New Computational Insights from Quantum Optics

Scott Aaronson Based on joint work with Alex Arkhipov

The Extended Church Turing Thesis (ECT)

Everything feasibly computable in the physical world is feasibly computable by a (probabilistic) Turing machine But building a QC able to factor n>>143 is

damn

hard! Can’t CS “meet physics halfway” on this one?

I.e., show computational hardness in more easily-accessible quantum systems?

Also, factoring is a “special” problem

Det

Our Starting Point

   

S n

  sgn  

i n

  1

a i

,  All I can say is, the bosons In

P

Per  

i n

  

S n

1

a i

, 

#P

-complete [Valiant]

FERMIONS BOSONS

So if n-boson amplitudes correspond to permanents…

Can We Use Bosons to Calculate the Permanent?

That sounds way too good to be true—it would let us solve

NP

-complete problems and more using QC!

Explanation:

Amplitudes aren’t directly observable.

To get a reasonable estimate of Per(A), you might need to repeat the experiment exponentially many times

Basic Result:

Suppose there were a polynomial-time classical randomized algorithm that took as input a description of a noninteracting-boson experiment, and that output a

sample

over n-boson states.

from the correct final distribution

Then P #P = BPP NP and the polynomial hierarchy collapses.

Motivation:

Compared to (say) Shor’s algorithm, we get “stronger” evidence that a “weaker” system can do interesting quantum computations

Crucial step we take: switching attention to sampling problems P

#P P ERMANENT

SampBQP BQP BPP

NP

PH



X



Y … F ACTORING 3SAT

BPP SampP

Related Work

Valiant 2001, Terhal-DiVincenzo 2002, “folklore”:

A QC built of noninteracting fermions can be efficiently simulated by a classical computer

Knill, Laflamme, Milburn 2001:

plus

adaptive measurements

Noninteracting bosons yield universal QC

Jerrum-Sinclair-Vigoda 2001:

Fast classical randomized algorithm to approximate Per(A) for

nonnegative

A

Gurvits 2005:

O(n 2 /  2 ) classical randomized algorithm to approximate Per(A) to additive error  |A| n

A. 2011:

Generalization of Gurvits’s algorithm that gives better approximation if A has repeated rows or columns

Bremner, Jozsa, Shepherd 2011:

Commuting Hamiltonians

The Quantum Optics Model

A rudimentary subset of quantum computing, involving only

non-interacting bosons

, and

not

based on qubits

Classical counterpart:

Galton’s Board, on display at the Boston Museum of Science

Using only pegs and non interacting balls, you probably can’t build a universal computer— but you

can

do some interesting computations, like generating the binomial distribution!

The Quantum Version

Let’s replace the balls by

identical single photons

, and the pegs by

beamsplitters

Then we see strange things like the

Hong-Ou-Mandel dip

    1 1 2 2  1 2 1 2    The two photons are now correlated, even though they never interacted!

Explanation involves

destructive interference of amplitudes

: Final amplitude of non-collision is 1 2 1 2  1 2 1 2  0

Getting Formal

The basis states have the form |S  =|s 1 ,…,s m  , where s i is the number of photons in the i th “mode” We’ll never create or destroy photons. So s 1 +…+s m =n is constant.

For us, m=poly(n) Initial state: |I  =|1,…,1,0,……,0 

U

You get to apply any m  m unitary matrix U If n=1 (i.e., there’s only one photon, in a superposition over the m modes) , U acts on that photon in the obvious way

M

:   

m



n

 1   distribute n identical photons into m modes U induces an M  M unitary  (U) on the n-photon states as follows:   

S

,

T

 Per

s

1 !



s m

 

S

,

T

!

t

1 !



t m

!

Here U S,T is an n  n submatrix of U (possibly with repeated rows and columns) , obtained by taking s i row of U and t j copies of the j th copies of the i column for all i,j th

OK, so why is it hard to sample the distribution over photon numbers classically?

Given

any

matrix A  C n  n , we can construct an m  m unitary U (where m  2n) as follows:

U

   

A C D

  Suppose we start with |I  =|1,…,1,0,…,0  (one photon in each of the first n modes), apply U, and measure.

Then the probability of observing |I  again is 2

p

: 

I



I

  2

n

Per 2

Claim 1:

p is

#P

-complete to estimate (up to a constant factor)

Idea:

Valiant proved that the P ERMANENT is

#P

-complete.

Claim 2:

Suppose we had a fast classical algorithm for boson sampling. Then we could estimate p in

BPP NP

Can use a classical reduction to go from a multiplicative approximation of |Per(A)| 2 to Per(A) itself.

p

: 

I

  

I

2

Idea:

sampling algorithm, and let r be its randomness. Use approximate counting to estimate   Let M be our classical 2

n

Pr

r M

Per   outputs   for boson sampling. Then

P #P

=

BPP NP

.

2

I



The Elephant in the Room

The previous result hinged on the difficulty of estimating a

single, exponentially-small probability

p—but what about noise and error?

The “right” question:

can a classical computer efficiently sample a distribution with

1/poly(n) variation distance

from the boson distribution?

Our Main Result:

Suppose it can. Then there’s a

BPP NP

algorithm to estimate |Per(A)| 2 , with high probability over a Gaussian matrix

A

~

N

 

n



n C

Our Main Conjecture

Estimating |Per(A)| 2 , for most Gaussian matrices A, is a

#P

-hard problem If the conjecture holds, then even a

noisy

n-photon experiment could falsify the Extended Church Thesis, assuming

P #P



BPP NP

!

Most of our paper is devoted to giving evidence for this conjecture We can prove it if you replace “estimating” by “calculating,” or “most” by “all”

First step: understand the distribution of |Per(A)|

2

for random A

Conjecture:

There exist constants C,D and  >0 such that for all n and  >0,

X

~

N

Pr 1

n C



n

 Per    

n

!

 

Cn D

  Empirically true!

Also, we can prove it with determinant in place of permanent

The Equivalence of Sampling and Searching

[A., CSR 2011]

[A.-Arkhipov] gave a

“sampling problem”

solvable using quantum optics that seems hard classically—but does that imply anything about more traditional problems?

Recently, I found a way to convert any sampling problem into a

search problem

of “equivalent difficulty”

Basic Idea:

Given a distribution D, the search problem is to find a string x in the support of D with large

Kolmogorov complexity

Using Quantum Optics to Prove that the Permanent is

#P

-Hard

[A., Proc. Roy. Soc. 2011]

Valiant famously showed that the permanent is

#P

-hard— but his proof required strange, custom-made gadgets We gave a new, more transparent proof by combining three facts: (1) n-photon amplitudes correspond to n  n permanents (2) Postselected quantum optics can simulate universal quantum computation

[Knill-Laflamme-Milburn 2001]

(3) Quantum computations can encode

#P

-hard quantities in their amplitudes

Experimental Issues

Basically, we’re asking for the n-photon generalization of the Hong-Ou-Mandel dip, where n = big as possible Our results suggest that such a generalized HOM dip would be evidence against the Extended Church-Turing Thesis

Physicists:

“Sounds hard! But not as hard as full QC”

Remark:

If n is

too

large, a classical computer couldn’t even verify the answers! Not a problem yet though… Several experimental groups (Bristol, U. of Queensland) are currently working to do our experiment with 3 photons. Largest challenge they face: reliable single-photon sources

Summary

Thinking about quantum optics led to:

- A new experimental quantum computing proposal - New evidence that QCs are hard to simulate classically - A new classical randomized algorithm for estimating permanents - A new proof of Valiant’s result that the permanent is

#P

-hard - (Indirectly) A new connection between sampling and searching

Open Problems

Prove our main conjecture: that Per(A) is

#P

-hard to approximate for a matrix A of i.i.d. Gaussians ($1,000)!

Can our model solve classically-intractable

decision

problems?

Fault-tolerance in the quantum optics model?

Find more ways for quantum complexity theory to “meet the experimentalists halfway”

Bremner, Jozsa, Shepherd 2011:

QC with commuting Hamiltonians can sample hard distributions