Splitting and Parallelizing of Quantum Convolutional Neural Networks for
  Learning Translationally Symmetric Data
The quantum convolutional neural network (QCNN) is a promising quantum
machine learning (QML) model that is expected to achieve quantum advantages in
classically intractable problems. However, the QCNN requires a large number of
measurements for data learning, limiting its practical applications in
large-scale problems. To alleviate this requirement, we propose a novel
architecture called split-parallelizing QCNN (sp-QCNN), which exploits the
prior knowledge of quantum data to design an efficient model. This architecture
draws inspiration from geometric quantum machine learning and targets
translationally symmetric quantum data commonly encountered in physics and
quantum computing science. By splitting the quantum circuit based on
translational symmetry, the sp-QCNN can substantially parallelize the
conventional QCNN without increasing the number of qubits and improve the
measurement efficiency by an order of the number of qubits. To demonstrate its
effectiveness, we apply the sp-QCNN to a quantum phase recognition task and
show that it can achieve comparable classification accuracy to the conventional
QCNN while considerably reducing the measurement resources required. Due to its
high measurement efficiency, the sp-QCNN can mitigate statistical errors in
estimating the gradient of the loss function, thereby accelerating the learning
process. These results open up new possibilities for incorporating the prior
data knowledge into the efficient design of QML models, leading to practical
quantum advantages.
Koki Chinzei, Quoc Hoan Tran, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato
Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data

###### Abstract

A quantum convolutional neural network (QCNN) is a promising quantum machine learning (QML) model to achieve quantum advantages in classically intractable problems. However, QCNN requires a large number of measurements for data learning, limiting its practical applications for large-scale problems. To relieve this requirement, we propose a novel architecture called split-parallelizing QCNN (sp-QCNN), which exploits the prior knowledge of quantum data for designing efficient circuits. This architecture draws inspiration from geometric quantum machine learning and targets translationally symmetric quantum data commonly encountered in condensed matter physics. By splitting the quantum circuit based on translational symmetry, sp-QCNN substantially parallelizes conventional QCNN without increasing the number of qubits and further improves the measurement efficiency by an order of the number of qubits. To demonstrate its effectiveness, we apply sp-QCNN to a quantum phase recognition task and show that it can achieve similar performance to conventional QCNN while considerably reducing the measurement resources required. Due to its high measurement efficiency, sp-QCNN can mitigate statistical errors in estimating the gradient of the loss function, thereby accelerating the learning process. These results open up new possibilities for incorporating the prior knowledge of data into the efficient design of QML models, leading to practical quantum advantages.

## I Introduction

Quantum computing is an innovative technology that solves classically intractable problems and opens up new frontiers in scientific research and technological advancements [1]. Quantum machine learning (QML) is one of the central research fields in quantum computing, allowing us to solve various tasks such as classification, regression, and clustering by discovering relationships and patterns between data using quantum computers [2]. Recent studies demonstrated quantum speedups in QML beyond classical machine learning for specific artificially engineered tasks, suggesting the potential of QML [3; 4]. It is conjectured that achieving quantum speedups or advantages in QML requires encoding the prior knowledge of the problem into the learning models [5; 6]. However, how to use the prior knowledge in real and practical applications to harness the potential quantum advantages remains unclear.

A quantum neural network (QNN) is a promising QML model that combines the principles of quantum information processing and artificial neural networks to enhance the capabilities of data-driven technologies [7; 8; 9; 10; 11]. A QNN is represented by a parametrized quantum circuit, which is optimized via training data to solve a given task [12]. Since efficiently simulating quantum circuits is generally impossible with classical computers, QNN can learn the complex features of data that are classically unwieldy [13; 14]. Among QNN architectures, a quantum convolutional neural network (QCNN) is a leading one that enables classification tasks [15; 16] [Fig. 1(a)]. For instance, QCNN can classify the phases of matter in quantum many-body systems, an important research object in the broad field of physics [17; 18; 19]. Due to its good trainability and feasibility [20], QCNN is particularly suitable for noisy intermediate-scale quantum (NISQ) devices with a limited number of possible gate operations [21].

The high resource requirement of the measurement process remains a practical barrier for QNNs, including QCNN, to learn data on real quantum computers [12]. During the learning process of QNN, a predefined loss function is minimized by adjusting the circuit's variational parameters. This loss function is computed from a training dataset by measuring specific observables in the parameterized quantum circuit. Therefore, the number of measurements scales with the number of parameters to be optimized and the amount of data to be processed. This situation presents a significant bottleneck when considering large-scale QML applications and the potential of practical quantum advantages [22]. To mitigate the measurement requirement, a possible solution is the multi-programming of quantum computation, enabling the execution of multiple circuits in parallel on different regions of a quantum processor [23; 24; 25; 26]. Although this parallelization reduces the total run time, it increases the required qubit resources, which are limited in current devices.

We address this issue by proposing a novel QNN architecture called split-parallelizing QCNN (sp-QCNN). This architecture targets translationally symmetric data, such as solid states in condensed matter physics, and exploits data symmetry as prior knowledge to substantially parallelize QCNN without increasing qubits and improve the measurement efficiency [Fig. 1(b)]. The circuit of sp-QCNN consists of two elements: translational sym
metrization on each layer and circuit splitting. First, we impose translational symmetry on the convolutional and fully-connected layers to maintain the symmetry of the input state. Second, we split the circuit (rather than discard some qubits) at the pooling layers and then perform the same unitary operations on each branch in parallel. The combination of this circuit structure and data symmetry substantially parallelizes conventional QCNN consisting of the same unitary layers, improving the measurement efficiency of local observables and their gradients with respect to the circuit parameters by \(\mathcal{O}(n)\) times (\(n\) is the number of qubits throughout this paper). For verification, we apply sp-QCNN to a quantum phase recognition task. The results show that sp-QCNN has sufficient expressivity, trainability, and generalization to recognize the symmetry-protected topological (SPT) phase [27, 28, 29, 30] and improves the measurement efficiency by a factor of \(\mathcal{O}(n)\). In training with limited measurement resources, sp-QCNN with high measurement efficiency can suppress statistical errors in estimating the gradient of the loss function to accelerate the learning process, compared with conventional QCNN.

Our architecture exploits the translational symmetry of data as prior knowledge to reduce measurement costs. Such symmetry-tailored architecture design, or more generally inductive bias, is considered crucial to take full advantage of QML capability [5, 6]. For instance, geometric quantum machine learning (GQML), in which the symmetry of a problem is prebuilt into the network structure, can enhance trainability and generalization [31, 32, 33, 34, 35, 36, 37, 38, 39] (e.g., permutation-equivariant QNNs do not suffer from barren plateaus, the exponential vanishing of the gradient in a loss function [40]). Meanwhile, our model exploits data symmetry to improve the measurement efficiency, opening up a new research direction on the architecture design of QNNs. Maximally using the prior knowledge of data is especially significant for near-term quantum devices lacking sufficient computational resources.

The remainder of this paper is organized as follows. First, Sec. II briefly reviews QCNN and discusses its computational cost. Section III introduces two key components of sp-QCNN, translational symmetrization and circuit splitting, and clarifies the similarities and differences between sp-QCNN and GQML. Section IV shows the advantage of sp-QCNN, i.e., the improvement of measurement efficiency for local observables and their gradients with respect to the circuit parameters, based on symmetry. For verification, Sec. V presents the application of sp-QCNN to a quantum phase recognition task, showing that it can solve the task with sufficient accuracy and improve the measurement efficiency by \(\mathcal{O}(n)\) times. Finally, Sec. VI summarizes this paper and discusses potential future research directions.

## II Review of QCNN

A convolutional neural network (CNN) is a celebrated classical machine learning model that solves classification tasks, such as image recognition [41, 42, 43]. A CNN consists of three different types of layers: convolutional, pooling, and fully connected layers. The convolutional layer filters input data to extract its local features, and the pooling layer coarse-grains the data to leave only relevant information. After the convolutional and pooling layers are alternately applied, the fully-connected transformation is applied to the remaining data to obtain a final output. In classification problems, for example, the output indicates which class the input data belongs to. For supervised learning tasks, CNN is trained to correctly classify training data.

A QCNN is a CNN-inspired QNN model that can treat quantum data whose dimension is exponentially larger than classical ones and is expected to achieve practical quantum advantages [15, 16]. Similar to CNN, QCNN consists of convolutional, pooling, and fully connected layers [Fig. 1(a)]. The convolutional layers apply local unitary gates to extract the local features of input data, and the pooling layers discard some qubits to coarse-grain the quantum information. After alternately applying the two layers, we perform the fully connected unitary, measure the remaining qubits, and obtain an output indi

Figure 1: Basic structures of (a) conventional and (b) sp-QCNNs. (a) In conventional QCNN, some qubits are discarded at each pooling layer, and only one of the remaining qubits is measured in the end to classify the quantum data. (b) In sp-QCNN, the translational symmetry of data is used as prior knowledge to design an efficient QML model. The circuit of sp-QCNN (the left circuit) consists of translationally symmetric layers and splitting structures, allowing us to substantially parallelize nonsplitting QCNN (the right circuit) to improve the measurement efficiency.
cating the data class. In QCNN, the quantum circuit is characterized by variational parameters, which are optimized to correctly classify training data. Such a variational algorithm is central in the NISQ era because it works even in a relatively shallow circuit [21].

A QCNN is prospective for quantum advantages in NISQ devices because of its two significant features. One is its high feasibility. In QCNN, since the number of qubits decreases exponentially in each pooling layer, the circuit depth is \(\mathcal{O}(\log n)\). This logarithmic depth is advantageous for NISQ devices in which the number of possible gate operations is limited. The other feature of QCNN is its high trainability. In many variational quantum algorithms, the exponential vanishing of the gradient in a loss function, known as the barren plateau phenomenon, prevents scalable optimization [44, 45, 46]. Meanwhile, Ref. [20] proved that QCNN does not suffer from the barren plateaus due to the logarithmic depth and the locality of unitary operations and observables. This property leads to the high trainability of QCNN, which is crucial for achieving quantum advantages in QML tasks.

The high resource requirement of measurements for optimization presents practical difficulties in QNNs, including QCNN [12]. Let us estimate the required measurement cost in QCNN. First, we suppose that half of the qubits are discarded at each pooling layer and the number of variational parameters is \(\mathcal{O}(n)+\mathcal{O}(n/2)+\mathcal{O}(n/4)+\cdots\sim\mathcal{O}(n)\) in total [in common QCNNs, gates acting in parallel share the same parameters, thus, the number of independent parameters is \(\mathcal{O}(\log n)\), but measuring the gradient of the loss function requires \(\mathcal{O}(n)\) cost (see Sec. IV.2 for details)]. We also let \(N_{\rm train},N_{\rm epoch}\), and \(N_{\rm shot}\) denote the number of training data, maximum epoch (one epoch refers to a complete iteration through a dataset), and the number of measurement shots used per observable, respectively. Then, the total required number of shots during training is \(\mathcal{O}(nN_{\rm train}N_{\rm epoch}N_{\rm shot})\). In terms of practicality, QCNN is not easy to implement for large-scale problems in which many qubits and a large dataset are necessary. Below, we introduce a new architecture of QCNN that can ideally decrease the required number of shots by \(\mathcal{O}(1/n)\) times, bringing QCNN close to realization.

## III Split-parallelizing QCNN

In this section, we describe the two key components of sp-QCNN, translational symmetrization on each layer and circuit splitting, and discuss the relationship between sp-QCNN and QQML through symmetry. Although sp-QCNN can be easily generalized to arbitrary-dimensional lattices, we focus on one-dimensional cases for simplicity.

### Translational symmetry

We exploit data symmetry as prior knowledge to design an efficient QML model. The target of sp-QCNN is translationally symmetric data, which is represented by a density matrix \(\rho_{i}\) with the following property:

\[T\rho_{i}T^{\dagger}=\rho_{i}, \tag{1}\]

where \(T\) is the translation operator by one qubit (e.g., \(T\ket{100\cdots}=\ket{010\cdots}\)). The most relevant field for sp-QCNN application is condensed matter physics, in which translationally symmetric materials such as solids are the largest research topic [47]. In Sec. V, we will demonstrate that sp-QCNN can detect the quantum phases of translationally symmetric many-body states.

To ensure the equivalence of outputs in the parallel computation, we also impose translational symmetry on each of the convolutional and fully connected layers, whose unitary is denoted by \(V_{i}\), as follows:

\[TV_{i}T^{\dagger}=V_{i}. \tag{2}\]

An example of such a unitary is given by (Fig. 2)

\[V_{i}=\prod_{k=1}^{d}R_{ZZ}^{\rm sym}(\delta_{k})R_{X}^{\rm sym}(\gamma_{k})R_ {Z}^{\rm sym}(\beta_{k})R_{X}^{\rm sym}(\alpha_{k}), \tag{3}\]

where we have defined

\[R_{X}^{\rm sym}(\theta) =\prod_{j=1}^{n}e^{-i\theta X_{j}},\;\;R_{Z}^{\rm sym}(\theta)= \prod_{j=1}^{n}e^{-i\theta Z_{j}}, \tag{4}\] \[R_{ZZ}^{\rm sym}(\theta) =\prod_{j=1}^{n}e^{-i\theta Z_{j}Z_{j+1}}, \tag{5}\]

with the periodic boundary condition \(Z_{j+n}=Z_{j}\). The rotation angles, \(\alpha_{k},\beta_{k},\gamma_{k}\), and \(\delta_{k}\), are variational parameters to be optimized via training and do not depend on the qubit position. By construction, \(V_{i}\) is symmetric by one-qubit translation: \([V_{i},T]=0\). In contrast, the convolutional layer of conventional QCNN, \(V_{i}^{\rm conv}\), is symmetric by two or more-qubit translation: \([V_{i}^{\rm conv},T]\neq 0\)

Figure 2: Example of translationally symmetric unitary layer. Single qubit rotations are applied in parallel, followed by ZZ rotations on the nearest neighboring qubits. These procedures are repeated \(d\) times. The rotation angles are translationally symmetric, and thus the number of independent parameters is \(4d\).
but, e.g., \([V_{i}^{\text{conv}},T^{2}]=0\) [ Fig. 1(a)]. As shown later, the translational symmetries of sp-QCNN contribute to the parallel computation.

### Circuit splitting

Another crucial component in sp-QCNN is circuit splitting. In conventional QCNN, the pooling layer discards some qubits to coarse-grain the quantum data. By contrast, in sp-QCNN, we split the circuit at the pooling layers rather than discarding the qubits, as shown in Fig. 1(b). After splitting, we perform the same operations on each branch and finally measure all the qubits in the computational basis. In some types of quantum computers, such as superconducting [48] and ion-trap devices [49], unitary operations can be performed in parallel, and thus this splitting does not significantly increase the run time.

We illustrate a concrete way of splitting a circuit. With \(n\) as the number of qubits, we choose a prime factor of \(n\), denoted by \(p\), and define \(q=n/p\). We, then, introduce splitting in which \(n=pq\) qubits are split into \(p\) branches (Fig. 3). First, we divide the qubits into \(q\) miniblocks each comprising \(p\) qubits in order from the top. Next, we split the circuit such that the \(j\)th qubit of the \(i\)th miniblock is connected to the \(i\)th qubit of the \(j\)th branch. By repeating this procedure on each new branch until the number of qubits becomes one, we obtain the entire sp-QCNN circuit.

Due to the translational symmetry of \(V_{i}\) and circuit splitting, sp-QCNN substantially parallelizes nonsplitting QCNN that consists of the same \(V_{i}\) [Fig. 1(b)]. For convenience, we define \(\left\langle A\right\rangle_{\text{ns}}\) and \(\left\langle A\right\rangle\) as the expectation values of an operator \(A\) in nonsplitting and sp-QCNNs. In nonsplitting QCNN, we measure one of the remaining qubit in the computational basis and consider its expectation value (i.e., \(\left\langle Z_{1}\right\rangle_{\text{ns}}\)) as the output of QCNN. On the other hand, in sp-QCNN, we measure all the qubits and regard the average of the \(n\) expectation values (i.e., \(\left\langle Z_{\text{avg}}\right\rangle=\sum_{j}\left\langle Z_{j}\right\rangle /n\)) as the output. In the next section, we will discuss the mechanism and validity of this parallelization in more detail.

### Relation with geometric quantum machine learning

We look at sp-QCNN from the viewpoint of GQML or equivariant QNN [31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. The concept of GQML has recently emerged as a potential solution to some critical QML issues associated with trainability and generalization. It leverages the symmetry of a problem as inductive bias and provides a problem-tailored circuit architecture. For example, let us consider a classical task that recognizes whether an image depicts a cat. If an image represents a cat, then its rotated image should also represent a cat. Thus, this task has rotational symmetry. In GQML, such symmetry is prebuilt into the network architecture. Formally, given a symmetry operation \(S\) and an output function \(f(\rho)\), the \(S\)-invariance of GQML is defined as follows:

\[f(\rho)=f(S\rho S^{\dagger})\ \ \forall\rho. \tag{6}\]

In other words, the symmetry operation \(S\) on input data never changes the output of GQML. In GQML, the neural network is usually designed based on the equivariant circuit to satisfy this invariance. In theory, GQML significantly enhances the capability of machine learning in several tasks [33, 40].

The circuit of sp-QCNN presents the same invariant property as GQML. Let us consider the unitary transformation \(U\) of the entire sp-QCNN. Due to the translational symmetry of each \(V_{i}\) and the splitting structure, \(U\) itself is translationally symmetric:

\[TUT^{\dagger}=U. \tag{7}\]

This symmetry leads to an equivariant relation between input and output, \(U(T\rho T^{\dagger})U^{\dagger}=T(U\rho U^{\dagger})T^{\dagger}\). That is, the translational operation applied on the input is identical to that on the output. We also define \(f(\rho)=\text{tr}(U\rho U^{\dagger}Z_{\text{avg}})\) with an observable \(Z_{\text{avg}}=\sum_{j}Z_{j}/n\). Then, the equivariant relation and \([Z_{\text{avg}},T]=0\) result in

\[f(\rho)=f(T\rho T^{\dagger})\ \ \forall\rho, \tag{8}\]

which is the \(T\)-invariance of GQML (6). In this sense, sp-QCNN can be viewed as an application of GQML to

Figure 3: Concrete way of splitting a circuit. We first divide the qubits into \(q\) miniblocks consisting of \(p\) qubits and split the circuit such that the \(j\)th qubit of the \(i\)th miniblock is connected to the \(i\)th qubit of the \(j\)th branch.
QCNN. This insight suggests that sp-QCNN can be used to raise QML capability in tasks where translational operation should not change the data output.

Our work offers a new direction for exploiting data symmetry to enhance QML potential. A critical difference between sp-QCNN and GQML is that input data itself is symmetric in our approach (see Eq. (1)), but not in GQML (e.g., the cat's picture is not rotation-invariant). Hence, the tasks to which each technique can be applied are distinct. In addition, each approach brings different benefits. Although the usual GQML improves trainability and generalization, our method reduces measurement costs through substantial parallelization. Thus, sp-QCNN is especially advantageous for near-term quantum devices in which computational resources are limited.

## IV Measurement efficiency in sp-QCNN

In this section, we describe the parallelization in sp-QCNN and show that it can improve the measurement efficiency of local observables and their gradients with respect to the circuit parameters. We also analytically prove that the improvement rate is \(\mathcal{O}(n)\) times for a random input state.

### Measurement efficiency of local observable

First, we show that the translational symmetry of \(V_{i}\) and circuit splitting allow for parallel computation and improve the measurement efficiency of local observables. A key property of sp-QCNN is the equivalence of expectation values for all the qubits. Let us recall that the unitary transformation of the entire sp-QCNN, \(U\), is translationally symmetric (see Eq. (7)). This symmetry leads to

\[\left\langle Z_{1}\right\rangle =\operatorname{tr}\left(U\rho U^{\dagger}Z_{1}\right)\] \[=\operatorname{tr}\left(U(T^{\dagger})^{j-1}\rho T^{j-1}U^{ \dagger}Z_{1}\right)\] \[=\operatorname{tr}\left(U\rho U^{\dagger}Z_{j}\right)\] \[=\left\langle Z_{j}\right\rangle, \tag{9}\]

where \(\rho\) is an input state satisfying Eq. (1), and we have used \(\rho=(T^{\dagger})^{j-1}\rho T^{j-1}\) and \(T^{j-1}Z_{1}(T^{\dagger})^{j-1}=Z_{j}\). This equation indicates the equivalence of the expectation values for all the qubits, i.e., \(\left\langle Z_{i}\right\rangle=\left\langle Z_{j}\right\rangle\) for any \(i\) and \(j\). This argument can be applied to other single-qubit Pauli operators, leading to \(\left\langle X_{i}\right\rangle=\left\langle X_{j}\right\rangle\) and \(\left\langle Y_{i}\right\rangle=\left\langle Y_{j}\right\rangle\). Figure 4(a) graphically illustrates this equivalence, which can also be proved by translating the circuit.

This equivalence tells us that sp-QCNN substantially parallelizes nonsplitting QCNN that consists of the same \(V_{i}\), as shown in Fig. 1(b). As mentioned above, we regard the average of the expectation values for all the qubits, \(\left\langle Z_{\text{avg}}\right\rangle=\sum_{j}\left\langle Z_{j}\right\rangle/n\), as the output in sp-QCNN. Meanwhile, we consider the expectation value for only one qubit, \(\left\langle Z_{1}\right\rangle_{\text{ns}}\), as the output in nonsplitting QCNN. Given the equivalence in Eq. (9), nonsplitting and sp-QCNNs produce the same results if statistical errors are absent:

\[\left\langle Z_{1}\right\rangle_{\text{ns}}=\left\langle Z_{\text{avg}} \right\rangle. \tag{10}\]

Here we have used \(\left\langle Z_{1}\right\rangle_{\text{ns}}=\left\langle Z_{1}\right\rangle\), which can be proved by noticing that nonsplitting QCNN is a part of sp-QCNN. In sp-QCNN, we estimate the output from \(T\) measurement shots as follows:

\[\left\langle Z_{\text{avg}}\right\rangle_{\text{est}}=\frac{1}{T}\sum_{\ell= 1}^{T}z_{\text{avg}}^{(\ell)}=\frac{1}{nT}\sum_{\ell=1}^{T}\sum_{j=1}^{n}z_{j} ^{(\ell)}. \tag{11}\]

Here, \(z_{j}^{(\ell)}=\pm 1\) is the \(\ell\)th measurement outcome at the \(j\)th qubit, and we have defined the average of the \(\ell\)th measurement outcomes as \(z_{\text{avg}}^{(\ell)}=\sum_{j}z_{j}^{(\ell)}/n\). The value of \(z_{\text{avg}}^{(\ell)}\) can be \(a/n\) (\(a\in\{-n,-n+2,\cdots,n\}\)), corresponding to the measurement outcome of \(Z_{\text{avg}}\). We note that the number of outcomes in sp-QCNN is \(n\) times greater than that in nonsplitting QCNN, in which the output is estimated as \(\sum_{\ell=1}^{T}z_{1}^{(\ell)}/T\). Therefore, sp-QCNN can reduce the required number of shots to achieve a certain estimation accuracy. Since this argument only relies on the symmetry property of data, sp-QCNN is general and can be applied to broad tasks with translationally symmetric data.

It is worth noting that sp-QCNN may not improve the measurement efficiency by exactly \(n\) times. This is because, in each shot, \(n\) measurement outcomes are correlated to each other via quantum entanglement. For example, if the output state is the GHZ state \(\left|\psi\right\rangle=(\left|000\cdots\right\rangle+\left|111\cdots\right\rangle )/\sqrt{2}\), then sp-QCNN does not improve the measurement efficiency at all because the \(n\) outcomes are completely correlated and can only provide one bit of information. In contrast, if the output state is the W state \(\left|\psi\right\rangle=(\left|100\cdots 00\right\rangle+\left|010\cdots 00\right\rangle+ \cdots+\left|000\cdots 01\right\rangle)/\sqrt{n}\), then the exact expectation value can be obtained with only one shot by measuring all the qubits in sp-QCNN, whereas many measurements are required in nonsplitting QCNN. Therefore, how well sp-QCNN improves the measurement efficiency depends on the details of the problem, such as input data and circuit structure. Later, we will analytically prove that sp-QCNN can improve the measurement efficiency by \(\mathcal{O}(n)\) times for a typical random input state.

The advantage of sp-QCNN is illustrated in Fig. 5(a). In actual experiments, we cannot obtain the exact expectation value \(\left\langle Z_{1}\right\rangle_{\text{ns}}\) or \(\left\langle Z_{\text{avg}}\right\rangle\) because of statistical errors. Therefore it is usually estimated from the mean value of a finite number of measurement outcomes. In nonsplitting QCNN, the estimated value is generally drawn from the Gaussian distribution with a variance of \(\mathcal{O}(1/N_{\text{shot}})\) in accordance with the central limited theorem. In sp-QCNN, we obtain \(n\) measurement outcomes at once and
thus expect that the variance scales as \(\mathcal{O}(1/nN_{\text{shot}})\), indicating the \(\mathcal{O}(n)\) times improvement of measurement efficiency.

To quantify the effectiveness of sp-QCNN, we introduce the relative measurement efficiency:

\[r\equiv\left(\frac{\sigma_{0}}{\sigma_{\text{sp}}}\right)^{2}. \tag{12}\]

Here \(\sigma_{0}\) and \(\sigma_{\text{sp}}\) are the standard deviations (i.e., square root of variance) of the Gaussians followed by an estimated expectation value in nonsplitting and sp-QCNNs with the same number of shots. This quantity means that the shot number required to achieve a certain estimation accuracy using sp-QCNN is \(1/r\)-times fewer than that using nonsplitting QCNN. In the next section, we will demonstrate the efficiency of sp-QCNN for a concrete task using this quantity.

### Measurement efficiency of gradient

In general, the most costly part of machine learning is the optimization of neural networks using a training dataset, in which the loss function is minimized by tuning the network parameters. In classical machine learning, gradient-based methods are often used for optimization and work well for large-scale problems. Even in QML, gradient-based optimizers are important and powerful tools. However, many measurements are necessary to estimate the gradient in quantum computing [22]. Our architecture makes such gradient measurements efficient.

We first describe a conventional way of measuring the gradient of \(\left\langle Z_{1}\right\rangle\). In many QCNNs, including sp-QCNN, multiple quantum gates share a single variational parameter \(\theta\). Here, let \(m_{\theta}\) be the number of gates sharing \(\theta\) in a branch. To calculate the gradient by \(\theta\), we suppose that the \(m_{\theta}\) gates have different variational parameters from each other, \(\theta_{j}\) (\(j=1,\cdots,m_{\theta}\)). Thereby, we calculate the gradient with the chain rule as \(\partial\left\langle Z_{1}\right\rangle/\partial\theta=\sum_{j}(\partial \theta_{j}/\partial\theta)(\partial\left\langle Z_{1}\right\rangle/\partial \theta_{j})=\sum_{j}\partial\left\langle Z_{1}\right\rangle/\partial\theta_{j}\), where we have used \(\partial\theta_{j}/\partial\theta=1\). When each gate is parametrized as \(e^{-\mathrm{i}\theta_{j}P}\) (\(P\) is a Pauli operator), \(\partial\left\langle Z_{1}\right\rangle/\partial\theta_{j}\) can be measured using the parameter-shift rule, \(\partial\left\langle Z_{1}\right\rangle/\partial\theta_{j}=\left\langle Z_{1 }\right\rangle_{\theta_{j}=\theta+\pi/4}-\left\langle Z_{1}\right\rangle_{ \theta_{j}=\theta-\pi/4}\)[9; 50]. Thus, in sp-QCNN, we can compute the gradient as follows:

\[\frac{\partial\left\langle Z_{1}\right\rangle}{\partial\theta}=\sum_{j=1}^{m_ {\theta}}\mathrm{tr}\left(\tilde{U}_{j+}\rho\tilde{U}_{j+}^{\dagger}Z_{1} \right)-\mathrm{tr}\left(\tilde{U}_{j-}\rho\tilde{U}_{j-}^{\dagger}Z_{1} \right), \tag{13}\]

where \(\tilde{U}_{j\pm}\) is the unitary transformation of sp-QCNN in which \(\theta_{j}=\theta\) is replaced with \(\theta_{j}=\theta\pm\pi/4\). This formula

Figure 4: Mechanism of parallelization in sp-QCNN. (a) In sp-QCNN, the expectation value of a local observable is equivalent for all the qubits. This can be proved by virtually translating the entire circuit. The translation does not change the input state and quantum circuit due to their translational symmetry but shifts the position of the measured qubit, showing the equivalence of expectation values at different qubits. (b) The gradient measurement can be parallelized in sp-QCNN. In accordance with the chain rule, the gradient is the sum of several derivatives, \(\partial\left\langle Z_{1}\right\rangle/\partial\theta=\sum_{j}\partial\left \langle Z_{1}\right\rangle/\partial\theta_{j}\). For example, we suppose that the parameter \(\theta\) is in the first convolutional layer as shown in the figure (the red boxes denote \(\partial/\partial\theta_{2}\) and \(\partial/\partial\theta_{1}\)). Then translating the circuit proves \(\partial\left\langle Z_{1}\right\rangle/\partial\theta_{j}=\partial\left\langle Z _{j-2}\right\rangle/\partial\theta_{1}\) and thus \(\partial\left\langle Z_{1}\right\rangle/\partial\theta=\sum_{j}\partial\left \langle Z_{j}\right\rangle/\partial\theta_{1}\), which can be computed with only two circuits by measuring all the qubits.
has \(2m_{\theta}\) terms, each of which is usually measured in a different circuit.

The circuit splitting and translational symmetry in sp-QCNN allow us to compute \(\partial\left\langle Z_{1}\right\rangle/\partial\theta_{j}\) in parallel, improving the gradient measurement efficiency. For simplicity, we suppose that each \(V_{i}\) has the form shown in Fig. 2 and that \(\theta\) is in the first convolutional layer [Fig. 4(b)]. By translating the entire circuit, we can rewrite each term in Eq. (13) as

\[\operatorname{tr}\left(\tilde{U}_{j\pm}\rho\tilde{U}_{j\pm}^{ \dagger}Z_{1}\right) =\operatorname{tr}\left(\tilde{U}_{j\pm}T^{j-1}\rho(T^{\dagger})^ {j-1}\tilde{U}_{j\pm}^{\dagger}Z_{1}\right)\] \[=\operatorname{tr}\left(\tilde{U}_{1\pm}\rho\tilde{U}_{1\pm}^{ \dagger}Z_{2-j}\right). \tag{14}\]

Here we have used \(\rho=T^{j-1}\rho(T^{\dagger})^{j-1}\), \((T^{\dagger})^{j-1}Z_{1}T^{j-1}=Z_{2-j}\), and \((T^{\dagger})^{j-1}\tilde{U}_{j\pm}T^{j-1}=\tilde{U}_{1\pm}\). This relation tells us that the derivative of \(Z_{1}\) by \(\theta_{j}\) is identical to that of \(Z_{2-j}\) by \(\theta_{1}\), as illustrated in Fig. 4(b). Thereby, Eq. (13) is reduced to

\[\frac{\partial\left\langle Z_{1}\right\rangle}{\partial\theta}=\sum_{j=1}^{m_ {\theta}}\operatorname{tr}\left(\tilde{U}_{1+}\rho\tilde{U}_{1+}^{\dagger}Z_{ j}\right)-\operatorname{tr}\left(\tilde{U}_{1-}\rho\tilde{U}_{1-}^{\dagger}Z_{j} \right), \tag{15}\]

where we have replaced \(Z_{j-2}\) with \(Z_{j}\) in the summation. According to this equation, we can obtain the gradient \(\partial\left\langle Z_{1}\right\rangle/\partial\theta\) with just two circuits \(\tilde{U}_{1\pm}\) by measuring all the qubits, instead of using \(2m_{\theta}\) circuits that are conventionally necessary.

By generalizing this argument and using the equivalence \(\left\langle Z_{\text{avg}}\right\rangle=\left\langle Z_{1}\right\rangle\), we estimate the gradient of the output as follows:

\[\left(\frac{\partial\left\langle Z_{\text{avg}}\right\rangle}{\partial \theta}\right)_{\text{est}}=\frac{m_{\theta}}{nT}\sum_{\ell=1}^{T}\sum_{j=1}^{ n}\left[z_{j+}^{(\ell)}-z_{j-}^{(\ell)}\right], \tag{16}\]

where \(z_{j\pm}^{(\ell)}\) is the \(j\)th qubit measurement outcome of the \(\ell\)th shot in the parameter-shifted circuit with \(\theta_{1}=\theta\pm\pi/4\). In our ansatz (Fig. 2), the factor \(m_{\theta}/n\) appears when \(\theta\) is in the second or later layer. We emphasize that sp-QCNN enables us to execute \(n\) parallel computations even for the gradient estimation, thus accelerating the gradient-based training. Similar to the previous case, the relative measurement efficiency \(r\) for the gradient depends on the details of the problem due to the entangled property of the output state.

### Measurement efficiency for random state

How well sp-QCNN improves the measurement efficiency depends on the details of the problem. Here, we analytically prove that the efficiency is improved by \(\mathcal{O}(n)\) times for a typical state randomly chosen from the \(T\)-invariant Hilbert subspace in the limit of \(n\to\infty\).

Let us begin by considering nonsplitting QCNN, where we measure \(Z_{1}\) and obtain an outcome \(s=\pm 1\) for every measurement. In the limit of \(n\to\infty\), the probability of obtaining an outcome \(\pm 1\) is almost \(1/2\) for a typical random state because the statistical fluctuations by randomness are negligible due to the exponentially large Hilbert space [this probability distribution is depicted in the left panel of Fig. 5(b)]. Given its Bernoulli distribution, the estimation accuracy of the expectation value is

\[\sigma_{0}\sim\mathcal{O}\left(\frac{1}{\sqrt{N_{\text{shot}}}}\right), \tag{17}\]

where \(N_{\text{shot}}\) is the number of shots.

In sp-QCNN, we measure all the qubits in the computational basis and regard the mean of the \(n\) measurement outcomes as the output of QCNN (see Eq. (11)). In other words, we measure \(Z_{\text{avg}}=\sum_{j}Z_{j}/n\) rather than \(Z_{1}\) and obtain one of the eigenvalues \(s\) (\(=\pm 1,\pm(n-2)/n,\cdots\)) as an outcome. Also, given that the full unitary transformation \(U\) is translationally symmetric, the output state

Figure 5: (a) Quantification of measurement efficiency. In actual experiments, statistical errors arise in estimating the expectation value of an observable. This figure shows the probability distribution of the estimated expectation value. Here, we define the relative measurement efficiency \(r\) as the ratio of the variances in sp-QCNN and nonsplitting QCNN. (b) Number of eigenstates of \(Z_{1}\) and \(Z_{\text{avg}}\) with an eigenvalue \(s\). While the possible measurement outcome is \(\pm 1\) in nonsplitting QCNN (left panel), it is widely distributed in the range of \(-1\) to \(1\) with a width of \(\mathcal{O}(1/\sqrt{n})\) in sp-QCNN (right panel).
of sp-QCNN has the same symmetry. The right panel of Fig. 5(b) shows the number of eigenstates of \(Z_{\rm avg}\) with an eigenvalue \(s\), \(D_{n}(s)\), on the \(T\)-invariant Hilbert subspace. In the limit of \(n\to\infty\), \(D_{n}(s)\) approaches the following asymptotic form (see Appendix A for derivation):

\[D_{n}(s)\sim\frac{C_{n}}{(1+s^{2})^{n/2}}, \tag{18}\]

where \(C_{n}\) is a constant independent of \(s\). The width of \(D_{n}(s)\) in \(s\) is \(\mathcal{O}(1/\sqrt{n})\), which finally gives rise to a small estimation error.

Here, we assume that when measuring \(Z_{\rm avg}\) for a typical state randomly chosen from the \(T\)-invariant subspace, the probability of obtaining an outcome \(s\) is proportional to \(D_{n}(s)\). This assumption would be justified in the limit of \(n\to\infty\), where \(D_{n}(s)\) is sufficiently large, and the statistical fluctuations are insignificant. Considering that the width of \(D_{n}(s)\) is \(\mathcal{O}(1/\sqrt{n})\), we can estimate the expectation value from \(N_{\rm shot}\) experiments with an accuracy

\[\sigma_{\rm sp}\sim\mathcal{O}\left(\frac{1}{\sqrt{nN_{\rm shot}}}\right). \tag{19}\]

From the quantification in Eq. (12), the relative measurement efficiency of sp-QCNN is

\[r=\left(\frac{\sigma_{0}}{\sigma_{\rm sp}}\right)^{2}\sim\mathcal{O}(n). \tag{20}\]

This result indicates the \(\mathcal{O}(1/n)\) times reduction in the number of experiments required to achieve a certain accuracy.

The scaling argument in Eq. (20) is valid in situations where the output state is a random quantum state. Therefore, it may arise in the early stage of the training process when the parameters of QCNN are randomly initialized and the output state is approximately random. In Sec. V, we will show that sp-QCNN exhibits \(\mathcal{O}(n)\) scaling for a concrete task in the early stage of training and, remarkably, even in the final stage.

## V Application to quantum phase recognition

In this section, we apply sp-QCNN to a quantum phase recognition task investigated in Ref. [16] and verify its effectiveness. For the remainder of this paper, we simulate the quantum circuit with Qulacs, an open-source quantum circuit simulator [51].

### Formulation of problem

Let us consider a one-dimensional cluster Ising model with the periodic boundary condition, whose Hamiltonian is given by

\[H=-\sum_{j=1}^{n}Z_{j}X_{j+1}Z_{j+2}-h_{1}\sum_{j=1}^{n}X_{j}-h_ {2}\sum_{j=1}^{n}X_{j}X_{j+1}, \tag{21}\]

where \(n\) is the number of qubits, and \(X_{j},Y_{j},\) and \(Z_{j}\) are the Pauli operators at the \(j\)th qubit. This Hamiltonian exhibits SPT [27; 28; 29; 30], paramagnetic (PM), and antiferromagnetic (AFM) phases on the \(h_{1}\)-\(h_{2}\) plane. The SPT phase is protected by \(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\) symmetry characterized by \(X_{\rm even(odd)}=\prod_{j\in{\rm even(odd)}}X_{j}\). The ground state of \(H\), an input state in our task, is translationally symmetric because of \(THT^{\dagger}=H\).

Our task is to recognize the SPT phase using sp-QCNN. Quantum phase recognition is one of the main applications of QCNN, and many studies have been conducted with the aim of practical quantum advantages [17; 18; 19]. In this task, sp-QCNN can be applied because the input data (i.e., the ground state of \(H\)) is translationally symmetric. For training data, we use \(M=20\) ground states of \(H\), \(|\phi_{i}\rangle\), evenly located on the line of \(h_{2}=0\) from \(h_{1}=0.05\) to \(1.95\). Using the Jordan-Wigner transformation [52], we can analytically obtain the exact ground state for \(h_{2}=0\), which transits from the SPT to PM phases at \(h_{1}=1\). To verify the generalization, we predict the entire phase diagram on the \(h_{1}\)-\(h_{2}\) plane using trained sp-QCNN and compare it with the true one computed with the density matrix renormalization group (DMRG) [53; 54; 55; 56]. In this work, we prepare the input data with exact diagonalization for simplicity. Yet, in actual experiments, other preparation methods must be applied, such as variational quantum eigensolver on a quantum computer and analog-digital transduction from a quantum experiment.

We adopt the following mean squared error as the loss function:

\[\mathcal{L}=\frac{1}{2M}\sum_{i=1}^{M}\left(\langle\phi_{i}|U^{\dagger}Z_{\rm avg }U|\phi_{i}\rangle-y_{i}\right)^{2}, \tag{22}\]

where \(|\phi_{i}\rangle\) and \(y_{i}\) are the training data and its corresponding label, and \(U\) is the total unitary of the circuit. Here, we set \(y_{i}\) as \(1\) if \(|\phi_{i}\rangle\) belongs to the SPT phase and \(0\) if it does not. We optimize the loss function using the stochastic gradient descent (SGD) method [57]. In SGD, we update the parameters as \(\vec{\theta}^{(t+1)}=\vec{\theta}^{(t)}-\eta^{(t)}\nabla\mathcal{L}\), where \(\vec{\theta}^{(t)}\) is the parameter vector at optimization step \(t\), and \(\nabla\mathcal{L}\) is calculated from only one of the training data at each step. We also decrease the learning rate as \(\eta^{(t)}=\eta_{0}/t\) to stabilize the training and set \(\eta_{0}=200\). Meanwhile, to investigate the statistical properties of sp-QCNN, we simulate the same circuits with \(N_{p}\) different random initial parameter sets and set \(N_{p}=200\) in Secs. V.2 and V.3 and \(N_{p}=50\) in Sec. V.4.
### Expressivity, trainability, and generalization

We show that our ansatz has sufficient expressivity, trainability, and generalization to recognize the quantum phase transition despite the constraint of translational symmetry before examining the measurement efficiency. To this end, we simulate the quantum circuit with infinite shots, i.e., no statistical errors in estimating expectation values. As an ansatz circuit, we use the translational unitary \(V_{i}\) shown in Fig. 2 with \(d=10\), where each layer has \(4\times 10=40\) independent parameters. We split the circuit such that the number of qubits in a branch varies as \(8\to 4\to 2\to 1\), \(12\to 6\to 3\to 1\), \(16\to 8\to 4\to 2\to 1\), and \(18\to 9\to 3\to 1\) for \(n=8,12,16\), and \(18\), respectively.

We first investigate the expressivity and trainability of our ansatz. Figure 6(a) shows the changes in the loss function during training for several problem sizes (i.e., the numbers of qubits) \(n\). We observe that the final loss after 200 epochs becomes smaller as \(n\) increases. This trend is consistent with the nature of phase transition that occurs in the thermodynamic limit (\(n\to\infty\)), suggesting that it is possible to achieve a lower training error in larger problem sizes. In Sec. V.4, we show that the final value of the loss function reached is comparable with that of conventional QCNN, indicating the high expressivity of our ansatz. In addition, the initial drop of the loss function becomes more rapid as \(n\) increases, and the statistical error by the randomness of initial parameters is modest even for \(n=18\). These results imply that our ansatz has high trainability and that sp-QCNN can be trained well even for larger problems.

We also examine the generalization of sp-QCNN. Figure 6(b) is the phase diagram predicted by trained sp-QCNN for \(n=16\) qubits. It coincides with the phase boundary computed by DMRG (dashed lines) well. In particular, sp-QCNN can detect the SPT-AFM phase transition that is not present in the training dataset. This result shows that sp-QCNN has enough generalization for the quantum phase recognition task.

### Measurement efficiency

As discussed above, the measurement efficiency of sp-QCNN depends on the details of the problem, such as input data and circuit structure. Here we show that sp-QCNN improves the measurement efficiency by \(\mathcal{O}(n)\) times in the quantum phase recognition task. To this end, we calculate how the efficiency changes during the training process examined in Sec. V.2. The measurement efficiency is quantified by the ratio of the variances in splitting and nonsplitting circuits, as shown in Fig. 5 (we assume that the nonsplitting circuit consists of the same unitary \(V_{i}\) as the splitting one). We investigate the efficiency for three typical input states: SPT, PM, and AFM states, which are the eigenstates of \(H\) for \((h_{1},h_{2})=(0,0),(+\infty,0)\), and \((0,-\infty)\), respectively. We also explore the efficiency of measuring the loss gradient for the first parameter.

Figures 7(a)-(d) show the changes in the relative measurement efficiency \(r\) during training for the three inputs and loss gradient. For the PM and AFM states ((b) and (c)), the efficiency \(r\) is high at the beginning of learning and does not significantly decreases during training. For the SPT state and loss gradient ((a) and (d)), \(r\) is initially high but decreases as training, finally converging to a small value (\(r=2\)-5). These results imply that the improvement rate of measurement efficiency strongly depends on the input data, what we measure, and the stage of learning. Even for the SPT state and loss gradient, the final efficiency is higher than one, indicating that the measurement in sp-QCNN is more efficient than that in nonsplitting QCNN.

Figures 8(a) and (b) show the relative measurement efficiency with varying the number of qubits \(n\) at 0 and 200 epochs for the four cases (cf. Fig. 7). At 0 epoch

Figure 6: (a) Changes in loss function for several numbers of qubits \(n\). The lines and shaded areas depict the median and the 10th–90th percentiles for 200 sets of initial parameters, respectively. (b) Phase diagram predicted by trained sp-QCNN for \(n=16\) qubits. The color denotes the average magnitude of \(\langle Z_{\text{avg}}\rangle\) for 200 sets of initial parameters. The gray dots and dashed green lines denote our training data and phase boundaries computed by DMRG, respectively. In these simulations, we set the depth of each layer as \(d=10\).
(a), the data points are nearly aligned on a straight line. This result supports that measurement efficiency is improved by \(\mathcal{O}(n)\) times in the early stages of learning and is consistent with the previous argument based on randomness. Even at 200 epoch (b), we can fit the data points in straight lines within their error bars, suggesting that the efficiency is also improved by \(\mathcal{O}(n)\) times in the final stages of learning. In other words, compared with nonsplitting QCNN, sp-QCNN can reduce the number of shots required to achieve a certain estimation accuracy of expectation values by a factor of \(\mathcal{O}(1/n)\) throughout the learning process.

We also investigate the measurement efficiency for predicting the phase diagram by trained sp-QCNN with \(n=8\) and 16 qubits (see Figs. 8(c) and (d)). By comparing these figures, we notice that the efficiency \(r\) for \(n=16\) is more than twice that for \(n=8\) in most areas. This result implies an improvement of \(\mathcal{O}(n)\) times for prediction. We also observe that the efficiency is low in the SPT phase but relatively high in the PM and AFM phases, a trend evident in Fig. 7 as well. We infer that this phenomenon is due to the following reason. For the SPT state, the expectation value of \(Z_{j}\) after training is almost one because we have assigned the label as \(y_{i}=1\) for the SPT phase in the loss function, which means that \(U\left|\phi_{\text{SPT}}\right\rangle\sim\left|0\cdots 0\right\rangle\). Given that the splitting circuit has no advantages for measuring \(\left|0\cdots 0\right\rangle\), the measurement efficiency is not significantly improved. For complete understanding, additional analyses must be conducted in future works.

### Training with limited measurement resources

So far, we have shown that sp-QCNN makes the measurement efficient and the efficiency \(r\) depends on the input state and the learning stage. In actual experiments, we cannot know \(r\) in advance, and controlling the shot number every epoch by feeding back measurement outcomes might be technically difficult. Therefore, we investigate how sp-QCNN enhances the performance of machine learning in training with a fixed small num

Figure 8: (a), (b) Relative measurement efficiency \(r\) with varying the number of qubits \(n\) for SPT, PM, AFM, and loss gradient at (a) 0 and (b) 200 epoch in Fig. 7. The four straight lines fit the corresponding types of data points. The error bars denote the standard deviations for 200 sets of initial parameters. (c), (d) Relative measurement efficiency \(r\) on \(h_{1}\)-\(h_{2}\) plane after 200 epochs for (c) \(n=8\) and (d) 16 qubits. The color denotes the magnitude of the relative measurement efficiency. The gray dots and dashed green lines denote our training data and phase boundaries computed by DMRG, respectively.

Figure 7: Changes in relative measurement efficiency during training. (a)–(c) show the efficiency \(r\) for different inputs, SPT, PM, and AFM states, whereas (d) shows the efficiency for measuring loss gradient by the first parameter. At each epoch, we simulate experiments with 1000 shots 10000 times, estimate \(\sigma_{0}\) and \(\sigma_{\text{sp}}\), and calculate the efficiency \(r=(\sigma_{0}/\sigma_{\text{sp}})^{2}\). The solid lines and shaded areas depict the mean values and standard deviations, respectively, for 200 sets of initial parameters. Except for evaluating the efficiency, we optimize the circuit using the exact expectation value (i.e., without statistical errors).
ber of shots. In situations with limited measurement resources, statistical errors in estimating the gradient of a loss function would disturb learning and degrade classification performance.

As the sp-QCNN ansatz, we use the unitary layer shown in Fig. 2 with \(d=5\), where the total number of parameters is \(60\) for \(n=8\) and \(80\) for \(n=16\). We also compare sp-QCNN with conventional QCNN depicted in Fig. 1(a), where the convolutional and fully-connected layers consist of two-qubit unitary gates parametrized as \(\prod_{j=1}^{15}e^{-i\theta_{j}P_{j}}\) (\(P_{j}=IX,IY,\cdots,ZZ\)). Since the gates acting in parallel share the same parameters, the number of independent parameters in conventional QCNN is \(75\) for \(n=8\) and \(105\) for \(n=16\). When measuring the gradient in the simulation, we match the shot number per parameter in conventional and sp-QCNNs for each layer. Here, we use \(2m_{\theta}N_{\text{shot}}\) shots for parameter \(\theta\) in sp-QCNN and set \(N_{\text{shot}}=5\).

We show that sp-QCNN suppresses statistical errors and accelerates the learning process. Figure 9 displays the changes in loss function with and without statistical errors in conventional and sp-QCNNs. In the absence of statistical errors (dashed lines), the loss function quickly converges for both QCNNs (the final value of the loss function in sp-QCNN) is comparable, or even superior, compared with that of the conventional one, indicating that our ansatz has sufficient expressivity). In the presence of statistical errors (solid lines), the loss convergence becomes substantially slow in conventional QCNN but is relatively modest in sp-QCNN. This fast convergence in sp-QCNN stems from its high measurement efficiency. Whereas significant statistical errors disturb the rapid and stable optimization in conventional QCNN, the high measurement efficiency in sp-QCNN suppresses the statistical errors, stabilizing and accelerating the optimization. As shown in the figure, this improvement effect is more pronounced for \(n=16\) than for \(n=8\) because the measurement efficiency is improved by a factor of \(\mathcal{O}(n)\). The fast convergence of training is highly effective for near-term quantum devices where a long optimization run is impractical due to limited computational resources.

## VI Conclusions

In this study, we have proposed a new QNN architecture, sp-QCNN, which reduces measurement costs by exploiting the translational symmetry of data as prior knowledge. In sp-QCNN, we symmetrize and split the QCNN circuit to parallelize the computation, thus improving the measurement efficiency. We have demonstrated the advantage of sp-QCNN for the quantum phase recognition task: it has high classification performance for this task and can improve the measurement efficiency by \(\mathcal{O}(n)\) times. In a realistic setting where measurement resources are limited, sp-QCNN can enhance the speed and stability of the learning process. These results present a new possibility for the symmetry-based architecture design of QNN and bring us one step closer to achieving the quantum advantages of QCNN in near-term quantum devices.

This work offers some research directions for the future. First, finding practical, applicable problems is crucial for quantum advantages since sp-QCNN can be used only for translationally symmetric data. The most promising application of sp-QCNN is the research of solids, where it could shed light on unsolved mysteries in condensed matter physics. The second direction is further studies of symmetry-based architecture design to reduce measurement costs. Although this work has provided a new approach for QML, its coverage is limited to data with translational symmetry. Hence, generalization to other symmetries, such as the space group, is intriguing and fruitful and may be applied to chemical molecules as well as solid-state materials. The third direction is to find a better ansatz. Although this work establishes the basis of sp-QCNN, the best \(V_{i}\) for a given problem remains unclear. Finding a more compact and expressive ansatz would be helpful to experimentally realize sp-QCNN.

Finally, we provide several open issues on sp-QCNN. This work has shown that sp-QCNN has sufficient expressivity, trainability, and generalization to solve the phase recognition task. However, whether it can solve other complicated tasks remains unclear. In particular, the

Figure 9: How statistical errors affect the learning process in conventional and sp-QCNNs for \(n=8\) (top) and \(16\) qubits (bottom). The orange (blue) solid and dashed lines denote the loss function with and without statistical errors in sp-QCNN (conventional QCNN), respectively. The shaded areas are the \(10\)th–\(90\)th percentiles of the loss function for \(50\) random initial parameters at each epoch. We match the number of shots per parameter to obtain the gradient in both QCNNs.
translational symmetry of \(V_{i}\) could suppress expressivity and limit solvable tasks. Uncovering the possibilities and limitations of sp-QCNN is an important open issue. For trainability, elucidating whether barren plateaus exist in sp-QCNN is crucial. In conventional QCNN, barren plateaus do not appear due to its unique architecture: the logarithmic circuit depth and the locality of unitary operations and observables [44, 45, 46, 20]. Considering that sp-QCNN shares these properties with conventional QCNN, we think no barren plateaus will appear even in sp-QCNN. The results in this paper show that the training of sp-QCNN works well up to \(n=18\) qubits, supporting our hypothesis. More thorough analyses are necessary for complete verification.

## Acknowlegments

Fruitful discussions with Masatoshi Ishii, Tomochika Kurita, Yuichi Kamata, and Yasuhiro Endo are gratefully acknowledged.

## Appendix A The number of eigenstates of \(Z_{\text{avg}}\)

Here, we derive Eq. (18), where the number of eigenstates of \(Z_{\text{avg}}\) for translationally symmetric states is

\[D_{n}(s)\sim\frac{1}{(1+s^{2})^{n/2}} \tag{19}\]

with an eigenvalue \(s\). In sp-QCNN, we measure \(Z_{\text{tot}}=\sum_{j}Z_{j}\) whose eigenvalues are \(\pm n,\pm(n-2),\cdots\) and obtain one of the eigenvalues every shot (for convenience, we consider \(Z_{\text{tot}}\) rather than \(Z_{\text{avg}}=Z_{\text{tot}}/n\)). In addition, the output state is translationally symmetric in sp-QCNN. Hence, for simplicity, we now focus on the \(T\)-invariant eigenspace of \(Z_{\text{tot}}\) with an eigenvalue \(z\), \(V_{z}\) (i.e., \(T\left|\phi\right\rangle=\left|\phi\right\rangle\) and \(Z_{\text{tot}}\left|\phi\right\rangle=z\left|\phi\right\rangle\) for any \(\left|\phi\right\rangle\in V_{z}\)). Below, we investigate the dimension of \(V_{z}\).

To this end, we introduce a cyclic group generated by \(T\),

\[G_{n}=\{I,T,T^{2},\cdots,T^{n-1}\}. \tag{20}\]

Let \(M_{z}\) be the set of the eigenstates of \(Z_{\text{tot}}\) with an eigenvalue \(z\) in the computational basis (e.g., \(M_{n-2}=\{\left|10\cdots 00\right\rangle,\cdots,\left|00\cdots 01\right\rangle\}\)). Then, we define an equivalence relation \(\sim\) by \(G_{n}\) in \(M_{z}\): for \(\left|a\right\rangle,\left|b\right\rangle\in M_{z}\), \(\left|a\right\rangle\sim\left|b\right\rangle\) holds if and only if \(g\left|a\right\rangle=\left|b\right\rangle\) with \({}^{\exists}g\in G_{n}\). We also define the equivalence class of \(\left|a\right\rangle\in M_{z}\) as \(\left|a\right\rangle=\left|\left\langle x\right\rangle\in M_{z}\left|\left. \left|x\right\rangle\sim\left|a\right\rangle\right\right\}\) and the quotient set as \(M_{z}/G_{n}=\{\left|a\right|\left|a\right\rangle\in M_{z}\}\). The elements of \(M_{z}/G_{n}\) correspond one-to-one to the bases of \(V_{z}\), such that \(\left|\Psi_{i}\right\rangle=\sum_{\left|\phi\right\rangle\in\left|\phi\right\rangle }\left|\phi\right\rangle/\mathcal{N}\), where \(\left|\Psi_{i}\right\rangle\) is the base of \(V_{z}\), \(\left|\Phi_{i}\right\rangle\) is the element of \(M_{z}/G_{n}\), and \(\mathcal{N}\) is the normalization factor \((T\left|\Psi_{i}\right\rangle=\left|\Psi_{i}\right\rangle)\) can be easily checked). Therefore, \(\text{dim}V_{z}=\left|M_{z}/G_{n}\right|\) holds, where \(\left|A\right|\) is the number of elements in \(A\). Using Burnside's lemma [58], we have \(\text{dim}V_{z}\) as follows:

\[\text{dim}V_{z}=|M_{z}/G_{n}|=\frac{1}{\left|G_{n}\right|}\sum_{g\in G_{n}}|M _{z}^{g}|, \tag{21}\]

where \(M_{z}^{g}=\{\left|\phi\right\rangle\in M_{z}\mid g\left|\phi\right\rangle= \left|\phi\right\rangle\}\). Then, the following theorem holds.

**Theorem 1**.: _For \(z\neq\pm n\), the following relation holds in the limit of \(n\rightarrow\infty\):_

\[F_{z}\equiv\text{dim}V_{z}\bigg{/}\frac{1}{n}\binom{n}{\ell_{z}}\xrightarrow{ n\rightarrow\infty}1 \tag{22}\]

_with \(\ell_{z}=(n+z)/2\). Here \(\binom{\cdot}{\cdot}\) denotes the binomial coefficient. This theorem states that the asymptotic form of \(\text{dim}V_{z}\) is \(\binom{n}{\ell_{z}}/n\)._

Proof.: Since \(F_{z}=F_{-z}\) trivially holds, we focus on \(-n+2\leq z\leq 0\), or \(1\leq\ell_{z}\leq\lfloor n/2\rfloor\) (\(\lfloor\cdot\rfloor\) is the floor function).

Figure 10: Illustration for calculating \(|M_{z}^{g}|\) with \(n=12,\ell_{z}=4\), and \(g=T^{3}\). Each white (black) circle indicates a single qubit state of \(\left|0\right\rangle\) (\(\left|1\right\rangle\)), and \(n=12\) and \(\ell_{z}=4\) mean that there are eight (\(=n-\ell_{z}\)) white and four (\(=\ell_{z}\)) black circles in total. Given that the order of \(g\) is four (‘\(\cdot\)’ \(g^{4}=I\)), we first divide the qubits into four sets, each consisting of three (\(=n/\chi(g)\)) qubits. In all the sets, the configuration of white and black circles must be the same as each other because of the condition that \(g\) does not change the state. Therefore, each set has two white and one (\(=\ell_{z}/\chi(g)\)) black circles, and there are three (\(=\binom{n/\chi(g)}{\ell_{z}/\chi(g)}\)) possible configurations shown in the figure.
We first rewrite Eq. (17) as

\[\text{dim}V_{z} =\frac{M_{z}^{I}}{|G_{n}|}+\frac{1}{|G_{n}|}\sum_{g\in G_{n}\setminus \{I\}}|M_{z}^{g}|\] \[=\frac{1}{n}\binom{n}{\ell_{z}}+\frac{1}{n}\sum_{g\in G_{n}\setminus \{I\}}|M_{z}^{g}|. \tag{18}\]

Therefore, \(F_{z}\) is reduced to

\[F_{z}=1+\sum_{g\in G_{n}\setminus\{I\}}|M_{z}^{g}|\Bigg{/}\binom{n}{\ell_{z} }\;. \tag{19}\]

We will evaluate the second term in this equation.

To calculate \(|M_{z}^{g}|\), we define the order of \(g\in G_{n}\), \(\chi(g)\), as the number of elements in the subgroup generated by \(g\) (i.e., \(\{g^{0},g^{1},g^{2},\cdots,g^{k-1}\}\) with \(g^{k}=I\)). Note that \(\chi(g)\) is a divisor of \(n\). Thereby, \(|M_{z}^{g}|\) is written as follows:

\[|M_{z}^{g}|=\begin{cases}0&\ell_{z}/\chi(g)\notin\mathbb{Z}\\ \binom{n/\chi(g)}{\ell_{z}/\chi(g)}&\ell_{z}/\chi(g)\in\mathbb{Z}.\end{cases} \tag{20}\]

Figure 10 shows a graphical description of Eq. (20) as an example for \(n=12,\ell_{z}=4\), and \(g=T^{3}\).

Based on Eq. (20), one can straightforwardly show that the second term in Eq. (19) vanishes in the limit of \(n\to\infty\) for \(\ell_{z}=1,2\) by noticing that \(\chi(g)=1\) only for \(g=I\) and \(\chi(g)=2\) only for \(g=T^{n/2}\). Thus, we focus on \(3\leq\ell_{z}\leq\lfloor n/2\rfloor\). Because of \(\chi(g)\geq 2\) for \(g\neq I\), we have

\[|M_{z}^{g}|\leq\binom{\lfloor n/2\rfloor}{\lfloor\ell_{z}/2\rfloor}. \tag{21}\]

This inequality can be shown by considering the properties of binomial coefficients: \(\binom{a}{b}<\binom{a^{\prime}}{b}\) (\(a<a^{\prime}\)) and \(\binom{a}{0}<\binom{a}{1}<\cdots<\binom{a}{\lfloor a/2\rfloor}\) (note that \(3\leq\ell_{z}\leq\lfloor n/2\rfloor\)). Using Eq. (21), the second term in Eq. (19) is bounded as follows:

\[0\leq\sum_{g\in G_{n}\setminus\{I\}}|M_{z}^{g}|\Bigg{/}\binom{n}{\ell_{z}}\; \leq\;n\binom{\lfloor n/2\rfloor}{\lfloor\ell_{z}/2\rfloor}\Bigg{/}\binom{n} {\ell_{z}}\;\equiv\;A_{\ell_{z}}. \tag{22}\]

The right-hand side of this inequality, \(A_{\ell_{z}}\), approaches zero in \(n\to\infty\) for \(3\leq\ell_{z}\leq\lfloor n/2\rfloor\), which can be proven by showing \(A_{3}\xrightarrow{n\to\infty}0\) and \(0<A_{\lfloor n/2\rfloor}<\cdots<A_{4}<A_{3}\) by definition of \(A_{\ell_{z}}\). Therefore, we have

\[\sum_{g\in G_{n}\setminus\{I\}}|M_{z}^{g}|\Bigg{/}\binom{n}{\ell_{z}}\; \xrightarrow{n\to\infty}0. \tag{23}\]

As mentioned above, this limit holds true even for \(\ell_{z}=1,2\). Substituting Eq. (23) to Eq. (19), we obtain

\[F_{z}\xrightarrow{n\to\infty}1. \tag{24}\]

with \(\ell_{z}\neq 0,n\).

This theorem states that \(\text{dim}V_{z}\) asymptotically approaches \(\binom{n}{\ell_{z}}/n\) (except for \(\text{dim}V_{n}=\text{dim}V_{-n}=1\)). Using Stirling's formula (\(n!\sim\sqrt{2\pi n}(n/e)^{n}\)), we have

\[\text{dim}V_{z}\sim 2^{n}\sqrt{\frac{2}{\pi n^{3}}}D_{n}(s), \tag{25}\]

where we have defined

\[D_{n}(s)\equiv\left[(1+s)^{1+s+\frac{1}{n}}(1-s)^{1-s+\frac{1}{n}}\right]^{-n /2} \tag{26}\]

with \(s=z/n\). For large \(n\), \(D_{n}(s)\) rapidly decreases to vanish away from the origin. Therefore, we expand the denominator of \(D_{n}(s)\) in \(s\), obtaining

\[D_{n}(s) =\left(1+(1+\mathcal{O}(1/n))s^{2}+\mathcal{O}(s^{4})\right)^{-n/2}\] \[\sim\frac{1}{(1+s^{2})^{n/2}}, \tag{27}\]

for sufficiently small \(s\). The width of \(D_{n}(s)\) is \(\mathcal{O}(1/\sqrt{n})\) in \(n\to\infty\), leading to the \(\mathcal{O}(n)\) times improvement of measurement efficiency in sp-QCNN (see Sec. IV.3). Finally, we remark that this discussion is approximately valid for large but finite \(n\) while this appendix considers the limit of \(n\to\infty\). In fact, in Sec. V.3, we have observed the clear \(\mathcal{O}(n)\) scaling for \(n=18\) at the beginning of training where the output state is almost random.
