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oai:dspace.mit.edu:1721.1/1637252025-11-18T06:27:51Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023

Natarajan, Anand Ma, Henry Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science 2025-11-17T19:09:37Z 2025-11-17T19:09:37Z 2025-05 2025-08-14T19:32:19.881Z https://hdl.handle.net/1721.1/163725 A major goal of quantum complexity theory is to understand which computational problems can be solved with access to certain quantum resources. The subfield of Hamiltonian complexity specifically considers computational problems that ask about properties of local Hamiltonians, which are of critical importance in quantum complexity because they can be viewed as quantum generalizations of classical constraint satisfaction problems. In this work, we study the complexity of certain restricted variants of the Quantum-k-Sat problem, a quantum analog of the NP-complete k-Sat problem. We introduce new variants of Quantum-k-Sat which place a basis restriction on the input Hamiltonian H = Σᵢ hᵢ . Each variant is defined by a fixed collection of bases B₁, . . . , Bᵣ of n-qubit space. We require that each Hamiltonian term hi must be diagonal in one of these bases. Our results resolve the complexity of certaim basis-restricted variants of Quantum-k-Sat. First we show the Quantum-6-Sat problem with Hamiltonian terms restricted to be diagonal in an X/Z mixed basis is QMA₁-complete. Second, we combine basis restriction with the restriction of commutativity, and show the following easiness result, which applies generally to higher-level quantum systems (qudits) and bases Q and R (which are real-valued and satisfy an overlap condition): The commmuting Quantum-Sat problem on qudits, where Hamiltonian terms are either diagonal in the Q basis, the R basis, or a single mixed Q/R basis, is in NP. S.M. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright retained by author(s) https://rightsstatements.org/page/InC-EDU/1.0/ Complexity of Basis-Restricted Local Hamiltonians Thesis application/pdf Master Master of Science in Electrical Engineering and Computer Science Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="e4cae771-99a6-4a99-a6ab-af6276ed31c2"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Complexity of Basis-Restricted Local Hamiltonians</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2025-05</PublicationDate> <Authors> <Author> <DisplayName>Ma, Henry</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>https://rightsstatements.org/page/InC-EDU/1.0/</License> <Abstract>A major goal of quantum complexity theory is to understand which computational problems can be solved with access to certain quantum resources. The subfield of Hamiltonian complexity specifically considers computational problems that ask about properties of local Hamiltonians, which are of critical importance in quantum complexity because they can be viewed as quantum generalizations of classical constraint satisfaction problems. In this work, we study the complexity of certain restricted variants of the Quantum-k-Sat problem, a quantum analog of the NP-complete k-Sat problem. We introduce new variants of Quantum-k-Sat which place a basis restriction on the input Hamiltonian H = Σᵢ hᵢ . Each variant is defined by a fixed collection of bases B₁, . . . , Bᵣ of n-qubit space. We require that each Hamiltonian term hi must be diagonal in one of these bases. Our results resolve the complexity of certaim basis-restricted variants of Quantum-k-Sat. First we show the Quantum-6-Sat problem with Hamiltonian terms restricted to be diagonal in an X/Z mixed basis is QMA₁-complete. Second, we combine basis restriction with the restriction of commutativity, and show the following easiness result, which applies generally to higher-level quantum systems (qudits) and bases Q and R (which are real-valued and satisfy an overlap condition): The commmuting Quantum-Sat problem on qudits, where Hamiltonian terms are either diagonal in the Q basis, the R basis, or a single mixed Q/R basis, is in NP.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>