Scott Aaronson

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Le Principe de la Postselection

• Scott Aaronson
• Institut pour l'Étude Avançée

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Could you ever learn enough about a person to
predict his or her future behavior reliably?
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Examples
Good novels dont just put their characters in
random situationsthey repeatedly subject the
characters to crucial tests that reveal aspects
of their personalities we didnt already know
(I guess)
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The Karp-Lipton Theorem (1982)
Suppose NP-complete problems were solvable in
polynomial time, but only nonuniformlythat is,
with polynomial-size circuits Then we could use
those circuits to collapse the Polynomial-Time
Hierarchy down to the seocnd level, NPNP This
would be almost as shocking as if PNP! If pigs
could whistle, then donkeys could fly
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We want to exploit a small circuit for solving
NP-complete problems but all we know is that it
exists!
Does there exist a circuit C of size nk, such
that for all Boolean formulas ? of size n, C
correctly decides whether ? is satisfiable, and C
outputs yes on whatever problem we wanted to
solve originally?
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But why should we care?
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Theorem (Kannan 1982) For every k, there exists
a language in NPNP that does not have circuits of
size nk
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Bshouty et al.s Improvement (1994)
If a function f0,1n?0,1 has a
polynomial-size circuit, then we can find the
circuit in ZPPNP, provided we can somehow compute
f (ZPP Zero-Error Probabilistic
Polynomial-Time) Idea Iterative learning.
Repeatedly find an input xt such that, among the
circuits that correctly compute f on x1,,xt-1,
at least a 1/3 fraction get xt wrong This process
cant continue for long!
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But what about quantum anthropic computing?
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PostBQP
I hereby define a newcomplexity class
(Postselected BQP)
Class of languages decidable by a bounded-error
polynomial-time quantum computer, if at any time
you can measure a qubit that has a nonzero
probability of being 1?, and assume the outcome
will be 1?
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Another Important Animal PP
Class of languages decidable by a
nondeterministic poly-time Turing machine that
accepts iff the majority of its paths do
PSPACE
PPPPP
PP
NP
P
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Theorem (A., 2004)PostBQP PP
Unexpectedly, this theorem turned out to have an
implication for classical complexity the
simplest known proof of the Beigel-Reingold-Spiel
man Theorem, that PP is closed under intersection
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Detour
The maximally mixed state In is just the
uniform distribution over n-bit strings
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Quantum Proofs
QMA (defined by Kitaev and Watrous) is the
quantum version of NP Does there exist a
quantum state ?? accepted by such-and-such a
circuit with high probability?
Unlike NP, QMA doesnt seem to be
self-reduciblewe dont know how to construct
?? given an oracle for QMA problems
But we can construct ?? in PostBQP. (Why?)
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Quantum Advice
Mike Ike We know that many systems in Nature
prefer to sit in highly entangled states of
many systems might it be possible to exploit
this preference to obtain extra computational
power?
BQP/qpoly Class of languages decidable by
polynomial-size, bounded-error quantum circuits,
given a polynomial-size quantum advice state ?n?
that depends only on the input length n
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How powerful is quantum advice?
Could it let us solve problems that are not even
computable given classical advice of similar
size?!
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Limitations of Quantum Advice
NP ? BQP/qpoly relative to an oracle(Uses direct
product theorem for quantum search)
BQP/qpoly ? PostBQP/poly ( PP/poly)
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Alices Classical Advice
Bob, suppose you used the maximally mixed state
in place of your quantum advice. Then x1 is the
lexicographically first input for which youd
output the right answer with probability less
than ½. But suppose you succeeded on x1, and used
the resulting reduced state as your advice. Then
x2 is the lexicographically first input after x1
for which youd output the right answer with
probability less than ½...
x1
x2
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But how many inputs must Alice specify?
We can boost a quantum advice state so that the
error probability on any input is at most (say)
2-100n then Bob can reuse the advice on as many
inputs as he likes
We can decompose the maximally mixed state on
p(n) qubits as the boosted advice plus 2p(n)-1
orthogonal states
Alice needs to specify at most p(n) inputs
x1,x2,, since each one cuts Bobs total success
probability by at least half, but the probability
must be at least 2-p(n) by the end
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PPP Does Not Have Quantum Circuits of Size nk
?
?
Does U accept x0 w.p. ? ½?If yes, set x0?LIf
no, set x0?L
Conditioned on deciding x0 correctly, does U
accept x1 w.p. ? ½?If yes, set x1?LIf no, set
x1?L
Conditioned on deciding x0 and x1 correctly, does
U accept x2 w.p. ? ½?If yes, set x2?LIf no, set
x2?L
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For any k, defines a language L that does not
have quantum circuits of size nk
On the other hand, clearly L ? PPP
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Quantum Karp-Lipton Theorem
Also PP does not have quantum circuits of size
nk PEXP requires quantum circuits of size f(n),
where f(f(n))?2n
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Concluding Thought What Makes Science Possible?
That which we can observe, we can understand
That which we can observe, and then observe in a
new situation where we cant predict what it will
do even given the earlier observation, and so on
for a polynomial number of steps, we can
understand (provided we can postselect a
description consistent with our observations)
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To Show PP ? PostBQP
Given a Boolean function f0,1n?0,1,let
sx f(x)1. Need to decide if sgt2n-1
Goal Decide if these amplitudes have the same or
opposite signs
Prepare ?0????1?H?? for some ?,?.Then
postselect on second qubit being 1?
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To Show PP ? PostBQP
On the other hand, if 2n-2s is negative, then we
wont. QED
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Beigel, Reingold, Spielman 1990 PP is closed
under intersectionSolved a problem that was
open for 18 years
Observation PostBQP is trivially closed under
intersection ? PP is too Given L1,L2?PostBQP, to
decide if x?L1 and x?L2, postselect on both
computations succeeding, and accept iff they both
accept
Other classical results proved with quantum
techniques Kerenidis de Wolf, A., Aharonov
Regev,