基于含辅助比特的量子线路迭代方案实现量子模拟

      近年来，量子信息科学取得了飞速的发展，引起了社会各界的广泛关注。2022年的诺贝尔物理学奖就颁给了量子信息领域的三位科学家。量子计算是建立在量子力学基础上的交叉学科，涉及到许多其他领域的知识和技术。人们的研究目标是利用量子力学中的一些特性，如叠加、纠缠等，构建出一种计算能力远超经典计算机的新型计算机——量子计算机。然而，要实现真正的量子计算机还有很长的路要走，目前量子计算的发展仍处于探索阶段。在这个阶段，人们摸索出了关于量子计算的各种应用，其中量子模拟就是量子计算的重要应用领域之一。量子模拟简单来说是利用可控制的量子系统来模拟和研究其他复杂的量子系统。然而，许多需要模拟的量子系统非常复杂，在实际操作中会遇到许多困难和挑战。例如，有些量子系统演化操作是非幺正的，而有些复杂的量子系统难以用少量的量子逻辑门来模拟实现。为了解决这些在量子模拟中遇到的难题，我们发现，可以通过添加辅助比特和迭代运算的方法，来实现复杂哈密顿量的量子模拟。在研究生期间，我基于这个思路，在两个方面进行了量子模拟方面的工作，并验证了该方法的有效性。

      第一个工作是关于非厄米系统量子模拟的方法。目前对于非厄米系统的研究日益增多，因为非厄米哈密顿量驱动的量子系统可以展现出厄米系统所没有的独特特性，例如非厄米拓扑带、趋肤效应等。然而要想在量子模拟器上对非厄米系统进行量子模拟是非常困难的，因为非厄米系统对应的演化操作是非幺正的，而目前绝大部分量子处理器施加的量子逻辑门都是幺正操作。为了解决这个问题，我们可以通过添加一个辅助比特来将非幺正操作构造成幺正操作，并且通过多层量子线路迭代运算这个构造后复杂的厄米操作，从而实现非厄米哈密顿量的量子模拟。最后我们将其运用到了带有非厄米微扰的Ising模型上，该模型用于研究非零温度下的量子相变。数值模拟结果与理论预测高度一致。这证明了我们提出的方法相比传统的量子模拟方法可以有效的在实验上模拟复杂的非厄米量子系统。

      第二个工作是关于量子主成分分析的量子模拟。机器学习在过去几十年迅速发展并得到广泛应用，近年来量子信息与机器学习结合，提出了量子机器学习概念，并相应提出了众多量子算法。这些算法大多需要通过量子模拟进行实验验证，以证实它们的实用性。其中，主成分分析算法是经典机器学习中很著名的算法，常被用于数据降维和分类。对应的量子版本，量子主成分分析算法于2014年被提出，但由于其实现条件严苛，少有人可以在实验上实现量子主成分分析。为了实现量子主成分分析算法的实验，我们设计了迭代线路，通过反复制备量子态进行迭代运算，实现了密度矩阵的指数化，并添加了辅助比特来提取本征值和本征态。我们还将量子主成分分析运用于肺炎CT图的识别中，并成功识别出CT图的阴性和阳性类别。

      总之，我们的研究表明，基于含辅助比特的量子线路迭代方法，可以有效地实现复杂哈密顿量的量子模拟，并且可以在实验上实现量子主成分分析算法。这些成果有望对量子计算和量子模拟等领域的发展提供一定的帮助。

In recent years, quantum information science has experienced rapid development, attracting widespread attention from various sectors of society. The 2022 Nobel Prize in Physics was awarded to three scientists in the field of quantum information. Quantum computing is an interdisciplinary field built upon the principles of quantum mechanics, encompassing knowledge and techniques from various other domains. The research objective is to leverage certain quantum mechanical properties, such as superposition and entanglement, to construct a new type of computer called a quantum computer, which surpasses the computational capabilities of classical computers. However, the realization of a fully functional quantum computer still faces significant challenges, and the current progress in quantum computing is still in its early stages. During this stage, researchers have explored various applications of quantum computing, with quantum simulation being one of the significant areas of application. Quantum simulation, in simple terms, utilizes controllable quantum systems to simulate and study other complex quantum systems. However, many quantum systems that require simulation are highly complex, presenting numerous challenges and difficulties during practical implementation. For instance, some quantum systems involve non-unitary evolution operations, and certain complex quantum systems are difficult to simulate using a small number of quantum logic gates. To address these challenges encountered in quantum simulation, we have discovered that the addition of auxiliary qubits and iterative operations can enable the simulation of complex Hamiltonians. During my graduate studies, based on this approach, I conducted research in two aspects of quantum simulation and successfully validated the effectiveness of this method.

The first work is about quantum simulation methods for non-Hermitian systems. Recent years have witnessed ongoing interests in exploring non-Hermitian phenomena. Quantum systems driven by non-Hermitian Hamiltonians can lead to unconventional and exclusive features, such as non-Hermitian topological bands and skin effects. However, experimentally investigating non-Hermitian physics is very challenging, particularly for achieving long-time dynamics in many-body systems, because non-Hermiticity usually originates from particle loss and decoherence assigned by the environmental, which is hard to manipulate. To bypass this difficulty, we can add an ancilla qubit to construct the non-unitary operation as a unitary operation, and then simulate the non-Hermitian Hamiltonian by iterating the complex Hermitian operator using multiple layers of quantum circuits. Finally, we applied this method to the Ising model with non-Hermitian perturbations, which is used to study quantum phase transitions at non-zero temperatures. The numerical simulation results are highly consistent with theoretical predictions, which demonstrates that our proposed method can effectively simulate complex non-Hermitian quantum systems in experiments compared to traditional quantum simulation methods.

The second work is about quantum simulation of quantum principal component analysis (PCA). Principal component analysis is a popular paradigm in the field of machine learning (ML) that boasts a variety of applications, ranging from data compression to image recognition. Its quantum counterpart, known as quantum principal component analysis (qPCA), offers the potential to exponentially speed up the algorithm for larger datasets. However, due to the limitations imposed by current experimental conditions, qPCA implementation based on quantum gates has yet to be satisfactorily realized experimentally. In order to demonstrate the quantum principal component analysis algorithm in experiments, we designed an iterative circuit, which iteratively operates on the quantum state by repeating it, to achieve density matrix exponentiation, and added an ancilla qubit to extract eigenvalues and eigenvectors. We also applied quantum principal component analysis to the recognition of pneumonia CT images, successfully identifying negative and positive categories of CT images.

In summary, our research shows that by iterating the quantum circuits with ancilla qubits, complex Hamiltonians can be effectively simulated, and quantum principal component analysis algorithm can be experimentally realized. These achievements are expected to provide some help to the development of quantum computing and quantum simulation.

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