--- _id: '8161' abstract: - lang: eng text: The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP. author: - first_name: Sevag full_name: Gharibian, Sevag id: '71541' last_name: Gharibian orcid: 0000-0002-9992-3379 - first_name: Miklos full_name: Santha, Miklos last_name: Santha - first_name: Jamie full_name: Sikora, Jamie last_name: Sikora - first_name: Aarthi full_name: Sundaram, Aarthi last_name: Sundaram - first_name: Justin full_name: Yirka, Justin last_name: Yirka citation: ama: 'Gharibian S, Santha M, Sikora J, Sundaram A, Yirka J. Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In: Potapov I, Spirakis P, Worrell J, eds. 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018). Vol 117. Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:58:1-58:16. doi:10.4230/LIPIcs.MFCS.2018.58' apa: Gharibian, S., Santha, M., Sikora, J., Sundaram, A., & Yirka, J. (2018). Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In I. Potapov, P. Spirakis, & J. Worrell (Eds.), 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018) (Vol. 117, p. 58:1-58:16). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2018.58 bibtex: '@inproceedings{Gharibian_Santha_Sikora_Sundaram_Yirka_2018, place={Dagstuhl, Germany}, series={Leibniz International Proceedings in Informatics (LIPIcs)}, title={Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}, volume={117}, DOI={10.4230/LIPIcs.MFCS.2018.58}, booktitle={43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)}, publisher={Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, author={Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin}, editor={Potapov, Igor and Spirakis, Paul and Worrell, James}, year={2018}, pages={58:1-58:16}, collection={Leibniz International Proceedings in Informatics (LIPIcs)} }' chicago: 'Gharibian, Sevag, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2).” In 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), edited by Igor Potapov, Paul Spirakis, and James Worrell, 117:58:1-58:16. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. https://doi.org/10.4230/LIPIcs.MFCS.2018.58.' ieee: 'S. Gharibian, M. Santha, J. Sikora, A. Sundaram, and J. Yirka, “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2),” in 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), Liverpool, UK, 2018, vol. 117, p. 58:1-58:16, doi: 10.4230/LIPIcs.MFCS.2018.58.' mla: Gharibian, Sevag, et al. “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2).” 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), edited by Igor Potapov et al., vol. 117, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018, p. 58:1-58:16, doi:10.4230/LIPIcs.MFCS.2018.58. short: 'S. Gharibian, M. Santha, J. Sikora, A. Sundaram, J. Yirka, in: I. Potapov, P. Spirakis, J. Worrell (Eds.), 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, p. 58:1-58:16.' conference: location: Liverpool, UK name: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) date_created: 2019-03-01T11:29:44Z date_updated: 2023-02-28T11:01:03Z department: - _id: '623' - _id: '7' doi: 10.4230/LIPIcs.MFCS.2018.58 editor: - first_name: Igor full_name: Potapov, Igor last_name: Potapov - first_name: Paul full_name: Spirakis, Paul last_name: Spirakis - first_name: James full_name: Worrell, James last_name: Worrell external_id: arxiv: - '1805.11139' intvolume: ' 117' keyword: - Complexity Theory - Quantum Computing - Polynomial Hierarchy - Semidefinite Programming - QMA(2) - Quantum Complexity language: - iso: eng main_file_link: - open_access: '1' url: http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=9640 oa: '1' page: 58:1-58:16 place: Dagstuhl, Germany publication: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) publication_identifier: unknown: - 978-3-95977-086-6 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik series_title: Leibniz International Proceedings in Informatics (LIPIcs) status: public title: Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2) type: conference user_id: '71541' volume: 117 year: '2018' ... --- _id: '8160' abstract: - lang: eng text: An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following results. The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete. P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] \subseteq PP. This improves the containment QMA \subseteq PP from Kitaev and Watrous. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in Ambainis' prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. author: - first_name: Sevag full_name: Gharibian, Sevag id: '71541' last_name: Gharibian orcid: 0000-0002-9992-3379 - first_name: Justin full_name: Yirka, Justin last_name: Yirka citation: ama: 'Gharibian S, Yirka J. The Complexity of Simulating Local Measurements on Quantum Systems. In: Wilde M, ed. 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Vol 73. Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:2:1-2:17. doi:10.4230/LIPIcs.TQC.2017.2' apa: Gharibian, S., & Yirka, J. (2018). The Complexity of Simulating Local Measurements on Quantum Systems. In M. Wilde (Ed.), 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017) (Vol. 73, p. 2:1-2:17). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TQC.2017.2 bibtex: '@inproceedings{Gharibian_Yirka_2018, place={Dagstuhl, Germany}, series={Leibniz International Proceedings in Informatics (LIPIcs)}, title={The Complexity of Simulating Local Measurements on Quantum Systems}, volume={73}, DOI={10.4230/LIPIcs.TQC.2017.2}, booktitle={12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, publisher={Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, author={Gharibian, Sevag and Yirka, Justin}, editor={Wilde, Mark}, year={2018}, pages={2:1-2:17}, collection={Leibniz International Proceedings in Informatics (LIPIcs)} }' chicago: 'Gharibian, Sevag, and Justin Yirka. “The Complexity of Simulating Local Measurements on Quantum Systems.” In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), edited by Mark Wilde, 73:2:1-2:17. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. https://doi.org/10.4230/LIPIcs.TQC.2017.2.' ieee: 'S. Gharibian and J. Yirka, “The Complexity of Simulating Local Measurements on Quantum Systems,” in 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), Paris, France, 2018, vol. 73, p. 2:1-2:17, doi: 10.4230/LIPIcs.TQC.2017.2.' mla: Gharibian, Sevag, and Justin Yirka. “The Complexity of Simulating Local Measurements on Quantum Systems.” 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), edited by Mark Wilde, vol. 73, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018, p. 2:1-2:17, doi:10.4230/LIPIcs.TQC.2017.2. short: 'S. Gharibian, J. Yirka, in: M. Wilde (Ed.), 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, p. 2:1-2:17.' conference: location: Paris, France name: 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017) date_created: 2019-03-01T11:25:27Z date_updated: 2023-02-28T11:00:48Z department: - _id: '623' - _id: '7' doi: 10.4230/LIPIcs.TQC.2017.2 editor: - first_name: Mark full_name: Wilde, Mark last_name: Wilde external_id: arxiv: - '1606.05626' intvolume: ' 73' keyword: - Complexity theory - Quantum Merlin Arthur (QMA) - local Hamiltonian - local measurement - spectral gap language: - iso: eng main_file_link: - open_access: '1' url: http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=8577 oa: '1' page: 2:1-2:17 place: Dagstuhl, Germany publication: 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017) publication_identifier: unknown: - 978-3-95977-034-7 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik series_title: Leibniz International Proceedings in Informatics (LIPIcs) status: public title: The Complexity of Simulating Local Measurements on Quantum Systems type: conference user_id: '71541' volume: 73 year: '2018' ...