# Design of Quantum error correcting code for biased error on heavy-hexagon
structure

Younghun Kim<EMAIL_ADDRESS>Jeongsoo Kang<EMAIL_ADDRESS>Younghun Kwon<EMAIL_ADDRESS>Department of Applied Physics, Center for
Bionano Intelligence Education and Research, Hanyang University, Ansan 15588,
Republic of Korea

###### Abstract

Surface code is an error-correcting method that can be applied to the
implementation of a usable quantum computer. At present, a promising candidate
for a usable quantum computer is based on superconductor-specifically
transmon. Because errors in transmon-based quantum computers appear biasedly
as Z type errors, tailored surface and XZZX codes have been developed to deal
with the type errors. Even though these surface codes have been suggested for
lattice structures, since transmons-based quantum computers, developed by IBM,
have a heavy-hexagon structure, it is natural to ask how tailored surface code
and XZZX code can be implemented on the heavy-hexagon structure. In this
study, we provide a method for implementing tailored surface code and XZZX
code on a heavy-hexagon structure. Even when there is no bias, we obtain
$0.231779\%$ as the threshold of the tailored surface code, which is much
better than $0.210064\%$ and $0.209214\%$ as the thresholds of the surface
code and XZZX code, respectively. Furthermore, we can see that even though a
decoder, which is not the best of the syndromes, is used, the thresholds of
the tailored surface code and XZZX code increase as the bias of the Z error
increases. Finally, we show that in the case of infinite bias, the threshold
of the surface code is $0.264852\%$, but the thresholds of the tailored
surface code and XZZX code are $0.296157\%$ and $0.328127\%$ respectively.

##  I Introduction

Quantum computers contain special features such as superposition and
entanglement, which are different from those of classical computers. Those
features help quantum computers find prime numbersref:shor_algorithm  or
perform database searchesref:grover_algorithm ; ref:j.bae  at a significantly
higher speed compared to classical computers.
A physical system of a superconductor or ion trap is used to construct a
quantum computer. The quantum hardware of the present generation is called
Noisy Intermediate-Scale Quantum (NISQ) hardwareref:nisq . This is because in
the quantum system, errors such as error of interaction between qubits, error
of constructing qubits, and error due to the external environment. Therefore,
these errors should be handled appropriately. One method to do this is to use
a quantum error correcting code(QEC code)ref:intro ; ref:majorana ;
ref:surface ; ref:anyon .
The quantum error-correcting code consists of a series of steps, such as
preparation of a logical state, measurement of a syndrome, and the use of a
decoder to find a correction operator from measurement values. At present, the
most well-known quantum error-correcting code is the surface code, which
constructs a logical qubit using a stabilizer made of four adjacent qubits on
the lattice structureref:stabilizer ; ref:google .
A promising candidate among the NISQ hardware is a quantum computer using a
superconductor. In particular, transmon-based quantum computer is known as a
best candidate for constructing realistic quantum computersref:crgate ;
ref:chow ; ref:takita ; ref:corcoles ; ref:krinner . In NISQ hardware
including superconducting quantum computers, errors of the Z operator occur
more frequently than those of other types. Because of this fact, there have
been quantum error correcting codes dealing with the bias of the Z
errorref:asymmetric ; ref:asymmetric2 ; ref:azad , which are Tailored Surface
coderef:tailored ; ref:tailored_symmetry ; ref:tailored_fragile  and XZZX code
ref:xzzx ; ref:cat_xzzx . When the properties of bias and a symmetry of the
syndrome are used, those codes provide a better threshold compared to the
surface code, even though they have the identical form of assignment in qubits
on the lattice structure.
However, the present quantum computers made by IBM, restrict the number of
connections to perform a low error rate operation. Therefore, it uses a heavy-
hexagon structure instead of a lattice structureref:heavy-hexagon . In a
heavy-hexagon structure, the maximum number of adjacent qubits is three,
instead of four.
Therefore, one can ask how a quantum error-correction code can be constructed
in a heavy-hexagon structure. Constructing a measurement circuit for the
syndrome is crucial to implementing a quantum error-correcting code in the
heavy-hexagon structure. Because heavy-hexagon structures have a low-degree
connection, the connection restriction requires the use of flag qubitref:fault
; ref:bridge ; ref:flag ; ref:ibm_bridge ; ref:surfmap . Moreover, as
explained earlier, the characteristic of existing errors, which implies that
errors of the Z operator occur more frequently compared to other types, should
be considered. Therefore, in this study, we propose a method to construct
Tailored Surface code and XZZX code on a heavy-hexagon structure. In
particular, we provide a measurement circuit for the syndrome in the tailored
surface code and the XZZX code on the heavy-hexagon structure.
Furthermore, we perform simulations on the proposed measurement circuit for
the syndrome to obtain the threshold of the surface code, tailored surface
code, and XZZX code not only in the lattice structure but also in the heavy-
hexagon structure. First, we can see that when there is no bias in the Z
error, the threshold of the tailored surface code is $0.231779\%$, which shows
better performance than the thresholds of the surface code and XZZX code being
$0.210064\%$ and $0.209214\%$, respectively. In addition, we show that even
when using a decoder, which is not most suitable for the syndrome, as the bias
of the Z error increases, the thresholds of the tailored surface code and XZZX
code increase. Finally, we can see that when the infinite bias is considered,
the thresholds of the tailored surface code and XZZX code are $0.296157\%$ and
$0.328127\%$, respectively. These values are larger than the threshold of the
surface code of $0.264852\%$.

This study is organized as follows. First, we explain the assignment of data
qubits, flag qubits, and syndrome qubits in the heavy-hexagon structure. Next,
we describe the design of the measurement circuit for the syndrome in the
proposed qubit structure. Then, we obtain the thresholds of the surface code,
tailored surface code, and XZZX code in the lattice and heavy-hexagon
structure.

##  II  Result

###  II.1 Tailored Surface code & XZZX code measurement circuit

To build QEC codes, errors should be detected by measuring syndrome qubits. In
this section, we provide a quantum circuit to implement the stabilizer
measurement of the tailored surface code and XZZX code in a heavy-hexagon
structure. The tailored surface code and XZZX code have the same qubit
structure despite the different stabilizers compared to the surface code.
In the following subsection, we first provide the positions of the data, flag,
and syndrome qubits in a heavy-hexagon structure. Second, we explain the
quantum circuit of stabilizer measurement to measure the syndrome using data
qubits, flag qubits, and syndrome qubits.

####  II.1.1  Heavy-hexagon architecture

Usually, QEC codes based on surface code need to check the parity of the four
data qubits. The lattice structure is designed such that each qubit is
connected to four different neighboring qubits. Meanwhile, in a heavy-hexagon
structure, qubits have less connectivity than in lattice structures. In other
words, each qubit of the heavy-hexagon structure cannot be entangled with four
qubits directly. To solve the issue, we use more auxiliary qubits so-called
flag qubits, so that some qubits can interact indirectly with four different
qubitsref:surfmap . Therefore, three kinds of qubits, such as data qubits to
build logical qubit state, syndrome qubits to detect errors, and flag qubits
to connect data qubits and the syndrome qubits are required to implement a
surface code in a heavy-hexagon structure. Fig.1 shows the arrangement of data
qubits, flag qubits, and syndrome qubits for an error-correcting code of
[[13,1,3]] in the Ithaca system with a heavy-hexagon structure of 65
superconducting qubits of IBMref:demonstration. Four distinguishable data
qubits are linked to the syndrome qubits along the flag qubits at each
stabilizer patches.

Figure 1:  [[13,1,3]] Surface code in the heavy-hexagon structure. The data
qubits(flag qubits) are represented by blue dots(red dots). Meanwhile, the
syndrome qubits are denoted by orange and green dots.

In Fig.1, the blue and red dots denote the data and flag qubits, respectively.
$S_{1}$ and $S_{2}$ indicate the stabilizers. The black edges indicate that
two lined-up qubits can be entangled. The implementation consists of 13 data
qubits, 36 flag qubits, and 12 syndrome qubits in the Ithaca system. Distance
3 surface-code based QEC codes, which can correct at least a single error, use
the same number of the data qubits and syndrome qubits except for the flag
qubits in the lattice structure. The tailored surface code, XZZX code, and
surface code have identical qubits arrangements regardless of the type of
stabilizer, but different stabilizer measurement circuits. The logical qubit
state $|{\psi}\rangle_{L}$ is defined as a data qubit state that has $S_{1}$
and $S_{2}$ stabilizers whose eigenvalues are equal to $+1$. An eigenvalue of
$-1$ is occured when there are some errors that anti-commute with the
corresponding stabilizer.

####  II.1.2  Measurement circuit of Tailored Surface code

The tailored surface code uses a Y stabilizer instead of a Z stabilizer. The Y
stabilizer checks the parity of the X and Z errors simultaneously, and the X
stabilizer checks the parity of the Z errors.

$\hat{X}_{a}\hat{X}_{b}\hat{X}_{c}\hat{X}_{d},\hat{Y}_{a}\hat{Y}_{b}\hat{Y}_{c}\hat{Y}_{d}$
(1)

The stabilizer for the four data qubits a,b,c, and d can be described as
above. The stabilizers are used to check the parity of the error from the
ancilla qubits in the stabilizer measurement circuit. The Y stabilizers
provide the result of the parity of the X or Z error. Two stabilizer types can
detect and correct X and Z errors. In particular, the parity information of
the Z errors from both stabilizers can improve the error-correcting
performance when the Z errors are dominantref:tailored_symmetry . Here $S_{1}$
($S_{2}$) denotes the group of Y stabilizers(X stabilizers). Each stabilizer
consists of at least three Pauli operators.

$\displaystyle S_{1}$
$\displaystyle=\\{\hat{Y}_{1}\hat{Y}_{16}\hat{Y}_{28},\hat{Y}_{5}\hat{Y}_{16}\hat{Y}_{20}\hat{Y}_{32},\hat{Y}_{9}\hat{Y}_{20}\hat{Y}_{36},\hat{Y}_{28}\hat{Y}_{44}\hat{Y}_{55},\hat{Y}_{32}\hat{Y}_{44}\hat{Y}_{48}\hat{Y}_{59},\hat{Y}_{36}\hat{Y}_{48}\hat{Y}_{63}\\}$
(2) $\displaystyle S_{2}$
$\displaystyle=\\{\hat{X}_{1}\hat{X}_{5}\hat{X}_{16},\hat{X}_{5}\hat{X}_{9}\hat{X}_{20},\hat{X}_{16}\hat{X}_{28}\hat{X}_{32}\hat{X}_{44},\hat{X}_{20}\hat{X}_{32}\hat{X}_{36}\hat{X}_{48},\hat{X}_{44}\hat{X}_{55}\hat{X}_{59},\hat{X}_{48}\hat{X}_{59}\hat{X}_{63}\\}$
(3)

Figure 2:  Y Stabilizer using six flag qubits. (a) Connection among qubits.
(b) Measurement circuit. The blue dots denote data qubits, and the red dots
denote flag qubits. H is a syndrome qubit. For a measurement of the Y
syndrome, the quantum states of the flag qubits(syndrome qubits) are prepared
as $|{0}\rangle$($|{+}\rangle$).

Fig.2 shows the Y stabilizer measurement circuit, where four data qubits, six
flag qubits, and one Y syndrome qubit are displayed as blue, red, and green
dots, respectively. In the Y stabilizer measurement circuit, the interaction
between the data qubit and flag qubit is performed through a controlled Y
operator, which is different from the CNOT operator used in the Z and X
stabilizer measurement circuits. If the Y syndrome qubit is measured in the X
basis and the result of measurement of Y stabilizer becomes
$|{+}\rangle$($|{-}\rangle$), the eigenvalue is +1(-1). When the controlled Y
is used for the Y stabilizer measurement, the qubit error to the data qubit or
flag qubit produces various results in the Y stabilizer measurement.

Fig.3(a) shows the case in which data qubit has an X error. Fig3(b) shows how
the measurement result is changed in the Y stabilizer measurement circuit when
there is an X error in the data qubit. The X error in the data qubit causes
the measurement result to flip and $|{-}\rangle$ is measured. If an X error
occurs additionally in different data qubit, the measurement result of the
syndrome flips twice and $|{+}\rangle$ is measured. Therefore, if the X errors
may be occurred even(odd) times, we get $|{+}\rangle$($|{-}\rangle$) in the Y
stabilizer measurement.
It can be observed that when there is an X error in the data qubit the parity
to the error can be checked by the measurement result of the Y syndrome;
however, there is no change in the measurement result for the flag qubit. It
should be noted that the error in the data qubit causes a propagation of the Z
error after an interaction between the data qubit and flag qubit. However,
$|{0}\rangle$ is robust with respect to the Z error. Therefore, when Y
stabilizer measurement circuit is prepared, the initial state of flag qubits
needs to be $|{0}\rangle$ in order that an error in a data qubit should not
affect the measurement of the flag qubits.

Figure 3: (a) Case where an X error occurs in Y stabilizer measurement circuit
(b) Change in the measurement when an X error occurs in Y stabilizer
measurement circuit

Fig.4 shows the case in which an Z error occurs in the data qubit. When an Z
error occurs in the data qubit, Fig.4(a) shows how the measurement circuit of
the Y stabilizer is performed and Fig.4(b) shows how the error affects the
measurement circuit of the Y stabilizer. The blue color denotes the flip of
the measurement result in the Y syndrome qubit, which changes from
$|{+}\rangle$ to $|{-}\rangle$. Both the X and Z errors in the data qubit
affect the measurement results of the Y syndrome qubit. Therefore, the Y error
in the data qubit causes it to flip twice and does not affect the measurement
result of the Y syndrome qubit.

Figure 4:  (a) Case where an Z error of a data qubit occurs in Y stabilizer
measurement circuit (b) Change in the measurement when an Z error of a data
qubit occurs in Y stabilizer measurement circuit Figure 5: (a) Case where an X
error of a flag qubit occurs in Y stabilizer measurement circuit (b) Changes
in the data qubit and the measurement when an X error of a flag qubit occurs
in Y stabilizer measurement circuit

Not only the data qubit but also the flag qubit may have an error in the Y
stabilizer measurement circuit. The error in the flag qubit may be propagated
to the data qubit. In order to detect the error propagated from the flag qubit
to the data qubit, every initial state of the flag qubit needs to be set as
$|{0}\rangle$. The CNOT is constructed by choosing the flag qubit adjacent to
the syndrome qubit as the target qubit and the syndrome qubit as the control
qubit.

Fig.5 shows the case in which an X error occurs in the flag qubit. The X error
of the flag qubit causes a Y error propagation to the data qubit. Error
propagation to the data qubit by the error of the flag qubit can be checked
through the measurement result of the flag qubit. That is, the Y stabilizer
measurement circuit enables checking not only the parity to X or Z error of
the data qubit but also the error propagation to the data qubit by the error
of the flag qubit. The tailored surface code also requires X stabilizer
measurement circuit, as shown in Fig.6(a). Similar to the Y stabilizer
measurement circuit, four data qubits(six flag qubits) are denoted by blue
dots(red dots) and one X syndrome qubit is displayed by orange dot. In the X
stabilizer measurement circuit, the interaction between the data qubit and
flag qubit is performed by the CNOT operation, where the flag qubit is used as
the control qubit and the corresponding the data qubit is used as the target
qubit.

Figure 6:  X Stabilizer using six flag qubits. (a) Connection among qubits.
(b) Measurement circuit. The blue dots denote data qubits, and the red dots
denote flag qubits. H is a syndrome qubit. For a measurement of the X
syndrome, the quantum states of flag qubits(syndrome qubits) are prepared as
$|{0}\rangle$($|{+}\rangle$).

Fig.6(b) shows how the parity to check the Z error of the data qubit can be
obtained in the X stabilizer measurement circuit. The initialization of each
flag qubit is prepared as $|{0}\rangle$ and the initialization of the X
syndrome qubit as $|{+}\rangle$. The difference between the X and Y
stabilizers exists when there is an X error in the data qubit and it does not
affect the X stabilizer’s syndrome result but the Y stabilizer’s syndrome
result. This is because of the difference between the CNOT and Controlled-Y
operators when constructing the entanglement between the flag qubits and the
data qubits.

####  II.1.3  Measurement Circuit of XZZX code

Every stabilizer of XZZX Code has only one type. The XZZX stabilizer consists
of operators to four data qubits and the XZZX stabilizer corresponding to
a,b,c, and d qubits is given as follows:

$\hat{X}_{a}\hat{Z}_{b}\hat{Z}_{c}\hat{X}_{d}$ (4)

Fig.7 shows the measurement circuit of the XZZX stabilizer. And the following
$S_{1}$ and $S_{2}$ are the group of XZZX stabilizers.

$\displaystyle S_{1}$
$\displaystyle=\\{\hat{X}_{1}\hat{Z}_{16}\hat{X}_{28},\hat{X}_{5}\hat{Z}_{16}\hat{Z}_{20}\hat{X}_{32},\hat{X}_{9}\hat{Z}_{20}\hat{X}_{36},\hat{X}_{28}\hat{Z}_{44}\hat{X}_{55},\hat{X}_{32}\hat{Z}_{44}\hat{Z}_{48}\hat{X}_{59},\hat{X}_{36}\hat{Z}_{48}\hat{X}_{63}\\}$
(5) $\displaystyle S_{2}$
$\displaystyle=\\{\hat{Z}_{1}\hat{Z}_{5}\hat{X}_{16},\hat{Z}_{5}\hat{Z}_{9}\hat{X}_{20},\hat{X}_{16}\hat{Z}_{28}\hat{Z}_{32}\hat{X}_{44},\hat{X}_{20}\hat{Z}_{32}\hat{Z}_{36}\hat{X}_{48},\hat{X}_{44}\hat{Z}_{55}\hat{Z}_{59},\hat{X}_{48}\hat{Z}_{59}\hat{Z}_{63}\\}$
(6)

The measurement circuit of the XZZX stabilizer is composed of four data
qubits, six flag qubits, and one syndrome qubit. The interaction between the
data qubit and flag qubit is important in the measurement circuit of the XZZX
stabilizer, because the measurement result of the flag qubit is used to detect
error propagation. It should be noted that Hadamard operator is required to
check the error propagation to the data qubit by the error of the flag qubit.
The Hadamard operator can transform a bit-flip error into a phase-filp error.
The circuit is designed such that all flag qubits are initialized as
$|{0}\rangle$, and Hadamard gates are applied such that data errors do not
disturb the flag qubit states. Therefore, one can detect the data qubit errors
with stabilizer measurement results identically as in the case of lattice-
structure-based XZZX code.

Figure 7: XZZX Stabilizer using six flag qubits. (a) Connection among qubits.
(b) Measurement circuit. The blue dots denote data qubits, and the red dots
denote flag qubits. For a measurement of the XZZX syndrome, the quantum states
of the flag qubits(syndrome qubits) are prepared as
$|{0}\rangle$($|{+}\rangle$). The interaction between the data qubit and the
flag qubit is established by CNOT and H operators.

The surface code, tailored surface code and XZZX code have two stabilizer
groups. We measure two stabilizer group syndrome simultaneously in the lattice
structure. However, we check them individually in the heavy-hexagon structure,
because the two stabilizer group measurement circuits share partial flag
qubits. Therefore, we do not need to consider the hook error in a heavy-
hexagon structure.

###  II.2  Physical Qubit Error Model

####  II.2.1  Depolarization Error Model

The X,Y and Z errors of a single qubit occur equally in the depolarization
error model. Therefore, when the probability of error in a qubit is p and
$p_{X},p_{Y},p_{Z}$ denote the error probabilities of the X,Y, and Z errors
respectively, the following relation holds:

$\frac{p}{3}=p_{X}=p_{Y}=p_{Z}$ (7)

The depolarization error model is an error channel on the qubit, where the
probabilities of the X,Y, and Z errors are identical.

####  II.2.2  Biased Dephasing Error Model

Real quantum computer hardware, such as a transmon-based quantum computer, is
susceptible to the Z errors, which implies that the Z errors are more frequent
than the X or Y errors. This situation can be described using a biased
dephasing error model. The biased dephasing error model is expressed by the
degree of bias $\eta$ which shows the different error probabilities in $\
p_{X},p_{Y}$, and $p_{Z}$. These satisfy the following relationship:

$\begin{array}[]{cc}&\frac{p}{2(\eta+1)}=p_{X}=p_{Y}\\\ \\\
&\frac{p\eta}{\eta+1}=p_{Z}\end{array}$ (8)

Therefore, the depolarization error model corresponds to the case of $\
\eta=0.5$ in the biased dephasing error model. Therefore, if $\eta=\infty$
then only the Z error will occur.

$\begin{array}[]{cc}p_{X}+p_{Y}+p_{Z}=p\\\ \end{array}$ (9)

In the error channel of Pauli operators, the sum of all types of errors is
$p$. The details of the biased dephasing error model are as follows:

  1. 1.

In a single gate, the qubit on which the gate acts, passes through an
independent error channel in the error probabilities of $p_{X},p_{Y}$, and
$p_{Z}$ at $\\{X,Y,Z\\}$ operators.

  2. 2.

In a CNOT gate, the two qubits, which the gate acts on, pass independently and
uniformly through the error channel of
$\\{I,X,Y,Z\\}^{\otimes{2}}\backslash\\{I\otimes I\\}$ in the error
probability of $p$.

  3. 3.

The qubits are prepared as the state of $\hat{X}|{0}\rangle=|{1}\rangle$ in a
probability of $p_{X}+p_{Y}$.

  4. 4.

A flip error occurs in every measurement result on a single qubit, with a
probability of $p_{X}+p_{Y}$.

  5. 5.

In a stabilizer measurement circuit, the data qubits pass independently
through an error channel in the error probabilities of $p_{X},p_{Y}$, and
$p_{Z}$ at $\\{X,Y,Z\\}$ operators..

Here the error models of the controlled-Y operator and CNOT operators are
treated in the same manner. In addition, it is assumed that every error has an
independent and identical distribution.

###  II.3  Simulation results

In this section we evaluate the thresholds of the tailored surface code, XZZX
code, and surface code in terms of the biased dephasing error model, including
the depolarization error model. In particular, when the bias is in
$\eta=\\{0.5,1,10,100,1000,\infty\\}$, we obtain the thresholds of the
tailored surface code, XZZX code, and surface code on the lattice and heavy-
hexagon structure. To evaluate the thresholds, we use the measurement circuit
for the syndrome described in the previous section and the stim coderef:stim
package. In addition, to obtain the correction operator, we use
PyMatchingref:pymatching , considering the X and Y errors independently in the
tailored surface code and the X and Z errors independently in the XZZX code
and surface code. In the decoder, we do not consider the correlation between
the errors. In the cases of the surface code and XZZX code, a logical X error
and a logical Z error exist. In the tailored surface code, the logical X and Y
errors exist. We obtain the logical failure by evaluating the similarity
between the logical state to initialized state of the data qubit and the final
state applied by the correction operator. For instance, in the surface code,
the logical X error rate is evaluated by checking whether the corrected state
is the same as $|{0}\rangle_{L}$ when the initialized state is prepared as
$|{0}\rangle_{L}$. Meanwhile, the logical Z error rate is evaluated by
checking whether the corrected state is the same as $|{+}\rangle_{L}$ when the
initialized state is prepared as $|{+}\rangle_{L}$. In QEC code, to consider
every error of two logical operators, $E_{total}$ is obtained by the logical
error rates $E_{X}$ and $E_{Z}$ as followsref:logic_error :

$\begin{array}[]{cc}E_{total}=1-(1-E_{X})\times(1-E_{Z})\\\ \end{array}$ (10)

The following figures show the thresholds in terms of error probabilities $p$
and $\eta$. Here, we perform simulations using at least 1200000 samples and
obtain results for $d=\\{3,5,7,9,11\\}$. Here, to correct the measurement, we
perform the error correction after measuring the syndrome in the repetition of
d round.

Figure 8: The thresholds of the surface code, tailored surface code, and XZZX
code in the lattice structure are displayed according to the bias. The red
circle displays the thresholds of the surface code, and the green triangle
displays the thresholds of the tailored surface code. The blue rectangle
displays the thresholds of the XZZX code.

Fig.8 shows the behavior of the threshold in all QEC codes on a lattice
structure. The threshold of the surface code is $0.543786\%$ at $\eta=0.5$,
which implies no bias. As $\eta$ increases to $\eta=\infty$, the threshold of
the surface code becomes $0.478029\%$. In addition, the threshold of the
tailored surface code becomes $0.534635\%$ at $\eta=0.5$ and diminishes to
$0.480034\%$ at $\eta=\infty$. In the XZZX code, the threshold at $\eta=0.5$
becomes $0.407061\%$, which is the lowest value. In addition, as the bias
increases, the threshold also increases. In the case of $\eta=\infty$, the
threshold becomes the largest value of $0.541467\%$, compared to the
thresholds of other codes.

Figure 9: Thresholds of the surface code, and tailored surface code, XZZX code
in the heavy-hexagon structure, according to the bias. The red circle displays
the thresholds of the surface code, and the green triangle displays the
thresholds of the tailored surface code. The blue rectangle displays the
thresholds of the XZZX code.

Finally, the thresholds of the QEC codes for the logical error are
investigated on the heavy-hexagon structure. Fig.9 shows the results. First,
in the case of the tailored surface code, the threshold for the logical error
becomes $0.231779\%$ at $\eta=0.5$. Second, in the case of the surface code
code, the threshold for the logical error becomes $0.210064\%$ at $\eta=0.5$.
In addition, the threshold of the XZZX code for the logical error has the
lowest value of $0.209214\%$ at $\eta=0.5$, compared to the thresholds of the
tailored surface code and surface code. When there is no bias, the threshold
of the tailored surface code becomes the highest value. When $\eta=\infty$,
the threshold of the XZZX code to the logical error becomes $0.328127\%$. The
threshold of the tailored surface code in the lattice structure decreases as
the bias increases, but the threshold of the tailored surface code in the
heavy-hexagon structure increases. Threshold of the tailored surface code
becomes $0.296157\%$ at $\eta=\infty$ which is higher than the threshold of
the surface code $0.264852\%$ at $\eta=\infty$. Therefore, in the heavy-
hexagon structure, the threshold of the tailored surface code is higher than
that of the surface code in all regions of bias. The tailored surface code has
a higher threshold value than the XZZX code when the bias is low. In the
heavy-hexagon structure, the thresholds of the QEC codes do not change rapidly
after the value of $\eta=100$. It should be noted that in the heavy hexagon
structure the threshold of the XZZX code provides a larger value than those of
the other error-correcting codes in the range of $\eta$ which is larger than
$\eta=10$.
Fig.10 shows the thresholds of the surface code, tailored surface code, and
XZZX code in the heavy-hexagon structure when the bias is 100. The thresholds
of each code are $0.263515\%$, $0.292152\%$, and $0.324207\%$ respectively.

(a) Surface code

(b) Tailored Surface code

(c) XZZX code

Figure 10: (a) Thresholds of surface code in the heavy-hexagon structure (b)
Thresholds of tailored surface code in the heavy-hexagon structure (c)
Thresholds of XZZX Code in the heavy-hexagon structure. The thresholds of
surface code, tailored surface code, and XZZX code in the heavy-hexagon
structure are displayed when the bias is 100.

##  III Conclusion

In this work, we provided the design of quantum error correcting codes in the
heavy-hexagon structure which is used in the hardware of IBM quantum
computers. First, we showed the method to assigning data qubits, syndrome
qubits, and flag qubits in the heavy-hexagon structure to construct a
measurement circuit for the syndrome of quantum error correcting code. Second,
we built a measurement circuit for the syndrome by considering the
entanglement among the three types of qubits. In the tailored surface code, we
used the Controlled-Y operator to create a measurement circuit for the the Y
syndrome. In the XZZX code, we used the Hadamard operator to detect the data
qubit errors.
Furthermore, we compared the thresholds of quantum error-correcting codes
designed in the heavy-hexagon structure with those of quantum error correcting
codes in the lattice structure. Our results showed that the flag qubits and
syndrome qubits might affect the performance of quantum error correcting codes
in a heavy-hexagon structure. Because the heavy-hexagon structure requires a
restriction on the connection, the flag qubits to overcome the restriction are
indispensable. It should be noted that because the number of the flag qubits
in the heavy-hexagon structure is more than half of the total number of
qubits, the errors of the flag qubits can have a sensitive effect on the
performance of quantum error correcting codes. Therefore, our results imply
that the quantum states of the flag qubits can be an important element in the
performance of quantum error-correcting codes in a heavy-hexagon structure. In
a future work, we will consider building a decoder using the correlation
obtained from the result of measurement of the flag qubits in the tailored
surface and XZZX codes.

## Acknowledgements

This work is supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education, Science and Technology (NRF2018R1D1A1B07049420 and
NRF2022R1F1A1064459) and Institute of Information and Communications
Technology Planning and Evaluation (IITP) grant funded by the Korean
government (MSIT) (No. 2020001343, Artificial Intelligence Convergence
Research Center (Hanyang University ERICA).

## Data availability

All data generated or used during the study appear in the submitted article.

##  Appendix A Measurement circuit of Z stabilizer on heavy-hexagon structure

To construct a surface code in a heavy-hexagon structure, the measurement
circuits of the X and Z stabilizers should be built. Because the measurement
circuits of the X stabilizer are discussed in the main text, in the Appendix,
we explain the measurement circuits of the Z stabilizer.

Figure 11: Z Stabilizer using six flag qubits in the heavy-hexagon structure.
(a) Connection among qubits. (b) Measurement circuit. The blue dots denote
data qubits, and the red dots denote flag qubits. H is a syndrome qubit. For a
measurement of the Z syndrome, the quantum states of the syndrome qubits(flag
qubits) are prepared as $|{+}\rangle$($|{0}\rangle$).

Fig.11.shows the assignment of qubits and gates to construct a measurement
circuit for the Z syndrome in the heavy-hexagon structure. In the measurement
circuit of the Z stabilizer, the X error is passed to the flag qubit through
CNOT operator to evaluate the parity of the X error of the data qubit.

The parity of the X error, which passed through the flag qubit to the syndrome
qubit, is checked in the syndrome qubit. It should be noted that the quantum
state of the flag qubits should be prepared as $|{0}\rangle$ and applied
Hadamard operators, in order for the X error of the data qubit not to affect a
flag qubit. The quantum state of the syndrome qubit is prepared as
$|{+}\rangle$, to check the parity of the X error. In addition, in the
measurement circuit of the Z stabilizer the CNOT operator is constructed using
the flag qubit adjacent to a data qubit as the target qubit and the flag qubit
adjacent to the syndrome qubit as the control qubit. The CNOT gate between a
data qubit and a flag qubit is built using the data qubit as the control qubit
and the flag qubit as the target qubit.

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11institutetext: 1 Beihang University, Beijing, China
2 Shanghai Artificial Intelligence Laboratory, Shanghai, China
3 Zhongguancun Laboratory, Beijing, China
11email: {shengxu, yanjingli, bohanzeng<EMAIL_ADDRESS>
# IDa-Det: An Information Discrepancy-aware Distillation for 1-bit Detectors

Sheng Xu1† Yanjing Li1† Bohan Zeng1† Teli Ma2 Baochang Zhang1,3∗ Xianbin Cao1
Peng Gao2 Jinhu Lü1,3

###### Abstract

Knowledge distillation (KD) has been proven to be useful for training compact
object detection models. However, we observe that KD is often effective when
the teacher model and student counterpart share similar proposal information.
This explains why existing KD methods are less effective for 1-bit detectors,
caused by a significant information discrepancy between the real-valued
teacher and the 1-bit student. This paper presents an Information Discrepancy-
aware strategy (IDa-Det) to distill 1-bit detectors that can effectively
eliminate information discrepancies and significantly reduce the performance
gap between a 1-bit detector and its real-valued counterpart. We formulate the
distillation process as a bi-level optimization formulation. At the inner
level, we select the representative proposals with maximum information
discrepancy. We then introduce a novel entropy distillation loss to reduce the
disparity based on the selected proposals. Extensive experiments demonstrate
IDa-Det’s superiority over state-of-the-art 1-bit detectors and KD methods on
both PASCAL VOC and COCO datasets. IDa-Det achieves a 76.9% mAP for a 1-bit
Faster-RCNN with ResNet-18 backbone. Our code is open-sourced on
https://github.com/SteveTsui/IDa-Det.

###### Keywords:

1-bit detector, Knowledge distillation, Information discrepancy

††footnotetext: $\dagger$ Equal contribution.††footnotetext: $*$ Corresponding
author.

##  1 Introduction

Recently, the object detection task [6, 19] has been greatly promoted due to
advances in deep convolutional neural networks (DNNs) [11]. However, DNN
models comprise a large number of parameters and floating-point operations
(FLOPs), restricting their deployment on embedded platforms. Techniques such
as compact network design [14, 23], network pruning [12, 15, 37], low-rank
decomposition [5], and quantization [25, 32, 35] have been developed to
address these restrictions and accomplish an efficient inference on the
detection task. Among

Figure 1: Input images and saliency maps following [9]. Images are randomly
selected from VOC test2007. Each row includes: (a) input images, saliency maps
of (b) Faster-RCNN with ResNet-101 backbone (Res101), (c) Faster-RCNN with
ResNet-18 backbone (Res18), (d) 1-bit Faster-RCNN with ResNet-18 backbone
(BiRes18), respectively.

these, binarized detectors have contributed to object detection by
accelerating the CNN feature extracting for real-time bounding box
localization and foreground classification [33, 30, 34]. For example, the
1-bit SSD300 [20] with VGG-16 backbone [27] theoretically achieve the
acceleration rate up to 15$\times$ with XNOR and Bit-count operations using
binarized weights and activations as described in [30]. With extremely high
energy-efficiency for embedded devices, they are able to be installed directly
on next-generation AI chips. Despite these appealing features, 1-bit
detectors’ performance often deteriorates to the point, which explains why
they are not widely used in real-world embedded systems.

The recent art [34] employs fine-grained feature imitation (FGFI) [29] to
enhance the performance of 1-bit detectors. However, it neglects the intrinsic
information discrepancy between 1-bit detectors and real-valued detectors. As
shown in Fig. 1, we demonstrate that saliency maps of real-valued Faster-RCNN
of the ResNet-101 backbone (often used as the teacher network) and the
ResNet-18 backbone, in comparison with 1-bit Faster-RCNN of the ResNet-18
backbone (often used as the student network) from top to bottom. They show
that knowledge distillation (KD) methods like [29] are effective for
distilling real-valued Faster-RCNNs, only when their teacher model and their
student counterpart share small information discrepancy on proposals, as shown
in Fig. 1 (b) and (c). This phenomenon does not happen for 1-bit Faster-RCNN,
as shown in Fig. 1 (b) and (d). This might explain why existing KD methods are
less effective in 1-bit detectors. A statistic on COCO and PASCAL VOC datasets
in Fig. 2 show that the discrepancy between proposal saliency maps of Res101
and Res18 (blue) is much smaller than that of Res101 and BiRes18 (orange).
That is to say, the smaller the distance is, the less the discrepancy is.
Briefly, conventional KD methods show their effectiveness on distilling real-
valued detectors but seem to be less effective on distilling 1-bit detectors.

(a) VOC trainval0712

(b) VOC test2007

(c) COCO trainval35k

(d) COCO minival

Figure 2: The Mahalanobis distance of the gradient in the intermediate neck
feature between Res101-Res18 (blue) and Res101-BiRes18 (orange) in various
datasets.

In this paper, we are motivated by the above observation and present an
information discrepancy-aware distillation for 1-bit detectors (IDa-Det),
which can effectively address the information discrepancy problem, leading to
an efficient distillation process. As shown in Fig. 3, we introduce a
discrepancy-aware method to select proposal pairs and facilitate distilling
1-bit detectors, rather than only using object anchor locations of student
models or ground truth as in existing methods [29, 34, 9]. We further
introduce a novel entropy distillation loss to leverage more comprehensive
information than the conventional loss functions. By doing so, we achieve a
powerful information discrepancy-aware distillation method for 1-bit detectors
(IDa-Det). Our contributions are summarized as:

  * •

Unlike existing KD methods, we distill 1-bit detectors by fully considering
the information discrepancy into optimization, which is simple yet effective
for learning 1-bit detectors.

  * •

We propose an entropy distillation loss further to improve the representation
ability of the 1-bit detector and effectively eliminate the information
discrepancy.

  * •

We compare our IDa-Det against state-of-the-art 1-bit detectors and KD methods
on the VOC and large-scale COCO datasets. Extensive results reveal that our
method outperformas the others by a considerable margin. For instance, on VOC
test2007, the 1-bit Faster-RCNN with ResNet-18 backbone achieved by IDa-Det
obtains 76.9% mAP, achieving a new state-of-the-art.

##  2 Related Work

1-bit Detectors. By removing the foreground redundancy, BiDet [30] fully
exploits the representational capability of the binarized convolutions. In
this way, the information bottleneck is introduced, which limits the amount of
data in high-level feature maps, while maximizing the mutual information
between feature maps and object detection. The performance of the Faster R-CNN
detector is significantly enhanced by the ASDA-FRCNN [33] which suppresses the
shared amplitude between the real-value and binary kernels. LWS-Det [34]
novelly proposes a layer-wise searching approach, minimizing the angular and
amplitude errors for 1-bit detectors. Also, FGFI [29] is used by LWS-Det to
distill the backbone feature map further.

Figure 3: Overview of the proposed information discrepancy-aware distillation
(IDa-Det) framework. We first select representative proposal pairs based on
the information discrepancy. Then we propose the entropy distillation loss to
eliminate the information discrepancy.

Knowledge Distillation. Knowledge distillation (KD), a significant subset of
model compression methods, aims to transfer knowledge from a well-trained
teacher network to a more compact student model. The student is supervised
using soft labels created by the teacher, as firstly proposed by [1].
Knowledge distillation is redefined by [13] as training a shallower network
after the softmax layer to approximate the teacher’s output. Object detectors
can be compressed using knowledge distillation, according to numerous recent
papers. Chen et al. [2] distill the student through all backbone features,
regression head, and classification head, but both the imitation of whole
feature maps and the distillation in classification head fail to add attention
to the important foreground, potentially resulting in a sub-optimal result.
Mimicking [16] distills the features from sampled region proposals. However,
just replicating the aforementioned regions may lead to misdirection, because
the proposals occasionally perform poorly. In order to distill the student,
FGFI [29] introduces a unique attention mask to create fine-grained features
from foreground object areas. DeFeat [9] balances the background and
foreground object regions to efficiently distill the student.

In summary, existing KD frameworks for object detection can only be employed
for real-valued students having similar information as their teachers. Thus,
they are often ineffective in distilling 1-bit detectors. Unlike prior arts,
we identify that the information discrepancy between real-valued teacher and
1-bit students are significant for distillation. We first introduce
Mahalanobis distance to identify the information discrepancy and then
accordingly distill the features. Meanwhile, we propose a novel entropy
distillation loss to prompt the discrimination ability of 1-bit detectors
further.

##  3 The Proposed Method

In this section, we describe our IDa-Det in detail. Firstly, we overview the
1-bit CNNs. We then describe how we employ the information discrepancy method
(IDa) to select representative proposals. Finally, we describe the entropy
distillation loss to delicately eliminate the information discrepancy between
the real-valued teachers and the 1-bit students.

###  3.1 Preliminaries

In a specific convolution layer, ${\bf w}\in\mathbb{R}^{C_{out}\times
C_{in}\times K\times K}$, ${\bf a}_{in}\in\mathbb{R}^{C_{in}\times
W_{in}\times H_{in}}$, and ${\bf a}_{out}\in\mathbb{R}^{C_{out}\times
W_{out}\times H_{out}}$ represent its weights and feature maps, where $C_{in}$
and $C_{out}$ represents the number of channels. $(H,W)$ are the height and
width of the feature maps, and $K$ denotes the kernel size. We then have

$\small{\bf a}_{out}={\bf a}_{in}\otimes{\bf w},$ (1)

where $\otimes$ is the convolution operation. We omit the batch normalization
(BN) and activation layers for simplicity. The 1-bit model aims to quantize
${\bf w}$ and ${\bf a}_{in}$ into ${\bf b}^{\bf
w}\in\\{-1,+1\\}^{C_{out}\times C_{in}\times K\times K}$ and ${\bf b}^{{\bf
a}_{in}}\in\\{-1,+1\\}^{C_{in}\times H\times W}$ using the efficient XNOR and
Bit-count operations to replace full-precision operations. Following [3, 4],
the forward process of the 1-bit CNN is

$\small{{\bf a}_{out}}=\bm{\alpha}\circ{{\bf b}^{{\bf a}_{in}}}\odot{{\bf
b}^{\bf w}},$ (2)

where $\odot$ is the XNOR, and bit-count operations and $\circ$ denotes the
channel-wise multiplication.
$\bm{\alpha}=[\alpha_{1},\cdots,\alpha_{C_{out}}]\in\mathbb{R}_{+}$ is the
vector consisting of channel-wise scale factors. ${\bf b}={\rm sign(\cdot)}$
denotes the binarized variable using the sign function, which returns 1 if the
input is greater than zero, and -1 otherwise. It then enters several non-
linear layers, e.g., BN layer, non-linear activation layer, and max-pooling
layer. We omit these for simplification. And then, the output ${\bf a}_{out}$
is binarized to ${\bf b}^{{\bf a}_{out}}$ via the sign function. The
fundamental objective of BNNs is calculating ${\bf w}$. We want it to be as
close as possible before and after binarization, such that the binarization
effect is minimized. Then, we define the reconstruction error as

$\small{L}_{R}({\bf w},\bm{\alpha})={\bf w}-\bm{\alpha}\circ{\bf b}^{\bf w}.$
(3)

###  3.2 Select proposals with Information Discrepancy

To eliminate the large magnitude scale difference between the real-valued
teacher and the 1-bit student, we introduce a channel-wise transformation for
the proposals††In this paper, the proposal denotes the neck/backbone feature
map patched by the region proposal of detectors. of the intermediate neck. We
first apply a transformation $\varphi(\cdot)$ on a proposal
$\tilde{R}_{n}\in\mathbb{R}^{C\times W\times H}$ and have

${R}_{n;c}(x,y)=\varphi(\tilde{R}_{n;c}(x,y))=\frac{\operatorname{exp}(\frac{\tilde{R}_{n;c}(x,y)}{\mathcal{T}})}{\sum_{(x^{\prime},y^{\prime})\in(W,H)}\operatorname{exp}(\frac{\tilde{R}_{n;c}(x^{\prime}y^{\prime})}{\mathcal{T}})},$
(4)

Figure 4: Illustration for the generation of the proposal pairs. Each single
proposal in one model generates a counterpart feature map patch in the same
location of the other model.

where $(x,y)\in(W,H)$ denote a specific spatial location $(x,y)$ in spatial
range $(W,H)$, and $c\in\\{1,\cdots,C\\}$ is the channel index.
$n\in\\{1,\cdots,N\\}$ is the proposal index. $N$ denotes the number of
proposals. $\mathcal{T}$ denotes a hyper-parameter controlling the statistical
attributions of the channel-wise alignment operation††In this paper, we set
$\mathcal{T}=4$.. After the transformation, features in each channel of a
proposal projected into the same feature space [28] and follow a Gaussian
distribution as

${p}({R}_{n;c})\sim\mathcal{N}(\mu_{n;c},{\sigma^{2}_{n;c}}).$ (5)

We further evaluate the information discrepancy between the proposals of the
teacher and the student. As shown in Fig. 4, the teacher and student have
$N_{T}$ and $N_{S}$ proposals, respectively. Each single proposal in one model
generates a counterpart feature map patch in the same location of the other
model, thus total $N_{T}+N_{S}$ proposal pairs are considered. To evaluate the
information discrepancy, we introduce the Mahalanobis distance of each
channel-wise proposal feature and measure the discrepancy as

$\displaystyle{\varepsilon}_{n}=\sum_{c=1}^{C}||({R}^{t}_{n;c}-{R}^{s}_{n;c})^{T}{\Sigma^{-1}_{n;c}}({R}^{t}_{n;c}-{R}^{s}_{n;c})||_{2},$
(6)

where $\Sigma_{n;c}$ denotes the covariance matrix of the teacher and student
in the $c$-th channel of the $n$-th proposal pair. The Mahalanobis distance
takes both the pixel-level distance between proposals and the statistical
characteristics differences in proposal pairs into account.

To select representative proposals with maximum information discrepancy, we
first define a binary distillation mask $m_{n}$ as

$\displaystyle m_{n}=\left\\{\begin{aligned} &1,\;\;\text{if pair
$(R_{n}^{t},R_{n}^{s})$ is selected}\\\
&0,\;\;\text{otherwise}\end{aligned}\right.$ (7)

where $m_{n}=1$ denotes distillation will be applied on such proposal pair,
otherwise remain unchanged. For each proposal pair, only when their
distribution is quite different, the student model can learn from the teacher
counterpart, where a distillation process is needed.

Based on the derivation above, discrepant proposal pairs will be optimized
through distillation. For distilling the selected pairs, we resort to maximize
conditional probability $p({R}_{n}^{s}|{R}_{n}^{t})$. That is to say, after
distillation or optimization, feature distributions of teacher proposal and
student counterpart become similar with each other. To this end, we define
$p({R}_{n}^{s}|{R}_{n}^{t})$ with $m_{n},n\in\\{1,\cdots,N_{T}+N_{S}\\}$ in
consideration as

$\small
p({R}_{n}^{s}|{R}_{n}^{t};m_{n})\sim{m_{n}}\mathcal{N}(\mu_{n}^{t},{\sigma_{n}^{t}}^{2})+(1-m_{n})\mathcal{N}(\mu_{n}^{s},{\sigma_{n}^{s}}^{2}).$
(8)

Subsequently, we introduce a bi-level optimization formulation to solve the
distillation problem as

$\displaystyle\mathop{\max}_{{R}_{n}^{s}}$ $\displaystyle
p({R}_{n}^{s}|{R}_{n}^{t};{\bf
m^{*}}),\;\;\forall\;n\in\\{0,\cdots,N_{T}+N_{S}\\},$ (9) $\displaystyle{\rm
s.t.}$ $\displaystyle{\bf m^{*}}=\mathop{\arg\max}_{{\bf
m}}\sum^{N_{T}+N_{S}}_{n=1}m_{n}\varepsilon_{n},$

where ${\bf m}=[m_{1},\cdots,m_{N_{T}+N_{S}}]$ and $||{\bf
m}||_{0}=\gamma\cdot(N_{T}+N_{S})$. $\gamma$ is a hyper-parameter. In this
way, we select $\gamma\cdot(N_{T}+N_{S})$ pairs of proposals containing the
most representative information discrepancy for distillation. $\gamma$
controls the proportion of discrepant proposal pairs, further validated in
Sec. 4.2.

For each iteration, we first solve the inner-level optimization, i.e.,
proposal selection, via exhaustive sorting [31]; and then solve the upper-
level optimization, distilling the selected pair, based on entropy
distillation loss discussed in Sec. 3.3. Considering that there are not too
many proposals involved, the process is relatively efficient for the inner-
level optimization.

###  3.3 Entropy distillation loss

After selecting a specific number of proposals, we crop the feature based on
the proposals we obtained. Most of the SOTA detection models are based on
Feature Pyramid Networks (FPN) [18], which can significantly improve the
robustness of multi-scale detection. For the Faster-RCNN framework in this
paper, we resize the proposals and crop the features from each stage of the
neck feature maps. We generate the proposals from the regression layer of the
SSD framework and crop the features from the feature map of maximum spatial
size. Then we formulate the entropy distillation process as

$\mathop{\rm{max}}_{R_{n}^{s}}\;p({R_{n}^{s}}|{R}_{n}^{t}).$ (10)

Here is the upper level of the bi-level optimization, where $\bm{m}$ is solved
and therefore omitted. We rewrite Equ. 10 and further achieve our entropy
distillation loss as

$L_{P}({\bf w},\bm{\alpha};{\gamma})=({{R}_{n}^{s}}-{R}_{n}^{t})+{\rm
Cov}({{R}_{n}^{s}},{{R}_{n}^{t}})^{-1}({{R}_{n}^{s}}-{R}_{n}^{t})^{2}+{\rm
log}({\rm Cov}({{R}_{n}^{s}},{{R}_{n}^{t}})),$ (11)

where ${\rm
Cov}({{R}_{n}^{s}},{{R}_{n}^{t}})=\mathbb{E}({{R}_{n}^{s}}{{R}_{n}^{t}})-\mathbb{E}({{R}_{n}^{s}})\mathbb{E}({{R}_{n}^{t}})$
denotes the covariance matrix.

Hence, we trained the 1-bit student model end-to-end, the total loss for
distilling the student model is defined as

$\small L=L_{GT}({\bf w},\bm{\alpha})+\lambda L_{P}({\bf
w},\bm{\alpha};{\gamma})+\mu L_{R}({\bf w},\bm{\alpha}),$ (12)

where $L_{GT}$ is the detection loss derived from the ground truth label and
$L_{R}$ is defined in Equ. 3.

##  4 Experiments

On two mainstream object detection datasets, i.e, PASCAL VOC [6] and COCO
[19], extensive experiments are undertaken to test our proposed method. First,
we go through the implementation specifics of our IDa-Det. Then, in the
ablation studies, we set different hyper-parameters and validate the
effectiveness of the components as well as the convergence of our method.
Moreover, we illustrate the superiority of our IDa-Det by comparing it to
previous state-of-the-art 1-bit CNNs and other KD approaches on 1-bit
detectors. Finally, we analyze the deploy efficiency of our IDa-Det on
hardware.

###  4.1 Datasets and Implementation Details

PASCAL VOC. Natural images from 20 different classes are included in the VOC
datasets. We use the VOC trainval2012 and VOC trainval2007 sets to train our
model, which contain around 16k images, and the VOC test2007 set to evaluate
our IDa-Det, which contains 4952 images. We utilize the mean average precision
(mAP) as the evaluation matrices, as suggested by [6].

COCO. All our experiments on COCO dataset are conducted on the COCO 2014 [19]
object detection track in the training stage, which contains the combination
of 80k images and 80 different categories from the COCO train2014 and 35k
images sampled from COCO val2014, i.e., COCO trainval35k. Then we evaluate our
method on the remaining 5k images from the COCO minival. We list the average
precision (AP) for IoUs$\in[0.5:0.05:0.95]$, designated as mAP$@[.5,.95]$,
using COCO’s standard evaluation metric. For further analyzing our method, we
also list AP50, AP75, APs, APm, and APl.

Implementation Details. Our IDa-Det is trained with two mainstream object
detectors, i.e., two-stage Faster-RCNN††In this paper, Faster-RCNN denotes the
Faster-RCNN implemented with FPN neck. [26] and one-stage SSD [20]. In Faster-
RCNN, we utilize ResNet-18 and ResNet-34 [11] as backbones. And we utilize
VGG-16 [27] as the backbone of SSD framework. PyTorch [24] is used for
implementing IDa-Det. We run the experiments on 4 NVIDIA GTX 2080Ti GPUs with
11 GB memory and 128 GB of RAM. We use ImageNet ILSVRC12 to pre-train the
backbone of a 1-bit student, following [22]. The SGD optimizer is utilized and
the batch size is set as 24 for SSD and 4 for Faster-RCNN, respectively.

We keep the shortcut, first layer, and the last layer (the 1$\times$1
convolution layer of RPN and a FC layer of the bbox head) in the detectors as
real-valued on Faster-RCNN framework after implementing 1-bit CNNs following
[22]. Following BiDet [30], the extra layer is likewise retained as real-
valued for the SSD framework. Following [30] and [8], we modify the network of
ResNet-18/34 with an extra shortcut and PReLU [10].

The architecture of VGG-16 is modified with extra residual connections,
following [30]. The lateral connection of the FPN [18] neck is replaced with
3$\times$3 1-bit convolution for improving performance. This adjustment is
implemented in

(a) Effect of $\mu$.

(b) Effect of $\lambda$ and $\gamma$.

Figure 5: On VOC, we (a) select $\mu$ on raw detector and different KD methods
including Hint [2], FGFI [29], and IDa-Det; (b) select $\lambda$ and $\gamma$
on IDa-Det with $\mu$ set as $1e\\!\\!-\\!\\!4$.

all of the Faster-RCNN experiments. For Faster-RCNN, we train the model with
two stages. Only the backbone is binarized at the first stage. Then we
binarize all layers in the second stage. Each stage counts 12 epochs. The
learning rate is set as 0.004 and decays by multiplying 0.1 in the 9-th and
11-th epochs. We use a loss coefficient set as 5 and multi-scale training
method. For fair comparison, all the methods in this paper are implemented
with the same training setup. The real-valued counterparts in this paper are
also trained for 24 epochs for fair comparison. For SSD, the model is trained
for 24 epochs with a learning rate of 0.01, which decays to 0.1 at the 16-th
and 22-nd epochs by multiplying 0.1.

We select real-valued Faster-RCNN with ResNet101 backbone (81.9% mAP on VOC
and 39.8% mAP on COCO) and real-valued SSD300 with VGG16 backbone (74.5% mAP
on VOC and 25.0% mAP on COCO) as teacher network.

###  4.2 Ablation Study

Selecting the hyper-parameter. As mentioned above, we select hyper-parame
ters $\lambda$, $\gamma$, and $\mu$ in this part. We first select the $\mu$,
which controls the binarization process. As plotted in Fig. 5 (a), we first
fine-tune the hyper-parameter $\mu$ controlling the binarization process in
four situations: raw BiRes18, and BiRes18 distilled via Hint [2], FGFI [29],
and our IDa-Det, respectively. Overall, the performances increase first and
then decrease when increasing the value of $\mu$. On raw BiRes18 and IDa-Det
BiRes18, the 1-bit student achieves the best performance when $\mu$ is set as
1$e$-4. And $\mu$ valued 1$e$-3 is better for the Hint, and the FGFI distilled
1-bit student. Thus, we set $\mu$ as 1$e$-4 for an extended ablation study.
Fig. 5 (b) shows that the performances increase first and then decrease with
the increase of $\lambda$ from left to right. In general, the IDa-Det obtains
better performances with $\lambda$ set as 0.4 and 0.6. With varying value of
$\gamma$, we find $\\{\lambda,\gamma\\}$ = {0.4, 0.6} boost the performance of
IDa-Det most, achieving 76.9% mAP on VOC test2007. Based on the ablative study
above, we set hyper-parameters $\lambda$, $\gamma$, and $\mu$ as 0.4, 0.6, and
1$e$-4 for the experiments in this paper.

Effectiveness of components. We first compare our information discrepancy-
aware (IDa) proposal selecting method with other methods to select proposals:
Hint [2] (using the neck feature without region mask) and FGFI [29]. We show
the effectiveness of IDa on two-stage Faster-RCNN in Tab. 1. On the Faster-
RCNN, the introducing of IDa achieves improvements of the mAP by 2.5%, 2.4%,
and 1.8% compared to non-distillation, Hint, and FGFI, under the same student-
teacher framework. Then we evaluate the proposed entropy distillation loss
against the conventional $\ell_{2}$ loss, inner-product loss, and cosine
similarity loss. As depicted in Tab. 1, our entropy distillation loss improves
the distillation performance by 0.4%, 0.3%, and 0.4% with Hint, FGFI, and IDa
method compared with $\ell_{2}$ loss. Compared with inner-product and cosine
similarity loss, entropy loss outperforms them by 2.1% and 0.5% in mAP on our
framework, which further reflects the effectiveness of our method.

Model |  | Proposal
---
selection
| Distillation
---
method
mAP
Res18 | ✗ | ✗ | 78.6
BiRes18 | ✗ | ✗ | 74.0
Res101-BiRes18 | Hint | $\ell_{2}$ | 74.1
Res101-BiRes18 | Hint | Entropy loss | 74.5
Res101-BiRes18 | FGFI | $\ell_{2}$ | 74.7
Res101-BiRes18 | FGFI | Entropy loss | 75.0
Res101-BiRes18 | IDa | Inner-product | 74.8
Res101-BiRes18 | IDa | Consine similarity | 76.4
Res101-BiRes18 | IDa | $\ell_{2}$ | 76.5
Res101-BiRes18 | IDa | Entropy loss | 76.9
Table 1: The effects of different components in IDa-Det with Faster-RCNN model on PASCAL VOC dataset. Hint [2] and FGFI[29] are used to compare with our information discrepancy-aware proposal selection (IDa). IDa and Entropy loss denote main components of the proposed IDa-Det. Framework | Backbone |  | Quantization
---
Method
| KD
---
Method
W/A |  | Memory Usage
---
(MB)
| OPs
---
($\times 10^{9}$)
mAP
Faster-RCNN | ResNet-18 | Real-valued | ✗ | 32/32 | 112.88 | 96.40 | 78.8
DoReFa-Net | ✗ | 4/4 | 21.59 | 27.15 | 73.3
ReActNet | ✗ | 1/1 | 16.61 | 18.49 | 69.6
Ours | ✗ | 74.0
LWS-Det | 73.2
3-4 | Ours | FGFI | 74.7
| Ours | DeFeat | 74.9
3-4 | IDa-Det | 76.9
| ResNet-34 | Real-valued | ✗ | 32/32 | 145.12 | 118.80 | 80.0
| DoReFa-Net | ✗ | 4/4 | 29.65 | 32.31 | 75.6
| ReActNet | ✗ | 1/1 | 24.68 | 21.49 | 72.3
| Ours | ✗ | 75.0
| Ours | FGFI | 75.4
| Ours | DeFeat | 75.7
3-4 | LWS-Det | 75.8
3-4 | IDa-Det | 78.0
SSD | VGG-16 | Real-valued | ✗ | 32/32 | 105.16 | 31.44 | 74.3
DoReFa-Net | ✗ | 4/4 | 29.58 | 6.67 | 69.2
ReActNet | ✗ | 1/1 | 21.88 | 2.13 | 68.4
Ours | ✗ | 69.5
LWS-Det | 69.5
3-4 | Ours | FGFI | 70.0
| Ours | DeFeat | 71.4
3-4 | IDa-Det | 72.5
Table 2: We report memory usage, FLOPs, and mAP ($\%$) with state-of-the-art
1-bit detectors, other KD methods on VOC test2007. The best results are bold.

###  4.3 Results on PASCAL VOC

With the same student framework, we compare our IDa-Det with the state-of-the-
art 1-bit ReActNet [22] and other KD methods, such as FGFI [29], DeFeat [9],
and LWS-Det [34], in the task of object detection with the VOC datasets. The
detection results of the multi-bit quantization method DoReFa-Net [36] is also
reported. We use the input resolution following [34], i.e. $1000\times 600$
for Faster-RCNN and $300\times 300$ for SSD.

Tab. 2 lists the comparison of several quantization approaches and detection
frameworks in terms of computing complexity, storage cost, and the mAP. Our
IDa-Det significantly accelerates computation and reduces storage requirements
for various detectors. We follow XNOR-Net [25] to calculate memory usage,
which is estimated by adding 32$\times$ the number of full-precision kernels
and 1$\times$ the number of binary kernels in the networks. The number of
float operations (FLOPs) is calculated in the same way as Bi-Real-Net [21].
The current CPUs can handle both bit-wise XNOR and bit-count operations in
parallel. The number of real-valued FLOPs plus $\frac{1}{64}$ of the number of
1-bit multiplications equals the OPs following [21].

(a) False positives

(b) Missed detection

Figure 6: Qualitative results on the gain from information discrepancy-aware
distilling. The top row shows IDa-Det student’s output. The bottom row images
are raw student model’s output without information discrepancy-aware
distilling.

Faster-RCNN. We summarize the experimental results on VOC test2007 of 1-bit
Faster-RCNNs from lines 2 to 17 in Tab. 2. Compared with raw ReActNet [22],
our baseline binarization method achieves 4.4%, and 2.7% mAP improvement with
ResNet-18/34 backbone respectively. Compared with other KD methods on the
1-bit detector with the same train and test settings, our IDa-Det surpasses
FGFI and DeFeat distillation method in a clear margin of 2.2%, 2.0% with
ResNet-18 backbone, and 2.6%, 2.3% with ResNet-34 backbone. Our IDa-Det
significantly surpasses the prior state-of-the-art, LWS-Det, by 3.7% in
ResNet-18 backbone, and 2.2% in ResNet-34 backbone with the same FLOPs and
memory usage. All of the improvements have impacts on object detection.

Compared with the raw real-valued detectors, the proposed IDa-Det surpasses
real-valued Faster-RCNN with ResNet-18/34 backbone ({76.9%, 78.0%} vs.{76.4%,
77.8%}) by obviously computation acceleration and storage savings by
5.21$\times$/5.53$\times$ and 6.80$\times$/5.88$\times$. The above results are
of great significance in the real-time inference of object detection.

Framework | Backbone |  | Quantization
---
Method
| KD
---
Method
W/A |  | mAP
---
@$[.5,.95]$
AP50 | AP75 | APs | APm | AP1
Faster-RCNN | ResNet-18 | Real-valued | ✗ | 32/32 | 32.2 | 53.8 | 34.0 | 18.0 | 34.7 | 41.9
FQN | ✗ | 4/4 | 28.1 | 48.4 | 29.3 | 14.5 | 30.4 | 38.1
ReActNet | ✗ | 1/1 | 21.1 | 38.5 | 20.5 | 9.7 | 23.5 | 32.1
Ours | ✗ | 26.8 | 46.1 | 27.9 | 14.7 | 28.4 | 36.0
LWS-Det | 26.9 | 44.9 | 27.7 | 12.9 | 28.7 | 38.3
3-4 | Ours | FGFI | 27.5 | 46.5 | 28.8 | 15.2 | 28.7 | 37.5
| Ours | DeFeat | 27.9 | 46.9 | 29.3 | 15.8 | 29.0 | 38.2
3-4 | IDa-Det |  | 29.3 | 48.7 | 30.9 | 16.7 | 29.8 | 39.9
| ResNet-34 | Real-valued | ✗ | 32/32 | 35.8 | 57.6 | 38.4 | 21.1 | 39.0 | 46.1
| FQN | ✗ | 4/4 | 31.8 | 52.9 | 33.9 | 17.6 | 34.4 | 42.2
| ReActNet | ✗ | 1/1 | 23.4 | 43.3 | 24.4 | 10.7 | 25.9 | 35.5
| Ours | ✗ | 29.0 | 47.7 | 30.9 | 16.6 | 30.5 | 39.0
| Ours | FGFI | 29.4 | 48.4 | 30.3 | 17.1 | 30.7 | 39.7
| Ours | DeFeat | 29.8 | 48.7 | 30.9 | 17.6 | 31.4 | 40.5
3-4 | LWS-Det | 29.9 | 49.2 | 30.1 | 15.1 | 32.1 | 40.9
3-4 | IDa-Det | 30.5 | 49.2 | 31.8 | 17.7 | 31.3 | 40.6
SSD | VGG-16 | Real-valued | ✗ | 32/32 | 23.2 | 41.2 | 23.4 | 5.3 | 23.2 | 39.6
DoReFa-Net | ✗ | 4/4 | 19.5 | 35.0 | 19.6 | 5.1 | 20.5 | 32.8
ReActNet | ✗ | 1/1 | 15.3 | 30.0 | 13.2 | 5.4 | 16.3 | 25.0
Ours | ✗ | 17.2 | 31.5 | 16.8 | 3.2 | 18.2 | 31.3
LWS-Det | 17.1 | 32.9 | 16.1 | 5.5 | 17.4 | 26.7
3-4 | Ours | FGFI | 17.7 | 32.3 | 17.3 | 3.3 | 18.9 | 31.8
| Ours | DeFeat | 18.1 | 32.8 | 17.9 | 3.3 | 19.4 | 32.6
3-4 | IDa-Det | 19.4 | 34.5 | 19.3 | 3.7 | 21.1 | 35.0
Table 3: Comparison with state-of-the-art 1-bit detectors and KD methods on
COCO minival. Optimal results are bold.

SSD. On the SSD300 framework with a VGG-16 backbone, our IDa-Det can
accelerate computation and save storage by 14.76$\times$ and 4.81$\times$
faster than real-valued counterparts, respectively, as illustrated in the
bottom section of Tab. 2. The drop in the performance is relatively minor
(72.5% vs. 74.3%). Also, our method surpasses other 1-bit networks and KD
methods by a sizable margin. Compared to 1-bit ReActNet, our raw 1-bit model
can achieve 1.1% higher mAP with the same computation. Compared with FGFI,
DeFeat, and LWS-Det, our IDa-Det exceeds them by 3.0%, 2.5%, and 1.1%,
respectively.

As shown in Fig. 6, BiRes18 achieved by IDa-Det effectively eliminates the
false positives and missed detections compared with raw BiRes18. In summary,
we achieve new state-of-the-art performance on PASCAL VOC compared to previous
1-bit detectors and KD methods on various frameworks and backbones. We also
achieve competitive results, demonstrating the IDa-Det’s superiority.

###  4.4 Results on COCO

Because of its diversity and scale, the COCO dataset presents a greater
challenge in the object detection task, compared with PASCAL VOC. On COCO, we
compare our proposed IDa-Det with the state-of-the-art 1-bit ReActNet [22], as
well as the KD techniques FGFI [29], DeFeat [9], and LWS-Det [34]. We also
report the detection result of the 4-bit quantization method FQN [17] and the
DoReFa-Net [36] for reference. Following [34], the input images are scaled to
$1333\times 800$ for Faster-RCNN and $300\times 300$ for SSD.

The mAP, AP with different IoU thresholds, and AP of objects with varying
scales are all reported in Tab. 3. Due to the constraints in the width of
page, we do not report the FLOPs and memory use in Tab. 3. We conduct
experiments on Faster-RCNN and SSD detectors, the results of which are
presented in the folllowing two parts.

Faster-RCNN. Comparing to the state-of-the-art 1-bit ReActNet, our baseline
binarized model achieves a 5.7% improvement on mAP$@[.5,.95]$ with the
ResNet-18 backbone. Compared to state-of-the-art LWS-Det, FGFI, and DeFeat,
our IDa-Det prompts the mAP$@[.5,.95]$ by 2.4%, 1.8%, and 1.4%, respectively.
With the ResNet-34 backbone, the proposed IDa-Det surpasses FGFI, DeFeat, and
LWS-Det by 1.1%, 0.7%, and 0.6%, respectively. IDa-Det, nevertheless, has
substantially reduced FLOPs and memory use.

SSD. On the SSD300 framework, our IDa-Det achieves 19.4% mAP$@[.5,.95]$ with
the VGG-16 backbone, surpassing LWS-Det, FGFI, and DeFeat by 2.3%, 1.7%, and
1.3% mAP, respectively.

To summarize, our method outperforms baseline quantization methods and other
KD methods in terms of the AP with various IoU thresholds and the AP for
objects of varied sizes on COCO, indicating IDa-Det’s superiority and
generality in many application settings.

Framework | Network | Method | W/A | Latency (ms) | Acceleration
---|---|---|---|---|---
Faster-RCNN | ResNet-18 | Real-valued | 32/32 | 12043.8 | -
IDa-Det | 1/1 | 2474.4 | 4.87$\times$
ResNet-34 | Real-valued | 32/32 | 14550.2 | -
IDa-Det | 1/1 | 2971.3 | 4.72$\times$
SSD | VGG-16 | Real-valued | 32/32 | 2788.7 | -
IDa-Det | 1/1 | 200.5 | 13.91$\times$
Table 4: Comparison of time cost of real-valued and 1-bit models (Faster-RCNN
and SSD) on hardware (single thread).

###  4.5 Deployment Efficiency

We implement the 1-bit models achieved by our IDa-Det on ODROID C4, which has
a 2.016 GHz 64-bit quad-core ARM Cortex-A55. With evaluating its real speed in
practice, the efficiency of our IDa-Det is proved when deployed into real-
world mobile devices. We use the SIMD instruction SSHL on ARM NEON, for making
inference framework BOLT [7] compatible with our IDa-Det. We compare our IDa-
Det to the real-valued backbones in Tab. 4. We utilize VOC dataset for testing
the model. For Faster-RCNN, the input images were scaled to $1000\times 600$
and $300\times 300$ for SSD. We can plainly see that IDa-Det’s inference speed
is substantially faster with the highly efficient BOLT framework. For example,
the acceleration rate achieves about 4.7$\times$ on Faster-RCNN, which is
slightly lower than the theoretical acceleration rate discussed in Sec. 4.3.
Furthermore, IDa-Det achieves 13.91$\times$ acceleration with SSD. All
deployment results in the object detection are significant on real-world edge
devices.

##  5 Conclusion

This paper presents a novel method for training 1-bit detectors with knowledge
distillation to minimize the information discrepancy. IDa-Det employs a
maximum entropy model to select the proposals with maximum information
discrepancy and proposes a novel entropy distillation loss to supervise the
information discrepancy. As a result, our IDa-Det significantly boosts the
performance of 1-bit detectors. Extensive experiments show that IDa-Det
surpasses state-of-the-art 1-bit detectors and other knowledge distillation
methods in object detection.

#### Acknowledgement.

This work was supported in part by the National Natural Science Foundation of
China under Grant 62076016, 92067204, 62141604 and the Shanghai Committee of
Science and Technology under Grant No. 21DZ1100100.

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