Download BQP and the Polynomial Hierarchy

BQP und 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 beast…
Scott Aaronson
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Quantum Computing: Where Does It Fit?
P#P
PH
AM
NP
PP
BQP
BPP
Factoring, discrete log, etc.:
In BQP
Not known to be in BPP
But in NPcoNP
Could there be a
problem in BQP\PH?
P
2
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? Standard correspondence between relativized PH and
AC0: replace ’s by OR gates, ’s by AND gates, and the oracle
string by an input of size 2n
Relativization is just the “obvious” way to address the BQP vs.
PH question, not some woo-woo thing
People who claim they don’t like oracle results really just
don’t understand them
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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 RFSMA
Natural conjecture: RFSPH
Alas, we can’t even prove RFSAM!
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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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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 LinialNisan Conjecture
Original Linial-Nisan Conjecture was proved by Braverman 2009,
after being open for 20 years
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Relational Problems
FBPP: Class of relations, R{0,1}*{0,1}*, for which there
exists a BPP machine that, given any x, outputs a y such that
Prx, y  R  1  o1
FBQP: Same but with quantum
We’ll produce separations where the FBQP machine succeeds
with probability 1-1/exp(n), while the FBPPPH machine
succeeds with probability at most (say) 99%
Note: Amplification not obvious; constant could actually matter!
If we compared FBQP to FPPH, a separation would be trivial!
“Output an n-bit string with Kolmogorov complexity  n/2”
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Fourier Sampling Problem
Given oracle access to a random Boolean function
f : 0,1   1,1
n
The Task:
Output strings z1,…,zn, at least 75% of which satisfy
and at least 25% of which satisfy
where
ˆf z  : 1
n/2
2
fˆ zi   2
 1
z x
x0,1n
fˆ zi   1
f x 
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FOURIER SAMPLING Is In BQP
|0
Algorithm:
H
|0
H
|0
H
H
f
H
H
Repeat n times;
output whatever
you see
Distribution over Fourier coefficients
Distribution over Fourier coefficients
output by quantum algorithm
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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 fˆ s 
Obvious problem: How do we know C will output the
particular s we’re interested in, thereby revealing anything
about fˆ s  ?
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!
That just adds more layers to the AC0 circuit
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Starting Point for Reduction
Suppose each bit of an N-bit string is 1 with independent
probability p. Then any depth-d circuit to decide whether
p=½ or p=½+ (with constant bias) must have size

2
 1/  1 /  d 2 

If you’re here, you can prove this
We’ll take a circuit that outputs slightly-larger-than-average
Fourier coefficients of f, and get a circuit for detecting  bias
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The Fourier Guessing Game
Sends truth
f
Key Theorem:
table of f to Bob
Regardless of Bob’s strategy,

s,b secret
Keeps

e e
Prz  s   n1
f
2n e
Alice: Chooses s{0,1} and
In other
words,atif random
>1.1, Bob
b{0,1}
uniformly
outputs
then“true”
For each
x{0,1}
, sets s with
Bob: Must output a z
such that ˆ
f z   
probability noticeably more than
s x  b
n


1 to
avoid
1 it!
1/2 … even
he tries
1 if w.p.

f x  : 
2
2n / 2

 1 otherwise
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Finishing the Proof
Let A be a random oracle
View A as encoding a random Boolean function
fn:{0,1}n{-1,1} for each n
Let R be the relational problem where, on input 0n, you’re
asked to output z1,…,zn, at least 75% of which satisfy fˆn  zi   1
and at least 25% of which satisfy fˆn zi   2
Clearly


Pr R  FBQP  1
A
A
On the other hand, standard diagonalization tricks imply

Pr R  FBPP
A
PH A
 0
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Decision Version: FOURIER CHECKING
Given oracle access to two Boolean functions
f , g : 0,1   1,1
n
Decide whether
(i) f,g are drawn from the uniform distribution U, or
(ii) f,g are drawn from the following “forrelated” n
distribution F: pick a random unit vector v  2 ,
then let
f x  : sgn vx , g x  : sgn vˆx 
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FOURIER CHECKING Is In BQP
|0
H
|0
H
|0
H
H
f
H
H
H
g
H
H
Probability of observing |0n:
2
n



2
if f,g are random
1 
x y

f  x  1 g  y   

3n 

2  x , y0,1n
 1 if f,g are forrelated
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Intuition: FOURIER CHECKING
Shouldn’t Be In PH
Why?
• For any individual s, computing the Fourier
coefficient fˆ s  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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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,
1
PrD C   PrU C  
2
k
Crucial Definition: A distribution D is -almost k-wise
independent if for all k-terms C,
PrD C 
1  
 1 
PrU C 
Approximation is
multiplicative, not additive
… that’s important!
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
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Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us:
n o 1
Let f:{0,1}n{0,1} be computed by a circuit of size 2 and
depth O(1). Then for all n(1)-wise independent distributions D,
Pr  f x   Pr n  f x   o1.
x~ D
x0,1
Razborov’08
Finally,Bazzi’07
Braverman’09
dramatically
Alas,
proved
we need
proved
the
simplified
depth-2
the…
the whole
Bazzi’s
case thing
proof
“Generalized Linial-Nisan Conjecture”: Let f be computed
n o 1
by a circuit of size 2
and depth O(1). Then for all
1/n(1)-almost n(1)-wise independent distributions D,
Pr  f x   Pr n  f x   o1.
x~ D
x0,1
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n{0,1} be
“Low-Fat Sandwich Conjecture”:o 1Let
f:{0,1}
n 
computed by a circuit of size 2 and depth O(1). Then there
exist polynomials pl,pu:RnR, of degree no(1), such that
p x   f x   pu x  x
(i) Sandwiching.
(ii) Approximation.

pu x   p x   o1
x0,1
E
n
(iii) Low-Fat. pl,pu can be written as
p x  
 C x,
C
pu x  
Terms C
Theorem (Bazzi):
C
Terms C
C
o 1

2

n
,
where  C
C
  C x 
C
o 1

2

n
 C
C
Low-Fat Sandwich Conjecture
 Generalized Linial-Nisan Conjecture
(Without the low-fat condition, Sandwich Conjecture
 Linial-Nisan Conjecture)
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We know how to prove constant-depth lower bounds! So
why is BQPAPHA so much harder than (say) PPAPHA?
Because 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
And this is a problem because…
Lemma (Beals et al. 1998): Every Boolean function f that has
a T-query quantum algorithm, also has a degree-2T real
polynomial p such that |p(x)-f(x)| for all x{0,1}n
Example: The following degree-4 polynomial distinguishes the
uniform distribution over f,g from the forrelated one:
1
p f , g   3n
2


x y
  f  x  1 g  y 


n
 x , y0,1

2
20
But this polynomial solves FOURIER CHECKING only by exploiting
“massive cancellations” between positive and negative terms
(Not coincidentally, the central feature of quantum algorithms!)
You might conjecture that if fAC0, 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
And that’s exactly what the Low-Fat Sandwich Conjecture says!
0 circuits would
2 be
Such a “low-fat” approximation
of
AC


1
x y
useful forpindependent
inf learning

 f , g   reasons
x  1 theory
g  y 


2  x , y0,1n
3n


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Open Problems
Prove the Generalized Linial-Nisan Conjecture!
$200
Yields an oracle A such that BQPAPHA
Prove Generalized L-N even for the special case of DNFs.
$100
Yields an oracle A such that BQPAAMA
Is there a Boolean function f:{0,1}n{-1,1} that’s wellapproximated 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?
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