Scott Aaronson (MIT)

1
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 lastbut 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

• Scott Aaronson (MIT)

2
Quantum Computing Where Does It Fit?
PP
Factoring, discrete log, etc. In BQP Not known
to be in BPP But in NP?coNP
PH
AM
NP
Could there be a problem in BQP\PH?
BQP
BPP
P
3
First question can we at least find an oracle A
such that BQPA?PHA?
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. Simons alg) was also oracle evidence then
Shor found a similar oracle that could be
instantiated by Factoring
4
BQP vs. PH A Timeline
1990
1995
2000
2005
2010
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 RFS?MA Natural conjecture
RFS?PH Alas, we cant even prove RFS?AM!
5
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
therebesides NP-intermediate problems like
factoring? Could quantum computers provide
exponential speedups even if PNP? Would a fast
quantum algorithm for NP-complete problems
collapse the polynomial hierarchy?
6
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 problemFourier
   Checkingthats 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

7
Fourier Sampling Problem
Given oracle access to a random Boolean function
The Task Output strings z1,,zn, at least 75 of
which satisfy and at least 25 of which satisfy
where
8
Fourier Sampling Is In BQP
Repeat n times output whatever you see
Algorithm
Distribution over Fourier coefficients
Distribution over Fourier coefficients output by
quantum algorithm
9
Fourier Sampling Is Not In PH
Key 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 were interested in, thereby revealing
anything about ? We dont! (Indeed,
theres 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
10
Decision Version Fourier Checking
Given oracle access to two Boolean functions

• Decide whether
• ?f,g? are drawn from the uniform distribution U,
  or
• ?f,g? are drawn from the following forrelated
  distribution F pick a random unit vector

then let
11
Fourier Checking Is In BQP
H
H
0?
H
H
H
f
0?
g
H
H
H
0?
H
Probability of observing 0??n
12
Intuition Fourier Checking Shouldnt 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?

13
A k-term is a product of k literals of the form
xi or 1-xi
A distribution D over 0,1N is k-wise
independent if for all k-terms C,
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
14
Bazzi07 proved the depth-2 case
Razborov08 dramatically simplified Bazzis proof
Finally, Braverman09 proved the whole thing
Alas, we need the
15
Low-Fat Sandwich Conjecture Let f0,1n?0,1
be computed by a circuit of size and
depth O(1). Then there exist polynomials
pl,puRn?R, of degree no(1), such that
(i) Sandwiching.
(ii) Approximation.
(iii) Low-Fat. pl,pu can be written as
where
Theorem (Bazzi) Low-Fat Sandwich
Conjecture ? Generalized Linial-Nisan
Conjecture (Without the low-fat
condition, Sandwich Conjecture ? Linial-Nisan
Conjecture)
16
Known techniques for showing a function f has no
small constant-depth circuits, also involve
(directly or indirectly) showing that f isnt
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
Our conjecture says that if f?AC0, then f is
approximated not merely by a low-degree
polynomial, but by a reasonable,
classical-looking onewith 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
But this polynomial solves Fourier Checking only
by exploiting massive cancellations between
positive and negative terms (Not coincidentally,
a central feature of quantum algorithms!)
17
Open Problems
Prove the Generalized Linial-Nisan
Conjecture!Yields an oracle A such that
BQPA?PHA Prove Generalized L-N even for the
special case of DNFs.Yields an oracle A such
that BQPA?AMA Is there a Boolean function
f0,1n?-1,1 thats 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 BQP?PH in the real world?