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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.