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Quantum Polynomial Time and the

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					    Quantum Polynomial Time
    and the Human Condition



                         / 2
       Scott Aaronson (UC Berkeley)
                     ―Computers are useless.
                       They only give you
                           answers.‖
                        –Pablo Picasso


Not merely a false statement, but a pompous
and asinine one
Computers CAN
ask questions
Answers are not always useless
(Picasso would say I‘m not addressing
the meat of his objection)
Computers led to some of the
deepest questions ever asked

 If you can recognize good ideas, can
 Can quantum parallelism be harnessed
  Could a machine be (Does P=NP?)
 you also have them? conscious?
 to solve astronomically hard problems?


 Goal of Talk: Show that this question is
 ‗useful‘ in Picasso‘s sense
These movies don‘t take their premise to its
logical conclusion. Why can‘t you learn from 2n-1
alternate realities instead of just one?
“Let a computer smear—with 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
The Popularizers Have Spoken
       Ridiculous!
Nature couldn’t 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

 [T]he 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
         Ridiculous!
Nature couldn’t possibly allow this!
 [i]ndeed 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
    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
                                           AmpP

   I hereby                                                 Circuit
   propose a
  complexity
                                                Tree
                       P
theory of pure                                  TSH
    quantum          OTree
     states n
    H2                                                    Vidal

                 MOTree

                                  2                   2
 one of whose
   goals is to                    1                   1
study possible
  Sure/Shor
                     Strict containment
                     Containment
  separators.       Non-containment      Classical
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)

                      00  01  10  11  
Example:              1
                      2
                                  +
                        1                1

                                               
  1     +   1           1     +   1      |11       |12
   2         2           2         2

 |01       |11       |02       |12     | has tree size  6
      Actual Technical Result
                    A., 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      n1/ 3 n then with high probability
                2        ,
                           1
                    C 
                            C
                                x
                                xC

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 Shor‘s algorithm
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
                    0   0   1   0   1
                    0   1   1   1   0
                NOTE1 FOR
                      0 0   1 1

             PHYSICISTS:
                      1 0 1 0 0
                      1 0 0 1 1

           I only care about
Explicit (non-random) coset states, obtained by
concatenating Reed-Solomon and Hadamard codes,
have tree size qubit states
               nclog n
Exponential lower bounds on ―manifestly orthogonal‖
tree size.
                     So…
Unless there‘s a clear, consistent dividing line
between what we‘ve seen and what QM predicts
we‘ll 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?
    BBBV’97 Hybrid Argument
   Can 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 aDUDE!!!
        small total probability of being queried
Everyone! The
Mark that item and rerun the algorithm

marked item!                   Someone must be
 Over here!!!              screaming about a marked
                            item… too bad quantum
                               mechanics is linear

1|1 2|2 3|3 4|4 5|5 6|6 7|7 8|8 9|9
    Actual Technical Result II
                 A., quant-ph/0402095
Is there some initial state—even a highly entangled,
not efficiently preparable one—that 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
                  Proof Idea
Can be reduced to showing a direct product
theorem for quantum search:


Given N items, K of which are marked, if we don‘t
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.
   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
 Nonlinear Quantum Computing
Abrams & Lloyd 1998: We could solve NP-
complete problems efficiently given a 1-qubit
nonlinear gate that acts as follows:

                      1
                             0 1
                                2



                                    0
 Nonlinear Quantum Computing
Abrams & Lloyd 1998: We could solve NP-
complete problems efficiently given a 1-qubit
nonlinear gate that acts as follows:

                      1
                             0 1
                                2



                                    0
 Nonlinear Quantum Computing
Abrams & Lloyd 1998: We could solve NP-
complete problems efficiently given a 1-qubit
nonlinear gate that acts as follows:

                      1
                             0 1
                                2



                                    0
 Nonlinear Quantum Computing
Abrams & Lloyd 1998: We could solve NP-
complete problems efficiently given a 1-qubit
nonlinear gate that acts as follows:

                      1
                             0 1
                                2



                                    0
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.
Weinberg‘s) subject to small environmental error?
Abrams and Lloyd‘s 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!)
Conclusion: The Garden of Forking Paths




            P
       Determinism                        BPP
                                       Randomness



                                                /     2
        NP / PP
      Postselection                      BQP
                                      Quantumness
―Computers are useless. They only give you answers.‖ –Picasso

				
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