# Urban Epidemic Hazard Index for Chinese Cities: Why Did Small Cities Become
Epidemic Hotspots?

Tianyi Li
System Dynamics Group, Sloan School of Management, MIT

Jiawen Luo
Institute of Geophysics, ETH Zurich

Cunrui Huang
School of Public Health, Sun Yat-sen University

tianyil<EMAIL_ADDRESS>L and J. L contribute equally to this
<EMAIL_ADDRESS>
###### Abstract

Multiple small- to middle-scale cities, mostly located in northern China,
became epidemic hotspots during the second wave of the spread of COVID-19 in
early 2021. Despite qualitative discussions of potential social-economic
causes, it remains unclear how this pattern could be accounted for from a
quantitative approach. Through the development of an urban epidemic hazard
index (EpiRank), we came up with a mathematical explanation for this
phenomenon. The index is constructed from epidemic simulations on a multi-
layer transportation network model on top of local SEIR transmission dynamics,
which characterizes intra- and inter-city compartment population flow with a
detailed mathematical description. Essentially, we argue that these
highlighted cities possess greater epidemic hazards due to the combined effect
of large regional population and small inter-city transportation. The proposed
index, dynamic and applicable to different epidemic settings, could be a
useful indicator for the risk assessment and response planning of urban
epidemic hazards in China; the model framework is modularized and can be
adapted for other nations without much difficulty.

Despite the nation-wide successful implementation of control measures against
COVID-19 (Kraemer et al., 2020; Lai et al., 2020; Tian et al., 2020), multiple
small- to middle-scale cities in China became local epidemic hotspots during
the new wave of the pandemic in early 2021; the list includes Tonghua,
Songyuan, Suihua, Qiqihar, Heihe, and Xingtai etc. (Yew, 2021; Steven, 2021).
Unlike their nearby capital cities (e.g., Shijiazhuang, Changchun, Harbin),
these small cities are probably not known to many Chinese before they are
enlisted as “high-risk regions” after the local bursts, and it is indeed an
unexpected phenomenon that they are highlighted among the over 340 prefectural
administrations in China. It is implied that the likelihood of epidemic
hazards is high in these regions. Many social-economic factors may account for
this fact (Niu et al., 2020; Qiu et al., 2020): for example, social scientists
may observe that these cities are all located in the northeast part of China,
where local economies are often underdeveloped, and local residents are often
more behavioral active than they are supposed to be in face of the epidemic
(e.g., Shelach, 2006); other conjectures may attend to the fact that since
these are neither coastal cities nor metropolitans where imported cases are
more common, local control measures and regulations are different and somewhat
relaxed, which led to heedlessness of early signals (e.g., Ye et al., 2020; Li
et al., 2021), or that these northern regions have cold winters and also less
residential housing space than the south, hence the hazard of severe
infections was harbored (e.g., Cao et al., 2020). Although these arguments are
sound, it is desired that quantitative explanations could be addressed to
decipher why such small cities in China stand out as epidemic hotspots.

This points to the significant necessity of the quantification of urban
regions’ epidemic hazards, desirably via a constructed index that assesses the
extent of potential risk. Such indices of potential epidemic risk can be
useful for effective decision analysis during epidemic response and planning
(Morse et al., 2012; Hamele et al., 2018), critical to the mitigation of
sudden and potentially catastrophic impacts of infectious diseases on society
(Shearer et al., 2020). Indeed, although the exit of COVID-19 is still on the
fly, both methodologically and practically (Thompson et al., 2020), it is
nevertheless prudent to start getting prepared for the next pandemic (Liu et
al., 2018; Neumann and Kawaoka, 2019; Simpson et al., 2020), both economically
and ecologically (Di Marco et al., 2020; Dobson et al., 2020), through
comprehensive pandemic risk management synthesis (Studzinski and Pasteur,
2020) and the upgraded implementation of digital technologies (Budd et al.,
2020)

Many successful attempts have been made to develop such epidemic risk indices
from various angles. According to US CDC, the preparedness for influenza
pandemics can be assessed with the Influenza Risk Assessment Tool (Trock et
al., 2012; Cox et al., 2014; Burke and Trock, 2018) through the Pandemic
Severity Assessment Framework (Reed et al., 2013; Holloway et al., 2014); for
the same purpose, the Tool for Influenza Pandemic Risk Assessment (WHO-TIPRA,
2016) is recommended by WHO; and there are miscellaneous other tools developed
for national-level pandemic planning, through either mathematical simulations
(e.g., Eichner et al., 2007) or scoring systems, in which case various social-
economic factors are being considered (e.g., McKay et al., 2019; Oppenheim et
al., 2019; Grima et al., 2020).

However, it is suggested that currently a suitable public health evaluation
framework for the assessment of epidemic risk and response is still not full-
fledge in scale (Warsame et al., 2020). Most proposed tools and frameworks are
subject to a few shortcomings: (1) assessments are in most cases from the
supply side (i.e., the preparedness) instead of the demand side (i.e., the
actual risk); (2) assessment of pandemic potential is often virus-specific
(i.e., pathological), not sufficiently general-purpose, as the risk potential
is also determined by various societal factors (transportation, population
(Copiello and Grillenzoni, 2020) etc.); (3) many indices rely on expert
scoring systems that often depend largely on subjectivity, and the calling for
mathematical models and algorithms for risk assessment and pandemic planning
is compelling (Wu and Cowling, 2011); (4) finally, many models focus on
nation-wide evaluation, and there is relatively little concentration on sub-
nation (e.g., city) level analysis, except for a few interesting attempts
(e.g., Prieto et al., 2020; Boyce and Katz, 2002; Zhu et al., 2020).

To deal with this problem, in this study we develop a novel epidemic hazard
index for Chinese cities, which quantifies the potential risk of epidemic
spread at the over 340 Chinese prefectural administrations. The index relies
on a simulation model which integrates intra-city compartment dynamics and a
detailed mathematical description of inter-city multi-channel transportation.
Calculation of the hazard index is based on this dynamic system that simulates
the domestic epidemic spread for user-specified diseases. In the model, intra-
city evolution is governed by the SEIR dynamics assuming no epidemic response
taken place, such that the constructed index serves as an early-warning
indicator at early periods of epidemics, before the incidence of any
structural change in the population flow upon policy intervention (e.g.,
Schlosser et al., 2020; Li, 2021); inter-city transportation is modeled with a
multi-layer bipartite network (Li, 2020), which make explicit considerations
of various events during inter-city population flow, including transit, cross-
infection due to path overlap, as well as the different transmissivities on
different transportation media.

Such a highlight on transportation (i.e., spacial patterns) is core to the
city-specific risk assessment of epidemics (Smith et al., 2002) and natural
hazards in general (Greiving et al., 2006). Indeed, over the course of the
still on-going pandemic, it is acknowledged that transportation, at both the
global and the domestic level, plays a critical role during the spread of
viruses and to a large extent may determine the severity of the disease at
different geological divisions (Christidis and Christodoulou, 2020; Liu et
al., 2021). Essentially, compared to regression models (Bo et al., 2014) or
the machine-learning approach (Feng et al., 2021), the highlight of
transportation asks for epidemic risk analysis from a network perspective
(e.g., Chu et al., 2020; Ge et al., 2020; So et al., 2020), upon which the
risk scores could then be computed from quantitative approaches (So et al.,
2020). An important precedent is the Global Epidemic and Mobility (GLEaM)
model, which integrates sociodemographic and population mobility data in a
spatially structured stochastic disease approach to simulate the spread of
epidemics at the worldwide scale (Balcan et al., 2010). GLEaM considers the
commuting on the airport network on top of local disease transmission, where
transportation is modeled via an effective operator; our model adopts a
similar methodology yet constructs a more realistic multi-layer mathematical
description of inter-city transportation, comparing also to various recent
studies in the same line of research (Jia et al., 2020; Wang et al., 2020;
Zhang et al., 2020; Chang et al., 2021).

## Model

The model is developed in Li (2020) and is briefed summarized here. Assume a
bi-partite graph with cities (nodes) classified as either central cities or
peripheral cities. The network is multi-layer $G=(V,E_{A/B/R/S})$, specifying
four means of inter-city transportation (i.e., different layers have different
connectivites between nodes): Air (A), Bus (B), Rail (R) and Sail (S). At each
node, the local urban population is divided into four compartments:
Susceptible (S), Exposed (E), Infected (I), Recovered (R), and the intra-city
epidemic spread follows the standard SEIR dynamics (e.g., Sterman, 2000;
Newman, 2010). We track the in- and out-flow of the exposed, susceptible and
recovered population at each node on the transportation network, which
determines the open-system SEIR dynamics (for a specific city $i$):

$\left\\{\begin{aligned}
\dot{S}_{i}&=-\frac{S_{i}}{P_{i}}(\frac{R_{0}}{D_{I}}I_{i}+z_{i})+\Delta
S_{i}^{in}-\Delta S_{i}^{out}\\\
\dot{E}_{i}&=\frac{S_{i}}{P_{i}}(\frac{R_{0}}{D_{I}}I_{i}+z_{i})-\frac{E_{i}}{D_{E}}+\Delta
E_{i}^{in}-\Delta E_{i}^{out}\\\
\dot{I}_{i}&=\frac{E_{i}}{D_{E}}-\frac{I_{i}}{D_{I}}\\\
\dot{R}_{i}&=\frac{I_{i}}{D_{I}}+\Delta R_{i}^{in}-\Delta
R_{i}^{out}.\end{aligned}\right.$ (1)

Epidemiological parameters $R_{0}$, $D_{E}$, $D_{I}$ are the basic
reproduction number, the incubation period, the infection period;
$z_{i}=z_{i}(t)$ is the zoonotic force, i.e., the seed of the disease, with
$z_{i}\neq 0$ only at node(s) that is the disease epicenter(s). The in- and
out-flows of each city are determined via the flowmaps
$F^{q}=\\{f_{i,j}^{q}\\}$ between pairs of cities $(i,j)$ specified for each
means of transportation $q$. During the inter-city flow, transit events are
considered, where only a certain proportion of flow ($TR_{c/p}$ for
central/peripheral cities, the same for each layer) enter the local
population, and the rest are directed to other destinations. Moreover, cross-
infections during inter-city travels are modeled, which take place between a
susceptible person and an exposed person who share an overlapped travel path
with the same destination. The strength of the cross-infection spillover
effect $R_{T}^{q}$ various on different transportation media (e.g., Hu et al.,
2020).

At each city $i$, the exposed inflow ($\Delta E_{i}^{in}(t)$) is given by

$\Delta
E_{i}^{in}(t)=\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}\overline{f_{j,i}^{q}(t)\mu_{j}(t)}(1-TR_{i}),$
(2)

where

$\overline{f_{j,i}^{q}\mu_{j}}=f_{j,i}^{q}\mu_{j}+\underset{k}{\overset{p^{q}(i,k)\cap
p^{q}(i,j)\neq
0}{\sum}}f_{k,i}^{q}\mu_{k}(R_{T}^{q}-1)\frac{f_{j,i}^{q}(1-\mu_{j}-\eta_{j})\text{min}(d^{q}_{j,i},d^{q}_{k,i})}{\underset{l}{\overset{p^{q}(i,k)\cap
p^{q}(i,l)\neq
0}{\sum}}f_{l,i}^{q}(1-\mu_{l}-\eta_{l})\text{min}(d^{q}_{l,i},d^{q}_{k,i})}$
(3)

is the adjusted exposed flow from city $j$ to $i$ by means $q$, taking care of
cross-infections ($d^{q}_{i,j}$ represents the shortest path distance between
$i$ and $j$ on layer $q$), and

$\mu_{i}(t)=\frac{\Delta
E_{i}^{out}(t)+\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}\overline{f_{j,i}^{q}(t-1)\mu_{j}(t)}TR_{i}^{q}}{\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{i,j}^{q}(t)},$
(4) $\eta_{i}(t)=\frac{\Delta
R_{i}^{out}(t)+\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{j,i}^{q}(t)\eta_{j}(t-1)TR_{i}^{q}}{\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{i,j}^{q}(t)},$
(5)

are the time-stamped proportion of the exposed and recovered population among
the total outflow population from city $i$; thus $(1-\mu_{i}-\eta_{i})$ is the
proportion of the susceptible population among the total outflow from city
$i$. The recovered inflow is tracking all the recovered people upon arrival
(via $\eta_{j}$): $\Delta
R_{i}^{in}(t)=\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{j,i}^{q}(t)(1-TR_{i}^{q})\eta_{j}(t-1)$,
and according to flow balance, the susceptible inflow is $\Delta
S_{i}^{in}(t)=\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{j,i}^{q}(t)(1-TR_{i}^{q})-\Delta
E_{i}^{in}(t)-\Delta R_{i}^{in}(t)$. The outflow population from city $i$’s
population $P_{i}$ is the total outbound flow minus the transferred inbound
flow, contributed by the $S,E,R$ compartments (with the proper assumption that
$I$ stay local, i.e., infected people do not participate in inter-city
travels). Proportionally, the outflows are:

$\Delta
X_{i}^{out}(t)=X_{i}(t)\frac{\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{i,j}^{q}(t)-\underset{q\in{Q}}{\sum}\underset{j\in{V}}{\sum}f_{j,i}^{q}(t)TR_{i}^{q}}{S_{i}(t)+E_{i}(t)+R_{i}(t)}.$
(6)

with $X$ being $S$, $E$ or $R$.

Figure 1: Multi-channel transportation network model with detailed flow
description. (a) Four layers of inter-city transportation (Air, Bus, Sail,
Rail) between Chinese prefectural administrations. (b) Bi-partite
categorization of nodes (central cities vs. peripheral cities). (c) Cross-
infection during travel due to path overlaps. (d) Inbound/outbound flow upon
city arrival.

Overall, the multi-layer network model is summarized in Figure 1 (see more
details in (Li, 2020)).

## EpiRank

With the constructed model framework, essentially a simulator of the spread of
epidemics, we are able to simulate imaginary diseases of arbitrary
epidemiological features originated from arbitrary epicenters. Suppose an
epidemic initiated at node $i$, with a certain set of epidemiological
parameters $R_{0},D_{E},D_{I}$. It is important to quantify the intensity of
this epidemic, i.e., the extent of its spread on the domestic scale. This
points to a centrality measure of the node that characterizes nodes’ ability
in spreading an epidemic. Borrowing from the idea of PageRank (Page et al.,
1999; Langville and Meyer, 2011), we construct this new centrality score in a
way similar to the eigenvalue centrality and term it as the EpiRank.

Start the simulator with the disease seeded at node $i$ upon a given zoonotic
force, specified with $z(t)$ during period $t_{z}^{s}$ to $t_{z}^{e}$.
Consider a constant force $z_{0}$ over such a period (whose length is $\Delta
t_{z}$), and the overall zoonotic force $Z$ is:

$Z=\int_{t_{z}^{s}}^{t_{z}^{e}}z(t)dt\sim\sum_{t_{z}^{s}}^{t_{z}^{e}}z(t)=(t_{z}^{e}-t_{z}^{s})z_{0}=\Delta
t_{z}z_{0}.$ (7)

The simulation proceeds, during which the local disease spreads to the entire
nation via the transportation network. After $\tau$ time steps, we obtain the
number of infected cases at city $j$, denoted as $I_{ij}^{\tau}$, with the
first subscript indicating the epicenter. Similarly we obtain $R_{ij}^{\tau}$,
$E_{ij}^{\tau}$ etc. We define the normalized total infection at city $j$ as
$U_{ij}^{\tau}=(I_{ij}^{\tau}+R_{ij}^{\tau})/Z$. $U_{ij}^{\tau}$ is then used
to compute the epidemic score EpiRank for node $i$:

$h_{i}^{\tau}=\alpha\underset{j\in{G},j\neq
i}{\sum}\frac{f(U_{ij}^{\tau})}{\underset{j\in{G}}{\sum}f(U_{ij}^{\tau})}h_{j}^{\tau}+(1-\alpha)U_{ii}^{\tau}.$
(8)

$f(U_{ij}^{\tau})$ represents a specific function of $U_{ij}^{\tau}$ to
indicate the relative weights of each city $j$ contributing to the score for
city $i$. Here we simply allow $f(U_{ij}^{\tau})=U_{ij}^{\tau}$ but further
considerations could be made, for example, applying a cutoff
$f(U_{ij}^{\tau})=max(U_{ij}^{\tau}-U_{0},0)$.

This score $h_{i}$ thus indicates the spreading intensity of epidemics at any
node $i$, which is contributed by (1) the city’s local severity of the
epidemic, and (2) its ability of spreading the disease to other cities, with
the intensity scores of other cities contributing to its own score at
particular weights. $\alpha\in[0,1]$ is the modulating parameter weighing over
these two effects: a small $\alpha$ concerns more on the city-wide local
spread of the epidemic ($\alpha=0$ corresponds to a complete local index),
while a large $\alpha$ puts more weight on the city’s capability of spreading
the disease out.

Write
$\bm{W}=\\{W_{ij}\\}=\\{U_{ij}^{\tau}/\underset{j\in{G}}{\sum}U_{ij}^{\tau}\\}$,
then $\bm{h}=\\{h_{i}^{\tau}\\}$ could be calculated with:

$\left\\{\begin{aligned} &h_{i}^{\tau}=\alpha\underset{j\in{G},j\neq
i}{\sum}\frac{U_{ij}^{\tau}}{\underset{j\in{G}}{\sum}U_{ij}^{\tau}}h_{j}^{\tau}+(1-\alpha)U_{ii}^{\tau}\\\
&\Longrightarrow(1+\alpha\frac{U_{ii}^{\tau}}{\underset{j\in{G}}{\sum}U_{ij}^{\tau}})h_{i}^{\tau}=\alpha\underset{j\in{G}}{\sum}\frac{U_{ij}^{\tau}}{\underset{j\in{G}}{\sum}U_{ij}^{\tau}}h_{j}^{\tau}+(1-\alpha)U_{ii}^{\tau}\\\
&\Longrightarrow(1+\alpha
W_{ii})h_{i}^{\tau}=\alpha\underset{j\in{G}}{\sum}W_{ij}h_{j}^{\tau}+(1-\alpha)U_{ii}^{\tau}\\\
&\Longrightarrow[\bm{I}-\alpha(\bm{W}-\text{diag}(\bm{W}))]\bm{h}=(1-\alpha)\text{diag}(\bm{U})\mathbbm{1}\\\
&\Longrightarrow\bm{h}=(1-\alpha)[\bm{I}-\alpha(\bm{W}-\text{diag}(\bm{W}))]^{-1}\text{diag}(\bm{U})\mathbbm{1}.\\\
\end{aligned}\right.$ (9)

When $\alpha=0$, $\bm{h}=\text{diag}(\bm{U})\mathbbm{1}$. When $\alpha=1$,
$\bm{h}$ is the eigenvector of [$\bm{W}-\text{diag}(\bm{W})$] of eigenvalue 1,
in which case we may impose a value for $\text{max}(\bm{h})$; wlog, we
consider $\alpha\neq 1$. Note $\bm{h}=\bm{h}(\Delta t_{z},z_{0})$ but not
$\bm{h}=\bm{h}(Z)$, since $\bm{U}=\bm{U}(\Delta t_{z},z_{0})\neq\bm{U}(Z)$,
i.e., the spread patterns are different under different distributions of the
same overall zoonotic force.

## Results

Connectivities of each layer (i.e., transportation routes) are determined from
public datasets and empirical considerations; city information (population,
GDP etc.) is obtained from public datasets; transportation parameters
(flowmap, transfer rate, cross-infection strength) are determined through
fitting the early spread of COVID-19 in China in January-February 2020, where
the multi-parameter inversion is conducted via a smart gradient method (see
(Li, 2020) and Supplemental Materials). Transfer rate at central/peripheral
cities are $TR_{c}=0.4$ and $TR_{p}=0.05$; the cross-infection strength are
$\\{R_{T}^{A},R_{T}^{R},R_{T}^{S},R_{T}^{B}\\}=\\{1.2,1.5,1.5,3\\}$ on the
four transportation media; inter-city flows are different for different types
of city pairs (central-central, central-peripheral, peripheral-peripheral),
and are determined at $f_{cc/cp/pp}^{A}=1000/500/0$,
$\hat{f}_{cc/cp/pp}^{R}=2000/200/500$, $f_{cc/cp/pp}^{S}=100/100/100$,
$f_{cc/cp/pp}^{B}=0/3000/1000$. These transportation parameters well fit the
early spread of COVID-19 in China; they are independent of epidemiological
concerns and are fixed throughout the simulations.

For epidemiological parameters, the zoonotic force $z$ is assumed to be 5
persons/day at Day 1 and zero afterwards, at a single epicenter (the simulator
nevertheless allows for simultaneous bursts at multiple epicenters). The base-
case disease is fixed at $R_{0}=2.5$, $D_{E}=6\ days$ and $D_{I}=3\ days$,
i.e., a mild reproduction of virus and a medium-range infection duration,
close to the clinical parameters of COVID or SARS (e.g., Wu et al., 2020).
Unintervened spread of this seeded disease to all Chinese prefectural
districts is simulated for 30 days, after which the ever-infected population
($I_{i}+R_{i}$) in each city $i$ are recorded and are used to compute the
hazard index $h$.

Figure 2: Urban Epidemic Hazard Index (EpiRank) of Chinese Cities (with
population greater than 1 million). Assuming a simulated disease with
$R_{0}=2.5$, $D_{E}=6\ days$, $D_{I}=3\ days$, seeded by a 1-day zoonotic
force at the strength of $5\ persons/day$; $\alpha=0.25$. Table: Top 15 ranks.
Among 300 cities, the six ground-true cities are successfully indicated at
high rankings (four in top 5$\%$ and all six in top 10$\%$).

Cities having population larger than 1 million (300 out of 347) participate in
the calculation of $\bm{h}$ under equation (9). For a value of $\alpha=0.25$,
the determined rankings of cities’ epidemic hazards are shown in Figure 2 (top
40 in the graph and top 15 in the table). One sees that, quite strikingly, the
six small cities where the new bursts of COVID took place (Tonghua, Songyuan,
Suihua, Qiqihar, Heihe, Xingtai) are successfully highlighted by the computed
hazard index. All six cities rank within or near top 10$\%$ in the list,
including four cities ranking within the top 15. Tests suggest that the result
is robust; the high ranks of the six denoted ground-true cities are largely
invariant to fluctuations in both transportation parameters and
epidemiological parameters.

The hazard rankings are computed at different $\alpha$, and correlation of the
ranks is demonstrated via the Spearman’s correlation coefficient (Figure 3;
comparing top 30 entries of each rank). A stable ranking at small values of
$\alpha$ is identified, along with a second invariance at the larger end
(Figure 3 center). Indeed, the ranking is almost completely different at, for
example, $\alpha=0.1$ vs. $\alpha=0.8$, with the latter having a new set of
cities ranked top in the list which are mostly located in the middle of China.
This is consistent with our theory, as a small/large $\alpha$ points to either
of the two end-members of epidemic hazards: the city-wide local spread, or the
capability of spreading the disease to other locations. Therefore, cities
ranking high at small $\alpha$ are regions with mass population but relatively
small transportation means, in which case local epidemic bursts are severely
harbored (Iacus et al., 2020) but not much spillovered to other cities. On the
opposite, cities ranking high at large $\alpha$ are regions where inter-city
transportation is sufficiently viable with respect to the humble local
population; in this case, when seeded a virus, the city is less likely to
become a closed epidemic cluster than to enormously spread the disease out to
other regions. The epidemic hazard index thus also implies the condition of a
city’s inter-city transportation infrastructures.

Figure 3: Spearman’s correlation of hazard rankings (top 30 in the list) at
different $\alpha$ (each sub-figure), for different sets of epidemiological
parameters: reproduction number $R_{0}=1.5/2.5/4$ and infection duration
$(D_{E},D_{I})=(2,2)/(6,3)/(9,10)$ days. Base case (center sub-figure) is
$R_{0}=2.5$ and $(D_{E},D_{I})=(6,3)$.

We initiated simulations for different sets of epidemiological parameters of
the assumed disease (Figure 3), with the combination of low/medium/high
infectivity ($R_{0}=1.5/2.5/4$) and short/medium/long infection duration
($(D_{E},D_{I})=(2,2)/(6,3)/(9,10)$ days). The invariance at small $\alpha$ is
largely maintained across the experiments, expect for a very severe virus with
high infectivity and short duration ($R_{0}=4.0,(D_{E},D_{I})=(2,2)$). In this
case, 30 days is sufficient for most population in most cities to get
infected, and thus top rankings lean instead on densely populated cities. The
second invariance around high $\alpha$ is also identifiable, although not as
clear as the first one. In some cases there is a third cluster at intermediate
values of $\alpha$, but its significance is not as high as the first two which
have well-grounded interpretations. Overall, one is able to conclude that the
two end-options of the epidemic hazard index, using small or large $\alpha$,
hold meaningful across different scenarios of epidemic onset.

Figure 4: Population/daily outflow as a simplified epidemic hazard indicator
(showing top 40 cities). Ranking compared to EpiRank (results in Figure 2);
differences marked with green, red or black arrows and numbers. The ratio is
effective but less accurate than EpiRank in highlighting the six epicenters
(with star marks). Shear population or city GDP does not provide the same
indication.

## Discussion

From the model and results one deduces that, the high epidemic hazard of these
small-scale cities computed at low $\alpha$, in which case $h_{i}$ draws
heavily on a city’s own infection $U_{ii}$, derives from the combined effect
of two factors: a relatively large local population, and a small inter-city
transportation flow. Intuitively, a serious epidemic cluster at the regional
scale is going to develop, when the region is sufficiently populated, and not
much inter-city outflow is dispersing the infection out of the epicenter. This
inspires the idea that alternatively, we could compute the population/outflow
ratio of each city and use this quantity to indicate cities’ epidemic hazard.
Results show that (Figure 4) similar to EpiRank, this ratio does serve as a
good hazard indicator, under which the six denoted cities are listed with high
ranks; furthermore, by contrast, shear population or city GDP, arguably two
most considered social-economic indicators of urban regions (Niu et al.,
2020), are not valid to reflect the ground-true ranking. Conceptually,
analysis on EpiRank help us pin down these two quantities among various
social-economic factors in establishing a promising mathematical explanation
of the observed phenomenon. For robust tests, we proportionally increased and
decreased values on the flowmaps; results suggest that the effectiveness of
this ratio (and certainly the effectiveness of EpiRank) in highlighting the
six epicenters is largely invariant to changes in absolute flow strength.

Nevertheless, it is seen that the simple population/outflow ratio, although
still effective and easy to compute, is not as accurate and informative as
EpiRank. This time the six ground-true cities are overall lower ranked, with
only 2 out 6 in top 5$\%$. This is because this ratio only considers a city’s
own population and transportation condition, whereas EpiRank takes a full
account of the regional and then the entire national picture, under the
networked dynamics approach. Indeed, it is not empirically inconsistent to
argue that the six high-epidemic-risk cities are all located in the north, not
only because they themselves have large population and small inter-city
outflow, but also because it is exactly that cities in northern China, with
which the six cities exchange most outflow population, all tend to have such
features and therefore the effect of local clusters is further locked in. The
advantage of EpiRank is implied; certainly, the simple ratio is also not able
to reveal cities’ ability of spreading the disease out, as EpiRank can shed
light on with high $\alpha$.

Although a promising quantitative explanation for the researched phenomenon is
developed, it is yet indiscreet to conclude that EpiRank is by any means a
sufficiently accurate index of urban epidemic hazards. The current dynamic
network model draws little besides the two aspects, urban population and
inter-city transportation, and too many real-world factors are left out.
Validation of EpiRank results is also difficult to be conducted in a
systematic and rigorous way, besides using the six epicenters as the ground
truth. Despite a mathematically consistent and empirically effective approach,
the proposed simulation framework and the constructed EpiRank index needs
further analysis and extensive tests in various settings (e.g., to investigate
the situation in the US (e.g., Chang et al., 2021)), before their powers and
shortcomings could be substantially uncovered; this study only serves as a
first attempt.

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# Controllable tunnelling of single flux-quanta mediated by quantum phase-slip
in disordered superconducting loops

J.A. Potter<EMAIL_ADDRESS>
Present address: National Physical Laboratory, Teddington, TW11 0LW, United
Kingdom J.C. Fenton  P.A. Warburton  London Centre for Nanotechnology, UCL,
17-19 Gordon Street, London WC1H 0AH, United Kingdom

###### Abstract

Quantum phase-slip (QPS) is the exact dual to the well-known Josephson effect.
Although there are numerous proposals for applications of QPS devices,
experimental work to develop these remains in the relatively early stages.
Significant barriers to exploiting QPS nanowires for useful technologies still
exist, such as establishing robust nanowire fabrication methods that allow
coupling to low-loss circuits, and demonstrating control over the QPS process
with an experimenter-controlled external bias. Here we report experiments
which show that both of these barriers have been overcome. We present
measurements at 300 mK of NbN coplanar waveguide (CPW) resonators embedded
with nanowires fabricated using a neon focused ion-beam. The internal quality
factor exceeds $2\times 10^{4}$ — significantly higher than previously
reported in comparable experiments. The resonator frequency tunes periodically
with an applied magnetic field, revealing tunnelling of the order parameter
that always occurs at half-integer values of the applied flux. In contrast to
previous studies of single QPS, the order-parameter tunnelling is shown to be
adiabatic, demonstrating improved control over energy dissipation in nanowire
QPS circuits. Our results highlight a promising pathway towards realising low-
loss nanowire-based QPS devices.

##  I Introduction

Quantum circuits based on superconducting materials are currently the leading
candidate for the implementation of a scalable quantum computer, already
beginning to tackle relevant computation and simulation problems [1, 2] and
recently demonstrating ‘quantum advantage’ [3]. The Josephson junction is near
ubiquitous in these circuits, providing the necessary nonlinearity. A quantum
phase-slip nanowire is the flux-charge dual to the Josephson junction [4], and
in theory every Josephson junction-based circuit has a QPS dual. As well as
being proposed as a qubit with favourable properties over traditional
Josephson-based technology [5, 6], the QPS nanowire’s dual property of a
nonlinear quantum capacitance enables potential applications such as novel
qubit-qubit couplers [7], parametric amplification for qubit readout, and a
primary quantum current standard [8]. As yet, the huge potential of QPS
nanowires remains to be fully exploited, and two key reasons for this stand
out. Firstly, consistent and reliable fabrication of materials and nanowires
with the requisite properties remains challenging; secondly, full control over
individual QPS events in a nanowire has not yet been demonstrated.

The QPS phenomenon is most pronounced in quasi-one-dimensional superconducting
nanowires, by which we mean that the cross-sectional dimensions of the
nanowire are comparable to the coherence length $\xi$ [9]. In these nanowires,
quantum (or indeed thermal) fluctuations can lead to complete suppression of
the superconducting order parameter over the cross-section of the wire. This
in turn leads to a sudden change of $2\pi$ in the phase difference between the
two ends of the wire, accompanied by the transfer of a quantised amount of
magnetic flux equal to the magnetic flux quantum $\Phi_{0}=h/2e$ through the
wire. This tunnelling of a flux quantum can be coherent [10, 11, 12] or
incoherent [13], depending on the relative scales of the phase-slip energy
$E_{\mathrm{S}}$ and the inductive energy $E_{\mathrm{L}}$ of the nanowire.
When $E_{\mathrm{S}}/E_{\mathrm{L}}<1$, the magnetic flux quantum number is
well-defined, and incoherent transfer of individual flux-quanta is observable.
This is in direct analogy to small Josephson junctions with large charging
energy, where tunnelling of single Cooper pairs can be observed [14]. It is
the incoherent QPS regime that is the focus of this paper. Incoherent QPS
occur probabilistically, with a frequency characterised by the phase-slip rate
$\Gamma_{\mathrm{S}}\equiv E_{\mathrm{S}}/h$. However if one waits for a time
$\tau$ that is much longer than $1/\Gamma_{\mathrm{S}}$, then flux-quantum
tunnelling will occur with extremely high likelihood. A device displaying
deterministic transfer of quantised amounts of magnetic flux may find useful
applications in tasks such as energy-efficient classical digital logic
processing [15, 16, 17, 18, 19]. Development of such a device is a key
enabling technology for control of superconducting quantum processors at
technologically-useful scales.

Historically, phase slips were studied in externally connected, current-biased
nanowires, where the collective effect of many phase slips manifests as a
resistance below $T_{\mathrm{c}}$ [20, 21, 22, 23, 24, 25]. However, in order
to isolate a single phase-slip, it is necessary to incorporate the nanowire
into a flux-biased superconducting loop. The flux-dependent energy states of a
continuous superconducting loop are described by a set of parabolas (see Fig.
1(a)), where each parabola corresponds to a unique value $N$ of the phase
winding number, or equivalently the number of flux quanta in the loop. A
single phase-slip corresponds to a transition between neighbouring parabolas,
and if no lower-energy state is available at a particular external flux bias,
then phase slips are forbidden. When sweeping the externally applied flux,
tunnelling of a single flux-quantum becomes allowed at
$\Phi_{\mathrm{app}}=(N+1/2)\Phi_{0}$ (known as the degeneracy point,
highlighted in red in Fig. 1(a)) [26, 27]. Tunnelling will occur when the
system passes the degeneracy point if the rate of phase slips is much greater
than the rate at which the flux is swept. However, if the flux is swept slowly
with respect to $\Gamma_{\mathrm{S}}$, tunnelling will cause the system to
enter a metastable state and it will then undergo irreversible relaxation to
the ground state at some later time. A number of recent experiments [28, 29,
30, 31] have demonstrated relaxation via QPS from a higher-energy metastable
state, but controlled single-flux-quantum tunnelling when passing through the
degeneracy point has not previously been demonstrated.

In this paper, we demonstrate for the first time single-flux-quantum
tunnelling occurring at the degeneracy point in a continuous superconducting
loop. The flux quanta tunnel through NbN nanowires embedded in the loop and
this is read out via coupling to a high-quality coplanar waveguide (CPW)
resonator. An important innovation in our fabrication technique is that the
nanowires were fabricated by neon focused-ion-beam (FIB) milling. The FIB
process enables the repeatable fabrication of nanowires with width ${w\approx
25\text{ nm}}$ ensuring a large phase-slip rate, while introducing minimal
losses to the host resonator. FIB milling has previously been shown to be
compatible with low-dissipation superconducting circuits [32, 33]; however,
this is the first report of the use of FIB to fabricate a device in which
quantum phase-slip behaviour has been measured. Our results show flux-periodic
tuning of the resonant frequency $\nu_{0}$ while maintaining a high intrinsic
quality factor $Q_{\mathrm{i}}$ at all values of applied flux. We show that a
single quantum phase-slip always occurs when adjacent winding-number states
become degenerate. This is ensured by a phase-slip rate — we estimate
${\Gamma_{\mathrm{S}}=35\text{ MHz}}$ — which is large in comparison to
$1/\tau_{\mathrm{E}}$, where $\tau_{\mathrm{E}}$ is the experimental
timescale. However, the phase-slip rate is less than the inductive energy
($E_{\mathrm{L}}/h>1\text{ THz}$), and less than the thermal energy
($k_{\mathrm{B}}T/h\approx 6$ GHz), which excludes the possibility of avoided
crossings associated with coherent QPS.

Our demonstration of the ability to control the tunnelling of single flux-
quanta represents important progress towards applications of QPS devices in
quantum information processing that have been proposed elsewhere. The low loss
in our device suggests the potential for high tunnelling rates in QPS devices
without a significant increase in $T_{1}$-type decoherence. In addition to
this, there is potential for a QPS digital logic processing device, based on
the deterministic transfer of single flux-quanta [15, 16, 17, 18]. Utilising
quantum tunnelling of flux in such a device should enable significant
reduction in the heat dissipation associated with each gate [34], a reduction
that will be necessary for the scaling up of systems beyond the 1,000-qubit
level.

##  II Fabrication and Experimental Details

Figure 1: (a) Flux-dependent energy spectrum of a continuous superconducting
loop, with blue dashed line highlighting the ground state. A single flux-
quantum may tunnel into the loop at the degeneracy point — highlighted in red.
$N$ is the winding number and $\Phi_{0}$ the flux quantum. (b) Optical (main
image; blue contrast) and SEM (inset; grey contrast) images of the NbN
nanowire-embedded loop located at the short-circuited termination of the CPW
resonator. The nanowires in this device are 25-nm wide and 200-nm long. The
lower and upper leads to the loop are connected to the CPW centre conductor
and superconducting ground plane respectively.

We fabricated nanowire-embedded resonators from 10-nm-thick films of
superconducting NbN. The NbN was deposited on $c$-axis oriented sapphire
substrates by dc magnetron sputtering of a 99.99$\%$ pure niobium target in a
1:1 Ar:N2 atmosphere at a pressure of $5\times 10^{-3}$ mbar and a power of
150 W. The resulting film was measured to have critical temperature
$T_{\mathrm{c}}=8.55$ K and sheet resistance $R_{\mathrm{sq}}=1.2$
k$\Omega$/sq.

Figure 2: (a) Upper panel: Single-tone spectroscopy of nanowire-embedded
resonator measured at $T=305$ mK and $\langle n\rangle\approx 50$. $|S_{21}|$
is plotted as a function of frequency and applied magnetic field. The top axis
shows the applied magnetic flux $\Phi_{\mathrm{app}}$ seen by the nanowire
loop, which is inferred from the periodicity of the resonator tuning. Dashed
white lines show resonant frequencies corresponding to the calculated energy
states of the loop. Lower panel: Magnetic-field dependence of the measured
intrinsic quality factor. (b) Calculated energy states of the loop and values
extracted from the measured resonant frequency.

Quarter-wavelength resonators were patterned by electron-beam lithography
(EBL) into a 300-nm layer of PMMA resist. Multiple resonators on each chip are
capacitively coupled to a common feedline and are patterned with narrow loops
galvanically coupled at the short-circuited end. At this stage, the loops
contain ‘precursor’ nanowires designed with a width of 400 nm. The pattern was
transferred into the NbN film by reactive ion etching (RIE) using a 2:5 volume
ratio of CHF3 and SF6 at 100 W and 100 mbar.

Nanowires are then patterned into the loops using a neon focused-ion-beam,
whereby a beam of Ne ions is accelerated from an atomically-defined tip onto
the sample surface with spot size down to 2 nm and sufficient energy to
sputter the metal film [35, 36]. The precursor wires were milled to a width of
25 nm (see Fig. 1(c)) using a 15-keV Ne beam. A Ne dose of 0.5 nC/$\mu$m-2 was
sufficient to clear the 10-nm film.

The sample was wire-bonded to a copper printed-circuit-board (PCB) and
enclosed within an ECCOSORB-lined brass sample box. This was cooled to a
temperature of $T\sim 300$ mK using a 3He refrigerator. Measurement of the rf
response of the device was made using a Vector Network Analyser (VNA) via an
input line with ${60\text{ dB}}$ of attenuation to reduce thermal noise from
room temperature. Signals in the output line were amplified with a high-
electron-mobility-transistor (HEMT) amplifier. Global magnetic field was
applied perpendicular to the plane of the loop using a superconducting
solenoid and a precision current source. The lines supplying current to the
coil were filtered at room temperature with an upper cut-off frequency of 9.2
kHz.

##  III Results and Discussion

A. Flux Dependence of Resonant Frequency

In this paper, we present results on a single NbN nanowire-embedded CPW
resonator (see Supplemental Information for comparison of multiple devices).
We measured the forward transmission ($S_{21}$) through the on-chip feedline,
where the $\lambda/4$ resonators appear as a notch-type resonance. The upper
panel of Fig. 2(a) shows the main result of this work: under an applied
magnetic field, the resonance tuning shows discontinuous changes of gradient
at periodic values of the applied field. As we will demonstrate in the
remainder of this paper, these discontinuities occur when two stable states
with winding number differing by one become degenerate and are due to single-
flux-quantum tunnelling mediated by quantum phase-slips in the nanowire [37].

The magnetic-field periodicity is 153 $\mu$T, which corresponds to a single
flux-quantum in our loop assuming a flux-focusing factor $\mathcal{F}=1.7$
[38]. The data in Fig. 2(a) corresponds to a single direction of magnetic
field sweep, but sweeps in the opposite direction were found to give the same
result. We also observe a non-periodic, parabolic decrease of the resonant
frequency as the magnitude of the applied field is increased. This is the
expected [39, 40] kinetic-inductance tuning of the NbN resonator, and can be
parametrised by a phenomenological field-scale ${B_{\star}=8\text{ mT}}$.

Figure 2(a) also shows periodic variation of $Q_{\mathrm{i}}$ of the resonance
as a function of applied field with the same field period as the resonant
frequency. Quality factors were obtained by an analytical fit [41] to the
equation

$|S_{21}(\nu)|=ae^{i\alpha}e^{-i2\pi\nu\tau}\Bigg{[}1-\frac{\Big{(}\frac{Q_{\mathrm{l}}}{|Q_{\mathrm{c}}|}\Big{)}e^{i\phi}}{1+2iQ_{\mathrm{l}}(\nu/\nu_{0}-1)}\Bigg{]},$
(1)

where $\nu$ is the probe frequency and $\nu_{0}$ is the resonance frequency.
$Q_{\mathrm{c}}$ and $Q_{\mathrm{l}}$ are the coupling and loaded quality
factors respectively, and obey the relationship
$1/Q_{\mathrm{l}}=1/Q_{\mathrm{i}}+1/Q_{\mathrm{c}}$. $\phi$ accounts for the
effect of impedance mismatches in the circuit, the scale factor $a$ represents
the change in amplitude due to any attenuation/amplification in the
measurement chain, $\alpha$ describes any initial phase offset of the signal,
and $\tau$ accounts for frequency-dependent cable delay. We find that
$Q_{\mathrm{i}}$ decreases approximately quadratically from $4\times 10^{3}$
at $\delta\Phi\equiv(\Phi_{\mathrm{app}}-N\Phi_{0})/\Phi_{0}=0$ to $3.4\times
10^{3}$ at $\delta\Phi=1/2$. We attribute this small change to non-equilibrium
quasiparticles excited by the induced screening current providing an extra
loss mechanism [42]. We observe no sharp decrease in $Q_{\mathrm{i}}$ at
$\delta\Phi=1/2$, which suggests that the heat dissipated by the quantum
phase-slip itself is not large enough to cause extra losses in the resonator.
We note that the intrinsic quality factor exceeds any currently reported in
the literature for QPS devices, and discuss this further in the Supplemental
Information.

The periodic tuning of the resonance is well fitted by a model of an inductive
superconducting loop remaining in its ground state (see Fig. 2(b)), where the
system is allowed to move between adjacent parabolas by transferring a single
flux-quantum through the nanowire at ${\Phi_{\mathrm{app}}=(N+1/2)\Phi_{0}}$.

The loop is made up of a wide section and a narrow section (as shown in Fig.
1(b)), and so can be modelled as two nonlinear kinetic inductances in series.
The flux-dependent kinetic inductance of the loop $L_{\mathrm{k}}(\Phi)$ is
therefore

$L_{\mathrm{k}}(I)=L_{\mathrm{k,1}}(0)\Bigg{[}1+\Bigg{(}\frac{I_{\mathrm{s}}}{I_{\star,1}}\Bigg{)}^{2}\Bigg{]}+L_{\mathrm{k,2}}(0)\Bigg{[}1+\Bigg{(}\frac{I_{\mathrm{s}}}{I_{\star,2}}\Bigg{)}^{2}\Bigg{]},$
(2)

where $I_{\star,1}$ and $I_{\star,2}$ are known to be of the order of the
critical current in the wide and narrow section respectively [43]. Since the
screening current is $I_{\mathrm{s}}=\Phi/L_{\mathrm{k}}$, we can insert this
into Eq. 2 and solve for $L_{\mathrm{k}}(\Phi)$.

The input impedance $Z_{\mathrm{in}}$ of a $\lambda/4$ CPW resonator
terminated by an inductive load, as a function of frequency and load
impedance, is

$Z_{\mathrm{in}}(\nu,Z_{\mathrm{L}})=Z_{0}\frac{Z_{\mathrm{L}}+iZ_{0}\tan{\big{(}\frac{2\pi\nu
l}{c}\big{)}}}{Z_{0}+iZ_{L}\tan{\big{(}\frac{2\pi\nu l}{c}\big{)}}},$ (3)

where $Z_{0}$ is the characteristic impedance of the resonator,
$Z_{L}(\Phi)=i2\pi\nu L_{\mathrm{k}}(\Phi)$ is the impedance of the inductive
load, $c$ is the speed of light in the resonator, and $l$ is its length. At
resonance, $\mathrm{Im}\\{Z_{\mathrm{in}}\\}\rightarrow\infty$, so given
$L_{\mathrm{k}}(\Phi)$ one can numerically calculate $\nu_{0}(\Phi)$, or given
$\nu_{0}(\Phi)$ one can numerically calculate $L_{\mathrm{k}}(\Phi)$.

One can also calculate the flux-dependent free energy of the loop $E(\Phi)$
from $L_{\mathrm{k}}(\Phi)$, using the relation

$L_{\mathrm{k}}^{-1}=\frac{d^{2}E}{d\Phi^{2}}.$ (4)

To obtain the free energy, we simply numerically integrate the inverse of the
inductance twice with respect to flux.

We calculated $L_{\mathrm{k}}(\Phi)$ for our device from Eq. 2 using only
independently determined parameters. A critical current density of
$J_{\mathrm{c}}=4.4\times 10^{5}$ Acm-2 was obtained from a dc measurement of
a track etched into the NbN film, and a sheet kinetic inductance of
${L_{\mathrm{sq}}=0.34\text{ nH/sq}}$ was inferred from the zero-field
$\nu_{0}$ of the resonator. The geometry of the loop was measured by SEM and
this was used to calculate $L_{\mathrm{k}}(0)$ and $I_{\star}$ (we set
$I_{\star}=I_{\mathrm{c}}$). We then calculated the white dashed lines in Fig.
2(a) using Eq. 3, and the solid black lines in Fig. 2(b) from Eq. 4. The blue
points in Fig. 2(b) were extracted from the measured $\nu_{0}$ using Eqs. 3
and 4.

Figure 3: $|S_{21}|$ measured at $\Phi_{\mathrm{app}}=0$ and
$\Phi_{\mathrm{app}}=\Phi_{0}/2$ along with fit (black line) to a linear
resonance model. This shows that at $\Phi_{\mathrm{app}}=\Phi_{0}/2$ the
response is linear, so the current in the nanowire is well below the critical
current.

B. Mechanisms for Flux Quantum Transfer

The periodic tuning of the resonator and the associated fit to the calculated
energy states of the loop constitute strong evidence that the loop remains in
its ground state for all values of $\Phi_{\mathrm{app}}$, and this is made
possible by a single flux-quantum transferring into or out of the loop at
$\delta\Phi=1/2$. It is important to establish the mechanism by which the flux
is able to enter the loop, so we now turn our attention to the transitions
between flux states. Across multiple devices on multiple chips, we found an
onset of flux-periodic tuning in devices containing nanowires with $w\lesssim
35$ nm. This dependence of the behaviour on nanowire width suggests that the
flux tunnelling occurs in the nanowires, and not in the wider part of the
loop. We can now examine some possible physical processes that could occur in
the nanowires and see how the data fits with them.

-Does the nanowire current exceed its critical current $I_{\mathrm{c}}^{\mathrm{nw}}$? Niobium-nitride resonators commonly exhibit a nonlinear $S_{21}$ response when they are driven with a high microwave power [44] as a result of the current-induced nonlinear kinetic inductance. As we see in Eq. 2, the kinetic inductance is quadratically dependent on $(I/I_{\mathrm{c}})^{2}$, and so the nonlinearity must be dominated by the part of the conductor with the lowest $I_{\mathrm{c}}$. This is confirmed by measurements of our circuits, where we find that nanowire-embedded resonators show a much higher degree of nonlinearity in their $S_{21}$ response than bare resonators (see Supplemental Information). The quadratic nature of the nonlinearity suggests that a strongly nonlinear response is a consequence of the magnitude of the rf current $I_{\mathrm{res}}$ in the resonator reaching a significant fraction of the nanowire critical current $I_{\mathrm{c}}^{\mathrm{nw}}$. Our NbN resonator readout method therefore gives us an indirect readout of whether the current in the nanowire is close to $I_{\mathrm{c}}^{\mathrm{nw}}$.

Figure 3 shows the $S_{21}$ response of the nanowire-embedded resonator at
$\delta\Phi=0$ and $\delta\Phi=1/2$, both measured in the low-power limit with
an estimated resonator photon population of $\langle n\rangle\approx 50$. In
both cases, the response is linear and well fitted by Eq. 1. We calculate,
using the relation $I_{\mathrm{s}}=dE(\Phi)/d\Phi$ that the maximum induced
screening current in the nanowires $I_{\mathrm{s}}(\delta\Phi=1/2)$ is 120 nA,
an order of magnitude less than $I_{\mathrm{c}}^{\mathrm{nw}}$. Crucially, the
lack of nonlinearity of the resonance at $\delta\Phi=1/2$, along with the fact
that $Q_{\mathrm{i}}$ remains a significant fraction of its zero-field value,
means that the nanowires are not being driven close to their critical current
by the applied flux. By a similar argument, we know that the nanowires are not
being driven through $T_{\mathrm{c}}$ by a local heating process, as this
would also result in nonlinearity of the resonance at $\delta\Phi=1/2$ due to
dissipation in the highly resistive normal metal.

-Is the nanowire a constriction Josephson-junction? ‘Dayem-bridge’ Josephson-junction SQUIDs are commonly embedded within CPW resonators [45, 46] to provide a flux-tunable resonant frequency. However, when the SQUID loop has a large inductance, one observes hysteretic tuning, characterised by the parameter ${\beta_{L}=2LI_{\mathrm{c}}/\Phi_{0}}$. When $\beta_{L}\gtrsim 1$, the SQUID behaviour becomes hysteretic with applied flux and the resonator will exhibit discontinuous jumps in the resonant frequency, as observed in [47, 48]. Our device does not undergo any discontinuous jumps, and the tuning over a single flux-quantum is symmetric, so $\beta_{L}<1$. Given this and the known loop inductance, we can set an upper bound on the critical current of any Josephson junction of ${I_{\mathrm{c}}^{\mathrm{JJ}}<100\text{ nA}}$. This bound is $10\times$ smaller than the expected transport critical current of our nanowires and also less than $I_{\mathrm{s}}(\delta\Phi_{0})$. Therefore, it is unrealistic to conclude that the flux-tuning we observe is a consequence of a dc SQUID formed of Dayem-bridge Josephson junctions. We also note that the closeness of the fit shown in Fig. 2 suggests there is no contribution to the flux-dependent inductance from a Josephson junction, which would add a $1/\cos{\Phi}$ term to Eq. 2.

Figure 4: Normalised tunnelling flux $\Delta\phi_{N}^{+}$ (defined in the main
text), as a function of the winding number $N$. The solid line corresponds to
half a flux quantum, $\Delta\phi_{N}^{+}=0.5$. Inset: Magnetic-field
dependence of resonant frequency of the nanowire-embedded resonator up to an
applied field of 2.4 mT. The black dotted lines mark transitions between
winding number states ($\phi_{N}^{+}$), and the red dotted lines mark the
energy minimum of each winding number state ($\phi_{{\mathrm{min}},N}$).

-Is this a thermal or quantum process? Figure 4 shows that tunnelling always occurs at degeneracy. Following Petković  _et al._ [30], $\Delta\phi_{N}^{+}$ is defined as $\Delta\phi_{N}^{+}=\phi_{N}^{+}-\phi_{{\mathrm{min}},N}$, where $\phi_{N}^{+}$ is the normalised flux $\phi=\Phi/\Phi_{0}$ at which tunnelling from state $N$ to state $N+1$ occurs, and $\phi_{{\mathrm{min}},N}$ is the flux that minimises the loop free-energy for a particular winding number $N$. Our data shows the periodicity, defined in this way, to be half a flux quantum for all values of the winding number except $N=1$ (we attribute the latter exception to enhanced flux-focusing at low magnetic field). This is in contrast to [30], where a characteristic dependence of $\Delta\phi_{N}^{+}$ on $N$ is shown to be a defining feature of thermally-activated phase-slips. By following the method of [30], we calculate that $\Delta\phi_{N}^{+}\approx 300$ would be required in order for the energy barrier to phase slips in our nanowire to be tuned to $\lesssim k_{\mathrm{B}}T$. Correspondingly, we can estimate (see Supplemental Information) that ${\Gamma_{\mathrm{QPS}}=35\text{ MHz}}$ for our nanowire, and we calculate a temperature-dependent $\Gamma_{\mathrm{TAPS}}$ that is below ${1\text{ Hz}}$ for ${T<1.5\text{ K}}$. Therefore, at our measurement temperature, quantum tunnelling of flux is overwhelmingly more likely than a thermal transition. For comparison, the inverse experimental timescale is ${1/\tau_{\mathrm{E}}\approx 0.2\text{ Hz}}$ since, for each setpoint of the magnetic field, it takes the VNA approximately $5\text{ s}$ to collect $S_{21}$ data across the resonance. Because ${k_{B}T/h>\Gamma_{\mathrm{QPS}}\gg 1/\tau_{\mathrm{E}}}$ — and to our knowledge this is the first reported study in this regime — when our device bias is swept through the degeneracy point, a single quantum phase-slip always occurs before we are able to observe the system in a higher-energy metastable state.

##  IV Conclusions

We have used a Ne FIB to fabricate NbN nanowires with widths down to 25 nm
embedded within CPW resonators. We observe periodic modulation of resonant
frequency and intrinsic quality factor, which is consistent with quantum
tunnelling of individual flux quanta mediated by quantum phase-slip, occurring
when states of different winding number become degenerate. This behaviour has
been observed in resonators with intrinsic quality factor, $Q_{\mathrm{i}}$,
up to $2.7\times 10^{4}$ at 300 mK, which to our knowledge is the highest
quality factor measured in quantum phase-slip experiments — note that the
losses here are significantly lower than suggested by comparable reports [10].
We estimate that the QPS rate is of the order 10–100 MHz, which means that the
tunnelling of a single flux-quantum is effectively deterministic on the
timescale of microseconds. This result shows the suitability of the Ne FIB
process for fabricating QPS devices. We also suggest that an incoherent QPS
device with a high QPS rate such as ours could be promising for classical
digital logic processing applications, where the quantum nature of the flux
tunnelling implies a reduction in heat dissipation compared with current
state-of-the-art devices, opening up a route to resolving an important
roadblock to the upscaling of qubit control electronics.

###### Acknowledgements.

The authors thank O. W. Kennedy for useful discussions. The authors gratefully
acknowledge funding from the United Kingdom Engineering and Physical Sciences
Research Council, Grant Nos. EP/L015242/1, EP/J017329/1, and EP/T001062/1.

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