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

DUNCE

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

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

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 it—quantum 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

The Popularizers Have Spoken

[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!

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

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

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

Exactly what property separates the Sure States we know we can prepare, from the

Shor States that suffice for factoring?

DIV

IDIN

G L

INE

Crucial Question for Me

I hereby propose a complexity theory of

pure quantum

states

one of whose goals is to

study possible

Sure/Shor separators.

2nH

Classical

Vidal

Circuit

AmpP

MOTree

OTree

TSH

Tree

P

1

2

1

2

Strict containmentContainmentNon-containment

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:

+

|11 |12

++

|01 |11 |02 |12

1

2

1

2

1

2

1

2

11

| has tree size 6

100 01 10 11

2

Actual Technical ResultA., quant-ph/0311039

If C = {x | Axb(mod 2)}, where A is chosen uniformly at random from then with high probability

requires trees of size nclog n even to approximate well

Proof uses recent breakthrough of Ran Raz

Conjecture: Same lower bound holds for states arising in Shor’s algorithm

1

x C

C xC

1/ 3

2 ,n n

Codewords of random stabilizer codes have superpolynomial tree size

Recent (as in last week) Developments2-D cluster states (as proposed by Briegel and Raussendorf) have tree size nclog n. Not true for 1-D

Explicit (non-random) coset states, obtained by concatenating Reed-Solomon and Hadamard codes, have tree size nclog 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

NOTE FOR PHYSICISTS:

I only care about qubit

states

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 PNP, date a supermodel, etc., is it enough for it to be possible that you achieve these things?

BBBV’97 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

1|1 2|2 3|3 4|4 5|5 6|6 7|7 8|8 9|9

DUDE!!! Everyone! The marked item! Over here!!!

Someone must be screaming about a marked

item… too bad quantum mechanics is linear

Actual Technical Result IIA., 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 IdeaCan 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

THAT’S IT?

Nonlinear Quantum Computing

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

0

10 1

2

Nonlinear Quantum Computing

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

0

10 1

2

Nonlinear Quantum Computing

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

0

10 1

2

Nonlinear Quantum Computing

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

0

10 1

2

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

Determinism Randomness

Postselection Quantumness

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“Computers are useless. They only give you answers.” –Picasso

P BPP

NP / PP BQP