Advice Coins

1
Advice Coins

• Scott Aaronson

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• 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/coinALL?

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• Hellman-Cover 1970 To distinguish a p1/2 coin
  from a p1/2? coin with constant bias, you need
  a probabilistic finite automaton with ?(1/?)
  states
• I.e. you cant detect a less than 1/exp(n) change
  in p without more than poly(n) bits to record the
  statisticsregardless of how many times you flip
  the coin
• Seems to answer our question! Except that it
  doesnt

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• 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
  p1/2 from p1/2? for any ?gt0
• So question stands PSPACE/coinALL?
  BQPSPACE/coinALL?

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• 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.