Parameterized Quantum Circuits for Machine Learning

Can quantum computers be used for implementing machine learning models that are better than traditional methods, and are such methods suitable for today’s noisy quantum hardware? In this thesis we made a Python framework for implementing machine learning models based on parameterized quantum circuits that are evaluated on quantum hardware. The framework is capable of implementing quantum neural networks (QNNs) and quantum circuit networks (QCNs), and train them using gradient-based method. To calculate the gradient of quantum circuit networks, we developed a backpropagation algorithm based on the parameters shift rule that leverage both classical and quantum hardware. We performed a numerical study where we sought to characterize how dense neural networks (DNNs), QNNs and QCNs behave as a function of model architecture. We focus on investigating the vanishing gradient phenomenon, and quantifying the models trainability and expressivity using the empirical fisher information matrix (EFIM) and trajectory length, respectively. We also test the performance of the models by training them on artificial data, as well as on real-world data sets. Due to the multi-circuit nature of QCNs, large models can be constructed by using multiple layers of small circuits. For shallow circuits with few qubits, the local gradients of the individual circuits can be easily estimated on noisy quantum hardware. This is contrary to single-circuit QNNs that are deep and consist of many qubits, whose gradient is difficult to estimate on quantum hardware due to the vanishing gradient phenomenon. However, when the gradients of QCNs are calculated with backpropogation on classical hardware using the local gradients, the gradient tends to vanish exponentially fast as the number of layers increase. We showed that the vanishing gradient of QCNs manifests itself as a loss landscape that is very flat in most directions, with strong distortions in a single direction. This characteristic loss landscape is typical for DNNs, and is known to cause slow optimization. However, for a conservative number of qubits and layers, QCNs had significantly less distorted loss landscape than similar DNNs. We also showed that during training of QCNs and DNNs, the former models required two orders of magnitude fewer epochs in order to become exponentially expressive. Finally, we showed that QCNs of few qubits and layers trained faster than both DNNs and QNNs on the artificial data. QCNs also trained and generalize better on some real-world data sets, using both ideal and noisy simulation. This shows that QCNs may have merit for some data sets, even on noisy hardware, but not all