[{"title":"Oracle complexity classes and local measurements on physical Hamiltonians","language":[{"iso":"eng"}],"type":"conference","abstract":[{"text":"The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of\r\nestimating ground state energies of local Hamiltonians. Perhaps surprisingly,\r\n[Ambainis, CCC 2014] showed that the related, but arguably more natural,\r\nproblem of simulating local measurements on ground states of local Hamiltonians\r\n(APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that\r\nAPX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable\r\nby a P machine making a logarithmic number of adaptive queries to a QMA oracle.\r\nIn this work, we show that APX-SIM is P^QMA[log]-complete even when restricted\r\nto more physical Hamiltonians, obtaining as intermediate steps a variety of\r\nrelated complexity-theoretic results.\r\n We first give a sequence of results which together yield P^QMA[log]-hardness\r\nfor APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,\r\nand QMA oracles, a logarithmic number of adaptive queries is equivalent to\r\npolynomially many parallel queries. These equalities simplify the proofs of our\r\nsubsequent results. (2) Next, we show that the hardness of APX-SIM is preserved\r\nunder Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a\r\nbyproduct, we obtain a full complexity classification of APX-SIM, showing it is\r\ncomplete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians\r\nemployed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete\r\nfor any family of Hamiltonians which can efficiently simulate spatially sparse\r\nHamiltonians, including physically motivated models such as the 2D Heisenberg\r\nmodel.\r\n Our second focus considers 1D systems: We show that APX-SIM remains\r\nP^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional\r\nqudits. This uses a number of ideas from above, along with replacing the \"query\r\nHamiltonian\" of [Ambainis, CCC 2014] with a new \"sifter\" construction.","lang":"eng"}],"date_updated":"2019-12-20T12:19:48Z","publication":"Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)","_id":"13226","status":"public","author":[{"last_name":"Gharibian","orcid":"0000-0002-9992-3379","id":"71541","first_name":"Sevag","full_name":"Gharibian, Sevag"},{"last_name":"Piddock","first_name":"Stephen","full_name":"Piddock, Stephen"},{"first_name":"Justin","last_name":"Yirka","full_name":"Yirka, Justin"}],"oa":1,"user_id":"71541","publication_status":"accepted","page":"38","date_created":"2019-09-16T07:41:31Z","main_file_link":[{"url":"https://arxiv.org/abs/1909.05981","open_access":"1"}],"external_id":{"arxiv":["1909.05981"]},"year":"2020","citation":{"apa":"Gharibian, S., Piddock, S., & Yirka, J. (n.d.). Oracle complexity classes and local measurements on physical Hamiltonians. In *Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)* (p. 38).","mla":"Gharibian, Sevag, et al. “Oracle Complexity Classes and Local Measurements on Physical Hamiltonians.” *Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)*, p. 38.","ieee":"S. Gharibian, S. Piddock, and J. Yirka, “Oracle complexity classes and local measurements on physical Hamiltonians,” in *Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)*, p. 38.","ama":"Gharibian S, Piddock S, Yirka J. Oracle complexity classes and local measurements on physical Hamiltonians. In: *Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)*. ; :38.","bibtex":"@inproceedings{Gharibian_Piddock_Yirka, title={Oracle complexity classes and local measurements on physical Hamiltonians}, booktitle={Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, author={Gharibian, Sevag and Piddock, Stephen and Yirka, Justin}, pages={38} }","short":"S. Gharibian, S. Piddock, J. Yirka, in: Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020), n.d., p. 38.","chicago":"Gharibian, Sevag, Stephen Piddock, and Justin Yirka. “Oracle Complexity Classes and Local Measurements on Physical Hamiltonians.” In *Proceedings of the 37th Symposium on Theoretical Aspects of Computer Science (STACS 2020)*, 38, n.d."},"department":[{"_id":"596"}]}]