{"title":"Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower  bounds","publication":"arXiv:2401.01633","abstract":[{"lang":"eng","text":"The Polynomial-Time Hierarchy ($\\mathsf{PH}$) is a staple of classical\r\ncomplexity theory, with applications spanning randomized computation to circuit\r\nlower bounds to ''quantum advantage'' analyses for near-term quantum computers.\r\nQuantumly, however, despite the fact that at least \\emph{four} definitions of\r\nquantum $\\mathsf{PH}$ exist, it has been challenging to prove analogues for\r\nthese of even basic facts from $\\mathsf{PH}$. This work studies three\r\nquantum-verifier based generalizations of $\\mathsf{PH}$, two of which are from\r\n[Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings\r\n($\\mathsf{QCPH}$) and quantum mixed states ($\\mathsf{QPH}$) as proofs, and one\r\nof which is new to this work, utilizing quantum pure states\r\n($\\mathsf{pureQPH}$) as proofs. We first resolve several open problems from\r\n[GSSSY22], including a collapse theorem and a Karp-Lipton theorem for\r\n$\\mathsf{QCPH}$. Then, for our new class $\\mathsf{pureQPH}$, we show one-sided\r\nerror reduction for $\\mathsf{pureQPH}$, as well as the first bounds relating\r\nthese quantum variants of $\\mathsf{PH}$, namely $\\mathsf{QCPH}\\subseteq\r\n\\mathsf{pureQPH} \\subseteq \\mathsf{EXP}^{\\mathsf{PP}}$."}],"language":[{"iso":"eng"}],"date_updated":"2024-01-07T20:09:58Z","type":"preprint","citation":{"short":"A. Agarwal, S. Gharibian, V. Koppula, D. Rudolph, ArXiv:2401.01633 (2024).","ama":"Agarwal A, Gharibian S, Koppula V, Rudolph D. Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower  bounds. <i>arXiv:240101633</i>. Published online 2024.","ieee":"A. Agarwal, S. Gharibian, V. Koppula, and D. Rudolph, “Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower  bounds,” <i>arXiv:2401.01633</i>. 2024.","bibtex":"@article{Agarwal_Gharibian_Koppula_Rudolph_2024, title={Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower  bounds}, journal={arXiv:2401.01633}, author={Agarwal, Avantika and Gharibian, Sevag and Koppula, Venkata and Rudolph, Dorian}, year={2024} }","chicago":"Agarwal, Avantika, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. “Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower  Bounds.” <i>ArXiv:2401.01633</i>, 2024.","mla":"Agarwal, Avantika, et al. “Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower  Bounds.” <i>ArXiv:2401.01633</i>, 2024.","apa":"Agarwal, A., Gharibian, S., Koppula, V., &#38; Rudolph, D. (2024). Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower  bounds. In <i>arXiv:2401.01633</i>."},"date_created":"2024-01-07T20:09:32Z","external_id":{"arxiv":["2401.01633"]},"_id":"50273","status":"public","author":[{"first_name":"Avantika","full_name":"Agarwal, Avantika","last_name":"Agarwal"},{"orcid":"0000-0002-9992-3379","id":"71541","first_name":"Sevag","last_name":"Gharibian","full_name":"Gharibian, Sevag"},{"first_name":"Venkata","last_name":"Koppula","full_name":"Koppula, Venkata"},{"full_name":"Rudolph, Dorian","last_name":"Rudolph","first_name":"Dorian"}],"year":"2024","user_id":"71541"}