The geometry of quantum computation
Abstract
Whether the class Quantum Merlin Arthur is equal to QMA_1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a quantum oracle relative to which QMA\neqQMA_1. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such trivial containments as BQP\subseteq{ZQEXP}.
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Geodesic deformation derives an equation for the derivative of H_q(0) with respect to q and integrates it from q=1; this procedure is slow.
“This can be done by deriving an equation for the derivative of H q ð0Þ with respect to q, and then integrating this equation starting at q ¼ 1, a method known as “geodesic deformation” [36].This procedure is slow, however, and a much faster one, that we will refer to as “q jumping,” is as follows.”
From Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems (2015) · cites this paper as [36]
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Wang, X., Allegra, M., Jacobs, K., Lloyd, S., Lupo, C., & Mohseni, M. (2015) · Physical Review Letters · doi:10.1103/physrevlett.114.170501 · Cited by 111 · see citing works
- Authors
- Dowling, M. R., & Nielsen, M. A.
- Affiliations
- The University of Queensland (AU)
- Venue
- Quantum Information and Computation
- Publisher
- Rinton Press
- Published
- 2008-11
- Volume
- 8
- Issue
- 10
- Pages
- 861-899
- ISSN
- 1533-7146 (ISSN-L)
- OpenAlex
- W2950872758
- Cited by
- 137
- References
- 0
- Citations by year
- 2026: 4 · 2025: 20 · 2024: 20 · 2023: 15 · 2022: 14 · 2021: 14 · 2020: 11 · 2019: 12 · 2018: 8 · 2017: 5 · 2015: 2 · 2013: 1
- FWCI
- 4.75
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Quantum Information and Computation (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Complexity and Algorithms in Graphs
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- QMA, quantum oracle, relativization, quantum complexity classes, BQP, one-sided error, real analysis
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