Advice Coins for Classical and Quantum Computation

TL;DR

This paper explores the power of advice coins in classical and quantum computation, revealing quantum automata's heightened sensitivity to coin bias changes and establishing that advice coin classes equal PSPACE/poly.

Contribution

It demonstrates quantum automata's increased sensitivity to advice coins and characterizes advice coin classes as equivalent to PSPACE/poly, using advanced algebraic and quantum techniques.

Findings

Quantum automata are highly sensitive to small bias changes in advice coins.

Advice coin classes BPPSPACE/coin and BQPSPACE/coin are equal to PSPACE/poly.

Classical automata have bounded sensitivity to advice coin bias, unlike quantum automata.

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.