Randomness-enhanced expressivity of quantum neural networks
As a hybrid of artificial intelligence and quantum computing, quantum neural
networks (QNNs) have gained significant attention as a promising application on
near-term, noisy intermediate-scale quantum (NISQ) devices. Conventional QNNs
are described by parametrized quantum circuits, which perform unitary
operations and measurements on quantum states. In this work, we propose a novel
approach to enhance the expressivity of QNNs by incorporating randomness into
quantum circuits. Specifically, we introduce a random layer, which contains
single-qubit gates sampled from an trainable ensemble pooling. The prediction
of QNN is then represented by an ensemble average over a classical function of
measurement outcomes. We prove that our approach can accurately approximate
arbitrary target operators using Uhlmann's theorem for majorization, which
enables observable learning. Our proposal is demonstrated with extensive
numerical experiments, including observable learning, R\'enyi entropy
measurement, and image recognition. We find the expressivity of QNNs is
enhanced by introducing randomness for multiple learning tasks, which could
have broad application in quantum machine learning.
Yadong Wu, Juan Yao, Pengfei Zhang, Xiaopeng Li
# Randomness-enhanced expressivity of quantum neural networks

###### Abstract

As a hybrid of artificial intelligence and quantum computing, quantum neural networks (QNNs) have gained significant attention as a promising application on near-term, noisy intermediate-scale quantum (NISQ) devices. Conventional QNNs are described by parametrized quantum circuits, which perform unitary operations and measurements on quantum states. In this work, we propose a novel approach to enhance the expressivity of QNNs by incorporating randomness into quantum circuits. Specifically, we introduce a random layer, which contains single-qubit gates sampled from an trainable ensemble pooling. The prediction of QNN is then represented by an ensemble average over a classical function of measurement outcomes. We prove that our approach can accurately approximate arbitrary target operators using Uhlmann's theorem for majorization, which enables observable learning. Our proposal is demonstrated with extensive numerical experiments, including observable learning, Renyi entropy measurement, and image recognition. We find the expressivity of QNNs is enhanced by introducing randomness for multiple learning tasks, which could have broad application in quantum machine learning.

_Introduction.-_ In recent years, significant breakthroughs have been made in the field of artificial intelligence. Among various machine learning algorithms, neural networks have played a vital role, thanks to their universal expressivity for deep architectures. As a quantum generalization of neural networks, quantum neural networks (QNNs) have been proposed based on parameterized quantum circuits. QNNs use quantum states instead of classical numbers as inputs[1, 2, 3, 4]. However, the evolution of the input quantum states is constrained to be unitary, which limits the expressivity of QNNs. For physical observables, which are linear functions of the input quantum states or density matrices, QNNs can achieve high accuracy only if the target operator shares the same eigenvalues with the measurement operator. For a general situation, it requires introducing auxiliary qubits, as proposed in [5]. To expresse non-linear functions of the input density matrices, such as purities, traditional approaches introduce multiple replicas, which is unfavorable on near-term, noisy intermediate-scale quantum (NISQ) devices with a limited number of logical qubits. Previous studies have also reported moderate accuracy for more general machine learning tasks, including image re-organization [6, 7, 8, 9, 10].

In this work, we propose a universal scheme to overcome the expressivity obstacle without the need for additional replicas. Our main inspiration comes from the recent development of the randomized measurement toolbox for quantum simulators [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In all of these protocols, a measurement is performed after a random unitary gate, and the desired property is predicted through a classical computer after collecting sufficient measurement outcomes. In particular, the random measurement has been experimentally realized in [35, 36, 37, 38, 39, 40]. These developments unveil that randomness plays a central role in extracting information from complex quantum systems efficiently. From a machine learning perspective, this implies that introducing random unitaries may enhance the expressivity of QNNs. This naturally leads to the concept of randomized quantum neural networks, where we collect measurement outcomes from an ensemble of parametrized quantum circuits to make final predictions. Analogous to the different types of layers in classical neural networks, randomized QNNs consist of deterministic layers and random layers. In deterministic layers, the quantum gates contain parameterized quantum gates as in traditional QNNs, while in random layers, they are sampled from trainable ensembles of single-qubit gates. This

Figure 1: An illustration is provided for the proposed architecture of randomized quantum neural networks. In this example, the circuit contains two deterministic layers \(\hat{U}_{1(2)}\) and one random layer \(\hat{U}_{r}\) in between, with the final measurement performed on two qubits. As demonstrated in this work, this architecture shows randomness-enhanced expressivity for a variety of general learning tasks.
is illustrated in FIG. 1. We demonstrate the high expressivity of the proposed architecture using several different tasks, including both linear and nonlinear functions of the input density matrix. Our results pave the way towards realizing the universal expressivity ability for QNNs.

_Architecture.-_ We begin with a detailed description of randomized QNNs. To be concrete, we focus on the architecture illustrated in FIG. 1 for \(N_{\text{sys}}=5\) qubits, which comprises two deterministic layers, namely \(\hat{U}_{1}\) and \(\hat{U}_{2}\), with a random single qubit gate layer \(\hat{U}_{r}\) in between.

Each deterministic layer \(\hat{U}_{l_{d}}\) (\(l_{d}=1,2\)) contains a number of units \(\hat{V}_{l_{d}}^{l}(\mathbf{\theta}_{l_{d}}^{l})\) (\(l\in\{1,2,...,L_{l_{d}}\}\)) and each deterministic layer is constructed as

\[\hat{U}_{l_{d}}=\hat{V}_{l_{d}}^{L_{l_{d}}}(\mathbf{\theta}_{l_{d}}^{L_{l_{d}}})... \hat{V}_{i}^{2}(\mathbf{\theta}_{l_{d}}^{2})\hat{V}_{l_{d}}^{1}(\mathbf{\theta}_{l_{d} }^{1}), \tag{1}\]

where \(\{\mathbf{\theta}_{l_{d}}^{l}\}\) are the parameters of the deterministic layers. In general, the arrangement of two-qubit gates in each deterministic layer allows for a large degree of freedom. In this work, we focus on the standard brick wall architecture with spatial locality. Each unit \(\hat{V}_{l_{d}}^{l}\) contains \(N_{\text{sys}}-1\) two qubit gates and each two qubit gate is a SU(4) matrix which can be parameterized as \(\exp(\sum_{j}c_{j}\hat{g}_{j})\). Here \(\hat{g}_{j}\) is the generator of SU(4) group and \(\{\mathbf{\theta}_{l_{d}}^{l}\}\) denotes parameters \(\{\mathbf{c}\}\) of all two qubit gates [41]. Nonetheless, alternative choices for each deterministic layer have the potential to enhance the expressivity of QNNs for a fixed number of gates [6].

For the sake of experimental convenience, the random layer \(\hat{U}_{r}\) comprises a tensor product of single-qubit gates, denoted by \(\hat{u}_{1}\otimes\hat{u}_{2}...\otimes\hat{u}_{N_{\text{sys}}}\). These gates are sampled from an ensemble

\[\mathcal{E}=\{(w_{i},\hat{U}_{r,i}=\hat{u}_{1}^{i}(\mathbf{\alpha}_{i}^{1})\otimes \hat{u}_{2}^{i}(\mathbf{\alpha}_{i}^{2})...\otimes\hat{u}_{N_{\text{sys}}}^{i}(\bm {\alpha}_{i}^{N_{\text{sys}}}))\}, \tag{2}\]

where \(i=1,2,...,N_{r}\) labels different elements and \(w_{i}\) is the corresponding weight with \(\sum_{i}w_{i}=1\). Each single qubit gate is parametrized by generators of SU(2) with 3 dimensional real vector \(\mathbf{\alpha}_{i}^{q}\) (\(q\in\{1,2,...,N_{\text{sys}}\}\)). Both \(\{w_{i}\}\) and \(\{\mathbf{\alpha}_{i}^{q}\}\) are trainable parameters. It is also straightforward to introduce multiple random layers into the full architecture of QNNs. Importantly, it is worth noting the differences between our definition and typical random measurement protocols. Firstly, our random layer can be added at any point in the quantum circuit, not necessarily before the final measurement. Secondly, our definition of \(\mathcal{E}\) allows for non-trivial correlations between single-qubit gates on different sites, which is typically absent in random measurement protocols. Both features are necessary for achieving a high expressivity in QNNs.

We consider a dataset \(\{(|\psi_{m}\rangle,\mathcal{T}_{m})\}\), in which \(m\in\{1,2,...,N_{D}\}\) labels different data and \(\mathcal{T}_{m}\) is the target information for the corresponding state \(|\psi_{m}\rangle\). For each unitary \(\hat{U}_{r,i}\) in the ensemble \(\mathcal{E}\), we perform projective measurements in the computational basis for \(k\sim O(1)\) qubits. In FIG. 1, we set \(k=2\), and the measurement yields the probability distribution given by:

\[p_{i,m}^{ss^{\prime}}=\langle\psi_{m}|\hat{U}_{1}^{\dagger}\hat{U}_{r,i}^{ \dagger}\hat{U}_{2}^{\dagger}(\hat{P}_{s}^{2}\otimes\hat{P}_{s^{\prime}}^{3}) \hat{U}_{2}\hat{U}_{r,i}\hat{U}_{1}|\psi_{m}\rangle, \tag{3}\]

where the projection operator \(\hat{P}_{s}^{q}=\frac{1+ss^{\prime}}{2}\) for \(s=\pm 1\). Due to the constraint \(\sum_{ss^{\prime}}p_{i,m}^{ss^{\prime}}=1\), there are only \(3\) non-trivial components of \(p_{i,m}^{ss^{\prime}}\), denoted by the vector \(\mathbf{p}_{i,m}\). We then use a classical computer to apply a general function \(f_{\mathbf{\theta}}(\cdot)\), parametrized by \(\mathbf{\beta}\), to the probability distribution \(p_{i,m}^{ss^{\prime}}\), which yields a single outcome denoted by \(\mathcal{P}_{i,m}=f_{\mathbf{\beta}}(\mathbf{p}_{i,m})\). The classical function can be described by elementary functions in the simplest setting, but is more generally described by classical neural networks. We further average the outcome over the ensemble \(\mathcal{E}\) to obtain the final prediction for the input state \(|\psi_{m}\rangle\) as:

\[\mathcal{P}_{m}=\sum_{i=1}^{N_{r}}w_{i}\mathcal{P}_{i,m}=\sum_{i=1}^{N_{r}}w_ {i}f_{\mathbf{\beta}}(\mathbf{p}_{i,m}). \tag{4}\]

We use the mean square error (MSE) as the loss function \(\mathcal{L}=\frac{1}{N_{D}}\sum_{m}(\mathcal{P}_{m}-\mathcal{T}_{m})^{2}\) with data size \(N_{D}\) during the training process. We apply the gradient descent algorithm to optimize the parameters \(\{\mathbf{\theta}_{l_{d}}^{l},w_{i},\mathbf{\alpha}_{i}^{q},\mathbf{\beta}\}\) to minimize the loss function \(\mathcal{L}\). In the following sections, we focus on the demonstration of high expressivity for randomized QNNs. Our examples range from simple physical tasks including observable learning and Renyi entropy measurement, to standard machine learning tasks such as image recognization.

_Observable learning.-_ To show the high expressivity of randomized QNNs, let us consider a simple scenario where the target, \(\mathcal{T}_{m}\), is an expectation of a physical observable \(\hat{O}\) with \(\mathcal{T}_{m}=\langle\psi_{m}|\hat{O}|\psi_{m}\rangle\). For simplicity, focusing on single-qubit measurement with \(k=1\), we first investigate whether the randomized QNNs as proposed as in FIG. 1 can approximate the target function \(\mathcal{T}_{m}\) as accurate as possible for sufficiently deep circuit structures with sufficiently large \(N_{r}\). As physical observables are linear in density matrices, a linear function \(f_{\mathbf{\beta}}(x)=\beta_{0}+\beta_{1}x\) will be applied to the measurement result. Explicitly, we introduce \(\hat{U}_{\text{tot},i}\) for a random realization \(i\) of the quantum circuit. As an example, we have \(\hat{U}_{\text{tot},i}=\hat{U}_{2}\hat{U}_{r,i}\hat{U}_{1}\). An accurate prediction of the target function requires that

\[\sum_{i=1}^{N_{r}}w_{i}\;\hat{U}_{\text{tot},i}^{\dagger}(\beta_{0}\hat{ \sigma}_{0}^{1}+\beta_{1}\hat{\sigma}_{z}^{1})\hat{U}_{\text{tot},i}=\hat{O}, \tag{5}\]

where \(\hat{\sigma}_{0}\) is the identity operator and Pauli matrix \(\hat{\sigma}_{z}\) is the single-qubit's measurement operator.

For the case of \(N_{r}=1\) and \(w_{1}=1\), our setup reduces to the traditional QNN without randomness. In this scenario, Eq.(5) requires that \((\beta_{0}\hat{\sigma}_{0}^{1}+\beta_{1}\hat{\sigma}_{z}^{1})\) and \(\hat{O}\) be related by a unitary transformation. Since the unitary transformation preserves the eigenvalues of the operator, the requirement cannot be satisfied for a general operator \(\hat{O}\). When \(N_{r}>1\), Eq.(5) can be expressed as \(\Phi(\hat{\Sigma})=\hat{O}\), where \(\hat{\Sigma}\equiv\beta_{1}\hat{\sigma}_{z}^{1}+\beta_{0}\hat{\sigma}_{0}^{1}\) and \(\Phi(\hat{X})\) is a mixed-unitary channel [42]. For sufficiently complex circuit structures, we expect \(\Phi\) to be generic. In comparison to the \(N_{r}=1\) case, there is no constraint from unitarity. However, we still need to ask whether Eq.(5) can be satisfied for
an arbitrary operator \(\hat{O}\). In the following, we prove that the answer to this question is affirmative. The proof consists of three steps:

**Step 1.** Mathematically, if there exists a mixed-unitary channel \(\Phi\) such that \(Y=\Phi(X)\), we say that \(X\) majorizes \(Y\), denoted by \(Y\prec X\)[43]. Thus, for a randomized QNNs which can accurately predict any observable \(\hat{O}\), we need to find values of \(\beta_{0}\) and \(\beta_{1}\) such that \(\hat{O}\prec\hat{\Sigma}\) for any \(\hat{O}\).

**Step 2.** According to Uhlmann's theorem for majorization [43, 44], \(\hat{O}\prec\hat{\Sigma}\) if and only if \(\mathbf{\lambda}_{\hat{O}}\prec\mathbf{\lambda}_{\hat{\Sigma}}\), where \(\mathbf{\lambda}_{\hat{X}}\) is the list of eigenvalues for the operator \(\hat{X}\) in the descending order. Here the majorization between two real vectors \(\mathbf{y}\prec\mathbf{x}\) is defined as (i) \(\sum_{j=1}^{q}x_{j}\geq\sum_{j=1}^{q}y_{j}\) for arbitrary \(1\leq q<\mathcal{D}\) and (ii) \(\sum_{j=1}^{\mathcal{D}}x_{j}=\sum_{j=1}^{\mathcal{D}}y_{j}\). Here \(\mathcal{D}\) is the dimension of the vectors. Noting that condition (ii) takes into account the trace-preserving property of mixed-unitary channels.

**Step 3.** We can always find \(\beta_{0}\) and \(\beta_{1}\) such that \(\mathbf{\lambda}_{\hat{O}}\prec\mathbf{\lambda}_{\hat{\Sigma}}\). Assuming \(\beta_{1}>0\), the first or last \(\mathcal{D}/2\) components of \(\mathbf{\lambda}_{\hat{\Sigma}}\) correspond to the values \(\beta_{0}+\beta_{1}\) or \(\beta_{0}-\beta_{1}\), respectively. The constant term \(\beta_{0}\) can then be determined using condition (ii), which gives \(\beta_{0}=\mathcal{D}^{-1}\sum_{j=1}^{\mathcal{D}}\lambda_{\hat{O},j}\). Moreover, condition (i) can always be satisfied for sufficiently large \(\beta_{1}\). This proves the existence of \(\beta_{0}\) and \(\beta_{1}\) such that \(\hat{O}\prec\hat{\Sigma}\).

Although randomized QNNs have the potential to express arbitrary operators, it is difficult to determine an upper bound or a required value for \(N_{r}\) in practical learning tasks. At the same time, it is unfavorable to have large \(N_{r}\) or a large number of random layers, especially in NISQ devices. Therefore, we turn to numerical simulations of the randomized QNNs, and investigate practical requirements on \(N_{r}\) and the number of random layers. Since the basis change can be efficiently captured by the deterministic layer \(\hat{U}_{1}\), we focus on observables \(\hat{O}\) that are diagonal in the computational basis. For simplicity, we further set \(\hat{U}_{1}=\hat{I}\) and \(\hat{U}_{2}\) composed by \(L_{2}\) units of a brick wall structure [41]. For each system size \(N_{\text{sys}}\), we test whether a random diagonal operator \(\hat{O}\) can be predicted accurately for different values of \(N_{r}\) by monitoring the training loss for a sufficiently large dataset. As an example, we plot the training loss as a function of the training epoch for \(N_{\text{sys}}=2\) in FIG. 2 (a). The curves are averaged over 10 operators with random eigenvalues from the uniform distribution \([-2.5,2.5]\). When we increase \(N_{r}\) from 1 to 3, there is a rapid decrease in the training loss for large training epochs. The result shows that \(N_{r}=3\) is sufficient for learning general operators for \(N_{\text{sys}}=2\) where the mean absolute error \(\mathbf{E}[\mathcal{P}-\mathcal{T}]=\frac{1}{N_{D}}\sum_{m}|\mathcal{P}_{m}- \mathcal{T}_{m}|\) can be decreased to \(10^{-5}\). More detail explanations of the trained random QNNs are given in the supplementary materials [41]. We further extend the system size \(N_{\text{sys}}\) to study how it affects the number of required random gates. The results are shown in FIG. 2 (b). Although we are limited to small system sizes \(N_{\text{sys}}\in\{2,3,4,5\}\), the results clearly show weak dependence of \(N_{r}\) on \(N_{\text{sys}}\). The training results show that \(N_{r}=3\) already gives highly accurate predictions for \(N_{\text{sys}}=5\). This guarantees the application of our proposal for randomized QNNs on NISQ devices for observable learning tasks.

_Renyi entropy measurement.-_ We have established the universal expressivity of observables, and now we will consider targets that are non-linear functions of density matrices. One example of such targets is the Renyi entropy, which is also of experimental interest. To compute the Renyi entropy for a subsystem \(A\) consisting of the central \(N_{\text{sub}}\) qubits, we first calculate the reduced density matrix \(\hat{\rho}_{A}\) of an input state \(|\psi_{m}\rangle\) by tracing out the degrees of freedom of the complementary subsystem \(\bar{A}\). We then choose the target as

\[\mathcal{T}_{m}=\text{Tr}_{A}[\hat{\rho}_{A}^{n}], \tag{6}\]

which is related to the n-th Renyi entropy through \(S_{A}^{(n)}=-\frac{1}{n-1}\ln(\mathcal{T}_{m})\). Since we are directly measuring a local property of the input wavefunction, it is reasonable to fix \(\hat{U}_{1}\) and \(\hat{U}_{2}\) to the identity matrix \(\hat{I}\) and focus on the random layer \(\hat{U}_{r}\) with \(k=N_{\text{sub}}\). This approach provides a lower bound on the expressivity of randomized QNNs. Because the target \(\mathcal{T}_{m}\) is proportional to \(\rho^{n}\), we choose the function \(f_{\mathbf{\beta}}(\mathbf{x})\) to be a polynomial up to the \(n\)-th order. However, it is worth noting that lower order polynomials may also work in certain cases [45]. We prepare a dataset with random states \(|\psi_{m}\rangle\), the detailed description of which is provided in the supplementary materials [41]. The numerical results for \(n=2\), \(N_{\text{sys}}=5\) and \(N_{\text{sub}}=1,2\) are shown in FIG. 3. To make accurate predictions, we need \(N_{r}=3\) for \(N_{\text{sub}}=1\) and \(N_{r}=9\) for

Figure 2: _Observable learning by QNN with a random layer._ (a) The training mean absolute loss is shown as a function of the training epoch for observable learning with \(N_{\text{sys}}=2\). The solid lines are averaged over the training process for 10 different random target operators with random initializations, and the shaded region represents the standard deviation. The dashed lines are the validation loss with the dataset containing 200 samples. (b) Mean absolute loss of training dataset for \(N_{\text{sys}}\in\{2,3,4,5\}\) and \(N_{r}\in\{1,2,3,4\}\). Markers are averaged over 10 different random target operators with random initializations, and error bars are the standard deviation.
\(N_{\text{sub}}=2\). We have also checked that the threshold of \(N_{r}\) does not change if we instead consider \(n=3\), whose results are shown in the supplementary information [41].

It is interesting to compare our results to the proposed random measurement protocol for Renyi entropies. Our results indicate that \(N_{r}\) scales as \(3^{N_{\text{sub}}}\) when measuring Renyi entropies. In comparison, the previous protocol required each single-qubit gate \(\hat{u}_{q}^{i}\) to be sampled from the circular unitary ensemble [39; 18]. For \(n=2\), the circular unitary ensemble can be replaced by unitary 2-designs, which are known to be the Clifford group. Since the single-qubit Clifford group contains 24 elements, the total number of unitary matrices \(\hat{U}_{r,i}\) would naively scale as \(24^{N\text{sub}}\). However, in practice, this can be significantly reduced because randomized measurement protocols only require \(N_{\text{s}}\) snapshots sampled from the full ensemble. The theoretical bound of \(N_{\text{s}}\) for measuring general linear observables in a subsystem with \(N_{\text{sub}}\) qubits using random Pauli measurements up to an error \(\epsilon\) is given by \(N_{\text{s}}\gtrsim 3^{N_{\text{sub}}}/\epsilon^{2}\)[19; 18]. Consequently, in this quantum neural network structure, the number of unitary matrices \(\hat{U}_{r,i}\) that contribute is at most \(3^{N_{\text{sub}}}\), as in the protocol learned by the randomized QNNs.

_Image recognition.-_ We consider image recognition, a more practical machine learning task, to demonstrate the enhanced expressivity of randomized QNNs. In this case, we use Google's 'Street View of House Number (SVHN)' dataset as an example [46]. Each image in the dataset corresponds to an integer number. For demonstration purposes, we select two categories of images containing the numbers '1' and '4'. Initially, we compress each image into an \(8\times 8\) pixel format, resulting in a \(64\)-dimensional real vector, which can be equivalently represented as a \(32\)-dimensional complex vector. Subsequently, we encode the image into the input wave function using \(N_{\text{sys}}=5\) qubits [41]. Unlike previous tasks, the mapping between the input and the output is highly complex and non-local, lacking a simple understanding. Consequently, we allow both \(\hat{U}_{1}\) and \(\hat{U}_{2}\) to be trainable. After measuring a single qubit, we choose a \(5^{\text{th}}\)-order polynomial for the function \(f_{\mathbf{\beta}}(\mathbf{x})\). Since the image recognition is a two category classification task, after obtaining the final ensemble average prediction \(\mathcal{P}_{m}\), we apply a logistic-sigmoid function to restrict the prediction in the region (0,1) with \(\mathcal{G}_{m}=1/(1+\exp(-\mathcal{P}_{m}))\). And we use the cross-entropy as the loss function \(\mathcal{L}=\frac{1}{N_{D}}\sum_{m}-\mathcal{T}_{m}\log(\mathcal{G}_{m})-(1- \mathcal{T}_{m})\log(1-\mathcal{G}_{m})\) to optimize the parameters in the randomized QNN. The accuracy \(F=\frac{1}{N_{D}}\sum_{m}|[\text{sign}(\mathcal{G}_{m}-0.5)+1]/2-\mathcal{T} _{m}|\) for \(N_{r}=1\) and \(N_{r}=4\) are shown in FIG. 4. For \(N_{r}=1\), the accuracy saturates at approximately 0.8 after a large number of epochs, while the averaged accuracy for the test dataset reaches \(69.8\%\). The introduction of a single random layer with \(N_{r}=4\) significantly enhances the accuracy of the predictions. In this case, the training dataset achieves an accuracy higher than \(90\%\), and the average accuracy for the test dataset is \(82.29\%\). The utilization of a non-trivial random layer with \(N_{r}=4\) demonstrates a significant improvement in the prediction capabilities of QNNs, indicating the enhanced expressivity of our randomized QNN architecture.

_Outlook.-_ This work introduces the concept of randomized quantum neural networks, which include random layers where quantum gates are selected from an ensemble of unitary matrices. It is proven that these random layers provide universal expressivity for general physical observables using Uhlmann's theorem for majorization. Numerical simulations further show that this ar

Figure 4: _Image recognition by QNN with a random layer._ The training loss is shown as a function of the training epoch for the image recognition task. The results are averaged over the training process for 10 different random initializations, and the shaded region represents the standard deviation.

Figure 3: _Renyi entropy measurements by QNN with a random layer._ The training loss is shown as a function of the training epoch for purity with \(N_{\text{sys}}=5\) and (a) \(N_{\text{sub}}=1\) or (b) \(N_{\text{sub}}=2\). The results are averaged over the training process for 10 different random initializations, and the shaded region represents the standard deviation. The dashed lines are the validation loss with the dataset containing 200 samples.
pressivity for non-linear functions of the density matrix, such as Renyi entropies and image recognition, with small ensemble sizes \(N_{r}\). These results indicate that the proposed method has potential for broad applications in NISQ devices. We further highlight the differences between our architecture and the proposal presented in a very recent paper [47]. Their work also incorporates a series of parameterized quantum circuits, where the circuit consists of multiple parametrized (controlled-) rotations that share the same parameter. In contrast, our architecture features only a few random layers described by a tensor product of single-qubit gates, making its training process more efficient.

While the focus of this work is on parameterized quantum circuits with brick wall structures, it is straightforward to combine this novel architecture with other proposals to further improve expressivity or learning efficiency. For instance, it is possible to add ancilla qubits and explore more sophisticated architectures for the deterministic layers. Additionally, it would be interesting to investigate the impact of random layers on other quantum machine learning algorithms beyond traditional quantum neural networks [48, 49, 50, 51, 52, 53], such as quantu autoencoders [48, 49].

_Acknowledgement.-_ We thank Yingfei Gu, Ning Sun, Ce Wang, Hai Wang, and Yi-Zhuang You for helpful discussions. YW and XL are supported by National Program on Key Basic Research Project of China (Grant No. 2021YFA1400900), National Natural Science Foundation of China (Grants No. 11934002), Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01). YW is supported by the National Natural Science Foundation of China (Grant No. 12174236). JY is supported by the National Natural Science Foundation of China (Grant No. 11904190).
