# Characterization of errors in a CNOT between surface code patches.

Bálint Domokos  Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary  Áron Márton  Department of Theoretical Physics, Institute
of Physics, Budapest University of Technology and Economics, Műegyetem rkp.
3., H-1111 Budapest, Hungary  János K. Asbóth  Department of Theoretical
Physics, Institute of Physics, Budapest University of Technology and
Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  HUN-REN Wigner
Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

###### Abstract

As current experiments already realize small quantum circuits on error
corrected qubits, it is important to fully understand the effect of physical
errors on the logical error channels of these fault-tolerant circuits. Here,
we investigate a lattice-surgery-based CNOT operation between two surface code
patches under phenomenological error models. (i) For two-qubit logical Pauli
measurements – the elementary building block of the CNOT – we optimize the
number of stabilizer measurement rounds, usually taken equal to $d$, the size
(code distance) of each patch. We find that the optimal number can be greater
or smaller than $d$, depending on the rate of physical and readout errors, and
the separation between the code patches. (ii) We fully characterize the two-
qubit logical error channel of the lattice-surgery-based CNOT. We find a
symmetry of the CNOT protocol, that results in a symmetry of the logical error
channel. We also find that correlations between X and Z errors on the logical
level are suppressed under minimum weight decoding.

##  1 Introduction

Fault-tolerant quantum computation will be the basis of future large-scale
quantum computers. To protect logical quantum information from environmental
noise quantum error correction is essential. Among the array of promising
quantum error-correcting codes the surface code [1, 2] stands out as a favored
choice due to its high threshold, scalability, and planar connectivity.
Recently, surface code memory experiments have been realized on various
platforms [3, 4, 5, 6, 7], and the experimental investigation of fault-
tolerant quantum gates has started [8, 9, 10, 11].

For fault-tolerant universal quantum computation with the surface code a
universal set of fault-tolerant gates is required. Single-qubit Clifford
gates, such as the Hadamard $H$ and the phase gate $S$, can be implemented by
braiding the corners of surface code patches [12]. Although transversal multi-
qubit logical Clifford operations exist between surface code patches [13], and
have been demonstrated recently with neutral atoms [8], this requires high
connectivity which is unachievable on several platforms. Another possible way
to implement multi-qubit Clifford gates is via measurements of multi-qubit
Pauli observables, an example for the CNOT gate [14] is shown in Fig. 1.
Multi-qubit Pauli measurements can be done fault tolerantly with lattice
surgery protocols [15, 16], requiring no more than two-dimensional nearest-
neighbour connectivity.

Figure 1: The measurement-based realization of the CNOT operation. This
circuit realizes a CNOT gate between the control (top) and target (bottom)
qubits, while an ancilla qubit is initialized in the $\ket{+}$ state and at
the end of the circuit, measured in the X-basis. Based on the outcomes of mid-
circuit measurements Pauli corrections may be performed on the control and
target qubits.

Non-Clifford operations can be achieved with Clifford gates and "magic
ancillas". Notably, the fault-tolerant preparation of magic states in surface
codes involves magic state distillation protocols [17], which are quite
costly. Logical magic states with fidelities higher than single qubit magic
state fidelities have been distilled with a superconducting architecture [18].
Beyond distillation, numerous other approaches exist to realize universal
fault-tolerant quantum computation [19], including higher-dimensional codes
[20], encoding information into islands of qudits [21], using gauge color
codes instead of surface codes [22], and implementing non-Clifford gates
through small coherent rotations [23]. The costs of these methods have to be
compared with state distillation protocols [24].

Notably, there exist more complex lattice surgery protocols, involving twist
defects, or the measurement of more than two logical operators [25, 16], but
these schemes are beyond the scope of our work.

In this work, we investigate the lattice-surgery-based fault-tolerant CNOT
gate between two surface code patches under phenomenological noise model. This
model includes Pauli noise on the physical level and phenomenological readout
errors during stabilizer measurements. (i) We show that for finite error rates
the optimal number of stabilizer measurement rounds during a ZZ (or XX)
measurement can differ slightly from the code distance if the probability of
random Pauli X (or Z) errors and the probability of readout errors are the
same. We also show that this is not the case for biased physical and readout
error rates the optimal number can heavily differ from the code distance. (ii)
We fully characterize the logical errors during a CNOT and show that the
symmetries of the logical CNOT protocol appear in the two-qubit logical error
channel, enforcing the channel to have a special structure. Our findings
contribute to a deeper understanding of the noise structure in surface code-
based logical operations, and are of interest for experimental implementations
of lattice surgery protocols.

The rest of this paper is structured as follows: In Sec. 2 we introduce the
surface code. We define check generators and the check graph to describe
multiple stabilizer measurement rounds. Moreover, we show how to visualize
logical protocols with multiple measurement rounds through spacetime diagrams.
In Sec. 3 we introduce lattice surgery protocols for two-qubit logical Pauli
measurements. We optimize the parameters of these protocols under random Pauli
and phenomenological readout noise. In Sec. 4 we show that the symmetries of
the lattice-surgery-based logical CNOT protocol enforce the logical Pauli
channel to have a special structure, characterized by 3 logical error
parameters. This last result is exact for the independent $X$ and $Z$ errors,
and a numerically well-supported approximation for depolarizing noise under
minimum weight perfect matching decoding.

##  2 The surface code and spacetime diagrams

In this paper we investigate the impact of errors on quantum operations
between surface code patches – concepts that we briefly introduce in this
Section.

###  2.1 A single patch of the surface code

A (rotated) surface code patch [26] is a square grid of $n=d^{2}$ physical
qubits, with $d$ an odd integer, as depicted in Fig. 2. A single logical qubit
is encoded in the collective quantum state of the $n$ physical qubits, in the
$+1$ eigensubspace of the $Z$\- and $X$\- stabilizers (more precisely
stabilizer generators), $B_{f}$ and $A_{f}$,

$\displaystyle B_{f}$ $\displaystyle=\prod_{j\in\partial f}Z_{j};$
$\displaystyle A_{f}$ $\displaystyle=\prod_{j\in\partial f}X_{j}.$ (1)

acting on the qubits at the corners of the corresponding face, $f$. The
logical states of the code are superpositions of the logical basis states,
$\ket{\overline{\psi}}=\alpha\ket{\overline{0}}+\beta\ket{\overline{1}}$, with

$\displaystyle\ket{\overline{0}}\\!$
$\displaystyle=\\!\prod_{f\in\text{light}}\\!\\!\\!\frac{1+A_{f}}{\sqrt{2}}\ket{0}^{\otimes
n};$ $\displaystyle\ket{\overline{1}}\\!$
$\displaystyle=\\!\prod_{f\in\text{light}}\\!\\!\\!\frac{1+A_{f}}{\sqrt{2}}\ket{1}^{\otimes
n}.$ (2)

Thus,
$A_{f}\ket{\overline{\psi}}=B_{f}\ket{\overline{\psi}}=\ket{\overline{\psi}}$
for all logical states $\ket{\overline{\psi}}$.

On the left and right (vertical) boundaries of the surface code patch, there
are only $X$-stabilizers, while on the top and bottom (horizontal) boundaries,
only $Z$ ones. Therefore, we call these $X$\- and $Z$-boundaries; these are
also known as smooth and rough boundaries.

Figure 2: The layout of a distance 5 surface code patch. $Z$\- ($X$-)
stabilizers are products of $Z$ ($X$) operators acting on the qubits at the
corners of dark (light) faces. A logical $Z$ ($X$) operator is a string of
single qubit $Z$ ($X$) operators, connecting bottom (left) and top (right)
boundaries depicted as a green (red) line.

Logical Pauli operators $\overline{X}$ and $\overline{Z}$ are realized by
strings of $X$ and $Z$ operators connecting different $X$\- and $Z$-boundaries
– examples are in Fig. 2. They commute with the stabilizers, moreover, these
operators multiplied by stabilizers are also logical operators, acting
equivalently in the logical subspace. The code distance $d$ of a patch is the
length (weight) of the shortest logical operator, equal to the linear size of
the patch, i.e., $d=\sqrt{n}$.

###  2.2 Errors, stabilizer measurements, check graphs

We consider two types of errors, against which the surface code protects the
encoded qubit: data qubit errors and phenomenological readout errors. We
restrict data qubit errors to probabilistic single-qubit Pauli $X$ and $Z$ (if
these occur on the same qubit, they result in Pauli $Y$). Phenomenological
readout errors are perfectly performed measurements with erroneously reported
results. Both types of errors can be detected by repeated measurements of the
stabilizer operators. We will refer to the errors as

$\displaystyle\begin{split}&\text{{Spacelike error}: Pauli error on a data
qubit;}\\\ &\text{{Timelike error}: Readout error on a
stabilizer.}\end{split}$

To handle these errors in the same manner we introduce _virtual qubits_
representing stabilizer measurements. As shown in Fig. 3, virtual qubits
(white dots) are placed between consecutive layers (time steps) of physical
qubits (black dots). Spacelike errors are located on physical qubits and
timelike errors are on virtual ones.

Errors are detected by _check generators_ : these are assigned to the faces
$f$ at time $t$, and are the products of the measurement outcomes of the
corresponding stabilizer right before and after $t$. (The definition of check
generators slightly differ at the boundaries which we will discuss later.)
Thus, an odd number of data qubit/measurement errors result in a check
generator value of $-1$, we call this a check violation. The locations of
check violations contain the classical information that we gain from
stabilizer measurements, this is called the _syndrome_.

The _check graph_ contains the check generators as nodes and qubits (both
physical and virtual) as edges. The check graph consists of two disconnected
pieces: the $X$-check graph contains $X$-check generators (comparing
measurement outcomes of $X$-stabilizers), and the $Z$-check graph contains
$Z$-check generators. $X$-type errors can violate $Z$-check generators, while
$Z$-type errors can violate $X$ ones. Notably, spacelike errors can be
correlated due to Pauli $Y$ noise. However, timelike errors are not
correlated, because readout errors on $Z$-stabilizers ($X$-type timelike
errors) and readout errors on $X$-stabilizers ($Z$-type timelike errors) are
independent.

To infer errors from the measurement data (the syndrome) a decoding algorithm
has to be run on the check graph; we took the minimum-weight perfect matching
(MWPM) [27, 28] approach. This produces a correction string for any given
syndrome, which has both timelike and spacelike corrections. We used the
efficient Blossom algorithm [29] to find the minimum-weight string, as
implemented in the PyMatching software package [30, 31]. This solves the
decoding on $X$\- and $Z$-check graphs independently, with proper weighting of
the edges to account for different data qubit and readout error rates. Our
codes and the numerical data are available at [32]

###  2.3 Timelike and spacelike boundaries of check graphs

Boundaries are important features of check graphs. The $X$\- and
$Z$-boundaries, defined above for a single surface code patch, can be
generalized for a check graphs – examples are shown in Fig. 3. _Spacelike
boundaries_ can be defined as surfaces containing the same boundary types in
each stabilizer measurement round. Differently put, spacelike $X$\- ($Z$-)
boundaries are vertical surfaces, where only $X$\- ($Z$-) check generators are
present. _Timelike boundaries_ can be defined in a similar manner, timelike
$X$\- ($Z$-) boundaries are horizontal surfaces, where only $X$\- ($Z$-) check
generators are present. We also define _perfect timelike boundaries_ , where
both $X$\- and $Z$-check generators are present.

While timelike $X$\- and $Z$-boundaries represent fault-tolerant
initialization and measurement protocols, perfect timelike boundaries are just
theoretical constructions. These are needed to describe the fault-tolerant
logical operation itself, excluding errors occurring during initialization and
during the final measurements.

Figure 3: Spacetime diagram segments and the corresponding check graphs
representing the three different timelike boundaries. Black (white) dots:
physical (virtual) qubits, blue (orange) volumes: $Z$\- ($X$-) check
generators. Top row: upper timelike $Z$-boundary, with a) spacetime diagram,
b) $Z$-check graph, c) $X$-check graph. Here only $Z$-check generators are
present at the top boundary. Middle row: upper timelike $X$-boundary, with d)
spacetime diagram, e) $Z$-check graph, f) $X$-check graph. Here only $X$-check
generators are present at the top boundary. Bottom row: upper perfect timelike
boundary, with g) spacetime diagram, h) $Z$-check graph, j) $X$-check graph.
These both contain check generators at the top boundary.

Initializing physical quantum bits in $\ket{0}$ ($\ket{+}$) generates a
timelike $Z$\- ($X$-) boundary: in this case, the values of $Z$\- ($X$-)
stabilizers are predetermined before the first round of stabilizer
measurements. Therefore, $Z$\- ($X$-) check generators at such a boundary
contain physical qubits from the "zeroth" time step and virtual qubits from
the first stabilizer measurement round. In contrast, the measurement outcomes
of $X$\- ($Z$-) stabilizers are random, therefore there is a missing row of
$X$\- ($Z$-) check generators at this boundary. Similarly, measurement of
physical qubits in the $Z$\- ($X$-) basis generates timelike $Z$\- ($X$-)
boundaries. We illustrate the three different timelike boundaries in Fig. 3.

Error strings can terminate at boundaries (both spacelike and timelike)
without violating check generators. $X$-type strings can terminate at $X$
boundaries while $Z$-type strings at $Z$ boundaries. We define logical errors
through the boundaries as well: an error string is a logical error string (can
consist of both spacelike and timelike errors) if it connects two disconnected
boundaries, from the same type, without violating any check generators. We
also define the so-called _fault distance_ of a logical protocol, which is the
length of the shortest logical error string. This is a straightforward
generalization of code distance for logical protocols.

###  2.4 Spacetime diagrams: visualization of protocols

Logical protocols can often be visualized by so-called _spacetime diagrams_
[33]. These do not include all details of the check graph representation, but
contain the important features such as boundaries and twist defects, which
separate different boundary types; examples are shown in Fig. 3. Spacetime
diagrams are useful tools to identify different logical error strings and
follow their propagation through the protocol.

We note that spacetime diagrams which represent general logical protocols can
have a richer repertoire of topological features [34, 33, 35]. If so-called
domain walls are present twists can have other functions than just separating
different boundary types, and logical error strings are somewhat harder to
define. Generally, spacetime diagrams can represent all surface code-based
logical Clifford operations [33].

##  3 Lattice-surgery-based ZZ-measurements: optimization of the number of
measurement rounds

Lattice surgery [15, 36, 37] is one of the leading approaches to fault-
tolerant quantum computation using surface code patches. The basic two-qubit
operation of lattice surgery is the measurement of the logical
$\bar{Z}\bar{Z}$ or the $\bar{X}\bar{X}$ operator. In this Section we review
how the measurement of the $\bar{Z}\bar{Z}$ operator is performed, and
investigate numerically how to optimize the number of measurement rounds for
this two-qubit logical operation. The results for the $\bar{X}\bar{X}$
measurement can be obtained by straightforward generalization.

We consider two surface code patches of distance $d$ with $X$-boundaries
facing each other, separated by a "bridge" of $w$ columns of data qubits
(coupling qubits). In practice, $w$ can vary, depending on the physical
distance of the two patches on the quantum computer; with some patch
definitions $w=0$ is also feasible [25].

The steps of the $\bar{Z}\bar{Z}$-measurement protocol are: 1) Initialize all
coupling qubits in $\ket{+}$; 2) Merge the patches, by measuring the
stabilizers of the extended rectangular patch; 3) Measure the coupling qubits
in the $X$-basis.

Without any errors, the outcome of the $\bar{Z}\bar{Z}$ measurement is the
product of the measurement outcomes of the newly defined intermediate
$Z$-stabilizers.

Measurement errors on the intermediate stabilizers might lead to errors in the
result of the $\bar{Z}\bar{Z}$ measurement. Thus, for fault tolerance,
multiple stabilizer measurement rounds are needed. The criterion for ensuring
a fault-tolerant measurement with a fault distance $d$ is to perform at least

$h_{2}=d$ (3)

rounds of stabilizer measurements between merge and split operations.

The spacetime diagram of the $\bar{Z}\bar{Z}$ measurement is shown in Fig. 4.
This diagram has perfect timelike boundaries at the top and at the bottom, and
timelike $X$-boundaries in the middle. It consists of 4 disconnected $X$\- and
2 disconnected $Z$-boundaries.

Figure 4: The spacetime diagram of the lattice-surgery-based $\bar{Z}\bar{Z}$
measurement. Parameters are the linear size $d$ (in data qubits) of each
patch, and the length $w$ of the "bridge" between them; and the number of
measurement rounds $h_{1}$ before/after, and $h_{2}$ during the phase where
the patches are merged.

###  3.1 Optimization of the number of measurement rounds in a two-qubit Pauli
measurement

Figure 5: Errors of the $\bar{Z}\bar{Z}$ measurement as a function of $h_{2}$,
the measurement rounds in the merged phase, with $d=7$, $h_{1}=h_{3}=1$,
$w=1$, and data qubit error rate $p=0.6\%$. We find that Eq.(6) (encircled
values) is a decent, but not optimal choice, for readout error (a) $q=p$, (c)
$q=p/6$, and (d) $q=3p$. (b): The fraction of timelike logical errors for
different $h_{2}$ values for $q=p$.

What is the optimal number of measurement rounds of the stabilizers in the
phase when the two patches are merged, $h_{2}$? Increasing $h_{2}$ suppresses
the contribution of timelike errors to the probability of a logical error, at
the cost of an overhead: longer logical gate operation, and a slight increase
in the probability of spacelike errors leading to a logical error. If $h_{2}$
is quite long, the contribution of timelike errors to the logical error is
already tiny, and thus, no big gains are expected from increasing $h_{2}$ (the
increased probability of spacelike errors can make the logical error rate even
slightly worse). Thus, the optimal number of rounds will be the value of
$h_{2}$ where the logical error rate saturates (no more practical advantage in
increasing $h_{2}$). This optimal number of rounds is often taken to be
$h_{2}\approx d$, but this can depend on the geometrical parameters and the
error rates as well. We aim to find how this depends on, e.g., the patch size
$d$, the length of the bridge $w$, and on the error rates.

We investigated the optimal value of $h_{2}$, with qubit noise that is
maximally biased in $X$, with spacelike error probability $p$:

$\varepsilon(\rho)=(1-p)\rho+p(X\rho X),$ (4)

and timelike error probability $q$.

We first estimate the optimal value of $h_{2}$ analytically, by a series
expansion treating the error rates as small parameters. Taking into account
only the lowest order terms in spacelike and timelike error rates $p$ and $q$,
the logical error rate is

$P_{L}(p,q)=Ap^{(d+1)/2}+Bq^{(h_{2}+1)/2}+\ldots.$ (5)

Here $A$ ($B$) is the multiplicity of the shortest spacelike (timelike) error
strings that can lead to a logical error. Higher order terms are strongly
suppressed in the case of small error rates. We see that $h_{2}$ reaches its
optimal value when the competing terms in Eq. (5) are equal. From that on it
is not that beneficial to further increase $h_{2}$, because the spacelike
error mechanisms are already the dominant sources.

Thus, the optimal value for $h_{2}$ in the small $p,q$ limit is

$h_{2}\approx(d+1)\dfrac{\ln(p)}{\ln(q)}-1,$ (6)

where we omitted the term

$\frac{2\text{ln}(A/B)}{\text{ln}(q)},$ (7)

assuming that in the $q\rightarrow 0$ limit it is negligible. We note that
this is analogous to arguments showing that elongated rather than square
shaped surface code patches are advantageous for biased noise [38].

Figure 6: The fraction of timelike logical errors as a function of the number
of measurement rounds during the merged phase($h_{2}$), for different bridge
lengths $w$. Here, $d=7$, and $p=q=0.006$. The is shifted from $h_{2}$=7 to
smaller values for $w=1$, higher values for large $w$, due to the higher
multiplicity of timelike errors.

We sampled logical errors via Monte Carlo simulations, by running minimum
weight perfect matching decoder (PyMatching) [30, 31] on the $X$\- and
$Z$-check graphs of the $\bar{Z}\bar{Z}$ measurement. Our results show that
the analytical approximation of Eq. (6) slightly overestimates the optimal
value of $h_{2}$. We simulated the $\bar{Z}\bar{Z}$ measurement with
geometrical parameters $d=7,w=1,h_{1}=h_{3}=1$ for three cases: $p=q$, $p=6q$,
$3p=q$. As shown in Fig. 5, in all three cases, we see that the decrease of
the logical error rate slows down significantly for smaller $h_{2}$ values
than Eq.6 suggests. For a more detailed picture, we also show the ratio of
timelike to logical errors, which transitions from 1 to 0 as $h_{2}$ is
increased. The critical value of $h_{2}$ where the transition happens is
smaller than expected based on the $p,q\to 0$ limit. This suggests that the
correction in Eq.(7) remains significant with these geometrical
($w=h_{1}=1,d=7$) and error parameters ($p,q\approx 10^{-3}$).

We found that increasing the separation $w$ of the patches the optimal value
of $h_{2}$ for the $\bar{Z}\bar{Z}$ measurement increases slightly, as shown
in Fig. 6. This is due to the increasing multiplicity of the shortest timelike
logical error string, while the multiplicity of the spacelike logical error
string does not change. The optimal value of $h_{2}$ is where the fraction of
timelike logical errors drops. Note that the increase in the optimal value of
$h_{2}$ is quite small even for distant patches ($w\approx 10d$): previous
estimates predicted a logarithmic growth [39].

In case $Z$ errors are included in the error model we expect qualitatively
similar results, but the optimum should come at an even lower $h_{2}$ than in
Fig. 5. Here the multiplicity of spacelike logical $Z$ error strings will be
much higher than that of other spacelike error strings. Moreover, the
multiplicity of spacelike and timelike logical error stings increase together
in the case of an increasing $w$, hence we expect no shift of the transition
in the fraction of timelike logical errors.

##  4 Investigation of lattice-surgery-based CNOT protocol

Figure 7: Spacetime diagram of the lattice-surgery-based CNOT. Control (left)
and target (right) patches have perfect lower and upper timelike boundaries,
ancilla patch (middle) has lower timelike $X$\- and upper timelike
$Z$-boundaries: initialized in $\ket{+}$, measured in $Z$. The $X$\- ($Z$-)
boundaries consist of three disconnected parts labeled $A,B,C$ ($D,E,F$).

In this Section we fully characterize the two-qubit Pauli logical error
channel of the standard lattice-surgery-based CNOT protocol [15], assuming
single qubit Pauli noise on the data qubits and phenomenological readout
errors. This characterization amounts to analytical and numerical estimates
for the probabilities of the 15 possible two-qubit logical Pauli errors that
can occur during a CNOT. Note that this CNOT gate follows the logical circuit
of Fig. 1.

The spacetime diagram of this CNOT, Fig. 7, has three disconnected
$X$-boundaries, labeled by $A$, $B$, $C$, and three disconnected $Z$\-
boundaries: $E$, $F$, $G$.

Any string operator connecting two disconnected boundaries of the same kind
leads to a Pauli error on the logical level. Since any of these errors,
squared, gives the identity, there are $16$ distinct connection classes based
on the parity of string endings at boundaries $A$, $B$, $C$, $D$, $E$, $F$.
Note that a string connecting $BC$ is equivalent to a pair of strings
connecting $AB$ and $BC$, and likewise for $EF$, $FG$, and $EG$. These
connection classes correspond to the 16 inequivalent two-qubit logical Paulis
(includng the error-free case) incurred during the CNOT. The correspondence
between string ending parities, connection classes and logical Paulis is
summarized in Table 1.

$N_{A}$ | $N_{B}$ | $N_{C}$ | Connection | Logical Pauli
---|---|---|---|---
0 | 0 | 0 | $\emptyset_{ABC}$ | 1
1 | 1 | 0 | A - B | $X_{1}$
1 | 0 | 1 | A - C | $X_{1}X_{2}$
0 | 1 | 1 | B - C | $X_{2}$
$N_{D}$ | $N_{E}$ | $N_{F}$ | Connection | Logical Pauli
0 | 0 | 0 | $\emptyset_{DEF}$ | 1
1 | 1 | 0 | D - E | $Z_{1}$
1 | 0 | 1 | D - F | $Z_{2}$
0 | 1 | 1 | E - F | $Z_{1}Z_{2}$
Table 1: Correspondence between logical Pauli errors and connection classes of
the boundaries. First column: the number of error string endings modulo 2 at
each boundary specifies the connection class – the sum of every row must be
even, since each string has two ends. Second column: The minimally required
connections between boundaries in each class, $\emptyset_{ABC}$
($\emptyset_{DEF}$) signals no connections between $X$\- ($Z$-) boundaries.
Third column: the corresponding logical Paulis. The 16 logical Pauli error
classes can be obtained by multiplying 1 row from the top and 1 from the
bottom half of the table.

We examine two error models: independent $X$ and $Z$ errors and depolarizing
noise, in both cases combined with phenomenological readout errors with rate
$q$. In the first case, we set the three error rates equal:

$\displaystyle\varepsilon(\rho)$ $\displaystyle=(1-p)^{2}\rho+p(1-p)(X\rho
X+Z\rho Z)+p^{2}Y\rho Y;$ $\displaystyle q$ $\displaystyle=p.$ (8)

In the second case we set $q$ to a value that ensures equal timelike and
spacelike error rates on both the $X$ and $Z$ check graphs,

$\displaystyle\varepsilon(\rho)$ $\displaystyle=(1-p)\rho+\dfrac{p}{3}(X\rho
X+Z\rho Z+Y\rho Y);$ $\displaystyle q$ $\displaystyle=\dfrac{2}{3}p.$ (9)

###  4.1 Symmetry of the CNOT protocol and its consequences on the logical
errors

Figure 8: Symmetry transformation of the spacetime diagram of the CNOT (a), in
3 steps. First, time reversal, resulting in (b); Then, mirror reflection,
resulting in (c); finally, swapping the $X$ and $Z$ labels (colors), resulting
in (d), identical to (a).

The CNOT protocol has a symmetry, evident from the spacetime diagram in Fig.
8, which constrains the possible logical error rates. The symmetry is composed
of: 1) a time reversal, 2) a mirror reflection on a vertical symmetry plane of
the "L" shape, and 3) flipping the $X$ and $Z$ labels. As a result the
original check graph effectively does not change. The order of the three steps
is interchangeable. If this symmetry transformation also leaves the spatial
distribution of error probabilities invariant (as in the case of independent
and depolarizing noise), the logical Pauli error strings which transform into
each other under the symmetry have to have the same probabilities. As a
consequence, symmetry partners among logical Pauli errors have the same
probabilities, resulting in a two-qubit logical error channel shown in Fig.
9a).

Due to the symmetry of the CNOT protocol, in case of independent $X$ and $Z$
errors with equal rate, the probabilities of the different error classes
factorize. Therefore, all error class probabilities can be written using the
probabilities of three possible classes of connections of different boundaries
of one kind via error + correction strings:

$\displaystyle p_{1}$ $\displaystyle=\mathbb{P}(A-C)=\mathbb{P}(E-F);$ (10a)
$\displaystyle p_{2}$ $\displaystyle=\mathbb{P}(B-C)=\mathbb{P}(D-E);$ (10b)
$\displaystyle p_{3}$ $\displaystyle=\mathbb{P}(A-B)=\mathbb{P}(D-F),$ (10c)

using the notation of Table 1. The probability that all $Z$-boundaries have an
even number of $Z$-string endings (equal to the corresponding probability for
all $X$-boundaries) can also be written using $p_{1}$, $p_{2}$, and $p_{3}$,
as

$p_{0}=\mathbb{P}(\emptyset_{ABC})=\mathbb{P}(\emptyset_{DEF})=1-p_{1}-p_{2}-p_{3}.$
(11)

The table of probabilities of logical Pauli errors building on this connection
is shown in Fig. 9a).

Figure 9: The structure of the two-qubit logical error channel of the lattice-
surgery-based CNOT for independent noise. a) Theoretical expectation for the
two-qubit logical Pauli error probabilities based on the connection parameters
$p_{1},p_{2},p_{3}$; non-white colors indicate symmetry partners. b) A
numerically obtained two-qubit Pauli error channel that shows the expected
symmetry pattern; colors indicate error rate as also written in the boxes.

###  4.2 Numerical characterization of the logical error channel

Figure 10: Logical Pauli error rates in CNOT as a function of the physical
noise parameter for patch size $d=7$, with (a) independent Pauli $X$ and $Z$
errors with equal error rates, and (b) depolarizing errors on the physical
qubits. Numerically obtained error rates are shown with symbols with
errorbars. Rates of symmetry partners have the same color; they are in most
cases equal to good precision. Expectations for these rates, calculated from
the fit parameters $p_{1},p_{2},p_{3}$, as described in the main text, are
shown with dashed lines, with shading indicating expected uncertainty.

With extensive numerical work, we fully characterized the logical error
channel of the CNOT for a wide range of code distances and physical error
rates. We constructed the $X$\- and $Z$-check graphs of the logical CNOT gate,
and solved the decoding problem with minimum weight perfect matching decoder
(PyMatching) [30, 31] under independent and depolarizing noise. To obtain
logical error probabilities $p_{IX},\ldots,p_{XZ},\ldots$, we ran Monte Carlo
simulations, where in each round we identified a logical Pauli error based on
the parities of error + correction string endings at boundaries using Table.
1.

We chose $h_{2}=d$ and $h_{1}=w=1$ to ensure the fault distance to be $d$ with
minimal resources. This choice might not the optimal for finite physical error
rates as we showed in the previous section.

We found in all cases with independent noise that the logical error channel
has the symmetry property expected based on the spacetime diagram, as per Fig.
9, to a good precision. A numerical example for patch size $d=5$ is also shown
in Fig. 9. For a more systematic exploration, we used a two-step procedure: 1)
we first extracted the string connection probability parameters $p_{1}$,
$p_{2}$, $p_{3}$ for each physical error probability by fitting a function in
the form of Fig. 9: two-qubit Pauli errors as arguments, and $p_{1}$, $p_{2}$
and $p_{3}$ as fitting parameters (obtaining the logical channel with the
structure of Fig. 9 closest to the numerical values); 2) we then compared the
numerically obtained values of the frequency of two-qubit errors with the
values obtained from the fitted model. We show this comparison for patch size
$d=7$ in Fig. 10, and for other parameter sets in Appendix A.

We found that also with depolarizing errors, the logical error channel of the
CNOT can be described with $p_{1},p_{2},p_{3}$ in the same way as per Fig.9.
We did not expect this, since the connection of Sec. 4.1 between the error
string probabilities and Pauli errors holds for independent $X$ and $Z$ errors
with equal rates, while for depolarizing noise these error are not
independent. One possible reason for this factorized structure could be that
the independent decoding of $X$ and $Z$ check graphs makes the logical error
channel effectively more and more independent as the bulk size grows. We
discuss this in more detail in Appendix B.

###  4.3 Threshold behaviour

Figure 11: The logical error rates as the function of physical error rates for
different fault distances. (a) Independent noise described in Eq. (4). (b)
Depolarizing noise described in Eq, (4). In both cases we see that the
lattice-surgery-based CNOT protocol shows threshold behaviour. The parameters
of the simulation are $d=h_{2}$, $h_{1}=1$, $w=1$.

The surface code, when used as a quantum memory, has a threshold behaviour.
There is a critical physical error rate, e.g., $p_{\text{th}}\approx 3\%$ for
independent noise [40], below which the logical error rates can be suppressed
exponentially by increasing the code distance.

Any lattice-surgery-based logical quantum circuit should also have an error
threshold, with the same value as the threshold defined above. Thus, when
scaling up the size (fault distance) of a logical circuit, the logical error
rates should decrease whenever the physical error rates are below threshold:
in the $d\longrightarrow\infty$ limit the classical statistical mechanical
model describing the decoding procedure [40] is independent of the details of
the boundaries as is dominated by the bulk.

We calculated numerically the threshold of the lattice-surgery-based CNOT
gate, and found good agreement with the known threshold of the surface code,
as shown in Fig. 11.

Although the threshold value is promising for fault-tolerant quantum
computation, the error rates here are much higher than in a simple memory
experiment. This is due to the higher number of possible error classes, as has
been observed for several other logical operations in [33].

The threshold value we obtain for depolarizing noise is around $4.5\%$, which
is $1.5$ times the threshold for independent noise. We believe that this is a
consequence of the independent decoding of $X$ and $Z$ syndrome graphs. We
discuss this in more detail in Appendix.B.

##  5 Conclusion

In this work, we investigated the performance of the lattice-surgery-based
logical CNOT protocol between two surface code patches under phenomenological
noise. We optimized the number of stabilizer measurement rounds during a
lattice surgery protocol realizing the measurement of a two-qubit logical
Pauli operator. We found that for equally strong timelike and spacelike errors
the optimal number can slightly differ from the naively expected $d$ (code
distance) rounds. This is because for small but finite error rates the
$1/\text{log}$ correction - coming from the entropy difference of logical
error classes - cannot be neglected. We also found that for biased timelike
and spacelike errors, the optimal number of stabilizer measurement rounds can
heavily differ from $d$. Our findings suggest that it is beneficial to tailor
the number of stabilizer measurement rounds during a lattice surgery protocol
to the noise model.

We fully characterized the two-qubit Pauli noise channel of the lattice-
surgery-based CNOT protocol and found that the symmetry of this protocol
appears in the noise channel. We identified symmetry partners among two-qubit
logical Pauli errors, which have to have the same probability in the case of
equally strong $X$ and $Z$ errors on the physical level. Moreover, the logical
error channel can be described with just three logical error parameters. This
last result is exact for independent ($X$ and $Z$ errors are uncorrelated)
noise, and a pretty good approximation for depolarizing noise.

A natural next step of our work would be circuit-level simulation of the
lattice-surgery-based CNOT protocol with a specific syndrome extraction
circuit. With error models close to experimental reality the symmetry of the
spacetime diagram will be broken, but only at the boundaries, thus we expect
that the logical error channel will be approximately symmetric. Bias in the
physical noise (e.g., larger $Z$ than $X$ error rates) will definitely break
the symmetry; it is an open question whether symmetrized (e.g., XZZX [38])
surface code restores the symmetry structure. The more general relations
between symmetries of logical protocols and the structure of the logical noise
also remained unexplored.

Our study can help to understand the structure of the logical noise in surface
code-based fault-tolerant quantum computation. Thus, it may have important
implications, which have to be considered during the designing of fault-
tolerant quantum algorithms.

##  6 Acknowledgement

This research was supported by the Ministry of Culture and Innovation and the
National Research, Development and Innovation Office within the Quantum
Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004). Supported by the Horizon Europe research and
innovation programme of the European Union through the HORIZON-CL4-2022-
QUANTUM01-SGA project 101113946 OpenSuperQPlus100 of the EU Flagship on
Quantum Technologies. We acknowledge the support of the Wigner Research Centre
for Physics’ trainee program.

## References

  * [1] A.Yu. Kitaev.  “Fault-tolerant quantum computation by anyons”.  Annals of Physics 303, 2–30 (2003).
  * [2] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland.  “Surface codes: Towards practical large-scale quantum computation”.  Physical Review A86 (2012).
  * [3] Rajeev Acharya et al.  “Suppressing quantum errors by scaling a surface code logical qubit”.  Nature 614, 676 – 681 (2022).
  * [4] Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin.  “A quantum processor based on coherent transport of entangled atom arrays”.  Nature 604, 451–456 (2022).
  * [5] J. F. Marques, B. M. Varbanov, M. S. Moreira, H. Ali, N. Muthusubramanian, C. Zachariadis, F. Battistel, M. Beekman, N. Haider, W. Vlothuizen, A. Bruno, B. M. Terhal, and L. DiCarlo.  “Logical-qubit operations in an error-detecting surface code”.  Nature Physics 18, 80–86 (2021).
  * [6] Sebastian Krinner, Nathan Lacroix, Ants Remm, Agustin Di Paolo, Elie Genois, Catherine Leroux, Christoph Hellings, Stefania Lazar, Francois Swiadek, Johannes Herrmann, et al.  “Realizing repeated quantum error correction in a distance-three surface code”.  Nature 605, 669–674 (2022).
  * [7] Youwei Zhao et al.  “Realization of an error-correcting surface code with superconducting qubits”.  Physical Review Letters129 (2022).
  * [8] Dolev Bluvstein et al.  “Logical quantum processor based on reconfigurable atom arrays”.  Nature (2023).
  * [9] Jiaxuan Zhang, Zhao-Yun Chen, Yun-Jie Wang, Bin-Han Lu, Hai-Feng Zhang, Jia-Ning Li, Peng Duan, Yu-Chun Wu, and Guo-Ping Guo.  “Demonstrating a universal logical gate set on a superconducting quantum processor” (2024).  arXiv:2405.09035.
  * [10] Bence Hetényi and James R Wootton.  “Creating entangled logical qubits in the heavy-hex lattice with topological codes” (2024).  arXiv:2404.15989.
  * [11] Daniel Honciuc Menendez, Annie Ray, and Michael Vasmer.  “Implementing fault-tolerant non-clifford gates using the [[8, 3, 2]] color code” (2023).  arXiv:2309.08663.
  * [12] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton.  “Poking holes and cutting corners to achieve Clifford gates with the surface code”.  Physical Review X7 (2017).
  * [13] P.W. Shor.  “Fault-tolerant quantum computation”.  In Proceedings of 37th Conference on Foundations of Computer Science.  Pages 56–65.  (1996).
  * [14] CWJ Beenakker, DP DiVincenzo, C Emary, and M Kindermann.  “Charge detection enables free-electron quantum computation”.  Physical Review Letters 93, 020501 (2004).
  * [15] Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter.  “Surface code quantum computing by lattice surgery”.  New Journal of Physics 14, 123011 (2012).
  * [16] Daniel Litinski.  “A game of surface codes: Large-scale quantum computing with lattice surgery”.  Quantum 3, 128 (2019).
  * [17] Sergey Bravyi and Alexei Kitaev.  “Universal quantum computation with ideal Clifford gates and noisy ancillas”.  Physical Review A71 (2005).
  * [18] Riddhi S. Gupta, Neereja Sundaresan, Thomas Alexander, Christopher J. Wood, Seth T. Merkel, Michael B. Healy, Marius Hillenbrand, Tomas Jochym-O’Connor, James R. Wootton, Theodore J. Yoder, Andrew W. Cross, Maika Takita, and Benjamin J. Brown.  “Encoding a magic state with beyond break-even fidelity”.  Nature 625, 259–263 (2024).
  * [19] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot.  “Roads towards fault-tolerant universal quantum computation”.  Nature 549, 172–179 (2017).
  * [20] Benjamin J. Brown.  “A fault-tolerant non-Clifford gate for the surface code in two dimensions”.  Science Advances 6, eaay4929 (2020).
  * [21] Katharina Laubscher, Daniel Loss, and James R. Wootton.  “Universal quantum computation in the surface code using non-Abelian islands”.  Physical Review A100 (2019).
  * [22] Héctor Bombín.  “Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes”.  New Journal of Physics 17, 083002 (2015).
  * [23] Hyeongrak Choi, Frederic T. Chong, Dirk Englund, and Yongshan Ding.  “Fault tolerant non-Clifford state preparation for arbitrary rotations” (2023).  arXiv:2303.17380.
  * [24] Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore.  “Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes”.  PRX Quantum2 (2021).
  * [25] Daniel Litinski and Felix von Oppen.  “Lattice surgery with a twist: Simplifying Clifford gates of surface codes”.  Quantum 2, 62 (2018).
  * [26] H. Bombin and M. A. Martin-Delgado.  “Optimal resources for topological two-dimensional stabilizer codes: Comparative study”.  Physical Review A76 (2007).
  * [27] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill.  “Topological quantum memory”.  Journal of Mathematical Physics 43, 4452–4505 (2002).
  * [28] Austin G. Fowler.  “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $o(1)$ parallel time” (2014).  arXiv:1307.1740.
  * [29] Jack Edmonds.  “Paths, trees, and flowers”.  Canadian Journal of Mathematics 17, 449–467 (1965).
  * [30] Oscar Higgott.  “Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching”.  ACM Transactions on Quantum Computing 3, 1–16 (2022).
  * [31] Oscar Higgott and Craig Gidney.  “Sparse blossom: correcting a million errors per core second with minimum-weight matching” (2023).  arXiv:2303.15933.
  * [32] “Numerical data and Python codes”.  https://doi.org/10.5281/zenodo.11079788.
  * [33] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts.  “Logical blocks for fault-tolerant topological quantum computation”.  PRX Quantum4 (2023).
  * [34] Craig Gidney.  “Inplace access to the surface code y basis”.  Quantum 8, 1310 (2024).
  * [35] György P. Gehér, Campbell McLauchlan, Earl T. Campbell, Alexandra E. Moylett, and Ophelia Crawford.  “Error-corrected Hadamard gate simulated at the circuit level” (2023).  arXiv:2312.11605.
  * [36] Christopher Chamberland and Earl T. Campbell.  “Universal quantum computing with twist-free and temporally encoded lattice surgery”.  PRX Quantum3 (2022).
  * [37] Christopher Chamberland and Earl T. Campbell.  “Circuit-level protocol and analysis for twist-based lattice surgery”.  Physical Review Research4 (2022).
  * [38] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown.  “The XZZX surface code”.  Nature Communications12 (2021).
  * [39] Daniel Litinski and Felix von Oppen.  “Braiding by Majorana tracking and long-range CNOT gates with color codes”.  Physical Review B96 (2017).
  * [40] Chenyang Wang, Jim Harrington, and John Preskill.  “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory”.  Annals of Physics 303, 31–58 (2003).

##  Appendix A Numerical results for the logical CNOT gate

To investigate the $\bar{Z}\bar{Z}$-measurement and the logical CNOT
operation, we have performed Pauli frame simulations. We have constructed
check graphs associated with the spacetime diagrams of the experiments testing
the $\bar{Z}\bar{Z}$-measurement and the CNOT gate which we could put into
Pymatching [30, 31]. With the correction operator and the randomly generated
error history in hand, we could identify the specific logical error mechanism
that occurred during the logical operation by checking the parities of error +
correction string endings at boundaries. To obtain a statistically significant
data on these logical error mechanisms multiple Monte Carlo rounds are needed.
For the CNOT gate, we collected data from $10^{4}$ Monte Carlo rounds. In case
of the $\bar{Z}\bar{Z}$ measurement, we investigated lower error rates,
therefore $10^{4}$ rounds would not have been enough. So we simulated until we
reached 10 erroneous logical operations for which depending on the size of the
check graph $10^{5}-10^{7}$ rounds were enough.

We collected Monte Carlo data for CNOT with geometrical parameters
$d=5,7,9,11,13$, $w=1$, $h_{1}=h_{3}=1$, $h_{2}=d$ under independent noise or
depolarizing noise with various error rates. For each simulation setting we
determined the $4\times 4$ table representing the two qubit logical Pauli
channel. We depict the numerically obtained values in Fig. 12 in the same way
as we did for $d=7$ in Fig. 10 for all other patch sizes. The data is
available at [32]. Our findings for $d=7$ also hold for other patch sizes.

In Fig. 12 one may notice that the $IX$ and $ZI$ outcomes have much higher
probabilities than the others. This is because the corresponding logical error
strings connect the two largest $X$ or $Z$ boundaries (while there is no
connection between separated boundaries on the other check graph). Therefore
these classes of connections have high multiplicities, which leads to them
dominating the logical errors for small error rates.

Figure 12: Logical Pauli error rates in CNOT as a function of the physical
noise parameter for various patch sizes $d$, with independent Pauli $X$ and
$Z$ errors with equal error rates, or depolarizing errors on the physical
qubits, as indicated on the plots. Numerically obtained error rates are shown
with symbols with errorbars. Rates of symmetry partners have the same color;
they are in most cases equal to good precision. Expectations for these rates,
calculated from the fit parameters $p_{1},p_{2},p_{3}$, as described in the
main text, are shown with dashed lines, with shading indicating expected
uncertainty. The other geometrical parameters of the procedure for all patch
sizes are: $w=1$, $h_{1}=h_{3}=1$, $h_{2}=d$.

##  Appendix B Correlations in the logical error channel under minimum weight
perfect matching decoding

In Sec. 4 we saw that the two-qubit logical Pauli channel for independent
noise with equal $X$ and $Z$ error rates can be fully described by the three
string connection probability parameters $p_{1}$, $p_{2}$, $p_{3}$. The
numerics reported in Sec.4 has also shown that this decomposition also works
for the depolarizing noise channel – surprising, since here the $X$ and $Z$
errors are correlated. We think that this is due to the use of the minimum
weight perfect matching decoding, under which the correlations between $X$ and
$Z$ errors are suppressed on the logical level.

The minimum weight perfect matching decoder finds the most probable $X$-type
and $Z$-type error histories independently, that are consistent with the given
syndrome. This error history will not be the most probable error history
necessarily if correlations between $X$ and $Z$ errors are present. Therefore,
we assume that for low enough error rates these correlation will not affect
the final logical error channel.

We find that the correlations in the logical noise channel are disappearing as
we are scaling up the system size. To characterize the strength of
correlations in the logical error channel of single logical qubit we introduce
the following measure:

$M=\frac{|P_{I}P_{Y}-P_{X}P_{Z}|}{P_{X}+P_{Y}+P_{Z}}\ ,$ (12)

where the logical error channel is defined as:

$\varepsilon(\rho)=P_{I}\rho+P_{Z}\bar{Z}\rho\bar{Z}+P_{Y}\bar{Y}\rho\bar{Y}+P_{X}\bar{X}\rho\bar{X}.$
(13)

(a)

(b)

Figure 13: Correlations between the possible Pauli outcomes of a memory
experiment under depolarizing noise with minimum weight decoder. a) Surface
code with error correction after one round of stabilizer measurements
($d\times d\times 1$). b) Surface code with error correction after $d$ round
of stabilizer measurements ($d\times d\times d$). As the bulk size grows the
logical error channel is becoming more and more similar to independent noise
due to the independence in the decoding protocol.


[1] [cor1]Corresponding author<EMAIL_ADDRESS>Image Processing
and Machine Vision Lab, Faculty of Electrical and Computer Engineering,
University of Tabriz, 29 Bahman Blvd., Tabriz, Iran.
*Corresponding author2: Seyyedhossein Shafiei, Orthopedic Surgery Research Centre, Sina University Hospital, Tehran University of Medical sciences, Tehran, Iran.

[1]

# Estimation of Acetabular Version from Anteroposterior Pelvic Radiograph
Employing Deep Learning

Ata Jodeiri<EMAIL_ADDRESS>Hadi Seyedarabi<EMAIL_ADDRESS>aFaculty of Electrical and Computer Engineering, University of Tabriz, Tabriz,
Iran  Fatemeh Shahbazi<EMAIL_ADDRESS>bSchool of Electrical & Computer
Engineering, University of Tehran, Tehran, Iran  Seyed Mohammad Mahdi Hashemi
<EMAIL_ADDRESS>Seyyedhossein Shafiei<EMAIL_ADDRESS>cOrthopedic Surgery Research Centre, Sina University Hospital, Tehran
University of Medical sciences, Tehran, Iran

###### Abstract

Background and Objective: The Acetabular version, an essential factor in total
hip arthroplasty, is measured by CT scan as the gold standard. The dose of
radiation and expensiveness of CT make anterior-posterior pelvic radiograph an
appropriate alternative procedure. In this study, we applied a deep learning
approach on anteroposterior pelvic X-rays to measure anatomical version,
eliminating the necessity of using Computed tomography scan.
Methods: The right and left acetabular version angles of the hips of 300
patients are computed using their CT images. The proposed deep learning model,
Attention on Pretrained-VGG16 for Bone Age, is applied to the AP images of the
included population. The age and gender of these people are added as two other
inputs to the last fully connected layer of attention mechanism. As the
output, the angles of both hips are predicted.
Results: The angles of hips computed on CT increase as people get older with
the mean values of 16.54 and 16.11 (right and left angles) for men and 20.61
and 19.55 for women in our dataset. The predicted errors in the estimation of
right and left angles using the proposed method of deep learning are in the
accurate region of error (<=3 degrees) which shows the ability of the proposed
method in measuring anatomical version based on AP images.
Conclusion: The suggested algorithm, applying pre-trained vgg16 on the AP
images of the pelvis of patients followed by an attention model considering
age and gender of patients, can assess version accurately using only AP
radiographs while obviating the need for CT scan. The applied technique of
estimation of anatomical acetabular version based on AP pelvic images using DL
approaches, to the best of authors’ knowledge, has not been published yet.

###### keywords:

Acetabular Version Deep Learning (DL) Anteroposterior Pelvic (AP) Radiograph
Artificial Intelligence (AI) Hip.

##  1 Introduction

The Acetabular version, defined as operative anteversion, sharp angle, and
anatomical anteversion [1]-[2], refers to the orientation related to the
opening of the acetabulum [3]. These definitions are based on the measurements
performed on the sagittal, coronal, and horizontal planes [1]-[2]. Natural
anatomical anteversion is 20.7 $\pm$ 3.8 [4], while it is recommended for
component positioning between $0^{\circ}$ and $30^{\circ}$ in various studies
[5] and suggested by Lewinnek et al. to be $15^{\circ}$ [5]. Anteversion is
defined as an important changing morphological feature in the biomechanics [6]
and pathologies of the hip joint [7]. The motion of the hip in both internal
and external is proved to be affected by femoral and acetabular versions [6].
Version abnormalities are also shown to be associated with hip pain [6],
acetabular fractures, osteoarthritis [7], dysplasia, and femoroacetabular
impingement [8]. Moreover, in hip surgeries such as total hip arthroplasty
(THA), which is an attempt to soothe pain, improve functional activity, and
enhance quality of life [9], determination of acetabulum version is a crucial
factor in both predicting and following up the outcome of the operation.
Position of the cup in THA is a significant issue in which inappropriate
positioning entails impingement, dislocation, severe wear, and reduction in
the range of motion [9]-[11] which may force patients for revision surgery
[10]. Assessment of the orientation of acetabulum has been done by 3-D imaging
systems (CT-scan and MRI) and anterior-posterior (AP) pelvic X-rays [11].

Klasan et al. in 2019 studied the whole-body CT scans of 404 patients (aged 16
to 40 years) and reported a higher average acetabular version in women (19.31
$\pm$ 5.04 compared to 16.46 $\pm$ 4.42 in men) while the version increases
with age. The anatomical measurement was done on cranial, central and caudal
levels [7]. In 2018, in a study by Lerch et al. [6], femoral and acetabular
versions of 462 symptomatic patients with hip pain were examined between
desired subgroups and compared with a control group. The results showed
significant differences in acetabular version in some of the subgroups
compared to the control group. The researchers in this study observed that the
femoral version’s abnormalities in the subjects suffering from hip pain, who
are qualified for hip preservation surgery, were more prevalent than in the
control group. As a result, it is recommended that the femoral version and
acetabular version should be assessed in youth with painful hips using CT or
MRI imaging techniques. A comparison was made based on the assessment of
anteroposterior radiograph of the selected cohort using computer software
approach and CT of the whole body of the control subjects. Two different
acetabular version computing radiographic methods, ischiolateral and Woo &
Morrey, were conducted by Raj et al. [9] on the AP images of patients with
Total Hip Replacement (THR) surgery. Comparing the results with the gold
standard CT using regression line, the values of Woo & Morrey were reported
close to CT values except in hip stiffness in the contralateral hip. AP
radiograph of hip and pelvis of 153 dysplasia patients were examined in order
to assess the prevalence of acetabular retroversion [3]. Snijders et al. [11]
measured radiographic anteversion based on ten different mathematical
approaches using patients’ AP pelvic radiography. Finally, they compared the
results of these methods with the gold standard on CT images. The linear
correlation of the measurements was in the [0.556 0.747] interval in which CT
scan was recommended as the reliable method for anteversion measurement [11].
Lots of researchers evaluated cup versions to assess the influence of cup
malposition in THA surgery using CT or lateral-pelvis radiographs [10]. In
some studies, lateral radiographs were proposed for anteversion calculations
though the others did not acknowledge them in distinguishing anteversion from
retroversion. Finally, the version was recommended to be measured accurately
using standard CT by many scholars.

Though CT is introduced as the gold standard method of acetabular version
measurement [9], AP radiograph images are the first assessment in patients
with pain in the hip [8] due to their availability, less radiation, and cost-
effectiveness [9]-[10]. Typically, 20$\%$ to 50$\%$ of high-tech imaging
methods, including CT, are declared unnecessary [12], where considering the
dose of radiation in CT, anteroposterior pelvis radiographs can be considered
as an appropriate alternative procedure. As a result, a low-risk and low-cost
method for accurate assessment of version is regarded as a significant need.
In this study, we used AP radiographs for assessment of the orientation of
acetabulum.

Here we propose an algorithm based on artificial intelligence (AI) in the
anatomical acetabular version measurement using AP images. New technologies in
orthopedics have received much attention in such a way that we can see
considerable developments in preoperative training, preoperative planning, and
intraoperative navigation to assist orthopaedists [13]. Nowadays, AI has
attracted more attention using deep learning (DL) as a class of machine
learning (ML) applications in the diagnosis of medical images [14]-[15].
Radiological image interpretation and diagnoses applying ML method have been
considered recently [16]. ML has also been applied in different fields,
including: segmentation of pulmonary embolism using angiography, colon polyp
detection with CT scans, diagnosing and detecting breast cancer in mammography
[16]. Diagnosis of fractures and estimation of pediatric bone age are among
the cases in that DL has been proved to be capable in orthopedics [14].
Jodeiri et al., 2020, proposed a deep learning approach to estimate pelvis
sagittal inclination. His CNN-based method on the AP images of 475 patients
with THR surgery, eliminated the necessity for taking CT scans [17].

Convolutional neural networks (CNN) are a well-known architecture of DL, which
is severely used in radiology [15]. Many studies have used CNN-based
procedures in musculoskeletal radiology listed as lesion detection (fractures,
cartilage abnormalities, ACL, and meniscal tears), classification
(osteoarthritis quantification, grading of spinal stenosis, and bone age
assessment), and segmentation [15]. Among the different methods of DL, VGG-16
was used in various applications as in detecting intertrochanteric hip
fractures [18], vertebral body localization (L3) [19], determining sex by hand
radiographs [20], and segmentation of knee MRI [21]. VGG16 network, a CNN used
in 2014 competition of ImageNet Large Scale Visual Recognition Competition
(ILSVRC), is trained on ImageNet [16] (a 1000 class labels of 1.28-M training
images).

In this study, our objectives, besides introducing a deep learning approach
for estimation of anatomical acetabular version using AP X-rays, are
summarized as follows: 1) to evaluate the correlation of acetabulum angle
determined by the proposed method and by CT-scan as the gold standard, 2) to
determine the mean value of acetabular version in the study population based
on their age, 3) to determine the mean value of acetabular version in the
study population based on their gender, and 4) to determine the mean value of
acetabular version in the study population. Applying DL method in AP
radiographs to estimate the anatomical version of the acetabulum, to the best
of authors’ knowledge, is the first study of this kind.

The methodology section, section 2, investigates AP pelvis radiographs and CT
scans of 300 patients, the algorithm of DL method, statistical analysis, and
cross-validation strategy. The achieved results of the above-mentioned
purposes using DL approach are reported in section 3. Discussion and
conclusion are the last parts in our study represented in sections 4 and 5,
respectively.

##  2 Materials and Method

###  2.1 Dataset

In order to collect our dataset, we searched the patient archiving and
communications system (PACS) of Sina University Hospital (Tehran University of
Medical Sciences, Tehran, Iran) for multiple trauma-listed patients
retrospectively. The Patients whose anterior-posterior pelvic (supine)
radiographs along with abdominopelvic CT scans were available in the PACS
system had been selected. The selected cohort contains 300 subjects (244 men
and 56 women ranged 13 to 92 years old) who aged $\geq$ 13 years and their
both AP radiographs and abdominal and pelvic CT scans were recorded
simultaneously in Sina Hospital’s emergency room during the desired time of
September 2018-September 2020. We excluded subjects who: i) their CT images
showed fractures in their acetabulum, femoral head and neck, and pelvic ring,
ii) had previous surgery or fracture of the pelvic bone or proximal femur, and
iii) their AP pelvic images were of poor quality or non-standard (including
tilt or rotation). The distribution of the studied population according to
their sex together with the histogram of the distribution for the population’s
age are presented in Fig.1. In tables 1 and 2, the information of angle of
both hips for both sexuality populations and three age groups studied in our
research are reported, respectively.

Figure 1: The distribution of the studied population according to their sex & the histogram of the distribution for the population’s age. Table 1: The information of angle of both hips for both sexuality populations. Gender | n | age | Right angle | Left angle
---|---|---|---|---
Male | 244 | 37.72$\pm$16.01 | 16.54$\pm$5.28 | 16.11$\pm$5.43
Female | 56 | 46.95$\pm$16.39 | 20.61$\pm$5.70 | 19.55$\pm$5.20
All | 300 | 39.44$\pm$16.47 | 17.30$\pm$5.58 | 16.75$\pm$5.55
Table 2: The information of angle of both hips for three age groups. Age (years) | n | Right angle | Left angle
---|---|---|---
$\leq$45 | 191 | 16.10$\pm$5.33 | 15.50$\pm$5.30
45-65 | 75 | 19.09$\pm$5.39 | 18.36$\pm$5.30
$\geq$65 | 34 | 20.06$\pm$5.52 | 20.23$\pm$5.15

The technique used for anterior-posterior pelvic radiographs is a standard
view [22], acquired in the supine position, at 120 cm as the distance of tube-
to-image and a photon beam centered midway between the pubic symphysis and the
top of the iliac crests without any rotation in the patient’s pelvis. A
Philips Ingenuity Flex 16 slice CT-scan system was used to acquire spiral CT
images of cases in the supine position, and 1.5 mm slices in axial plane were
used. A medical student under direct supervision of a hip surgeon (SHS)
estimated hip angles of both sides using CT scans of patients and recorded
them as the gold standard.

###  2.2 Model

The Attention on Pretrained-VGG16 for Bone Age model [23] proposed by Mader is
the DL approach used in our study. He added an attention mechanism with
convolutional layers to the pre-trained VGG16 [24]. Convolution layers,
assuming as feature extractors, are the main blocks in CNNs [19]. When we
stack several convolution layers, each of them extracting new features from
previous ones, the hierarchical features from our input will be built
[19],[25]. This particular architecture is reported as a success in lots of
tasks in computer vision [25].

Patient privacy regulations and highly trained human experts for image
annotation make medical images rare and inaccessible in order to train neural
networks [25]. To overcome this challenge, the transfer learning has become
the center of attention [19],[25]. So the model, VGG16 here, is trained using
a vast labeled dataset, ImageNet in this model, attention mechanism is added
and then fine-tuned using medical images to be adapted for our target of
acetabular version estimation. The good performance of this solution, training
on a vast labeled dataset and fine-tuning using medical images, is reported in
[26]-[27]. Moreover, data shortage as well as model generalization are also
addressed using different techniques of data augmentation similarly height and
width shifts and zooming, all with the range of 0.05 and the filling mode is
set to nearest [17], all are performed during the training stage. For our
purpose, rotation cannot be used as an augmentation technique since right and
left angles of hips may be considered inaccurately. We also mirror our AP
images as well as the angles. This approach is performed for the training
subset in which during the training step, one of the original AP or the
mirrored ones for each subject is selected randomly.

As our AP radiographs are in different sizes we rescale them to 1024$\ast$1024
in the preprocessing stage and then perform normalization. These grayscale
radiographs are sent to the model as the input. In the next stage we
concatenate three of each input image to make a $1024\ast 1024\ast 3$ image
for the input to CNN of pre-trained VGG16. As shown in the diagram of the
study (Fig.2), batch normalization is done after performance of VGG16. The
attention mechanism is added since the whole model focuses on the most
relevant regions of radiographs. Before pooling, the gap’s pixels become
on/off, followed by the lambda layer to rescale the pixels. The outputs, right
and left angles, are then max normalized. At this point, the gender and age of
patients are as two other inputs sent to two different neurons, with gender as
1 for male and 0 for female and ages of subjects are normalized between 0-1.
At the last stage, the right and left angles are predicted. It is worth
mentioning that for the first time, here the predicted output is anatomical
version, and not computed from radiograph images.

Figure 2: Block diagram of the proposed model.

Our model is trained over 1000 epochs. Every 30 epochs, if the loss function
doesn’t decrease meaningfully, the learning rate (lr) would decrease from the
base value ($10^{-3}$) by a factor of 0.8. At the end of the training stage,
we observed that lr reached to $10^{-4}$. Nesterov-accelerated Adaptive
Moment, NADAM is selected as the optimizer with mean square error (MSE) as
loss function. Employing an optimization algorithm (NADAM) explained as a kind
of improved ADAM (Adaptive Movement Estimation) which adds NAG (Nesterov’s
Accelerated Gradient), sometimes called Nesterov momentum, is considered to be
an improved type of momentum presented by Dozat [28].

###  2.3 Cross-Validation

Here, in order to assess the performance of the proposed algorithm, the model
should be trained on the train subset and then be tested on the test subset,
which is not used in the training process. The employed strategy for this
stage is five-fold cross-validation, where the whole dataset is split into
five equal folds. Then in five different experiments, the network is trained
using only three folds, validated in one of another folds, and the last one is
kept to test the network, whereupon in all the five steps, the test dataset is
unique and different. This step confirms the independency of the method on the
selected test subsample.

For this reason, we used a Core i7 processor system empowered by a CPU of 3.4
GHz and RAM of 32 GB besides Nvidia GeForce GTX 1080 GPU by the library of
Tensorflow.

Mean, SD and frequency are reported for all the quantitative continuous and
qualitative nominal variable parameters.

##  3 Results

In the studied population, the values of both hips’ angles computed from CT
images increased as people aging, with the mean values of 16.54 and 16.11
(right and left angles) for men and 20.61 and 19.55 degrees for women. As our
output was the angle of right and left hips, scatter plots of both angles of
ground truth (CT scan) and predicted by proposed methods for men and women are
depicted in Fig.3 and Fig.4, respectively. The average error in each plot is
also reported with less than 3 degrees as accurate, between 3 to 6 degrees as
moderate, and more than 6 degrees as poor.

Figure 3: The scatter plots of both angles of ground truth (CT scan) and
predicted by proposed methods for men aged 13-92 years old. Figure 4: The
scatter plots of both angles of ground truth (CT scan) and predicted by
proposed methods for women aged 13-92 years old.

In both measurements of left and right angles for the male population, the
average error proves our accurate prediction (green range of error). As shown
in Fig.1 and table 1 the values of angles in our dataset are limited for women
and so the results are situated in the moderate range. Also Fig.5 shows the
scatter plot of all the population with an accurate range of error in both
hips.

Figure 5: The scatter plots of both angles of ground truth (CT scan) and
predicted by proposed methods for all the population aged 13-92 years old.

These results are also reported in table 3. It can be seen that the low error
of prediction in the accurate region is evidence of the power of the proposed
method in angle prediction of both hips in average.

Table 3: The computed error in both hips angle prediction for 5 different folds and on average, for male, female and both gender. |  |  | Male | Female | Both Gender
---|---|---|---|---|---
| #Men | #Women | Left-Angle | Right-Angle | Left-Angle | Right-Angle | Left-Angle | Right-Angle
|  |  | -Error | -Error | -Error | -Error | -Error | -Error
Fold1 | 51 | 9 | 3.16$\pm$2.60 | 3.10$\pm$2.50 | 3.21$\pm$1.98 | 2.59$\pm$2.71 | 3.17$\pm$2.50 | 3.02$\pm$2.51
Fold2 | 48 | 12 | 3.14$\pm$2.33 | 2.81$\pm$2.71 | 3.59$\pm$2.96 | 4.15$\pm$4.41 | 3.23$\pm$2.44 | 3.08$\pm$3.12
Fold3 | 50 | 10 | 2.93$\pm$2.22 | 3.23$\pm$2.55 | 1.74$\pm$1.21 | 3.10$\pm$1.85 | 2.73$\pm$2.12 | 3.21$\pm$2.43
Fold4 | 48 | 12 | 2.78$\pm$3.07 | 2.70$\pm$2.54 | 3.24$\pm$3.04 | 2.78$\pm$3.30 | 2.87$\pm$3.04 | 2.72$\pm$2.68
Fold5 | 47 | 13 | 2.69$\pm$2.31 | 2.63$\pm$2.12 | 3.11$\pm$3.47 | 3.08$\pm$2.99 | 2.78$\pm$2.58 | 2.73$\pm$2.32
Average |  |  | 2.94$\pm$2.51 | 2.89$\pm$2.48 | 3.01$\pm$2.72 | 3.14$\pm$3.05 | 2.96$\pm$2.55 | 2.31$\pm$2.61

The average error of right and left angles for the whole dataset is exhibited
in Fig.6. These displayed charts are also the proof of the ability of the
proposed method in recognition of the version in which the majority of errors
are less than 6 degrees.

Figure 6: The average error of right and left angles for the whole dataset.

The error of predicted angles for both hips and the average histogram of
angles in Fig.7 illustrates the ratio of frequency and average error of angles
in which the average error gets lower as the frequency of angles gets higher.
The color of bars in the histogram shows the average error of both hips
according to the mean average range.

Figure 7: The error of predicted angles for both hips and the average
histogram of angles.

As you can see in Fig.1, the majority of people are aged 13 to 57 years old.
Fig.8 reports the scatter plots of both angles of ground truth (CT scan) and
predicted by proposed methods for the whole population aged 23-55 and 60-92
years old. As we expected, the more number of people, the lower average error
is reported for both hips angles prediction.

Figure 8: The scatter plots of both angles of ground truth (CT scan) and
predicted by proposed methods for the whole population aged (a) 23-55 and (b)
60-92 years old.

All of our results (Figures 3-8 and table 3) prove the ability of the proposed
method for predicting right and left angles of hips, leads us to achieve the
acceptable error.

##  4 Discussion

The orientation of the acetabulum (version) is a significant feature in the
biomechanics [6] and pathologies of the hip joint. Determination of acetabulum
version is an essential factor in both predicting and following up the outcome
in THA in which malposition of the cup is associated with impingement,
dislocation, severe wear, and reduction in the range of motion, force patients
for revision surgeries.

Assessment of the orientation of acetabulum has been done by 3-D imaging
systems, CT-scan/MRI, and AP pelvic X-rays. The CT scan of the whole body is
presented as the gold standard method. The cheap and safe AP radiograph
systems are the initial choice of imaging tool in patients with hip pain. In
this study, we used AP radiographs as an excellent method for assessment of
the orientation of the acetabulum. Based on applying DL, the subset of AI, and
its development in image diagnosis problems including radiology, the optimum
performance of CNN-based algorithm is reported in a wide range of literature.
We also examined the performance of attention on pre-trained VGG16 for the
bone age model for our purpose of anatomical acetabular version measurement
using available AP images, reducing the need for CT scans.

The input of the model was AP pelvic radiographs of patients and as the output
the angle of the right and left hips were computed. Our reported results
demonstrate the ability of the model for version estimation. Thanks to AI
capability, acetabular version measurement can be done by only using anterior-
posterior pelvic X-rays, eliminating the need for high radiation and expensive
CT scans.

As a limitation in our study, despite the bold advantages (no need for CT
images, so anatomical version estimation can be done based on only cheap, low-
radiation, and accessible AP radiographs), the AP pelvic radiographs are not
really standard. To overcome the resulting error, we considered the minimum
properties of a standard photo, namely (i) symmetric obturator foramina, (ii)
symmetric iliac wings, and (iii) sacrum and symphysis pubic being at the same
line.

##  5 Conclusion

To conclude, in our current study, we proposed a method to estimate version in
hip disorders and THA. Our results show that suggested algorithm, applying
pre-trained vgg16 on the AP images of the pelvis of patients followed by the
attention model, can assess anatomical version accurately. We achieved it by
using only AP radiographs in which obviate the necessity for CT. The applied
technique we used for estimating anatomical acetabular version (based on AP
pelvic images using DL approaches), to the best of authors’ knowledge, has not
been published yet.

##  6 Conflict of interest statement

The authors declare that there are no conflicts of interest.

## References

  * [1] D.W. MURRAY, “THE DEFINITION AND MEASUREMENT OF ACETABULAR ORIENTATION”. J Bone Joint Surg [Br] 1993 ; Vol. 75-B, No. 2:228-32.
  * [2] D. Suzuki, S. Nagoya, H. Takashima, K. Tateda, T. Yamashita, “Three-dimensional orientation of the acetabulum”. Clin Anat 2017;30 (6):753e60.
  * [3] J.W. Mast, R.L. Brunner, J. Zebrack, “Recognizing acetabular version in the radiographic presentation of hip dysplasia”. Clinical Orthopaedics and Related Research. 2004 Jan (418):48-53.
  * [4] V. Krebs, S.J. Incavo, W.H. Shields, “The Anatomy of the Acetabulum: What is Normal?”. Clin Orthop Relat Res. 2009 Apr; 467(4): 868–875.
  * [5] A. Cardoso, M. Duarte, R. Viegas, F. Oliveira, P. Pinto, and P. Rego, “Measuring Acetabular Version after Total Hip Arthroplasty: A Comparison of Two Radiographic Methods”. Rev Bras Ortop. 2021 Mar; 56(4): 513–516.
  * [6] T.D. Lerch, I.A.S. Todorski, S.D. Steppacher, F. Schmaranzer, S.F. Werlen,K.A. Siebenrock, and M. Tannast, “Prevalence of Femoral and Acetabular Version Abnormalities in Patients With Symptomatic Hip Disease: A Controlled Study of 538 Hips”. Am J Sports Med, 2018. 46(1): p. 122-134.
  * [7] A. Klasan, T. Neri, C. Sommer, M.A. Leie, Ph. Dworschak, M.D. Schofer, T.J. Heyse, “Analysis of acetabular version: Retroversion prevalence, age, side and gender correlations”. Journal of Orthopaedic Translation, 2019. 18: p. 7-12.
  * [8] K.L. Welton, M.K. Jesse, M.J. Kraeutler, T. Garabekyan, O. Mei-Dan, “The Anteroposterior Pelvic Radiograph”, The Journal of Bone and Joint Surgery: January 3, 2018 - Volume 100 - Issue 1 - p 76-85
  * [9] M. Raj, A. Jaiman and R. Kumar Chopra, “Evaluation of version of acetabular component following total hip replacement on conventional radiograph and computed tomography”. Journal of Orthopaedics, Trauma and Rehabilitation. Vol. 27(2), December 2020:179-185.
  * [10] K.G. SEAGRAVE, A. TROELSEN, H. MALCHAU, H. HUSTED, and K. GROMOV, “Acetabular cup position and risk of dislocation in primary total hip arthroplasty”. Acta Orthop, 2017. 88(1): p. 10-17.
  * [11] T.E. Snijders, T.P.C. Schlösser, S.M. van Gaalen, R.M. Castelein, H. Weinans, A. de Gast, “Non-equivalent Results from Different Anteversion Measurements Methods for the Evaluation of the Acetabular Cup Orientation in Total Hip Arthroplasty”. Orthop Surg, 2019. 11(2): p. 241-247.
  * [12] W.R. Hendee, G.J. Becker, J.P. Borgstede, J. Bosma, W.J. Casarella, B.A. Erickson, C.D. Maynard, J.H. Thrall, P.E. Wallner, “Addressing Overutilization in Medical Imaging”. Radiology, 2010. 257(1): p. 240-245.
  * [13] M. Wang, D. Li, X. Shang, J. Wang, "A review of computer-assisted orthopaedic surgery systems". Int J Med Robot, 2020: p. e2118.
  * [14] P.H. Yi, J. Wei, T.K. Kima, H.I. Sair, F.K. Hui, G.D. Hager, J. Fritz, J.K. Oni, "Automated detection & classification of knee arthroplasty using deep learning". Knee, 2020. 27(2): p. 535-542.
  * [15] P. Chea, and J.C. Mandell, "Current applications and future directions of deep learning in musculoskeletal radiology". Skeletal Radiol, 2020. 49(2): p. 183-197.
  * [16] B.J. Erickson, P. Korfiatis, Z. Akkus, T.L. Kline, "Machine Learning for Medical Imaging". Radiographics, 2017. 37(2): p. 505-515.
  * [17] A. Jodeiri , R.A. Zoroofi , Y. Hiasa , M. Takao , N. Sugano , Y. Sato , Y. Otake, "Fully automatic estimation of pelvic sagittal inclination from anterior-posterior radiography image using deep learning framework". Comput Methods Programs Biomed, 2020. 184: p. 105282.
  * [18] T. Urakawa, Y. Tanaka1, S. Goto1, H. Matsuzawa, K. Watanabe, N. Endo, "Detecting intertrochanteric hip fractures with orthopedist-level accuracy using a deep convolutional neural network". Skelet Radiol. 2019; 48:239–44.
  * [19] S. Belharbi, C. Chatelain, R. Hérault, S. Adam, S. Thureau, M. Chastan, R. Modzelewski, "Spotting L3 slice in CT scans using deep convolutional network and transfer learning". Comput Biol Med. 2017 ;87:95–103.
  * [20] S. Yune, H. Lee, M. Kim, S.H. Tajmir, M.S. Gee, S, Do, "Beyond Human Perception: Sexual Dimorphism in Hand and Wrist Radiographs Is Discernible by a Deep Learning Model". J Digit Imaging [Internet]. 2018 [cited 2019 Apr 8];
  * [21] Z. Zhou, G. Zhao, R. Kijowski, F. Liu, "Deep convolutional neural network for segmentation of knee joint anatomy: Zhou et al". Magn Reson Med. 2018;80:2759–70.
  * [22] J.C. Clohisy, J.C. Carlisle, P.E. Beaulé, Y.J. Kim, R.T. Trousdale, R.J. Sierra, M. Leunig, P.L. Schoenecker, and M.B. Millis, "A Systematic Approach to the Plain Radiographic Evaluation of the Young Adult Hip", The Journal of Bone & Joint Surgery: November 2008 - Volume 90 - Issue Supplement-4 - p 47-66,
  * [23] K. Mader, :Attention on Pretrained-VGG16 for Bone Age.” Kaggle, https://www.kaggle.com/kmader/attention-on-pretrained-vgg16-for-bone-age.
  * [24] C. Kimmel, A. Ojjeh, and S. Safwan, "Predicting Bone Age from Hand Radiographs Using. Deep Convolutional Neural Networks". CS230: Depp Learning, Spring 2018 - Stanford University.
  * [25] H. Lee, S. Tajmir, J. Lee, M. Zissen, B.A. Yeshiwas, T.K. Alkasab, G. Choy, S. Do, "Fully Automated Deep Learning System for Bone Age Assessment". J Digit Imaging 30, 427–441 (2017).
  * [26] H.C Shin, H.R. Roth, M. Gao, L. Lu, Z. Xu, I. Nogues, J. Yao, D. Mollura, R.M. Summers, "Deep convolutional neural networks for computer-aided detection: CNN architectures, dataset characteristics and transfer learning". IEEE Trans Med Imaging. 2016;35: 1285–1298.
  * [27] N. Tajbakhsh, J.Y. Shin, S.R. Gurudu, R.T. Hurst, C.B. Kendall, M.B. Gotway, J. Liang, "Convolutional neural networks for medical image analysis: full training or fine tuning?" IEEE Trans Med Imaging. 2016;35: 1299–1312.
  * [28] T. Dozat. "Incorporating Nesterov Momentum into Adam". Workshop track - ICLR 2016.