We mention the $\mathtt{v}$-site mostly for completeness, as other references
work with $\mathtt{v}$-bundles. However, in view of theorem 1.1.1, we can work
with the quasi-pro-étale site instead.

We note that, by design, the quasi-pro-étale topos of a diamond is locally
perfectoid. Indeed, any diamond $X$ is covered by a perfectoid, and hence we
can pullback this cover to any $Y\in X_{\mathtt{qpro\acute{e}t}}$. In
particular, since perfectoids are locally weakly contractible [8, Def. 3.2.1],
the quasi-pro-étale topos of $X$ is locally weakly contractible and hence
_replete_ by [8, Prop. 3.2.3]. This will be important for us in the sequel.

There are natural canonical maps of ringed sites

${X_{\mathtt{v}}}$${X_{\mathtt{qpro\acute{e}t}}}$${X_{\mathtt{\acute{e}t}}}$$\scriptstyle{\lambda}$$\scriptstyle{\nu}$

given by the inclusion functors. These functors will turn out to be an
essential part of the correspondence. We note that the pullback via $\lambda$
and $\nu$ are fully faithful, and the cohomology of étale sheaves agree on all
three sites. [24, Props. 14.7, 14.8]

######  Definition B.0.9 ([24, Def. 15.5]).

Let $X$ be an analytic adic space over $\mathbf{Z}_{p}$. We define the
_diamond associated to $X$_ to be the pro-étale sheaf

$X^{\diamond}\colon\mathsf{Perf}\to\mathsf{Set}\qquad
S\mapsto\left\\{(S^{\sharp},\iota),~{}f\colon S^{\sharp}\to X\right\\}/\cong,$

where $S^{\sharp}$ is a perfectoid space,
$\iota\colon(S^{\sharp})^{\flat}\xrightarrow{\sim}S$ is an isomorphism, and
$f$ is a morphism of adic spaces. This data is considered up to isomorphism of
such triples.

######  Theorem B.0.10 ([24, Lemma. 15.6]).

Let $X$ be an analytic adic space over $\operatorname{{Spa}}\mathbf{Z}_{p}$.
We have $|X^{\diamond}|=|X|$, and $X^{\diamond}$ is locally spatial (and
therefore spatial if $X$ is qcqs). Furthermore, $X^{\diamond}$ is a diamond,
and hence it is $\mathtt{qpro\acute{e}t}$ and $\mathtt{v}$ locally perfectoid.

The associated diamond functor preserves étaleness, and induces an equivalence
of (finite) étale sites $X_{\mathtt{\acute{e}t}}\cong
X^{\diamond}_{\mathtt{\acute{e}t}}$, $X_{\mathtt{f\acute{e}t}}\cong
X_{\mathtt{f\acute{e}t}}^{\diamond}$.

We note that when proving that when proving that $X^{\diamond}$ there are two
keys steps: the first is to show that this is indeed a pro-étale sheaf, which
reduces to showing that the functor

$(\operatorname{{Spa}}\mathbf{Z}_{p})^{\diamond}\colon
S\mapsto\left\\{(S^{\sharp},\iota)\mid\iota(S^{\sharp})^{\flat}\xrightarrow{\sim}S\right\\}/\cong$

parametrizing untilts of $S$ is a pro-étale sheaf. The next step is to show
that any analytic adic space over $\operatorname{{Spa}}\mathbf{Z}_{p}$ is the
quotient of a perfectoid by a pro-étale equivalence relation. This information
is crucial to the proof of our main theorem.

We also note that the procedure $X\mapsto X^{\diamond}$ does lose some
information, as this functor is not fully faithful. This procedure preserves
information of a more “topological” nature (such as the étale site).

Now if $X$ is a rigid analytic variety, then we have a structure sheaf
$\mathcal{O}_{X}$ on $X_{\mathtt{\acute{e}t}}$, which we can pullback to a
quasi-pro-étale/$\mathtt{v}$-sheaf. We can then define the following completed
version of the structure sheaf on these sites.

######  Definition B.0.11 (The completed structure sheaf).

Let $X$ be an analytic adic space over a non-archemidean field $K$. The
_completed structure sheaf_ , or the _quasi-pro-étale structure sheaf_ ,
$\widehat{\mathcal{O}}_{X}$ is defined to be the sheaf

$\widehat{\mathcal{O}}_{X}^{+}=\lim_{n}\nu^{-1}\mathcal{O}_{X}^{+}/p^{n};\quad\widehat{\mathcal{O}}_{X}=\widehat{\mathcal{O}}_{X}^{+}\left[\frac{1}{p}\right]$

where the limit is taken as quasi-pro-étale sheaves. A similar definition also
works within $X_{\mathtt{v}}$.

The cohomology of $\widehat{\mathcal{O}}_{X}$ captures interesting phenomena
of the quasi-pro-étale topology of rigid-analytic varieties. First, there is a
acyclicity phenomena for affinoid perfectoid.

######  Theorem B.0.12 ([24, Prop. 8.5 (iii)]).

Let $X$ be a rigid-analytic space over $K$ and
$Y=\operatorname{{Spa}}(R,R^{+})\in X_{\mathtt{qpro\acute{e}t}}$ an affinoid
perfectoid. Then
$\mathcal{O}_{X}(Y)\xrightarrow{\sim}\widehat{\mathcal{O}}_{X}(Y)$ and
$R\Gamma(Y,\widehat{\mathcal{O}}_{X})=\Gamma(Y,\mathcal{O}_{X})=R$.

In particular, if $X$ is a rigid-analytic variety, which from now on we always
see as a diamond, and $Y\to X$ is an affinoid perfectoid quasi-pro-étale cover
of $X$, then we can compute the cohomology
$R\Gamma(X,\widehat{\mathcal{O}}_{X})$ as the Čech nerve of this cover.

On the other hand, if $X$ is proper, then the cohomology of
$\widehat{\mathcal{O}}_{X}$ actually captures the étale cohomology of $X$.
First, note that We can consider $C$ as a sheaf of rings on the
$\mathtt{qpro\acute{e}t}$-site of $X$ by considering _continuous_ maps into
it; that is, we consider $C$ as the sheaf

$\underline{C}\colon
X_{\mathtt{qpro\acute{e}t}}^{\mathtt{op}}\to\mathsf{Rings},\quad
Y\mapsto\operatorname{Hom}(|Y|,C),$

where the morphisms are taken in the category of topological spaces. When it
is clear from context, we will denote the sheaf $\underline{C}$ by simply $C$.
We note that by repleteness of the quasi-pro-étale topos, we have that
$\underline{\mathcal{O}_{C}}={\operatorname{\mathit{R}}}\lim_{n}\underline{\mathcal{O}_{C}}/p^{n}$
[8, Prop. 3.1.10], and hence ${\operatorname{\mathit{R}}}\Gamma(X,C)$ computes
the étale cohomology of $X$ with $C$-coefficients.

######  Theorem B.0.13 (The generic version of the primitive comparision
theorem).

Let $X$ be a proper rigid analytic variety over $C$. Then the natural map
$C\to\widehat{\mathcal{O}}_{X}$ induces an equivalence of
$\mathbf{E}_{\infty}$-algebras

$R\Gamma(X,C)\xrightarrow{\sim}R\Gamma(X,\widehat{\mathcal{O}}_{X}),$

where we are taking global sections in $\mathtt{qpro\acute{e}t}$ site 777 Here
we can also use the $\mathtt{v}$-site using an analogous structure sheaf
$\check{\mathcal{O}}_{X}$ (see [19, Def. 2.1]). The relevance of this remark
is that the proof technically involves $\mathtt{v}$-cohomology, but this is
inessential [19, Lemma 2.9]. .

###### Proof.

The primitive comparison theorem [18, Cor. 3.9.24], says that

${\operatorname{\mathit{R}}}\Gamma(X,\mathcal{O}_{C}/p)\xrightarrow[a]{\sim}{\operatorname{\mathit{R}}}\Gamma(X,\widehat{\mathcal{O}}_{X}^{+}/p)$

is an almost quasi-isomorphism. The result now follows from a “almost derived
Nakayama” argument: from the exact sequences $0\to
p^{n}\mathcal{O}_{C}/p^{n+1}\to\mathcal{O}_{C}/p^{n+1}\to\mathcal{O}_{C}/p^{n+1}\to
0$ and $0\to
p^{n}\widehat{\mathcal{O}}_{X}^{+}/p^{n+1}\to\widehat{\mathcal{O}}_{X}^{+}/p^{n+1}\to\widehat{\mathcal{O}}_{X}^{+}/p^{n}\to
0$ we get by induction

${\operatorname{\mathit{R}}}\Gamma(X,\mathcal{O}_{C}/p^{n})\xrightarrow[a]{\sim}{\operatorname{\mathit{R}}}\Gamma(X,\widehat{\mathcal{O}}_{X}^{+}/p^{n}),$

so now the result follows by taking ${\operatorname{\mathit{R}}}\lim$, since
${\operatorname{\mathit{R}}}\Gamma$ commutes with it and by the fact that
$X_{\mathtt{qpro\acute{e}t}}$ is replete we have
$\mathcal{O}_{C}={\operatorname{\mathit{R}}}\lim_{n}\mathcal{O}_{C}/p^{n}$ and
$\widehat{\mathcal{O}}_{X}^{+}={\operatorname{\mathit{R}}}\lim_{n}\widehat{\mathcal{O}}_{X}^{+}/p^{n}$.
∎

######  Remark B.0.14.

This proof, as stated, follows from the development of a $6$-functor formalism
of $p$-torsion sheaves on diamonds. The original statement is due to Scholze
on [22, Thm. 1.3], using the flattened pro-étale topology.

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# Spin chain model for correlated quantum channels

Davide Rossini1, Vittorio Giovannetti2 and Simone Montangero2 1 International
School for Advanced Studies SISSA/ISAS, via Beirut 2-4, I-34014 Trieste, Italy
2 NEST-CNR-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7 , I-56126
Pisa, Italy 222URL: www.qti.sns.it

(September 2, 2024)

###### Abstract

We analyze the quality of the quantum information transmission along a
correlated quantum channel by studying the average fidelity between input and
output states and the average output purity, giving bounds for the entropy of
the channel. Noise correlations in the channel are modeled by the coupling of
each channel use with an element of a one dimensional interacting quantum spin
chain. Criticality of the environment chain is seen to emerge in the changes
of the fidelity and of the purity.

##  1 Introduction

The common scenario in quantum communication protocols is constituted by two
distant parties, Alice and Bob, who want to exchange information through a
quantum communication link. Due to unavoidable noise in the channel, this
cannot be perfectly accomplished, and some strategies aimed to reduce
communication errors have to be employed. These are based on complex
encoding/decoding operations and on suitably tailoring the physical system
that acts as a channel. In this context the effect of noise on the quantum
communication is typically quantified by the so called capacities of the
channel, that is the optimal rates at which (quantum or classical) information
can be reliably transmitted in the limit of infinite channel uses [1]. The
vast majority of the results obtained so far focused on the case of memoryless
quantum channels, where the noise acts independently for each channel use.
However, in real physical situations, correlations in the noise acting between
successive uses can be established. When this happens the communication line
is said to be a memory channel, or more precisely, a correlated channel. The
analysis of these setups is much more demanding than the memoryless case, and,
at present, only a restricted class of them has been solved [2, 3, 4, 5, 6, 8,
9, 7].

Recently, a physical model for representing correlated channels has been
proposed in Refs. [4, 10], which, in the context of Bosonic channels and qubit
channels respectively, has established a direct connection between these
systems and many-body physics. The setup discussed in these proposals is
depicted in Fig. 1. Here Alice sends her messages to Bob by encoding them into
a $n$-long sequence of information carriers $S$ (the red dots of the figure),
which model subsequent channel uses associated with $n$ independent Bosonic
modes [4] or $n$ independent spins [10]. The correlated noise of the channel
is then described by assuming that each carrier interacts independently with a
corresponding element of a $n$-party environment $E$ (sketched with the
connected black dots of the figure), which, in Refs. [4] and [10], represents
a multi-mode Gaussian state and a many-body spin state, respectively. Thus
given an input state $\rho_{S}$ of the $n$ carriers, the corresponding output
state associated with the channel is

${\cal E}_{n}(\rho_{S})=\mbox{Tr}_{E}[{\cal
U}\,(\rho_{S}\otimes\sigma_{E})\,{\cal U}^{\dagger}]\;,$ (1)

where $\sigma_{E}$ is the joint input state of $E$ and the partial trace is
performed over the environment. In this equation ${\cal U}$ represents the
unitary coupling between $S$ and $E$, which is expressed as

$\displaystyle{\cal U}=\bigotimes_{\ell=1}^{n}U^{(\ell)}\;,$ (2)

with $U^{(\ell)}$ being the interaction between the $\ell$-th carrier and its
environmental counterpart (in Ref. [4] these were beam-splitter couplings,
while in Ref. [10] they were phase-gate couplings). Within this framework,
memoryless channels ${\cal E}_{n}={\cal E}^{\otimes n}$ are obtained for
factorizable environmental input states, while correlated noise models
correspond to correlated environmental states $\sigma_{E}$. Interestingly
enough, in Ref. [10] it was shown that it is possible to relate the quantum
capacity [11] of some specific channels (1) to the properties of the many-body
environment $E$.

Figure 1: Model for memory channels: The red dots represent the channel uses
(ordered, for instance, starting from the left to the right). The channel
noise is modeled as a collection of local interactions between the channel
uses and the many-body environment $E$ (black dots).

In this paper we discuss a variation of the model (1), which allows us to
adapt some of the techniques used in Ref. [12] for characterizing the
decoherence effects induced by spin quantum baths, in order to analyze the
efficiency of a class of correlated qubits channels. To do so we consider a
unitary coupling ${\cal U}$ that does not factorize as in Eq. (2). Instead we
assume $E$ to be a spin chain characterized by a free Hamiltonian
$\mathcal{H}_{E}$, whose elements interact with the carriers $S$ through the
local Hamiltonian $\mathcal{H}_{\rm int}$. With this choice we write

$\displaystyle{\cal U}=\exp[-i\,(\mathcal{H}_{E}+\mathcal{H}_{\rm
int})\,t]\;,$ (3)

with the interaction time $t$ being a free parameter of the model. In
particular, as a chain Hamiltonian, we consider a spin-$1/2$ $XY$ model in a
transverse field, which can exhibit, in some parameters region, ground state
critical properties that greatly enhance spin correlations [13]. Therefore the
distance of the chain from criticality is non trivially related to memory
effects in the channel. In the second part of the paper, we generalize the
previous scheme by introducing a given number of $m$ extra spins between any
two consecutive qubits, as shown in Fig. 2. In this case, we can use the
number $m$ to modulate the memory effects.

Figure 2: Generalized model of spin chain memory channels. As an example, in
this figure we set $m=2$.

##  2 The Model

As the environment $E$ of the system in Fig. 1 we consider an interacting one-
dimensional quantum spin-$1/2$ chain described by an $XY$ exchange Hamiltonian
in a transverse magnetic field:

$\mathcal{H}_{E}=-\frac{J}{2}\sum_{j=1}^{n}\left[(1+\gamma)\sigma_{j}^{x}\sigma_{j+1}^{x}+(1-\gamma)\sigma_{j}^{y}\sigma_{j+1}^{y}+2\lambda\sigma_{j}^{z}\right]\;,$
(4)

where $\sigma^{\alpha}_{j}$ (with $\alpha=x,y,z$) are the Pauli matrices of
the $j$-th spin, $J$ is the coupling strength between neighboring spins, and
$\lambda$ is the external field strength 111 Hereafter we always use open
boundary conditions, therefore we assume
$\sigma^{\alpha}_{n}\sigma^{\alpha}_{1}=0$.. The model in Eq. (4) for
$0<\gamma\leq 1$ belongs to the Ising universality class, and has a critical
point at $\lambda_{c}=1$; for $\gamma=0$ it reduces to the $XX$ universality
class, that is critical for $|\lambda|\leq 1$ [13].

Following Ref. [12], we then assume that each carrier qubit is coupled to one
environmental spin element through the coupling Hamiltonian

$\mathcal{H}_{\rm int}(j)=-\varepsilon|e\rangle_{j}\langle
e|\otimes\sigma^{z}_{j}\;,$ (5)

where $\left|g\right\rangle_{j}$ and $\left|e\right\rangle_{j}$ respectively
represent the ground and the excited state of the $j$-th qubit. Hence the
total Hamiltonian $\mathcal{H}\equiv\mathcal{H}_{E}+\mathcal{H}_{\rm int}$ is
given by

$\mathcal{H}=-\frac{J}{2}\sum_{j=1}^{n}\left[(1+\gamma)\sigma_{j}^{x}\sigma_{j+1}^{x}+(1-\gamma)\sigma_{j}^{y}\sigma_{j+1}^{y}+2\lambda\sigma_{j}^{z}\right]-\varepsilon\sum_{j=1}^{n}|e\rangle_{j}\langle
e|\sigma_{j}^{z}.$ (6)

Finally, as in Refs. [10, 12], we suppose that at time $t=0$ the environment
chain is prepared in the ground state $\left|\varphi\right\rangle_{E}$ of
${\cal H}_{E}$. We then consider a generic input state
$\left|\psi\right\rangle_{S}$ of the $n$ qubit carriers of the system (i.e.,
the input state of the red dots in Fig. 1), and write it in the computational
basis:

$\left|\psi\right\rangle_{S}=\sum_{x}\alpha_{x}\left|x\right\rangle_{S}\,,$
(7)

where $\alpha_{x}$ are complex probability amplitudes and the sum runs over
$N=2^{n}$ possible choices of $x$, each of them being a binary string of $n$
elements in which the $j$-th element is represented as $g$ or $e$, according
to the state (ground or excited, respectively) of the corresponding $j$-th
qubit.

For each vector $\left|x\right\rangle_{S}$ we define ${\cal S}_{x}$ as the set
of the corresponding excited qubits (for instance, given $n=5$ and
$\left|x\right\rangle_{S}=\left|egeeg\right\rangle_{S}$, then ${\cal S}_{x}$
contains the $1^{\rm st}$, $3^{\rm rd}$ and $4^{\rm th}$ qubits). After a time
$t$, the global state of the qubits and the chain will then evolve into

$\left|\psi\right\rangle_{S}\otimes\left|\varphi\right\rangle_{E}\quad\stackrel{{\scriptstyle\cal
U}}{{\longrightarrow}}\quad\sum_{x}\alpha_{x}\left|x\right\rangle_{S}\otimes{\cal
U}_{x}\left|\varphi\right\rangle_{E}\;,$ (8)

where ${\cal U}$ is the global evolution operator of Eq. (3), while ${\cal
U}_{x}\equiv\exp[-i\,\mathcal{H}_{E}^{x}\,t]$ is associated to the following
chain Hamiltonian:

$\mathcal{H}_{E}^{x}\equiv\mathcal{H}_{E}-\varepsilon\sum_{j\in{\cal
S}_{x}}\sigma_{j}^{z}\,.$ (9)

According to Eq. (1), the channel output state is then described by the
density matrix

${\cal
E}_{n}(|\psi\rangle_{S}\langle\psi|)=\rho_{S}^{\prime}=\sum_{x,y}L_{xy}\;\alpha_{x}\alpha_{y}^{*}\;|x\rangle_{S}\langle
y|\;,$ (10)

where

$L_{xy}\equiv{{}_{E}\langle}\varphi|\,{\cal U}_{y}^{\dagger}\,{\cal
U}_{x}\left|\varphi\right\rangle_{E}\;,$ (11)

can be seen as a generalized Loschmidt echo, denoting the scalar product of
the input environment state $\left|\varphi\right\rangle_{E}$ evolved with
${\cal U}_{x}$ and ${\cal U}_{y}$, respectively [14]. These quantities can be
evaluated by first mapping the Hamiltonian (9) into a free-fermion model via a
Jordan Wigner transformation [15]

$c_{k}=\exp\Bigg{(}i\pi\sum_{j=1}^{k-1}\sigma_{j}^{+}\sigma_{j}^{-}\Bigg{)}\,\sigma_{k}^{-}\;,$
(12)

where $\sigma^{\pm}=(\sigma^{x}\pm i\sigma^{y})/2$, and then by diagonalizing
it with a Bogoliubov rotation of the Jordan Wigner fermions
$\\{c^{\dagger}_{k},c_{k}\\}_{k=1,\ldots,n}$. This allows one to find an
explicit expression of the Loschmidt echo in terms of the determinant of a
$2n\times 2n$ matrix (see Ref. [12] for details):

$L_{xy}={{}_{E}\langle}\varphi|e^{i\mathcal{H}_{E}^{x}t}e^{-i\mathcal{H}_{E}^{y}t}|\varphi\rangle_{E}={\rm
det}(\mathbb{I}-\rho_{0}+\rho_{0}e^{iH_{x}t}e^{-iH_{y}t})\,,$ (13)

where $\mathcal{H}_{k}=\sum_{ij}[H_{k}]_{ij}\Psi^{\dagger}_{i}\Psi_{j}$,
${\bf\Psi^{\dagger}}=\left(c_{1}^{\dagger}\ldots c_{N}^{\dagger}\,c_{1}\ldots
c_{N}\right)$, and
$[\rho_{0}]_{ij}={{}_{E}\langle}\varphi|\Psi_{i}^{\dagger}\Psi_{j}|\varphi\rangle_{E}$
are the two-point correlation functions of the chain.

##  3 The channel

The echoes (11) provide a complete characterization of the correlated channel
${\cal E}_{n}$. In particular, since $L_{xx}=1$ for all $x$, Eq. (10) shows
that the channel ${\cal E}_{n}$ is unital, i.e. it maps the completely mixed
state $\frac{1}{N}\sum_{x}|x\rangle_{S}\langle x|$ into itself. Furthermore,
the $N\times N$ matrix of elements $L_{xy}/N$ coincides with the Choi-
Jamiolkowski state [16] of the map. The latter is defined as the output
density matrix obtained when sending through the channel ${\cal E}_{n}$ half
of the canonical maximally entangled state
$|+\rangle_{SA}\equiv\frac{1}{\sqrt{N}}\sum_{x}|x\rangle_{S}\otimes|x\rangle_{A}$
of the $N$-level system $S$, i.e.

$\displaystyle J({\cal E}_{n})\equiv({\cal E}_{n}\otimes{\cal
I}_{A})(|+\rangle_{SA}\langle+|)=\sum_{x,y}\frac{L_{xy}}{N}\;|xx\rangle_{SA}\langle
yy|\;,$ (14)

with $A$ being a $N$-dimensional ancillary system and ${\cal I}_{A}$ being the
identity map. Similarly to the case analyzed in Ref. [10], this is a maximally
correlated state [17] whose 1-way distillable entanglement is known to
coincides with the “hashing bound” [17, 18, 19]:

$\displaystyle D_{1}(J({\cal E}_{n}))=H(J_{S}({\cal E}_{n}))-H(J({\cal
E}_{n}))=\log_{2}N-H(J({\cal E}_{n}))\;,$ (15)

where $J_{S}({\cal E}_{n})\equiv\mbox{Tr}_{A}[J({\cal E}_{n})]$ is the reduced
density matrix of $J({\cal E}_{n})$ associated with the system $S$, while
$H(\cdot)=\mbox{Tr}[(\cdot)\log_{2}(\cdot)]$ is the von Neumann entropy. At
least for the subclass of forgetful channels [8], the regularized version of
Eq. (15) can then be used [18, 10] to bound the quantum capacity [1, 11] of
${\cal E}_{n}$. This is 222The inequality (16) is a consequence of the fact
that the quantum capacity $Q$ of a channel does not increase if we provide the
communicating parties with a 1-way (from the sender to the receiver) classical
side communication line [18, 20]. It is derived by constructing an explicit
quantum communication protocol in which i) Alice sends through the channel
half of the maximally entangled state $|+\rangle_{SA}$ to Bob, ii) the
resulting state $J({\cal E}_{n})$ is then 1-way distilled obtaining
$D_{1}(J({\cal E}_{n}))$ Bell pairs which, finally, iii) are employed to
teleport Alice messages to Bob. It is worth noticing that for the channel
analyzed in Ref. [10] the right hand side of Eq. (15) was also an upper bound
for $Q$.

$\displaystyle Q({\cal
E}_{n})\geqslant\lim_{n\rightarrow\infty}\frac{D_{1}(J_{S}({\cal
E}_{n}))}{n}=1-\lim_{n\rightarrow\infty}\frac{H(J({\cal E}_{n}))}{n}\;.$ (16)

The quantity $H(J({\cal E}_{n}))$ corresponds to the entropy of the channel
${\cal E}_{n}$ of Ref. [21], which can be used as an estimator of the channel
noise. In our case it has also a simple interpretation in terms of the
properties of the many-body system $E$: it measures the entropy of the ground
state $|\varphi\rangle_{E}$ after it has evolved through a random application
of the perturbed unitaries ${\cal U}_{x}$ 333 This is a trivial consequence of
the fact that $J({\cal E}_{n})$ is the reduced density matrix of the pure
state $\frac{1}{\sqrt{N}}\sum_{x}|xx\rangle_{SA}\otimes{\cal
U}_{x}|\varphi\rangle_{E}$ tracing out the environment, and of the fact that
the von Neumann entropies of the reduced density matrices of a pure bipartite
system coincide., i.e.

$\displaystyle H(J({\cal
E}_{n}))=H(\sigma_{E}^{\prime})\;,\qquad\mbox{with}\qquad\sigma_{E}^{\prime}\equiv\frac{1}{N}\sum_{x}{\cal
U}_{x}\;|\varphi\rangle_{E}\langle\varphi|\;{\cal U}^{\dagger}_{x}\;.$ (17)

Unfortunately, for large $n$ the computation of the von Neumann entropy of the
state $J({\cal E}_{n})$ is impractical both analytically and numerically,
since it requires to evaluate an exponential number of $L_{xy}$ elements.
Interestingly enough, however, we can simplify our analysis by considering the
fidelity between $J({\cal E}_{n})$ and its input counterpart $|+\rangle_{SA}$
(see Eq. (14)). As discussed in the following section, this is a relevant
information theoretical quantity, since it is directly related to the average
fidelity between input and output state of the channel ${\cal E}_{n}$ and
provides us an upper bound for $H(J({\cal E}_{n}))$. Similarly we can compute
the purity of $J({\cal E}_{n})$ which, on one hand, gives a lower bound for
$H(J({\cal E}_{n}))$, while, on the other hand, it is directly related to the
average channel output purity of the map ${\cal E}_{n}$.

Figure 3: Averaged channel fidelity as a function of the interaction time, for
different values of the transverse field $\lambda$: from right to left
$\lambda=0.25,\,0.5,\,0.75,\,0.9,\,1,\,1.1,\,1.5,\,2$. Here we simulated a
channel of $n=50$ qubits coupled to an Ising chain, and set an interaction
strength $\varepsilon=0.05$; the fidelity has been evaluated by sampling over
$N_{av}=5\times 10^{4}$ randomly chosen initial conditions. In the inset we
plot the averaged fidelity at a fixed interaction time $Jt^{*}=5$, as a
function of $\lambda$.

##  4 Average transmission fidelity

According to Eq. (14), the fidelity between the Choi-Jamiolkowski state
$J({\cal E}_{n})$ and its input counterpart coincides with the average value
of the Loschmidt echoes $L_{xy}$, i.e.

$\displaystyle{\cal F}\equiv{}_{SA}\langle+|J({\cal
E}_{n})|+\rangle_{SA}=\frac{1}{N^{2}}\sum_{x,y}L_{xy}\;.$ (18)

Even without computing all the $L_{xy}$, this quantity can be numerically
evaluated by performing a sampling over $N_{av}$ randomly chosen couples
($x,y$) of initial conditions, and averaging over them 444 We numerically
checked the convergence of ${\cal F}$ with $N_{av}$. We first considered a
situation with a few number of qubits ($n\leq 10$), such to compare sampled
averages, ${\cal F}_{av}$, with exact averages over all possible events,
${\cal F}_{ex}={\cal F}$. We found that, already at $N_{av}=10^{4}$, absolute
differences $|{\cal F}_{av}-{\cal F}_{ex}|$ are always less than $2\times
10^{-2}$, while at $N_{av}=5\times 10^{4}$ the error is less than $5\times
10^{-3}$, independently of the values of the interaction time $t$, the
transverse field $\lambda$ and the system size $n$. In a second time, we
simulated systems with definitely larger sizes ($n\approx 50$) and simply
check the convergence of ${\cal F}_{av}$ with $N_{av}$. Differences between
fidelities with $N_{av}=10^{4}$ and $N_{av}=5\times 10^{4}$ are of the same
order as the deviation of the curve with $N_{av}=10^{4}$ from the exact one
for small sizes. Therefore we can reliably affirm that fidelity results with
$N_{av}=5\times 10^{4}$ are exact, up to an absolute error of order $5\times
10^{-3}$. :

${\cal F}\approx{\cal F}_{av}\equiv\frac{1}{N_{av}}\sum_{(x,y)=1}^{N_{av}}{\rm
Re}[L_{xy}]\,,$ (19)

(where we used the fact that $L_{yx}=L_{xy}^{*}$). The quantity ${\cal F}$
provides an upper bound for $H(J({\cal E}_{n}))$ through the quantum Fano
inequality [22], i.e.

$\displaystyle H(J({\cal E}_{n}))\leqslant H_{2}({\cal F})+(1-{\cal
F})\log_{2}(4^{n}-1)\leqslant H_{2}({\cal F})+2n(1-{\cal F})\;,$ (20)

where $H_{2}(\cdot)=-(\cdot)\log_{2}(\cdot)-[1-(\cdot)]\log_{2}[1-(\cdot)]$ is
the binary entropy function555Equation (20) can be easily derived by noticing
that $H(J({\cal E}_{n}))$ and ${\cal F}$ coincide, respectively, with the
exchange entropy and entanglement fidelity of the channel ${\cal E}_{n}$
associated with the maximally mixed state $\mathbb{I}_{S}/2^{n}$ of $S$..
Furthermore ${\cal F}$ is directly related to the average transmission
fidelity $\langle F\rangle$ of the map ${\cal E}_{n}$. For a given pure input
state (7), the transmission fidelity is

$F(\psi)\equiv{{}_{S}\langle\psi}|{\cal
E}_{n}(|\psi\rangle_{S}\langle\psi|)\left|\psi\right\rangle_{S}=\sum_{x,y}L_{xy}|\alpha_{x}|^{2}|\alpha_{y}|^{2}\;.$
(21)

Taking the average with respect to all possible inputs, we get

$\langle F\rangle=\sum_{x,y}L_{xy}\,p_{xy}\,,$ (22)

where $p_{xy}=\langle|\alpha_{x}|^{2}|\alpha_{y}|^{2}\rangle$ with
$\langle...\rangle$ being the average with respect to the uniform Haar
measure.

Figure 4: Fidelity for a channel coupled to an Ising chain with $\lambda=1$
and different qubit numbers $n$: from right to left
$n=4,\,6,\,8,\,10,\,16,\,30,\,50$; the interaction strength is kept fixed at
$\varepsilon=0.05$; data are averaged over $N_{av}=5\times 10^{4}$
configurations. Inset: ${\mathcal{F}}_{av}$ at a fixed time $t^{*}$, as a
function of $n$.

The probability distribution $p_{xy}$ can be computed by using simple
geometrical arguments [23]. As shown in A, this yields
$p_{xy}=\frac{1+2\,\delta_{x,y}}{N(N+2)}$ and hence:

$\langle F\rangle=\left(\frac{2}{N(N+2)}\sum_{x>y}{\rm
Re}[L_{xy}]\right)+\frac{3}{N+2}\,,$ (23)

where we used the fact that $L_{xx}=1$ and $L_{xy}=L_{yx}^{*}$. Therefore,
from Eq. (18) we get

$\displaystyle\langle F\rangle=\frac{N}{N+2}\;{\cal F}+\frac{2}{N+2}\;.$ (24)

The fidelities ${\cal F}$ and $\langle F\rangle$ are not directly related to
the channel quantum capacity, nonetheless, as in Eq. (20), they can be used to
derive bounds for $Q$ 666In particular from Eq. (20) and Eq. (16) one gets
$Q\geqslant 1-2\lim_{n\rightarrow\infty}{\cal F}$.. More generally, values
near to unity of the fidelity between the output channel states and their
corresponding input states, are indicative of a fairly noiseless communication
line. On the contrary, values of the transmission fidelities close to zero,
while indicating output states nearly orthogonal to their input counterparts,
do not necessarily imply null or low capacities, since such huge discrepancies
between inputs and outputs could still be corrected by a proper encoding and
decoding strategy (e.g. consider the case of a channel which simply rotate the
system states).

Figure 5: Short-time Gaussian decay rate $\alpha$ as a function of the
transverse field $\lambda$. We rescaled $\alpha$ with the system size
according to the scaling of Fig. 4. In the inset we plot the first derivative
of the same curves in the main panel, with respect to $\lambda$.

Equation (13) allows us to numerically compute the averaged transmission
fidelity (19): numerical results in this and in the next sections are given
for the case $\gamma=1$, i.e. we study the correlated quantum channel defined
by the Ising model. The behavior of the averaged channel fidelity
${\mathcal{F}}_{av}$, defined in Eq. (19) and related to the memory channel
scheme of Fig. 1, with respect to the interaction time $t$ (the free model
parameter) is shown in Fig. 3: the plots are given for $n=50$ qubits, each of
them coupled to one spin of an Ising chain with a coupling strength
$\varepsilon=0.05$. Different curves stand for different values of the
transverse magnetic field $\lambda$: as it can be clearly seen, the fidelity
${\mathcal{F}}_{av}$ decays as a Gaussian in time, irrespective of the field
strength $\lambda$ 777 For times longer than those in the scales of Figs. 3
and 4, revivals of the fidelity are present. See B for details. . The
signature of criticality in the environment chain can be identified by
studying the function ${\mathcal{F}}_{av}(\lambda,t)$ for fixed interaction
time $t^{*}$: the inset of Fig. 3 displays a non analytic behavior for the
derivative of ${\mathcal{F}}_{av}(\lambda,t^{*})$ with respect to $\lambda$ at
the critical point $\lambda_{c}=1$. This non analyticity will be clearer in
the following, where the size scaling will be considered. In Fig. 4 we study
the behavior of ${\mathcal{F}}_{av}$ with the number $n$ of qubits, at a fixed
value of transverse magnetic field $\lambda=1$. As it is shown in the inset,
at a given interaction time $t^{*}$, the fidelity
${\mathcal{F}}_{av}(\lambda_{c},t^{*})$ decays exponentially with $n$ (the
same behavior is found when $\lambda\neq 1$). This dependence of the decay
rate implies that the average fidelity ${\mathcal{F}}_{av}$ can be fitted by:

${\mathcal{F}}_{av}\sim e^{-\alpha t^{2}}\qquad\textrm{ with
}\quad\alpha\propto n\,;$ (25)

in other words, the Gaussian decay rate is extensive. Indeed, since the
fidelity is a global quantity that describes the evolution of the state of the
whole $n$-body system, it should start decaying as a Gaussian (at least at
small times) [24], with a reasonably extensive decay ratio. This prediction is
confirmed by the results of Fig. 5, where we report the decay rate as a
function of the transverse magnetic field for different system sizes $n$. In
proximity of the critical point, the decay rate undergoes a sudden change,
which becomes more evident when increasing the system size. The signature of
criticality at $\lambda_{c}=1$ and the finite size effects can be better
analyzed by looking at the derivative of the decay rate with respect to the
transverse field: The inset of Fig. 5 clearly show that
$\partial_{\lambda}\alpha$ exhibits a non analytic behavior at the critical
point $\lambda_{c}$, at the thermodynamical limit. Notice also that, due to
finite size effects, the maximum of $\partial_{\lambda}\alpha$ does not
coincide exactly with the critical point, that can be rigorously defined only
at the thermodynamical limit, but occurs at a slightly smaller value of
$\lambda$. However, we checked that a finite size scaling gives the right
prediction of the critical point located at $\lambda=1$.

##  5 Average output purity

Another quantity that can be evaluated with relatively little numerical effort
is the purity of the Choi-Jamiolkowski state $J({\cal E}_{n})$, i.e.

$\displaystyle{\cal P}_{2}\equiv\mbox{Tr}[J({\cal
E}_{n})^{2}]=\frac{1}{N^{2}}\sum_{xy}|L_{xy}|^{2}\;.$ (26)

As in the case of ${\cal F}$, this can be computed by approximating the
summation with a random sampling, i.e.

${\cal P}_{2}\approx{\cal
P}_{av}\equiv\frac{1}{N_{av}}\sum_{(x,y)=1}^{N_{av}}|L_{xy}|^{2}\,.$ (27)

The quantity (26) provides us two important pieces of information. First of
all, it yields a useful bound on the channel entropy $H(J({\cal E}_{n}))$.
This follows from the inequality [25]

$\displaystyle H(J({\cal E}_{n}))\geqslant H_{2}(J({\cal
E}_{n}))=-\log_{2}{\cal P}_{2}\;,$ (28)

with $H_{2}(\cdot)\equiv-\log_{2}\mbox{Tr}[(\cdot)^{2}]$ being the Rényi
entropy of order 2. Furthermore ${\cal P}_{2}$ is directly related to the
average output purity $\langle P_{2}\rangle$ of the channel ${\cal E}_{n}$.
This is obtained by averaging over all possible inputs $|\psi\rangle_{S}$ the
purity of the output state ${\cal E}_{n}(|\psi\rangle_{S}\langle\psi|)$, i.e.

$\langle P_{2}\rangle\equiv\langle{\rm Tr}\left[({\cal
E}_{n}(|\psi\rangle_{S}\langle\psi|))^{2}\right]\rangle=\sum_{x,y}|L_{xy}|^{2}\,\langle|\alpha_{x}|^{2}|\alpha_{y}|^{2}\rangle=\sum_{x,y}|L_{xy}|^{2}\,p_{xy}\,,$
(29)

where we used Eq. (10) and where $p_{xy}$ are the probabilities defined in Eq.
(32). According to Eq. (26) this yields,

$\langle P_{2}\rangle=\frac{N}{N+2}{\cal P}_{2}+\frac{2}{N+2}\;.$ (30)

Figure 6: Averaged purity of the channel output state as a function of the
interaction time, for various transverse field strengths: from left to right
$\lambda=0.25$ (black circles), 0.5 (red squares), 0.75 (green diamonds), 0.9
(brown triangles up), 1 (blue triangles down), 1.1 (magenta triangles left),
1.5 (orange triangles right), 2 (violet crosses). Here we simulated a channel
of $n=30$ qubits coupled to an Ising chain, set an interaction strength
$\varepsilon=0.05$, and averaged over $N_{av}=5\times 10^{4}$ random initial
conditions. In the inset we plot the short-time Gaussian decay rate $\beta$ as
a function of $\lambda$.

The average purity is a rather fair indicator of the noise induced by the
coupling to the environment: if the carrier qubits get strongly entangled with
the environment, $P_{2}$ is greatly reduced from the unit value (for large $n$
it will tend to zero); on the other hand, a channel which simply unitarily
rotates the carrier states has a unit purity. However, we should stress that
also the purity may intrinsically fail as a transmission quality quantifier:
there are strongly noisy channels with very high output purity (consider, for
example, the channel which maps each input state into the same pure output
state).

Then, we study the average channel purity ${\mathcal{P}}_{av}$ as a function
of the model free parameter, the interaction time $t$. The results are
reported in Fig. 6 for a chain of $n=30$ qubits and different values of the
transverse field $\lambda$. We notice qualitatively different behaviors
depending on the values of the transverse field $\lambda$: If
$\lambda<\lambda_{c}$ the averaged purity oscillates in time and
asymptotically tends to an average constant value; as far as the critical
point is approached, ${\mathcal{P}}_{av}$ drops to smaller values (revivals
are again due to finite system size effects), reflecting the fact that at
criticality correlations between the qubits and the environment are stronger.
Crossing the critical point, in the $z$-ordered phase ($\lambda>\lambda_{c}$)
the purity is generally higher and asymptotically takes values very close to
the unit value. This can be easily understood in the limit $\lambda\to\infty$:
in this case the spins in the chain are “freezed” along the field direction
and they cannot couple with anything else, resulting in a watchdog-like effect
[24]. Independently of the transverse field value, the average purity
${\mathcal{P}}_{av}$ decays as a Gaussian ${\mathcal{P}}_{av}\sim e^{-\beta
t^{2}}$ in the short time limit. As for the averaged fidelity
${\mathcal{F}}_{av}$, we have then analyzed the decay rate $\beta$ as a
function of $\lambda$: as before, $\beta$ exhibits a signature of criticality
via a divergence, at the thermodynamical limit, in its first derivative with
respect to $\lambda$ (see the inset of Fig. 6).

Figure 7: Upper panel: average channel fidelity for the generalized model in
Fig. 2, with $n=12$ qubits, $\varepsilon=0.05$ (averages have been performed
over $N_{av}=10^{4}$ initial states). The various curves are for different
numbers $m$ of spins between two consecutive qubits, and different values of
transverse magnetic fields $\lambda=0.25,\,1,\,2$. Lower panel: absolute
differences in the fidelities between configurations at various $m$, as a
function of $\lambda$ and at a fixed interaction time $Jt^{*}=10$. Figure 8:
Gaussian decay rate of the averaged fidelity for model in Fig. 2, as a
function of the transverse field $\lambda$. The various curves stand for
different numbers of spins $m$ between two consecutive qubits (here $n=12$,
$\varepsilon=0.05$, $N_{av}=10^{4}$). In the inset we show the first
derivative with respect to $\lambda$ of the curves in the main panel.

##  6 Generalized model

We finally concentrate on the generalized model depicted in Fig. 2, where a
certain number $m$ of environment spins are present between two consecutive
spins coupled to the qubits. The richness of the model, that is characterized
by a large number of parameters, and by a global size which grows both with
$n$ and $m$, requires a huge numerical effort in order to simulate it,
therefore we decided to analyze only the average channel fidelity
${\mathcal{F}}_{av}$. In the upper panel of Fig. 7 we show
${\mathcal{F}}_{av}$ as a function of the interaction time $t$ for different
values of $m$ and three values of the transverse field $\lambda=0.25,\,1,\,2$;
we fix a number of qubits $n=12$ and an interaction strength
$\varepsilon=0.05$. Hereafter we will concentrate on this case as a typical
result, as we performed some checks with larger numbers of channel uses
($n=30,\,50$ and $m=0,\,1,\,2$), and found qualitatively analogous results. We
immediately observe that differences for various $m$ are tiny, even if the
fidelity generally tends to increase when increasing $m$; the sensibility with
$m$ suddenly enhances at criticality ($\lambda_{c}=1$), where correlations in
the environment decay much slower than in the other cases. On the other hand,
when $\lambda$ is far from $\lambda_{c}$, differences between fidelities upon
a variation of $m$ are greatly suppressed, and the generalized model mostly
behaves as the model in Fig. 1. Again, this reflects the fact that, out of
criticality, each qubit is mostly influenced only by the spin that is coupled
to, as the spin does not exchange correlations with the other environmental
spins. The resulting channel properties are then defined only by the local
properties of the chain. On the contrary, at criticality, the spins are
correlated and then the resulting channel properties are influenced by the
distance of the spins coupled with the qubits. In the lower panel of Fig. 7 we
explicitly plot the differences in the fidelities for various $m$ as a
function of $\lambda$, and for a fixed interaction time; a peak in proximity
of $\lambda_{c}$ is clearly visible. We point out that, as already noted at
the end of Sec. 4 concerning the size scaling of the fidelity, the maximum in
the differences does not occur exactly at the critical point.

The sensitivity to criticality is again demonstrated by the averaged fidelity
Gaussian decay rate $\alpha$ as a function of $\lambda$, as shown in Fig. 8
for different values of $m$: the first derivative in the inset has a maximum
in correspondence of a value that approaches the critical point $\lambda_{c}$
at the thermodynamical limit. Indeed, increasing $m$ is equivalent to
approaching the thermodynamical limit of the chain, thus resulting in an
increase of the quantum phase transition effects. A double check of this comes
from the cyan triangles-down curve of Fig. 8: in this case we take $m=4$, but
we break one of the links between two intermediate spins. The environment is
then formed by disconnected chains, each of them made up by $5$ spins,
therefore the system cannot undergo a phase transition in the limit
$n\to\infty$: the signature of criticality has completely disappeared.

##  7 Conclusions

In conclusion we have introduced and characterized a class of correlated
quantum channels, and we have given bounds for its entropy by means of the
averaged channel fidelity ${\mathcal{F}}_{av}$ and purity
${\mathcal{P}}_{av}$. Even though in general these bounds might not be strict,
we give a characterization of the channel in terms of quantities that have a
clear meaning from the point of view of the many body model we have
introduced.

In the case of an environment defined by a quantum Ising chain, we have shown
that the averaged channel purity and the fidelity depend on the environment
parameters and are strongly influenced by spin correlations inside it, in
particular by the fact whether the environment is critical or not. We expect
that some different environment models, such as, for example, the $XY$ spin
chain, will behave qualitatively similarly of what found in this work, as it
belongs to the same universality class. This might not be the case for other
models, like the Heisenberg chain, which will be object of further study in
the near future.

## Acknowledgments

We thank R. Fazio for discussions and support, F. Caruso for comments, and D.
Burgarth for pointing out Ref. [21]. This work have been supported by the
“Quantum Information Program” of Centro De Giorgi of Scuola Normale Superiore.

## Appendix A

The probability $p_{xy}$ of Eq. (22) can be computed as follows: we first
define $r_{x}\equiv|\alpha_{x}|$ and convert the string $x$ into a decimal
number from $1$ to $N=2^{n}$ by trivially identifying $g\equiv 0$ and $e\equiv
1$. The average over a uniform distribution of all pure input states on the
Bloch hypersphere for $n$ qubits is

$p_{xy}=C_{N}\int_{0}^{1}{\rm d}r_{1}\cdots\int_{0}^{1}{\rm
d}r_{N}\>r_{x}^{2}\,r_{y}^{2}\,\delta(1-{\bf r}^{2})\,,$ (31)

where ${\bf r}^{2}=r_{1}^{2}+\cdots+r_{N}^{2}$, and
$C_{N}^{-1}\equiv\int_{0}^{1}{\rm d}r_{1}\cdots\int_{0}^{1}{\rm
d}r_{N}\,\delta(1-{\bf r}^{2})$ is a normalization constant. Changing the
limits of integration due to the delta function and using the property of the
Gamma function $\Gamma(z)=\int_{0}^{+\infty}y^{z-1}e^{-y}{\rm d}y$, it is easy
to show that $C_{N}=2^{N}\pi^{-N/2}\Gamma(N/2)$ and

$p_{xy}=\frac{1+2\,\delta_{x,y}}{N(N+2)}\,.$ (32)

## Appendix B

For times longer than those in the scales of Figs. 3 and 4, time revivals of
the fidelity are present, i.e. the fidelity increases back towards the unit
value periodically, due to the finite system size. For finite values of the
transverse field revivals are not perfect, that is ${\mathcal{F}}_{av}(t)\neq
1$ for $t>0$. Anyway, as far as $\lambda$ increases, the revivals are stronger
and happen with period $t^{R}$ which does not depends on the system size $n$.
This can be understood in the limit $\lambda\to+\infty$, where the ground
state of the environment $\left|\varphi\right\rangle_{E}$ is a fully
$z$-polarized state, thus being an eigenstate of the chain Hamiltonian
$\mathcal{H}^{x}_{E}$ in Eq. (9): the generalized Loschmidt echo of Eq. (11)
is then given by $L_{xy}=e^{-i\varepsilon\mathcal{N}_{x-y}t}$, where
$\mathcal{N}_{x-y}$ is the number of excited qubits in the sequence $x$ minus
the one in the sequence $y$. It is easy to see that there are
$2^{n}\times{n\choose k}$ different possibilities to choose two sequences
$x,y$ such that the corresponding states differ in the state of $k$ qubits,
then $\mathcal{N}_{x-y}=\pm j$ with $j=0,2,\ldots,k$ (if $k$ is even) or
$j=1,3,\ldots,k$ (if $k$ is odd). Therefore, when averaging over input states,
each term contributes with $p_{xy}\,e^{\pm i\,\varepsilon jt}$. Noting that
$({\cal U}_{y}^{\dagger}\,{\cal U}_{x})^{\dagger}={\cal
U}_{x}^{\dagger}\,{\cal U}_{y}$, we have

$\langle F\rangle=2^{n}p_{xx}+p_{xy}\left(c_{0}+c_{1}\cos(\varepsilon
t)+c_{2}\cos(2\varepsilon t)+\ldots+c_{n}\cos(n\varepsilon t)\right)\;,$ (33)

It follows a perfect revival for the fidelity at times $t^{R}$ such that
$\varepsilon t^{R}=2\pi\;[{\rm mod}\;2\pi].$

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††thanks: 0000-0001-9980-0363

# Random site percolation thresholds on square lattice for complex
neighborhoods containing sites up to the sixth coordination zone

Krzysztof Malarz<EMAIL_ADDRESS>AGH University of Science and Technology,
Faculty of Physics and Applied Computer Science,
al. Mickiewicza 30, 30-059 Kraków, Poland

(March 18, 2023)

###### Abstract

The site percolation problem is one of the core topics in statistical physics.
Evaluation of the percolation threshold, which separates two phases (sometimes
described as conducting and isolating), is useful for a lot of problems
ranging from core condensed matter to interdisciplinary application of
statistical physics in epidemiology or any other transportation or
connectivity problems. In this paper—with Newman–Ziff fast Monte Carlo
algorithm and finite-size scaling theory—we compute the random site
percolation thresholds $p_{c}$ for a square lattice with complex neighborhoods
containing sites from the sixth coordination zone. Complex neighborhoods are
those that contain sites from various coordination zones (which are not
necessarily compact ones). We present the source codes of the appropriate
procedures (written in C) and the results of their usage in computations of
the percolation threshold on the square lattice for neighborhoods containing
sites up to the sixth coordination zone. We conclude that similarly to the
honeycomb lattice, the percolation thresholds for complex neighborhoods on a
square lattice follow the power law $p_{c}(\zeta)\propto\zeta^{-\gamma_{2}}$
with $\gamma_{2}=0.5454(60)$, where $\zeta=\sum_{i}z_{i}r_{i}$ is the weighted
distance of sites in complex neighborhoods ($r_{i}$ and $z_{i}$ are the
distance from the central site and the number of sites in the coordination
zone $i$, respectively).

Monte Carlo simulation; finite-size scaling; non-compact neighborhoods;
universal formula for percolation thresholds

##  I Introduction

Percolation Broadbent and Hammersley (1957); Hammersley (1957) is one of the
core problems in statistical physics with many interdisciplinary applications
ranging from materials science Cheng  _et al._ (2020), through studies of
polymer composites Zhang  _et al._ (2020), forest fires Malarz  _et al._
(2002), agriculture Ramírez  _et al._ (2020), oil and gas exploration
Ghanbarian  _et al._ (2020), diseases propagation Ziff (2021), transportation
networks Dong  _et al._ (2020), quantifying urban areas Cao  _et al._ (2020),
to Bitcoins transfer Bartolucci  _et al._ (2020) (see References Li  _et al._
, 2021; Saberi, 2015 for reviews).

The percolating system undergoes a (purely geometrical) phase transition (in
terms of the conductivity or transportation properties of the system) from the
phase corresponding to an insulator (for low system connectivity $p<p_{c}$) to
the phase of the conductor (for high system connectivity $p>p_{c}$). The
critical connectivity of the system $p_{c}$ (called the percolation threshold)
separates these two phases and depends on the dimension of the system $d$, the
topology of the lattice, the number $z$ of sites in the assumed neighborhood,
the type of percolation (that is, the bond dilution site), etc. Stauffer and
Aharony (1994); Wierman (2014).

Percolation thresholds were initially estimated for the nearest-neighbor
interactions Dean (1963); Dean and Bird (1967); Suding and Ziff (1999) but
later also complex neighborhoods (termed also extended for compact
neighborhoods) were studied for various lattices embedded in:

  * •

$d=2$ (for a square Dalton  _et al._ (1964); Domb and Dalton (1966); Gouker
and Family (1983); Malarz and Galam (2005); Galam and Malarz (2005); Majewski
and Malarz (2007); Xun  _et al._ (2021), a triangular Dalton  _et al._ (1964);
Domb and Dalton (1966); d’Iribarne  _et al._ (1999); Malarz (2020, 2021), a
honeycomb Dalton  _et al._ (1964); Malarz (2022) and other Archimedean
Lebrecht  _et al._ (2021); Xun  _et al._ (2022) lattices);

  * •

$d=3$ (for a simple cubic Kurzawski and Malarz (2012); Malarz (2015); Xun  _et
al._ (2021) lattice);

  * •

$d=4$ (for a simple hypercubic Kotwica  _et al._ (2019); Zhao  _et al._ (2022)
lattice)

dimensions.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: (Color online). Shapes of basic neighborhoods on square lattice.
1(a) sq-1, $r^{2}=1$, 1(b) sq-2, $r^{2}=2$, 1(c) sq-3, $r^{2}=4$, 1(d) sq-4,
$r^{2}=5$, 1(e) sq-5, $r^{2}=8$, 1(f) sq-6, $r^{2}=9$. The number $r$ is the
radius of (orange) circle indicating equidistant sites marked by solid (black)
circles to the central one marked with open (red) circle

Simultaneously with the estimation of percolation thresholds for various
lattices, some effort went into searching an analytical formula allowing for
the prediction of the percolation threshold position based on lattice
characteristics. For example, Xun  _et al._ Xun  _et al._ (2022) estimated the
site and bond percolation thresholds for the 11 Archimedean lattices with
complex and compact (extended) neighborhoods containing sites up to the tenth
coordination zone. For the site percolation problem, the critical site
occupation probability $p_{c}$ follows asymptotically

$p_{c}(z)=a/z$ (1)

with the total number $z$ of sites in the neighborhood and $a\approx 4.51235$.
This dependence should be reached exactly for the percolation of compact
neighborhoods with a large number $z$ of sites that make up the neighborhood
(for example, for discs). To take into account the he finite-$z$ effect an
additional term $b$ in the denominator of Equation 1

$p_{c}(z)=c/(z+b)$ (2)

should be included Xu  _et al._ (2021). For the two-dimensional lattices $b=3$
Xun  _et al._ (2022). The third universal scaling studied in Reference Xun
_et al._ , 2022 was

$p_{c}(z;d)=1-\exp(d/z)$ (3)

proposed by Koza  _et al._ Koza  _et al._ (2014); Koza and Poła (2016).

Much earlier Galam and Mauger Galam and Mauger (1996, 1997) proposed a
universal formula for site percolation problem

$p_{c}(z;d)=\dfrac{p_{0}}{[(d-1)(z-1)]^{a}},$ (4)

where $z$ and $d$ are the total number of sites in the neighborhood and the
dimension of the system space, respectively. They recognized two classes of
systems (two sets of $(p_{0},a)$ parameters) Galam and Mauger (1996). Their
paper Galam and Mauger (1996) was immediately criticized by van der Marck van
der Marck (1997) who showed ‘an example of two networks, where $d$ and $z$ are
equal, but the percolation thresholds differ’.

For complex neighborhoods, the situation is even worse, since for a given
lattice topology (and thus fixed $d$) there are many neighborhoods with
exactly the same total number $z$ of sites in the neighborhood but different
percolation thresholds $p_{c}$ (see: Table 1 and Figure 4 in Reference
Majewski and Malarz, 2007 for the square lattice; Table 1 in Reference Malarz,
2020 and Table 1 and Figure 3(a) in Reference Malarz, 2021 for the triangular
lattice; and Table 1 and Figure 4(a) in Reference Malarz, 2022 for the
honeycomb lattice).

To solve the above-mentioned problems of $p_{c}(z)$ degeneration the index

$\xi=\sum_{i}z_{i}r_{i}^{2}/i$ (5)

was proposed Malarz (2021). The $z_{i}$ and $r_{i}$ are the number of sites
and their distance from the central site in the neighborhood in the $i$-th
coordination zone. The index $\xi$ allowed for a successful distinguishing
between neighborhoods and cancel $p_{c}(z)$ degeneration for the triangular
lattice with complex neighborhoods containing sites up to the fifth
coordination zone. The percolation thresholds dependence

$p_{c}(\xi)\propto\xi^{-\gamma_{1}}$ (6)

was well fitted with the power law with $\gamma_{1}(\textsc{tr})\approx
0.710(19)$. Unfortunately, this dependence does not hold for the honeycomb
lattice (see Figure 4(b) in Reference Malarz, 2022). Thus, another index

$\zeta=\sum_{i}z_{i}r_{i}$ (7)

was introduced to simultaneously resolve the problem of $p_{c}(z)$
degeneration and to distinguish among various complex neighborhoods for the
honeycomb lattice Malarz (2022). For honeycomb lattice and complex
neighborhoods up to the fifth coordination zone

$p_{c}(\zeta)\propto\zeta^{-\gamma_{2}}$ (8)

with $\gamma_{2}(\textsc{hc})\approx 0.4981(90)$ Malarz (2022).

In this paper—using the fast Monte Carlo Newman–Ziff algorithm Newman and Ziff
(2001)—we calculate critical occupation probabilities $p_{c}$ (percolation
thresholds) for random site percolation in a square lattice and neighborhoods
combined with basic neighborhoods presented in Figure 1. The basic
neighborhoods contain sites from the first coordination zone (sq-1, Figure
1(a)) up to the sixth coordination zone (sq-6, Figure 1(f)). These complex
neighborhoods are presented in Figure 4 in Appendix B. Calculations of
percolation thresholds are based on the finite-size scaling hypothesis Privman
(1990); Stauffer and Aharony (1994); Landau and Binder (2009).

The second aim of this paper is to check if Equations 6 and 8 holds for a
square lattice with complex neighborhoods and, if so, which of them performs
better.

The rest of the paper is organized as follows. The details of the calculations
are presented in the following Section II. The results of the calculations are
given in Section III. The article is summarized and concluded in Section V.
The Appendices contain: written in C boundaries procedures (Appendix A),
graphical presentation of neighborhood shapes (Appendix B); and figures
presenting results of finite-size scaling analysis (Appendix C).

##  II Computations

Our calculations of the percolation thresholds $p_{c}$ are based on finite-
size analyses of the probability $\mathcal{P}_{\text{max}}$ that the randomly
selected site belongs to the larges cluster of occupied sites.

According to the finite-size hypothesis Privman (1990); Stauffer and Aharony
(1994); Landau and Binder (2009), in the vicinity of a phase transition
(marked by a critical point $x_{c}$), many quantities $\mathcal{A}$
characterizing the system obey a scaling relation

$\mathcal{A}(x;L)=L^{-\varepsilon_{1}}\mathcal{F}\left((x-x_{c})L^{\varepsilon_{2}}\right),$
(9)

where $x$ measures the level of system disorder (temperature for the Ising or
Potts model, site/bond occupation probability for percolation problem), $L$ is
the linear size of the system, $\mathcal{F}$ is a scaling function (usually
analytically unknown) and $\varepsilon_{1}$ and $\varepsilon_{2}$ are scaling
exponents. In other words, there exists a function $\mathcal{F}$, that for
properly assumed values of $x_{c}$, $\varepsilon_{1}$ and $\varepsilon_{2}$
the dependencies of $\mathcal{A}(x)$ collapse into a single curve
independently of the (finite) system size $L$. This also provides an elegant
way to predict the value of the critical parameter $x_{c}$ as

$\mathcal{A}(x;L)L^{\varepsilon_{1}}=\mathcal{F}\left((x-x_{c})L^{\varepsilon_{2}}\right),$
(10)

which for $x=x_{c}$ yields

$\mathcal{A}(x;L)L^{\varepsilon_{1}}=\mathcal{F}\left(0\right).$ (11)

In other words, we expect the curves $L^{\varepsilon_{1}}\mathcal{A}(x)$
plotted for various sizes of linear systems $L$ to intercept each other at
$x=x_{c}$.

For our purposes, we assume that $\mathcal{A}\equiv\mathcal{P}_{\text{max}}$
(the probability that a randomly selected site belongs to the large cluster)
and $x\equiv p$ (the probability of occupation of the sites). For the problem
of site percolation, the critical values of the exponents $\varepsilon_{1}$
and $\varepsilon_{2}$ are known exactly (Stauffer and Aharony, 1994, p. 54) as
$\varepsilon_{1}=\frac{5}{36}/\frac{4}{3}=\frac{5}{48}$ and
$\varepsilon_{2}=1/\frac{4}{3}=\frac{3}{4}$, respectively.

To compute the probability of belonging to the largest cluster

$\mathcal{P}_{\max}(p;L)=\mathcal{S}_{\text{max}}(p;L)/N$ (12)

we first need to calculate the sizes of the largest cluster $S_{\text{max}}$
and $N=L^{2}$ is the number of all sites available in the system.

To that end, we use three concepts presented in Reference Newman and Ziff,
2001.

  * •

The first is the fast system construction scheme (known as the Newman–Ziff
algorithm). The efficiency of this approach is based on the recursive
construction of the system with $n$ occupied sites with the addition of only
one occupied site to the system containing $(n-1)$ already occupied sites.

  * •

The second concept is the way of transforming the $\bar{\mathcal{A}}(n;N)$
dependence on the integer number of occupied sites $n$ into the dependence
$\mathcal{A}(p;N)$ on the probability of the site occupation $p$

$\mathcal{A}(p;N)=\sum_{n=0}^{N}\bar{\mathcal{A}}(n;N)\mathcal{B}(n;N,p),$
(13)

where

$\mathcal{B}(n;N,p)=\binom{N}{n}p^{n}(1-p)^{N-n}$ (14)

are values of the binomial (Bernoulli) probability distribution.

  * •

The third concept allows us to efficiently construct the binomial coefficients
(14).

The applied scheme defined in Equation 13 together with the construction of
the binomial distribution coefficients mentioned above is presented in
Algorithm 1.

Algorithm 1 Conversion $\bar{\mathcal{A}}(n)$ to $\mathcal{A}(p)$ Newman and
Ziff (2001)

1:$p_{1}$, $p_{2}$, $\Delta p$, $N$, $\bar{\mathcal{A}}(n)$ $\triangleright$
$n\in\\{0,1,\cdots,N-1,N\\}$

2:$\mathcal{A}(p)$ $\triangleright$ for $p$ from $p_{1}$ to $p_{2}$ every
$\Delta p$

3:$p\leftarrow p_{1}$

4:while $p\leq p_{2}$ do

5: $n_{\text{max}}=pN$ $\triangleright$ store $\mathcal{B}(N,n,p)$ to
$\hat{B}$

6: $B(n_{\text{max}})=pN$

7: for $n=n_{\text{max}}+1,N$ do

8: $B(n)=B(n-1)\cdot\dfrac{(N-n+1)p}{n(1-p)}$

9: end for

10: for $n=n_{\text{max}}-1,0,-1$ do

11: $B(n)=B(n+1)\cdot\dfrac{(n+1)(1-p)}{p(N-n)}$

12: end for

13: $c\leftarrow\sum\hat{B}$

14: $\hat{B}\leftarrow\hat{B}/c$

15: $\mathcal{A}(p)=0$

16: for all $n$ do

17: $\mathcal{A}(p)\leftarrow\mathcal{A}(p)+B(n)\bar{\mathcal{A}}(n)$

18: end for

19: return $p,\mathcal{A}(p)$

20: $p\leftarrow p+\Delta p$

21:end while

##  III Results

In Figure 2 examples of the results (for neighborhood sq-1,2,3,4,5,6 and
various sizes of linear systems $L=128$, 256, 512, 1024, 2048, and 4096) of
the computations obtained with the procedure described in Section II are
presented.

Figure 2(a) shows dependencies of the larger cluster size
$\mathcal{S}_{\text{max}}$ (normalized to the system size $L^{2}$) vs. the
number of occupied sites $n$ (also normalized to the system size $L^{2}$).
With increasing system linear size $L$ the dependence
$\mathcal{S}_{\text{max}}(n)$ becomes stepper and stepper.

Figure 2(b) shows $L^{\varepsilon_{1}}\mathcal{P}_{\text{max}}(p)$ for $p$
ranging from 0.134 to 0.152 estimated for every $\Delta p=10^{-4}$. Figure
2(c) shows close-up of Figure 2(b) in the vicinity of the percolation
threshold (for $p$ from 0.1430 to 0.1435 for every $\Delta p=10^{-5}$). The
finite-size effects in $L^{\varepsilon_{1}}\mathcal{P}_{\text{max}}(p)$ vanish
at the percolation threshold $p_{c}$, resulting in a common point of
$L^{\varepsilon_{1}}\mathcal{P}_{\text{max}}(p)$ plotted for various linear
sizes $L$ of the systems. The analogous dependencies for all other complex
neighborhoods—presented in Figure 4 in Appendix B—containing sites from the
sixth coordination zone are shown in Figure 7 in Appendix C. The results are
averaged over the realizations of the $R=10^{5}$ system. The obtained
percolation thresholds $p_{c}$ are gathered in Table 1.

(a)

(b)

(c)

Figure 2: (Color online). An example of results obtained for sq-1,2,3,4,5,6 neighborhood and various linear system sizes $L=128$, 256, 512, 1024, 2048, and 4096. 2(a) The size of the largest cluster $\mathcal{S}_{\text{max}}$ vs. the number of occupied sites $n$. Both quantities are normalized to the system size $L^{2}$. 2(b) Dependence of $L^{\varepsilon_{1}}\mathcal{P}_{\text{max}}(p)$ for $p$ from 0.134 to 0.152 every $\Delta p=10^{-4}$. 2(c) Close-up on Figure 2(b) in the vicinity of the percolation threshold $p_{c}$ for $p$ from 0.1430 to 0.1435 every $\Delta p=10^{-5}$ Table 1: Percolation thresholds $p_{c}$ for a square lattice with complex neighborhoods (and their characteristics $z$, $\zeta$, $\xi$) containing sites from the sixth coordination zone lattice | $z$ | $\zeta$ | $\xi$ | $p_{c}$
---|---|---|---|---
sq-1,2,3,4,5,6 | 28 | 58.8591 | 136 | 0.143251110.142 d’Iribarne  _et al._ (1999), 0.143255 Xun  _et al._ (2021)
sq-2,3,4,5,6 | 24 | 54.8591 | 132 | 0.14576
sq-1,3,4,5,6 | 24 | 53.2022 | 128 | 0.14801
sq-1,2,4,5,6 | 24 | 50.8591 | 120 | 0.15223
sq-1,2,3,5,6 | 20 | 40.9705 | 96 | 0.16661
sq-1,2,3,4,6 | 24 | 47.5453 | 104 | 0.16133
sq-3,4,5,6 | 20 | 49.2022 | 124 | 0.15221
sq-2,4,5,6 | 20 | 46.8591 | 116 | 0.15844
sq-2,3,5,6 | 16 | 36.9705 | 92 | 0.17601
sq-2,3,4,6 | 20 | 43.5453 | 100 | 0.16529
sq-1,4,5,6 | 20 | 45.2022 | 112 | 0.15816
sq-1,3,5,6 | 16 | 35.3137 | 88 | 0.18007
sq-1,3,4,6 | 20 | 41.8885 | 96 | 0.16675
sq-1,2,5,6 | 16 | 32.9705 | 80 | 0.18216
sq-1,2,4,6 | 20 | 39.5453 | 88 | 0.17409
sq-1,2,3,6 | 16 | 29.6568 | 64 | 0.20135
sq-4,5,6 | 16 | 41.2023 | 108 | 0.16819
sq-3,5,6 | 12 | 31.3137 | 84 | 0.19868
sq-3,4,6 | 16 | 37.8885 | 92 | 0.17288
sq-2,5,6 | 12 | 28.9706 | 76 | 0.20036
sq-2,4,6 | 16 | 35.5454 | 84 | 0.18454
sq-2,3,6 | 12 | 25.6569 | 60 | 0.21503
sq-1,5,6 | 12 | 27.3137 | 72 | 0.19936
sq-1,4,6 | 16 | 33.8885 | 80 | 0.18143
sq-1,3,6 | 12 | 24.0000 | 56 | 0.22577
sq-1,2,6 | 12 | 21.6568 | 48 | 0.23076
sq-5,6 | 8 | 23.3137 | 68 | 0.24422
sq-4,6 | 12 | 29.8885 | 76 | 0.19800
sq-3,6 | 8 | 20.0000 | 52 | 0.25673
sq-2,6 | 8 | 17.6568 | 44 | 0.26600
sq-1,6 | 8 | 16.0000 | 40 | 0.27309
sq-6222equivalent to sq-1 | 4 | 12.0000 | 36 | 0.592753330.592746 (Stauffer and Aharony, 1994, p. 17), 0.59274621(13) Newman and Ziff (2000), 0.59274(5) Tencer and Forsberg (2021) for sq-1

(a)

(b)

(c)

Figure 3: (Color online). 3(a) Degeneracy of $p_{c}(z)$ for a square lattice
with complex neighborhoods ranging from sq-1 up to sq-1,2,3,4,5,6. 3(b)
Dependence $p_{c}(\xi)$ for various neighborhoods containing sites up to the
sixth coordination zone. Orange crosses show equivalent neighborhoods
(sq-1$\equiv$sq-2$\equiv$sq-3$\equiv$sq-5$\equiv$sq-6, sq-2,5$\equiv$sq-1,3
and sq-2,3,5$\equiv$sq-1,2,3). 3(c) Dependence $p_{c}(\zeta)$. Inflated
neighborhoods corresponding to higher indexes (i.e., sq-2, sq-3, sq-5, sq-6,
sq-2,5 and sq-2,3,5) are excluded from the fitting procedure. The solid
(violet) line indicates Equation 8 with $\gamma_{2}(\textsc{sq})=0.5454(60)$

In Figure 3 the dependencies of the percolation threshold $p_{c}$ on the total
number $z=\sum_{i}z_{i}$ of sites in the complex neighborhoods containing the
sites of the $i$-th coordination zone and the indexes $\xi$ and $\zeta$ are
presented.

Figure 3(a) shows the dependence of the percolation threshold $p_{c}$ on $z$.
The percolation thresholds for neighborhoods containing sites up to the fifth
coordination zone are taken from References Malarz and Galam, 2005; Majewski
and Malarz, 2007 and those for neighborhoods containing sites from the sixth
coordination zone presented here in Table 1. It is clear that the number $z$
cannot differentiate between the various shapes of the neighborhoods or the
percolation thresholds associated with them.

In Figure 3(b) the dependence (6) on the percolation threshold $p_{c}$ for
complex neighborhoods on the square lattice on the index $\xi$ is presented.
Similarly to the earlier observation for the honeycomb lattice Malarz (2022),
some deviations from the straight line in Equation 6 are observed for
$32\leq\xi\leq 64$.

Figure 3(c) shows the dependence (8) of the percolation threshold $p_{c}$ for
complex neighborhoods on the square lattice on the index $\zeta$. The
percolation thresholds for neighborhoods containing sites up to the fifth
coordination zone are taken from References Malarz and Galam, 2005; Majewski
and Malarz, 2007, those for neighborhoods containing sites from the sixth
coordination zone presented here in Table 1 and those for compact
neighborhoods containing sites from the seventh to the tenth coordination
zones (i.e., sq-1,2,$\cdots$,6,7, sq-1,2,$\cdots$,7,8, sq-1,2,$\cdots$,8,9 and
sq-1,2,$\cdots$,9,10) are taken from References Xun  _et al._ , 2021 and
d’Iribarne  _et al._ , 1999.

##  IV Discussion

The percolation thresholds $p_{c}$ obtained in simulations range from 0.59275
(for sq-6) to 0.14325 (for sq-1,2,3,4,5,6). The latter agrees in five
significant digits with its earlier estimate
$p_{c}(\textsc{sq-1,2,3,4,5,6})=0.143255$ Xun  _et al._ (2021). The sq-6
neighborhood is topologically equivalent to sq-1 (but for a three-time larger
lattice constant), resulting in identical percolation thresholds
$p_{c}(\textsc{sq-6})=p_{c}(\textsc{sq-1})$. For the sq-6 neighborhood we deal
with several simultaneous independent percolation problems on several
identically shaped lattices. The latter reduces effective system size, but our
results show, that this effect is perfectly compensated by effective increase
of number of samples.

Similarly to the honeycomb lattice Malarz (2022), the power law (8) also holds
for a square lattice with complex neighborhoods with
$\gamma_{2}(\textsc{sq})=0.5454(60)$ given by the least-squares method.
Inflated neighborhoods sq-2, sq-3, sq-5, sq-6, sq-2,5 and sq-2,3,5
(corresponding to sq-1, sq-1, sq-1, sq-1, sq-1,3 and sq-1,2,3, respectively)
were excluded from the fitting procedure.

Among the neighborhoods that contain sites up to the sixth coordination zone,
there are six pairs of various neighborhoods with exactly the same $\zeta$
index, namely $\zeta(\textsc{sq-1,2,3,4,5})=\zeta(\textsc{sq-2,4,5,6})\approx
46.86$, $\zeta(\textsc{sq-4,5,6})=\zeta(\textsc{sq-1,3,4,5})\approx 41.20$,
$\zeta(\textsc{sq-2,5,6})=\zeta(\textsc{sq-1,2,3,5})\approx 28.97$,
$\zeta(\textsc{sq-5,6})=\zeta(\textsc{sq-1,3,5})\approx 23.31$,
$\zeta(\textsc{sq-2,4,6})=\zeta(\textsc{sq-1,2,3,4})\approx 35.55$ and
$\zeta(\textsc{sq-2,6})=\zeta(\textsc{sq-1,2,3})\approx 17.66$. The
differentiate power of a scalar index $\zeta$ is still better than the
differentiate power of an index $\xi$ and both are much better than the
differentiate power of the total number of sites in the neighborhood $z$.

##  V Conclusion

In this paper with Newman and Ziff effective Monte Carlo algorithm we
calculated 32 percolation threshold for complex neighbourhoods (containing
sites from the sixth coordination zone) on square lattice.

As scalar indexes $\xi$ (5) and $\zeta$ (7) allow simultaneously for (more or
less effective) distinguishing between various neighbourhoods and accordingly
fitting $p_{c}$ to the inverse power-low on $\xi$ (6) or $\zeta$ (8) (with
various efficiency depending on the underlying lattice shape) searching for
another index seems to be still an open task. The index may involve various
powers of $z_{i}$, $r_{i}$ and $i$, where $i$ stands for the number of
coordination zone from which sites constituting neighbourhood come from and
$z_{i}$ and $r_{i}$ are the number and the distance of sites in this
coordination zone to the central site in the neighbourhood, respectively.

The results obtained in this paper may be helpful in further searching for the
universal formula, in the spirit of Equation 4 Galam and Mauger (1994), for
percolation threshold $p_{c}$, also for complex neighborhoods, but
independently of the underlying two-dimensional lattice shape. Also further
studies on the topic presented here may result in finding universal formula
for $p_{c}$ not only for two-dimensional lattices but in higher dimensions
(including nonphysical dimensions, like on four- Kotwica  _et al._ (2019);
Zhao  _et al._ (2022) and five-dimensional simple hyper-cubic Mertens and
Moore (2018) lattices).

###### Acknowledgements.

I am grateful to Małgorzata J. Krawczyk for a fruitful discussion on boundary
conditions for neighborhoods containing sites up-to the sixth coordination
zone on the square lattice.

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##  Appendix A Boundaries procedures

Below, we present a set of boundaries() functions (written in C) to be
replaced in the Newman–Ziff program published in Reference Newman and Ziff,
2001 to obtain the single realization of $\mathcal{S}_{\max}(n;L)$ for the
neighborhoods presented in Figures 1(a), 1(b), 1(c), 1(d), 1(e) and 1(f).

###  A.1 SQ-1

boundaries() function for the sq-1 neighborhood (originally presented in
Reference Newman and Ziff, 2001).

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 1nn core:

6 nn[i][0] = (N+i +1)%N;

7 nn[i][1] = (N+i -1)%N;

8 nn[i][2] = (N+i +L)%N;

9 nn[i][3] = (N+i -L)%N;

10// 1nn left border:

11 if (i%L==0) nn[i][1] = (N+i+L -1)%N;

12// 1nn right border:

13 if ((i+1)%L==0) nn[i][0] = (N+i-L +1)%N;

14 }

15}

###  A.2 SQ-2

boundaries() function for the sq-2 neighborhood.

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 2nn core:

6 nn[i][0] = (N+i +L+1)%N;

7 nn[i][1] = (N+i +L-1)%N;

8 nn[i][2] = (N+i -L+1)%N;

9 nn[i][3] = (N+i -L-1)%N;

10// 2nn left border:

11 if(i%L==0) {

12 nn[i][1] = (N+i+L +L-1)%N;

13 nn[i][3] = (N+i+L -L-1)%N; }

14// 2nn right border:

15 if((i+1)%L==0) {

16 nn[i][0] = (N+i-L +L+1)%N;

17 nn[i][2] = (N+i-L -L+1)%N; }

18 }

19}

###  A.3 SQ-3

boundaries() function for the sq-3 neighborhood.

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 3nn core:

6 nn[i][0] = (N+i +2*L)%N;

7 nn[i][1] = (N+i -2*L)%N;

8 nn[i][2] = (N+i +2)%N;

9 nn[i][3] = (N+i -2)%N;

10// 3nn left border:

11 if(i%L==0 || i%L==1)

12 nn[i][3] = (N+i+L -2)%N;

13// 3nn right border:

14 if((i+1)%L==0 || (i+2)%L==0)

15 nn[i][2] = (N+i-L +2)%N;

16 }

17}

###  A.4 SQ-4

boundaries() function for the neighborhood sq-4. The preprocesor directive
#define Z 4 in source code in Reference Newman and Ziff, 2001 requires
replacing to #define Z 8.

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 4nn core:

6 nn[i][0] = (N+i +L+2)%N;

7 nn[i][1] = (N+i +L-2)%N;

8 nn[i][2] = (N+i -L+2)%N;

9 nn[i][3] = (N+i -L-2)%N;

10 nn[i][4] = (N+i +2*L+1)%N;

11 nn[i][5] = (N+i +2*L-1)%N;

12 nn[i][6] = (N+i -2*L+1)%N;

13 nn[i][7] = (N+i -2*L-1)%N;

14// 4nn left border:

15 if(i%L==0 || i%L==1 ) {

16 nn[i][1] = (N+i+L +L-2)%N;

17 nn[i][3] = (N+i+L -L-2)%N;

18 nn[i][5] = (N+i+L +2*L-1)%N;

19 nn[i][7] = (N+i+L -2*L-1)%N; }

20// 4nn right border:

21 if((i+1)%L==0 || (i+2)%L==0) {

22 nn[i][0] = (N+i-L +L+2)%N;

23 nn[i][2] = (N+i-L -L+2)%N;

24 nn[i][4] = (N+i-L +2*L+1)%N;

25 nn[i][6] = (N+i-L -2*L+1)%N; }

26 }

27}

###  A.5 SQ-5

boundaries() function for the sq-5 neighborhood.

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 5nn core:

6 nn[i][0] = (N+i +2*L+2)%N;

7 nn[i][1] = (N+i +2*L-2)%N;

8 nn[i][2] = (N+i -2*L+2)%N;

9 nn[i][3] = (N+i -2*L-2)%N;

10// 5nn left border:

11 if(i%L==0 || i%L==1) {

12 nn[i][1] = (N+i+L +2*L-2)%N;

13 nn[i][3] = (N+i+L -2*L-2)%N; }

14// 5nn right border:

15 if((i+1)%L==0 || (i+2)%L==0) {

16 nn[i][0] = (N+i-L +2*L+2)%N;

17 nn[i][2] = (N+i-L -2*L+2)%N; }

18 }

19}

###  A.6 SQ-6

boundaries() function for the sq-6 neighborhood.

⬇

1void boundaries()

2{

3 int i,j;

4 for (i=0; i<N; i++) {

5// 6nn core:

6 nn[i][0] = (N+i +3*L)%N;

7 nn[i][1] = (N+i -3*L)%N;

8 nn[i][2] = (N+i +3)%N;

9 nn[i][3] = (N+i -3)%N;

10// 6nn left border:

11 if(i%L==0 || i%L==1 || i%L==2)

12 nn[i][3] = (N+i+L -3)%N;

13// 6nn right border:

14 if((i+1)%L==0 || (i+2)%L==0 || (i+3)%L==0)

15 nn[i][2] = (N+i-L +3)%N;

16 }

17}

##  Appendix B Neighborhoods shapes

In Figure 4 the shapes of all complex neighborhoods containing sites from the
sixth coordination zone are presented.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

(u)

(v)

(w)

(x)

(y)

(z)

(aa)

(ab)

(ac)

(ad)

(ae)

Figure 4: (Color online). Shapes of neighborhoods on square lattice combined
with basics neighborhoods presented in Figure 1 and containing sites from the
sixth coordination zone (Figure 1(f)). 4(a) sq-1,6, 4(b) sq-2,6, 4(c) sq-3,6,
4(d) sq-4,6, 4(e) sq-5,6, 4(f) sq-1,2,6, 4(g) sq-1,3,6, 4(h) sq-1,4,6, 4(i)
sq-1,5,6, 4(j) sq-2,3,6, 4(k) sq-2,4,6, 4(l) sq-2,5,6, 4(m) sq-3,4,6, 4(n)
sq-3,5,6, 4(o) sq-4,5,6, 4(p) sq-1,2,3,6, 4(q) sq-1,2,4,6, 4(r) sq-1,2,5,6,
4(s) sq-1,3,4,6, 4(t) sq-1,3,5,6, 4(u) sq-1,4,5,6, 4(v) sq-2,3,4,6, 4(w)
sq-2,3,5,6, 4(x) sq-2,4,5,6, 4(y) sq-3,4,5,6, 4(z) sq-1,2,3,4,6, 4(aa)
sq-1,2,3,5,6, 4(ab) sq-1,2,4,5,6, 4(ac) sq-1,3,4,5,6, 4(ad) sq-2,3,4,5,6,
4(ae) sq-1,2,3,4,5,6

##  Appendix C Results

Figure 7 presents the dependencies of $\mathcal{P}_{\max}\cdot L^{\beta/\nu}$
on the probability of occupation $p$ for neighborhoods ranging from sq-6 to
sq-1,2,3,4,5,6 for various linear system sizes $L=128$, 256, 512, 1024, 2048,
and 4096.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(a)

(b)

Figure 7: (Color online). $L^{\varepsilon_{1}}\mathcal{P}_{\text{max}}$ vs.
$p$ for various complex neighborhoods and various values of $L$ indicated in
the title of the figures. The results are averaged over the realizations of
the $R=10^{5}$ system. The probability of occupation $p$ is scanned with
$\Delta p=10^{-5}$ separation step. 5(a) sq-6, 5(b) sq-1,6, 5(c) sq-2,6, 5(d)
sq-3,6, 5(e) sq-4,6, 5(f) sq-5,6, 5(g) sq-1,2,6, 5(h) sq-1,3,6, 5(i) sq-1,4,6,
5(j) sq-1,5,6, 5(k) sq-2,3,6, 5(l) sq-2,4,6, 5(m) sq-2,5,6, 5(n) sq-3,4,6,
5(o) sq-3,5,6, 6(a) sq-4,5,6, 6(b) sq-1,2,3,6, 6(c) sq-1,2,4,6, 6(d)
sq-1,2,5,6, 6(e) sq-1,3,4,6, 6(f) sq-1,3,5,6, 6(g) sq-1,4,5,6, 6(h)
sq-2,3,4,6, 6(i) sq-2,3,5,6, 6(j) sq-2,4,5,6, 6(k) sq-3,4,5,6, 6(l)
sq-1,2,3,4,6, 6(m) sq-1,2,3,5,6, 6(n) sq-1,2,4,5,6, 6(o) sq-1,3,4,5,6, 7(a)
sq-2,3,4,5,6, 7(b) sq-1,2,3,4,5,6