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Numerical Ranges and Applications in Quantum Information

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dc.contributor.advisor Kribs, David
dc.contributor.advisor Zeng, Bei Cao, Ningping 2021-08-09T19:38:44Z 2021-08-09T19:38:44Z 2021-07 2021-07-22 2019-11-29
dc.description.abstract The numerical range (NR) of a matrix is a concept that first arose in the early 1900’s as part of efforts to build a rigorous mathematical framework for quantum mechanics and other challenges at the time. Quantum information science (QIS) is a new field having risen to prominence over the past two decades, combining quantum physics and information science. In this thesis, we connect NR and its extensions with QIS in several topics and apply the knowledge to solve related QIS problems. We utilize the insight offered by NRs and apply them to quantum state inference and Hamiltonian reconstruction. We propose a new general deep learning method for quantum state inference in chapter 3 and employ it to two scenarios – maximum entropy estimation of unknown states and ground state reconstruction. The method manifests high fidelity, exceptional efficiency, and noise tolerance on both numerical and experimental data. A new optimization algorithm is presented in chapter 4 for recovering Hamiltonians. It takes in partial measurements from any eigenstates of an unknown Hamiltonian H, then provides an estimation of H. Our algorithm almost perfectly deduces generic and local generic Hamiltonians. Inspired by hybrid classical-quantum error correction (Hybrid QEC), the higher rank (k : p)-matricial range is defined and studied in chapter 5. This concept is a new extension of NR. We use it to study Hybrid QEC, also to investigate the advantage of Hybrid QEC over QEC under certain conditions. en_US
dc.language.iso en en_US
dc.publisher University of Guelph en_US
dc.subject quantum information en_US
dc.subject numerical range en_US
dc.subject quantum error correction en_US
dc.subject quantum state tomography en_US
dc.title Numerical Ranges and Applications in Quantum Information en_US
dc.type Thesis en_US Mathematics and Statistics en_US Doctor of Philosophy en_US Department of Mathematics and Statistics en_US
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dcterms.relation Xin, T., Lu, S., Cao, N., Anikeeva, G., Lu, D., Li, J., Long, G., & Zeng, B. (2019). Local-measurement-based quantum state tomography via neural networks. Npj Quantum Information, 5(1), 1–8. en_US
dcterms.relation Cao, N., Kribs, D. W., Li, C. K., Nelson, M. I., Poon, Y. T., & Zeng, B. (2021). Higher rank matricial ranges and hybrid quantum error correction. Linear and Multilinear Algebra, 69(5), 827-839. en_US
dcterms.relation Hou, S. Y., Cao, N., Lu, S., Shen, Y., Poon, Y. T., & Zeng, B. (2020). Determining system Hamiltonian from eigenstate measurements without correlation functions. New Journal of Physics, 22(8), 083088. en_US University of Guelph en_US

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