Quantum Phase Recognition using Quantum Tensor Networks

Abstract

The current generation of quantum devices fall in the noisy intermediate scale quantum (NISQ) era and are afflicted with qubit decoherence issues, scalability problems, circuit trainability and much more. Keeping these issues in mind, quantum-classical hybrid algorithms have been developed that break the problem into tasks for both classical and quantum processors. The classical processor is leveraged for computational tasks like arithmetic operations and optimizations which are easier for classical computers to handle, while computational tasks where quantum processors give better results like solving systems of linear equations come in the "quantum part" of the algorithm. A special class of quantum-classical hybrid algorithms is the variational quantum algorithms (VQAs) which has been used extensively to solve quantum chemistry problems, particle physics problems, optimization problems and much more as they provide a general framework for solving problems. Classical machine learning has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than what we currently have. With this motivation, this thesis examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. We first start with an introductory chapter explaining the focus of our works and our contributions. We also discuss the applications of quantum computing in natural sciences. Then we dive into the necessary background of our works in detail. This includes sections on the basics of quantum computing, the NISQ era, variational quantum algorithms, tensor network ansatz and a literature review of the use of tensor networks in quantum computing. We introduce the circuits used by us and explain the tasks tackled in this work. We first look at a preliminary classical image classification task on the FashionMNIST dataset and study the effect of repeating tensor network layers on the ansatz’s expressibility and performance. Finally, we take on the problem of quantum phase recognition for the Transverse Ising and Heisenberg spin models in one and two dimensions, where we were able to reach ≥ 98% test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits. Finally, we analyze our results and end with a discussion on the scope of future works.