Advice Coins for Classical and Quantum Computation

  • Scott Aaronson
  • Andrew Drucker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


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.


BQP finite automata finite-precision arithmetic PSPACE Quantum Computation root-finding 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Scott Aaronson
    • 1
  • Andrew Drucker
    • 1
  1. 1.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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