# Constant-overhead quantum error correction with thin planar connectivity

Maxime A. Tremblay  Institut quantique & Département de physique, Université
de Sherbrooke, Sherbrooke, Qc, Canada, J1K 2R1  Nicolas Delfosse  Michael E.
Beverland  Microsoft Quantum & Microsoft Research, Redmond, WA 98052, USA

###### Abstract

Quantum LDPC codes may provide a path to build low-overhead fault-tolerant
quantum computers. However, as general LDPC codes lack geometric constraints,
naïve layouts couple many distant qubits with crossing connections which could
be hard to build in hardware and could result in performance-degrading
crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their
Tanner graphs into a small number of planar layers. Each layer contains long-
range connections which do not cross. For any CSS code with a degree-$\delta$
Tanner graph, we design stabilizer measurement circuits with depth at most
$(2\delta+2)$ using at most $\lceil\delta/2\rceil$ layers. We observe a
circuit-noise threshold of 0.28% for a positive-rate code family using 49
physical qubits per logical qubit. For a physical error rate of $10^{-4}$,
this family reaches a logical error rate of $10^{-15}$ using fourteen times
fewer physical qubits than the surface code.

Figure 1:  A crossing-free planar layout for hypergraph product codes with
four layers. Stabilizer generators are measured using circuits built from
single-qubit operations and CNOT gates between qubits connected by an edge.
The top two layers have edges connecting qubits in the same row as shown in
the zoom, while the lower two layers have edges connecting qubits in the same
column. Each layer is planar, such that no pair of edges cross. For
comparison, the lower right box shows the non-planar connections passing
through the lower-left corner before the decomposition, which exhibits many
crossings.

Quantum error correction (QEC) is typically implemented by measuring Pauli
operators called stabilizer generators of a QEC code to detect faults. In
quantum low density parity check (LDPC) codes, the stabilizer generators have
low weight, making them easier to implement than general codes. A further
appealing property possessed by some quantum LDPC codes is positive-rate,
allowing them to achieve arbitrarily low logical error rates with a constant
ratio of tens of physical qubits per logical qubit. For large computations,
this can correspond to more than an order of magnitude lower qubit overhead
than alternative codes with vanishing rate such as the surface code. However,
these positive-rate LDPC codes have non-local stabilizer generators [1, 2]
making them somewhat daunting to implement in hardware. In this work, we seek
a practical implementation of positive-rate quantum LDPC codes which performs
well in a full circuit-level noise analysis.

To clarify our discussion, we define the connectivity graph of a quantum
circuit, with vertices corresponding to qubits and edges connecting vertices
corresponding qubits which undergo entangling operations in the circuit. We
can further define a layout as a specification of the physical locations of
the connectivity graph’s qubits and connections. In this paper we are
interested in circuits which measure the stabilizer generators of a QEC code.
Given a family of codes, we focus on constant-depth stabilizer measurement
circuits to avoid a build-up of errors which could spoil any fault-tolerance
threshold.

A natural first question is if the non-local stabilizer generators of
positive-rate quantum LDPC codes can be measured using a circuit with local
connectivity in a 2D qubit layout. In recent work [3], we show that although a
single non-local stabilizer can be measured in constant depth, the full set of
stabilizer generators cannot be collectively measured in constant depth
without the ratio of logical to physical qubits vanishing.

Given that a 2D qubit layout with local connectivity is excluded for positive-
rate quantum LDPC codes, we consider quantum hardware equipped with some set
of long-range connections. With unrestricted connections, one could simply lay
out the code qubits in a 2D grid along with an ancilla for each stabilizer
generator that has connections to the code qubits in that stabilizer’s
support. However, this typically results in an unbounded number of crossing
connections [4, 5]; see Fig. 1. In many hardware platforms, in addition to
being challenging to implement, crossing connections can spread error through
crosstalk [6, 7, 8, 9]. This raises the following question.

Can we construct high-threshold constant-depth stabilizer measurement circuits
for positive-rate quantum LDPC codes in a 2D qubit layout without crossings in
the two-qubit gate connectivity?

In this work, we provide a positive answer to this question through the design
of an $l$-planar layout, consisting of qubits placed in a 2D grid, with edges
separated into $l=O(1)$ planar layers, with no crossings in each layer; see
Figure 1. Furthermore, we provide a circuit construction consistent with this
layered planar architecture to measure the stabilizer generators of any CSS-
type [10, 11] quantum LDPC code in constant depth.

######  Theorem 1.

Let $Q$ be a CSS code such that each stabilizer generator has weight at most
$\delta$ and each qubit is involved in at most $\delta$ stabilizer generators.
Then, one can implement the measurement of all the stabilizer generators of
$Q$ with a circuit with depth $2\delta+2$ using a
$\left\lceil\delta/2\right\rceil$-planar layout.

Below, after proving this theorem, we refine these results by specializing to
a family of quantum codes known as hypergraph product (HGP) codes [12] which
are constructed from a pair of input graphs. In this case, we find a low-depth
stabilizer measurement circuit which reduces to the standard circuit in the
case of surface codes111Surface codes are HGP codes formed when the input
graphs are the Tanner graphs of a pair of repetition codes [13]. Moreover, we
prove in Lemma 1 that the circuit depth can be reduced to $\delta+2$ when the
vertices of the input graphs of the HGP code admit a balanced ordering.

Lastly, we numerically explore the performance of these circuits for a family
of HGP codes using the decoding routine of Grospellier and Krishna [14], but
replacing their idealized noise model by circuit noise. Using our layered
planar connectivity, we obtain a circuit-noise threshold of $2.8(2)\times
10^{-3}$, providing strong evidence that these codes can offer a significant
advantage over surface codes which have a comparable threshold.

Quantum LDPC codes — All the quantum codes we consider in this work are CSS
codes [10, 11]. Recall that a CSS code with length $n$ is defined by a set of
commuting stabilizer generators
$s_{X,1},\dots,s_{X,r_{X}},s_{Z,1},\dots,s_{Z,r_{Z}}$ with
$s_{X,i}\in\\{I,X\\}^{n}$ and $s_{Z,j}\in\\{I,Z\\}^{n}$. The $X$ Tanner graph
is the bipartite graph $T_{X}=(V,E)$ whose vertex set is $V=V_{q}\cup V_{X}$
where $V_{q}=\\{q_{1},\dots,q_{n}\\}$ is the qubit set and
$V_{X}=\quantity{s_{X,1},\dots,s_{X,r_{X}}}$. There is an edge between $q_{i}$
and $s_{X,j}$ iff $s_{X,j}$ acts non-trivially on qubit $q_{i}$. The $Z$
Tanner graph $T_{Z}$ is defined similarly from the $Z$ stabilizer generators,
and the overall Tanner graph is their union $T=T_{X}\cup T_{Z}$. The code is a
quantum LDPC code if the Tanner graph has bounded degree. Quantum error
correction works by measuring all the stabilizer generators and applying a
correction based on the outcomes observed. Our goal is to design practical
stabilizer measurement circuits for quantum LDPC codes.

We first show that quantum circuits with a low-degree connectivity graph can
be implemented with a layered planar connectivity using a small number of
layers. This result applies to any quantum circuit (not just stabilizer
measurement circuits) and in particular to all low-depth quantum circuits.

######  Proposition 1.

Let $C$ be a circuit made with single-qubit and two-qubit operations whose
connectivity graph has degree at most $\delta$. Then, $C$ can be implemented
with a $\lceil\delta/2\rceil$-planar layout.

###### Proof.

This follows directly from the fact that for any graph with degree at most
$\delta$, the smallest edge partition such that each subgraph is planar
involves at most $\lceil\delta/2\rceil$ subgraphs [15, 16]. Furthermore, since
any planar graph can be drawn with arbitrary vertex location [17], we can fix
the position of each qubit across layers. ∎

We now introduce the coloration circuit associated with an edge coloration of
a Tanner graph that can be used for any CSS code. An edge coloration of a
graph is a coloration of the edges such that incident edges support distinct
colors. We will often consider a minimum edge coloration, that is an edge
coloration with a minimum number of colors.

input : A minimum edge coloration $\mathcal{C}_{X}$ of $T_{X}$.

output : The measurement outcome of all $X$ stabilizer generators.

1 Prepare an ancilla in $\ket{+}$ for each generator $s_{X,i}$.

2 for  _color $c\in\mathcal{C}_{X}$_ do

3 Simultaneously apply all gates $\operatorname{CNOT}_{i\to j}$ from the $i$th
ancilla to the $j$th data qubit supported on an edge $\\{i,j\\}$ with color
$c$.

4

5Measure each ancilla in the $X$ basis.

Algorithm 1 Coloration circuit

######  Proposition 2.

Let $Q$ be a CSS code with $X$ Tanner graph $T_{X}$. Then, the coloration
circuit measures all the $X$ stabilizer generators of $Q$ in depth
$\deg(T_{X})+2$.

###### Proof.

The CNOTs applied in step 3 can be applied simultaneously because they
correspond to edges with the same color, guaranteeing they have disjoint
support. We see the circuit measures the $X$ generators by rearranging the
CNOTs, which commute, to form a sequence of single-generator measurement
circuits. Each such circuit prepares an ancilla in $\ket{+}$, applies CNOTs
from the ancilla to the generator’s support, then measures the ancilla in the
$X$ basis. The depth of the coloration circuit is $\deg(T_{X})+2$ because the
Tanner graph, which is bipartite, admits an edge coloration with $\deg(T_{X})$
colors [18]. ∎

Swapping the roles of $X$ and $Z$ provides a $Z$ stabilizer measurement
circuit with depth $\deg(T_{Z})+2$. We are now equipped to prove Th. 1.

###### Proof of Th. 1.

By Proposition 2, a circuit extracting both $X$ and $Z$ syndromes with depth
$\deg(T_{X})+\deg(T_{Z})+2$ is formed by running the $X$ then the $Z$ circuit
with two overlapping time steps. The connectivity graph of the cardinal
circuit has degree $\deg(T)$. Therefore Prop. 1 proves the existence of a
$\lceil\deg(T)/2\rceil$-planar layout. ∎

For some codes, the depth can be further reduced by interleaving $X$ and $Z$
stabilizer measurements. However, the design of an interleaved $X/Z$
stabilizer measurement circuit is non-trivial because the CNOT gates involved
in $X$ and $Z$ measurements do not commute. Now we specialize to HGP codes for
which we provide an interleaved $X/Z$ stabilizer measurement circuit.

Figure 2:  (a) The Tanner graph of a HGP code is the Cartesian product of two
bipartite input graphs. Here qubits are displayed according to their label,
which does not correspond to their physical location in the 2D layout. (b) The
four possible configurations in which an $X$ and a $Z$ stabilizer can overlap.
Red arrows show the cardinal circuit ordering. (c) The commutation of a pair
of CNOT gates.

Hypergraph product codes — The HGP code $\operatorname{HGP}(G_{1},G_{2})$ [12]
is defined from the Cartesian product $G_{1}\times G_{2}$ of two bipartite
graphs (Fig. 2(a)). For $m\in\\{1,2\\}$, let $V_{m}=B_{m}\cup C_{m}$ be the
vertex set of $G_{m}$ and $E_{m}$ be its edge set. Edges of $E_{m}$ connect a
vertex of $B_{m}$ with a vertex of $C_{m}$. We assume that each vertex of
$G_{m}$ is given by a label $i=1,\dots,|V_{m}|$. Each pair $(i,j)\in
B_{1}\times B_{2}\cup C_{1}\times C_{2}$ represents a data qubit of the HGP
code while each pair $(i,j)$ in $B_{1}\times C_{2}$ (resp. $C_{1}\times
B_{2}$) corresponds to an $X$ (resp. $Z$) stabilizer generator. The stabilizer
generator with label $(i,j)$, denoted $S_{(i,j)}$ is supported on the qubits
with label $(i^{\prime},j)$ where $\\{i^{\prime},i\\}\in E_{1}$ and
$(i,j^{\prime})$ with $\\{j,j^{\prime}\\}\in E_{2}$. To avoid confusion with
the coordinates introduced later which specify the physical locations of
qubits in a 2D layout, we refer to the pair $(i,j)$ as a data qubit label or
stabilizer generator label.

We associate a direction ${\bf N},{\bf S},{\bf E},{\bf W}$ with each edge. An
edge between stabilizer vertex $(i,j)$ and qubit vertex $(i,j^{\prime})$ with
$j^{\prime}=j+\ell\pmod{|V_{1}|}$ has direction ${\bf N}$ if
$0<\ell\leq|V_{1}|/2$ and direction ${\bf S}$ otherwise. We define the
directions ${\bf E}$ and ${\bf W}$ similarly for edges between stabilizer and
qubit vertices $(i,j)$ and $(i^{\prime},j)$. For each direction ${\bf
D}\in\\{{\bf N},{\bf S},{\bf E},{\bf W}\\}$, we consider the subgraph $T_{\bf
D}$ of the Tanner graph $T$ induced by the edges with direction ${\bf D}$.

The following circuit interleaves $X$ and $Z$ measurements and can achieve a
depth lower than the coloration circuit.

input : A minimum edge coloration $\mathcal{C}_{\bf D}$ of $T_{\bf D}$.

output : The outcome of the measurement of all the $X$ and $Z$ stabilizer
generators.

1 Prepare an ancilla in $\ket{+}$ for each $X$ stabilizer generator and an
ancilla in $\ket{0}$ for each $X$ stabilizer generator.

2 for  _direction ${\bf D}\in\quantity{{\bf E},{\bf N},{\bf S},{\bf W}}$_ do

3 for  _color $c\in\mathcal{C}_{\bf D}$_ do

4 Simultaneously apply all $\operatorname{CNOT}$ gates supported on an edge of
$T_{\bf D}$ with color $c$.

5

6

7Measure each $X$ and $Z$ ancilla in the $X$ and $Z$ basis respectively.

Algorithm 2 Cardinal circuit

Note that the $\operatorname{CNOT}$ is either aligned or anti-aligned with an
edge depending on the type of the stabilizer. The control qubit of the
$\operatorname{CNOT}$ is the ancilla for $X$ stabilizers, and it is the data
qubit for $Z$ stabilizers.

######  Proposition 3.

Let $Q$ be a hypergraph product code with Tanner graph $T$. Then, the cardinal
circuit implements the measurement of all the stabilizer generators of $Q$ in
depth $\deg(T_{\bf N})+\deg(T_{\bf S})+\deg(T_{\bf E})+\deg(T_{\bf W})+2$.

###### Proof.

It is easy to check that the depth of the cardinal circuit is $\deg(T_{\bf
N})+\deg(T_{\bf S})+\deg(T_{\bf E})+\deg(T_{\bf W})+2$. This is because the
bipartite graph $T_{\bf D}$ admits an edge coloration with $\deg(T_{\bf D})$
colors [18].

We now prove by induction that the cardinal circuit measures the stabilizer
generators. Denote by $C_{m}$ the circuit obtained by applying the cardinal
circuit construction to the subset of stabilizer generators
$s_{1},\dots,s_{m}$. Clearly, for $m=1$, the circuit $C(s_{1})$ measures the
stabilizer generator $s_{1}$. We will show that the concatenation of $C_{m}$
and $C(s_{m+1})$, that we denote $C_{m}C(s_{m+1})$, has the same action as
$C_{m+1}$.

If all the $\operatorname{CNOT}$ gates of $C(s_{m+1})$ commute with all the
$\operatorname{CNOT}$s of $C_{m}$, we can simply reorder the
$\operatorname{CNOT}$s of the circuit $C_{m}C(s_{m+1})$ to obtain the cardinal
circuit $C_{m+1}$. Assume now that some $\operatorname{CNOT}$ gates of
$C(s_{m+1})$ do not commute with the gates of $C_{m}$. Again, we would like to
put the $\operatorname{CNOT}$ of $C_{m}C(s_{m+1})$ in the cardinal order, but
swapping these $\operatorname{CNOT}$s produces extra $\operatorname{CNOT}$s as
one can see in Fig. 2(d). We will show that these extra $\operatorname{CNOT}$s
cancel out.

If $s_{m+1}$ is a $Z$ stabilizer, the corresponding $\operatorname{CNOT}$s
only fail to commute with $\operatorname{CNOT}$s associated with the previous
$X$ stabilizer generators $s_{i}$ that overlap with $s_{m+1}$. Swapping these
$\operatorname{CNOT}$s produces an extra gate
$\operatorname{CNOT}_{s_{i}\rightarrow s_{m+1}}$ as shown in Fig. 2(c). By the
hypergraph product construction, if $s_{m+1}$ overlaps with an $X$ stabilizer
generator $s_{i}$, their overlap contains exactly two qubits, in one of the
four possible configurations of Fig. 2(b). To bring the $\operatorname{CNOT}$s
into cardinal order, we need to perform either 0 or 2 swaps between the
$\operatorname{CNOT}$s of $s_{m+1}$ and $s_{i}$. This results in two
consecutive $\operatorname{CNOT}$ gates $\operatorname{CNOT}(s_{i},s_{m+1})$
which cancel out. Thus, reordering the CNOT gates in $C_{m}C(s_{m+1})$ to
produce $C_{m+1}$ preserves the action of the circuit. A similar argument
applies when $s_{m+1}$ is an $X$ stabilizer generator. Applying this
inductively starting with a single stabilizer generator, we reach the
cardinally ordered circuit proving that it has the same action as the initial
stabilizer measurement circuit. ∎

From Prop. 3 we see that the cardinal circuit only has a lower depth than the
coloration circuit if the Tanner subgraphs $T_{D}$ have sufficiently low
degree. To ensure this, we must order the vertices $(i,j)$ of the Tanner graph
to distribute the edges more equally between the four directions around each
vertex. This motivates the notion of balanced ordering that we introduce now.

A balanced ordering for a graph $G=(V,E)$ is a labeling of the vertices by
integers $i=1,\dots,|V|$ such that for each vertex $i$ we have
$\delta_{+}(i)-\delta_{-}(i)=(\delta(i)\mod 2)$ where $\delta_{+}(i)$ is the
number of vertices connected to $i$ of the form $(i+\ell)\pmod{|V|}$ with
$0<\ell\leq|V|/2$ and $\delta_{-}(i)=\delta(i)-\delta_{+}(i)$. When the graph
is not clear from the context, we will use the notation
$\delta_{\pm}(G,i)=\delta_{\pm}(i)$. The following lemma is proven in the
Supplemental Material.

######  Lemma 1.

Let $Q=\operatorname{HGP}(G_{1},G_{2})$ be a hypergraph product code. Then, we
have

$\deg(T)\leq\sum_{{\bf D}={\bf N},{\bf S},{\bf E},{\bf W}}\deg(T_{\bf D})\leq
2\deg(T)\cdot$

Moreover, if $G_{1}$ and $G_{2}$ have only even degree vertices and admit a
balanced ordering, then the lower bound is tight.

For HGP codes based on even, balanced graphs, this lemma, combined with Prop.
3, proves that the depth of the cardinal circuit is about half that of the
coloration circuit.

Figure 3:  The failure rate per round averaged over 10 successive rounds of
error correction. The dashed lines are obtained using
$P_{L}(p,k)=c_{1}\quantity(p/p_{t})^{c_{2}k^{c_{3}}}$ finding fitting
constants $c_{1}=0.64$, $c_{2}=1.3$, $c_{3}=0.21$ and the threshold
$p_{t}=2.8(2)\times 10^{-3}$.

Numerical results — We use the standard circuit noise model. We simulate the
performance of a family of HGP codes with parameters $[[25s^{2},s^{2}]]$, i.e
encoding $k=s^{2}$ logical qubits into $n=25s^{2}$ physical qubits. The
syndrome extraction is performed with the cardinal circuit using $24s^{2}$
ancilla qubits. More details about the noise model and the codes can be found
in the Supplementary Material.

To decode and correct faults, we use the approach set out in Ref. [19], which
is based on Belief Propagation (BP) [20, 21] and Small Set Flip (SSF) [22].
After each round of stabilizer extraction, BP is iterated until there is a
local minimum of the number of violated stabilizer measurements. To probe the
overall performance after $T$ rounds of error correction, a perfect stabilizer
measurement round is applied and BP and SSF are alternated until the SSF
decoder converges to a correction. If either no correction is found, or if the
net effect of all noise and corrections is to introduce a logical operator, we
say that a failure has occurred by round $T$.

By sampling over many realizations of this procedure with outcomes extracted
using the cardinal circuit, we estimate the logical failure rate of a family
of HGP codes over a range of physical failure rates as shown in Fig. 3. We use
a simple extrapolation of the data to estimate a threshold of
$p_{t}=2.8(2)\times 10^{-3}$ and compare the qubit overhead with that of the
surface code in Table 1. Based on our simulation, we propose the heuristic
formula

$\displaystyle P_{L}(p,k)=c_{1}\quantity(p/p_{t})^{c_{2}k^{c_{3}}}$ (1)

where $c_{1}=0.64$, $c_{2}=1.3$, $c_{3}=0.21$ and $k=24s^{2}$ to provide an
estimate of the logical failure rate per round with physical error rate
$p<p_{t}$. Further details on the code construction and our numerical approach
can be found in the Supplementary Material.

Logical failure rate | $10^{-9}$ | $10^{-12}$ | $10^{-15}$
---|---|---|---
Logical qubits | $1600$ | $6400$ | $18\,496$
Surface code physical qubits | $387\,200$ | $2\,880\,000$ | $13\,354\,112$
HGP code physical qubits | $78\,400$ | $313\,600$ | $906\,304$
Improvement using HGP codes |  $4.94$$\times$ |  $9.18$$\times$ |  $14.73$$\times$
Table 1: Total number of data and ancilla qubits required to achieve specific
logical failure rates with a physical error rate of $10^{-4}$ using surface
codes and HGP codes. The HGP code data is estimated from Eq. 1, while the
surface code data is estimated using the formula
$P^{\prime}_{L}(p,k,d)=ak\quantity(p/p^{\prime}_{t})^{(d+1)/2}$ with
optimistic values of $p^{\prime}_{t}=0.011$ and $a=0.03$ from Ref. [13] and
Ref. [23].

Outlook and hardware challenges — We have shown that syndrome extraction can
be implemented in constant depth using a planar layout for any CSS quantum
LDPC code, including HGP codes, but also many other families of interest
including hyperbolic codes [24] and homological product codes [25]. Our design
simultaneously seeks to minimize the depth of the stabilizer measurement
circuit which achieves faster quantum error correction and reduce the time
allowed for error buildup, while also avoiding crossings of the connections
that couple qubits which is expected to improve fabrication and reduce cross-
talk. Further improvements could come from optimizing the circuit to minimize
the spreading of errors [26] using better decoders [27, 28, 29, 30], or by
leveraging improved planar graph algorithms.

We hope that these significant quantum error-correction advantages will
motivate experimental teams to overcome the challenges to build quantum
hardware in planar layouts. We foresee two major obstacles. Firstly, our
design requires a number of long range links within each layer. Significant
experimental progress has been made in that direction using for instance
photonic couplings to establish long range connections [31, 32, 32, 33, 34,
35, 36, 37, 38] but it is unclear which of these approaches could be scaled to
larger systems. Secondly, there is a tension between the need for insulation
between the layers to reduce crosstalk and the fact that data qubits must
participate in all layers.

###### Acknowledgements.

Acknowledgments — The authors would like to thank David Poulin for his
encouragements in the early stage of this project, Jeongwan Haah for his
comments on a preliminary version of this work and Nouédyn Baspin for
insightful discussions.

##  I Supplemental Material

###  I.1 Tanner subgraphs for HGP codes

Here we prove the bound stated in Lemma 1 on the degree of sub graphs of the
Tanner graphs of HGP codes.

###### Proof.

Edges of the Tanner graph of the form $\\{(i,j),(i^{\prime},j)\\}$ are called
horizontal edges and the edges $\\{(i,j),(i,j^{\prime})\\}$ are vertical
edges. The lower bound is trivial because the four types of edges form a
partition of the edge set of $T$. Consider the subgraph $T_{{\bf h}}$ (resp.
$T_{{\bf v}}$) of $T$ induced by horizontal (resp. vertical) edges. By
definition, $T_{\bf E}$ and $T_{\bf W}$ are subgraphs of $T_{\bf h}$ which
implies $\deg(T_{\bf E}),\deg(T_{\bf W})\leq\deg(T_{\bf h})$. Similarly, we
obtain $\deg(T_{\bf S}),\deg(T_{\bf N})\leq\deg(T_{\bf v})$. Moreover, due the
product structure of the graph, we have $\deg(T)=\deg(T_{\bf h})+\deg(T_{\bf
v})$, which leads to the upper bound.

Assume that $G_{1}$ and $G_{2}$ have only even degree vertices and are
equipped with a balanced ordering and let us prove that the lower bound is
tight. Then, we have $\delta_{\pm}(G_{m},i)=\deg_{G_{m}}(i)/2$. Let $(i,j)$ be
a maximum degree vertex in $T_{\bf N}$. If $(i,j)$ is a stabilizer vertex,
then by definition of the direction ${\bf N}$, the degree of $(i,j)$ in the
graph $T_{\bf N}$ is given by $\delta_{+}(G_{2},j)$. If $(i,j)$ is a qubit
vertex, its degree is given by $\delta_{-}(G_{2},j)$. Therefore, we always
have

$\displaystyle\deg(T_{\bf N})$
$\displaystyle\leq\max(\delta_{+}(G_{2},j),\delta_{-}(G_{2},j))$
$\displaystyle=\deg_{G_{2}}(j)/2$ $\displaystyle\leq\deg(G_{2})/2.$

By the same reasoning, we obtain $\deg(T_{\bf S})\leq\deg(G_{2})/2$ and
combining both equations, we find

$\displaystyle\deg(T_{\bf S})+\deg(T_{\bf N})\leq\deg(G_{2}).$ (2)

Applying the same argument to horizontal edges, we get

$\displaystyle\deg(T_{\bf E})+\deg(T_{\bf W})\leq\deg(G_{1}).$ (3)

Using Eq. (2) and (3), we obtain

$\displaystyle\sum_{{\bf D}={\bf N},{\bf S},{\bf E},{\bf W}}\deg(T_{\bf D})$
$\displaystyle\leq\deg(G_{1})+\deg(G_{2})$

which concludes the proof of the lemma because
$\deg(G_{1})+\deg(G_{2})=\deg(T)$. ∎

###  I.2 Numerical simulations

In our numerical studies we consider HGP codes constructed from the product of
random (3, 4)-regular Tanner graphs with $4s$ bit nodes and $3s$ check nodes
with girth at least 8. The codes are generated using the procedure of Ref.
[19] and for each $s$, we pick the best code from a few hundred samples by
decoding i.i.d noise with the SSF decoder. The actual number of samples is
based on the decoding time and the computing resources. Depending on the code
length, this ranges from 50 to 1000 samples. For all these graphs, we found a
balanced vertex ordering using a greedy Metropolis-Hasting inspired algorithm
by defining a measure of how close to balanced is a vertex ordering of a
graph.

We use a circuit noise model in which each operation, including an identity
gate, is faulty with probability $p$. If a unitary operation is faulty, we
apply a random uniform non-trivial Pauli error on the support of the
operation. If a single-qubit measurement is faulty, its outcome is flipped.

In our simulations, we estimate the logical failure rate by averaging over $T$
successive rounds of error correction. Based on Fig. 4 we can assume that the
logical error rate after $T$ rounds behaves roughly like $P_{L}(T)=c+qT$ for
some constants $c$ and $q$ that depend on the noise rate and the code. For
simplicity, we estimate the logical error rate per round as $P_{L}=P_{L}(T)/T$
for $T=10$ rounds of error correction. This value leads to a pessimistic
estimate of the logical failure rate after $T$ rounds for any $T\geq 10$. This
is a reasonable model because we expect these code blocks to be kept alive for
many logical cycles during a quantum computation because they encode hundreds
of thousands of logical qubits. Indeed, one could conceivably run a full large
scale quantum algorithm using only the logical qubits encoded in a single
block of the codes simulated in Fig. 3.

Figure 4:  Failure rate after an increasing number of error correcting rounds
for codes of length 5625 and 10000 with $p=10^{-3}$. We observe that after 6
rounds, the increment per round is roughly constant.

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# Curriculum for Crowd Counting - Is it Worthy?

Muhammad Asif Khan1 , Hamid Menouar1 and Ridha Hamila2
1Qatar Mobility Innovations Center, Qatar University, Doha, Qatar
2Department of Electrical Engineering, Qatar University, Doha, Qatar
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
https://orcid.org/0000-0003-2925-8841 https://orcid.org/0000-0002-4854-909X
https://orcid.org/0000-0002-6920-7371

###### Abstract

Recent advances in deep learning techniques have achieved remarkable
performance in several computer vision problems. A notably intuitive technique
called Curriculum Learning (CL) has been introduced recently for training deep
learning models. Surprisingly, curriculum learning achieves significantly
improved results in some tasks but marginal or no improvement in others.
Hence, there is still a debate about its adoption as a standard method to
train supervised learning models. In this work, we investigate the impact of
curriculum learning in crowd counting using the density estimation method. We
performed detailed investigations by conducting 112 experiments using six
different CL settings using eight different crowd models. Our experiments show
that curriculum learning improves the model learning performance and shortens
the convergence time.

##  1 Introduction

Crowd counting is an interesting problem in computer vision research [Khan et
al., 2022, Fan et al., 2022, Gouiaa et al., 2021]. Though several methods [Li
et al., 2008, Topkaya et al., 2014, Chen et al., 2012, Chan and Vasconcelos,
2009] have been proposed earlier to estimate crowd in an image, the defacto
state-of-the-art approach for crowd counting is using density estimation.
Density estimation employs a deep learning model such as a convolution neural
network (CNN) to estimate the crowd density in an image. The ground truths to
train the model are density maps of crowd images. A density map is generated
from the dot annotation map where each dot (representing the head position of
a person) is convolved with a Gaussian function.

Over the years, several models have been proposed to improve the accuracy
performance over benchmark datasets. Notably, CrowdCNN was the first CNN-based
model proposed in [Zhang et al., 2015]. CrowdCNN is a single-column 6-layer
CNN network using density map prediction. Due to the single column, CrowdCNN
does not capture the scale variations present in head sizes in crowd images.
CrowdNet [Boominathan et al., 2016] and MCNN [Zhang et al., 2016] propose
multi-column architectures to cope with the scale variations. The CrowdNet
model uses a shallow and a deep network to predict different crowd densities.
The MCNN model used three columns of CNN layers with different sizes of
convolution kernels in each layer to efficiently capture the scale variations.
Though, these multi-column networks achieve better accuracy, their performance
is poor on highly congested scenes mainly due to two reasons. First, the
model’s ability to capture scale variations is limited by the number of
columns. Second, these shallow models become quickly saturated due to the
small number of neurons. To solve the scale variation problem, improved model
architectures such as encoder-decoder e.g., TEDnet [Jiang et al., 2019],
SASNet [Song et al., 2021], and pyramid structure using multi-scale modules
e.g., MSCNN [Zeng et al., 2017], SANet [Cao et al., 2018] are proposed. For
crowd estimation in highly congested scenes, models using transfer learning
from pre-trained models such as VGG-16 [Simonyan and Zisserman, 2015], ResNet
[He et al., 2016], MobileNet [Sandler et al., 2018], and Inception [Szegedy et
al., 2015] achieved best results. Few notable models using transfer learning
include CSRNet [Li et al., 2018], C-CNN [Shi et al., 2020], BL [Ma et al.,
2019] (VGG), MobileCount [Gao et al., 2019] (MobileNet), MMCNN [Peng et al.,
2020], MFCC [Gu and Lian, 2022] (ResNet), SGANet [Wang and Breckon, 2022]
(Inception) etc.

Recently, Curriculum Learning (CL) has gained significant attention as an
alternative method to improve performance in various deep-learning tasks.
Curriculum learning refers to the set of techniques to train deep learning
models by imitating human curricula. In a CL strategy, the training samples
are organized in a specific order (typically by increasing or decreasing
difficulty) before feeding to the model. CL was first formalized in [Bengio et
al., 2009] inspired by the fact that humans learn better when the tasks are
presented in a meaningful order i.e., typically in the order of increasing
complexity (or difficulty). CL potentially brings two benefits: (i) faster
convergence and (ii) improved accuracy.

CL has been applied in several supervised learning applications including
object localization [Ionescu et al., 2016, Shi and Ferrari, 2016, Tang et al.,
2018], object detection [Chen and Gupta, 2015, Li et al., 2017, Sangineto et
al., 2019] and machine translation [Kocmi and Bojar, 2017, Platanios et al.,
2019, Wang et al., 2019]. CL has also been successfully applied in
reinforcement learning [Narvekar et al., 2020]. Although CL applied in several
problems achieved improved training, faster convergence, and performance
gains; authors in [Wu et al., 2021] show CL could not benefit the accuracy
performance of image classification on CIFAR10 dataset [Krizhevsky, 2009].
Some recent works also applied CL in crowd density estimation [Khan et al.,
2023b].

Figure 1: Curriculum learning framework.

This paper aims to perform a rigorous evaluation of CL in yet another
important task in computer vision i.e., crowd counting. Crowd counting employs
density estimation using pixel-wise regression from crowd images. Recently,
very few works have applied CL in crowd density estimation reporting potential
benefits in some scenarios. As the results reported in these works are only
incremental, this paper aims to extensively investigate the potential of CL in
crowd density estimation. The contribution of this paper is as follows:

  * •

We conducted $\sim 112$ experiments using eight mainstream crowd-counting
models and six different CL settings over two benchmark crowd datasets.

  * •

The models’ performance is evaluated using two widely used metrics for crowd
counting and results are compared to understand when and to which extent CL
outperforms standard learning.

  * •

Conclusions are drawn for prospective researchers in the area of crowd
counting.

##  2 Background

Curriculum learning (CL) is defined as ”training criteria $\mathcal{C}$ over
$T$ training steps: $\mathcal{C}=<Q_{1},...Q_{t},...Q_{T}>$ such that each
criterion $Q_{t}$ is a reweighting of the target training distribution
$P(z)$”:

$Q_{t}(z)=W_{t}(z)P(z)\quad\forall z\in\text{training \; set}\;D$ (1)

In eq: 1, (i) the entropy of $P(z)$ gradually increases i.e.,
$H(Q_{t})<H(Q_{t+1})$, (ii) the weight of any example increases i.e.,
$W_{t}(z)\leq W_{t+1}(z)$, or (iii) $Q_{T}(z)=P(z)$ [Wang et al., 2022b].

A formal description of the curriculum learning method is presented in
Algorithm 1.

Require: pacing function $g$, scoring function $f$, data $X$

Result: mini-batches [$B_{1},B_{2},...B_{M}$]

1

2$results$ = sort $X$ using $f$

3

4for  _$i=1,\cdots M$_ do

5 $size\leftarrow g(i)$

6 $X_{i}=X[1,...,size]$

7 uniformly sample $B_{i}$ to $results$

8 append $B_{i}$ to $result$

9

10 end for

return _$results$_

Algorithm 1 Curriculum Learning

There are two main parts of curriculum learning, a scoring function, and a
pacing function. The scoring function is used to organize the training samples
in a specific meaningful order while the pacing function samples the amount of
data exposed to the model in each training step. Fig. 1 depicts the curriculum
learning process.

###  2.1 Scoring Function

A scoring function ($f$) is a function that sorts the training data in the
order of increasing or decreasing difficulty (in curriculum versus anti-
curriculum learning, respectively). For a given scoring function
$f:X\rightarrow R$, ($x_{i},y_{i}$) is more difficult than ($x_{j},y_{j}$) if
$f(x_{,}y_{i})>f(x_{j},y_{j})$. A scoring function can be defined in two ways;
(i) self-taught, or (ii) transfer-scoring. In self-taught scoring function,
the network is trained on uniformly-sampled (randomly ordered) batches to
compute the score (difficulty) for each training sample. In transfer-scoring
function, a pre-trained model is used to compute the score for each training
sample.

###  2.2 Pacing Function

A pacing function $g$ is a function that determines a subset of training data
fed to the model in a particular iteration. For training data $X$ of size $N$,
$g:[M]\rightarrow[N]$ finds subsets $X_{1},X_{2},...X_{M}\subset X$. From each
$X$, mini-batches $\\{B_{i}\\}_{i=1}^{M}$ are uniformly sampled. Fig. 2
depicts six different pacing functions used in curriculum learning.

Figure 2: Various pacing functions applied to ShanghaiTech Part B dataset with
a total number of samples (N) = 400 and batch size = 8.

In the more native form of CL, the training examples are organized in order of
increasing difficulty. However, some works suggest anti-curriculum, in which
the training examples are organized in the order of decreasing difficulty.

##  3 Related Work

Recently, curriculum learning has been adopted in a few crowd-counting works.
In [Li et al., 2021], authors propose TutorNet to improve density estimation
in crowd counting. A main network that generates density maps for crowd images
is supervised at the training stage by a TutorNet network. The network
produces a weight map of the same shape as the density map. Each value in the
weight map then represents the per-pixel learning rate of the model error.
Thus, this weight map is used as a curriculum to train the main network.
TutorNet uses ResNet as a frontend to extract features. The work also uses
scaling of the density map pixel values. TutorNet is evaluated over
ShanghaiTech [Zhang et al., 2016] and Fudan-ShanghaiTech dataset (FDST) [Fang
et al., 2019]. The results show major improvement using density map scaling
with further improvement using TutorNet (pixel-level curriculum). The work did
not consider TutorNet alone, hence providing little intuition on the efficacy
of CL. The authors in [Wang et al., 2022a] followed a similar approach to
implement CL in crowd-counting tasks. A weight is assigned to each pixel in a
crowd image which indicates the per-pixel difficulty. More specifically, a
region-aware density map (RAD) is first generated through an average pooling
operation and then the Gaussian function is applied to RAD to produce an
attention map. In the attention maps, the simple pixels are assigned higher
weights. A modified loss function is proposed to use the attention maps during
the model training. The learning performance is evaluated on ShanghaiTech
[Zhang et al., 2016], UCF-QNRF [Idrees et al., 2018], WorlExpo’10 [Zhang et
al., 2015], and GCC [Qi et al., 2020] datasets.

The computation of pixel-wise curricula can be a more expensive task as
compared to sample-wise curricula in many works on CL. A recent study [Khan et
al., 2023a] on curriculum learning integrated with dataset pruning to improve
the learning performance and convergence time supports the efficacy of sample-
wise curriculum learning.

A review of the aforementioned works provides limited hints on the efficacy of
curriculum learning in crowd-counting tasks. However, whether the performance
gains in these works are the results of the curriculum learning, the
underlying crowd model, or of both together? Whether curriculum learning can
improve the performance of any kind of crowd models e.g., shallow, deep,
multi-column, encoder-decoder, and multi-scale, with and without transfer
learning? This work aims to further investigate to answer these questions.

##  4 Experiments and Evaluation

We conducted more than 112 experiments using eight (8) mainstream crowd models
and two well-known datasets to investigate the efficacy of curriculum learning
in crowd counting.

###  4.1 Datasets

The two datasets chosen are ShanghaiTech Part A and ShanghaiTech Part B
datasets, both published in [Zhang et al., 2016]. The datasets contain cross-
scene crowd images with varying crowd densities and have been extensively used
for benchmarking in numerous studies on crowd counting and density estimation.

###  4.2 Baseline Crowd Models

We choose eight (8) different crowd models to use in our experiments. These
include MCNN [Zhang et al., 2016], CMTL [Sindagi and Patel, 2017], MSCNN [Zeng
et al., 2017], CSRNet [Li et al., 2018], SANet [Cao et al., 2018], TEDnet
[Jiang et al., 2019], Yang et al. [Yang et al., 2020], and SASNet [Song et
al., 2021]. The eight crowd models are chosen such that they vary in terms of
model size, complexity, and design.

###  4.3 Curriculum Settings

We consider six different types of pacing functions: linear, quadratic,
exponential, root, logarithmic, and step. These pacing functions are
calculated using Eq. 2.

$g=\begin{cases}Nb+aNb&(linear)\\\\[5.0pt]
Nb+N\frac{1-b}{aT}t^{P}-p=1/2,1,2&(quad)\\\\[5.0pt]
Nb+\frac{N\left(1-b\right)}{e^{10}-1}\left(exp\left(\frac{10t}{aT}-1\right)\right)&(exp)\\\\[5.0pt]
Nb+N\left(1-b\right)\left(1+\frac{1}{10}log\left(\frac{t}{aT}+e^{-10}\right)\right)&(log)\\\\[5.0pt]
Nb+N\left[\frac{x}{aT}\right]&(step)\\\\[5.0pt] \end{cases}$ (2)

For each dataset, we kept the value of $b$ fixed, while choosing the value of
$a=[0.2,0.4,0.6,0.8]$. Thus, in each experiment, we take a fraction ($b$) of
full training data and incrementally add batches according to the pacing
function (with parameter $a$). The pacing functions used are plotted in Fig.
2.

(a) Linear pacing function (b) Quadratic pacing function (c) Exponential
pacing function

(d) Root pacing function (e) Log pacing function (f) Step pacing function

Figure 3: Pacing functions used in our experiments.

###  4.4 Evaluation Metrics

We used two commonly used metrics to test the performance in the crowd-
counting task i.e., mean absolute error (MAE) and mean squared error (MSE).
MAE and MSE can be calculated using Eq. 3 and Eq. 4, respectively.

$MAE=\frac{1}{N}\sum_{1}^{N}{(e_{n}-g_{n})}$ (3)

$MSE=\frac{1}{N}\sum_{1}^{N}{(e_{n}-g_{n})}$ (4)

where $N$ represents the total number of examples in the dataset, $g_{n}$ is
the actual count of people in the $n^{th}$ crowd image, and ${e_{n}}$ is the
estimated count (computed as the sum of pixel values in the predicted density
map for the same image). Other less frequently used metrics are grid average
mean error (GAME) for more localized counting, structural similarity index
(SSIM), and peak signal-to-noise ratio (PSNR) for the quality of the predicted
density maps.

###  4.5 Training Details

Each crowd model is trained first on the ShanghaiTech Part B dataset using
standard training. The same model is trained (from scratch) using curriculum
learning with a single pacing function. Since there are six pacing functions
used in this study, the same model is trained six times with a different
pacing function. In each experiment, we carefully selected the pacing function
parameter $\alpha$ to define reasonable subsets in each iteration by examining
the relative performance over a few epochs (Fig. 3). Thus, we conduct a total
of seven (7) complete training for a single model and an overall 56
experiments on the ShanghaiTech Part B dataset. The same number of experiments
are then repeated for the ShanghaiTech Part A dataset. All models are trained
using the PyTorch framework on two RTX-8000 GPUs. In all experiments, we use
Adam optimizer with an initial learning rate of $1\times 10^{-2}$ and a
ReduceLROnPlateau learning rate decay function based on MAE values.

##  5 Results and Analysis

The trained model in each experiment is evaluated over the two metrics (i.e.,
MAE and MSE). We carefully chose the values of parameters $a$ and $b$ in the
pacing functions to achieve reasonable subsets of training data in curriculum
learning settings. In standard training, the batches are uniformly sampled
from the full training dataset. The best-achieved results in each experiment
are depicted in Table 1 and 2.

Table 1: A comparison of standard training versus curriculum learning (using six different pacing functions) over ShanghaiTech Part B dataset using two metrics (MAE and MSE). The bold text shows the lowest error values. Model | Standard | Curriculum Learning
---|---|---
Random | Linear | Log | Quadratic | Exponential | Step | Root
MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE
MCNN | 26.4 | 41.3 | 19.2 | 32.2 | 23.8 | 38 | 21.4 | 33.8 | 23.1 | 37.4 | 23.4 | 37.4 | 22.8 | 37.1
CMTL | 20.0 | 31.1 | 19.6 | 30.6 | 19.8 | 30.7 | 18.8 | 30.4 | 20.0 | 31.6 | 20.2 | 32.0 | 19.4 | 30.5
MSCNN | 17.7 | 30.2 | 16.9 | 29.2 | 17.6 | 29.9 | 17.2 | 29.4 | 17.8 | 30.2 | 17.8 | 30.1 | 16.8 | 28.8
CSRNet | 10.6 | 16.0 | 7.8 | 14.2 | 10.2 | 16 | 8.2 | 14.6 | 9.4 | 15.3 | 9.8 | 15.8 | 8.6 | 14.9
SANet | 8.4 | 13.6 | 8.4 | 13.5 | 8.8 | 14.0 | 8.1 | 13.3 | 8.5 | 13.8 | 8.2 | 13.4 | 8.6 | 13.7
TEDnet | 8.2 | 12.8 | 7.6 | 12.2 | 8.2 | 12.7 | 8.3 | 13 | 7.7 | 12.4 | 8.1 | 12.6 | 7.8 | 12.4
Yang et al. | 12.3 | 21.2 | 11.8 | 20.4 | 11.4 | 20.0 | 10.6 | 18.3 | 12.2 | 21.4 | 12.6 | 21.6 | 12.0 | 20.8
SASNet | 6.4 | 9.9 | 6.6 | 10 | 6.3 | 9.6 | 6.8 | 10.5 | 7.2 | 11.2 | 6.9 | 10.6 | 6.4 | 9.8

We note several interesting observations in the results. First, curriculum
learning clearly brings significant improvements in some cases. For instance,
on the ShanghaiTech Part B dataset, the MAE for MCNN was reduced from 26.4 to
19.2 using the linear pacing function (best case) and 21.4 using the quadratic
pacing function (second best case). Similarly, the MAE for CSRNet was reduced
from 10.6 to 7.8 using the linear pacing function. On the ShanghaiTech Part A
dataset, the MAE for MCNN was reduced from 110.2 to 102.4 (linear pacing
function), and for CSRNet the MAE was reduced from 68.2 to 58.4.

Second, curriculum learning brings marginal improvement in most cases. This is
evident from the MAE values of all models for several selections of pacing
functions. Third, curriculum learning could not improve or underperforms the
standard training in some cases (indicated with red font color in both
tables). This observation highlights the importance of the pacing function as
an important hyperparameter in curriculum learning. Fourth, the benefit of
curriculum learning is evident for all models on both datasets. However, the
level of improvement varies among the models which raise the logical question
of what makes curriculum learning outperform standard training. The last
observation is that the benefits of each pacing function have been consistent
to a certain level. For instance, the best results were produced by linear
function for MCNN [Zhang et al., 2016], CSRNet [Li et al., 2018], and TEDnet
[Jiang et al., 2019] on both datasets. Similarly, SASNet [Song et al., 2021]
produces better results using the log pacing function on both datasets. The
Step function almost underperformed all other pacing functions except for
SANet [Cao et al., 2018] and TEDnet [Jiang et al., 2019].

Table 2: A comparison of standard training versus curriculum learning (using six different pacing functions) over ShanghaiTech Part A dataset using two metrics (MAE and MSE). The bold text shows the lowest error values. Model | Standard | Curriculum Learning
---|---|---
Random | Linear | Log | Quadratic | Exponential | Step | Root
MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE
MCNN | 110.2 | 173.2 | 102.3 | 150.8 | 109.6 | 172.4 | 108.8 | 170.6 | 106.7 | 169.0 | 112.4 | 177.6 | 102.4 | 151.2
CMTL | 101.3 | 152.4 | 96.7 | 142.4 | 100.6 | 151.8 | 98.2 | 149.2 | 103.5 | 156.2 | 105.4 | 160.3 | 100.2 | 150.4
MSCNN | 83.8 | 127.4 | 82.3 | 122.6 | 83.4 | 128.3 | 82.6 | 125.8 | 83.8 | 129.2 | 85.8 | 133.5 | 81.2 | 120.8
CSRNet | 68.2 | 115.0 | 58.4 | 102.5 | 62.0 | 105.4 | 60.6 | 105.2 | 62.2 | 106.1 | 62.0 | 105.5 | 62.3 | 106.4
SANet | 67.0 | 104.5 | 66.9 | 103.8 | 68.4 | 105.4 | 66.2 | 100.8 | 66.4 | 101.2 | 70.2 | 118.4 | 67.2 | 105.0
TEDnet | 64.2 | 109.1 | 60.2 | 102.9 | 65.8 | 114.2 | 66.4 | 115.2 | 65.0 | 111.3 | 63.4 | 106.4 | 63.8 | 106.9
Yang et al. | 104.6 | 145.2 | 102.3 | 139.0 | 104.4 | 143.7 | 98.4 | 134.2 | 104.8 | 145.8 | 105.6 | 147.2 | 99.1 | 134.8
SASNet | 53.6 | 88.4 | 51.2 | 81.0 | 51.4 | 81.4 | 52.8 | 86.2 | 53.2 | 87.0 | 55.8 | 90.4 | 52.3 | 84.6

Besides the potentially significant results discussed previously, Fig. 4
depicts the clear benefit of curriculum learning in terms of convergence time.
The y-axis shows the MSE loss of the model during the training phase, whereas
the top and bottom x-axes show the number of samples seen by the model during
the curriculum learning and standard training settings. The loss drops too
quickly in curriculum learning as compared to standard training highlighting
the faster convergence of curriculum learning.

Figure 4: Illustration of model convergence using standard training versus
curriculum learning.

##  6 Conclusions

This article presents a detailed experimental analysis of curriculum learning
in crowd-counting. Although curriculum learning has been effective in
reinforcement learning, its efficacy in supervised learning is not fully
evident due to the lack of detailed investigations. We performed an extensive
set of experiments on eight mainstream crowd-counting models to evaluate the
performance of curriculum learning. The results show significant improvements
in some cases, marginal improvements in most cases with no improvement in a
few cases. Through the detailed analysis of the results, we conclude that
curriculum learning can potentially improve the performance of deep learning
models by carefully choosing the pacing function and its parameters. Moreover,
given the short training time budget, curriculum learning is a good choice to
cut the convergence time. For future work, we suggest extending the
investigation to other computer vision tasks using large datasets and
different scoring functions appropriate to the task.

## Acknowledgement

This publication was made possible by the PDRA award PDRA7-0606-21012 from the
Qatar National Research Fund (a member of The Qatar Foundation). The
statements made herein are solely the responsibility of the authors.

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# Consistency test of the fine-structure constant from the whole ionization
history

Ke Wang1<EMAIL_ADDRESS>and Lu Chen2<EMAIL_ADDRESS>333Corresponding author 1 Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, School of Physical
Science and Technology, Lanzhou University, Lanzhou 730000, China
Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China
2 School of Physics and Electronics,
Shandong Normal University, Jinan 250014, China

###### Abstract

In cosmology, the fine-structure constant can affect the whole ionization
history. However, the previous works confine themselves to the recombination
epoch and give various strong constraints on the fine-structure constant. In
this paper, we also take the reionization epoch into consideration and do a
consistency test of the fine-structure constant from the whole ionization
history. We find that the constraints from these two epochs are not consistent
with each other by $4.64\sigma$. The straightforward explanation for this
inconsistency is a varying fine-structure constant. Another explanation is
that the data we used are not good enough. We prefer the latter explanation
and a need for calibration of some important parameters involved in
reconstructing the reionization history.

###### pacs:

???

##  I Introduction

The fine-structure constant $\alpha_{\text{EM}}=e^{2}/4\pi\epsilon_{0}\hbar c$
is a fundamental constant in physics, which indicates the strength of
electromagnetic interaction. Its value has been measured with various
experiments in laboratory, such as the measurements of the neutron de Broglie
wavelength, the quantum Hall effect, the electron anomalous magnetic moment,
local atomic clock measurements and so forth Kinoshita:1996vz ; Lange:2020cul
. The full 2018 CODATA Committee on Data for Science and Technology
$https://physics.nist.gov/$  gives the numerical value of
$\alpha_{\text{EM}}=1/137.035999084(21)$, its relative standard uncertainty is
$1.5\times 10^{-10}$. In astrophysical and cosmological systems, we also can
get relatively precise measurements of the fine-structure constant with high-
resolution astrophysical spectroscopy measurements Webb:2010hc ;
Murphy:2017xaz ; Wilczynska:2020rxx ; Milakovic:2020tvq ; Murphy:2021xhb ;
daFonseca:2022qdf , big bang nucleosynthesis Deal:2021kjs  and cosmic
microwave background (CMB) Planck:2014ylh ; Hart:2017ndk ; Hart:2019dxi ;
Hart:2021kad  respectively. For reviews of the other measurements, see
Uzan:2010pm . Since these measurements may come from different epochs of our
universe, it is possible for people to probe a varying fine-structure constant
Hart:2021kad ; daFonseca:2022qdf ; Liu:2021mfk . Comparing all above
measurements, in fact, one can find that these measurements are consistent
with each other and there is no evidence to support a varying fine-structure
constant now.

However, there is a caveat for measurements with CMB Planck:2014ylh ;
Hart:2017ndk ; Hart:2019dxi ; Hart:2021kad  that these previous constraints on
the fine-structure constant are only obtained from recombination epoch
although the fine-structure constant can affect the whole ionization history.
For recombination epoch, the story is simple. The fine-structure constant
influences the the Thomson scattering cross-section and the energy levels of
hydrogen and helium directly. As a result, its derived effects on the
visibility function can leave footprints on the CMB power spectra, which can
be researched with cosmological observations. As for the reionization epoch,
the situation is complicated. Because there is a lack of knowledge about the
reionization, such as its sources and its outset. Here we suppose that the
star-forming galaxies are the major sources of the reionization. Therefore,
there are two important parameters for reconstruting the reionization history:
the ionizing photon emission rate of a star-forming galaxy $\xi_{\rm ion}$ and
the escape fraction of ionizing photons $f_{\rm esc}$. To obtain the
constraints on the fine-structure constant from the reionization epoch, there
are two essential prerequisites. On the one hand, we should know the values of
these two parameters with respect to $\alpha_{\text{EM}}=1/137.035999084(21)$.
On the other hand, we should know the dependence of these two parameters on
$\alpha/\alpha_{\rm EM}$, where $\alpha$ is the fine-structure constant in
question.

In this work, we make the fine-structure constant free during the whole
ionization history and constrain its values with observational data in two
models. In the $\Lambda$CDM$+\alpha/\alpha_{\text{EM}}$ model, the fine-
structure constant remains a constant $\alpha$ at all redshifts. In the
$\Lambda$CDM$+\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rei}}/\alpha_{\text{EM}}$
model, there is a constant $\alpha_{\text{rec}}$ at the recombination epoch
and another constant $\alpha_{\text{rei}}$ at the reionization epoch. In the
latter model, we can do a consistency test of the fine-structure constant from
the whole ionization history and have a glance at the redshift-varying fine-
structure constant.

This paper is organized as follows. In section II, the ionization history of
our universe is reconsidered with the rescaled fine-structure constant. In
subsection II.1, we introduce the recombination calculation related to
$\alpha_{\text{rec}}$. In subsection II.2, modifications of the reionization
history are presented. In section III, we put constraints on free parameters
in previous models with the same data combination including CMB measurement
from Planck 2018 Aghanim:2018eyx , SFR density from UV data and IR data
Madau:2014bja , $Q_{\text{HII}}$ constraints from observations of quasars
Bouwens:2015vha ; Fan:2005es ; Schenker:2014tda , baryon acoustic oscillation
(BAO) data Beutler:2011hx ; Ross:2014qpa ; Alam:2016hwk ;
deSainteAgathe:2019voe ; Ata:2017dya ; Hou:2018yny , as well as the
measurement of Type Ia supernova Scolnic:2017caz . Finally, a brief summary is
included in section IV.

##  II Effects of the Fine-Structure Constant on Ionization History

In this section, the effects of varying fine-structure constant on the
ionization history are introduced and the evolution of ionized fraction are
shown in Fig. 1. Note we denote the fine-structure constant as $\alpha$
uniformly for both recombination epoch and reionization epoch.

Figure 1: The evolution of ionized fraction $x_{e}(z)$ in the
$\Lambda$CDM$+\alpha/\alpha_{\text{EM}}$ model. The black solid curve
indicates the fiducial case of $\alpha/\alpha_{\text{EM}}=1.00$. The red
dashed curve corresponds to $\alpha/\alpha_{\text{EM}}=0.95$ and the green
dotted one illustrates $\alpha/\alpha_{\text{EM}}=1.05$. Values of the other
necessary parameters are set to their mean values as Tab. I shows.

###  II.1 Effects of the Fine-Structure Constant on Recombination

The evolution of ionized fraction $x_{\text{e}}=x_{\text{p}}+x_{\text{HeII}}$
are described by a set of differential equations during the recombination
epoch Planck:2014ylh :

$\displaystyle\dfrac{dx_{\text{p}}}{dz}$ $\displaystyle=$
$\displaystyle\frac{C_{\text{H}}}{H_{0}(1+z)E(z)}\left[x_{\text{e}}x_{\text{p}}\langle
n_{\text{H}}\rangle\alpha_{\text{H}}-\beta_{\text{H}}(1-x_{\text{p}})e^{-h\nu_{\text{H2s}}/kT_{\rm
M}}\right],$ (1) $\displaystyle\dfrac{dx_{\text{HeII}}}{dz}$ $\displaystyle=$
$\displaystyle\frac{C_{\text{HeI}}}{H_{0}(1+z)E(z)}\left[x_{\text{e}}x_{\text{HeII}}\langle
n_{\text{H}}\rangle\alpha_{\text{HeI}}-\beta_{\text{HeI}}(f_{\text{He}}-x_{\text{HeII}})e^{-h\nu_{\text{HeI},2^{1}{\rm
s}}/kT_{\rm M}}\right],$ (2) $\displaystyle\dfrac{dT_{\rm M}}{dz}$
$\displaystyle=$ $\displaystyle\frac{8\sigma_{\rm T}a_{\rm
R}T^{4}}{3H_{0}E(z)(1+z)m_{\rm e}c}(T_{\rm M}-T)+\frac{2T_{\rm M}}{1+z}.$ (3)

Here $\alpha_{i}$ and $\beta_{i}$ ($i=\text{H, HeI}$) are the effective
recombination rates and photoionization rates, respectively. The coefficients
$C_{i}$ are

$\displaystyle C_{\text{H}}$ $\displaystyle=$
$\displaystyle\frac{1+K_{\text{H}}\Lambda_{\text{H}}\langle
n_{\text{H}}\rangle(1-x_{\text{p}})}{1+K_{\text{H}}(\Lambda_{\text{H}}+\beta_{\text{H}})\langle
n_{\text{H}}\rangle(1-x_{\text{p}})},$ (4) $\displaystyle C_{\text{HeI}}$
$\displaystyle=$
$\displaystyle\frac{1+K_{\text{HeI}}\Lambda_{\text{HeI}}\langle
n_{\text{H}}\rangle(f_{\text{He}}-x_{\text{HeII}})\,\hbox{e}^{h\nu_{\rm
ps}/kT_{\rm
M}}}{1+K_{\text{HeI}}(\Lambda_{\text{He}}+\beta_{\text{HeI}})\langle
n_{\text{H}}\rangle(f_{\text{He}}-x_{\text{HeII}})\,\hbox{e}^{h\nu_{\rm
ps}/kT_{\rm M}}},$ (5)

where the so-called “$K$-quantities” $K_{i}$ represent the effective
transition rates for the main resonances of hydrogen and helium. $\Lambda_{i}$
describe the two-photon decay rates.

As mentioned above, the fine-structure constant $\alpha$ implies the strength
of electromagnetic interaction. Therefore it affects the energy levels of
hydrogen and helium atoms directly as
$E_{i}\propto(\alpha/\alpha_{\text{EM}})^{2}$. This leads to two main sequent
changes. Firstly, the transition frequencies $\nu_{i}$ are altered, such as
the atomic transition rates of H 2s$\to$1s, He $2^{1}p\to 1^{1}s$ and
$2^{3}p\to 1^{1}s$. Likewise, values of $\Lambda_{i}$, $\alpha_{i}$,
$\beta_{i}$, and $K_{i}$ are also rescaled. Secondly, this leads to the
effective temperature $T_{\text{eff}}\propto(\alpha/\alpha_{\text{EM}})^{-2}$
when we calculate $\alpha_{i}$ and $\beta_{i}$. In addition, the Thomson
scattering cross-section $\sigma_{T}$ ought to be modified. Overall, above
modifications can be summarized as following Hart:2017ndk :

$\begin{split}\Lambda_{i}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{8},\
\ \alpha_{i}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{2},\ \
\beta_{i}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{5},\ \
K_{i}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{-6},\ \
T_{\text{eff}}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{-2},\ \
\sigma_{T}\propto\left(\dfrac{\alpha}{\alpha_{\text{EM}}}\right)^{2}.\end{split}$
(6)

As Fig. 1 shows, recombination begins earlier with higher $\alpha$.

###  II.2 Effects of the Fine-Structure Constant on Reionization

Previous researches reveal stars and galaxies contributed the majority of
ionizing photons during the epoch of reionization Madau:2014bja ;
Robertson:2015uda . In this work, we reconstruct the reionization history with
star formation history, instead of the instantaneous reionization model.

The ionized fraction $Q_{\text{HII}}$ (equivalent to $x_{\text{p}}$
physically) can be calculated by the following differential equation
Robertson:2015uda :

$\displaystyle\dot{Q}_{\text{HII}}=-\dfrac{Q_{\text{HII}}}{t_{\text{rec}}}+\dfrac{\dot{n}_{\text{ion}}}{\langle
n_{\text{H}}\rangle}.$ (7)

Here $t_{\text{rec}}$ is the recombination time defined by Robertson:2015uda ;
Bouwens:2015vha

$\displaystyle t_{\text{rec}}$ $\displaystyle=$
$\displaystyle\left[C_{\text{HII}}\alpha_{\text{H}}\left(1+\dfrac{Y_{\text{p}}}{4x_{\text{p}}}\right)\langle
n_{\text{H}}\rangle(1+z)^{3}\right]^{-1}$ (8) $\displaystyle=$ $\displaystyle
0.88\ \text{Gyr}\left(\dfrac{1+z}{7}\right)^{-3}\left(\dfrac{T_{0}}{2\times
10^{4}\text{K}}\right)^{-0.7}\left(\dfrac{C_{\text{HII}}}{3}\right)^{-1},$

where $T_{0}\sim 2\times 10^{4}$ K is the temperature of the ionizing hydrogen
gas and $C_{\text{HII}}\sim 3$ is the clumping factor of ionized hydrogen. The
second term of Eq. 7 represents the effect of SFR. The growth rate of number
density for ionized photons is Robertson:2015uda

$\displaystyle\dot{n}_{\text{ion}}=f_{\text{esc}}\xi_{\text{ion}}\rho_{\text{SFR}},$
(9)

where $f_{\text{esc}}$ is the escaping fraction, which indicates the fraction
of the number of escaping Lyman continuum photons to that of Lyman continuum
photons produced in a galaxy. $\xi_{\text{ion}}$ transforms a UV luminosity
density into the Lyman continuum photon emission rate of a star-forming
galaxy. We adopt the widely used values $f_{\text{esc}}=0.2$ and
$\log_{10}\xi_{\text{ion}}=53.14\
[\text{Lyc}\cdot\text{photons}\cdot\text{s}^{-1}\cdot\text{M}_{\odot}^{-1}\cdot\text{yr}]$.
$\rho_{\text{SFR}}$ is the SFR density, which reads in units of
$\text{M}_{\odot}\cdot\text{yr}^{-1}\cdot\text{Mpc}^{-3}$ as follows
Robertson:2015uda ; Chen:2021gwj :

$\displaystyle\rho_{\text{SFR}}=\dfrac{a_{\text{p}}(1+z)^{b_{\text{p}}}}{1+[(1+z)/c_{\text{p}}]^{d_{\text{p}}}}-\dfrac{a_{\text{p}}(1+22)^{b_{\text{p}}}}{1+[(1+22)/c_{\text{p}}]^{d_{\text{p}}}}.$
(10)

$a_{\text{p}}$, $b_{\text{p}}$, $c_{\text{p}}$ and $d_{\text{p}}$ are four
undetermined coefficients. We add the second term because there is almost no
21 cm absorption signal before $z\sim 22$ in the 21 cm absorption profiles
Bowman:2018yin .

Considering a varying fine-structure constant in star formation, the
recombination time $t_{\text{rec}}$ should be rescaled as
$t_{\text{rec}}\propto\alpha_{\text{H}}^{-1}\propto(\alpha/\alpha_{\text{EM}})^{-2}$.
The escaping rate of ionizing photons $f_{\text{esc}}$ would be inversely
propotional to the Thomson cross-section, which implies
$f_{\text{esc}}\propto\sigma_{T}^{-1}\propto(\alpha/\alpha_{\text{EM}})^{-2}$.
The ionizing photon production rate $\xi_{\text{ion}}$ would be rescaled as
$\xi_{\text{ion}}\propto(\alpha/\alpha_{\text{EM}})^{-2}$ because the Lyman
limits shift shortward by $(\alpha/\alpha_{\text{EM}})^{2}$. We show the
ionized fraction during reionization epoch in Fig. 1 for different
$\alpha/\alpha_{\text{EM}}$.

##  III Results

In this section, we put constraints on the fine-structure constant and other
cosmological parameters by reconstructing the ionization history of our
universe from observational data. We apply the following data combination: the
latest measurements of CMB power spectra—Planck 2018 TT,TE,EE$+$lowE$+$lensing
Aghanim:2018eyx , the SNIa Pantheon sample Scolnic:2017caz , the BAO
measurements at redshifts $z=0.106,0.15,0.32,0.57,1.52,2.34$ (named 6dFGS
Beutler:2011hx , MGS Ross:2014qpa , DR12 Alam:2016hwk , DR14
deSainteAgathe:2019voe ; Ata:2017dya ; Hou:2018yny  respectively), the SFR
density from UV and IR data Madau:2014bja , as well as the $Q_{\text{HII}}$
constraints between $5.0\leq z\leq 8.0$ from observations of Gunn-Peterson
optical depth and Ly-$\alpha$ emission in galaxies Bouwens:2015vha ;
Fan:2005es ; Schenker:2014tda . In this work, we use the SFR data below $z\sim
8$ to reconstruct the reionization history. Certainly, some researches, such
as Ref. Bouwens:2015vha ; Ishigaki_2018 , have provided the SFR density
extending to $z\sim 10$. However, the data at high-$z$ vary dramatically with
different truncation magnitude $M_{\text{trunc}}$ and their errors also get
larger with higher redshifts. So, our choice would not impact the results
significantly. We modify CAMB Lewis:1999bs  and CosmoRec Shaw:2011ez  packages
to calculate the optical depth and CMB power spectra. Then the Markov Chain
Monte Carlo sampler—CosmoMC Lewis:2002ah ; Lewis:2013hha  is applied to
constrain the free parameters. In the first model, named the
$\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ model thereafter, there are ten free
parameters:
$\\{\Omega_{\text{b}}h^{2},\Omega_{\text{c}}h^{2},100\theta_{\text{MC}},\text{ln}\left(10^{10}A_{\text{s}}\right),n_{\text{s}},a_{\text{p}},b_{\text{p}},c_{\text{p}},d_{\text{p}},\alpha/\alpha_{\text{EM}}\\}$.
Five of them are basic parameters in the $\Lambda$CDM model:
$\Omega_{\text{b}}h^{2}$ and $\Omega_{\text{c}}h^{2}$ are the density of
baryons and cold dark matter today respectively, $100\theta_{\text{MC}}$ is
100 times the ratio of angular diameter distance to the large scale structure
sound horizon, $A_{\text{s}}$ is the amplitude of the power spectrum of
primordial curvature perturbations, and $n_{\text{s}}$ is the scalar spectrum
index. Four of them, $a_{\text{p}},b_{\text{p}},c_{\text{p}},d_{\text{p}}$ are
parameters of SFR density as described in section II.2. Lastly,
$\alpha/\alpha_{\text{EM}}$ indicates the ratio of fine-structure constant
$\alpha$ to its fiducial value $\alpha_{\text{EM}}$. It remains unchanged
during the whole ionization history. In the second model, named the
$\Lambda$CDM$+$ $\alpha_{\text{rec}}/\alpha_{\text{EM}}$$+$
$\alpha_{\text{rei}}/\alpha_{\text{EM}}$ model, there are eleven free
parameters:
$\\{\Omega_{\text{b}}h^{2},\Omega_{\text{c}}h^{2},100\theta_{\text{MC}},\text{ln}\left(10^{10}A_{\text{s}}\right),n_{\text{s}},a_{\text{p}},b_{\text{p}},c_{\text{p}},d_{\text{p}},\alpha_{\text{rec}}/\alpha_{\text{EM}},\alpha_{\text{rei}}/\alpha_{\text{EM}}\\}$.
We use $\alpha_{\text{rec}}$ and $\alpha_{\text{rei}}$ to denote the fine-
structure constant during recombination and reionization epoch respectively.

Our results are summarized in Tab. I. We have listed the 68$\%$ limits of the
free parameters and necessary derived parameters in these two models. The
ratio of fine-structure constant reads
$\alpha/\alpha_{\text{EM}}=1.000042^{+0.001983}_{-0.001972}$ at 68$\%$C.L. in
the $\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ model, it consists well with $1$.
However, in the
$\Lambda$CDM+$\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rei}}/\alpha_{\text{EM}}$
model, our results show
$\alpha_{\text{rec}}/\alpha_{\text{EM}}=1.001494^{+0.002041}_{-0.002063}$ and
$\alpha_{\text{rei}}/\alpha_{\text{EM}}=0.8540342^{+0.031678}_{-0.027209}$ at
68$\%$C.L.. It indicates $\alpha_{\text{rec}}$ is inconsistent with
$\alpha_{\text{rei}}$ by 4.64$\sigma$. The straightforward explanation for
this inconsistency is that the fine-structure constant may vary with redshift.
However, the value at low redshift is much smaller than $\alpha_{\text{EM}}$.
Therefore, we prefer another conservative explanation that the observational
data about SFR we used are not good enough and a calibration of some
parameters about SFR is necessary. Since $f_{\rm esc}$ and $\xi_{\rm ion}$
appear in product form, for example, above inconsistency can be solved by
scaling this product to $f_{\rm esc}\xi_{\rm ion}\left(\frac{\alpha_{\rm
rei}}{\alpha_{\rm EM}}\right)^{-4}$. In fact, this calibration is acceptable.
Because the reasonable constraints on $f_{\rm esc}$ and $\xi_{\rm ion}$ at the
epoch of reionization have not been obtained so far Ishigaki_2018 .

Moreover, we find the optical depth $\tau=0.0573_{-0.0006}^{+0.0005}$ at
68$\%$C.L. in the $\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ model and
$\tau=0.0559_{-0.0076}^{+0.0069}$ in the
$\Lambda$CDM+$\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rei}}/\alpha_{\text{EM}}$
model, which is a little higher than $\tau=0.054\pm 0.007$ given by Planck
2018 Aghanim:2018eyx . Also, we list values of Hubble constant,
$H_{0}=67.80\pm 0.67$ km $\cdot$ s-1 $\cdot$ Mpc-1 in the
$\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ model and $H_{0}=68.17\pm 0.70$ km
$\cdot$ s-1 $\cdot$ Mpc-1 in the
$\Lambda$CDM+$\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rei}}/\alpha_{\text{EM}}$
model. Therefore, the varying fine-structure constant can relieve the so-
called Hubble tension slightly. Furthermore, we show the triangular plots of
several parameters in previous two models in Fig. 2 and Fig. 3. We find that
there is an expected strong degeneracy between $100\theta_{\rm MC}$ and the
fine-structure constant during the recombination as well as an expected
degeneracy between $\tau$ and the fine-structure constant during the
reionization.

Table 1: The 68$\%$ limits for the cosmological parameters in the $\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ and $\Lambda$CDM$+$ $\alpha_{\text{rec}}/\alpha_{\text{EM}}$$+$ $\alpha_{\text{rei}}/\alpha_{\text{EM}}$ model.  |  $\Lambda$CDM+$\alpha/\alpha_{\text{EM}}$ |  $\Lambda$CDM$+$ $\alpha_{\text{rec}}/\alpha_{\text{EM}}$$+$ $\alpha_{\text{rei}}/\alpha_{\text{EM}}$
---|---|---
$\Omega_{\text{b}}h^{2}$ |  $0.02244\pm 0.00014$ |  $0.02241\pm 0.00014$
$\Omega_{\text{c}}h^{2}$ |  $0.1191\pm 0.0013$ |  $0.1198\pm 0.0013$
$100\theta_{\text{MC}}$ |  $1.04107\pm 0.00272$ |  $1.04305_{-0.00284}^{+0.00280}$
$\ln(10^{10}\text{A}_{\text{s}})$ |  $3.0484\pm 0.0057$ |  $3.0492_{-0.0068}^{+0.0064}$
$n_{s}$ |  $0.9670_{-0.0064}^{+0.0063}$ |  $0.9630\pm 0.0064$
$a_{\text{p}}$[$\text{M}_{\odot}\cdot\text{yr}^{-1}\cdot\text{Mpc}^{-3}$] |  $0.01772_{-0.00071}^{+0.00070}$ |  $0.01633_{-0.00071}^{+0.00070}$
$b_{\text{p}}$ |  $2.968\pm 0.121$ |  $2.897_{-0.134}^{+0.124}$
$c_{\text{p}}$ |  $2.554_{-0.101}^{+0.088}$ |  $2.736_{-0.112}^{+0.098}$
$d_{\text{p}}$ |  $5.125\pm 0.097$ |  $5.791\pm 0.140$
$\alpha/\alpha_{\text{EM}}$ |  $1.000042_{-0.001972}^{+0.001983}$ |  -
$\alpha_{\text{rec}}/\alpha_{\text{EM}}$ |  - |  $1.001494_{-0.002063}^{+0.002041}$
$\alpha_{\text{rei}}/\alpha_{\text{EM}}$ |  - |  $0.854034_{-0.027209}^{+0.031678}$
$\tau$ |  $0.0573_{-0.0006}^{+0.0005}$ |  $0.0559_{-0.0076}^{+0.0069}$
$10^{4}Q_{\text{HII}}(z=22)$ |  $2.133\pm 0.019$ |  $2.125\pm 0.020$
$H_{0}$[km $\cdot$ s-1 $\cdot$ Mpc-1] |  $67.80\pm 0.67$ |  $68.17\pm 0.70$
Figure 2: The triangular plot of
$100\theta_{\text{MC}},\tau,H_{0},\alpha/\alpha_{\text{EM}}$ in the
$\Lambda$CDM$+\alpha/\alpha_{\text{EM}}$ model from observational data. Figure
3: The triangular plot of
$100\theta_{\text{MC}},\tau,H_{0},\alpha_{\text{rec}}/\alpha_{\text{EM}},\alpha_{\text{rei}}/\alpha_{\text{EM}}$
in the
$\Lambda$CDM$+\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rec}}/\alpha_{\text{EM}}$
model from observational data.

##  IV Summary and discussion

In this paper, we investigate how the fine-structure constant $\alpha$ affect
the whole ionization history of our universe, including both the recombination
and reionization epoch. $\alpha$ influences the Thomson scattering cross-
section and atomic energy levels, then modify the ionization history, which
are reflected in the CMB power spectra. Therefore we constrain the fine-
structure constant and other free parameters in two $\alpha$-related models
from data combination of Planck 2018, BAO data, PANTHON samples, SFR density
from UV and IR measurements, and the $Q_{\text{HII}}$ constraints, using
CosmoRec$+$CAMB$+$CosmoMC packages. In the
$\Lambda$CDM$+\alpha/\alpha_{\text{EM}}$ model, the fine-structure constant
behaves as a constant and we find its value consists well with the standard
one. In the
$\Lambda$CDM$+\alpha_{\text{rec}}/\alpha_{\text{EM}}+\alpha_{\text{rei}}/\alpha_{\text{EM}}$
model, the fine-structure constant has different redshift-independent values
at the recombination and reionization epochs. Our results show the value of
$\alpha_{\text{rei}}$ deviates from $\alpha_{\text{rec}}$ by 4.64$\sigma$. A
simple explanation is a redshift-varying fine-structure constant. But we
prefer another explanation that the data we used needs more refined
calibration, especially at the reionization epoch.

Acknowledgments

We acknowledge the use of HPC Cluster of Tianhe II in National Supercomputing
Center in Guangzhou. Ke Wang is supported by grants from NSFC (grant No.
12005084) and grants from the China Manned Space Project with NO. CMS-
CSST-2021-B01. Lu Chen is supported by grants from NSFC (grant No. 12105164).
This work has also received funding from project ZR2021QA021 supported by
Shandong Provincial Natural Science Foundation.

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