{"doi":"10.4230/LIPIcs.ITCS.2022.75","_id":"27160","oa":"1","type":"conference","department":[{"_id":"623"},{"_id":"7"}],"title":"On polynomially many queries to NP or QMA oracles","author":[{"orcid":"0000-0002-9992-3379","last_name":"Gharibian","full_name":"Gharibian, Sevag","id":"71541","first_name":"Sevag"},{"last_name":"Rudolph","full_name":"Rudolph, Dorian","first_name":"Dorian"}],"date_created":"2021-11-05T08:08:29Z","publication":"13th Innovations in Theoretical Computer Science (ITCS 2022)","abstract":[{"lang":"eng","text":"We study the complexity of problems solvable in deterministic polynomial time\r\nwith access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$\r\nand $P^{QMA}$, respectively. The former allows one to classify problems more\r\nfinely than the Polynomial-Time Hierarchy (PH), whereas the latter\r\ncharacterizes physically motivated problems such as Approximate Simulation\r\n(APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by\r\nthe classes $P^{NP[\\log]}$ and $P^{QMA[\\log]}$, defined identically to $P^{NP}$\r\nand $P^{QMA}$, except that only logarithmically many oracle queries are\r\nallowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by\r\na $P^{NP}$ machine have a \"query graph\" which is a tree, then this computation\r\ncan be simulated in $P^{NP[\\log]}$.\r\n  In this work, we first show that for any verification class\r\n$C\\in\\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\\exp}\\}$, any $P^C$ machine with a query\r\ngraph of \"separator number\" $s$ can be simulated using deterministic time\r\n$\\exp(s\\log n)$ and $s\\log n$ queries to a $C$-oracle. When $s\\in O(1)$ (which\r\nincludes the case of $O(1)$-treewidth, and thus also of trees), this gives an\r\nupper bound of $P^{C[\\log]}$, and when $s\\in O(\\log^k(n))$, this yields bound\r\n$QP^{C[\\log^{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to\r\ncombine Gottlob's \"admissible-weighting function\" framework with the\r\n\"flag-qubit\" framework of [Watson, Bausch, Gharibian, 2020], obtaining a\r\nunified approach for embedding $P^C$ computations directly into APX-SIM\r\ninstances in a black-box fashion. Finally, we formalize a simple no-go\r\nstatement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear\r\npolynomial $p$ specified via an arithmetic circuit, if one can \"weakly\r\ncompress\" $p$ so that its optimal value requires $m$ bits to represent, then\r\n$P^{NP}$ can be decided with only $m$ queries to an NP-oracle."}],"user_id":"71541","date_updated":"2023-02-28T11:07:56Z","page":"1-27","intvolume":"       215","citation":{"short":"S. Gharibian, D. Rudolph, in: 13th Innovations in Theoretical Computer Science (ITCS 2022), 2022, pp. 1–27.","mla":"Gharibian, Sevag, and Dorian Rudolph. “On Polynomially Many Queries to NP or QMA Oracles.” <i>13th Innovations in Theoretical Computer Science (ITCS 2022)</i>, vol. 215, no. 75, 2022, pp. 1–27, doi:<a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">10.4230/LIPIcs.ITCS.2022.75</a>.","chicago":"Gharibian, Sevag, and Dorian Rudolph. “On Polynomially Many Queries to NP or QMA Oracles.” In <i>13th Innovations in Theoretical Computer Science (ITCS 2022)</i>, 215:1–27, 2022. <a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">https://doi.org/10.4230/LIPIcs.ITCS.2022.75</a>.","ieee":"S. Gharibian and D. Rudolph, “On polynomially many queries to NP or QMA oracles,” in <i>13th Innovations in Theoretical Computer Science (ITCS 2022)</i>, 2022, vol. 215, no. 75, pp. 1–27, doi: <a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">10.4230/LIPIcs.ITCS.2022.75</a>.","bibtex":"@inproceedings{Gharibian_Rudolph_2022, title={On polynomially many queries to NP or QMA oracles}, volume={215}, DOI={<a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">10.4230/LIPIcs.ITCS.2022.75</a>}, number={75}, booktitle={13th Innovations in Theoretical Computer Science (ITCS 2022)}, author={Gharibian, Sevag and Rudolph, Dorian}, year={2022}, pages={1–27} }","ama":"Gharibian S, Rudolph D. On polynomially many queries to NP or QMA oracles. In: <i>13th Innovations in Theoretical Computer Science (ITCS 2022)</i>. Vol 215. ; 2022:1-27. doi:<a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">10.4230/LIPIcs.ITCS.2022.75</a>","apa":"Gharibian, S., &#38; Rudolph, D. (2022). On polynomially many queries to NP or QMA oracles. <i>13th Innovations in Theoretical Computer Science (ITCS 2022)</i>, <i>215</i>(75), 1–27. <a href=\"https://doi.org/10.4230/LIPIcs.ITCS.2022.75\">https://doi.org/10.4230/LIPIcs.ITCS.2022.75</a>"},"year":"2022","language":[{"iso":"eng"}],"volume":215,"main_file_link":[{"url":"https://drops.dagstuhl.de/opus/frontdoor.php?source_opus=15671","open_access":"1"}],"status":"public","issue":"75"}