scott aaronson (mit) bqp and ph a tale of two strong-willed complexity classes… a 16-year-old...

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Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at last—but only for relational problems… The beast guarding the inner sanctum unmasked: the Generalized Linial-Nisan Conjecture… Where others flee in terror, a Braver Man attacks… A $200 bounty for slaughtering the wounded 1

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Page 1: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Scott Aaronson (MIT)

BQP and PHA tale of two strong-willed complexity

classes…A 16-year-old quest to find an oracle that

separates them…A solution at last—but only for relational

problems…The beast guarding the inner sanctum

unmasked: the Generalized Linial-Nisan Conjecture…

Where others flee in terror, a Braver Man attacks…

A $200 bounty for slaughtering the wounded beast…

1

Page 2: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Quantum Computing: Where Does It Fit?

PH

PBPP

AM

NP

P#P

BQP

2

Factoring, discrete log, etc.:In BQP

Not known to be in BPPBut in NPcoNP

Could there be a problem in BQP\PH?

Page 3: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

First question: can we at least find an oracle A such that BQPAPHA?

Essentially the same as finding a problem in quantum logarithmic time, but not AC0

Why? Well-known correspondence between relativized PH and AC0: interpret the ’s as OR gates, the ’s as AND gates, and the oracle string as an input of size 2n

Oracles are just the “obvious” way to address the BQP vs. PH question, not some woo-woo thing

Recall that the early evidence for BPP≠BQP (e.g. Simon’s alg) was also oracle evidence; then Shor found a similar oracle that could be “instantiated” by FACTORING

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Page 4: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

BQP vs. PH: A Timeline

Bernstein and Vazirani define BQP

They construct an oracle problem, RECURSIVE FOURIER SAMPLING, that has quantum query complexity n but classical query complexity n(log n) First example where quantum is superpolynomially better!

A simple extension yields RFSMA

Natural conjecture: RFSPH

Alas, we can’t even prove RFSAM!

19901995

20002005

2010

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Page 5: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Why do we care whether BQP PH?

Does simulating quantum mechanics reduce to search or approximate counting?

What other candidates for exponential quantum speedups are there—besides NP-intermediate problems like factoring?

Could quantum computers provide exponential speedups even if P=NP?

Would a fast quantum algorithm for NP-complete problems collapse the polynomial hierarchy?

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Page 6: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

This Talk

1. We achieve an oracle separation between the relational versions of BQP and PH (FBQP and FBPPPH)

2. We study a new oracle problem—FOURIER CHECKING—that’s in BQP, but not in BPP, MA, BPPpath, SZK...

3. We conjecture that FOURIER CHECKING is not in PH, and prove that this would follow from the Generalized Linial-Nisan ConjectureOriginal Linial-Nisan Conjecture was proved by Braverman 2009, after being open for 20 years

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Page 7: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Fourier Sampling ProblemGiven oracle access to a random Boolean function

1,11,0: nf

The Task:

Output strings z1,…,zn, at least 75% of which satisfy

and at least 25% of which satisfy

nx

xz

nxfzf

1,02/

12

1:ˆwhere

1ˆ izf

2ˆ izf

7

Page 8: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

FOURIER SAMPLING Is In BQP

Algorithm:

H

H

H

H

H

H

f

|0

|0

|0

Repeat n times; output whatever

you see

Distribution over Fourier coefficients

Distribution over Fourier coefficients output by quantum algorithm

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Page 9: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

FOURIER SAMPLING Is Not In PHKey Idea: Show that, if we had a constant-depth 2poly(n)-size circuit C for FOURIER SAMPLING, then we could violate a known AC0 lower bound, by “sneaking a MAJORITY problem” into the estimation of some random Fourier coefficient

Obvious problem: How do we know C will output the specific s we’re interested in, thereby revealing anything about ?

We don’t! (Indeed, there’s only a ~1/2n chance it will)

But we have a long time to wait, since our reduction can be nondeterministic! Just adds more layers to the AC0 circuit

Challenge: Show that w.h.p., C is forced to estimate eventually, even if it tries to avoid it

sf̂

sf̂

9

sf̂

Page 10: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Decision Version: FOURIER CHECKINGGiven oracle access to two Boolean functions

1,11,0:, ngf

Decide whether

if,g are drawn from the uniform distribution U, or

iif,g are drawn from the following “forrelated” distribution F: pick a random unit vector ,2nv

then let

xx vxgvxf ˆsgn:,sgn:

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Page 11: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

FOURIER CHECKING Is In BQP

H

H

H

H

H

H

f

|0

|0

|0

g

H

H

H

Probability of observing |0n:

forrelated are if1

random are if21

2

12

1,0,3 f,g

f,gygxf

n

yx

yx

nn

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Page 12: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Intuition: FOURIER CHECKING Shouldn’t Be In PH

Why?

• For any individual s, computing the Fourier coefficient is a #P-complete problem

• f and g being forrelated is an extremely “global” property: conditioning on a polynomial number of f(x) and g(y) values should reveal almost nothing about it

But how to formalize and prove that?

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sf̂

Page 13: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

1

Pr

Pr1

C

C

U

D

Crucial Definition: A distribution D is -almost k-wise independent if for all k-terms C,

Theorem: For all k, the forrelated distribution F is O(k2/2n/2)-almost k-wise independent

Proof: A few pages of Gaussian integrals, then a discretization step

A k-term is a product of k literals of the form xi or 1-xi

A distribution D over {0,1}N is k-wise independent if for all k-terms C,

kUD CC2

1PrPr

13

Approximation is multiplicative, not additive

… that’s important!

Page 14: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Bazzi’07 proved the depth-2 case

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Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us:

Let f:{0,1}n{0,1} be computed by a circuit of size and depth O(1). Then for all n(1)-wise independent distributions D,

.1PrPr1,0~

oxfxfnxDx

1

2on

“Generalized Linial-Nisan Conjecture”: Let f be computed by a circuit of size and depth O(1). Then for all 1/n(1)-almost n(1)-wise independent distributions D,

.1PrPr1,0~

oxfxfnxDx

1

2on

Razborov’08 dramatically simplified Bazzi’s proofFinally, Braverman’09 proved the whole thingAlas, we need the…

Page 15: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

“Low-Fat Sandwich Conjecture”: Let f:{0,1}n{0,1} be computed by a circuit of size and depth O(1). Then there exist polynomials pl,pu:RnR, of degree no(1), such that

15

xxpxfxp u

1E

1,0oxpxpu

x n

CCu

CC xCxpxCxp

Terms Terms

,

C

oCC

C

oCC nn 11 2,2

Theorem (Bazzi): Low-Fat Sandwich Conjecture Generalized Linial-Nisan Conjecture

(Without the low-fat condition, Sandwich Conjecture Linial-Nisan Conjecture)

1

2on

(i) Sandwiching.

(ii) Approximation.

(iii) Low-Fat. pl,pu can be written as

where

Page 16: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Known techniques for showing a function f has no small constant-depth circuits, also involve (directly or indirectly) showing that f isn’t approximated by a low-degree polynomial

But every function with a T-query quantum algorithm, is approximated by a degree-2T real polynomial! [Beals et al. 98]

Example: The following degree-4 polynomial distinguishes the uniform distribution over f,g from the forrelated one:

2

1,0,3

12

1,

nyx

yx

nygxfgfp

16

But this polynomial solves FOURIER CHECKING only by exploiting “massive cancellations” between positive and negative terms

(Not coincidentally, a central feature of quantum algorithms!)

Our conjecture says that if fAC0, then f is approximated not merely by a low-degree

polynomial, but by a “reasonable,” “classical-looking” one—with some bound on the

coefficients that prevents massive cancellations

Such a “low-fat” approximation of AC0 circuits would be useful for independent

reasons in learning theory

Page 17: Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at

Open ProblemsProve the Generalized Linial-Nisan Conjecture!Yields an oracle A such that BQPAPHA

Prove Generalized L-N even for the special case of DNFs.Yields an oracle A such that BQPAAMA

Is there a Boolean function f:{0,1}n{-1,1} that’s well-approximated in L2-norm by a low-degree real polynomial, but not by a low-degree low-fat polynomial?

Can we “instantiate” FOURIER CHECKING by an explicit (unrelativized) problem?

More generally, evidence for/against BQPPH in the real world?

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$100

$200