are for a linearly-connected qubit geometry (the pulse
schedules for the triangle geometry are not shown as they are similar to those
for the linear geometry). These pulse schedules are calibrated for the qubit
parameters shown in Table 2, where we use values typical for fixed-frequency
superconducting qubits. For the linear geometry, our calibrated pulses yield
average fidelities of $0.9896$ and $0.9888$ for the $ZZ$ and $XX$ stabilizers,
respectively. Similar calibrated pulses for the triangle geometry yield
average fidelities of $0.9870$ and $0.9832$ for the $ZZ$ and $XX$ stabilizers,
respectively.

Qubit frequencies |  Qubit-qubit couplings
---|---
$\omega_{q_{1}}/2\pi$ = 4.8 GHz |  $J_{1,2}/2\pi$ = 4 MHz (Triangle)
|  = 0 (Linear)
$\omega_{q_{2}}/2\pi$ = 5.2 GHz |  $J_{2,3}/2\pi$ = 4 MHz
$\omega_{q_{3}}/2\pi$ = 5 GHz |  $J_{1,3}/2\pi$ = 4 MHz
Relaxation times |  Dephasing times
$T_{1,1}$ = 100 $\mu$s |  $T_{\varphi,1}$ = 200 $\mu$s
$T_{1,2}$ = 100 $\mu$s |  $T_{\varphi,2}$ = 200 $\mu$s
$T_{1,3}$ = 100 $\mu$s |  $T_{\varphi,3}$ = 200 $\mu$s
Table 2: Qubit parameters: qubit frequencies $\omega_{q}$, qubit-qubit
coupling strengths $J$, relaxation times $T_{1}$ and pure dephasing times
$T_{\varphi}$.

To simulate the gate dynamics which incorporates the effects of dissipation
due to $T_{\varphi}$ dephasing and $T_{1}$ relaxation, we use the Lindblad
master equation:

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[\hat{H}(t),\rho(t)]+\sum_{j}\frac{1}{T_{\varphi,j}}\bigg{(}\hat{\sigma}_{z,j}\rho(t)\hat{\sigma}_{z,j}-\rho(t)\bigg{)}$
$\displaystyle\hskip
5.69046pt+\sum_{j}\frac{1}{T_{1,j}}\left(\hat{\sigma}_{-,j}\rho(t)\hat{\sigma}_{+,j}-\frac{1}{2}\left\\{\hat{\sigma}_{+,j}\hat{\sigma}_{-,j},\rho(t)\right\\}\right),$
(21)

where $\rho(t)$ is the density matrix of the whole system, $T_{\varphi,j}$ and
$T_{1,j}$ are the dephasing and relaxation times of qubit $j$, respectively.
Here $\\{\cdot,\cdot\\}$ denotes the anticommutator. The Lindblad master
equation simulation in this paper is done using the Python package QuTiP [43,
44].

From the Lindblad master equation simulation, we obtain the actual channel or
propagator $\mathcal{V}_{\mathrm{actual}}$ of the $ZZ$ and $XX$ stabilizers
which evolve the system from the beginning to the end of each stabilizer’s
protocol time. We then convert these propagators into their normal forms
$\hat{\mathcal{N}}_{\mathrm{actual}}$ using Eq. (4). Afterward, we convert
$\hat{\mathcal{N}}_{\mathrm{actual}}$ into the superoperator form and then
take the matrix logarithm of $\hat{\mathcal{N}}_{\mathrm{actual}}$ in order to
get the time-independent effective Lindbladians
$\hat{\widetilde{\mathcal{L}}}$ [Eq. (6)]. We then decompose these effective
Lindbladians $\hat{\widetilde{\mathcal{L}}}$ using the cluster expansion
described in Sec. III and exponentiate the decomposed Lindbladians to get the
normal-form of the approximate noise channels
$\hat{\mathcal{N}}_{\mathrm{approx}}$ as detailed in Sec. IV.1. To convert the
approximate noise channels of the $ZZ$ and $XX$ stabilizers into the standard
form $\hat{\mathcal{V}}_{\mathrm{approx}}$, we multiply the corresponding
$\hat{\mathcal{N}}_{\mathrm{approx}}$ by the ideal (noiseless) target
operation $\hat{\mathcal{U}}$, i.e.,

$\hat{\mathcal{V}}_{\mathrm{approx}}=\hat{\mathcal{N}}_{\mathrm{approx}}\,\hat{\mathcal{U}}.$
(22)

We use the following procedure to implement the $\llbracket 2,0,0\rrbracket$
code:

  1. 1.

Choose a Bell state for the input state $|\Psi_{\mathrm{in}}\rangle$ of the
data qubits.

  2. 2.

Reset the syndrome qubit to state $|0\rangle$.

  3. 3.

Apply a selected error channel, actual $\hat{\mathcal{V}}_{\mathrm{actual}}$
or approximate $\hat{\mathcal{V}}_{\mathrm{approx}}$ (for different gain
factors), of the $ZZ$ stabilizer.

  4. 4.

Apply a noiseless (perfect) projective measurement of the syndrome qubit in
the $Z$ basis.

  5. 5.

Apply a selected error channel, actual $\hat{\mathcal{V}}_{\mathrm{actual}}$
or approximate $\hat{\mathcal{V}}_{\mathrm{approx}}$ (for different gain
factors), of the $XX$ stabilizer.

  6. 6.

Apply a noiseless (perfect) projective measurement of the syndrome qubit in
the $Z$ basis.

  7. 7.

Compute the infidelities between the data-qubits density matrices obtained
from the noiseless (ideal) $\hat{\mathcal{V}}_{\mathrm{ideal}}$, actual
$\hat{\mathcal{V}}_{\mathrm{actual}}$ and approximate
$\hat{\mathcal{V}}_{\mathrm{approx}}$ channels.

  8. 8.

Repeat steps 2-7.

We apply the above procedure to implement the $\llbracket 2,0,0\rrbracket$
code for two different connectivities: linear and triangle geometries (see
Fig. 4). In order to understand the contribution of noise components with
different degrees of correlations, for both geometries we perform simulations
using two different approximate noise channels: second-order
($\hat{\mathcal{N}}_{\mathrm{approx},2}$) and third-order
($\hat{\mathcal{N}}_{\mathrm{approx},3}$) approximate noise channels [Eq.
(11)].

Figure 6: Results simulated for the linear connectivity using the second-order
$\hat{\mathcal{N}}_{\mathrm{approx},2}$ [upper panels: (a), (b), (c), and (d)]
and third-order $\hat{\mathcal{N}}_{\mathrm{approx},3}$ [lower panels: (e),
(f), (g), and (h)] approximate noise channels. They are calculated at the 8th
round implementation of the $\llbracket 2,0,0\rrbracket$ code for different
input data-qubits states $|\Psi_{\mathrm{in}}\rangle$; the state-averaged
results are denoted by $\langle\cdots\rangle_{|\Psi_{\mathrm{in}}\rangle}$ and
shown as solid purple lines. (a,b,e,f) Infidelity between the data-qubits
density matrices as a function of gain factor $g$: (a,e) Infidelity between
ideal and approximate density matrices, (b,f) Infidelity between actual and
approximate density matrices. (c,g) Honesty ratio as a function of gain factor
$g$. (d,h) Accuracy ratio as a function of gain factor $g$. Magenta arrow
denotes $g=g_{\mathrm{opt}}$, i.e., the optimal value of $g$ that maximizes
the state-averaged accuracy ratio subject to the honesty constraint [Eq.
(12)], where $g_{\mathrm{opt}}\simeq 1.14$ for the second-order and
$g=g_{\mathrm{opt}}\simeq 1.0005$ for the third-order approximate channel.
Shaded areas in panels (c), (d), (g), and (h) denote regimes where approximate
noise models are not honest (honesty ratio $<1$).

###  V.4 Results and discussion

Figure 6 shows the simulation results for the $\llbracket 2,0,0\rrbracket$
code, computed using the second-order (upper panels) and third-order (lower
panels) approximate noise channels. The results shown are for the 8th round
implementation of the code in a linear geometry using different input data
qubits states $|\Psi_{\mathrm{in}}\rangle$. Figures 6(a) and 6(e) show that
the infidelity between the ideal and approximate data-qubits density matrices
increases linearly with the gain factor $g$. This is because the gain factor
amplifies the noise in the approximate noise channels. Panels (b) and (f) of
Fig. 6 show that the infidelities between the resulting actual and approximate
data-qubits density matrices
$\mathcal{D}^{\mathrm{inf}}_{|\Psi_{\mathrm{in}}\rangle}(\rho_{\mathrm{actual}},\rho_{\mathrm{approx}})$
exhibit a nonmonotonic dependence on the gain factor $g$, where the minimum is
at $g\simeq 0.95-1.1$ and $g\simeq 1$ for the second-order and third-order
approximate noise channels, respectively.

From these infidelities, we calculate the honesty and accuracy ratios as shown
in Figs. 6 (c,g) and 6(d,h), respectively. The honesty ratio increases
linearly with the gain factor $g$ where the honesty constraint (honesty ratio
$\geq 1$) is satisfied for all the four input Bell states only when $g\gtrsim
1.14$ and $g\gtrsim 1$ for the second-order and third-order approximate
channels, respectively. On the other hand, the accuracy ratios have
nonmonotonic dependences on the gain factor $g$ with its maximum occurring at
$g\simeq 0.95-1.1$ for the second-order approximate noise channel and $g\simeq
1$ for the third-order approximate noise channel. The maximum state-averaged
accuracy ratios subject to the honesty constraint are obtained by choosing the
optimal $g=g_{\mathrm{opt}}$ in the honesty regime, where
$g_{\mathrm{opt}}\simeq 1.14$ and $g_{\mathrm{opt}}\simeq 1.0005$ for second-
and third-order approximations, respectively. As shown in Fig. 8 of Appendix
C, the results simulated for the triangle geometry display similar qualitative
behaviors as those for the linear geometry.

Figure 7: Results simulated by using the second-order
$\hat{\mathcal{N}}_{\mathrm{approx},2}(g_{\mathrm{opt}})$ [upper panels: (a),
(b), (c), and (d)] and third-order
$\hat{\mathcal{N}}_{\mathrm{approx},3}(g_{\mathrm{opt}})$ [lower panels: (e),
(f), (g), and (h)] approximate noise channels, where $g_{\mathrm{opt}}$ is the
optimal gain factor chosen to maximize the approximation accuracy subject to
the honesty criterion [Eq. (12)]. The values of $g_{\mathrm{opt}}$ for the
second-order approximate noise channels are $g_{\mathrm{opt}}\simeq 1.14$ for
linear and $g_{\mathrm{opt}}\simeq 1.2$ for triangle geometry. For the third-
order approximation, $g_{\mathrm{opt}}\simeq 1.0005$ for linear and
$g_{\mathrm{opt}}\simeq 1.002$ for triangle geometry. Panels (a,b,e,f): State-
averaged infidelity between the data-qubits density matrices simulated by
using (a,e) ideal and actual noise channels (solid lines), ideal and
approximate noise channels (dotted lines), (b,f) actual and approximate noise
channels (Inset of panel f: Semilog plot). Panels (c,d,g,h): (c,g) state-
averaged honesty ratio calculated using $g_{\mathrm{opt}}$, (d,h) maximum
state-averaged accuracy ratio calculated using $g_{\mathrm{opt}}$.

Based on the simulations of the infidelities at each round of the $\llbracket
2,0,0\rrbracket$ code, we plot the state-averaged results for the second-order
and third-order approximate noise channels as a function of round number in
the upper and lower panels of Fig. 7, respectively. The approximate density
matrices $\rho_{\mathrm{approx}}$ are calculated using the optimal gain factor
($g_{\mathrm{opt}}$) [Eq. (11)], which makes the approximate channels honest
and as accurate as possible. Here we show the results for both linear and
triangular connectivities.

Panels (a), (b), (e) and (f) of Fig. 7 show the state-averaged infidelities of
the data-qubits density matrices calculated after each round of the
$\llbracket 2,0,0\rrbracket$ code, where the averaging is performed over the
four different input Bell states. In Figs. 7(a) and 7(e), the infidelities
between the data-qubits density matrices of the ideal
($\rho_{\mathrm{ideal}}$) and actual ($\rho_{\mathrm{actual}}$) channels are
plotted using solid lines, while the infidelities between the data-qubits
density matrices of the ideal ($\rho_{\mathrm{ideal}}$) and approximate
($\rho_{\mathrm{approx}}$) channels are depicted by dashed lines. Figures 7(b)
and 7(f) show the infidelities between the data-qubits density matrices of the
actual ($\rho_{\mathrm{actual}}$) and approximate ($\rho_{\mathrm{approx}}$)
channels. These infidelities increase with the number of rounds because the
noise decoherence effect accumulates as the number of rounds increases, which
in turn increases the distance between the density matrices. The increase is
larger for the triangle geometry compared to the linear connectivity because
the triangle geometry has one more extra crosstalk channel due to the coupling
between qubits 1 and 2 (see Fig. 4).

The state-averaged honesty ratios for the approximate channels are plotted in
panels (c) and (g) of Fig. 7. The results are calculated using the optimal
gain factor ($g_{\mathrm{opt}}$) which is the gain factor that maximizes the
state-averaged accuracy ratios subject to the constraint that the approximate
channel must be honest (i.e., honesty ratio $\geq 1$ [Eq. (12)]) for all four
Bell input states. Using this optimal gain factor $g_{\mathrm{opt}}$, we
compute the state-averaged accuracy ratios, which are shown in panels (d) and
(h) of Fig. 7. As seen in Figs. 7(d) and 7(h), the accuracy of the
approximation first increases with the number of rounds and for a large enough
round number ($>$ 5 for the second-order and $>$ 3 for the third-order
approximation), the accuracy decreases with the number of rounds due to the
accumulation of the noise decoherence effect. The reason for the nonmonotonic
behavior of the accuracy ratio as a function of round number is because the
distance between $\rho_{\mathrm{ideal}}$ and $\rho_{\mathrm{actual}}$
increases roughly linearly with the number of rounds [Figs. 7(a) and 7(e)]
while the distance between $\rho_{\mathrm{actual}}$ and
$\rho_{\mathrm{approx}}$ grows exponentially with the number of rounds [Figs.
7(b) and 7(f)]. Note that in going from the second-order to the third-order
approximation, the accuracy ratio increases by $\sim 1-2$ orders of magnitude,
which implies that the three-body correlated noise has significant effects on
the actual performance of the code.

##  VI Conclusions and Outlook

We propose a generic perturbative series approach, based on the cluster-
expansion method, for decomposing the Lindbladian noise in a Markovian channel
into components based on the qubit-qubit correlation degrees. Furthermore, we
also present a systematic way to construct honest and accurate approximate
noise channels from the decomposed Lindbladians, where the accuracy
(efficiency) increases (decreases) as we include correlated noise components
with higher and higher correlation degrees. Our work provides a general
technique to characterize correlated errors,which goes beyond the Pauli noise
model [21, 22] and also the commonly assumed approximate noise acting on only
one or two qubits.

By applying the cluster expansion approach to a concrete physical setting,
namely, fixed-frequency superconducting qubits coupled via static always-on
interactions, we show that for realistic parameter sets common to this setup,
correlated noise beyond two qubits have significant effects on the performance
of QEC. While here we focus on superconducting qubits, our technique can
actually be applied to any kind of quantum computing platform. Moreover, our
method can be extended to more complex settings, such as multi-level qubit
systems, where effects due to noncomputational states (e.g., leakage) can be
taken into account, and also to larger number of qubits. Crucially, our
technique gives systematic characterization of the correlated-noise components
of higher and higher degrees as the system size increases. This systematic
order-by-order noise characterization allows us to quantify the smallest
maximum degree of noise correlation that needs to be taken into account for a
certain desired accuracy in the noise modeling of large quantum processors.
Our approach therefore paves the way towards constructing an accurate, honest
and scalable approximate simulation of large quantum devices.

##  Appendix A Superoperator representation

Quantum channels are completely positive and trace preserving (CPTP) linear
maps that act on a $d\times d$ density matrices
$\rho=\sum_{i,j=1}^{d}\rho_{ij}|e_{i}\rangle\langle e_{j}|$. We can represent
quantum channels and their generators (i.e., the Lindbladian for our case
here) in the superoperator representation where they become $d^{2}\times
d^{2}$ matrices acting on a vectorized $d^{2}\times 1$ density operator
$|\rho\rangle\rangle=\sum_{i,j=1}^{d}\rho_{ij}|e_{i}\rangle\otimes|e_{j}\rangle$.
Operators that act on the bra and ket now transform into operators acting on
different copies of the Hilbert space $\mathcal{H}\otimes\mathcal{H}$, i.e.,
$\hat{A}\rho\hat{B}\rightarrow\hat{A}\otimes\hat{B}^{T}|\rho\rangle\rangle$,
where the superscript $T$ indicates a matrix transpose. As a result, in the
superoperator representation, the Lindblad master equation [Eq.(2)] is written
as
$\partial_{t}|\rho(t)\rangle\rangle=\hat{\mathcal{L}}|\rho(t)\rangle\rangle$,
where we use the symbol $\;\hat{}\;$ to denote the superoperator
representation, e.g., $\mathcal{L}\rightarrow\hat{\mathcal{L}}$.

Using the superoperator representation, we write the Lindbladian as

$\displaystyle\hat{\mathcal{L}}$
$\displaystyle=-i(\hat{H}\otimes\mathds{1}-\mathds{1}\otimes\hat{H}^{T})$
$\displaystyle\hskip
14.22636pt+\sum_{j}\gamma_{j}\left(\hat{A}_{j}\otimes\hat{A}_{j}^{*}-\frac{1}{2}\hat{A}_{j}^{\dagger}\hat{A}_{j}\otimes\mathds{1}-\frac{1}{2}\mathds{1}\otimes\hat{A}_{j}^{T}\hat{A}_{j}^{*}\right).$
(23)

As a result, the action of a superoperator on a density matrix becomes simply
a matrix-vector multiplication. Furthermore, the composition of superoperators
is given by a product of the matrix representations of the superoperators. In
the superoperator representation, a quantum channel is therefore given by the
matrix exponentiation of its Lindbladian generator, i.e.,
$\hat{\mathcal{V}}=\hat{T}\left[\exp(\int_{0}^{t}\hat{\mathcal{L}}(t^{\prime})dt^{\prime})\right]$.
At any given time $t$, the state of the system is then given by
$|\rho(t)\rangle\rangle=\hat{\mathcal{V}}|\rho(0)\rangle\rangle$.

##  Appendix B Example applications of the cluster expansion approach

In this Appendix, we give two examples of the application of our cluster
expansion method. As the first example, we consider the simplest case, i.e., a
Lindbladian of a three-qubit quantum processor which consists of only a jump
operator acting on qubit 1:

$\hat{\widetilde{\mathcal{L}}}=\hat{\Lambda}^{\\{1\\}},$ (24)

where
$\Lambda^{\\{1\\}}(\rho)=\gamma_{1}\left(\hat{A}_{1}\rho\hat{A}_{1}^{\dagger}-\frac{1}{2}\hat{A}_{1}^{\dagger}\hat{A}_{1}\rho-\frac{1}{2}\rho\hat{A}_{1}^{\dagger}\hat{A}_{1}\right)$
contains only the jump operator $\hat{A}_{1}$ acting on qubit 1. Using the
cluster expansion in Eq. (III), we have the decomposed Lindbladian as

$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{23}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{1\\}}\otimes\frac{\mathds{1}_{\\{2,3\\}}}{4}\right)\right]\otimes\mathds{1}_{\\{2,3\\}}\right)$
$\displaystyle=\hat{\Lambda}^{\\{1\\}},$ (25a)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{13}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{2\\}}\otimes\frac{\mathds{1}_{\\{1,3\\}}}{4}\right)\right]\otimes\mathds{1}_{\\{1,3\\}}\right)$
$\displaystyle=0,$ (25b)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{3\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{12}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{3\\}}\otimes\frac{\mathds{1}_{\\{1,2\\}}}{4}\right)\right]\otimes\mathds{1}_{\\{1,2\\}}\right)$
$\displaystyle=0,$ (25c)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1,2\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{3}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{1,2\\}}\otimes\frac{\mathds{1}_{\\{3\\}}}{2}\right)\right]\otimes\mathds{1}_{\\{3\\}}\right)$
$\displaystyle\hskip
14.22636pt-\hat{\widetilde{\mathcal{L}}}^{\\{1\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}$
$\displaystyle=0,$ (25d)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1,3\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{2}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{1,3\\}}\otimes\frac{\mathds{1}_{\\{2\\}}}{2}\right)\right]\otimes\mathds{1}_{\\{2\\}}\right)$
$\displaystyle\hskip
14.22636pt-\hat{\widetilde{\mathcal{L}}}^{\\{1\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{3\\}}$
$\displaystyle=0,$ (25e)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{2,3\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{1}\left[e^{\hat{\Lambda}^{\\{1\\}}}\left(\mathds{1}_{\\{2,3\\}}\otimes\frac{\mathds{1}_{\\{1\\}}}{2}\right)\right]\otimes\mathds{1}_{\\{1\\}}\right)$
$\displaystyle\hskip
14.22636pt-\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{3\\}}$
$\displaystyle=0,$ (25f)
$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1,2,3\\}}$
$\displaystyle=\hat{\widetilde{\mathcal{L}}}-\hat{\widetilde{\mathcal{L}}}^{\\{1\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{3\\}}$
$\displaystyle\hskip
14.22636pt-\hat{\widetilde{\mathcal{L}}}^{\\{1,2\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{2,3\\}}-\hat{\widetilde{\mathcal{L}}}^{\\{1,3\\}}$
$\displaystyle=0.$ (25g)

As expected, the cluster expansion gives only a non-trivial Lindbladian acting
on qubit 1. In evaluating Eq. (25), we have used the fact that
$\mathrm{Tr}\left[e^{\hat{\Lambda}^{\\{1\\}}}\rho\right]=1$, since
$e^{\hat{\Lambda}^{\\{1\\}}}$ is a CPTP map which preserves the trace of
density matrices.

As a second example, let us consider the Lindbladian of a three-qubit quantum
processor which comprises a jump operator acting on both qubits 1 and 2, i.e.,

$\hat{\widetilde{\mathcal{L}}}=\hat{\Lambda}^{\\{1,2\\}},$ (26)

where

$\Lambda^{\\{1,2\\}}(\rho)=\gamma_{1,2}\left(\hat{A}_{1,2}\rho\hat{A}_{1,2}^{\dagger}-\frac{1}{2}\hat{A}_{1,2}^{\dagger}\hat{A}_{1,2}\rho-\frac{1}{2}\rho\hat{A}_{1,2}^{\dagger}\hat{A}_{1,2}\right)$
(27)

consists of a correlated jump operator $\hat{A}_{1,2}$ acting on both qubits 1
and 2. We decompose the Lindbladian using the cluster expansion [Eq. (III)] as

$\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{23}\left[e^{\hat{\Lambda}^{\\{1,2\\}}}\left(\mathds{1}_{\\{1\\}}\otimes\frac{\mathds{1}_{\\{2,3\\}}}{4}\right)\right]\otimes\mathds{1}_{\\{2,3\\}}\right)$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{2}\left[\frac{1}{2}e^{\hat{\Lambda}^{\\{1,2\\}}}\right]\right)\equiv\langle\hat{\Lambda}^{\\{1,2\\}}\rangle_{2},$
(28a) $\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{13}\left[e^{\hat{\Lambda}^{\\{1,2\\}}}\left(\mathds{1}_{\\{2\\}}\otimes\frac{\mathds{1}_{\\{1,3\\}}}{4}\right)\right]\otimes\mathds{1}_{\\{1,3\\}}\right)$
$\displaystyle=\mathrm{ln}\left(\mathrm{Tr}_{1}\left[\frac{1}{2}e^{\hat{\Lambda}^{\\{1,2\\}}}\right]\right)\equiv\langle\hat{\Lambda}^{\\{1,2\\}}\rangle_{1},$
(28b) $\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{1,2\\}}$
$\displaystyle=\hat{\Lambda}^{\\{1,2\\}}-\langle\hat{\Lambda}^{\\{1,2\\}}\rangle_{2}-\langle\hat{\Lambda}^{\\{1,2\\}}\rangle_{1},$
(28c) $\displaystyle\hat{\widetilde{\mathcal{L}}}^{\\{3\\}}$
$\displaystyle=\hat{\widetilde{\mathcal{L}}}^{\\{2,3\\}}=\hat{\widetilde{\mathcal{L}}}^{\\{1,3\\}}=\hat{\widetilde{\mathcal{L}}}^{\\{1,2,3\\}}=0.$
(28d)

The total Lindbladian is therefore decomposed into components that act on
qubit 1, qubit 2 and both qubits 1 and 2. The single-qubit components
$\hat{\widetilde{\mathcal{L}}}^{S_{1}}$ (e.g.,
$\hat{\widetilde{\mathcal{L}}}^{\\{1\\}},\hat{\widetilde{\mathcal{L}}}^{\\{2\\}}$)
of the Lindbladian can be thought of as parts of the Lindbladian that act on
each qubit irrespective of the other qubits states, and the two-qubit
components $\hat{\widetilde{\mathcal{L}}}^{S_{2}}$ (e.g.,
$\hat{\widetilde{\mathcal{L}}}^{\\{1,2\\}}$) of the Lindbladian describe pure
two-qubit correlations. Note that while the single-qubit components
$\hat{\widetilde{\mathcal{L}}}^{S_{1}}$ of the Lindbladian obtained from the
cluster expansion approach are guaranteed to be a physical Lindbladian, the
higher-order correlated components $\hat{\widetilde{\mathcal{L}}}^{S_{m>1}}$
are not guaranteed to be physical. However, in this work we find that we can
construct a CPTP approximate noise channel by choosing appropriate gain
factors for each of the Lindbladian components in Eq. (11).

##  Appendix C Simulation results for the triangle geometry

In this Appendix, we present results calculated for the 8th round of the
$\llbracket 2,0,0\rrbracket$ code in the triangle geometry. Figure 8 shows the
infidelities between different quantum channels where the approximate channels
are simulated using the second- and third-order approximations. The overall
qualitative behaviors of the results for the triangle geometry are similar to
those of the linear geometry shown in Fig. 6 of Sec. V.

Figure 8: Results simulated for the triangular connectivity using the second-
order $\hat{\mathcal{N}}_{\mathrm{approx},2}$ [upper panels: (a), (b), (c) and
(d)] and third-order $\hat{\mathcal{N}}_{\mathrm{approx},3}$ [lower panels:
(e), (f), (g) and (h)] approximate noise channels. They are calculated at the
8th round implementation of the $\llbracket 2,0,0\rrbracket$ code for
different input data-qubits states $|\Psi_{\mathrm{in}}\rangle$; the state-
averaged results are denoted by
$\langle\cdots\rangle_{|\Psi_{\mathrm{in}}\rangle}$ and shown as solid purple
lines. (a,b,e,f) Infidelity between the data-qubits density matrices as a
function of gain factor $g$: (a,e) Infidelity between ideal and approximate
density matrices, (b,f) Infidelity between actual and approximate density
matrices. (c,g) Honesty ratio as a function of gain factor $g$. (d,h) Accuracy
ratio as a function of gain factor $g$. Magenta arrow denotes
$g=g_{\mathrm{opt}}$, i.e., the optimal value of $g$ that maximizes the state-
averaged accuracy ratio subject to the honesty criterion [Eq. (12)], where
$g_{\mathrm{opt}}\simeq 1.2$ for the second-order and $g_{\mathrm{opt}}\simeq
1.002$ for the third-order approximate channel. Shaded areas in panels (c),
(d), (g), and (h) denote regimes where approximate noise models are not honest
(honesty ratio $<1$).

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