{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/267871"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/267871","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes","abstract":"This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ($p-d$) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\\textit{ab-initio}$ systems, including a fully periodic $\\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.","abstract_html":"This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ($p-d$) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as <span class=\"etd-inline-math\"><em>ab-initio</em></span> systems, including a fully periodic <span class=\"etd-inline-math\"><em>ab-initio</em></span> graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.","abstract_has_math":true,"creators":["Schwarz, Lauretta Rebecca"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Alavi, Ali"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-10-01","date_published":"2017-10-01","updated_at":"2026-07-22T22:24:10Z","subjects":["theoretical chemistry","electronic structure theory","Quantum Monte Carlo methods"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/30719c93-272a-4b6e-b2f8-acf1ef99dede/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.13792","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Alavi, Ali"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["EPSRC studentship"]},{"key":"dc:creator","label":"Author","values":["Schwarz, Lauretta Rebecca"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2017-10-01"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/267871"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["theoretical chemistry","electronic structure theory","Quantum Monte Carlo methods"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/30719c93-272a-4b6e-b2f8-acf1ef99dede/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.13792"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d7cd1898-4221-4bbe-8685-142a8acf5aba/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ($p-d$) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\\textit{ab-initio}$ systems, including a fully periodic $\\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["12ad190fa41f07442fbd114a105bce3d","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes"]}]}],"canonical_facts":{"dc:contributor.advisor":["Alavi, Ali"],"dc:contributor.sponsor":["EPSRC studentship"],"dc:creator":["Schwarz, Lauretta Rebecca"],"dc:date.issued":["2017-10-01"],"dc:description.abstract":["This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. 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The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\\textit{ab-initio}$ systems, including a fully periodic $\\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches."],"dc:format.checksum.md5":["12ad190fa41f07442fbd114a105bce3d","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.13792"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d7cd1898-4221-4bbe-8685-142a8acf5aba/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/267871"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/30719c93-272a-4b6e-b2f8-acf1ef99dede/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["theoretical chemistry","electronic structure theory","Quantum Monte Carlo methods"],"dc:title":["Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:10Z"}