Abstract
I will discuss the now-ancient question of where BQP, Bounded-Error Quantum Polynomial-Time, fits in among classical complexity classes. After reviewing some basics from the 90s, I will discuss the Forrelation problem that I introduced in 2009 to yield an oracle separation between BQP and PH, and the dramatic completion of that program by Ran Raz and Avishay Tal in 2018. I will then discuss very recent work, with William Kretschmer and DeVon Ingram, which leverages the Raz-Tal theorem, along with a new "quantum-aware" random restriction method, to obtain results that illustrate just how differently BQP can behave from BPP. These include oracles relative to which NP^{BQP} ̸ ⊂ BQP^{PH} - solving a 2005 open problem of Lance Fortnow - and conversely, relative to which BQP^{NP} ̸ ⊂ PH^{BQP}; an oracle relative to which ? = NP and yet BQP ≠ QCMA; an oracle relative to which NP ⊆ BQP yet PH is infinite; an oracle relative to which ? = NP≠ BQP = PP; and an oracle relative to which PP = PostBQP ̸ ⊂ QMA^{QMA^{…}}. By popular demand, I will also speculate about the status of BQP in the unrelativized world.
BibTeX - Entry
@InProceedings{aaronson:LIPIcs.FSTTCS.2021.1,
author = {Aaronson, Scott},
title = {{BQP After 28 Years}},
booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-215-0},
ISSN = {1868-8969},
year = {2021},
volume = {213},
editor = {Boja\'{n}czy, Miko{\l}aj and Chekuri, Chandra},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15512},
URN = {urn:nbn:de:0030-drops-155124},
doi = {10.4230/LIPIcs.FSTTCS.2021.1},
annote = {Keywords: quantum computing, complexity theory, oracle separations, circuit lower bounds}
}
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Keywords: |
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quantum computing, complexity theory, oracle separations, circuit lower bounds |
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Collection: |
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41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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29.11.2021 |
Creative Commons Attribution 4.0 International license (CC BY 4.0)