{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139241"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139241","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Improved Tools for Local Hamiltonians","abstract":"In this thesis we consider computational problems related to many-body spin systems with a structured energy operator, a local Hamiltonian. We begin with the most structured setting where the Hamiltonian has a spectral gap and spatial locality. This setting is widely studied using approximate ground space projectors (AGSPs). In chapter 1 we give an improved analysis of AGSPs in the setting of local Hamiltonians with a degenerate ground space. This implies a direct generalization of the AGSP⇒entanglement bound implication of [Arad, Landau, and Vazirani ’12] from unique to degenerate ground states. We use the improved analysis to give a particularly simple algorithm for frustration-free spin systems provided an AGSP with structure as a matrix product operator. We apply our tools to a recent 2D area law of [Anshu, Arad, and Gosset ’21], giving a sub-exponential-time classical algorithm to compute the ground states. This time complexity cannot be improved beyond sub-exponential assuming the randomized exponential time hypothesis, even for the special case of classical constraint satisfaction problems on the 2D grid. In chapter 2 we consider frustrated systems and extend results for spin chains to certain trees with intrinsic dimension β < 2. This condition is met for generic trees in the plane and for certain models of hyperbranched polymers in 3D. In chapter 3 we relax the conditions on the Hamiltonian and no longer require a spectral gap or geometric locality, and we consider an approximation problem for the spectrum of the local Hamiltonian. We give a simple proof of a Chernoff bound for the spectrum of a k-local Hamiltonian based on Weyl’s inequalities. The complexity of estimating the spectrum’s ϵ(n)-th quantile up to constant relative error thus exhibits the following dichotomy: For ϵ(n) = d −n the problem is NP-hard and maybe even QMA-hard, yet there exists constant a > 1 such that the problem is trivial for ϵ(n) = a −n.","abstract_html":"In this thesis we consider computational problems related to many-body spin systems with a structured energy operator, a local Hamiltonian. We begin with the most structured setting where the Hamiltonian has a spectral gap and spatial locality. This setting is widely studied using approximate ground space projectors (AGSPs). In chapter 1 we give an improved analysis of AGSPs in the setting of local Hamiltonians with a degenerate ground space. This implies a direct generalization of the AGSP⇒entanglement bound implication of [Arad, Landau, and Vazirani ’12] from unique to degenerate ground states. We use the improved analysis to give a particularly simple algorithm for frustration-free spin systems provided an AGSP with structure as a matrix product operator. We apply our tools to a recent 2D area law of [Anshu, Arad, and Gosset ’21], giving a sub-exponential-time classical algorithm to compute the ground states. This time complexity cannot be improved beyond sub-exponential assuming the randomized exponential time hypothesis, even for the special case of classical constraint satisfaction problems on the 2D grid. In chapter 2 we consider frustrated systems and extend results for spin chains to certain trees with intrinsic dimension β &lt; 2. This condition is met for generic trees in the plane and for certain models of hyperbranched polymers in 3D. In chapter 3 we relax the conditions on the Hamiltonian and no longer require a spectral gap or geometric locality, and we consider an approximation problem for the spectrum of the local Hamiltonian. We give a simple proof of a Chernoff bound for the spectrum of a k-local Hamiltonian based on Weyl’s inequalities. The complexity of estimating the spectrum’s ϵ(n)-th quantile up to constant relative error thus exhibits the following dichotomy: For ϵ(n) = d −n the problem is NP-hard and maybe even QMA-hard, yet there exists constant a &gt; 1 such that the problem is trivial for ϵ(n) = a −n.","abstract_has_math":false,"creators":["Abrahamsen, Nilin"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Shor, Peter W."],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:22:23Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/139241","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Shor, Peter W."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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We begin with the most structured setting where the Hamiltonian has a spectral gap and spatial locality. This setting is widely studied using approximate ground space projectors (AGSPs). In chapter 1 we give an improved analysis of AGSPs in the setting of local Hamiltonians with a degenerate ground space. This implies a direct generalization of the AGSP⇒entanglement bound implication of [Arad, Landau, and Vazirani ’12] from unique to degenerate ground states. We use the improved analysis to give a particularly simple algorithm for frustration-free spin systems provided an AGSP with structure as a matrix product operator. We apply our tools to a recent 2D area law of [Anshu, Arad, and Gosset ’21], giving a sub-exponential-time classical algorithm to compute the ground states. This time complexity cannot be improved beyond sub-exponential assuming the randomized exponential time hypothesis, even for the special case of classical constraint satisfaction problems on the 2D grid. In chapter 2 we consider frustrated systems and extend results for spin chains to certain trees with intrinsic dimension β < 2. This condition is met for generic trees in the plane and for certain models of hyperbranched polymers in 3D. In chapter 3 we relax the conditions on the Hamiltonian and no longer require a spectral gap or geometric locality, and we consider an approximation problem for the spectrum of the local Hamiltonian. We give a simple proof of a Chernoff bound for the spectrum of a k-local Hamiltonian based on Weyl’s inequalities. The complexity of estimating the spectrum’s ϵ(n)-th quantile up to constant relative error thus exhibits the following dichotomy: For ϵ(n) = d −n the problem is NP-hard and maybe even QMA-hard, yet there exists constant a > 1 such that the problem is trivial for ϵ(n) = a −n."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Improved Tools for Local Hamiltonians"]}]}],"canonical_facts":{"dc:contributor.advisor":["Shor, Peter W."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Abrahamsen, Nilin"],"dc:date.accessioned":["2022-01-14T14:58:51Z"],"dc:date.available":["2022-01-14T14:58:51Z"],"dc:date.issued":["2021-06"],"dc:description.abstract":["In this thesis we consider computational problems related to many-body spin systems with a structured energy operator, a local Hamiltonian. We begin with the most structured setting where the Hamiltonian has a spectral gap and spatial locality. This setting is widely studied using approximate ground space projectors (AGSPs). In chapter 1 we give an improved analysis of AGSPs in the setting of local Hamiltonians with a degenerate ground space. This implies a direct generalization of the AGSP⇒entanglement bound implication of [Arad, Landau, and Vazirani ’12] from unique to degenerate ground states. We use the improved analysis to give a particularly simple algorithm for frustration-free spin systems provided an AGSP with structure as a matrix product operator. We apply our tools to a recent 2D area law of [Anshu, Arad, and Gosset ’21], giving a sub-exponential-time classical algorithm to compute the ground states. This time complexity cannot be improved beyond sub-exponential assuming the randomized exponential time hypothesis, even for the special case of classical constraint satisfaction problems on the 2D grid. In chapter 2 we consider frustrated systems and extend results for spin chains to certain trees with intrinsic dimension β < 2. This condition is met for generic trees in the plane and for certain models of hyperbranched polymers in 3D. In chapter 3 we relax the conditions on the Hamiltonian and no longer require a spectral gap or geometric locality, and we consider an approximation problem for the spectrum of the local Hamiltonian. We give a simple proof of a Chernoff bound for the spectrum of a k-local Hamiltonian based on Weyl’s inequalities. The complexity of estimating the spectrum’s ϵ(n)-th quantile up to constant relative error thus exhibits the following dichotomy: For ϵ(n) = d −n the problem is NP-hard and maybe even QMA-hard, yet there exists constant a > 1 such that the problem is trivial for ϵ(n) = a −n."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139241"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Improved Tools for Local Hamiltonians"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:22:23Z"}