Quantum Polynomial Time and the Human Condition

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Quantum Polynomial Time and the Human Condition

• Scott Aaronson (UC Berkeley)

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• Computers are useless. They only give you
  answers.
• Pablo Picasso

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Computers CAN ask questions
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Answers are not always useless
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(Picasso would say Im not addressing the meat of
his objection)
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Computers led to some of the deepest questions
ever asked
If you can recognize good ideas, can you also
have them? (Does PNP?)
Can quantum parallelism be harnessed to solve
astronomically hard problems?
Could a machine be conscious?
Goal of Talk Show that this question is useful
in Picassos sense
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Wonderful Life
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These movies dont take their premise to its
logical conclusion. Why cant you learn from
2n-1 alternate realities instead of just one?
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Quarantine
Let a computer smearwith the right kind of
quantum randomness and you create, in effect, a
parallel machine with an astronomical number of
processors All you have to do is be sure that
when you collapse the system, you choose the
version that happened to find the needle in the
mathematical haystack. From a scene in which the
protagonist causes a computer to factor a huge
number, by using his newfound ability to
postselect quantum measurement outcomes
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Quarantine
The Popularizers Have Spoken
See, quantum computers, by taking advantage of
weird quantum phenomena which make no sense and
no one understands but the numbers work out so
shut the fuck up and take itquantum computers
are able to compute all possible computations at
the same time, by existing simultaneously in an
infinite number of parallel universes. Popular
Eschatology
Unlike a laboratory rat or an ordinary computer,
which must probe the pathways one at a time, the
quantum computer can simultaneously traverse
every twist and turn and immediately converge
upon the prize. George Johnson, Slate
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Quarantine
Ridiculous!Nature couldnt possibly allow this!
QC of the sort that factors long numbers seems
firmly rooted in science fiction The present
attitude would be analogous to, say, Maxwell
selling the Daemon of his famous thought
experiment as a path to cheaper electricity from
heat. Leonid Levin
The cost of such an operation or of maintaining
such vectors should be linearly related to the
amount of non-degeneracy of these vectors,
where the non-degeneracy may vary from a
constant to linear in the length of the vector.
Oded Goldreich
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Quarantine
Ridiculous!Nature couldnt possibly allow this!
indeed within the usual formalism one can
construct quantum computers that may be able to
solve at least a few specific problems
exponentially faster than ordinary Turing
machines. But particularly after my discoveries
I strongly suspect that even if this is
formally the case, it will still not turn out to
be a true representation of ultimate physical
reality Stephen Wolfram
It will never be possible to construct a quantum
computer that can factor a large number faster,
and within a smaller region of space, than a
classical machine would do, if the latter could
be built out of parts at least as large and as
slow as the Planckian dimensions. Gerard t Hooft
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Crucial Question for Me
Exactly what property separates the Sure States
we know we can prepare, from the Shor States that
suffice for factoring?
DIVIDING LINE
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AmpP
I hereby propose a complexity theory of pure
quantum states one of whose goals is to study
possible Sure/Shor separators.
Circuit
Tree
?
?P
TSH
OTree
Vidal
?
MOTree
?
?2
?2
?1
?1
Strict containmentContainmentNon-containment
?
Classical
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The tree size of an n-qubit state ?? is the
minimum size of a tree of linear combinations and
tensor products that represents ??. (Size
of leaf vertices) Example
?? has tree size ? 6
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Actual Technical ResultA., quant-ph/0311039
Codewords of random stabilizer codes have
superpolynomial tree size
If C x Ax?b(mod 2), where A is chosen
uniformly at random from then with high
probability requires trees of size ?nc?log n
even to approximate well Proof uses recent
breakthrough of Ran Raz Conjecture Same lower
bound holds for states arising in Shors algorithm
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Recent (as in last week) Developments
2-D cluster states (as proposed by Briegel and
Raussendorf) have tree size ?nc?log n. Not true
for 1-D Explicit (non-random) coset states,
obtained by concatenating Reed-Solomon and
Hadamard codes, have tree size ?nc?log
n Exponential lower bounds on manifestly
orthogonal tree size.
0 0 1 0 1
0 1 1 1 0
0 0 1 1 1
1 0 1 0 0
1 0 0 1 1
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So
Unless theres a clear, consistent dividing line
between what weve seen and what QM predicts
well see, we ought to worry now about the
quantum computing picture of reality
Could all paths of a maze be traversed
simultaneously? In order to win the lottery,
prove P?NP, date a supermodel, etc., is it enough
for it to be possible that you achieve these
things?
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BBBV97 Hybrid ArgumentCan a quantum algorithm
that makes fewer than ?N queries find 1 marked
item out of N?
In the case that no items are marked, some item
must have a small total probability of being
queriedMark that item and rerun the algorithm
?11?
?22?
?33?
?44?
?55?
?66?
?77?
?88?
?99?
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Actual Technical Result IIA., quant-ph/0402095
Is there some initial stateeven a highly
entangled, not efficiently preparable onethat
would let a quantum computer solve NP-complete
problems in polynomial time? After all, such a
state might encode information about every MAX
CLIQUE problem of size n! We would therefore
evade the BBBV conclusion
Theorem Relative to some oracle, NP ? BQP/qpoly
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Proof Idea
Can be reduced to showing a direct product
theorem for quantum search
Given N items, K of which are marked, if we dont
have enough queries to find even one marked item,
then the probability of finding all K of them
decreases exponentially in K. Klauck gave an
incorrect proof of this. I give the first correct
proof, using the polynomial method of Beals et
al. Recently improved by Klauck, Špalek, and de
Wolf.
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So How Should You Solve NP-Complete Problems?
Measure electron spins to guess a random
solution. If the solution is wrong, kill
yourself.
If the solution is right, destroy the human race.
If wrong, cause it to exist for billions of
years.
Actual Technical Result III (A.,
quant-ph/0401062) Let PostBQP be the class of
problems solvable in quantum polynomial time
using postselection. Then PostBQP PP
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Nonlinear Quantum Computing
Abrams Lloyd 1998 We could solve NP-complete
problems efficiently given a 1-qubit nonlinear
gate that acts as follows
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Nonlinear Quantum Computing
Abrams Lloyd 1998 We could solve NP-complete
problems efficiently given a 1-qubit nonlinear
gate that acts as follows
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Nonlinear Quantum Computing
Abrams Lloyd 1998 We could solve NP-complete
problems efficiently given a 1-qubit nonlinear
gate that acts as follows
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Nonlinear Quantum Computing
Abrams Lloyd 1998 We could solve NP-complete
problems efficiently given a 1-qubit nonlinear
gate that acts as follows
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Observation Given custom-designed 1-qubit
nonlinear gates, we could even solve
PSPACE-complete problems efficiently (but not
more)
But what about realistic nonlinear gates (e.g.
Weinbergs) subject to small environmental
error? Abrams and Lloyds claim to solve
NP-complete problems in this setting seems
incorrect
Open Problem The Two-Edged Sword Can we amplify
an exponentially small success probability
without also amplifying exponentially small
errors? (Maybe Gwyneth would be better off
without trans-universe communication!)
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Conclusion The Garden of Forking Paths
Determinism
Randomness
Postselection
Quantumness
Computers are useless. They only give you
answers. Picasso