)New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers(

אהרונסון סקוט)Scott Aaronson(

MIT

את לדמות שקשה חדשות עדויותמחשבים עם הקוונטים מכניקת

קלאסיים

In 1994, something big happened in the foundations of computer science, whose meaning

is still debated today…

Why exactly was Shor’s algorithm important?

Boosters: Because it means we’ll build QCs!

Skeptics: Because it means we won’t build QCs!

Me: For reasons having nothing to do with building QCs!

Shor’s algorithm was a hardness result for one of the central computational problems

of modern science: QUANTUM SIMULATION

Shor’s Theorem:

QUANTUM SIMULATION is not in

probabilistic polynomial time,

unless FACTORING is also

Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

Advantages of the new results:

Based on “generic” complexity assumptions, rather than the classical hardness of FACTORING

Give evidence that QCs have capabilities outside the entire polynomial hierarchy

Use only extremely weak kinds of QC (e.g. nonadaptive linear optics)—testable before I’m dead?

Today: New kinds of hardness results for simulating quantum mechanics

Disadvantages:

Most apply to sampling problems (or problems with many possible valid outputs), rather than decision problems

Harder to convince a skeptic that your QC is indeed solving the relevant hard problem

Problems not “useful” (?)

There exist black-box sampling and relational problems in BQP that are not in BPPPH

Assuming the “Generalized Linial-Nisan Conjecture,” there exists a black-box decision problem in BQP but not in PH

Original Linial-Nisan Conjecture was recently proved by Braverman, after being open for 20 years

Results (from arXiv:0910.4698)

Unconditionally, there exists a black-box decision problem that requires (N) queries classically ((N1/4) even using postselection), but only O(1) queries quantumly

Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then P#P=BPPNP, and hence PH collapses.

Indeed, even if such a distribution can be sampled in BPPPH, still PH collapses.

Suppose the output distribution of any linear-optics circuit can even be approximately sampled in BPP. Then a BPPNP machine can approximate the permanent of a matrix of independent N(0,1) Gaussians.

Conjecture: The above problem is #P-complete.

Results (from recent joint work with Alex Arkhipov)

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!

1990

1995

2000

2005

2010

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

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

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 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

sf̂

sf̂

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:

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

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: no polynomial number of f(x) and g(y) values should reveal much of anything about it

But how to formalize and prove that?

sf̂

1

Pr

Pr1

C

C

U

D

Key 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

Approximation is multiplicative, not additive

… that’s important!

Bazzi’07 proved the depth-2 case

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…

Coming back to our result for relational problems: what was surprising was that we showed hardness of a BQP sampling problem, using a nondeterministic reduction from MAJORITY—a “#P” problem!

This raises a question: is something similar possible in the unrelativized (non-black-box) world?

Indeed it is. Consider the following problem:

QSAMPLING: Given a quantum circuit C, which acts on n qubits initialized to the all-0 state. Sample from C’s output distribution.

Suppose QSAMPLINGBPP. Then P#P=BPPNP

(so in particular, PH collapses to the third level)

Result/Observation:

Why QSAMPLING Is Hard

2

1,022

1:

nxn

xfp

Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit:

H

H

H

H

H

H

f

|0

|0

|0

Then the probability of observing the all-0 string is

Claim 1: p is #P-hard to estimate (up to a constant factor)

Related to my result that PostBQP=PP

Proof: If we can estimate p, then we can also compute xf(x) using binary search and padding

Claim 2: Suppose QSAMPLINGBPP. Then we could estimate p in BPPNP

Proof: Let M be a classical algorithm for QSAMPLING, and let r be its randomness. Use approximate counting to estimate

Conclusion: Suppose QSAMPLINGBPP. Then P#P=BPPNP

nr

rM 0 outputs Pr

2

1,022

1:

nxn

xfp

Ideally, we want a simple, explicit quantum system Q, such that any classical algorithm that even approximately simulates Q would have dramatic consequences for classical complexity theory

We believe this is possible, using non-interacting bosons

nS

n

iiiaA

1,Per

BOSONS

nS

n

iiiaA

1,

sgn1Det

FERMIONS

There are two basic types of particle in the universe…

Their transition amplitudes are given respectively by…

All I can say is, the bosons got the harder job…

U

Our Result: Take a system of n identical photons, with m=O(n2) modes (basis states) each. Put each photon in a known mode, then apply a random mm scattering matrix U:

Let D be the distribution that results from measuring the photons. Suppose there’s an efficient classical algorithm that samples any distribution even 1/poly(n)-close to D in variation distance. Then in BPPNP, one can estimate the permanent of a matrix X of i.i.d. N(0,1) complex Gaussians, to additive error with high probability over X.

,!1On

n

Conjecture: This problem is #P-complete

The Permanent of Gaussians Conjecture (PGC)Given a matrix X of i.i.d, N(0,1) complex Gaussians, it is #P-complete to approximate Per(X) to within with 1-1/poly(n) probability over X ,poly

!

n

n

“But isn’t the permanent easy to approximate, by Jerrum-Sinclair-Vigoda?”

Yes—for nonnegative matrices. For general matrices, can get huge cancellations between positive and negative terms, and indeed even approximating the permanent is #P-complete in the worst case

Intuition for PGC: We know computing the permanent of a random matrix is #P-complete—over finite fields. “Merely” need to extend that result to the reals or complex numbers!

Basic difficulty: When doing LFKN-style interpolation, errors in the permanent estimates can blow up exponentially

PGCHardness of BOSONSAMPLINGIdea: Given a Gaussian random matrix X, we’ll “smuggle” X into the unitary transition matrix U for m=O(n2) bosons

Useful fact we rely on: given a Haar-random mm unitary matrix, an nn submatrix looks approximately Gaussian

!!

Per:

1

2

m

SS ss

Up

Neat Fact: The pS’s sum to 1

where US is an nn matrix containing si copies of the ith row of U (first n columns only)

Suppose that initially, modes 1,…,n contain one boson each while modes n+1,…,m are unoccupied. Then after applying U, we observe a configuration (list of occupation numbers) s1,…,sm, with probability

Problem: Bosons like to pile on top of each other!

Call a configuration S=(s1,…,sm) good if every si is 0 or 1 (i.e.,

there are no collisions between bosons), and bad otherwise

If bad configurations dominated, then our sampling algorithm might “work,” without ever having to solve a hard PERMANENT instance

Furthermore, the “bosonic birthday paradox” is even worse than the classical one!

,3

2box same in the land particlesboth Pr

rather than ½ as with classical particles

Fortunately, we show that with n bosons and mkn2 boxes, the probability of a collision is still at most (say) ½

Experimental ProspectsWhat would it take to implement the requisite experiment with photonics?• Reliable phase-shifters and beamsplitters, to implement a Haar-random unitary on m photon modes• Reliable single-photon sources• Reliable photodetector arraysBut crucially, no nonlinear optics or postselected measurements!

Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible

Prize 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

Prove the Permanent of Gaussians Conjecture!Would imply that even approximate classical simulation of linear-optics circuits would collapse PH

$100

$200

NIS500

More Open Problems (no prizes)

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

Can we use BOSONSAMPLING to solve any decision problem outside BPP?

Can you convince a skeptic (who isn’t a BPPNP machine) that your QC is indeed doing BOSONSAMPLING?

Can we get unlikely classical complexity consequences from P=BQP or PromiseP=PromiseBQP?

SummaryI like to say that we have three choices: either

(1)The Extended Church-Turing Thesis is false,

(2)Textbook quantum mechanics is false, or

(3)QCs can be efficiently simulated classically.For all intents and purposes?