CCF: Medium: Learning From Classical and Quantum Data: a Fourier Perspective

量子计算为包括机器学习、模拟和优化在内的各种应用提供了令人兴奋的新机会。量子计算机对机器学习特别有吸引力，因为它们能够高效地对复杂的学习模型进行编码。然而，经典计算机和量子计算机之间存在根本差异，这使得学习过程具有挑战性。量子系统使用量子比特来表示和处理信息，量子比特是比特的量子力学模拟，通过叠加对基本态的组合进行编码。量子系统可以利用纠缠等强大的特征，但必须遵守量子力学假设，如不可克隆和海森伯格测不准原理。此外，现有的量子计算机具有相对较少的量子比特数量，其处理高维数据的能力天生有限，因此需要有效的技术来识别输入数据中对学习任务重要的特征。量子学习模型显著增强的表现力还需要高效和可扩展的训练过程。此外，量子电路中测量的随机性使得训练过程本质上是概率的。提供这些问题的解决方案对于有效和高效的量子机器学习(QML)的实施至关重要，也是该项目的重点。与该项目相关的教育计划包括本科生的研究经验、教师培训计划、教学材料、暑期学校和研讨会，以及大量的在线工具和资源。扩大参与计算(BPC)计划包括准备面向高中生的可访问教育材料、为代表不足的群体的学生提供研讨会和暑期研究机会、从少数族裔服务机构招聘渠道，以及指导计划。该项目致力于在有噪声的中级量子(NISQ)设备上实现QML的关键挑战，并考虑以下问题：(I)当前的NISQ电路的量子比特输入能力有限。输入特征的选择是通过量子傅立叶分析(QFA)来实现的，目的是确定在QML任务中能够产生高精度的输入量子比特的小样本；(Ii)量子电路的不可克隆和测量假设使得计算电路输出的准确确定性估计是不可行的。然后利用一种新的随机梯度下降方法解决了样本/数据高效的QML模型的精确训练问题；(Iii)量子神经网络(QNN)等QML模型具有很高的模型复杂性，因为底层Hilbert空间的维度随着量子比特数的增加而指数增长，这导致学习模型中的参数数量呈指数级增长。因此，就所需的样本/迭代数量而言，这种模型的训练费用高得令人望而却步。该项目依赖于QFA来开发窄带量子感知器，以构建能够高效和准确地训练的QNN；以及(Iv)该项目的结果将在现实世界中的量子态识别问题上得到验证。总体而言，该项目调查了QML的进步对传统机器学习模型的影响，目的是提高经典培训过程的效率和普适性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Quantum computing presents exciting new opportunities for a wide variety of applications, including machine learning, simulation, and optimization. Quantum computers are particularly attractive for machine learning because of their ability to efficiently encode complex learning models. However, there are fundamental differences between classical and quantum computers that render the learning process challenging. Quantum systems represent and process information using qubits, a quantum mechanical analog of bits, which encode combinations of basic states through superposition. Quantum systems can use powerful features such as entanglement but must obey quantum mechanical postulates such as the no-cloning and the Heisenberg Uncertainty Principle. Moreover, existing quantum computers with a relatively small number of qubits are inherently limited in their ability to process high-dimensional data, and effective techniques are therefore needed to identify the features in the input data which are important to the learning tasks. The significantly increased expressive power of quantum learning models also requires highly efficient and scalable training procedures. Furthermore, the stochastic nature of measurements in quantum circuits makes the training process inherently probabilistic. Providing solutions to these issues is essential for the implementation of effective and efficient Quantum Machine Learning (QML), and is the focus of the project. The education program associated with this project comprises research experiences for undergraduate students, teacher-training programs, instructional material, summer schools and workshops, and a large number of online tools and resources. Broadening participation in computing (BPC) plans include the preparation of accessible educational material targeting high-school students, workshops, and summer research opportunities for students from underrepresented groups, recruiting pipelines from minority serving institutions, and mentoring programs. This project addresses critical challenges aimed at enabling QML on Noisy Intermediate-Scale Quantum (NISQ) devices, and considers the following problems: (i) Current NISQ circuits are limited in their qubit input capacity. The input feature selection is approached through Quantum Fourier Analysis (QFA) with the goal of identifying a small sample of input qubits that can yield high accuracy in QML tasks; (ii) The no-cloning and measurement postulates of quantum circuits make it infeasible to compute accurate deterministic estimates of circuit outputs. The problem of sample/data-efficient accurate training of QML models is then addressed with the help of a novel randomized gradient descent approach; (iii) QML models, such as Quantum Neural Networks (QNNs), have high model complexity, since the dimensionality of the underlying Hilbert space grows exponentially with the number of qubits, and this leads to an exponentially large number of parameters in the learning models. The training of such models is therefore prohibitively expensive in terms of the required number of samples/iteration. The project relies on QFA to develop narrow-band quantum perceptrons to build QNNs that can be efficiently and accurately trained; and (iv) The results of the project will be validated on real-world problems in quantum state discrimination. Overall the project investigates implications of advances in QML for conventional machine learning models with the goal of enhancing the efficiency and generalizability of classical training processes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.