1 / 5

Advice Coins

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?.

joycedominick
Télécharger la présentation

Advice Coins

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Advice Coins Scott Aaronson

  2. 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?

  3. 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

  4. 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?

  5. 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.

More Related