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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…

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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?


Factoring, discrete log, etc.:


Not known to be in BPP

But in NPcoNP




Could there be a problem in BQP\PH?





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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

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

  • We study a new oracle problem—Fourier Checking—that’s in BQP, but not in BPP, MA, BPPpath, SZK...

  • 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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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

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

The Task:

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

and at least 25% of which satisfy


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Fourier Sampling Is In BQP




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

Obvious problem: How do we know C will output the particular 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!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

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 table of f to Bob

Keeps s,b secret

Key Theorem:

Regardless of Bob’s strategy,

Alice: Chooses s{0,1}n and b{0,1} uniformly at random

Bob: Must output a z such that

For each x{0,1}n, sets

In other words, if >1.1, Bob outputs the “true” s with probability noticeably more than 1/2n … even if he tries to avoid it!

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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 satisfyand at least 25% of which satisfy


On the other hand, standard diagonalization tricks imply

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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

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Fourier Checking Is In BQP















Probability of observing |0n:

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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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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,

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

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,

Razborov’08 dramatically simplified Bazzi’s proof

Finally, Braverman’09 proved the whole thing

Bazzi’07 proved the depth-2 case

Alas, we need the…

“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,

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“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:RnR, of degree no(1), such that

(i) Sandwiching.

(ii) Approximation.

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


Theorem (Bazzi): 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:

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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 cancellationsAnd that’s exactly what the Low-Fat Sandwich Conjecture says!

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

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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 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 BQPPH in the real world?