# Containing misinformation spreading in temporal social networks

Wei Wang  Cybersecurity Research Institute, Sichuan University, Chengdu
610065, China  Yuanhui Ma  School of Mathematics, Southwest Jiaotong
University, Chengdu 610031, China  Tao Wu  School of Cyber Security and
Information Law, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China  Yang Dai<EMAIL_ADDRESS>School of Economics
and Management, Southwest Jiaotong University, Chengdu 610031, China  Xingshu
Chen<EMAIL_ADDRESS>Cybersecurity Research Institute, Sichuan University,
Chengdu 610065, China  Lidia A. Braunstein  Instituto de Investigaciones
Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias
Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes
3350, (7600) Mar del Plata, Argentina  Center for Polymer Studies and
Department of Physics, Boston University, Boston, Massachusetts 02215, USA

###### Abstract

Many researchers from a variety of fields including computer science, network
science and mathematics have focused on how to contain the outbreaks of
Internet misinformation that threaten social systems and undermine societal
health. Most research on this topic treats the connections among individuals
as static, but these connections change in time, and thus social networks are
also temporal networks. Currently there is no theoretical approach to the
problem of containing misinformation outbreaks in temporal networks. We thus
propose a misinformation spreading model for temporal networks and describe it
using a new theoretical approach. We propose a heuristic-containing (HC)
strategy based on optimizing final outbreak size that outperforms simplified
strategies such as those that are random-containing (RC) and targeted-
containing (TC). We verify the effectiveness of our HC strategy on both
artificial and real-world networks by performing extensive numerical
simulations and theoretical analyses. We find that the HC strategy greatly
increases the outbreak threshold and decreases the final outbreak threshold.

###### pacs:

89.75.Hc, 87.19.X-, 87.23.Ge

##  I Introduction

Many communications platforms, e.g., Twitter, Facebook, email, WhatsApp, and
mobile phones, allow numerous ways of sharing information Vosoughi  _et al._
(2018); Del Vicario  _et al._ (2016); Qiu  _et al._ (2017); Schmidt  _et al._
(2017); Castellano  _et al._ (2009); Zhang  _et al._ (2016). One task for
researchers is developing ways to distinguish between true and false
information, i.e., between “news” and “fake news” Kumar and Shah (2018). This
task is important because access to true information is essential in the
process of intelligent decision-making Funk  _et al._ (2009); Granell  _et
al._ (2013); Wang  _et al._ (2014). For example, when the Severe Acute
Respiratory Syndrome (SARS) spread across Guangzhou, China in 2003, the
Chinese Southern Weekly published a newspaper article entitled “There is a
Fatal Flu in Guangzhou.” This information was forwarded over 126 million times
by TV news and in other newspapers Tai and Sun (2007); Chow (2003).
Individuals receiving this true information could adopt simple, effective
protective measures against being infected (e.g., by staying at home, washing
hands, or wearing masks). Misinformation, on the other hand, encourages
irrational behavior and reckless decision-making, and its spread can undermine
societal well-being and sway the outcome of elections Allcott and Gentzkow
(2017); Soll (2016); Bessi  _et al._ (2015); Mocanu  _et al._ (2015). Bovet
and Makse Bovet and Makse (2018) analyzed 171 million tweets sent during the
five months prior to the 2016 US presidential election and found that
misinformation strongly affected the outcome of that election.

To contain the spread of misinformation we must understand the dynamic
information spreading mechanisms that facilitate it Pastor-Satorras  _et al._
(2015); de Arruda  _et al._ (2018); Wang  _et al._ (2018); Chen  _et al._
(2018); Young (2011); Weiss  _et al._ (2014); Iribarren and Moro (2009);
Alvarez-Zuzek  _et al._ (2019); Di Muro  _et al._ (2018); Valdez  _et al._
(2018); Yang  _et al._ (2018). Vosoughi et al. Vosoughi  _et al._ (2018)
examined true and fake information on Twitter from 2006 to 2017 and found that
misinformation spreads more quickly than true information. Using the spreading
mechanisms common in real-data analysis, researchers have proposed several
mathematical models to describe the spreading dynamics of true and fake
information Moreno  _et al._ (2004); Nekovee  _et al._ (2007); Zhou  _et al._
(2007); Isham  _et al._ (2010); Zhao  _et al._ (2012); Kwon  _et al._ (2013).
Moreno et al. Moreno  _et al._ (2004) developed mean-field equations to
describe the spread of classical misinformation on static scale-free networks
that enables a theoretical study not requiring extensive numerical
simulations. Borge-Holthoefer and Moreno Borge-Holthoefer and Moreno (2012)
found that although there are no influential spreaders in the classical
misinformation model presented in Ref. Moreno  _et al._ (2004), nodes with
high $k$-cores and ranking values are more likely to be the influential
spreaders of true information and also of infectious diseases Lü  _et al._
(2016a, b); Kitsak  _et al._ (2010); Morone and Makse (2015); Pei  _et al._
(2018). When we include the burst behavior of individuals in the
misinformation model, hubs emerge as influential nodes Borge-Holthoefer  _et
al._ (2013). Using real-world data, researchers found that social networks
evolve with time, and thus evolving temporal networks more accurately
represent the topology of real-world networks than static networks Holme and
Saramäki (2012, 2013); Holme (2015); Speidel  _et al._ (2016); Masuda and
Holme (2013); Masuda and Lambiotte (2016); Li  _et al._ (2017); Nadini  _et
al._ (2018); Liu  _et al._ (2016); Kalantar-Zadeh  _et al._ (2006); Starnini
_et al._ (2013); Speidel  _et al._ (2016).

Researchers have found that the temporal nature of networks strongly affect
their spreading dynamics. Perra et al. Perra  _et al._ (2012) found that in
susceptible-infected-susceptible (SIS) epidemic spreading a temporal network
behavior suppresses the spreading more effectively than a static integrated
network. Researchers have also found that the SIS and susceptible-infected-
recovered (SIR) models on temporal networks exhibit the same outbreak
threshold Valdano  _et al._ (2015, 2018); Onaga  _et al._ (2017). Nadini et
al. Nadini  _et al._ (2018) found that tightly connected clusters in temporal
networks inhibit SIR processes, but accelerate SIS spreading. Pozzana et al.
found that when node attractiveness—its role as a preferential target for
interactions—in temporal networks is heterogeneous, the contagion process is
altered Pozzana  _et al._ (2017). Karsai et al. Karsai  _et al._ (2014) found
that strong ties between individuals strongly inhibit the classical spreading
dynamics of misinformation.

Several strategies for containing the spread of misinformation in temporal
networks have been proposed Ogura and Preciado (2017); Liu  _et al._ (2014);
Lima  _et al._ (2015); Moinet  _et al._ (2018). Liu et al. Liu  _et al._
(2014) examined epidemic spreading on activity driven temporal networks and
developed mean-field based theoretical approaches for three different control
strategies, i.e., random, targeted, and egocentric. The egocentric strategy is
most effective. It immunizes a randomly selected neighbor of a node in the
observation window. Other effective approaches using extensive numerical
simulations have been proposed Holme and Liljeros (2014); Masuda  _et al._
(2013); Vestergaard  _et al._ (2014). For example, Holme and Liljeros Holme
and Liljeros (2014) take into consideration the time variation of nodes and
edges and propose a strategy for containing the outbreak of an epidemic based
on the birth and death of links.

Because there is still no theoretical approach to containing the spread of
misinformation in temporal networks, we here systematically examine its spread
in activity-driven networks. The rest of the paper is organized as follows.
Section II describes the misinformation spreading dynamics on temporal
networks and develops a theory to describe the spreading dynamics. Section IV
proposes three containment strategies. Section V describes the results of our
extensive numerical stochastic simulations, which show that our suggested
theory agrees with the numerical simulations. Section IV presents our
conclusions.

##  II Misinformation spreading in temporal networks

We here introduce our model for the spreading dynamics of misinformation in
temporal networks.

###  II.1 Mathematical descriptions for temporal networks

The widely used approaches mathematically describing temporal networks tend to
be either event-based or snapshot representations Masuda and Lambiotte (2016).
The event-based representation approach describes describes temporal networks
using ordered events $\\{u_{i},v_{i},t_{i},\Delta t_{i};i=1,2,\cdots\\}$,
where node $u_{i}$ and $v_{i}$ are connected at time $t_{i}$ in the time
period $\Delta t_{i}$. The snapshot approach describes temporal networks using
a discrete sequence of static networks
$\mathcal{G}=\\{\mathcal{G}(1),\mathcal{G}(2),\cdots,\mathcal{G}(t_{\rm
max})\\}$, where $\mathcal{G}(t)$ is the snapshot network at time $t$, and
$t_{\rm max}$ is the number of snapshots of the temporal network. Each
snapshot network $\mathcal{G}(t)$, contains $N$ nodes, where $N$ is fixed, and
$M_{t}$ edges. Thus the average temporal degree of snapshot network
$\mathcal{G}(t)$ is $\langle k_{t}\rangle=2M_{t}/N$. Using the adjacency
matrix, the temporal network is
$\mathcal{A}=\\{\mathcal{A}(1),\mathcal{A}(2),\cdots,\mathcal{A}(t_{\rm
max})\\}$, where $\mathcal{A}(t)$ is the adjacency matrix of network
$\mathcal{G}(t)$.

We here adopt the snapshot approach to describe temporal networks. As the
meaning of the adjacency matrix in static networks, $\mathcal{A}_{uv}(t)=1$
when nodes $u$ and $v$ are connected at time $t$, otherwise,
$\mathcal{A}_{uv}(t)=0$. Thus the degree $k_{u}(t)$ of node $u$ at time $t$ is
$k_{u}(t)=\sum_{v=1}^{N}\mathcal{A}_{uv}(t)$ for undirected temporal networks.
The average degree of node $u$ in the temporal network $\mathcal{G}$ is
$\langle k_{u}(t)\rangle=\frac{1}{t_{\rm max}}\sum_{t=1}^{t_{\rm
max}}k_{u}(t)$. Knowing the adjacency matrix $\mathcal{A}$ we obtain the
eigenvalues of ${\bf\mathcal{A}}$. Here $\Lambda_{1}({\bf\mathcal{A})}$
$\Lambda_{2}({\bf\mathcal{A}})$, $\cdots$, and $\Lambda_{N}({\bf\mathcal{A}})$
are the eigenvalues of ${\bf\mathcal{A}}$ in decreasing order. The spectral
radius is thus $\Lambda_{1}({\bf\mathcal{A}})$, which quantifies the threshold
outbreak of epidemics in temporal networks.

###  II.2 Activity-driven networks

We use the classical activity-driven network Perra  _et al._ (2012) to model a
temporal network with $N$ nodes. We build the activity-driven network using
the following steps.

  1. 1.

We assign to each node $i$ an activity potential value $x_{i}$ according to a
given probability density distribution $f(x)$. The activity of node $i$ is
$a_{i}=\eta x_{i}$, where $\eta$ is a rescaling factor, i.e., at each time
step, node $i$ is active with probability $a_{i}$. The higher the value of
$\eta$, the higher the average degree of the temporal network. The higher the
value of $a_{i}$, the higher the degree of node $i$. We assume that $f(x)$
follows a power-law function, i.e., $f(x)\sim x^{-\gamma}$, where $\gamma$ is
the potential exponent. We make this assumption in order to generate a
heterogeneous degree distribution temporal network. After further calculations
we find $f(x)=\frac{\gamma-1}{\epsilon^{1-\gamma}}x^{-\gamma}$, $\langle
k_{t}\rangle=2m\langle a\rangle$, and $\langle a\rangle=\eta\langle x\rangle$,
where $\langle x\rangle=\int_{\epsilon}^{\infty}xf(x)dx$, and $\epsilon$ is
the minimum value of the activity potential $x_{i}$. Each active node has $m$
edges, and each edge randomly links to a network node. An edge connects the
same pair of nodes with probability $m/N$. Note that in the thermodynamic
limit of a sparse temporal network, there are no multiple edges between nodes
and non-local loops

  2. 2.

At the end of time step $t$, we delete all edges in network $\mathcal{G}(t)$.

  3. 3.

We repeat steps (2) and (3) until $t_{\rm max}$ in order to generate temporal
network $\mathcal{G}$.

###  II.3 Misinformation spreading model

We use an ignorant-spreader-refractory model to describe the spreading
dynamics of misinformation Moreno  _et al._ (2004). Here nodes are classified
as either ignorant, spreader, or refractory. Ignorant nodes are unaware that
the information is false but are susceptible to adopting it. Spreader nodes
are aware that the information is false and are willing to transmit it to
ignorant nodes. Refractory nodes receive the misinformation but do not spread
it. The misinformation spreading dynamics on temporal networks evolves as
follows. We first randomly select a small fraction $\rho_{0}$ of spreader
nodes to be seeds in network $\mathcal{G}(t_{0})$, where $1\leq t_{0}\leq
t_{\rm max}$. We designate the remaining $1-\rho_{0}$ nodes to be ignorant. At
time step $t$ each spreader $i$ transmits with probability $\lambda$ the
misinformation to ignorant neighbors in network $\mathcal{G}(t)$. In addition,
each spreader $i$ becomes a refractory node with a probability

$\mu_{i}(t)=1-(1-\mu)^{n(t)},$ (1)

where $\mu$ is the intrinsic recovery probability, and $n(t)$ is the number of
nodes in the spreader and refractory states of node $i$. The dynamics evolve
until there are no spreader nodes. Note that when $t$ reaches $t_{\rm max}$,
the misinformation spreads on $\mathcal{G}(1)$ in the next time step. Figure 1
shows misinformation spreading on a temporal network.

Figure 1: Illustration of misinformation spreading on temporal networks.

##  III Theoretical analysis

We here develop a generalized discrete Markovian approach to describe the
misinformation spreading dynamics on temporal networks Valdano  _et al._
(2015). We denote $I_{i}(t)$, $S_{i}(t)$, and $R_{i}(t)$ to be the fraction of
nodes in the ignorant, spreader, and refractory states, respectively, at time
$t$. Because a node can only be in one of the three states,
$I_{i}(t)+S_{i}(t)+R_{i}(t)=1$.

An ignorant node $i$, becomes a spreader with probability
$p_{i}\left(t\right)$ at time $t$, where

$p_{i}\left(t\right)=1-\prod_{j}{\left[1-\lambda\mathcal{A}_{ij}\left(t\right)I_{j}\left(t\right)\right]}.$
(2)

Here
$\prod_{j}{\left[1-\lambda\mathcal{A}_{ij}\left(t\right)I_{j}\left(t\right)\right]}$
is the probability that node $i$ has not received any misinformation from
neighbors at time $t$ in $\mathcal{G}(t)$. At time step $t+1$, node $i$
remains ignorant with a probability

$I_{i}(t+1)=I_{i}(t)-I_{i}(t)p_{i}(t).$ (3)

The decrease of $I_{i}(t)$ is equal to the increase of $S_{i}(t)$, because an
ignorant node will become a spreader when it obtains the information from
neighbors in state $S$. In addition, spreader $i$ becomes refractory with a
probability
$\mu_{i}(t)=1-(1-\mu)^{1+\sum_{j=1}^{N}\mathcal{A}_{ij}(t)[I_{j}(t)+R_{j}(t)]}$.
Thus the evolution of node $i$ in the spreader state is

$S_{i}(t+1)=S_{i}(t)+I_{i}(t)p_{i}(t)-\mu_{i}(t)S_{i}(t).$ (4)

The evolution of node $i$ in the refractory state is

$R_{i}(t+1)=R_{i}(t)+\mu_{i}(t)S_{i}(t).$ (5)

Using Eqs. (3)-(5), we obtain the fraction of nodes at time $t$ in each state,

$H(t)=\frac{1}{N}\sum_{i=1}^{N}H_{i}(t),$ (6)

where $H\in\\{I,S,R\\}$. Note that in the steady state there are no spreader
nodes, only refractory and stifler nodes. The fraction of nodes that receive
the misinformation in the final state is $R(\infty)=R$. Here $R$ is the order
parameter of a continuous phase transition with $\lambda$. If the
misinformation transmission probability $\lambda$ is larger than the critical
threshold, i.e., $\lambda>\lambda_{c}$, the size of the global misinformation
is of the order of the system size. Otherwise, the global misinformation $R=0$
for $\lambda\leq\lambda_{c}$ is in the thermodynamic limit. At shorter times,
a vanishingly small fraction of nodes receive the misinformation, i.e.,
$S_{i}\left(t\right)\approx 0$ and
$R_{i}\left(t\right)=1-S_{i}\left(t\right)-I_{i}\left(t\right)\approx 0$. The
recovery probability in Eq. (1) of a spreader node $i$ is
$\mu_{i}\left(t\right)\approx\mu$, since node $i$ must connect to a spreader
that supplies the misinformation, and there is a low probability that it will
connect to other spreader or refractory neighbors. Thus Eq. (4) can be
rewritten

$S_{i}(t+1)\approx\sum_{j=1}^{N}[(1-\mu)+\lambda\mathcal{A}_{ij}(t)\delta_{ij}]S_{j}(t),$
(7)

where $\delta_{ij}$ is the Kronecker delta function, i.e., $\delta_{ij}=1$ if
$i=j$, and zero otherwise. We define the transmission tensor $\mathbf{M}$ to
be

$\mathbf{M}_{ij}^{tt^{\prime}}=\delta^{t,t^{\prime}+1}[(1-\mu)\delta_{ij}+\lambda\mathcal{A}_{ij}(t)].$
(8)

We mask the tensorial origin of the space through the map
$(i,t)\rightarrow\alpha=Nt+i$, where $1\leq\alpha\leq N\;t_{\rm max}$. Thus
$\mathbf{M}$ can be rewritten

$\mathbf{M}=\left(\begin{array}[]{ccccc}0&1-\mu+\lambda\mathcal{A}(1)&0&\cdots&0\\\
0&0&1-\mu+\lambda\mathcal{A}(2)&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&1-\mu+\lambda\mathcal{A}(t_{\rm max}-1)\\\
1-\mu+\lambda\mathcal{A}(t_{\rm max})&0&0&\cdots&0\\\ \end{array}\right).$ (9)

Inserting Eq. (9) into (7), we have

$\widehat{S}(\tau)=\mathbf{M}\widehat{S}(\tau-1),$ (10)

where $\widehat{S}(\tau)$ is the probability that a node is in the spreader
state at each time step $t$ during $[\tau t_{\rm max},(\tau+1)t_{\rm max}]$.
Here $\widehat{S}(\tau)$ increases exponentially if the largest eigenvalue of
$\mathbf{M}$, denoted $\Lambda_{1}$, is larger than $1$. Thus the
misinformation spreads, and the threshold condition is Valdano  _et al._
(2015)

$\Lambda_{1}(\mathbf{M})=1,$ (11)

In an unweighted undirected network $\mathcal{G}$, the largest eigenvalue
$\Lambda_{1}(\mathbf{M})$ of $\mathbf{M}$ is

$\Lambda_{1}(\mathbf{M})=\Lambda_{1}(\mathbf{P}^{1/t_{\rm max}}),$ (12)

where

$\mathbf{P}=\sum_{t=1}^{t_{\rm max}}(1-\mu+\lambda\mathcal{A}(t)).$ (13)

##  IV Misinformation containing strategies

Because misinformation spreading on social networks can induce social
instability, threaten political security, and endanger the economy, we propose
three strategies—random, targeted, and heuristic—for containing the spread of
misinformation in temporal networks using a given fraction of containing nodes
$f$. We first immunize a fraction of $f$ nodes using a static containment
strategy. The misinformation then spreads on the residual temporal network. If
node $i$ is “immunized,” it cannot be infected (transmit) by the
misinformation received from neighbors (the misinformation to neighbors).
Mathematically the immunized node set is $\mathcal{V}$, and the number of
immunized nodes equals the number of elements in $\mathcal{V}$, i.e.,
$|\mathcal{V}|=\lceil f\;N\rceil$. We set $v_{i}=1$ if node $i$ is immunized,
otherwise $v_{i}=0$. After immunization, Eqs. (2)–(4) can be written

$p_{i}\left(t\right)=1-\prod_{j}{\left[1-\lambda(1-v_{j})\mathcal{A}_{ij}\left(t\right)I_{j}\left(t\right)\right]},$
(14) $I_{i}(t+1)=I_{i}(t)-I_{i}(t)(1-v_{i})p_{i}(t),$ (15)

and

$S_{i}(t+1)=S_{i}(t)+I_{i}(t)(1-v_{i})p_{i}(t)-\mu_{i}(t)S_{i}(t),$ (16)

respectively. In an effective containing strategy the misinformation spreading
dynamics is suppressed for a given fixed fraction $f$ of immunized nodes,
i.e., the objective function is

$\min_{\mathcal{V}}{\lim_{t\rightarrow\infty}{\frac{1}{N}\sum_{i}R_{i}(t)}},$
(17)

where the constraint conditions are Eqs. (15)–(16) and
$V_{i}\,\,\in\left\\{0,1\right\\}$,
$S_{i}\left(t\right)\in\,\left[0,1\right]$,
$I_{i}\,\left(t\right)\in\,\left[0,1\right]$, and
$R_{i}\left(t\right)\in\,\left[0,1\right]$. Because the problem is NP-hard,
finding an accurate solution for large-scale temporal networks and a finite
immunization size $f$ is difficult. To address this problem we propose three
containment strategies.

  * •

Strategy I: Random containment (RC). The most used strategy for containing the
spread of misinformation is randomly immunizing a fraction of $f$ nodes Zuzek
_et al._ (2015).

  * •

Strategy II: Targeted containment (TC). Another intuitive way is to immunize
the nodes with highest average degree $\langle k\rangle$ in the temporal
network $\mathcal{G}$. Specifically, we first compute the average degree of
each node $i$ as $\langle k_{i}\rangle=\frac{1}{t_{\rm max}}\sum_{t=1}^{t_{\rm
max}}\sum_{j=1}^{N}\mathcal{A}_{ij}(t)$. We then rank all nodes in descending
order in the vector $\mathcal{W}$ according to the average degree of each
node. Finally we immunize the top $\lceil fN\rceil$ nodes of $\mathcal{W}$.

  * •

Strategy III: Heuristic containment (HC). Using the TC, we apply an HC
strategy. Because TC is much better than RC, we perform the HC strategy by
replacing the immunization nodes. When the repeat time is very large, the
immunized nodes reach an optimal value.

(i)

We initialize a vector $\mathcal{W}$ according to the descending order of the
average degree of nodes. The first $\lceil f\;N\rceil$ nodes of $\mathcal{W}$
are immunized, the final misinformation outbreak size is $R_{o}$, and
$\mathcal{W}_{0}$ is denoted a set. The remaining nodes
$\mathcal{W}_{1}=\mathcal{W}\backslash\mathcal{W}_{0}$ are denoted a set.

(ii)

We randomly select nodes in $\mathcal{W}_{0}$ and $\mathcal{W}_{1}$, denoted
$v_{0}$ and $v_{1}$, respectively. We switch their order in vector
$\mathcal{W}$ and denote the new vector $\mathcal{W}_{n}$. We immunize the
first $\lceil f\;N\rceil$ nodes $\mathcal{W}_{n}$ and compute the final
misinformation outbreak size $R_{n}$.

(iii)

When $R_{n}>R_{o}$, we update vector $\mathcal{W}$, i.e.,
$\mathcal{W}\rightarrow\mathcal{W}_{n}$. Otherwise, there is no change.

(iv)

We repeat steps (ii) and (iii) until
$\frac{1}{t_{s}}\sum_{i=1}^{t_{s}}|\mathcal{W}-\mathcal{W}_{n}|<\epsilon^{\prime}$.
In the simulations we set $t_{s}=100$ and $\epsilon^{\prime}=N^{-1}$.

##  V Numerical simulations

For the activity-driven network, we set $N=10^{3}$, $t_{\rm max}=20$,
$\eta=10$, $m=50$, $\gamma=2.1$, and $\epsilon=10^{-3}$. For real-world
networks, we use the data collected by the Sociopatterns group isella2011s,
which records the interactions among the participants at a conference. The
time resolution of the signal is 20 sec. Because the temporal network is
sparse, it is difficult for the information to spread in the original network.
We thus aggregate the temporal network using four windows, $w=30{\rm min}$,
$60{\rm min}$, $120{\rm min}$, and $240{\rm min}$. We average all simulation
results more than 1000 times.

We use variability to locate the numerical network-sized dependent outbreak
threshold Shu  _et al._ (2015, 2016),

$\chi=\frac{(\langle R^{2}\rangle-\langle R\rangle^{2})^{1/2}}{\langle
R\rangle},$ (18)

where $R$ is the relative size of misinformation spreading at the steady
state. At the outbreak threshold $\lambda_{c}$, $\chi$ exhibits a peak. When
$\lambda\leq\lambda_{c}$, the global misinformation does not break out, but
when $\lambda>\lambda_{c}$ the global misinformation does break out.

Figure 2: Misinformation spreading on activity-driven temporal networks. (a)
The final misinformation outbreak size $R$, and the numerical variability
$\chi$ versus the transmission probability $\lambda$ with different values of
recovery probability $\mu$. In (a), the symbols are the numerical simulation
results, and lines are the theoretical predictions.

Figure 2 shows the misinformation spreading on activity-driven networks. Note
that the final misinformation outbreak size $R$ increases with $\lambda$. The
larger the recovery probability $\mu$, the lower the values of $R$ because
spreader nodes are less likely to transmit the misinformation to stifler
neighbors [see Fig. 2(a)]. Note that our theoretical and numerical predictions
of the final misinformation outbreak size $R$ agree. Figure 2(b) shows the
variability $\chi$ as a function of $\lambda$. There is a peak at the
misinformation outbreak threshold $\lambda_{c}$. Figure 3 shows $\lambda_{c}$
versus $\mu$ in which $\lambda_{c}$ increases linearly with $\mu$. The
theoretical predictions of $\lambda_{c}$ obtained from Eq. (11) agree with the
stochastic simulations.

Figure 3: The outbreak threshold $\lambda_{c}$ versus $\mu$. The symbols
represent the stochastic simulation threshold, and line is the theoretical
prediction.  Figure 4: Misinformation spreading on real-world temporal
networks. The final misinformation outbreak size $R$ versus $\lambda$ on ACM
Hypertex conference data set with aggregating window (a) $w=30~{}{\rm min}$,
(b) $w=60~{}{\rm min}$, (c) $w=120~{}{\rm min}$ and (d) $w=240~{}{\rm min}$.
The symbols are the numerical simulation results, and lines are the
theoretical predictions.

Figure 4 shows misinformation spreading in real-world temporal networks. As in
Fig. 2, $R$ increases with $\lambda$ and decreases with $\mu$. As in SIR
epidemic spreading Ogura and Preciado (2017), the effective outbreak threshold
$(\lambda/\mu)_{c}$ is a constant value. In addition, when the value
aggregating window $w$ is small, there are fewer opportunities for spreaders
to transmit the misinformation to stifler neighbors, thus the misinformation
does not break out globally, i.e., there are smaller values of $R$ for smaller
$w$. Once again our theoretical results agree with the numerical simulations.

Figure 5: Containing misinformation spreading dynamics on activity-driven
temporal networks. The final misinformation outbreak size $R$ (a) and
variability $\chi$ (c) versus the transmission probability $\lambda$ under the
RC strategy. $R$ (b) and $\chi$ (d) versus $\lambda$ under the TC strategy. In
(a) and (b), the lines are the theoretical predictions, and symbols are the
numerical simulation results.

We next examine the performances of our proposed strategies for mitigating
misinformation spreading on artificial and real-world temporal networks.
Figure 5 shows $R$ versus $\lambda$ for different values of the fraction of
containing nodes $f$. Note that $R$ decreases with $f$ because no more nodes
receive the misinformation. Note also that the TC strategy performs much
better than the RC strategy because the higher degree nodes $\langle k\rangle$
are contained, and spreaders can no longer transmit the misinformation to
stiflers. Thus when we immunize the same fraction of containing nodes, the
values of $R$ for the TC strategy are smaller than those for the RC strategy.
For example, when $f=0.5$, using the RC strategy $\approx 30\%$ of the nodes
are informed by the misinformation, but using the TC strategy none are
informed by the misinformation. In addition, the outbreak threshold
$\lambda_{c}$ decreases with $f$. Figures 5(c)–5(d) show the numerical
predictions for $\lambda_{c}$.

Figure 6: Effectiveness of TC and HC strategies on activity-driven networks.
The final misinformation outbreak size $R$ versus $f$ for a given values of
$\lambda=0.2$ (a) and $\lambda=0.3$ (b). The lines and symbols are the
theoretical and the numerical predictions of $R$, respectively.

Figure 6 shows that the HC strategy performs better than the TC strategy. We
set To this end, we set $\lambda=0.2$ and 0.3 in 6(a) and 6(b), respectively,
and compare the final misinformation spreading size $R$ with $f$. We find that
the misinformation spreading dynamics are suppressed when we use the HC
strategy. For example, when $f=0.3$ and $\lambda=0.3$, few nodes are informed
by the misinformation, i.e., $R\approx 0.01$ when we use the HC strategy, but
when we use the TC strategy to contain the misinformation the fraction of
nodes informed by the misinformation is finite, i.e., $R\approx 0.25$. Our
theoretical results agree with the numerical simulation results.

Figure 7: The effective outbreak threshold $(\lambda/\mu)_{c}$ versus the
fraction of containing nodes $f$ on activity-driven networks for RC strategy
(a), TC strategy (b), HC strategy (c) and the comparision of the three
strategies (d). The lines and symbols are the theoretical and numerical
predictions of $(\lambda/\mu)_{c}$, respectively. The vertical line represents
the critical probability $f_{c}$.

An effective containing strategy with a fraction of immunized node $f$ and a
small outbreak threshold $\lambda_{c}$ greatly decreases the final
misinformation outbreak size $R$. Figure 7 shows the effective outbreak
threshold $(\lambda/\mu)_{c}$ versus $f$ on activity-driven networks for the
RC, TC, and HC strategies. Here $(\lambda/\mu)_{c}$ increases with $f$, and
$(\lambda/\mu)_{c}$ is the largest using the HC strategy when $f$ is fixed.
When $f$ is sufficiently large, no $\lambda$ value can induce a global
misinformation outbreak. We denote $f_{c}$ the critical probability that at
least a fraction of $f_{c}$ nodes must be containing to halt misinformation
speading in temporal networks. We find that the values of $f_{c}$ for the HC
containing strategy are the smallest of all containing strategies. The $f_{c}$
value for the RC strategy is 5 times the $f_{c}$ value for the TC strategy,
and the $f_{c}$ value for the TC strategy is 2.5 times the $f_{c}$ value for
HC strategy. Thus the HC strategy is the most effective.

Figure 8: Effectiveness of TC and HC strategies on ACM Hypertex conference
data set. $R$ versus $f$ with aggregating window (a) $w=30~{}{\rm min}$, (c)
$w=60~{}{\rm min}$, (e) $w=120~{}{\rm min}$ and (g) $w=240~{}{\rm min}$ for
$\lambda=0.2$. $R$ versus $f$ with (b) $w=30~{}{\rm min}$, (d) $w=60~{}{\rm
min}$, (f) $w=120~{}{\rm min}$ and (h) $w=240~{}{\rm min}$ for $\lambda=0.3$.
The lines and symbols are the theoretical and numerical predictions of $R$,
respectively.

We finally examine real-world networks to varify the effectiveness of our
proposed three strategies. Figure 8 compares the performances of the TC and HC
strategies by examining $R$ versus $f$ for given values of $\lambda$. As in
Fig. 6, the HC strategy most effectively contains the misinformation spreading
on temporal networks irrespective of the values of $\lambda$. In addition,
Fig. 9 shows that the effective outbreak threshold $(\lambda/\mu)_{c}$ is the
smallest when using the HC strategy. Thus our theory accurately predicts the
numerical simulation results.

Figure 9: Comparing the effectiveness of RC, TC and HC strategies on ACM
Hypertex conference data set with aggregating window (a) $w=30~{}{\rm min}$,
(b) $w=60~{}{\rm min}$, (c) $w=120~{}{\rm min}$ and (d) $w=240~{}{\rm min}$.
The lines and symbols are the theoretical and numerical predictions of
$(\lambda/\mu)_{c}$, respectively.

##  VI Conclusions

We have systematically examined the dynamics of misinformation spreading on
temporal networks. We use activity driven networks to describe temporal
networks, and use a discrete Markovian chain to describe the spreading
dynamics. We find that the global misinformation outbreak threshold correlates
with the topology of temporal networks. Using extensive numerical simulations,
we find that our theoretical predictions agree with numerical predictions in
both artificial and real-world networks.

To contain misinformation spreading on temporal networks, we propose three
strategies, random containing (RC), targeted containing (TC), and heuristic
containing (HC) strategies. We perform numerical simulations and a theoretical
analysis on both artificial and four real-world networks and find that the HC
strategy outperforms the other two strategies, maximizes the outbreak
threshold, and minimizes the final outbreak size. Our proposed containing
strategy expands our understanding of how to contain public sentiment and
maintain social stability.

###### Acknowledgements.

This work was partially supported by the China Postdoctoral Science Foundation
(Grant No. 2018M631073), and Fundamental Research Funds for the Central
Universities. LAB thanks UNMdP and CONICET (PIP 00443/2014) for financial
support.

## References

  * Vosoughi  _et al._ (2018) S. Vosoughi, D. Roy, and S. Aral, Science 359, 1146 (2018).
  * Del Vicario  _et al._ (2016) M. Del Vicario, A. Bessi, F. Zollo, F. Petroni, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Proceedings of the National Academy of Sciences 113, 554 (2016).
  * Qiu  _et al._ (2017) X. Qiu, D. F. Oliveira, A. S. Shirazi, A. Flammini, and F. Menczer, Nature Human Behaviour 1, 0132 (2017).
  * Schmidt  _et al._ (2017) A. L. Schmidt, F. Zollo, M. Del Vicario, A. Bessi, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Proceedings of the National Academy of Sciences 114, 3035 (2017).
  * Castellano  _et al._ (2009) C. Castellano, S. Fortunato, and V. Loreto, Reviews of Modern Physics 81, 591 (2009).
  * Zhang  _et al._ (2016) Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang, and Y.-C. Zhang, Physics Reports 651, 1 (2016).
  * Kumar and Shah (2018) S. Kumar and N. Shah, arXiv:1804.08559 (2018).
  * Funk  _et al._ (2009) S. Funk, E. Gilad, C. Watkins, and V. A. Jansen, Proceedings of the National Academy of Sciences 106, 6872 (2009).
  * Granell  _et al._ (2013) C. Granell, S. Gómez, and A. Arenas, Physical Review Letters 111, 128701 (2013).
  * Wang  _et al._ (2014) W. Wang, M. Tang, H. Yang, Y. Do, Y.-C. Lai, and G. Lee, Scientific Reports 4, 5097 (2014).
  * Tai and Sun (2007) Z. Tai and T. Sun, New Media & Society 9, 987 (2007).
  * Chow (2003) Y. Chow, Weekly Releases (1997–2007) 7, 2318 (2003).
  * Allcott and Gentzkow (2017) H. Allcott and M. Gentzkow, Journal of Economic Perspectives 31, 211 (2017).
  * Soll (2016) J. Soll, Politico Magazine 18 (2016).
  * Bessi  _et al._ (2015) A. Bessi, M. Coletto, G. A. Davidescu, A. Scala, G. Caldarelli, and W. Quattrociocchi, PLoS ONE 10, e0118093 (2015).
  * Mocanu  _et al._ (2015) D. Mocanu, L. Rossi, Q. Zhang, M. Karsai, and W. Quattrociocchi, Computers in Human Behavior 51, 1198 (2015).
  * Bovet and Makse (2018) A. Bovet and H. A. Makse, arXiv preprint arXiv:1803.08491 (2018).
  * Pastor-Satorras  _et al._ (2015) R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Reviews of modern physics 87, 925 (2015).
  * de Arruda  _et al._ (2018) G. F. de Arruda, F. A. Rodrigues, and Y. Moreno, arXiv:1804.08777 (2018).
  * Wang  _et al._ (2018) W. Wang, M. Cai, and M. Zheng, Physica A 499, 121 (2018).
  * Chen  _et al._ (2018) X. Chen, W. Wang, S. Cai, H. E. Stanley, and L. A. Braunstein, Journal of Statistical Mechanics: Theory and Experiment 2018, 053501 (2018).
  * Young (2011) H. P. Young, Proceedings of the National Academy of Sciences 108, 21285 (2011).
  * Weiss  _et al._ (2014) C. H. Weiss, J. Poncela-Casasnovas, J. I. Glaser, A. R. Pah, S. D. Persell, D. W. Baker, R. G. Wunderink, and L. A. N. Amaral, Physical Review X 4, 041008 (2014).
  * Iribarren and Moro (2009) J. L. Iribarren and E. Moro, Physical Review Letters 103, 038702 (2009).
  * Alvarez-Zuzek  _et al._ (2019) L. G. Alvarez-Zuzek, M. A. Di Muro, S. Havlin, and L. A. Braunstein, Physical Review E 99, 012302 (2019).
  * Di Muro  _et al._ (2018) M. A. Di Muro, L. G. Alvarez-Zuzek, S. Havlin, and L. A. Braunstein, New Journal of Physics 20, 083025 (2018).
  * Valdez  _et al._ (2018) L. D. Valdez, H. Rêgo Aragão, H. E. Stanley, S. Havlin, and L. A. Braunstein, New Journal of Physics 20, 125003 (2018).
  * Yang  _et al._ (2018) L.-X. Yang, P. Li, X. Yang, Y. Wu, and Y. Y. Tang, Nonlinear Dynamics 91, 1853 (2018).
  * Moreno  _et al._ (2004) Y. Moreno, M. Nekovee, and A. F. Pacheco, Physical Review E 69, 066130 (2004).
  * Nekovee  _et al._ (2007) M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili, Physica A 374, 457 (2007).
  * Zhou  _et al._ (2007) J. Zhou, Z. Liu, and B. Li, Physics Letters A 368, 458 (2007).
  * Isham  _et al._ (2010) V. Isham, S. Harden, and M. Nekovee, Physica A 389, 561 (2010).
  * Zhao  _et al._ (2012) L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, and H. Cui, Physica A 391, 2444 (2012).
  * Kwon  _et al._ (2013) S. Kwon, M. Cha, K. Jung, W. Chen, and Y. Wang, in  _2013 IEEE 13th International Conference on Data Mining_ (IEEE, 2013) pp. 1103–1108.
  * Borge-Holthoefer and Moreno (2012) J. Borge-Holthoefer and Y. Moreno, Physical Review E 85, 026116 (2012).
  * Lü  _et al._ (2016a) L. Lü, T. Zhou, Q.-M. Zhang, and H. E. Stanley, Nature Communications 7, 10168 (2016a).
  * Lü  _et al._ (2016b) L. Lü, D. Chen, X.-L. Ren, Q.-M. Zhang, Y.-C. Zhang, and T. Zhou, Physics Reports 650, 1 (2016b).
  * Kitsak  _et al._ (2010) M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse, Nature physics 6, 888 (2010).
  * Morone and Makse (2015) F. Morone and H. A. Makse, Nature 524, 65 (2015).
  * Pei  _et al._ (2018) S. Pei, F. Morone, and H. A. Makse, in  _Complex Spreading Phenomena in Social Systems_ (Springer, 2018) pp. 125–148.
  * Borge-Holthoefer  _et al._ (2013) J. Borge-Holthoefer, S. Meloni, B. Gonçalves, and Y. Moreno, Journal of Statistical Physics 151, 383 (2013).
  * Holme and Saramäki (2012) P. Holme and J. Saramäki, Physics Reports 519, 97 (2012).
  * Holme and Saramäki (2013) P. Holme and J. Saramäki,  _Temporal networks_ (Springer, 2013).
  * Holme (2015) P. Holme, The European Physical Journal B 88, 234 (2015).
  * Speidel  _et al._ (2016) L. Speidel, K. Klemm, V. M. Eguíluz, and N. Masuda, New Journal of Physics 18, 073013 (2016).
  * Masuda and Holme (2013) N. Masuda and P. Holme, F1000prime Reports 5 (2013).
  * Masuda and Lambiotte (2016) N. Masuda and R. Lambiotte,  _A Guidance to Temporal Networks_ (World Scientific, 2016).
  * Li  _et al._ (2017) A. Li, S. P. Cornelius, Y.-Y. Liu, L. Wang, and A.-L. Barabási, Science 358, 1042 (2017).
  * Nadini  _et al._ (2018) M. Nadini, K. Sun, E. Ubaldi, M. Starnini, A. Rizzo, and N. Perra, Scientific Reports 8, 2352 (2018).
  * Liu  _et al._ (2016) M.-X. Liu, W. Wang, Y. Liu, M. Tang, S.-M. Cai, and H.-F. Zhang, arXiv:1605.04557 (2016).
  * Kalantar-Zadeh  _et al._ (2006) K. Kalantar-Zadeh, N. Kuwae, D. Regidor, C. Kovesdy, R. Kilpatrick, C. Shinaberger, C. McAllister, M. Budoff, I. Salusky, and J. Kopple, Kidney International 70, 771 (2006).
  * Starnini  _et al._ (2013) M. Starnini, A. Baronchelli, and R. Pastor-Satorras, Physical Review Letters 110, 168701 (2013).
  * Perra  _et al._ (2012) N. Perra, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani, Scientific Reports 2, 469 (2012).
  * Valdano  _et al._ (2015) E. Valdano, L. Ferreri, C. Poletto, and V. Colizza, Physical Review X 5, 021005 (2015).
  * Valdano  _et al._ (2018) E. Valdano, M. R. Fiorentin, C. Poletto, and V. Colizza, Physical Review Letters 120, 068302 (2018).
  * Onaga  _et al._ (2017) T. Onaga, J. P. Gleeson, and N. Masuda, Physical Review Letters 119, 108301 (2017).
  * Pozzana  _et al._ (2017) I. Pozzana, K. Sun, and N. Perra, Physical Review E 96, 042310 (2017).
  * Karsai  _et al._ (2014) M. Karsai, N. Perra, and A. Vespignani, Scientific Reports 4, 4001 (2014).
  * Ogura and Preciado (2017) M. Ogura and V. M. Preciado, in  _Temporal Network Epidemiology_ (Springer, 2017) pp. 241–266.
  * Liu  _et al._ (2014) S. Liu, N. Perra, M. Karsai, and A. Vespignani, Physical Review Letters 112, 118702 (2014).
  * Lima  _et al._ (2015) A. Lima, M. De Domenico, V. Pejovic, and M. Musolesi, Scientific Reports 5, 10650 (2015).
  * Moinet  _et al._ (2018) A. Moinet, R. Pastor-Satorras, and A. Barrat, Physical Review E 97, 012313 (2018).
  * Holme and Liljeros (2014) P. Holme and F. Liljeros, Scientific Reports 4, 4999 (2014).
  * Masuda  _et al._ (2013) N. Masuda, K. Klemm, and V. M. Eguíluz, Physical Review Letters 111, 188701 (2013).
  * Vestergaard  _et al._ (2014) C. L. Vestergaard, M. Génois, and A. Barrat, Physical Review E 90, 042805 (2014).
  * Zuzek  _et al._ (2015) L. G. A. Zuzek, C. Buono, and L. A. Braunstein, in  _Journal of Physics: Conference Series_ , Vol. 640 (IOP Publishing, 2015) p. 012007.
  * Shu  _et al._ (2015) P. Shu, W. Wang, M. Tang, and Y. Do, Chaos 25, 063104 (2015).
  * Shu  _et al._ (2016) P. Shu, W. Wang, M. Tang, P. Zhao, and Y.-C. Zhang, Chaos 26, 063108 (2016).


††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Few-mode to mesoscopic junctions in gatemon qubits

Alisa Danilenko  Center for Quantum Devices, Niels Bohr Institute, University
of Copenhagen, 2100 Copenhagen, Denmark  Deividas Sabonis  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen,
Denmark  Georg W. Winkler  Microsoft Quantum, Microsoft Station Q, University
of California, Santa Barbara, California 93106-6105, USA  Oscar Erlandsson
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Peter Krogstrup  Center for Quantum Devices, Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Microsoft
Quantum Materials Lab–Copenhagen, 2800 Lyngby, Denmark  Charles M. Marcus
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark

###### Abstract

We investigate a semiconductor nanowire-based gatemon qubit with epitaxial Al
on two facets of the nanowire, allowing gate control of wire density. Two
segments have the Al removed, one forming a Josephson junction, and the other
operating as a transistor, providing in-situ switching between dc transport
and qubit operation. Gating the NW changes the bulk wire potential
distribution, while gating the Josephson junction changes the number of
junction modes. Both effects are revealed by the dependence of qubit frequency
on parallel magnetic field. A detailed model of the wire and junction yields
behavior consistent with experiment. In the multi-mode regime, fluctuations in
qubit frequency are considerably smaller than the theoretical “universal”
value, also smaller than numerics, and consistent with previous measurements
of fluctuating critical current.

Figure 1: (a) Scanning electron micrograph of Device 1 (cross section sketch
on the right). The NW is contacted on the right to the ground plane, the
capacitor island is connected in the center. The purpose of gate voltage
$V_{\rm C}$ is to control the Josephson junction, while $V$LP and $V$RP are
intended to tune the bulk wire. Direction of applied magnetic field $B$ is
shown. (b) Differential conductance
$g\equiv\frac{{\mathrm{d}}I_{\text{B}}}{{\mathrm{d}}V_{\text{B}}}$ as a
function of $V$B shows the superconducting gap of the junction in applied
field, with a cut (c) taken at $B$ = 0.08 T. $V$FET = +6 V, $V$LP = $V$RP = 0
V, $V_{\rm C}$ = $-3.6$ V. (d) Rabi oscillations at $B=0$. Demodulated
transmission $V$H measured as a function of drive duration $\tau$d at the
qubit frequency $f_{Q}$ = 4.37 GHz. Exponentially decaying sinusoidal fit
yields Rabi time $T^{\text{R}}_{2}~{}=~{}120$ ns. (e) Qubit relaxation: $V$H
measured as a function of wait time $\tau$w between drive and readout pulses.
The qubit is excited with a $\pi$ pulse calibrated from (d). Exponential fit
yields $T_{1}~{}=~{}1.8~{}\mu$s.

Recent materials advances Krogstrup  _et al._ (2015) have lead to a new
approach to Josephson qubit technology based on hybrid superconductor-
semiconductor nanowires (NWs) de Lange  _et al._ (2015); Larsen  _et al._
(2015) and comparable two-dimensional platforms Casparis  _et al._ (2018).
This approach allows voltage control of qubit operation and reduced
sensitivity to charge noise Kringhøj  _et al._ (2020a, b); Bargerbos  _et al._
(2020). Hybrid NWs can also form the basis of Andreev qubits Hays  _et al._
(2018); Tosi  _et al._ (2019); Hays  _et al._ (2021); Matute-Cañadas  _et al._
(2022), protected $0-\pi$ qubits Larsen  _et al._ (2020), systems to
investigate the presence of topological phases Sabonis  _et al._ (2020), and
voltage-controlled qubit couplers Casparis  _et al._ (2019). Because the
electron wavelength in the semiconductor is comparable to the NW diameter,
electronic states under the proximitizing superconductor typically occupy a
small number of transverse modes Kringhøj  _et al._ (2018). For NWs with
facets not covered by the superconductor, this mode structure can be altered
by electrostatic gating Splitthoff  _et al._ (2022).

In this Letter, we compare the magnetic field and gate-voltage dependence of
gatemon qubits fabricated from epitaxial InAs/Al NWs to a detailed numerical
model of the wire and Andreev bound states (ABSs) in the junction Winkler  _et
al._ (2019). To gather parameters for the model, we take advantage of another
feature of hybrid NWs by creating a local field-effect transistor (FET) that
allows in-situ switching between dc transport and circuit quantum
electrodynamics (cQED) configurations Kringhøj  _et al._ (2020a). Magnetic
field and gate-voltage dependences of qubit frequency $f_{Q}$ Kringhøj  _et
al._ (2021a) are in reasonable agreement with the model, and consistent with
gate-voltage Zuo  _et al._ (2017) and magnetic field dependences Gül  _et al._
(2014) of critical currents in NW junctions, here measurable in the same qubit
junction.

At gate voltages corresponding to several ABSs in the qubit junction,
mesoscopic (random, repeatable) fluctuations of qubit frequency as a function
of gate voltage were observed. Comparing experimental results with both
numerics and theoretical universal statistics for mesoscopic critical current
fluctuations Beenakker (1991), we find that the observed qubit-frequency
fluctuations, $\sigma_{f_{Q}}\sim\,$130 MHz, corresponding to critical-current
fluctuations $\sigma_{I_{c}}\sim\,$1 nA, are smaller than theoretical values
for a short junction Beenakker (1991), though consistent with previous
experimental values of critical current fluctuations Takayanagi  _et al._
(1995). (Mesoscopic fluctuations of $f_{Q}$ have not been reported previously,
to our knowledge). We tentatively ascribed the reduced fluctuation of $f_{Q}$
to non-ideal material interfaces Doh  _et al._ (2005); Takayanagi  _et al._
(1995) or a Fermi velocity mismatch between the Al-covered region and the bare
semiconductor junction Schäpers  _et al._ (1997) leading to normal reflection
competing with Andreev reflection within the junction interfaces.

Devices were fabricated on a high-resistivity silicon substrate covered with a
20 nm NbTiN film, deposited by sputtering. Each chip contains three gatemons
based on NWs , about 100 nm in diameter, with two or three of six facets
covered with Al Kringhøj  _et al._ (2021a), two with FET-switched dc transport
capabilities, each with individual resonator readout circuits. Overall, three
devices were measured. Data from two devices, yielding consistent results, are
reported here. Resonators, transmission line, and electrostatic gates were
fabricated using additional layers of sputtered NbTiN, patterned using
electron beam lithography and reactive ion etching. Before placing the NW on
the bottom gates, a lithographically patterned layer of HfO2 dielectric was
deposited by atomic layer deposition. A micrograph of one of the FET-switched
devices is shown in Fig. 1(a). The right side of the NW connects to the ground
plane, the left side to a dc contact through the FET, and the center to the
qubit island. The Josephson junction, seen inside the white box in Fig. 1(a),
is formed by wet etching $\sim 100$ nm of the Al shell. While the orientation
of the shell is not discernible during manual NW placement, scanning electron
micrographs taken afterwards can resolve the Al shell. Devices with the shell
on the up-facing half of the NW, which allow control of carrier density in the
NW by bottom gates, are then measured. Gate voltage $V_{\rm C}$, underneath
the junction, was used to tune $f_{Q}$, while voltages $V$LP and $V$RP were
used to tune the density in the bulk NW. All measurements were performed in a
dilution refrigerator with a base temperature of 20 mK using a 6-1-1 T vector
magnet.

Setting the FET in the conducting state by applying +6.0 V on the FET gate,
differential conductance
$g\equiv{\mathrm{d}}I_{\text{B}}/{\mathrm{d}}V_{\text{B}}$ of the junction was
measured as a function of voltage bias $V_{\text{B}}$. With the qubit junction
in the tunneling regime, $g$ can be used to measure the parent gap and ABS
features in the semiconductor, as shown in Figs. 1(b, c). The two higher-bias
peaks occur at the bias where coherence peaks from the Al gap, $\Delta\sim
250\,\mu$eV, on the two sides of the junctions align, $V_{\text{B}}\sim\pm
2\Delta/e\sim\pm 0.5\,$mV. The two lower-bias peaks reflect where the
coherence peak in one lead aligns with a subgap ABS in the other lead.
Extrapolating the field dependence of the gap gives a field of $\sim$1.4 T
where the gap closes. Following dc transport characterization of the junction,
the FET was switched to a non-conductive state (FET gate at $-6$ V) allowing
the device to be operated in cQED mode as a qubit. Setting
$V\textsubscript{C}=-1.7$ V, $V\textsubscript{LP, RP}=0$ V gives a qubit
frequency $f_{Q}=4.37$ GHz, measured via two-tone spectroscopy. Rabi
oscillations [Fig. 1(d)] were measured by applying a series of pulses of
duration $\tau\textsubscript{d}$ at $f_{Q}$ and plotting the demodulated
transmission $V\textsubscript{H}$ as a function of $\tau\textsubscript{d}$.
Fitting to an exponentially decaying sinusoid yields a Rabi time of 119 $\pm$
1 ns. Qubit relaxation [Fig. 1(e)] was measured by applying a $\pi$ pulse,
found using data in (d), at $f_{Q}$, then waiting $\tau\textsubscript{w}$
before applying a readout pulse at the resonator frequency, 5.46 GHz, giving a
$V$H signal that decreases with increasing $\tau\textsubscript{w}$. An
exponential fit yields a qubit lifetime of 1.81 $\pm$ 0.13 $\mu$s.
The dependence of $f_{Q}$ on axial magnetic field $B$ and gate voltage is
shown in Fig. 2(a), for Device 2, which is similar to Device 1 in material and
fabrication. For each value of $V\textsubscript{LP, RP}$, $V\textsubscript{C}$
was compensated to keep the zero-field frequency constant, to try to minimise
the effect of $V\textsubscript{LP, RP}$ on the junction, since the aim was to
tune the bulk wire density. This compensation had the added benefit of making
sure that $f_{Q}$ stayed in a measurable range at zero field. The changes made
to $V\textsubscript{C}$ are very small compared to its full range of
operation. In the measured field and frequency range, the qubit frequency
decreased monotonically with increasing $B$. The solid lines in Fig. 2(a) are
simple fits relating the qubit frequency expected closing of the
superconducting gap in field, $f_{Q}(B)=f_{Q}(0)\,[{1-(B/B_{c})^{2}}]^{1/4}$,
where $B_{c}$ is the critical field Tinkham (2004). As illustrated in Fig.
2(b), a trend in the dependence on $V\textsubscript{LP, RP}$ was observed: the
more positive the gate voltage applied to the NW, the more rapidly $f_{Q}$
decayed in field.

Figure 2: (a) Measured qubit frequency (normalized) as a function of parallel
magnetic field $B$ at a range of $V\textsubscript{LP, RP}$ values.
$V\textsubscript{C}$ was compensated to keep the zero field frequency constant
for all gate configurations. (b) Critical field $B_{c}$, extracted from
fitting the data for each gate configuration measured. Shown with a linear
fit.

To understand the effect of the gate voltage configuration on the magnetic
field dependence of qubit frequency, we perform numerical modeling of the
energy spectrum and qubit frequency using codes similar to Refs. Vaitiekėnas
_et al._ (2020); Kringhøj  _et al._ (2021b); Shen  _et al._ (2021) using a
self-consistent Thomas-Fermi approximation, including Zeeman and orbital
effects of the magnetic field, treating coupling of the superconductor to the
semiconductor in terms of a self-energy boundary condition (see Supplementary
Material). We find that the simulated qubit frequency follows roughly the
quadratic decay of the parent gap for small fields, as observed in the
experiment. Figure 2(b) inset shows the simulated dependence of $B_{C}$ on the
NW gate voltage. It should be noted that the lever arm in the simulation is
much larger than in experiment, hence the significant difference in absolute
gate voltage values. There is still a trend towards more rapidly decaying
$f_{Q}$ at more positive gate voltage in the simulation, but the numerical
results are less monotonic than what we observe in the experiment. At more
negative gate voltages, the wave function is pressed up away from the gates
and close to the superconducting shell, reducing the wave function cross-
section threaded by magnetic flux, leading to a more gradual decay of $f_{Q}$
in field. Within this interpretation, the effect is due to gating of the NW,
not the junction.

With increasing magnetic field, nonmonotonic evolution of $f_{Q}$ was observed
both in experiment, as shown in Fig. 3. The corresponding electrostatics of
the wire and junction, as well as the field dependence of ABSs in the
junction, are shown in Figs. 3(b,c). In this configuration, the junction was
fairly open, with $V\textsubscript{C}$ accumulating electrons, while the back
gates $V\textsubscript{LP, RP}$ were rather negative, pushing electrons
towards the superconductor.

The spectrum in Fig. 3(c) shows several ABS in the junction, a few of which
show an oscillatory behavior in field. The magnetic field where they have
their minimum corresponds roughly to half a magnetic flux quantum through the
cross section of the NW.

The energies and phase dependence of ABS is directly linked to the qubit
frequency. To understand the effect of ABS energies on the qubit frequency, we
simulate the supercurrent using the Kwant package Groth  _et al._ (2014) and
the analysis developed in Refs. Ostroukh  _et al._ (2016); Zuo  _et al._
(2017). The simulated critical current $I$c, yields a qubit frequency
$f_{\text{Q}}\approx\sqrt{8E_{J}E_{C}}/h=\sqrt{2E_{C}I_{c}/h\pi e}$ (valid for
$E_{J}\gg E_{C}$; in the experiment $E_{J}/E_{C}\sim 10$) , where
$E_{C}=2\,\mu$eV is estimated from electrostatic simulations of the qubit
island com  and $E_{J}=\hbar I_{c}/2e$.

In the simulated qubit frequency as a function of magnetic field [Fig. 3(d)],
one can see a lobe-like structure where the qubit frequency oscillates as a
function of magnetic field. The oscillation period of the ABS spectrum is
lined up with the qubit frequency. Therefore, we conclude that the oscillatory
behavior of the qubit frequency in field is linked to flux modulation of ABSs
which form in the junction. Based on these simulations, we propose that the
additional revival observed in the experiment is due to junction-based
physics, in contrast to the low field behavior of $f_{Q}$ in Fig. 2, which we
attribute to depletion within the wire.

Figure 3: Two-tone spectroscopy as a function of drive frequency, $f$d, and
parallel magnetic field, $B$, with $V$FET = $+6.0$ V, $V$LP, RP = 0 V, $V$C =
$-1.8$ V. At low field, the qubit frequency, $f_{Q}$, decreases rapidly,
disappearing from the measurement window at $B\sim 130$ mT. At higher field,
$f_{Q}$ recovers, re-entering the measurement window at $B\sim 220$ mT then
decreases, leaving the measurement window at $B\sim 400$ mT. (b) Numerical
electrostatic potential, $U$, near the junction on a vertical cut through the
NW, for a configuration with nonmonotonic $f_{Q}$. The superconductor (Al) is
indicated on the top (cyan), due to the positive band offset of 50 meV there
is an accumulation layer towards it. The left plunger, the cutter and the
right plunger gates are indicated on the bottom (dark blue). Both plungers are
set to the same voltage $V_{\mathrm{LP}}=V_{\mathrm{RP}}=0\,\mathrm{V}$.
$V_{\mathrm{C}}=+0.55\,\mathrm{V}$. (c) Simulated local density of states in
the junction region for this gate configuration as a function of parallel
magnetic field. A few low-energy ABSs showing flux-modulated oscillations can
be found inside the junction. (d) Simulated qubit frequency for the same gate
configuration. The flux-modulation of ABS results in an oscillation of the
qubit frequency. Figure 4: (a) Two-tone spectroscopy as a function of drive
frequency ($f_{\rm d}$) and gate voltage $V_{\rm C}$. The qubit frequency
exhibits mesoscopic fluctuations. The data is fitted with a smoothed spline
(dashed line), and the light blue shading indicates the standard deviation,
covering the area within $+/-\sigma$ of the fit. (b) Simulated trace of qubit
frequency. $f_{Q}$ as a function of gate voltage $V_{\rm C}$, with the
fluctuations analyzed in a similar manner to the experimental data.

Increasing the number of modes in the semiconductor as much as possible by
setting gate voltages $V_{\rm PL}$ and $V_{\rm PR}$ to +8 V results in random-
looking but repeatable mesoscopic fluctuations of $f_{Q}$ with $V_{\text{C}}$.
In this regime, $\frac{E_{J}}{E_{C}}$ is closer to $\sim~{}30$. These
fluctuations are closely related to mesoscopic fluctuations of critical
current in Josephson junctions Beenakker (1991); Takayanagi  _et al._ (1995);
Gunel  _et al._ (2012). In superconductor-normal-superconductor (SNS)
Josephson junctions with a distance $L$ between superconductors which is long
compared to the mean free path, $l$, in the N region and the superconducting
coherence length, $\xi$, mesoscopic fluctuations of critical current
$I_{\text{c}}$ are expected to have a non-universal magnitude, with standard
deviation $\sigma_{I_{\text{c}}}\sim ev_{F}l/L^{2}$, where $v_{F}$ is the
Fermi energy in the N region Altshuler and Spivak (1987). On the other hand,
in short, disordered junctions $l\ll L<\xi$, critical current fluctuations are
are expected to be “universal,” $\sigma_{I_{\text{c}}}\sim
e\Delta_{0}$/$\hbar$, independent of junction parameters Beenakker (1991). Our
junction length is $\sim$ 100 nm, which we can expect to be in the short
junction limit, consistent with previous work van Woerkom  _et al._ (2016);
Spanton  _et al._ (2017).

Fluctuation statistics were extracted from experimental data by sampling over
$V_{\text{C}}$ in Device 2. Figure 4(a) shows two-tone spectroscopy data as a
function of junction gate $V_{\rm C}$ at $B=0$. We perform these measurements
at much more positive gate voltages than in the previous section, such that
the number of modes in the junction is maximized. The qubit frequency
fluctuation is quantified by fitting a smoothed spline to the trace and
extracting a standard deviation, yielding $\sigma_{f_{Q}}\sim 130$ MHz. This
corresponds to critical current fluctuation $\sigma_{I_{c}}\sim(\pi
ehf_{Q}/E_{C})\sigma_{f_{Q}}\sim$ 1.0 nA. This is much smaller than the
theoretically predicted value Beenakker (1991) for a short junction, which
would give of order $\sim$ 50 nA, but is in closer agreement with experiments
measuring critical current oscillations Takayanagi  _et al._ (1995); Gunel
_et al._ (2012) which find fluctuations on the order of 1 nA.

Similar traces are simulated for a range of gate voltages, as shown in Fig.
4(b) for simulated $V_{\rm{LP,RP}}=0$. Changing the plunger gate voltages can
be viewed as a variation in the simulated properties of the junction, which
was not attempted in our experiment. The simulated fluctuation in $f_{Q}$ are
in the range 0.4–1 GHz, corresponding to critical current fluctuations in the
range 3–9 nA. These values are larger than what was observed experimentally,
but smaller than the universal value for a short junction.

In conclusion, we have investigated semiconductor based transmon qubits that
allow both dc transport and cQED operation, controlled by a field effect
transistor, and are magnetic field compatible. With the help of numerics, we
observe that the rate of qubit frequency decay in field is controllable by
gating of the bulk NW, while additional oscillatory behavior at higher fields
may be attributed to flux modulation of ABSs in the junction. In the many-
channel regime, mesoscopic fluctuations of the qubit frequency were
considerably smaller than expected from universal theory, somewhat smaller
than numerics, and comparable to previous corresponding results of
fluctuations in critical current.

## Acknowledgments

We thank Ruben Grigoryan, Bernard van Heck, Marina Hesselberg, Karthik
Jambunathan, Anders Kringhøj, Thorvald W. Larsen, Robert McNeil, Karolis
Parfeniukas, Karl Petersson, Lukas Splitthoff, Agnieszka Telecka, Shivendra
Upadhyay, Sachin Yadav, and David van Zanten for help with device fabrication
and valuable discussion. Research was supported by the Danish National
Research Foundation, Microsoft, a grant (Project 43951) from VILLUM FONDEN,
and the European Research Council under grant HEMs-DAM No.716655.

## References

  * Krogstrup  _et al._ (2015) P. Krogstrup, N. Ziino, W. Chang, S. Albrecht, M. Madsen, E. Johnson, J. Nygård, C. Marcus, and T. Jespersen, Nature Materials 14, 400 (2015).
  * de Lange  _et al._ (2015) G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo, Phys. Rev. Lett. 115, 127002 (2015).
  * Larsen  _et al._ (2015) T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nygård, and C. M. Marcus, Phys. Rev. Lett. 115, 127001 (2015).
  * Casparis  _et al._ (2018) L. Casparis, M. R. Connolly, M. Kjaergaard, N. J. Pearson, A. Kringhøj, T. W. Larsen, F. Kuemmeth, T. Wang, C. Thomas, S. Gronin, G. C. Gardner, M. J. Manfra, C. M. Marcus, and K. D. Petersson, Nature Nanotechnology 13, 915 (2018), 1711.07665 .
  * Kringhøj  _et al._ (2020a) A. Kringhøj, T. W. Larsen, B. van Heck, D. Sabonis, O. Erlandsson, I. Petkovic, D. I. Pikulin, P. Krogstrup, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 124, 056801 (2020a).
  * Kringhøj  _et al._ (2020b) A. Kringhøj, B. van Heck, T. W. Larsen, O. Erlandsson, D. Sabonis, P. Krogstrup, L. Casparis, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 124, 246803 (2020b).
  * Bargerbos  _et al._ (2020) A. Bargerbos, W. Uilhoorn, C.-K. Yang, P. Krogstrup, L. P. Kouwenhoven, G. de Lange, B. van Heck, and A. Kou, Phys. Rev. Lett. 124, 246802 (2020).
  * Hays  _et al._ (2018) M. Hays, G. de Lange, K. Serniak, D. J. van Woerkom, D. Bouman, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, Phys. Rev. Lett. 121, 047001 (2018).
  * Tosi  _et al._ (2019) L. Tosi, C. Metzger, M. F. Goffman, C. Urbina, H. Pothier, S. Park, A. L. Yeyati, J. Nygård, and P. Krogstrup, Phys. Rev. X 9, 011010 (2019).
  * Hays  _et al._ (2021) M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nygård, A. L. Yeyati, A. Geresdi, and M. H. Devoret, Science 373, 430 (2021), https://www.science.org/doi/pdf/10.1126/science.abf0345 .
  * Matute-Cañadas  _et al._ (2022) F. J. Matute-Cañadas, C. Metzger, S. Park, L. Tosi, P. Krogstrup, J. Nygård, M. F. Goffman, C. Urbina, H. Pothier, and A. L. Yeyati, Phys. Rev. Lett. 128, 197702 (2022).
  * Larsen  _et al._ (2020) T. W. Larsen, M. E. Gershenson, L. Casparis, A. Kringhøj, N. J. Pearson, R. P. G. McNeil, F. Kuemmeth, P. Krogstrup, K. D. Petersson, and C. M. Marcus, Physical Review Letters 125, 056801 (2020), 2004.03975 .
  * Sabonis  _et al._ (2020) D. Sabonis, O. Erlandsson, A. Kringhøj, B. van Heck, T. W. Larsen, I. Petkovic, P. Krogstrup, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 125, 156804 (2020).
  * Casparis  _et al._ (2019) L. Casparis, N. J. Pearson, A. Kringhøj, T. W. Larsen, F. Kuemmeth, J. Nygård, P. Krogstrup, K. D. Petersson, and C. M. Marcus, Phys. Rev. B 99, 085434 (2019).
  * Kringhøj  _et al._ (2018) A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nygård, K. D. Petersson, and C. M. Marcus, Physical Review B 97, 060508 (2018), 1703.05643 .
  * Splitthoff  _et al._ (2022) L. J. Splitthoff, A. Bargerbos, L. Grünhaupt, M. Pita-Vidal, J. J. Wesdorp, Y. Liu, A. Kou, C. K. Andersen, and B. van Heck, (2022), arXiv:2202.08729 [cond-mat.supr-con] .
  * Winkler  _et al._ (2019) G. W. Winkler, A. E. Antipov, B. van Heck, A. A. Soluyanov, L. I. Glazman, M. Wimmer, and R. M. Lutchyn, Phys. Rev. B 99, 245408 (2019).
  * Kringhøj  _et al._ (2021a) A. Kringhøj, T. W. Larsen, O. Erlandsson, W. Uilhoorn, J. Kroll, M. Hesselberg, R. McNeil, P. Krogstrup, L. Casparis, C. Marcus, and K. Petersson, Phys. Rev. Applied 15, 054001 (2021a).
  * Zuo  _et al._ (2017) K. Zuo, V. Mourik, D. B. Szombati, B. Nijholt, D. J. van Woerkom, A. Geresdi, J. Chen, V. P. Ostroukh, A. R. Akhmerov, S. R. Plissard, D. Car, E. P. A. M. Bakkers, D. I. Pikulin, L. P. Kouwenhoven, and S. M. Frolov, Phys. Rev. Lett. 119, 187704 (2017).
  * Gül  _et al._ (2014) Ö. Gül, H. Y. Günel, H. Lüth, T. Rieger, T. Wenz, F. Haas, M. Lepsa, G. Panaitov, D. Grützmacher, and T. Schäpers, Nano Letters 14, 6269 (2014).
  * Beenakker (1991) C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991).
  * Takayanagi  _et al._ (1995) H. Takayanagi, J. B. Hansen, and J. Nitta, Phys. Rev. Lett. 74, 166 (1995).
  * Doh  _et al._ (2005) Y.-J. Doh, J. Dam, A. Roest, E. Bakkers, L. Kouwenhoven, and S. Franceschi, Science (New York, N.Y.) 309, 272 (2005).
  * Schäpers  _et al._ (1997) T. Schäpers, A. Kaluza, K. Neurohr, J. Malindretos, G. Crecelius, A. Hart, H. Hardtdegen, and H. Lüth, Applied Physics Letters 71, 3575 (1997).
  * Tinkham (2004) M. Tinkham,  _Introduction to Superconductivity_ (Courier Corporation, 2004).
  * Vaitiekėnas  _et al._ (2020) S. Vaitiekėnas, G. W. Winkler, B. van Heck, T. Karzig, M.-T. Deng, K. Flensberg, L. I. Glazman, C. Nayak, P. Krogstrup, R. M. Lutchyn, and C. M. Marcus, Science 367 (2020).
  * Kringhøj  _et al._ (2021b) A. Kringhøj, G. W. Winkler, T. W. Larsen, D. Sabonis, O. Erlandsson, P. Krogstrup, B. van Heck, K. D. Petersson, and C. M. Marcus, Phys. Rev. Lett. 126, 047701 (2021b).
  * Shen  _et al._ (2021) J. Shen, G. W. Winkler, F. Borsoi, S. Heedt, V. Levajac, J.-Y. Wang, D. van Driel, D. Bouman, S. Gazibegovic, R. L. M. Op Het Veld, D. Car, J. A. Logan, M. Pendharkar, C. J. Palmstrøm, E. P. A. M. Bakkers, L. P. Kouwenhoven, and B. van Heck, Phys. Rev. B 104, 045422 (2021).
  * Groth  _et al._ (2014) C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New J. Phys. 16, 063065 (2014), the code is publicly available at https://kwant-project.org/.
  * Ostroukh  _et al._ (2016) V. P. Ostroukh, B. Baxevanis, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 94, 094514 (2016).
  * (31) COMSOL, Inc. [www.comsol.com].
  * Gunel  _et al._ (2012) Y. Gunel, I. Batov, H. Hardtdegen, K. Sladek, A. Winden, K. Weis, G. Panaitov, D. Gruetzmacher, and T. Schaepers, Journal of Applied Physics 112 (2012), 10.1063/1.4745024.
  * Altshuler and Spivak (1987) B. Altshuler and B. Spivak, Zh. Eksp. Teor. Fiz. 92 (1987).
  * van Woerkom  _et al._ (2016) D. van Woerkom, A. Proutski, B. Heck, D. Bouman, J. Väyrynen, L. Glazman, P. Krogstrup, J. Nygård, L. Kouwenhoven, and A. Geresdi, Nature Physics 13 (2016), 10.1038/nphys4150.
  * Spanton  _et al._ (2017) E. Spanton, M. Deng, S. Vaitiekenas, P. Krogstrup, J. Nygård, C. Marcus, and K. Moler, Nature Physics 13 (2017), 10.1038/nphys4224.