Summary: Advice Coins for Classical and Quantum Computation
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 clas-
sical 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 polynomi-
als 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.
1.1 The Distinguishing Problem