# Fault-tolerant error correction for a universal non-Abelian topological
quantum computer at finite temperature

Alexis Schotte<EMAIL_ADDRESS>IBM Quantum, IBM Almaden Research
Center, San Jose, CA 95120, USA  Lander Burgelman  Department of Physics and
Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium  Guanyu Zhu
<EMAIL_ADDRESS>IBM Quantum, IBM Almaden Research Center, San Jose, CA
95120, USA  IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA

###### Abstract

We study fault-tolerant error correction in a quantum memory constructed as a
two-dimensional model of Fibonacci anyons on a torus, in the presence of
thermal noise represented by pair-creation processes and measurement errors.
The correction procedure is based on the cellular automaton decoders
originating in the works of Gács [1] and Harrington [2]. Through numerical
simulations, we observe that this code behaves fault-tolerantly and that
threshold behavior is likely present. Hence, we provide strong evidence for
the existence of a fault-tolerant universal non-Abelian topological quantum
computer.

##  I Introduction

Anyons are emergent quasi-particles that exist in two-dimensional condensed
matter systems and whose exchange statistics generalize that of Bosons and
Fermions. These particles have spurred much interest due to their potential
applications for quantum computation. In particular, it was found that with
certain types of non-Abelian anyons, a universal quantum computation can be
performed by braiding and fusing these particles [3, 4, 5]. An intriguing
benefit of this paradigm is that, due to their topological nature,
computations are intrinsically robust to perturbations at zero temperature. At
non-zero temperature, however, thermal anyonic excitations can corrupt the
computation by performing non-trivial braids with the computational anyons.
Since systems exhibiting anyonic excitations have a spectral gap $\Delta$,
this source of errors can be suppressed to some extent at temperatures
$T\ll\Delta/k_{B}$ as the density of thermal anyons scales as
$e^{-\Delta/k_{B}T}$. Alas, this passive protection does not suffice, because
the presence of thermal anyons is unavoidable at non-zero temperatures when
scaling up the size of computations. Therefore, proficient active error
correction schemes for non-Abelian models are paramount for the realization of
topological quantum computers.

Besides their envisaged use for topological quantum computation, topologically
ordered systems (i.e., those that support anyonic excitations on top of their
ground space) are also of much interest for quantum error correction. In
particular, one of the characteristics of such systems is a robust ground
space degeneracy, which allows one to use their ground space as the code space
of an error correcting code. This realization led to the discovery of
topological quantum error correcting codes, which encode logical quantum
states in topologically ordered states of a system of qudits (typically
arranged on a two-dimensional lattice). Since their discovery in the 90s, most
research has focused exclusively on Abelian topological codes such as the
surface code and the color code [6, 7, 5, 8, 9, 10, 11, 12, 13, 14], which
admit an elegant characterization in terms of the stabilizer formalism [15].
Due to their geometrical locality and high error thresholds, these codes are
considered to be promising candidates for protecting quantum information from
noise in error-corrected quantum computers. One of the drawbacks of Abelian
topological codes, however, is that they do not allow one to execute a
universal set of logical gates in a protected fashion in two dimensions.
Hence, they must be supplemented with additional protocols such as magic state
distillation [16] or code switching to higher-dimensional codes [17, 18] ,
which introduce a large space-time overhead [19]. Alternatively, there exist
non-Abelian topological codes which do not suffer from this inherent
limitation, and are able to perform a universal gate set natively within their
code space in two dimensions [3]. The trade-off is that such codes go beyond
the stabilizer formalism and are therefore very hard to simulate classically.

While active error correction in Abelian anyon models and Abelian topological
codes has been studied extensively, quantum error correction based on non-
Abelian anyon models has not enjoyed the same focus. Nevertheless, important
progress has been made over the last decade, including both analytical proofs
and numerical demonstrations of threshold behavior for various non-Abelian
topological error correcting codes [20, 21, 22, 23, 24]. Moreover, syndrome
extraction circuits for such non-Abelian string-net codes have been developed
in recent years [25, 24]. In addition, state preparation for non-Abelian codes
based on the Kitaev quantum double models via measurements has also been
proposed recently for the experimental implementation on qubit lattices [26,
27], although further development is still needed in the context of fault-
tolerant state preparation. Notably, previous studies in this field already
include codes based on the Fibonacci anyon model, which is universal for
quantum computation [23, 24]. In particular, a quantum memory of qubits
supporting doubled Fibonacci anyonic excitations was found to have a threshold
that lies remarkably close to that of the surface code under similar
assumptions [24].

These results, however, all assume perfect syndrome measurements, which are
topological charge measurements in this context. As we aim to model more
realistic scenarios, we must take faulty measurements into consideration.
Again, much is known in the case of Abelian topological codes [2, 7, 28, 29,
30, 31]. For their non-Abelian counterparts, one key result stands out: in
Ref. [32] a proof was formulated that topological codes based on non-cyclic
anyon models admit a error correction thresholds with faulty topological
charge measurements. While this result is remarkable, non-cyclic anyon models
are not universal for quantum computation, and it remains an open question
whether similar claims can be made for universal models.

In this work, we take a step towards demonstrating that fault-tolerance is
indeed possible for universal non-Abelian topological codes. To this end, we
define a quantum memory constructed as a two-dimensional model of Fibonacci
anyons on a torus. We study active continuous quantum error correction on this
model in the presence of thermal noise represented by pair-creation processes,
and with faulty syndrome measurements. The correction procedure is based on
the cellular automaton decoders originating in the works of Gács [1] and
Harrington [2], and further studied in the context of non-Abelian models in
Ref. [32]. Through numerical simulations, we study how the average memory
lifetime changes with the error rate. The results indicate that this code is
indeed fault-tolerant, which is strong evidence for the existence of fault-
tolerant universal non-Abelian codes.

The structure of this work is as follows. In Sec. II we introduce the
topological Fibonacci code. We then describe the details of the noise model in
Sec. III and introduce the cellular automaton decoder in Sec. IV. We proceed
by giving an outline of the numerical simulations performed in this work in
Sec. V. Finally, we present the corresponding numerical results in Sec. VI and
conclude with a discussion in Sec. VII.

##  II The Fibonacci code

We consider a two-dimensional model comprised of hexagonal tiles laid out on
the surface of a torus. The resulting geometry can be represented as an
$L\times L$ hexagonal lattice with periodic boundary conditions in both
directions (Fig. 2). Each of these hexagonal tiles can contain an excitation
known as a _Fibonacci anyon_.

Anyons are point-like quasi-particle excitations which can be characterized
algebraically in terms of a unitary modular tensor category (UMTC). A thorough
description of anyon models using UMTCs goes beyond the scope of this work,
however, some details are given in Sec. A. For now, it is sufficient to state
that an anyon model specifies a set of anyon labels, also referred to as
particle types, which can fuse according to a specific set of fusion rules.
The Fibonacci anyon model considered in this work contains two labels,
$\mathbf{1}$ and $\tau$, which obey the fusion rules

$\mathbf{1}\times\mathbf{1}=\mathbf{1}\,,\quad\mathbf{1}\times\tau=\tau\times\mathbf{1}=\tau\,,\quad\tau\times\tau=\mathbf{1}+\tau\,.$
(1)

In general, one can associate a vector space to a given set of anyons, where
the basis vectors are labeled by the different ways in which the anyons can
fuse. This fusion space has a topological degeneracy, and can therefore be
used to robustly encode quantum information. In particular, for the Fibonacci
anyon model the anyonic vacuum on a two-dimensional torus has a twofold
degeneracy [33]. Starting from our two-dimensional model, we can therefore
define an error correcting code whose code space is identified with the
anyonic vacuum on the torus and which encodes a single logical qubit. A basis
for this code space can be defined using Wilson line operators along the
homologically non-trivial cycles $x$ and $y$ shown in Fig. 1:

$\begin{array}[]{l}W^{a}_{x}\ket{\mathbf{1}}_{x}=\frac{S_{a\mathbf{1}}}{S_{\mathbf{1}\mathbf{1}}}\ket{\mathbf{1}}_{x}\,,\\\
W^{a}_{x}\ket{\tau}_{x}=\frac{S_{a\tau}}{S_{\mathbf{1}\tau}}\ket{\tau}_{x}\,,\end{array}\qquad
a\in\\{\mathbf{1},\tau\\}\,,$ (2)

We note that a different basis, $\\{\ket{0}_{y},\ket{1}_{y}\\}$, can be
defined analogously by swapping the $x$ and $y$ labels above, where the two
bases are related through the modular $S$ matrix
$S_{ab}=\\!\\!\phantom{a}_{y}\\!\braket{a}{b}_{x}$. For the Fibonacci anyon
model its numerical values are

$S=\dfrac{1}{\sqrt{1+\phi^{2}}}\begin{pmatrix}1&\phi\\\
\phi&-1\end{pmatrix}\,,$ (3)

where $\phi=\dfrac{1+\sqrt{5}}{2}$.

Figure 1: The two homologically non-trivial cycles on a torus.

The action of the mapping class group on the anyonic vacuum then corresponds
to unitary operations on the code space. For the Fibonacci category, any
logical unitary operator can be realized in this way, up to arbitrary
precision [3]. Therefore, the quantum error correcting code defined above
natively supports universal quantum computation.

Errors in this code appear as spurious anyonic excitations, which can corrupt
the encoded information if their world lines between creation and re-
annihilation are topologically non-trivial, i.e., form a non-contractible
cycle 111Note that a pair of Fibonacci anyons can also fuse to a single non-
trivial anyon when one member of the pair has been transported along a non-
trivial cycle. Since the resulting state is no longer in the code space, this
does not constitute a logical operation on the encoded information. However,
one can show that the encoded information is irrevocably lost in case of such
an event [21].. The objective of error correction is then to systematically
remove these spurious excitations, without corrupting the quantum memory in
the process. This correction is performed in an active and continuous manner,
and can be broken down into a series of discrete steps. At each step, a
suitable recovery operation is performed based on a measured list of positions
and types of the excitations, called the error syndrome.

We conclude this section by noting that the numerical simulation of the error-
correction process requires the introduction of some additional manipulations
on fusion states of multiple Fibonacci anyons. As these are technical details
that do not contribute to the intuition of the procedure, we defer their
definition to Sec. A.

##  III Noise model and correctability

Having defined our model and code space, we now turn to the description of the
noise model used in our simulations. We model continuous active error
correction in our Fibonacci code as a sequence of time steps, where each time
step itself consists of three parts: the application of pair-creation noise,
faulty syndrome measurement, and error correction respectively.

At each time step, first, for each edge of the hexagonal lattice graph a pair
of anyons is created across this edge with a probability $p$. Immediately
after each pair creation event, the resulting charge in the two affected tiles
is sampled, effectively collapsing all superpositions of anyonic charge within
each tile to either $\mathbf{1}$ or $\tau$. After the pair creation noise has
been applied, faulty syndrome extraction is simulated by generating a list of
the anyon charge in all tiles, and flipping each outcome individually with a
probability $q$. In addition to the charges that are correctly detected, the
resulting faulty syndrome can contain both “ghost defects” (indicating a non-
trivial charge when none is truly present) and “missing defects” (failing to
report a true non-trivial charge). Finally, this faulty syndrome is passed to
a decoder, introduced in the following section, which then performs a set of
local operations based on the current (and past) syndrome information in an
attempt to move the system back towards the initial state.

After each time step, the current state of the system is copied and it is
checked whether it is still correctable. This is done by passing the copy to a
clustering decoder [22, 23, 24] and simulating a decoding procedure with
perfect syndrome measurements starting from this given initial state. If this
perfect decoding is successful, the memory is considered intact and the
simulation is continued. If perfect decoding is unsuccessful, the memory is
considered corrupted and the simulation is aborted. The memory lifetime is
then defined as the number of time steps after which a perfect clustering
decoder can no longer successfully restore the initial state.

(a)

(b)

Figure 2: (a) Pair-creation events creating anyonic excitations in neighboring
tiles. The dotted ellipse represents a collapse to the total charge of the
anyons it contains. A ghost defect is shown in blue, a missing defect is
highlighted in orange with a cross. (b) The outcome of this noise process
represented on the decoding graph. Note that the missing defect (highlighted
in orange) will (by definition) not be visible in the syndrome.

For a given pair of tiles which share an edge, the process of pair creation
across this edge corresponds to the matrix elements

$\bra{a^{\prime}\\!,b^{\prime}\\!;c^{\prime}}U_{\text{pc}}\ket{a,b;c}=\delta_{c,c^{\prime}}F^{aa1}_{\tau\tau
a^{\prime}}F^{a^{\prime}\\!\tau a}_{b\,c\,b^{\prime}}\,,$ (4)

where we have used the $F$-symbols of the Fibonacci category, given in (10).
Here, $\ket{a,b;c}$ represents the state where the affected tiles have anyon
charges $a$ and $b$, respectively, with total charge $c$. This then defines
the probability distribution according to which outcomes $a^{\prime}$ and
$b^{\prime}$ are sampled. Since our noise model does not allow any
superposition in the anyonic charge of individual tiles, it should be
considered semi-classical rather than fully quantum-mechanical. Note, however,
that this does not render our model completely classical. Indeed,
superpositions in the total charge $c$ of the affected tiles are an inherent
part of the state evolution that cannot be captured faithfully by any
classical probabilistic process. Furthermore, while the extreme decoherence
assumption for the anyon charge in individual tiles greatly simplifies the
numerical simulation outlined in this work, it was argued in Ref. [22] that
this decoherence is unlikely to have any tangible influence on the observed
memory lifetimes, as the essential topological nature of the noise processes
is still captured correctly.

We note that this type of noise can be understood as originating from the
connection to a thermal bath with inverse temperature $\beta=1/(k_{B}T)$
determined by the error rate $p$ through the relation

$\dfrac{p}{1-p}=\mathrm{e}^{-\beta\Delta}\,.$ (5)

Here, $\Delta$ represents the energy required to create a pair of anyonic
excitations and place them in neighboring tiles.

To conclude this section, we emphasize that in the case of non-Abelian error
correction, even decoding with perfect syndrome measurements is still an
inherently stochastic procedure due to the indeterminacy of anyonic charge
measurements. This means that perfect decoding can sometimes either be
successful or unsuccessful even starting from the same initial state. Our
definition of the memory lifetime therefore simply corresponds to a
statistical estimate of the actual memory lifetime. Furthermore, it is not
known which decoder is optimal for the Fibonacci code. Hence, the choice for
the clustering decoder to verify the correctability of states is, in a way, an
arbitrary one. This choice, however, is motivated by the recent discovery that
the clustering decoder yields high thresholds for a related error correcting
code exhibiting doubled Fibonacci anyonic excitations, and performed
significantly better in this context than decoders based on a perfect matching
strategy [24]. In any case, one should keep in mind that the memory lifetime
as defined above, does not represent the true memory lifetime. Instead the
sub-optimal verification process entails that it merely provides us with a
lower bound on the true value.

##  IV Harrington’s cellular automaton decoder

The model described above is paired with a decoder which is a straight-forward
adaptation of the cellular automaton decoder introduced in Ref. [2].
Previously, this decoder has also been used for a similar phenomenological
model of Ising anyons in Ref. [32], where the existence of an error correction
threshold was proven analytically. At each time step during the error
correction simulation, based on the reported measurement outcomes in the
faulty syndrome, the decoding algorithm will apply local transition rules to
fuse neighboring anyons or to move anyons to neighboring tiles.

Intuitively these transition rules work as follows. The lattice is divided
into square colonies of size $Q\times Q$. At each time step, the transition
rules will attempt to fuse neighboring non-trivial anyons, as observed in the
faulty syndrome. If a non-trivial anyon has no neighbors, the transition rules
will move it to the center of its colony. At larger timescales, higher-level
transition rules are applied on a renormalized lattice where anyons located at
colony centers will be fused with anyons at neighboring colony centers, or
moved toward the center of their respective super-colonies, which consist of
$Q\times Q$ colonies. This renormalization scheme is then continued at higher
levels until eventually the $Q^{n}\times Q^{n}$ super-colony covers the entire
lattice for some integer $n$. To ensure the latter is possible, we will always
assume that the linear lattice size satisfies $L=Q^{n}$ for some integer $n$.
An example of these processes is shown in Fig. 4.

To describe the action of the decoding algorithm more precisely, we will
define its action at different renormalization levels $k$. The level-0
transition rules are those already discussed above and are applied at every
time step based on the reported faulty syndrome obtained from the most recent
round of faulty measurements. The transition rules are applied to one location
at a time and take into consideration only the anyon content of that site and
of its eight neighbors. A detailed definition of these rules is given in App.
C. When an anyon is moved from a site $l$ to a neighboring site $l^{\prime}$,
the (true) anyon content of site $l$ is fused with that of site $l^{\prime}$
and the resulting charge is placed on site $l^{\prime}$ while the charge of
site $l$ is restored to the vacuum. This happens irrespective of whether or
not the syndrome for both sites was correct. Hence, when the decoder attempts
to move a ghost defect (a trivial charge misidentified as a non-trivial one)
to a neighboring site, this process does not create additional excitations.
This does not, however, mean that mistaking a trivial charge for a non-trivial
one has no negative consequences. Indeed, these wrong syndromes may cause the
decoder to stretch out existing errors.

The level-1 transition rules are not applied in every time step, but only when
$t$ is a multiple of a parameter called the _work period_ , which we will
denote by $U$. We require that $U=b^{2}$ for some positive integer $b$. One
should think of $U$ as the time scale at which a coarse-graining is performed.
Level-1 transition rules act at on a coarse-grained lattice where the sites
correspond to the centers of the level-0 colonies, and these are grouped into
level-1 colonies of size $Q^{2}\times Q^{2}$. Hence, the actions determined by
the level-1 transition rules involve pairs of level-0 colony centers separated
by a distance $Q$. An example of such a move is provided in Fig. 4(d). The
transition rules themselves are nearly identical to the level-0 rules, but are
based on two sets of level-1 syndromes $s_{1,c}$ and $s_{1,n}$ (defined below)
rather than one. For a site $l$ (which is a level-0 colony center), the
transition rules use $s_{1,c}(l)$ as the anyon content of site, while the
anyon content of its neighbors (that is, the neighboring level-0 colony
centers) is taken to be $s_{1,n}(l^{\prime})$.

The definitions of the level-1 syndromes $s_{1,c}$ and $s_{1,n}$ require a
pair of variables $f_{c},f_{n}\in[0,1]$. Intuitively these variables serve as
detection thresholds for the level-1 syndromes by determining the fraction of
measurements that must return a non-trivial outcome at a site before it
qualifies as a non-trivial level-1 syndrome. The proper definition, however,
is slightly more complicated and uses a coarse-grained counting method. Below,
we give the precise definition of $s_{1,c}$, the definition of $s_{1,n}$ is
entirely analogous (using $f_{n}$ instead of $f_{c}$). We start by dividing
the work period $U=b^{2}$, into $b$ intervals of $b$ time steps each. For each
of these intervals, we say a non-trivial syndrome is present at a colony
center $l$ if a non-trivial charge was reported there for at least $f_{c}b$ of
the $b$ time steps in the interval. When at least $f_{c}b$ of the $b$
intervals have a non-trivial syndrome, $s_{1,c}(l)$ is set to one. A visual
example of this coarse-grained counting procedure is shown in Fig. 3.

Figure 3: A visual example of the coarse-grained counting procedure to
determine the level-1 syndrome for a level-0 colony center. The row of dots
represent $U$ time steps, divided in $b$ intervals of size $b$. The time steps
during which a non-trivial measurement outcome was reported are indicated by
the colored dots. The crosses in the second row indicate in which intervals
the fraction of non-trivial measurement outcomes is equal to or higher than
$f_{c}$.

The motivation for using two types of syndromes for $k>0$ is as follows.
Suppose that an error spans across two neighboring colonies, which we will
label $\rho$ and $\rho^{\prime}$. The level-0 transition rules will transport
all resulting anyons to the respective colony centers, where they can now be
acted upon by level-1 transition rules at the end of the work period. Imagine
that a non-trivial anyon is now present at both colony centers. When
considering the level-1 transition rules acting on $\rho$, there are four
possible scenarios for the syndromes $s_{1,c}(\rho)$ and
$s_{1,n}(\rho^{\prime})$. In case $s_{1,c}(\rho)=0$, the transition rules act
trivially on $\rho$. If both $s_{1,c}(\rho)=1$ and $s_{1,n}(\rho^{\prime})=1$
then the transition rules will be applied correctly and the anyons will be
fused. However, if $s_{1,c}(\rho)=1$ but $s_{1,n}(\rho^{\prime})=0$, the
transition rules may move the anyon in $\rho$ away from $\rho^{\prime}$,
thereby increasing the weight of the error. Hence, it is desirable to set
$f_{c}>f_{n}$ to decrease the odds that when a level-k syndrome reports an
non-trivial anyon at a colony center, the level-k syndrome for its neighbors
are false negatives. We must be careful not to set $f_{c}$ too high or $f_{n}$
too low, however. If we choose $f_{c}$ to high, low-weight errors could cause
$s_{1,c}$ to never report any non-trivial charges, delaying any necessary
corrections. Similarly, setting $f_{n}$ too low will result in low-weight
error triggering many false positives for $s_{1,n}$, which can cause the
decoder to make wrong decisions.

Level-$k$ transition rules are applied when $t\mod U^{k}=0$. They operate on a
renormalized lattice that uses the centers of level-$(k-1)$ colonies as sites,
and groups these into level-$k$ colonies of size $Q^{k}\times Q^{k}$. The
level-$k$ syndromes $s_{k,c}$ and $s_{k,n}$ are determined by the coarse-
grained counting method described above, using $b^{k}$ intervals of $b^{k}$
time steps each. For linear system size $L$, k ranges from 0 to
$k_{\text{max}}=\log_{Q}(L)$.

It is important to note that non-Abelian anyons do not allow for instantaneous
moves. Indeed, while one can construct a unitary string operator for Abelian
anyons, no such operator can be constructed for the non-Abelian case. This
discrepancy can be traced back to the fact that fusion outcomes are non-
deterministic for non-Abelian anyons, implying it is not possible to move an
non-Abelian anyon by annihilating it with one member of a particle-
antiparticle pair (as is done in e.g., the surface code).

Therefore, the actions determined by level-$k$ transition rules, for $k>0$
cannot be applied withing a single time step. Instead, they will be broken up
into a sequence of moves involving only pairs of neighboring sites which will
be applied in $Q^{k}$ consecutive time steps. We further limit the model by
requiring that the number recovery operations affecting a single tile in the
lattice (or site in the decoding graph), is no greater than one in each time
step. This allows all recovery operations applied in one time step to be
performed in parallel. Hence, we must define a hierarchy determining which
actions (moves or fusions between neighboring tiles) get prioritized based on
the renormalization level from which they originated. In our case, it was
opted to always prioritize correction processes from the highest
renormalization level 222Note that if one were to prioritize the level-0
corrections, higher-level correction could never be completed, as they would
be undone immediately after their first action is applied.

It was argued in [32] that the prohibition of instantaneous corrections would
likely not influence the threshold behavior other than slightly lowering the
memory lifetimes relative to a hypothetical case where this restriction is
dropped. We explicitly verify this claim for our Fibonacci model below in Sec.
VI.

(a)

(b)

(c)

(d)

Figure 4: Illustration of the transition rules on the decoding graph. The gray
disks represent non-trivial syndromes, and the blue arrows represent the
actions suggested by the decoder. The blue dotted lines represent the $3\times
3$ colonies. (a-c) show a sequence of level-0 transition rules and possible
outcomes of those actions. In (d) non-trivial anyons have been transported to
two neighboring colony centers, the blue arrow represent a level-1 transition
which could be applied at the end of the work period.

(a)

(b)

(c)

Figure 5: (a) Level-0 colonies of size $Q\times Q$. (b) Level-1 colonies
defined as $Q\times Q$ level-0 colonies. (c) Renormalized lattice used for the
level-1 transition rules.

##  V Outline of the simulation

The goal of this work is to numerically determine a fault-tolerant error
threshold for the error correcting code defined in Sec. II with pair-creation
noise and measurement noise as outlined in Sec. III, and with the cellular
automaton decoder introduced in Sec. IV. This is achieved by performing Monte-
Carlo simulations to determine the average memory lifetime for a range of
system sizes and error rates. These results then allow one to estimate the
value of the error threshold.

A single Monte-Carlo sample (with some fixed values for the noise strength $p$
and the measurement error rate $q$) is obtained as follows. First, the state
of the system is initialized as a ground state (i.e.: containing no anyons).
Then a sequence of time steps is performed consisting of the application of
pair-creation noise with rate $p$, a round of faulty syndrome measurements
with error probability $q$, and finally a sequence of recovery operations. At
the end of each time step, it is verified whether or not the state is
considered correctable, according to the criteria specified in Sec. III. This
sequence of time steps is continued until one of the following three outcomes
occurs: (1) The largest connected group of anyons grows too large, rendering
its classical simulation intractable 333Note that such cases are likely to
correspond configurations in which the initial state cannot be recovered.; (2)
A noise process or recovery operation induces a logical error by fusing a pair
of anyons along a path that forms a non-contractible loop when combined with
their fusion tree; (3) The verification procedure at the end of a time step
fails. The memory lifetime is then set as the number of time steps that were
completed. The course of a single Monte-Carlo sample in the simulation is
summarized as pseudo-code in Alg. 1.

Algorithm 1 Numerical simulation

initialize state

$t=0$

while correctable with clustering decoder $\And$ no logical errors made do

$t\leftarrow t+1$

apply pair-creation noise

perform faulty measurements

for $k=0:k_{\text{max}}$ do

if $t\\!\mod U^{k}=0$ then

update level-$k$ syndromes

apply level-$k$ transition rules

end if

end for

end while

memory lifetime = t

##  VI Numerical results

The Monte Carlo simulation described above were performed for various system
sizes with $p=q$. The following parameters were used:

$\displaystyle Q$ $\displaystyle=3\,,$ $\displaystyle b$ $\displaystyle=7\,,$
$\displaystyle F_{c}$ $\displaystyle=0.7\,,$ $\displaystyle F_{n}$
$\displaystyle=0.2\,.$

The resulting average memory lifetimes for $L=3$, $L=9$ and $L=27$ are shown
below in Fig. 6(a).

(a) Average memory lifetime in function of the error strength, with $p=q$, for
various system sizes. Each data point represents the average over 1000 Monte
Carlo samples. The blue line shows the coherence time of a single physical
qubit. The average memory lifetimes for $p\leq 10^{-3}$ were fitted to a
function of the form $f(p)\sim p^{-a}$. The results for $L=3$, $L=9$ and
$L=27$ are shown as the green, yellow and red lines respectively.

(b) Average lifetime in function of the error strength with $p=q$ for various
system sizes and (unphysical) _instantaneous_ corrections. Each data point
represents the average over 1000 Monte Carlo samples. The blue line shows the
coherence time of a single physical qubit.

Figure 6:

These results clearly indicate that the code presented in this work is indeed
fault-tolerant. Furthermore, while the current data is not sufficient to
demonstrate a clear-cut fault-tolerant threshold, it still exhibits threshold
behavior and is remarkably similar to the results previously obtained for the
toric code [2] and the Ising topological code [32]. We estimate that the
fault-tolerant threshold for the Fibonacci topological with pair-creation
noise and measurement noise lies between $p=10^{-4}$ and $p=5\cdot 10^{-4}$,
which corresponds to an inverse temperature between $\beta=9.2/\Delta$ and
$\beta=7.6/\Delta$. This is comparable to the threshold found for the Ising
topological code [32], and only one order of magnitude below that for the
surface code under similar circumstances [2]. For physical error rates near
$p=q=10^{-4}$, corresponding to a temperature $1/\beta$ one order of magnitude
below the spectral gap, a code of linear size $L=27$ yields logical error
rates of the order $10^{-8}$.

A second round of simulations was performed to determine average memory
lifetimes with the assumption that all corrections happen instantaneously.
While this is akin to the Abelian topological codes, where distant anyons can
be fused using unitary string-operators, this scenario is unphysical for non-
Abelian anyons as they do not admit unitary string-operators. Nevertheless, it
is worth studying to which extend the results in Fig. 6(a) are influenced by
the restriction to non-instantaneous recovery operations. In Ref. [32] it was
conjectured that allowing instantaneous corrections does not significantly
change the qualitative behavior of the average memory lifetimes as a function
of the error rate, but mostly just increases the memory lifetimes. Our
results, shown in Fig. 6(b), confirm this hypothesis.

##  VII Discussion and outlook

The results presented in this work demonstrate that fault-tolerant error
correction is possible for non-Abelian topological quantum error correcting
codes supporting a universal logical gate set within their code space. For a
code consisting of Fibonacci anyons in hexagonal tiles on a two-dimensional
torus, subjected to pair creation noise and measurement noise, we demonstrated
that the cellular automaton decoder detailed in this work is fault-tolerant.
In particular, for physical error rates $p\leq 10^{-3}$, it was found that the
logical memory lifetime surpasses the physical coherence time for all system
sizes. When interpreting the pair-creation noise as resulting from a non-zero
temperature, this pseudo-threshold corresponds to an inverse temperature
$\beta=6.9/\Delta$, where $\Delta$ is the energy required to create a pair of
Fibonacci anyons. Furthermore, our results suggest that this code admits a
fault-tolerant quantum error correction threshold around $p=10^{-4}$, or
$\beta=9.2/\Delta$, which is similar to the fault-tolerant threshold found for
the Ising topological code [32].

Several future research directions present themselves. First, more research on
a possible fault-tolerant threshold is necessary. Wile the numeric results
presented in this work provide a strong indication that a fault-tolerant error
correction threshold exists, they do not conclusively prove its existence, nor
do they provide a precise estimate of its value. Hence, an important open
problem is the formulation of a mathematical proof of its existence. Such
proofs were previously formulated for the toric code [2] and for non-cyclic
non-Abelian anyon models such as in the Ising topological code [32]. Due to
the cyclic nature of Fibonacci anyons (or any universal anyon model), however,
the existing proofs are not sufficient.

Second, it would be interesting to study different decoders in an identical
setting. This includes both different cellular-automaton decoders such as
those in Refs. [34, 31], as well as new decoders tailored to the Fibonacci
topological code.

Third, while this work demonstrates that the Fibonacci topological code can be
operated fault-tolerantly as a quantum memory, results regarding its use for
fault-tolerant quantum computing are still lacking. We envisage that fault-
tolerant topological quantum computing at non-zero temperatures could be
achieved by combining the code and decoding procedure presented in this work
with the scheme for performing Dehn twists presented in Refs. [35, 36, 37].
Alternatively, one can also perform transversal logical gates in a folded
Fibonacci code [38].

Finally, it would be of great interest to expand the current results to
microscopic models for non-Abelian topological quantum error correction, such
as the Fibonacci Turaev-Viro code [24].

## Acknowledgments

The authors would like to thank Guillaume Dauphinais and Jim Harrington for
enlightening discussions on the cellular automaton decoder. The computational
resources (Stevin Supercomputer Infrastructure) and services used in this work
were provided by the Flemish Supercomputer Center (VSC), funded by Ghent
University, the Research Foundation Flanders (FWO), and the Flemish
Government. AS was supported by a fellowship of the Belgian American
Educational Foundation. LB was supported by a PhD fellowship from the FWO. GZ
was supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Co-design Center for Quantum
Advantage (C2QA) under contract number DE-SC0012704.

##  Appendix A Fibonacci anyons

Below, we give with a brief overview of the topological aspects of our model.
A thorough exposition of the theory of anyon models [39, 40, 41, 42] is beyond
the scope of this work, and we restrict to a basic description of the aspects
of the Fibonacci model that are required for the specific purpose of our
simulations. We refer to Ref. [33] for an in-depth discussion of anyonic
fusion states on the torus.

The Fibonacci anyon model has two particle types, $\mathbf{1}$ and $\tau$,
that satisfy the fusion rules

$\mathbf{1}\times\mathbf{1}=\mathbf{1}\,,\quad\mathbf{1}\times\tau=\tau\times\mathbf{1}=\tau\,,\quad\tau\times\tau=\mathbf{1}+\tau\,.$
(6)

It is a _non-Abelian_ anyon model, as the fusion of two $\tau$-anyons can
yield two distinct outcomes. In this case, the fusion space $V_{ab}^{c}$
associated to the fusion of anyons $a$ and $b$ to $c$ is one-dimensional, and
is spanned by the state vector $\ket{a,b;c}$ which we will represent
graphically as

$\ket{a,b;c}\rightarrow\raisebox{-19.91684pt}{\includegraphics[scale={.42}]{fig/fusion_state}}\,.$
(7)

When fusing several anyons, their total charge is a collective property of the
anyons that does not depend on the specific order in which they are fused.
Mathematically, this is expressed through the associativity of the fusion
rules,

$(a\times b)\times c=a\times(b\times c)\,.$ (8)

If we consider the case of three anyons $a$, $b$ and $c$ that fuse to a total
charge $d$ then this fusion may be carried out in two distinct ways, implying
the existence of two equivalent decompositions of the associated fusion space
$V_{abc}^{d}$ in terms of fusion states (7). These equivalent decompositions
are related through a unitary transformation called an $F$-move, which is
represented graphically as

$\raisebox{-25.6073pt}{\includegraphics[scale={.42}]{fig/F-move_comp_1.pdf}}\;=\sum_{f}F^{abe}_{cdf}\;\raisebox{-25.6073pt}{\includegraphics[scale={.42}]{fig/F-move_comp_2.pdf}}\,.$
(9)

The coefficients $F^{abe}_{cdf}$ in this expression are called _$F$ -symbols_
of the anyon model. For the Fibonacci model they are given by

$F^{\tau\tau 1}_{\tau\tau
1}=\frac{1}{\phi}\,,\quad\;F^{\tau\tau\tau}_{\tau\tau 1}=F^{\tau\tau
1}_{\tau\tau\tau}=\frac{1}{\sqrt{\phi}}\,,\quad\;F^{\tau\tau\tau}_{\tau\tau\tau}=-\frac{1}{\phi}\,,$
(10)

where all other $F$-symbols consistent with the fusion rules (6) are equal to
1, and 0 otherwise.

In addition to this recoupling one can also consider the exchange or
_braiding_ of pairs of anyons, which preserves their total charge. At the
level of the fusion space, such an exchange corresponds to a basis
transformation to a basis associated to a different linear ordering of the
anyons444As opposed to the more conventional view of braiding as an active
transformation that maps between different fusion spaces, we have opted for
the equivalent framework of braiding as a passive basis transformation, as the
latter is more appropriate in view of our specific model and numerical
simulations.. Within $V_{ab}^{c}$ there are two such possible basis
transformations,

$\raisebox{-19.91684pt}{\includegraphics[scale={.42}]{fig/fusion_state}}=R^{ab}_{c}\raisebox{-19.91684pt}{\includegraphics[scale={.42}]{fig/swap_cw.pdf}}=\left(R^{ba}_{c}\right)^{*}\raisebox{-19.91684pt}{\includegraphics[scale={.42}]{fig/swap_ccw.pdf}}\,,$
(11)

which we will refer to as a _clockwise_ and a _counterclockwise swap_
respectively. For the Fibonacci model the $R$-symbols appearing in these
expressions are given by

$R^{\tau\tau}_{1}=e^{\frac{4\pi i}{5}}\,,\qquad
R^{\tau\tau}_{\tau}=e^{-\frac{3\pi i}{5}}\,,$ (12)

where all other $R^{ab}_{c}$ allowed by the fusion rules are equal to 1, and 0
otherwise.

As we are dealing with a system that allows for an extensive amount of anyonic
excitations, we will be interested in fusion states of many anyons,
$a_{1},a_{2},...,a_{n}$, with some total charge $c$. This gives rise to an
exponentially large topological Hilbert space $V_{a_{1}a_{2}\cdots a_{n}}^{c}$
spanned basis states of the form

$\raisebox{-11.38092pt}{\includegraphics[scale={.42}]{fig/standard_basis_state}}\;.$
(13)

When dealing with anyonic fusion states on surfaces of higher genus, one must
take into account additional degrees of freedom in these states that are
related to the anyonic charge that runs along non-contractible cycles. On a
torus, this results in two distinct descriptions of fusion states, which are
related through a basis change. These are known as the _inside_ and _outside_
bases, and are depicted in Fig. 7. A detailed discussion can be found in Ref.
[33]. We will simply refer to these additional degrees of freedom as the
_handle labels_ of the state.

(a)

(b)

Figure 7: Two possible basis choices for anyons on a torus: a) the _inside_
basis, b) the _outside_ basis.

For our current purpose, we won’t need the full description of said handle
labels. It is sufficient for us to pick one basis, and subsequently set the
total charge of all anyons to the vacuum. We are left with only a single label
($\mathbf{1}$ or $\tau$) representing the anyonic charge flowing along the
non-contractible cycle associated with our basis choice. This is precisely the
origin of the twofold degeneracy of the anyonic vacuum on the torus, which we
have taken to be our code space.

In the this work we will always start from the code state corresponding to a
trivial handle label. As a change in handle label at any point during the
error correction procedure must be the consequence of a topologically non-
trivial process which constitutes a logical error, any simulation is aborted
at the occurrence of such an event, meaning that all handle labels can be
safely ignored in the remainder of our discussion.

A basic functionality required for the simulation of the error correction
procedure is the ability to correctly sample a measurement of the total charge
of a given set of anyons. Starting from a given fusion state, this can be
achieved by first transforming to a basis in which the relevant anyons are
fused sequentially. For a system of many anyons these reordering basis
transformations are obtained by combining Eqs. (9) and (11), giving rise to
the clockwise swap

$\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/cw_braid_diagram_initial.pdf}}\\\
=\sum_{b_{2}^{\prime}}B^{b_{1}a_{j}b_{2}}_{a_{j+1}b_{3}b_{2}^{\prime}}\;\;\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/cw_braid_diagram_crossing.pdf}}\;,$
(14)

and the counterclockwise swap

$\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/cw_braid_diagram_initial.pdf}}\\\
=\sum_{b_{2}^{\prime}}\left(B^{b_{1}a_{j}b_{2}}_{a_{j+1}b_{3}b_{2}^{\prime}}\right)^{*}\;\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/ccw_braid_diagram_crossing.pdf}}\,,$
(15)

where

$B^{b_{1}a_{j}b_{2}}_{a_{j+1}b_{3}b_{2}^{\prime}}=\sum_{c}F^{a_{j+1}a_{j}c}_{b_{3}b_{1}b_{2}^{\prime}}R^{a_{j}a_{j+1}}_{c}F^{b_{1}a_{j}b_{2}}_{a_{j+1}b_{3}c}\;.$
(16)

By performing a certain set of these basis transformations, the fusion order
can always be made consistent with the group of anyons of which we want to
measure the total charge. Subsequently, the fusion state is recoupled such
that the relevant group of anyons is connected to the rest of the state by a
single edge $c$. For a charge measurement of a pair of anyons this recoupling
takes the form

$\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/cw_braid_diagram_initial.pdf}}=\sum_{c}F^{b_{1}a_{j}b_{2}}_{a_{j+1}b_{3}c}\;\raisebox{-14.22636pt}{\includegraphics[scale={.38}]{fig/diagram_F-
move.pdf}}\,,$ (17)

and charge measurements of larger groups of anyons simply require consecutive
applications of the recoupling identity (9). Finally, the charge outcome is
obtained sampling the total charge $c$ from the probability distribution
corresponding to the resulting superposition of fusion states.

##  Appendix B Classical simulatbility

It is well known that Fibonacci anyons are universal for quantum computation
[3]. One might therefore be tempted to conclude that the classical simulation
of the topological Fibonacci code described in Sec. II with pair-creation
noise is unlikely to succeed. However, as was noted in Ref. [23], the
simulation of noise and error correction processes does not require the
simulation of _general_ anyon dynamics. In particular, individual noise
processes create distinct connected groups of anyons with vacuum total charge
(or extend such existing groups). These groups correspond to anyons that have
interacted at some point during their lifetime, and must thus only be merged
whenever a noise or error correction process involves two members from
disconnected groups. Since each connected group has a trivial total charge,
braiding between disconnected groups is trivial. Hence, the total fusion space
factorizes into a tensor product of fusion spaces of individual connected
groups, and we are only required to simulate anyon dynamics within each of
these groups separately. This factorization of the fusion space is illustrated
in Fig. 8.

(a)

(b)

(c)

Figure 8: (a) A set of noise processes. (b) A generic state in the full fusion
space of all anyons created by the noise processes involves superpositions in
the labels $x$ and $y$. (c) The factorized Hilbert space which is sufficient
to represent the state.

The creation and subsequent merging of disconnected groups of anyons by noise
and recovery processes can be thought of as a kind of percolation process.
Hence, below the percolation threshold, one expects that the size of the
largest connected group scales as $O(\log(L))$ (with variance $O(1)$), where
$L$ is the linear system size [43]. As this is a probabilistic statement,
there will be instances where the largest connected group has a size larger
than $O(\log(L)$, but the probability of such events is suppressed
exponentially with the system size $L$. This logarithmic scaling of the
largest cluster size $s=O(\log(L))$ counters the exponential scaling of the
dimension of the fusion space $d=O(\exp(s))$ for individual connected groups.
Therefore, the fusion spaces of individual connect groups will have dimension
$\text{dim}=O(\text{poly}(L))$, meaning that the dynamics within connected
groups can be simulated efficiently.

Exploiting the tensor product structure within the total fusion space,
requires the use of basis for the anyonic fusion space which reflects this
structure and can be updated dynamically to keep track of noise and recovery
processes. This is best achieved by using the framework of _curve diagrams_ ,
which were introduced in Ref. [23] (also see Ref. [44] for a more rigorous
treatment using the language of modular functors), and also discussed
extensively in Ref. [24]. Since these are merely a technical tool for keeping
track of the most appropriate basis during the numerical simulations, we will
refrain from discussing them here. Interested readers are referred to the
aforementioned references for details.

##  Appendix C Transition rules for $Q=3$

Below, we give a full definition of the transition rules for $Q=3$. It is
possible to define more general transition rules that apply for any colony
size, examples of such rules can be found in Refs. [2] and [32]. However,
since the numerical simulations performed in this work were performed to $Q=3$
we have taken the freedom to tailor the transition rules to this case
specifically.

  * •

North-West
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(1,0))\neq 0$, move south;
else if $s_{k,n}(\rho+(1,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,1))\neq 0$, do nothing;
else move south;

  * •

North
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,-1))\neq 0$, do nothing;
else move south;

  * •

North-East
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(-1,1))\neq 0$, move north-east;
else if $s_{k,n}(\rho+(1,1))\neq 0$, move south;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, move west;
else if $s_{k,n}(\rho+(1,0))\neq 0$, move south;
else if $s_{k,n}(\rho+(-1,-1))\neq 0$, do nothing;
else move south-west;

  * •

East
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(-1,1))\neq 0$, move north-east;
else if $s_{k,n}(\rho+(1,1))\neq 0$, move south;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(1,0))\neq 0$, do nothing;
else move west;

  * •

South-East
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(1,0))\neq 0$, move south;
else if $s_{k,n}(\rho+(-1,1))\neq 0$, move north-east;
else if $s_{k,n}(\rho+(1,-1))\neq 0$, move south-west;
else if $s_{k,n}(\rho+(1,1))\neq 0$, move south;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, move west;
else move north;

  * •

South
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(1,0))\neq 0$, move south;
else if $s_{k,n}(\rho+(1,-1))\neq 0$, move south-west;
else if $s_{k,n}(\rho+(1,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(0,1))\neq 0$, do nothing;
else move north;

  * •

South-West
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(1,0))\neq 0$, move south;
else if $s_{k,n}(\rho+(1,-1))\neq 0$, move south-west;
else if $s_{k,n}(\rho+(1,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, move north;
else if $s_{k,n}(\rho+(0,1))\neq 0$, move east;
else if $s_{k,n}(\rho+(-1,-1))\neq 0$, do nothing;
else move north-east;

  * •

West
if $s_{k,c}(\rho)=0$, do nothing;
else if $s_{k,n}(\rho+(0,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(1,0))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(1,-1))\neq 0$, do nothing;
else if $s_{k,n}(\rho+(-1,-1))\neq 0$, do nothing;
else move east;

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# Measurement of proton-induced radiation in animal tissue††thanks: Presented
at the XXXV Mazurian Lakes Conference on Physics, Piaski, Poland, September
3-9, 2017

P. Sękowski  I. Skwira-Chalot  T. Matulewicz  Faculty of Physics, University
of Warsaw, PL-02093, Warsaw, Poland

###### Abstract

Hadron therapy, because of the dosimetric and radiobiological advantages, is
more and more often used in tumour treatment. This treatment method leads also
to the radioactive effects induced by energetic protons on nuclei. Nuclear
reactions may lead to the production of radioactive isotopes. In the present
experiment, two animal (human-like) tissue samples were irradiated with 60 MeV
protons. Gamma-ray spectroscopy and lifetime measurements allowed identifying
isotopes produced during the irradiation, e.g. 18F and 34mCl.

10.5506/APhysPolB.49.681

##  1 Introduction

The present project is motivated by insufficient knowledge about long-lived
radioisotopes, which can be produced during proton therapy [1, 2]. Except for
the high linear transfer of energy, the efficiency of particle therapy can
also be augmented by induced radioactivity. During radioactive decays,
different particles (which deposit energy in surrounding tissues) are emitted
and synergistic effect can occur. The efficiency of tumour cell killing by
mixed radiation is higher than that for a separated radiation.

The main goal of the project is an appraisal of dose from the induced
radioactivity deposited in irradiated and surrounding tissues. Gamma-ray
spectroscopy and lifetime measurements can be used to determine the amount of
isotopes produced during irradiation. To provide the consistency of the
result, the Geant4/GATE [3, 4] simulations are used.

##  2 Materials and Methods

The experiment was performed at the Institute of Nuclear Physics of the Polish
Academy of Sciences in Cracow. The proton beam accelerated to 60 MeV (proton
energy for eye therapy) was provided by the AIC-144 isochronous cyclotron and
the samples were irradiated with doses in the range from 30 Gy to 500 Gy. To
achieve a homogeneous distribution of the dose in the sample, a technique
called Spread Out of the Bragg Peak [5] was used.

Liver and bone samples were irradiated. These samples are composed of not only
light nuclides like hydrogen, carbon or oxygen, but also of heavier ones, like
potassium or iron. Furthermore, those tissues have an ability to accumulate
much heavier elements. The pig liver was chosen because of its composition,
which is similar to the human one, and its easy availability. The samples of
bone were prepared from a beef meal with a few additional drops of water.
Liver and bone samples were frozen using liquid nitrogen so that they kept
their shape during irradiation by a horizontal beam.

##  3 Results

The energy spectrum of gamma rays emitted by the beef bone irradiated with the
dose equal to 250 Gy is presented in Fig. 1. The spectrum was measured using a
HPGe detector. There are several notable peaks: the 511 keV $\beta^{+}$
annihilation peak, 147 keV, 1157 keV and 2127 keV. Except for the 1157 keV
line $\left({}^{44}\textrm{Sc}\right)$, they originate from 34mCl [6].

Figure 1: Gamma-ray energy spectrum of a bone irradiated with a dose of 250
Gy, measured with a HPGe detector placed in a low-background lead shield. The
measurement started 2 hours after irradiation and lasted for 2 minutes.

Fig. 2 presents the gamma-ray energy spectrum of an irradiated pig liver
measured using a LaBr3 scintillator detector. There are three notable peaks:
the annihilation peak, 683 keV and 1460 keV. Three main sources of the
annihilation peak are $\beta^{+}$ decays of 11C, 13N and 18F, which is
confirmed by the time spectrum exhibiting the three decay constants of those
isotopes. The line at the energy of 1460 keV originates from the natural
background (40K). The peak with energy around 680 keV has no confirmed origin.
The most probable source of this $\gamma$-ray line is 204At because of the
energy and intensity of the gamma-ray line and a similar half-life. For the
further calculation, it was assumed that this isotope is the source of the
observed radiation.

Figure 2:  Gamma-ray energy spectrum of liver irradiated with a dose of 500 Gy
measured with a LaBr3 detector. The measurement started 10 minutes after
irradiation and lasted for 100 seconds.

To estimate the dose from proton-induced radioactive isotopes, Monte Carlo
simulations were performed using the Geant4/GATE package [3, 4]. In the
simulation, the radioactive isotopes were located in the centre of a water
sphere of 4 cm diameter. Fig. 3 presents examples of spatial dose
distributions obtained for two isotopes, 11C (left) and 34mCl (right).

Figure 3: Spatial projections of dose distribution in water from point-like
sources of 11C (left) and 34mCl (right).

In order to calculate the dose delivered to the surrounding tissue from the
observed radioisotopes, the irradiated sample volume and the total received
dose were taken into account (see Tab. 1).

Table 1: Dose from notable radioisotopes. Radioisotope | Dose [Gyisotope/Gytherapy cm3] | Tissue
---|---|---
11C | 8.7 $\cdot$10-9 | Liver
13N | 1.9 $\cdot$10-9 | Liver
18F | 4.3 $\cdot$10-11 | Liver
34mCl | 3.9 $\cdot$10-9 | Bone
204At | 7.8 $\cdot$10-10 | Liver

##  4 Conclusions

Based on the presented results, there is no indication that the induced
radioactivity, created during eye proton therapy, changes significantly the
global therapeutic effects. Some curious gamma-ray lines were observed, for
example that at 683 keV, that should be studied further. In order to test the
local effects of the induced radioactivity, radiobiological studies should be
performed.

##  5 Forthcoming Research

During particle therapy of deeply located tumours, a more energetic proton
beam is used. Therefore an extension of the present experimental activities to
higher beam energies is needed. The experiment will be continued also for
other types of therapeutic beams, like carbon ions and neutrons.

The last step of this project will explore the influence of radioactivity
induced in the irradiated tissue on the surrounding non-irradiated cells.

## Acknowledgements

The authors would like to thank Dr. Jan Swakoń and his team for performing
irradiations and general support. Several samples were measured in the
laboratory of Prof. Jerzy Mietelski. We are grateful to him and his
collaborators for kind cooperation.

## References

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  * [6] Nuclear Data Center at Korea Atomic Energy Research Institute, http://atom.kaeri.re.kr/ (accessed 1 Aug 2017)