Advice Coins Scott Aaronson
PSPACE/coin: Class of problems solvable by a PSPACE machine that can flip an “advice coin” (heads with probability p, tails with probability 1-p) as many times as it wants • Clear that PSPACE/poly PSPACE/coin • Other direction? Could PSPACE/coin=ALL?
Hellman-Cover 1970: To distinguish a p=1/2 coin from a p=1/2+ coin with constant bias, you need a probabilistic finite automaton with (1/) states • I.e. you can’t detect a less than 1/exp(n) change in p without more than poly(n) bits to record the statistics—regardless of how many times you flip the coin • Seems to answer our question! Except that it doesn’t
First problem: p could be unbelievably small (1/Ackermann(n)), and info could be stored in log(1/p) • Second problem: Hellman-Cover theorem is false for quantum finite automata! • I can give a QFA with 2 qubits that distinguishes p=1/2 from p=1/2+ for any >0 • So question stands: PSPACE/coin=ALL? BQPSPACE/coin=ALL?
Main Result: PSPACE/coin, BQPSPACE/coin are both contained in Something/poly • Main Idea: Limiting distribution (or quantum state) of an s-state automaton can be expressed in terms of degree-s rational functions of p. These can oscillate at most s times as p goes from 0 to 1. • Need to count and compare roots of real polynomials. If everything is doable in NC, then a PSPACE/poly upper bound follows.