# Optimizing Multidimensional Pooling for Variational Quantum Algorithms
## 1 Introduction  
For performing machine learning (ML) tasks on multidimensional data, convolutional neural networks (CNNs) often outperform other techniques, such as multi-layer perceptrons (MLPs), with smaller model sizes, shorter training times, and higher accuracies [1, 2]. One factor that contributes to the benefits of CNNs is the conservation of spatio-temporal data locality, allowing them to preserve only relevant data connections and remove extra-neous ones [1, 2]. CNNs are constructed using a sequence of convolution and pooling pairs followed by a fully connected layer [1]. In the convolution layer, filters are applied to input data for specific applications, and the pooling layers reduce the spatial dimensions in the generated feature maps [3]. The reduced spatial dimensions generated from the pooling layers reduce memory requirements, which is a major concern for resource-constrained devices [4, 5].  
Exploiting data locality in pooling could be beneficial in fields such as quantum-to-classical (Q2C) data decoding [6], audio-visual data compression [7, 8], and, in particular, quantum machine learning (QML) [1]. However, most QML algorithms do not consider the multidimensionality or data locality of input datasets by converting them into flattened 1-D arrays [9, 10]. Nevertheless, quantum computing has shown great potential to outperformtraditional, classical computing for specific machine learning tasks [11]. By exploiting quantum parallelism, superposition, and entanglement, quantum computers can accelerate certain computation tasks with exponential speedups. However, in the current era of noisy intermediate-scale quantum (NISQ) devices, the implementation of quantum algorithms is constrained by the number of quantum bits (qubits) and fidelity of quantum gates [12]. For contemporary QML techniques, this problem is addressed by a hybrid approach where only the highly parallel and computationally intensive part of the algorithm is executed in quantum hardware, and the remaining parts are executed using classical computers [13]. Such methods, known as variational quantum algorithms (VQAs), exploit a fixed quantum circuit structure with parameterized rotation gates, denoted as ansatz, whose parameters are optimized using classical backpropagation techniques such as gradient descent [13].  
In this work, we propose two generalized techniques for efficient pooling operations in QML, namely, the quantum Haar transform (QHT) for quantum average pooling and partial quantum measurements for two-norm/Euclidean pooling.  
We characterize their fidelity to the corresponding classical pooling operations using a state-of-the-art quantum simulator from IBM Quantum for a wide variety of real-world, high-resolution, multidimensional data.  
The rest of the paper is organized as follows. Section 2 covers necessary background information, including various quantum operations. Section 3 discusses existing related work. Section 4 introduces our proposed methodology, with great detail given to the constituent parts along with spatial complexity (depth) analysis of the corresponding circuits. Section 5 presents our experimental results, with an explanation of our verification metrics. Finally, Section 6 concludes our work and projects potential future directions.