[ { "id": "https://authors.library.caltech.edu/records/3hxfe-q5m49", "eprint_id": 99903, "eprint_status": "archive", "datestamp": "2023-08-19 03:07:40", "lastmod": "2023-10-18 18:54:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Fefferman-B", "name": { "family": "Fefferman", "given": "Bill" } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Pseudorandom generators and the BQP vs. PH problem", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Scott Aaronson, Yi-Kai Liu, and Emanuele Viola for helpful discussions.\n\n
Submitted - 1007.0305.pdf
", "abstract": "It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC\u2080, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof from [NW94]) can be avoided in this setting. This is a question that has been asked previously in the pseudorandomness literature [BSW03]. We then make three main contributions: (1) We show that our conjecture implies the existence of an oracle relative to which BQP is not in the PH. This entails giving an explicit construction of unitary matrices, realizable by small quantum circuits, whose row-supports are \"nearly-disjoint.\" (2) We give a simple framework (generalizing the setting of Aaronson [A10]) in which any efficiently quantumly computable unitary gives rise to a distribution that can be distinguished from the uniform distribution by an efficient quantum algorithm. When applied to the unitaries we construct, this framework yields a problem that can be solved quantumly, and which forms the basis for the desired oracle. (3) We prove that Aaronson's \"GLN conjecture\" [A10] implies our conjecture; our conjecture is thus formally easier to prove. The GLN conjecture was recently proved false for depth greater than 2 [A10a], but it remains open for depth 2. If true, the depth-2 version of either conjecture would imply an oracle relative to which BQP is not in AM, which is itself an outstanding open problem. Taken together, our results have the following interesting interpretation: they give an instantiation of the Nisan-Wigderson generator that can be broken by quantum computers, but not by the relevant modes of classical computation, if our conjecture is true.", "date": "2019-11-18", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20191118-130935010", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-130935010", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.1007.0305", "primary_object": { "basename": "1007.0305.pdf", "url": "https://authors.library.caltech.edu/records/3hxfe-q5m49/files/1007.0305.pdf" }, "resource_type": "monograph", "pub_year": "2019", "author_list": "Fefferman, Bill and Umans, Chris" }, { "id": "https://authors.library.caltech.edu/records/22gnh-3hk76", "eprint_id": 99881, "eprint_status": "archive", "datestamp": "2023-09-15 06:25:05", "lastmod": "2023-10-23 21:26:15", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Blasiak-J", "name": { "family": "Blasiak", "given": "Jonah" } }, { "id": "Church-T", "name": { "family": "Church", "given": "Thomas" } }, { "id": "Cohn-H", "name": { "family": "Cohn", "given": "Henry" } }, { "id": "Grochow-J-A", "name": { "family": "Grochow", "given": "Joshua A." } }, { "id": "Umans-C", "name": { "family": "Umans", "given": "Chris" } } ] }, "title": "Which groups are amenable to proving exponent two for matrix multiplication?", "ispublished": "unpub", "full_text_status": "public", "note": "J.B.: Supported by NSF grant DMS-1600391. All authors also thank AIM for hosting a SQuaRE, during which this work was developed. \n\nT.C.: Supported by NSF grant DMS-1350138, the Alfred P. Sloan Foundation, and the Frederick E. Terman Fellowship. \n\nJ.A.G.: Supported by a Santa Fe Institute Omidyar Fellowship and NSF grant DMS-1750319. \n\nC.U.: Supported by NSF grant CCF-1423544 and a Simons Foundation Investigator grant.\n\nSubmitted - 1712.02302.pdf
", "abstract": "The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding \u03c9 in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on \u03c9 and is conjectured to be powerful enough to prove \u03c9=2, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove \u03c9=2 in this framework, which ruled out a family of potential constructions in the literature. \n\nIn this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results:\n(1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove \u03c9=2 in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over (Z/pZ)^n that are degree d polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture.\n(2) We show that symmetric groups S_n cannot prove nontrivial bounds on \u03c9 when the embedding is via three Young subgroups---subgroups of the form S_(k\u2081)\u00d7S_(k\u2082)\u00d7\u22ef\u00d7S_(k\u2113)---which is a natural strategy that includes all known constructions in S_n.\nBy developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on \u03c9 and methods for ruling them out.", "date": "2019-11-18", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20191118-072853843", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191118-072853843", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1600391" }, { "agency": "NSF", "grant_number": "DMS-1350138" }, { "agency": "Alfred P. Sloan Foundation" }, { "agency": "Frederick E. Terman Fellowship" }, { "agency": "Santa Fe Institute" }, { "agency": "NSF", "grant_number": "DMS-1750319" }, { "agency": "NSF", "grant_number": "CCF-1423544" }, { "agency": "Simons Foundation" } ] }, "doi": "10.48550/arXiv.1712.02302", "primary_object": { "basename": "1712.02302.pdf", "url": "https://authors.library.caltech.edu/records/22gnh-3hk76/files/1712.02302.pdf" }, "resource_type": "monograph", "pub_year": "2019", "author_list": "Blasiak, Jonah; Church, Thomas; et el." } ]