Advice coins for classical and quantum computation
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Aaronson_Advice coins.pdf
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269.67 KB
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Author(s)
Aaronson, Scott
Drucker, Andrew Donald
Date Issued
June 2011
Journal
Automata, Languages and Programming
Publisher
Springer Berlin / Heidelberg
Citation
Aaronson, Scott, and Andrew Drucker. “Advice Coins for Classical and Quantum Computation.” Automata, Languages and Programming. Ed. Luca Aceto, Monika Henzinger, & Jiří Sgall. Vol. 6755. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. 61–72. Web. 27 June 2012. © Springer Berlin / Heidelberg
Version
Author's final manuscript
Abstract
We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coin’s bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coin’s bias is bounded by a classic 1970 result of Hellman and Cover.
Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomial’s roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/978-3-642-22006-7_6
