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THE QUANTUM BERNOULLI FACTORY
Terry Rudolph
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
Q-SAMPLE
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
Q-SAMPLE
Interesting?
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
Q-SAMPLE
Interesting?
90’s – Invasion of the Math Lords Invasion I
Provably rapidly mixing Markov chains
-permanents of +’ve matrices -integrating log-concave functions -volumes of convex bodies -ferromagnetic Ising (Jerrum, Sinclair, Vigoda, Applegate, Kannan, Dyer, Frieze,...)
Q-SAMPLE
Interesting?
Coogee question 1: How crazy is this?
90’s – Invasion of the Math Lords Invasion II
Exact sampling -”coupling from the past” (Propp, Wilson)
90’s – Invasion of the Math Lords Invasion II
Exact sampling -”coupling from the past” (Propp, Wilson)
From McKay, Information Theory
90’s – Invasion of the Math Lords Invasion II
Exact sampling -”coupling from the past” (Propp, Wilson)
From McKay, Information Theory
Coogee question 2: Can we “quantize” CFTP to make exact Q-samples?
90’s – Invasion of the Math Lords Invasion II
Exact sampling -”coupling from the past” (Propp, Wilson)
From McKay, Information Theory
Bernoulli Factory Problem: Given the ability to exactly sample a distribution, what other distributions can also be exactly sampled by suitable processing?
90’s – Invasion of the Math Lords Invasion II
Exact sampling -”coupling from the past” (Propp, Wilson)
From McKay, Information Theory
Bernoulli Factory Problem: Given the ability to exactly sample a distribution, what other distributions can also be exactly sampled by suitable processing?
Most attention given to: Given the ability to repeatedly sample a coin which is heads with probability p, for which functions f can we output a single new coin with probability of heads f(p)?
NOTE: unbounded input so not equivalent to Turing machine model
QUOIN
COIN
NOTE: unbounded input so not equivalent to Turing machine model
Outline of the theorem proof:
Imagine we have constructible bounding functions:
Outline of the theorem proof:
Imagine we have constructible bounding functions:
Define sequence of constructible functions:
Outline of the theorem proof:
Imagine we have constructible bounding functions:
Define sequence of constructible functions:
f(p) can be convexly decomposed:
First guess for L(p) and U(p), Bernstein polynomial approximants:
First guess for L(p) and U(p), Bernstein polynomial approximants:
0.0 0.2 0.4 0.6 0.8 1.0p
0.2
0.4
0.6
0.8
1.0fHpL
CONCLUSIONS
- Bernoulli factory mainly interesting because it leads to provable differences between classical info and quantum info
CONCLUSIONS
- Practical advantage?
- What the heck is the class of Q-sampleable distributions?
- For what set of functions f(p) can we implement:
Coogee questions 4-6:
- Bernoulli factory mainly interesting because it leads to provable differences between classical info and quantum info
“QUANTUM-QUANTUM-BERNOULLI FACTORY”
