Generating high-quality data (e.g. images or video) is one of the most exciting and challenging frontiers in unsupervised machine learning. Utilizing quantum computers in such tasks to potentially enhance conventional machine learning algorithms has emerged as a promising application, but poses big challenges due to the limited number of qubits and the level of gate noise in available devices. In this work, we provide the first practical and experimental implementation of a quantum-classical generative algorithm capable of generating high-resolution images of handwritten digits with state-of-the-art gate-based quantum computers. In our quantum-assisted machine learning framework, we implement a quantum-circuit based generative model to learn and sample the prior distribution of a Generative Adversarial Network. We introduce a multi-basis technique which leverages the unique possibility of measuring quantum states in different bases, hence enhancing the expressibility of the prior distribution. We train this hybrid algorithm on an ion-trap device based on 171 Yb + ion qubits to generate high-quality images and quantitatively outperform comparable classical Generative Adversarial Networks trained on the popular MNIST data set for handwritten digits.
Variational Quantum Algorithms (VQAs) are promising candidates for finding practical applications of near to mid-term quantum computers. There has been an increasing effort to study the intricacies of VQAs, such as the presence or absence of barren plateaus and the design of good quantum circuit ansätze. Many of these studies can be linked to the loss landscape that is optimized as part of the algorithm, and there is high demand for quality software tools for flexibly studying these loss landscapes. In our work, we collect a variety of techniques that have been used to visualize the training of deep artificial neural networks and apply them to visualize the high-dimensional loss landscapes of VQAs. We review and apply the techniques to three types of VQAs: the Quantum Approximate Optimization Algorithm, the Quantum Circuit Born Machine, and the Variational Quantum Eigensolver. Additionally, we investigate the impact of noise due to finite sampling in the estimation of loss functions. For each case, we demonstrate how our visualization techniques can verify observations from past studies and provide new insights. This work is accompanied by the release of the open-source Python package orqviz, which provides code to compute and flexibly plot 1D and 2D scans, Principal Component Analysis scans, Hessians, and the Nudged Elastic Band algorithm. orqviz enables flexible visual analysis of high-dimensional VQA landscapes and can be found at: github.com/zapatacomputing/orqviz.
Quantum two-level systems, i.e., qubits, form the basis for most quantum machine learning approaches that have been proposed throughout the years. However, higher dimensional quantum systems constitute a promising alternative and are increasingly explored in theory and practice. Here, we explore the capabilities of multi-level quantum systems, so-called qudits, for their use in a quantum machine learning context. We formulate classification and regression problems with the data re-uploading approach and demonstrate that a quantum circuit operating on a single qudit is able to successfully learn highly non-linear decision boundaries of classification problems such as the MNIST digit recognition problem. We demonstrate that the performance strongly depends on the relation between the qudit states representing the labels and the structure of labels in the training data set. Such a bias can lead to substantial performance improvement over qubit-based circuits in cases where the labels, the qudit states, and the operators employed to encode the data are well-aligned. Furthermore, we elucidate the influence of the choice of the elementary operators and show that a squeezing operator is necessary to achieve good performances. We also show that there exists a trade-off for qudit systems between the number of circuit-generating operators in each processing layer and the total number of layers needed to achieve a given accuracy. Finally, we compare classification results from numerically exact simulations and their equivalent implementation on actual IBM quantum hardware. The findings of our work support the notion that qudit-based algorithms exhibit attractive traits and constitute a promising route to increasing the computational capabilities of quantum machine learning approaches.
Quantum process learning is emerging as an important tool to study quantum systems. While studied extensively in coherent frameworks, where the target and model system can share quantum information, less attention has been paid to whether the dynamics of quantum systems can be learned without the system and target directly interacting. Such incoherent frameworks are practically appealing since they open up methods of transpiling quantum processes between the different physical platforms without the need for technically challenging hybrid entanglement schemes. Here we provide bounds on the sample complexity of learning unitary processes incoherently by analyzing the number of measurements that are required to emulate well-established coherent learning strategies. We prove that if arbitrary measurements are allowed, then any efficiently representable unitary can be efficiently learned within the incoherent framework; however, when restricted to shallow-depth measurements only low-entangling unitaries can be learned. We demonstrate our incoherent learning algorithm for low entangling unitaries by successfully learning a 16-qubit unitary on ibmq kolkata, and further demonstrate the scalabilty of our proposed algorithm through extensive numerical experiments.
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