equ[][]

# Fault-Tolerant Computing with Single Qudit Encoding

M. Mezzadri† Università di Parma, Dipartimento di Scienze Matematiche, Fisiche
e Informatiche, I-43124, Parma, Italy  Gruppo Collegato di Parma, INFN-Sezione
Milano-Bicocca, I-43124 Parma, Italy  A. Chiesa† Università di Parma,
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124, Parma,
Italy  Gruppo Collegato di Parma, INFN-Sezione Milano-Bicocca, I-43124 Parma,
Italy  UdR Parma, INSTM, I-43124 Parma, Italy  L. Lepori  Università di Parma,
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124, Parma,
Italy  Gruppo Collegato di Parma, INFN-Sezione Milano-Bicocca, I-43124 Parma,
Italy  S. Carretta<EMAIL_ADDRESS>Università di Parma,
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, I-43124, Parma,
Italy  Gruppo Collegato di Parma, INFN-Sezione Milano-Bicocca, I-43124 Parma,
Italy  UdR Parma, INSTM, I-43124 Parma, Italy

###### Abstract

We discuss stabilizer quantum-error correction codes implemented in a single
multi-level qudit to avoid resource escalation typical of multi-qubit codes.
These codes can be customized to the specific physical errors on the qudit,
effectively suppressing them. We demonstrate a Fault-Tolerant implementation
on molecular spin qudits, showcasing nearly exponential error suppression with
only linear qudit size growth. Notably, this outperforms qubit codes using
thousands of units. We also outline the required properties for a generic
physical system to Fault-Tolerantly implement these embedded codes.

A “useful” quantum computer should be universal, accurate and scalable [1].
Meeting these conditions on a noisy hardware requires a Fault-Tolerant quantum
error correction (QEC) approach, which allows for universal quantum
computation with an arbitrary error reduction, while remaining experimentally
achievable.
The standard way to tackle this problem relies on encoding the elementary unit
of computation (a logical qubit, LQ) into a collection of distinct physical
qubits [2, 3, 4, 5, 6]. Different codes were developed along these lines,
progressively increasing the maximum tolerated error on each elementary
operation (the so-called threshold) [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
However, all of them show an explosion in the number of physical qubits and
gates to achieve the error suppression required for a reliable computation
[12, 17, 18], which represents an important road-block to an actual
implementation.

We pursue an alternative approach, in which the elementary protected unit of
information is encoded in a single $d-$levels qudit. This idea of embedded
codes [19, 20, 21, 22, 23, 24] would dramatically reduce the resource overhead
in their physical implementation and hence has recently inspired proposals
based on systems with an intrinsic multi-level structure, such as photons [21,
23, 25] or molecular spins [22, 26, 27, 28, 29]. In some cases the break-even
point has been reached [30] and gates [31, 32, 33] or error detection [34]
without error propagation [35, 23] were proposed. Yet, a comprehensive Fault-
Tolerant implementation accommodating realistic noise models, which encompass
noisy ancillae, encodings, measurements and especially noise during gates, is
still missing [36, 23].

Here we develop the first fault-tolerant (FT) implementation of an embedded
stabilizer code, in which all quantum computing (QC) steps (i.e. a universal
set of logical operations, stabilization and correction) are transparent to
the errors handled by the code, i.e. errors do not propagate during any of
these operations.

We demonstrate FTQC on a physical hardware consisting of Molecular Spin Qudits
(MSQs), based on a physically motivated error model where pure dephasing is
largely dominant [22, 26, 27]. We first introduce a illustrative MSQ hardware
and its coupling with the environment [24]. Thanks to a clear hierarchy in the
error operators, a code subspace protected from the most relevant ones can be
derived. Then, we prove that all the QC procedures (one- and two-qubit logical
gates, stabilization and recovery) can be made Error Transparent (ET) by
proper pulse sequences.

We provide a threshold analysis based on numerical simulations, showing that
the corrected qubit outperforms the uncorrected one for reasonable duration of
the elementary gates and of the error rate. Moreover, we find an almost
exponential error reduction which comes only with a linear growth in the
number of qudit levels.

The basic ingredients which grant FTQC are (i) a well defined hierarchy in the
errors affecting the physical hardware and (ii) the high connectivity between
its eigenstates. This latter feature can be rather easily obtained in MSs,
thanks to their unparalleled tunability at the synthetic level [37, 27].
Nevertheless, the code can be extended to other qudit-based hardwares meeting
the requirements we have identified for FTQC.

This approach extremely simplifies the actual control of the hardware and
could make the qudit-embedded approach a viable path towards the physical
implementation of a “useful” quantum computing machine.

General protocol – Errors corrupting quantum information initially encoded in
the pure state $\rho_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$ are represented by
Kraus operators $E_{k}$, i.e. the density matrix after possible errors is
$\rho=\sum_{k}E_{k}\rho_{0}E_{k}^{\dagger}$. Protection against a set of
errors $\\{E_{k}\\}$ can be achieved by identifying code words $\ket{0_{L}}$
and $\ket{1_{L}}$ which satisfy Knill-Laflamme conditions (KLc) [38]:

$\displaystyle\bra{0_{L}}{E}_{k}^{\dagger}E_{j}\ket{0_{L}}$ $\displaystyle=$
$\displaystyle\bra{1_{L}}{E}_{k}^{\dagger}E_{j}\ket{1_{L}}$ (1a)
$\displaystyle\bra{0_{L}}{E}_{k}^{\dagger}E_{j}\ket{1_{L}}$ $\displaystyle=$
$\displaystyle 0.$ (1b)

By using $d$ dimensional qudits, at most $d/2$ different errors can be
corrected. We then consider the vector set
$\\{E_{k}\ket{0_{L}},E_{k}\ket{1_{L}}\\}_{k=0}^{d/2-1}$, and the orthonormal
basis set $\mathcal{A}$ obtained from its Gram-Schmidt orthogonalization. We
denote the set $\mathcal{A}$ as $\\{\ket{\ell,k}\\}$ with $\ell=0,1$ and
$k=0,\ldots,d/2-1$.

Formally, given a gate $G$ on a single qubit, the corresponding logical gate
$G_{L}$ that extends $G$ to the protected logical qubit is given by
$G_{L}\doteq G\otimes\mathrm{I}_{d/2}$. This means applying the gate $G$ to
each subspace $\ket{\ell,k}$ independently of $k$ and highlights the basic
idea to achieve FTQC, i.e. to perform logical operations _independently_ of
the error [31]. Two-qubit controlled logical gates $CG$ can be realized in a
similar way, by implementing $CG$ independently of the error $k$ on both the
control and target LQs (see SM).

Error detection translates into the stabilization of information through the
measurement of

$S=\sum_{k,\ell}\lambda_{k}\ket{\ell,k}\bra{\ell,k}$ (2)

where $\lambda_{k}\neq\lambda_{k^{\prime}}$ for $k\neq k^{\prime}$. This is an
extension of stabilizers codes on qubits, with the important simplification
that we have only one multi-valued stabilizer that gives directly the error
syndrome instead of multiple two-valued stabilizers [39, 40]. This reduces the
impact of measurement errors (see below). Even more importantly, this
completely removes the need for a syndrome decoder, which is usually an
important bottleneck for large qubit stabilizer codes [41, 42].

Stabilization is achieved by exploiting a $d/2-$levels ancilla linked to the
$d-$levels LQ and implementing a $k$-controlled operation $CU$ between the LQ
(control, initially in a generic state $\ket{\bar{\psi}}$) and the ancilla
(target, initialized in its ground state $\ket{0}$). The $CU$ gate acts as
follows: $\ket{\ell,k}\ket{0}\mapsto\ket{\ell,k}\ket{k}$,
$\ket{\ell,k}\ket{k}\mapsto-\ket{\ell,k}\ket{0}$ and the identity otherwise
111The relative phase between $\ket{\ell,k}\ket{0}$ and $\ket{\ell,k}\ket{k}$
is necessary to ensure that the corresponding generator has zero diagonal.
Since the ancilla is initialized in $\ket{0}$, this relative phase is
irrelevant.. Therefore, $CU$ maps each error $k$ to a specific eigenstate of
the ancilla. Hence, a final measurement of the ancilla in its eigenbasis
identifies the error syndrome $k$ and stabilizes information in the subspace
$\ket{\ell,k}$. Note that $CU$ is ET since it does not affect the state of the
LQ but only the state of the ancilla.

The recovery (correction) step $R^{L}_{k}$ simply consists in mapping back the
stabilized states $\ket{\ell,k}$ into $\ket{\ell,0}$ by a unitary
transformation depending on $k$. Since $R^{L}_{k}$ can be written as
$\mathrm{I}_{2}\otimes\mathrm{R_{k}}$, no evolution between subspaces with
different $\ell$ is performed.

Finally, readout of the LQ state corresponds to a measurement of $Z_{L}$,
which again translates into distinguishing the value of $\ell$ independently
of the error $k$. To simplify this step one can choose $\ell=0$ and $\ell=1$
logical subspaces disjoint in the eigenbasis. In this way, the readout simply
corresponds to summing the probabilities of finding the LQ into the
eigenstates belonging to each of the two subspaces.

At last, LQ encoding can be achieved analogously to standard qubit stabilizer
codes [9, 7]. Starting from the qudit into its ground eigenstate, we drive it
to the proper superposition of eigenstates defining
$\ket{0_{L}}\equiv\ket{0,0}$. Errors occurring during this step will bring the
system to a mixture
$\sum_{\ell\ell^{\prime}kk^{\prime}}\rho_{\ell,k}^{\ell^{\prime},k^{\prime}}\ket{\ell,k}\bra{\ell^{\prime},k^{\prime}}$.
Hence, we first perform a logical measurement of the qudit discarding it if we
get an $\ell=1$ outcome and we finally proceed with a stabilization step
discarding it for any $k\neq 0$.

Figure 1: (a) Diagonal elements of the error operators $E_{k}$, $k=0,1,2,3$
for $d=4$, in the basis of the eigenvectors $\ket{d_{k}}$. Scheme of single-
qubit gates (b), stabilization (c), and recovery (d) for a $d=4$ LQ, using the
code words reported in Table II of the SM. In each panel the histogram
represents the populations of the eigenstates in blue (orange) for
$\ell=0(1)$. The upper triangular matrix on the right is the driving
Hamiltonian in the rotating frame, with filled boxes indicating the set of
$\theta_{i}$ resonant pulses between pairs of eigenstates (arrows in the
histograms) which are simultaneously sent to the system. Different colors of
the arrows and boxes indicate different pulse frequencies, while phases are
indicated on top. For stabilization (c) the eigenstates of a two-level ancilla
($0,1$) are also reported.

Fault-Tolerant implementation – For a Fault-Tolerant implementation of the
general protocol discussed above the physical system used as a qudit must
fulfill two key properties.

First, we need the QEC code to propagate errors at a higher order than the
elementary one. This requires a well defined hierarchical structure in the
Kraus representation of the error channel, since only at most $d/2$ of them
can be handled by the code.

Second, each operation $U$ on the LQ must be realized in an ET [31] fashion,
i.e. in such a way that the evolution between the error spaces (with different
$\ell$) is exactly the same as that between the code spaces [31]. This is
achieved if $U$ is implemented in a single step by properly engineering the
driving Hamiltonian $\tilde{H}=i\log U$. The latter corresponds to the
coupling of the system with an external field, such as an oscillating
electromagnetic field for a spin system. The one-shot implementation of $U$
thus requires a specific connectivity between the qudit eigenstates, i.e. the
possibility to directly induce transitions between them via resonant pulses.
As detailed below, this in practice implies an all-to-all connectivity between
the qudit eigenstates. This requirement can be met by encoding a qudit into a
MSQ, thanks to the high degree of chemical tunability of the Hamiltonian in
this class of systems [37, 27, 44, 45, 46, 47, 48, 49, 50].

In particular, we apply the general formalism described above to a
illustrative 7-spin cluster, introduced in Refs. [24, 51]. The system
Hamiltonian is the following:

$H=\sum_{i>j}J_{i,j}{\bf s}_{i}\cdot{\bf s}_{j}+\sum_{i>j}D_{i,j}({\bf
s}_{i}\crossproduct{\bf s}_{j})_{z}+\mu_{B}B\sum_{i}g_{i}s_{i}^{z},$ (3)

where $s_{i}^{\alpha}$ are spin operators on sites $i=1,...,7$, arranged to
form a pair of corner-sharing tetrahedra as in the existing Ni7 molecule [52].
In this molecule, the central spin is $s_{7}=3/2$, while all the others are
$s_{i}=1/2$. In Hamiltonian (3), the first term is the dominant Heisenberg
exchange interaction between neighboring spin pairs with anti-ferromagnetic
couplings $J_{i,j}$, the second is the (axial) anti-symmetric exchange
coupling and the third is the Zeeman interaction with an external magnetic
field $B$ along $z$. By proper choice of the hierarchy between the different
parameters, this Hamiltonian gives rise to an energy spectrum consisting of
eight low-energy total spin $1/2$ doublets well separated from the first
excited $S=3/2$ (see SM). Moreover, the anti-symmetric exchange couples to
first order all the different low-energy multiplets. Hence, this molecule
provides up to 16 low-energy eigenstates whose mutual transitions can be
individually and directly addressed. This feature is specifically exploited in
the FT implementation of our protocol. It can be obtained also using simpler
MSQs, such as single-spin molecules in presence of transverse magnetic fields
and/or high-rank zero-field splitting anisotropy.

In MSQs (as in trapped ions [53, 54], atoms [55] and several other spin
architectures [56, 57]) pure dephasing (with decay time $T_{2}$) is the
leading error at low temperature, while spin relaxation is orders of magnitude
slower [58, 59, 60]. Therefore in the following we will focus on it. Pure
dephasing in MSQs arises from the coupling between the system spins and the
surrounding nuclear spins [61, 26]. In the secular and Born-Markov
approximations, one can derive a Lindblad equation for the evolution of the
density matrix [24], resulting in a decay only of off-diagonal terms of $\rho$
[$\rho_{ij}(t)=\rho_{ij}(0)e^{-\gamma_{ij}t}$] with rates $\gamma_{ij}=\sum
C_{kk^{\prime}}^{zz}[\bra{d_{i}}s_{k}^{z}\ket{d_{i}}\bra{d_{i}}s_{k^{\prime}}^{z}\ket{d_{i}}+\bra{d_{j}}s_{k}^{z}\ket{d_{j}}\bra{d_{j}}s_{k^{\prime}}^{z}\ket{d_{j}}-2\bra{d_{i}}s_{k}^{z}\ket{d_{i}}\bra{d_{j}}s_{k^{\prime}}^{z}\ket{d_{j}}]$.
These are essentially related to the structure of the eigenstates
$\ket{d_{i}},\ket{d_{j}}$, apart from the coefficients $C_{kk^{\prime}}^{zz}$
which depends on the molecular structure (see SM). From the solution of the
Lindblad equation we extract the corresponding diagonal Kraus operators
$E_{k}$ at a fixed time by state tomography [62]. Since they display a clear
hierarchical structure, we can focus on the $d/2$ most relevant ones and solve
the KLc (1). For instance, the $E_{k}$ operators for $d=4$ are reported in the
top panel of Fig. 1, in order of decreasing norm. $E_{0}$ and $E_{1}$ (red
box) are clearly leading and we can fulfill the KLc for them. The magnitude of
the components of the resulting code words in the eigenbasis is shown by the
bars in Fig. 1. We have tested with different model Hamiltonians and spatial
distributions of nuclear spins that the hierarchical structure of the errors
is rather general. The other $E_{k}$ operators and the corresponding code
words are reported in the SM, for different number qudit eigenstates ($d$ from
4 to 12) used to define the LQ. Note that by changing $d$ we are actually
changing the code and hence a threshold analysis for each case is needed (see
below).

Based on this molecular spin hardware, we now discuss the ET implementation of
logical gates. Let us start from a logical planar rotation
$R_{L}^{P}(\theta,\phi)=\exp\left[-i\left(\cos\phi\;Y_{L}-\sin\phi\;X_{L}\right)\theta/2\right]$,
as sketched in Fig. 1-(a). Its ET realization requires resonant pulses between
each eigenstate in the logical $\ell=0$ subspace [blue bars in Fig. 1-(b)] and
each one in the logical $\ell=1$ subspace (orange bars). Different pulses of
length $\theta_{j}$ are indicated by double arrows in the histogram and their
phase $\varphi_{j}$ is reported in the connectivity matrix $\tilde{H}$ nearby.
As detailed in the SM, one can show (with a generalized rotating frame
formalism [63]) that the set of simultaneous pulses
$\\{\theta_{j},\varphi_{j}\\}$ implements the unitary
$\exp\small[-i\tilde{H}\small]$, which corresponds to
$R_{L}^{P}(\theta,\phi)$. Here we assume all the energy gaps to be
spectroscopically distinguishable. Remarkably, since
$\theta,\phi\in\mathbb{R}$, the resulting set of gates
$R_{L}^{P}(\theta,\phi)$ is a universal set for one-body logical operations.
Along the same lines one can implement in a single step also a two-body ET
$C-\varphi$ logical gate (see below).

We now examine the stabilization process, reported in Fig. 1-(c). In this
case, the eigenstates of both the LQ and of a $d/2$ levels ancilla are shown
in the product basis $\ket{d_{j},k}$. We recall that the ancilla is
initialized in its ground state $\ket{0}$. $CU$ stabilization requires to
excite the ancilla iff the LQ is in $k=1$. This translates into the set of
simultaneous pulses indicated in Fig. 1-(b) between each qudit eigenstate
$\ket{d_{i}}$ with the ancilla in $\ket{0}$ and each qudit eigenstate
$\ket{d_{j}}$ with the ancilla in $\ket{1}$. Following the argument above, one
can derive the connectivity matrix with pulse lengths and phases by taking the
logarithm of the $CU$ gate (see SM). Remarkably, the ancilla does not need to
be encoded. Indeed, it is excited to the eigenstate $\ket{k}$ with the same
probability $p_{k}$ of having an error $E_{k}$. Hence, it suffers an error
$E_{k^{\prime}}$ with the conditional probability $p_{k}\,p_{k^{\prime}}$
which therefore results to be of a higher order than the error on the qudit.
Fig. 1-(d) also shows the recovery step from $k=1$ to $k=0$ (for the specific
case $d=4$), which is obtained by a pair of pulses within the two disjoint
$\ell=0,1$ subspaces 222This is due to the specific form of the error words
$\\{\ket{\ell,k}\\}$, in which differ the two $\ell=0,1$ subspaces are
disjoint for all $k$..
Summarizing, we have presented a MSQ characterized by a clear hierarchical
structure in the error operators and providing the capability to implement in
a single step any QC operation of the embedded QEC protocol. Hence, the
proposed implementation of the code is Fault-Tolerant.

Figure 2: Simulation of the Fault-Tolerant implementation of single qubit
logical gates on a MNM [24]. (a) Error $\mathcal{E}_{e}=1-\mathcal{F}_{e}^{2}$
as a function of $1/T_{2}$, where $\mathcal{F}_{e}$ is the entanglement
fidelity [65], averaged over different planar rotations
$R_{L}^{p}(\theta,\varphi)$ for
$(\theta,\phi)=(\frac{\pi}{4},\pi),(\frac{\pi}{2},\pi),(\frac{\pi}{2},-\frac{\pi}{2}),(\frac{\pi}{2},-\frac{\pi}{4}),(\frac{\pi}{2},-\frac{\pi}{8})$.
(b) Performance of the code as function of the number of levels for different
values of $T_{2}$. The same driving field amplitude is assumed for
manipulating the LQ and the uncorrrected qubit, represented by a spin 1/2 with
$g=2$.

Threshold Analysis – We now show the effectiveness of our scheme for the
correction of pure dephasing errors (with characteristic time $T_{2}$) in
MSQs. This is done by numerically integrating the Lindblad equation for the
system density matrix, subject both to pure dephasing and to the sequence of
pulses required to implement the desired operations (see SM). These include a
universal set of logical gates, stabilization ($CU$), recovery. We also
consider possible measurement errors (see below). Differently from standard
threshold analysis [12], we are not assuming discrete errors occurring with a
given probability but a continuous dephasing realistically describing the
system and acting during all the procedures, with elementary error rate
$1/T_{2}$. 333This allows us to compare the performance of a spin 1/2 qubit
(without EC) and of the LQ even though the duration of the elementary
transitions on them is different (and hence the related error probability).
Indeed, the elementary error probability for a two-level system is
$p=(1-\exp[-\tau/T_{2}])/2$ [62]. Since the duration $\tau$ of the transition
is much smaller for a spin 1/2, it would result in a much smaller $p$ compared
to the LQ, thus making the comparison unfair.).

We start by considering generic $R_{L}^{P}(\theta,\phi)$ rotations, followed
by stabilization and correction and we report [Fig. 2-(a)] the final logical
error $\bar{\mathcal{E}}_{e}$ as a function of the elementary error rate
$1/T_{2}$. Here $\mathcal{E}_{e}=1-\mathcal{F}_{e}^{2}$, where
$\mathcal{F}_{e}$ is the entanglement fidelity [65] of the procedure
444$\mathcal{F}_{e}$ is the average fidelity on the initial states
$\ket{0_{L}},\ket{1_{L}},$
$\frac{1}{\sqrt{2}}(\ket{0_{L}}+\ket{1_{L}}),\frac{1}{\sqrt{2}}(\ket{0_{L}}-\ket{1_{L}}),\frac{1}{\sqrt{2}}(\ket{0_{L}}+i\ket{1_{L}}),\frac{1}{\sqrt{2}}(\ket{0_{L}}-i\ket{1_{L}})$,
and we average on a universal set of $(\theta,\phi)$ values. The logical error
is obtained by full state tomography of the logical state and corresponds to
the failure probability of the QEC protocol, i.e. to errors which cannot be
corrected by the code.

We immediately note that the slope of the curves in Fig. 2-(a) in log-log
scale for the LQ (number of levels $\geq 4$) is larger than for the
uncorrected case (blue line). This means that $\bar{\mathcal{E}}_{e}$ is
propagated through the executed circuit to a higher order than the elementary
error, thus remarking the Fault Tolerance of all the procedures involved.
Notably, for reasonable gate duration ($90$ ns for the LQ, see [68, 69, 70]
and SM) the LQ with $d=4$ beats the uncorrected two-level system for
$T_{2}\gtrsim 2\;\mu$s, a perfectly achievable values in MSQs [61, 68, 60].
This corresponds to an error probability for the elementary transition between
a pair of qudit levels of $p\gtrsim\tau/2T_{2}\approx 2.2\%$. By increasing
$d$ (i.e. including more qudit eigenstates in the encoding) the codes beat the
spin 1/2 independently of $T_{2}$. Indeed, errors are reduced and propagated
to higher orders, as evidenced by the increase of the slope of the curves in
Fig. 2-(a).

Remarkably, we find an almost exponential suppression of the error with a
linear increase of $d$, as reported in Fig. 2-(b), independently of $T_{2}$.
This makes the proposed FTQC scheme much more efficient than standard qubit
concatenation techniques, where an exponential error suppression is obtained
only with an exponential growth of the Hilbert space [62]. To quantify this
gain, we note that the final error suppression is strikingly large when
compared to standard multi-qubit codes: here we achieve a logical error of
$10^{-12}$ by employing $12\times 6$ levels (LQ + ancilla) and
$T_{2}=10\;\mu$s, which should be compared to more than $5000$ qubits needed
by Floquet codes with the same elementary error $p\approx\tau/2T_{2}=0.45\%$
[18]. This remarkable performance, obtained exploiting qudits with all-to-all
connectivity, is here shown by focusing on the dominant family of errors of
molecular qudits.

Figure 3: Numerical simulation of the FT implementation of the two-qubit
logical $C$-$\varphi$. The logical error is shown as a function of $1/T_{2}$
at the end of a “logical gate - EC” cycle. Simulations are computationally
demanding and hence we limit to $d=4,6$. Inset: simulated circuit, with
external lines corresponding to LQs in a generic logical state
$\ket{\bar{\psi}}$, coupled by an error-corrected switch initialized in
$\ket{0_{L}}$ (middle line). Each unit is coupled to a $d/2-$levels ancilla
for stabilization.

Leakage errors could arise due to the finite duration of the pulses, but can
be largely reduced by pulse-shaping techniques [71] and/or by using longer
pulses. Indeed, increasing the length of the pulses simply shifts the curves
in Fig. 2 to the left without changing their slope. This will slightly
increase the threshold value of $T_{2}$, without compromising Fault-Tolerance.

Moreover, the depth of our circuit does not increase with $d$, since all
operations are implemented in parallel, again in contrast with standard codes.
Remarkably, here a single measurement is sufficient for syndrome extraction
and the corresponding error [72] can be suppressed by repeated measures as
shown by simulations in the SM.

We now consider a two-qubit logical gate, such as the controlled-phase
$C-\varphi$. In the MSQ architecture, this can be realized by considering a
molecule consisting of three interacting LQs, where the middle one acts as a
switch of the effective coupling between the other two [73, 74], $Q_{1}$ and
$Q_{2}$ (inset of Fig. 3). In presence of an interaction between the three
units, excitations of the switch are dependent on the state of both $Q_{1}$
and $Q_{2}$. This allows us to induce a $2\pi$ (semi-)resonant excitation of
the switch (initialized in $\ket{0_{L}}$) only for a specific 2-qubit logical
state, such as $\ket{1_{L}1_{L}}$. As a result, a phase is added only to the
$\ket{1_{L}1_{L}}$ component of the two-qubit logical state [75, 73, 74, 76],
i.e. a $C-\varphi$ gate is implemented (see also SM).
Since this involves a logical rotation $R_{L}^{P}(2\pi,0)$ of the switch, the
latter must also be encoded and corrected. Simulation of the two-qubit gate
followed by EC on all the three units is shown in Fig. 3 for $d=4,6$. The
logical error is suppressed for $T_{2}\gtrsim 25\;\mu$s ($4$-levels encoding)
and for $T_{2}\gtrsim 500$ ns ($6$-levels encoding), yielding
$p\gtrsim\tau/2T_{2}\approx 8\%$. We stress that these numbers are within
experimental capabilities [61, 77, 68].

Conclusions – We have shown a Fault-Tolerant protocol for quantum computation
with qudit embedded stabilizers codes in a molecular spin architecture [76].
This embedded protocol prevents the blowing up in the number of physical
units, while yielding a large error suppression already with a small number of
qudit levels, thus making the proposed route very appealing for an actual
implementation. Moreover, we have pinpointed the crucial requirements for a
physical system to achieve FTQC, thus potentially extending the strategies
proposed here to other qudit platforms showing a well defined hierarchy of
errors and proper connectivity between its eigenstates.

###### Acknowledgements.

†These Authors contributed equally to the work. We warmly thank P. Santini for
useful and stimulating discussions. This work received financial support from
European Union – NextGenerationEU, PNRR MUR project PE0000023-NQSTI, from the
European Union’s Horizon 2020 program under Grant Agreement No. 862893 (FET-
OPEN project FATMOLS), from Fondazione Cariparma and from Novo Nordisk
foundation under grant NNF21OC0070832 in the call “Exploratory
Interdisciplinary Synergy Programme 2021”. Project funded by the European
Union – NextGenerationEU under the National Recovery and Resilience Plan
(NRRP), Mission 4 Component 1 Investment 3.4 and 4.1. Decree by the Italian
Ministry n. 351/2022 CUP D92B22000530005.

## References

  * Chao and Reichardt [2018] R. Chao and B. W. Reichardt, Fault-tolerant quantum computation with few qubits, npj Quantum Inf. 4, 42 (2018).
  * Gottesman [2000] D. Gottesman, An introduction to quantum error correction (2000), arXiv:quant-ph/0004072 [quant-ph] .
  * Devitt  _et al._ [2013] S. J. Devitt, W. J. Munro, and K. Nemoto, Quantum error correction for beginners, Rep. Progr. Phys. 76, 076001 (2013).
  * Nigg  _et al._ [2014] D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Quantum computations on a topologically encoded qubit, Science 345, 302 (2014).
  * Terhal [2015] B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  * Campbell  _et al._ [2017] E. Campbell, B. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
  * Aliferis  _et al._ [2006] P. Aliferis, D. Gottesman, and J. Preskill, Quantum accuracy threshold for concatenated distance-3 codes, Quant. Inf. Comput. 6, 97–165 (2006).
  * Knill [2005] E. Knill, Quantum computing with realistically noisy devices, Nature 434, 39 (2005).
  * Cross  _et al._ [2009] A. W. Cross, D. P. Divincenzo, and B. M. Terhal, A comparative code study for quantum fault tolerance, Quantum Info. Comput. 9, 541–572 (2009).
  * Wang  _et al._ [2011] D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Surface code quantum computing with error rates over 1%, Phys. Rev. A 83, 020302(R) (2011).
  * Fowler  _et al._ [2009] A. G. Fowler, A. M. Stephens, and P. Groszkowski, High-threshold universal quantum computation on the surface code, Phys. Rev. A 80, 052312 (2009).
  * Fowler  _et al._ [2012a] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012a).
  * Stephens [2014] A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Phys. Rev. A 89, 022321 (2014).
  * Chamberland  _et al._ [2016] C. Chamberland, T. Jochym-O’Connor, and R. Laflamme, Thresholds for universal concatenated quantum codes, Phys. Rev. Lett. 117, 010501 (2016).
  * Marks  _et al._ [2017] J. Marks, T. Jochym-O’Connor, and V. Gheorghiu, Comparison of memory thresholds for planar qudit geometries, New J. Phys. 19, 113022 (2017).
  * Campbell [2014] E. T. Campbell, Enhanced fault-tolerant quantum computing in $d$-level systems, Phys. Rev. Lett. 113, 230501 (2014).
  * Suchara  _et al._ [2013] M. Suchara, J. Kubiatowicz, A. Faruque, F. T. Chong, C.-Y. Lai, and G. Paz, Qure: The quantum resource estimator toolbox, in  _2013 IEEE 31st International Conference on Computer Design (ICCD)_ (2013) pp. 419–426.
  * Paetznick  _et al._ [2023] A. Paetznick, C. Knapp, N. Delfosse, B. Bauer, J. Haah, M. B. Hastings, and M. P. da Silva, Performance of planar floquet codes with majorana-based qubits, PRX Quantum 4, 010310 (2023).
  * Pirandola  _et al._ [2008] S. Pirandola, S. Mancini, S. L. Braunstein, and D. Vitali, Minimal qudit code for a qubit in the phase-damping channel, Phys. Rev. A 77, 032309 (2008).
  * Cafaro  _et al._ [2012] C. Cafaro, F. Maiolini, and S. Mancini, Quantum stabilizer codes embedding qubits into qudits, Phys. Rev. A 86, 022308 (2012).
  * Michael  _et al._ [2016] M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
  * Chiesa  _et al._ [2020] A. Chiesa, E. Macaluso, F. Petiziol, S. Wimberger, P. Santini, and S. Carretta, Molecular nanomagnets as qubits with embedded quantum-error correction, J. Phys. Chem. Lett. 11, 8610 (2020).
  * Cai  _et al._ [2021] W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundamental Research 1, 50 (2021).
  * Chiesa  _et al._ [2022] A. Chiesa, F. Petiziol, M. Chizzini, P. Santini, and S. Carretta, Theoretical design of optimal molecular qudits for quantum error correction, J. Phys. Chem. Lett. 13, 6468 (2022).
  * Omanakuttan and Gross [2023] S. Omanakuttan and J. A. Gross, Multispin clifford codes for angular momentum errors in spin systems (2023), arXiv:2304.08611 [quant-ph] .
  * Petiziol  _et al._ [2021] F. Petiziol, A. Chiesa, S. Wimberger, P. Santini, and S. Carretta, Counteracting dephasing in molecular nanomagnets by optimized qudit encodings, npj Quantum Inf. 7, 133 (2021).
  * Carretta  _et al._ [2021] S. Carretta, D. Zueco, A. Chiesa, A. Gómez-León, and F. Luis, A perspective on scaling up quantum computation with molecular spins, Appl. Phys. Lett. 118, 240501 (2021).
  * Chizzini  _et al._ [2022a] M. Chizzini, L. Crippa, L. Zaccardi, E. Macaluso, S. Carretta, A. Chiesa, and P. Santini, Quantum error correction with molecular spin qudits, Phys. Chem. Chem. Phys. 24, 20030 (2022a).
  * Lim  _et al._ [2023] S. Lim, J. Liu, and A. Ardavan, Fault-tolerant qubit encoding using a spin-7/2 qudit (2023), arXiv:2303.02084 [quant-ph] .
  * Ni  _et al._ [2023] Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu, C.-L. Zou, L. Sun, S.-B. Zheng, Y. Xu, and D. Yu, Beating the break-even point with a discrete-variable-encoded logical qubit, Nature 616, 56 (2023).
  * Ma  _et al._ [2020] Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, Error-transparent operations on a logical qubit protected by quantum error correction, Nat. Phys. 16, 827–831 (2020).
  * Reinhold  _et al._ [2020] P. Reinhold, S. Rosenblum, W.-L. Ma, L. Frunzio, L. Jiang, and R. J. Schoelkopf, Error-corrected gates on an encoded qubit, Nat. Phys. 16, 822 (2020).
  * Puri  _et al._ [2020] S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias-preserving gates with stabilized cat qubits, Sci. Adv. 6, eaay5901 (2020).
  * Rosenblum  _et al._ [2018] S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. Frunzio, and R. J. Schoelkopf, Fault-tolerant detection of a quantum error, Science 361, 266 (2018).
  * Hu  _et al._ [2019] L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xua, W. Wang, Y. Wu, H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M. Duan, and L. Sun, Quantum error correction and universal gate set operation on a binomial bosonic logical qubit, Nat. Phys. 15, 503–508 (2019).
  * Grimsmo  _et al._ [2020] A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
  * Atzori and Sessoli [2019] M. Atzori and R. Sessoli, The second quantum revolution: Role and challenges of molecular chemistry, J. Am. Chem. Soc. 141, 11339 (2019).
  * Knill and Laflamme [1997] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  * Fowler  _et al._ [2012b] A. G. Fowler, A. C. Whiteside, and L. C. L. Hollenberg, Towards practical classical processing for the surface code, Phys. Rev. Lett. 108, 180501 (2012b).
  * Bravyi  _et al._ [2014] S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014).
  * Das  _et al._ [2022] P. Das, C. A. Pattison, S. Manne, D. M. Carmean, K. M. Svore, M. Qureshi, and N. Delfosse, Afs: Accurate, fast, and scalable error-decoding for fault-tolerant quantum computers, in  _2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA)_ (2022) pp. 259–273.
  * Sweke  _et al._ [2020] R. Sweke, M. S. Kesselring, E. P. L. van Nieuwenburg, and J. Eisert, Reinforcement learning decoders for fault-tolerant quantum computation, Machine Learning: Science and Technology 2, 025005 (2020).
  * Note [1] The relative phase between $\mathinner{|{\ell,k}\rangle}\mathinner{|{0}\rangle}$ and $\mathinner{|{\ell,k}\rangle}\mathinner{|{k}\rangle}$ is necessary to ensure that the corresponding generator has zero diagonal. Since the ancilla is initialized in $\mathinner{|{0}\rangle}$, this relative phase is irrelevant.
  * Lockyer  _et al._ [2021] S. J. Lockyer, A. Chiesa, G. A. Timco, E. J. L. McInnes, T. S. Bennett, I. J. Vitorica-Yrezebal, S. Carretta, and R. E. P. Winpenny, Chem. Sci. 12, 9104 (2021).
  * Lockyer  _et al._ [2022] S. J. Lockyer, A. Chiesa, A. Brookfield, G. A. Timco, G. F. S. Whitehead, E. J. L. McInnes, S. Carretta, and R. E. P. Winpenny, Five-spin supramolecule for simulating quantum decoherence of bell states, J. Am. Chem. Soc. 144, 16086 (2022).
  * [46] C. J. Rogers, D. Asthana, A. Brookfield, A. Chiesa, G. A. Timco, D. Collison, L. S. Natrajan, S. Carretta, R. E. P. Winpenny, and A. M. Bowen, Modelling conformational flexibility in a spectrally addressable molecular multi-qubit model system, Angew. Chem. Int. Ed. n/a, e202207947.
  * Nakazawa  _et al._ [2012] S. Nakazawa, S. Nishida, T. Ise, T. Yoshino, N. Mori, R. D. Rahimi, K. Sato, Y. Morita, K. Toyota, D. Shiomi, M. Kitagawa, H. Hara, P. Carl, P. Höfer, and T. Takui, A synthetic two-spin quantum bit: g-engineered exchange-coupled biradical designed for controlled-not gate operations, Angew. Chem. Int. Ed. 51, 9860 (2012).
  * Rugg  _et al._ [2019] B. K. Rugg, M. D. Krzyaniak, B. T. Phelan, M. A. Ratner, R. M. Young, and M. R. Wasielewski, Photodriven quantum teleportation of an electron spin state in a covalent donor–acceptor–radical system, Nat. Chem. 11, 981 (2019).
  * Olshansky  _et al._ [2020] J. H. Olshansky, J. Zhang, M. D. Krzyaniak, E. R. Lorenzo, and M. R. Wasielewski, Selectively addressable photogenerated spin qubit pairs in dna hairpins, Journal of the American Chemical Society 142, 3346 (2020).
  * Jacobberger  _et al._ [2022] R. M. Jacobberger, Y. Qiu, M. L. Williams, M. D. Krzyaniak, and M. R. Wasielewski, Using molecular design to enhance the coherence time of quintet multiexcitons generated by singlet fission in single crystals, Journal of the American Chemical Society 144, 2276 (2022).
  * Chizzini  _et al._ [2022b] M. Chizzini, L. Crippa, A. Chiesa, F. Tacchino, F. Petiziol, I. Tavernelli, P. Santini, and S. Carretta, Molecular nanomagnets with competing interactions as optimal units for qudit-based quantum computation, Phys. Rev. Res. 4, 043135 (2022b).
  * Garlatti  _et al._ [2012] E. Garlatti, S. Carretta, M. Affronte, E. C. Sañudo, G. Amoretti, and P. Santini, Magnetic properties and relaxation dynamics of a frustrated ni7 molecular nanomagnet, J. Phys.: Cond. Matt. 24, 104006 (2012).
  * Bruzewicz  _et al._ [2019] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Lett. 6, 021314 (2019).
  * Ringbauer  _et al._ [2022] M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, A universal qudit quantum processor with trapped ions, Nat. Phys. 18, 1053–1057 (2022).
  * Kuhr  _et al._ [2005] S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, Analysis of dephasing mechanisms in a standing-wave dipole trap, Phys. Rev. A 72, 023406 (2005).
  * Morello  _et al._ [2020] A. Morello, J. J. Pla, P. Bertet, and D. N. Jamieson, Donor spins in silicon for quantum technologies, Adv. Quantum Technol. 3, 2000005 (2020).
  * Chatterjee  _et al._ [2021] A. Chatterjee, P. Stevenson, S. De Franceschi, A. Morello, N. P. de Leon, and F. Kuemmeth, Semiconductor qubits in practice, Nat. Rev. Phys. 3, 157 (2021).
  * Fataftah  _et al._ [2016] M. Fataftah, J. M. Zadrozny, S. C. Coste, M. J. Graham, D. M. Rogers, and D. E. Freedman, Employing forbidden transitions as qubits in a nuclear spin-free chromium complex., J. Am. Chem. Soc. 138, 1344 (2016).
  * Bader  _et al._ [2016] K. Bader, M. Winkler, and J. van Slageren, Tuning of molecular qubits: Very long coherence and spin–lattice relaxation times, Chem. Commun. , 3623 (2016).
  * Ding  _et al._ [2016] Y.-S. Ding, Y.-F. Deng, and Y.-Z. Zheng, The rise of single-ion magnets as spin qubits, Magnetochem. 2, 40 (2016).
  * Bader  _et al._ [2014] K. Bader, D. Dengler, S. Lenz, B. Endeward, S.-D. Jiang, P. Neugebauer, and J. van Slageren, Room temperature quantum coherence in a potential molecular qubit, Nat. Comm. 5, 5304 (2014).
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge university press, 2000).
  * Leuenberger and Loss [2003] M. N. Leuenberger and D. Loss, Grover algorithm for large nuclear spins in semiconductors, Phys. Rev. B 68, 165317 (2003).
  * Note [2] This is due to the specific form of the error words $\\{\mathinner{|{\ell,k}\rangle}\\}$, in which differ the two $\ell=0,1$ subspaces are disjoint for all $k$.
  * Knill  _et al._ [2001] E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking quantum computers: The five-qubit error correcting code, Phys. Rev. Lett. 86, 5811 (2001).
  * Note [3] This allows us to compare the performance of a spin 1/2 qubit (without EC) and of the LQ even though the duration of the elementary transitions on them is different (and hence the related error probability). Indeed, the elementary error probability for a two-level system is $p=(1-\exp[-\tau/T_{2}])/2$ [62]. Since the duration $\tau$ of the transition is much smaller for a spin 1/2, it would result in a much smaller $p$ compared to the LQ, thus making the comparison unfair.
  * Note [4] $\mathcal{F}_{e}$ is the average fidelity on the initial states $\mathinner{|{0_{L}}\rangle},\mathinner{|{1_{L}}\rangle},$ $\frac{1}{\sqrt{2}}(\mathinner{|{0_{L}}\rangle}+\mathinner{|{1_{L}}\rangle}),\frac{1}{\sqrt{2}}(\mathinner{|{0_{L}}\rangle}-\mathinner{|{1_{L}}\rangle}),\frac{1}{\sqrt{2}}(\mathinner{|{0_{L}}\rangle}+i\mathinner{|{1_{L}}\rangle}),\frac{1}{\sqrt{2}}(\mathinner{|{0_{L}}\rangle}-i\mathinner{|{1_{L}}\rangle})$.
  * Atzori  _et al._ [2016a] M. Atzori, E. Morra, L. Tesi, A. Albino, M. Chiesa, L. Sorace, and R. Sessoli, Quantum coherence times enhancement in vanadium(iv)-based potential molecular qubits: the key role of the vanadyl moiety, J. Am. Chem. Soc. 138, 11234 (2016a).
  * Atzori  _et al._ [2016b] M. Atzori, L. Tesi, E. Morra, M. Chiesa, L. Sorace, and R. Sessoli, Room-temperature quantum coherence and rabi oscillations in vanadyl phthalocyanine: Toward multifunctional molecular spin qubits., J. Am. Chem. Soc. 138, 2154 (2016b).
  * Atzori  _et al._ [2021] M. Atzori, E. Garlatti, G. Allodi, S. Chicco, A. Chiesa, A. Albino, R. D. Renzi, E. Salvadori, M. Chiesa, S. Carretta, and L. Sorace, Radiofrequency to microwave coherent manipulation of an organometallic electronic spin qubit coupled to a nuclear qudit, Inorganic Chemistry 60, 11273 (2021).
  * Castro  _et al._ [2022] A. Castro, A. García Carrizo, S. Roca, D. Zueco, and F. Luis, Optimal control of molecular spin qudits, Phys. Rev. Appl. 17, 064028 (2022).
  * Kandala  _et al._ [2017] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242–246 (2017).
  * Ferrando-Soria  _et al._ [2016] J. Ferrando-Soria, E. Moreno-Pineda, A. Chiesa, A. Fernandez, S. A. Magee, S. Carretta, P. Santini, I. Vitorica-Yrezabal, F. Tuna, E. J. L. McInness, and R. E. P. Winpenny, A modular design of molecular qubits to implement universal quantum gates., Nat. Commun. 7, 11377 (2016).
  * Chiesa  _et al._ [2021] A. Chiesa, F. Petiziol, E. Macaluso, S. Wimberger, P. Santini, and S. Carretta, Embedded quantum-error correction and controlled-phase gate for molecular spin qubits, AIP Adv. 11, 025134 (2021).
  * Chiesa  _et al._ [2014] A. Chiesa, G. F. S. Whitehead, S. Carretta, L. Carthy, G. A. Timco, S. J. Teat, G. Amoretti, E. Pavarini, R. E. P. Winpenny, and P. Santini, Molecular nanomagnets with switchable coupling for quantum simulation., Sci. Rep. 4, 7423 (2014).
  * Chiesa  _et al._ [2023] A. Chiesa, S. Roca, S. Chicco, M. C. de Ory, A. Gómez-León, A. Gomez, D. Zueco, F. Luis, and S. Carretta, Blueprint for a molecular-spin quantum processor, Phys. Rev. Appl. 19, 064060 (2023).
  * Zadrozny  _et al._ [2015] J. M. Zadrozny, J. Niklas, O. G. Poluektov, and D. E. Freedman, Millisecond coherence time in a tunable molecular electronic spin qubit., ACS Cent. Sci. 1, 488 (2015).


# Linear Polycatenanes from Kinetoplast Edge Loops

Josh Ragotskie  Department of Physics and Astronomy, California State
University, Long Beach  Nathaniel Morrison  Department of Mathematics and
Statistics, California State University, Long Beach  Department of Physics,
University of California, Berkeley  Christopher Stackhouse  Department of
Physics and Astronomy, California State University, Long Beach  Ryan C. Blair
Department of Mathematics and Statistics, California State University, Long
Beach  Alexander R. Klotz  Department of Physics and Astronomy, California
State University, Long Beach  Corresponding author<EMAIL_ADDRESS>
###### Abstract

We use graph theory simulations and single molecule experiments to investigate
percolation properties of kinetoplasts, the topologically linked mitochondrial
DNA from trypanosome parasites. The edges of some kinetoplast networks contain
a fiber of redundantly catenated DNA loops, but previous investigations of
kinetoplast topology did not take this into account. Our graph simulations
track the size of connected components in lattices as nodes are removed,
analogous to the removal of minicircles from kinetoplasts. We find that when
the edge loop is taken into account, the largest component after the network
de-percolates is a remnant of the edge loop, before it undergoes a second
percolation transition and breaks apart. This implies that stochastically
removing minicircles from kinetoplast DNA would isolate large polycatenanes,
which is observed in experiments that use photonicking to stochastically
destroy kinetoplasts from Crithidia fasciculata. Our results imply
kinetoplasts may be used as a source of linear polycatenanes for future
experiments.

##  I Introduction

Catenanes are a form of mechanically interlocked molecule in which closed
macrocycles are connected topologically like links in a chain. A polycatenane
is a polymer chain in which the macrocycles serve as monomers and are
polymerized by topological catenantion. Catenanes and polycatenanes are part
of the growing field of mechanically interlocked molecules, which are of
interest to the materials community because of the prospect of controlling
physical properties, common chemistry and varied topology, and their potential
use in nanomachines erbas2015artificial . Polycatenanes have been produced
synthetically wu2017poly , and understanding the relationship between
molecular topology and material properties is central to the development of
catenated polymers. Single-molecule experiments are desirable for such a
purpose: linear polymer physics has benefited from the use of viral genomic
DNA molecules as model experimental systems marciel , and it would be ideal if
a similar system existed to study polycatenanes.

Kinetoplasts are complex DNA structures found in the mitochondria of
trypanosome parasites. Often described as molecular chainmail, they consist of
approximately 5,000 topologically linked circular molecules, known as
minicircles, and several dozen larger molecules called maxicircles
shapiro1995structure . The kinetoplast is part of a complex gene editing
apparatus: maxicircles contain the mitochondrial genes, which are transcribed
into messenger RNA that must be edited by guide RNA molecules transcribed from
the minicircles through the addition and removal of uracil
sturm1990kinetoplast . While the main reason for their study is to better
understand and disrupt the parasite life cycle, in recent years they have seen
use as a model system for two-dimensional polymer physics klotz2020equilibrium
; soh2020deformation . Typically it is their planar rather than their
catenated nature that serves in this capacity, and it is unclear how their
catenated molecular topology affects their mechanical properties, although
more recent studies have begun to investigate this relation yadav2023tuning ;
he2023single .

Most of what is known about kinetoplast topology comes from experiments on
Crithidia fasciculata (an insect parasite) kinetoplasts, in which minicircles
were linearized with restriction enzymes and gel electrophoresis was used to
determine the relative proportion of linear, circular, doubly-catenanted, and
triply-catenated molecules chen1995topology . It was concluded that each
minicircle was connected to three others on average and that the regular
lattice most consistent with the data was the honeycomb. High-resolution
images of kinetoplasts appear to have disordered minicircles
barker1980ultrastructure , and a more recent analysis using atomic force
microscopy supported the mean valence of 3, but not as a honeycomb lattice
he2023single . The maxicircles are catenated within the network and form their
own catenating network shapiro1993kinetoplast , but it is not known the extent
to which they contribute to the overall structure or topology of the network.
The relation between minicircle sequence and topology may not be random. For
example, it is known that approximately 90% of Crithidia fasciculata
minicircles have the same sequence, and a recent experiment examined the
networks after removal of that majority (as well as the maxicircles) using
restriction enzymes yadav2023tuning . The networks changed appearance but
remained intact, suggesting that some combination the remaining 10% of
minicircles form a percolating network.

A lesser-known feature of kinetoplasts is that they are surrounded by a fiber
of dense linked minicircles that serves a similar mechanical role as an
elastic band in a shower cap. This edge fiber is of unknown biological
function but becomes less prominent during kinetoplast replication. It is
present in Crithidia fasciculata, and has been observed across several
trypansome genera including in Pythomonas serpens (a plant parasite),
Leishmania mexicana (a mammal parasite), and Trypanosoma mega (an amphibian
parasite) barker1980ultrastructure  but does not appear to be prominent in the
human-infecting parasites Trypanosoma brucei and cruzi
guilbride1998replication . The degree of redundancy, the number of parallel
minicircles connecting hubs around the edge loop, is primarily estimated from
electron or atomic force microscopy of kinetoplasts networks that have been
spread on surfaces. The spreading procedures can degrade the networks, and as
such these estimates must be treated as lower bounds. Barker
barker1980ultrastructure  in 1980 stated that each hub contains 32
minicircles, with 16 minicircles connecting each hub to its neighbor on either
side. In a more recent AFM study, He et al he2023single . estimate based on
the height of the hubs that they contain an average of 4.7 minircles. If two
minicircles connect each hub to its two neighbors and one the interior, this
suggests a mean redundancy of 2. It is not known whether the minicircles
appearing on the edge have a specific sequence, although Kleisen et al.
indicate that they are cut by the same endonucleases as the rest of the
network kleisen .

Experiments that remove minicircles from kinetoplasts will eventually remove
enough such that the percolation threshold is passed. A network can be
represented as a graph of nodes (in this case, minicircles) connected by edges
(in this case Hopf linkages). If a small fraction of the edges between nodes
are removed, a single connected component will still span the network.
However, if a large fraction of edges are removed, the largest remaining
connected component will be much smaller than the system size. The critical
fraction of nodes such that a giant component exists is known as the
percolation threshold, and depends on the underlying lattice topology
van1997percolation . In a kinetoplast experiment such as that of Chen et al.
chen1995topology , linearizing a minicircle is equivalent to removing all the
edges connecting one node to its neighbors, and when the percolation threshold
is passed, there should no longer be a cohesive DNA structure, simply an
aggregate of formerly connected minicircles. If a kinetoplast has a purely
honeycomb lattice topology (which has a percolation threshold of 70%) the
network should de-percolate when 30% of minicircles are removed. Several of
Chen et al.’s restriction assays were below this fraction, but the large
connected components cannot pass through an agar gel and are not part of the
data analysis. We note that in independent experiments from Ibrahim et al.
ibrahim , Li et al. tevin , and Yadav et al. yadav2023tuning , there is still
DNA visible in the wells that likely corresponds to these remaining large
clusters, with Li et al. presenting images of “poly-interlocked” structures.
Beyond predicting the electrophoresis experiments for different lattices,
theoretical work on kinetoplast topology has examined the probability that
closely packed circles diao2015orientation  or ring polymers
michieletto2015kinetoplast  will form a percolating network and the
relationship between minicircle valence and percolation. No analysis to date
(that we are aware of) has included the effect of the edge loop on percolation
of a kinetoplast network.

In this manuscript we examine percolation phenomena in networks with redundant
edges, similar to the edge loop of kinetoplasts. Our purpose is to demonstrate
that kinetoplasts may be useful source of linear polycatenanes for single
molecular experiments. We support the findings from our simulations with
fluorescence microscopy experiments in which photochemical cleavage is used to
de-percolate Crithidia kinetoplasts and show that the edge loops remain after
the networks have largely been destroyed.

Figure 1: a. Schematic of the edge of a kinetoplast, in which a planar
catenated DNA network is surrounded by a linearized band of redundantly linked
minicircles. b. and c. show electron micrographs of the kinetoplasts from
Crithidia fasciculata, highlighting the edge loop and its features (used with
permission from Barker, 1980 barker1980ultrastructure ).

##  II Simulations

Following previous work on the topic chen1995topology ; ibrahim ; ferschweiler
, we represent kinetoplast networks as graphs in which each node represents a
minicircle, and an edge between two nodes represents a topological connection
between two minicircles. The removal of a node from a network is equivalent to
breaking a minicircle, either by restriction enzyme digestion or photonicking.
If a subgraph remains but all nodes surrounding it are removed, the subgraph
is no longer connected to the rest of the network. The computational portion
of this work involves measuring the size of connected subgraphs in networks
from which nodes have been removed.

As is standard in graph theory graph , we represent lattices as their
corresponding adjacency matrices. The adjacency matrices for undirected,
unweighted lattices are symmetric binary matrices where each node is
represented as both a row and a column, and a non-zero entry in row $x$ and
column $y$ indicates node $x$ shares an edge with node $y$. To mimic the
effect of the edge loop, we assign nodes on the edges an integer value $R\geq
1$ representing the number of parallel minicircles. Similarly, the non-zero
entries in the rows and columns of the adjacency matrix corresponding to nodes
on the edge are all $R$. Removing a node is equivalent either to removing the
corresponding row and column from the adjacency matrix or to reducing the non-
zero labels in row and column of an edge node from $R$ to $R-1$. For example,
if $R=5$, then the same node would have to be reduced five times in order to
“cut” it completely. To investigate broad features of redundantly-edged
lattices, we initialize networks of various topology and simulate the removal
of minicircles by zeroing nodes in the network. We determine which occupied
nodes are connected in the same subgraph, to calculate a list of the sizes of
all the connected components in the system.

Figure 2: Right: A portion of the kinetoplast in the hexagonal model near the
boundary where the boundary is modeled by 3-fold catenated parallel loops.
Center: The graph encoding the kinetoplast network for this portion of the
kinetoplast. Left: The simplified graph encoding the kinetoplast network where
the boundary of the portion of the planar honeycomb lattice is indicated with
thickened edges.

We investigate square, honeycomb, and “random” lattices. Random lattices are
generated from random points on a sparse square lattice and choosing a common
radius $r$ such that the probably of two nodes sharing an edge is proportional
to the overlap area between two circles of radius $r$ centered on each node.
The radius is chosen such that the mean valence of each node is approximately
3, consistent with what is expected of kinetoplast DNA.

Degradation of the kinetoplast is modeled by an iterated process of randomly
removing a vertex, and all edges incident to that vertex, from a spatial
graph. We call such a removal a dissolution. After each dissolution step, the
size of every connected component in the graph, down to individual isolated
vertices, is recorded. This process proceeds until every connected component
of the resulting graph has fewer vertices than a threshold value. For the
computations discussed in this paper, we chose this threshold value to be 100,
from an initial 5000.

###  II.1 Simulation Results

Many properties of percolating network manifest as size-independence or apply
in the limit of very large networks when edge effects are neglibible. However,
because we are studying biological networks with approximately 5000
components, and because we are explicitly studying an edge effect (that of the
catenated loop), we are not concerned with size-independence or the
asymptotically large limit. To identify percolation thresholds in our
simulations, we measure the second-largest cluster of the networks and observe
its local maximum secondlargest . While this metric has weak dependence on the
system size, it is suitable for our investigations.

The salient features of our model can be seen in the two supplemental videos,
which show the dissolution of two 70x70 networks on a square lattice, one
without edge redundancy and one with R=10, with snapshots shown in Fig. 3. The
largest component is displayed in yellow, the second largest (when unique) in
green, other components as light blue, and unoccupied sites in dark blue. As
each video plays, the charts below show the largest component size in blue and
the second-largest in red, scaled up for visibility. In the edgeless network,
the largest component ceases to span the network near the expected percolation
threshold and then rapidly decreases in size. In the edge-redundant network,
after the interior percolation transition, the largest component remains
localized to the perimeter of the lattice, until it begins to break into many
smaller fragments at the second percolation transition, at which point both
the largest and second-largest component lie on the edge. Experimentally, this
would manifest as several larger linear polycatenanes amidst a sea of smaller
fragments.

Figure 3: The dissolution of a 70x70 square lattice network with ten-fold edge
redundancy. The largest connected component is labelled in yellow, the second
largest in green, other components as light blue, and unoccupied sites in dark
blue. The first image shows a giant component with a few removed nodes. The
second is at the percolation threshold of the network interior, in which the
largest component still spans the entire system. In the third image, the
interior is no longer percolating but the edge of the system still contains
its largest component. In the fourth image, the edge has broken into two
components.

The basic features of edged network percolation can be seen in Fig. 4, where
data is presented from a square lattice simulation. For a network with no
redundant edge, below the percolation threshold (corresponding to the removal
of 0.4073 of nodes) the largest component is small compared to the entire
lattice. With redundant edges, there remains a significantly sized component
around the circumference of the network, leaving a largest component taking up
a non-negligible fraction of the network. The percolation threshold may be
identified by a local maximum in the second-largest component. This is
observed at the expected value for edgeless networks, but as the edge
redundancy increases a secondary peak emerges corresonding to a second
percolation transition, in which the edge breaks up into two or more
components. The waterfall plot shows this process as edge redundancy
increases, increasing the amplitude of the secondary percolation peak and
pushing it towards a higher fraction of removed minicircles. This implies, for
example that when 70% of minicircles are removed, an edgeless network will
have no large components, whereas a network with a sufficiently redundant edge
will still have a largely pristine edge fiber with an empty interior.

Figure 4: a. The size of the largest connected component in 70x70 square
networks with edge redundancy of 1, 10, and 20 as a function of the fraction
of dissolved minicircles. At a dissolution fraction that leaves an edgeless
lattice with a small largest component, the edged lattices have a larger
largest component around their edge. b. The size of the second largest
connected component in 70x70 square networks with edge redundancy of 1, 10,
and 20. A local maximum indicates a percolation transition. With a redundant
edge, a secondary transition arises when the edge breaks. The dashed line
shows the percolation threshold of a square lattice. c. Waterfall plots of the
second largest components for lattices with edge redundancy between 1 and 20.

We can compare the effect of lattice geometry by examining heatmaps showing
the effect of increasing edge redundancy on the second largest cluster size at
different removal fractions. Heatmaps in Figure 5 show the second largest
component size as a function of a the fraction of removed minicircles and the
edge redundancy, for the square, honeycomb, and random networks. The known
percolation thresholds for the square and honeycomb lattices (1-0.59=0.41 and
1-0.70=0.3 respectively) are recovered for the edgeless lattices, and the
random lattice has a higher percolation threshold. The higher percolation
threshold for the random lattice is likely due to the topological similarities
between the random lattice and lattices with defects, which have previously
been shown to have a higher percolation threshold spencerziff . As the
redundancy increases, the peak splits into two with the top peak,
corresponding to interior de-percolation, remaining at its edgeless location
and the second peak, corresponding to edge fragmentation, being found at a
removal fraction that increases with redundancy.

Figure 5: Heatmaps showing the second largest cluster size as a function of
the percentage of minicircles removed (from top to bottom) and edge redundancy
up to 20 (left to right) for square, honeycomb, and random networks. In all
cases, edge redundancy causes the peak to split into two, the upper
conrresponding to interior de-percolation and the the lower corresponding to
edge loop fragmentation.

Finally, to support our assertion that edge redundancy leads to the edge loop
being the last cohesive component of the network, we can vary the edge
redundancy, perform dissolution simulations, and measure whether the interior
de-percolates before the exterior breaks. Interior de-percolation is defined
as follows: a graph consisting of every non-peripheral node is analyzed for
connected components, and the coordinates of the largest component are
analyzed to determine whether it touches the edge on all four sides. If it
does not, the interior is considered de-percolated. The exterior is determined
to have broken if the value of any matrix element on the edge reaches zero
during the dissolution process. In Figure 6 we plot the probability that the
interior de-percolates before the exterior breaks, as a function of the edge
redundnacy for a square lattice with 4900 nodes and a hexagonal lattice with
5000 nodes. For $R\leq 3$ it is very unlikely for the edge to survive until
the interior de-percolates, whereas for $R\geq 6$ it is almost guaranteed that
the edge will be intact when the interior depercolates.

Figure 6: Probability that the interior of a network de-percolates before the
exterior breaks, as defined in the text, as a function of the edge redundancy
for a 70x70 square lattice and a 5000-node honeycomb lattice. When not
visible, standard error bars are smaller than points.

##  III Experiments

Figure 7: Schematic of our experiments in which intense light is used to
linearize minicircles and remove them from the network. Right side shows an
image of a kinetoplast before, during, and after the light intensity is
increased, with 5 micron scale bar.

Our experiments are meant to demonstrate that fibers from the edge loop are
the last cohesive remnants of destroyed kinetoplasts. We visualize
kinetoplasts from Crithidia fasciculata (Topogen, Inc.) using fluorescence
microscopy by staining them with YOYO-1 dye at a ratio of 4 base-pairs per
YOYO in 0.5x TBE buffer with 4% beta-mercaptoethanol according to procedures
described previously klotz2020equilibrium . The kinetoplast-containing buffer
are sealed between two glass cover slips with 100-micron thick double-sided
tape. The molecules are illuminated with an LED filtered to 490 nm (blue) and
visualized with a Leica DMI8 inverted fluorescence microscope. The combination
of YOYO-1, DNA, light, and dissolved oxygen can induce a reaction that breaks
DNA aakerman1996single . Typically this is undesirable and is prevented by the
additon of beta-mercaptoethanol or glucose oxidase to the buffer, by degassing
buffer solutions to remove dissolved oxygen, and by limiting the intensity of
the illumination persson2010dna . To promote photodisintegration of the DNA we
forewent the degassing phase and increased the illumination intensity far
beyond the recommended amount (Fig. 7), which causes minicircles to rapidly
break. The mechanism of YOYO-1 photocleavage of DNA, which happens when two
single-stranded breakages occur opposite one-another, is discussed by Akerman
and Tuite aakerman1996single  and examined on a single-molecule level by
Alazadehheidari et al. alizadehheidari2015nanoconfined .

Figure 8: Two montages of kinetoplasts undergoing photodisintegration as their
interior de-percolates. The top montage shows a free kinetoplast which loses
its interior connectivity, resulting in a thick circular polycatanane, which
continues to degrade. The bottom montage shows a kinetoplast adsorbed to a
surface, in which visible holes appear in the interior. Eventually the edge
loop breaks and the kinetoplast opens up.

Figure 8 shows two time lapses of a molecule undergoing photodisintegration.
The top montage shows a molecule in free solution that forms a visible hole in
its interior, leaving a contiguous boundary. Eventually the boundary becomes
more like a thick ring polymer, undergoing shape fluctuations that are not
observed in whole kinetoplasts klotz2020equilibrium . It has been shown that
kinetoplasts with a significant fraction of removed minicircles (while still
forming a percolating network) exhibit larger-amplitude fluctuations
yadav2023tuning , which we confirm. The bottom montage shows a kinetoplast
absorbed to a glass over slip. Dark patches appear in the interior and become
more prominent until the edge loop breaks and the molecule opens up, leaving a
large edge loop fragment and a distribution of smaller interior fragments.
Because kinetoplasts can be considerably larger than the focal plane of a high
numerical aperture lens, it can be difficult observing the whole process with
such clarity. Several more time of photodisintegration lapses can be seen in
Fig. 9, as well as some other examples of edge loops post-disintegration.

Figure 9: Four montages of kinetoplasts undergoing photodisintegration,
leaving behind a fibrous polycatenane. The bottom row shows the end stage of
four more degradations.

We attempted to observe the dissociated edge loops undergoing agar gel
electrophoresis with optical microscopy. We performed experiments in which 400
$\mu$m tall microfluidic channels (Ibidi GmbH) were filled with an agarose
solution which was allowed to gelify. Kinetoplasts were placed in the
microchannel reservoir and driven with an electric field towards the boundary
of the gel where their motion was arrested. After photodistintegration, the
remaining edge loops did not penetrate the gel. However, because the gel
typically contracted over the course of the experiment, there were continuous
gel domains separated by regions of fluid through which kinetoplasts could
pass. This often lead to situations in which kinetoplasts were trapped between
regions of gel with the electric field straining them in unexpected
directions. As kinetoplasts began to photodisintegrate, their released
fragments would translate away from them, and eventually their edge fibers
would be liberated from the interior and stretch dramatically as in Figure 10.
The circumference of a kinetoplast is approximately 16 microns, comparable to
the maximum observed length of the stretched fiber.

Figure 10: Montage from an experiment in which a kinetoplast is trapped in a
gel network under an applied field while being photodisintegrated. The edge
loop at this point is the only remaining cohesive structure and becomes
significantly stretched.

##  IV Discussion and Conclusions

We have investigated the effect of the kinetoplast edge loop on the
percolation properties of the network under random minicircle removal. We
sought to investigate the effect of the edge loop on the kinetoplast’s
percolation behavior, and found that the edge loop or fragments from it are
likely to be the last large remaining component when a kinetoplast is
destroyed by minicircle removal. To verify this, we destroyed kinetoplasts
using YOYO-1 photocleavage and observed similar features as those predicted in
simulations. This suggests that stochastically degraded kinetoplasts can serve
as a source of polycatenanes for future polymer experiments. It was not our
goal in this work to characterize the equilibrium or elastic properties of
these isolated edge loops, which is a topic of future study.

More precise measurements of the percolation threshold of Crithidia
fasciculata may provide insight into the network’s topology. In gel
electrophoresis experiments, the non-penetrating fraction of DNA may be
collected and visualized to determine if they primarily consist of holey
membranes (as in Yadav et al. yadav2023tuning ) or edge loops (as in this
work), and the disappearance of the non-penetrating segment may serve as
evidence that the second percolation threshold, that which breaks the edge
loop, has been surpassed. As seen in Fig. 4, the first percolation threshold
depends on the interior lattice topology while the second depends on the
degree of edge redundancy. Thus, an experiment that measures the enzyme
activity at which each threshold occurs could provide a new estimate of both
the lattice topology and the edge redundancy. Finally, our simulations that
suggest an edge redundancy of at least 4 is required for the loop to survive
depercolation, combined with our observations that the loop does survive
depercolation, indicate that the edge valence of 4.7 reported by He et al
he2023single . may be lower than the in vivo redundancy due to sample
preparation effects.

##  V Acknowledgements

This work was supported by the National Science Foundation (Grant No.
2105113).

## References

  * [1] Sundus Erbas-Cakmak, David A Leigh, Charlie T McTernan, and Alina L Nussbaumer.  Artificial molecular machines.  Chemical reviews, 115(18):10081–10206, 2015.
  * [2] Qiong Wu, Phillip M Rauscher, Xiaolong Lang, Rudy J Wojtecki, Juan J De Pablo, Michael JA Hore, and Stuart J Rowan.  Poly [n] catenanes: Synthesis of molecular interlocked chains.  Science, 358(6369):1434–1439, 2017.
  * [3] Amanda B Marciel and Charles M Schroeder.  New directions in single polymer dynamics.  Journal of Polymer Science Part B: Polymer Physics, 51(7):556–566, 2013.
  * [4] Theresa A Shapiro and Paul T Englund.  The structure and replication of kinetoplast dna.  Annual review of microbiology, 49(1):117–143, 1995.
  * [5] Nancy R Sturm and Larry Simpson.  Kinetoplast dna minicircles encode guide rnas for editing of cytochrome oxidase subunit iii mrna.  Cell, 61(5):879–884, 1990.
  * [6] Alexander R Klotz, Beatrice W Soh, and Patrick S Doyle.  Equilibrium structure and deformation response of 2d kinetoplast sheets.  Proc. Natl. Acad. Sci. U.S.A., 117(1):121–127, 2020.
  * [7] Beatrice W Soh and Patrick S Doyle.  Deformation response of catenated dna networks in a planar elongational field.  ACS Macro Lett., 9(7):944–949, 2020.
  * [8] Indresh Yadav, Dana Al Sulaiman, and Patrick S Doyle.  Tuning the topology of a two-dimensional catenated dna network.  Physical Review Research, 5(1):013141, 2023.
  * [9] Pinyao He, Allard J Katan, Luca Tubiana, Cees Dekker, and Davide Michieletto.  Single-molecule structure and topology of kinetoplast dna networks.  Physical Review X, 13(2):021010, 2023.
  * [10] Junghuei Chen, Carol A Rauch, James H White, Paul T Englund, and Nicholas R Cozzarelli.  The topology of the kinetoplast dna network.  Cell, 80(1):61–69, 1995.
  * [11] Douglas C Barker.  The ultrastructure of kinetoplast dna with particular reference to the interpretation of dark field electron microscopy images of isolated, purified networks.  Micron (1969), 11(1):21–62, 1980.
  * [12] Theresa A Shapiro.  Kinetoplast dna maxicircles: networks within networks.  Proceedings of the National Academy of Sciences, 90(16):7809–7813, 1993.
  * [13] D Lys Guilbride and Paul T Englund.  The replication mechanism of kinetoplast dna networks in several trypanosomatid species.  Journal of cell science, 111(6):675–679, 1998.
  * [14] Coen M KLEISEN, Piet BORST, and Peter J WEIJERS.  The structure of kinetoplast dna: 1. the mini-circles of crithidia luciliae are heterogeneous in base sequence.  European journal of biochemistry, 64(1):141–151, 1976.
  * [15] Steven C van der Marck.  Percolation thresholds and universal formulas.  Physical Review E, 55(2):1514, 1997.
  * [16] L Ibrahim, P Liu, M Klingbeil, Y Diao, and J Arsuaga.  Estimating properties of kinetoplast dna by fragmentation reactions.  Journal of Physics A: Mathematical and Theoretical, 52(3):034001, 2018.
  * [17] Tevin Li, Hao Zhang, Lianzhe Hu, and Fangwei Shao.  Topoisomerase-based preparation and afm imaging of multi-interlocked circular dna.  Bioconjugate Chemistry, 27(3):616–620, 2016.
  * [18] Yuanan Diao, Victor Rodriguez, Michele Klingbeil, and Javier Arsuaga.  Orientation of dna minicircles balances density and topological complexity in kinetoplast dna.  Plos one, 10(6):e0130998, 2015.
  * [19] Davide Michieletto, Davide Marenduzzo, and Enzo Orlandini.  Is the kinetoplast dna a percolating network of linked rings at its critical point?  Physical biology, 12(3):036001, 2015.
  * [20] Donald G Ferschweiler, Ryan Blair, and Alexander R Klotz.  Percolation and dissolution of borromean networks.  Physical Review E, 107(2):024304, 2023.
  * [21] Ronald L Rivest and Jean Vuillemin.  On recognizing graph properties from adjacency matrices.  Theoretical Computer Science, 3(3):371–384, 1976.
  * [22] A Margolina, HJ Herrmann, and D Stauffer.  Size of largest and second largest cluster in random percolation.  Physics Letters A, 93(2):73–75, 1982.
  * [23] Meryl A Spencer and Robert M Ziff.  Honeycomb lattices with defects.  Physical Review E, 93(4):042132, 2016.
  * [24] Björn Åkerman and Eimer Tuite.  Single-and double-strand photocleavage of dna by yo, yoyo and toto.  Nucleic acids research, 24(6):1080–1090, 1996.
  * [25] Fredrik Persson and Jonas O Tegenfeldt.  Dna in nanochannels—directly visualizing genomic information.  Chemical Society Reviews, 39(3):985–999, 2010.
  * [26] Mohammadreza Alizadehheidari, Erik Werner, Charleston Noble, Michaela Reiter-Schad, Lena K Nyberg, Joachim Fritzsche, Bernhard Mehlig, Jonas O Tegenfeldt, Tobias Ambjornsson, Fredrik Persson, et al.  Nanoconfined circular and linear dna: Equilibrium conformations and unfolding kinetics.  Macromolecules, 48(3):871–878, 2015.