 
Summary: Advice Coins for Classical and Quantum Computation
Scott Aaronson # Andrew Drucker +
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 spacebounded clas
sical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin,
of languages decidable by classical and quantum polynomialspace 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 Ne# for finding roots of polynomi
als in NC; a result from algebraic geometry that lowerbounds the separation of a polynomial's
roots; and a result on fixedpoints of superoperators due to Aaronson and Watrous, originally
proved in the context of quantum computing with closed timelike curves.
1 Introduction
1.1 The Distinguishing Problem
The fundamental task of mathematical statistics is to learn features of a random process from
