New Computational Insights from Quantum Optics

Scott AaronsonBased 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>>15 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

nS

n

iiiaA

1,Per

BOSONS

nS

n

iiiaA

1,

sgn1Det

FERMIONS

Our Starting Point

In P #P-complete [Valiant]All I can say is, the bosons

got the harder job

Can We Use Bosons to Calculate the Permanent?

Explanation: Amplitudes aren’t directly observable.To get a reasonable estimate of Per(A), you might need to repeat the experiment exponentially many times

That sounds way too good to be true—it would let us solve NP-complete problems and more using QC!

So if n-boson amplitudes correspond to permanents…

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 from the correct final distribution over n-boson states.Then P#P=BPPNP 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

BQP

P#P

BPP

BPPNP

PH

FACTORING

PERMANENT

3SAT

XY…

SampP

SampBQP

Valiant 2001, Terhal-DiVincenzo 2002, “folklore”: A QC built of noninteracting fermions can be efficiently simulated by a classical computer

Related Work

A. 2011: Generalization of Gurvits’s algorithm that gives better approximation if A has repeated rows or columns

Knill, Laflamme, Milburn 2001: Noninteracting bosons plus adaptive measurements yield universal QC

Jerrum-Sinclair-Vigoda 2001: Fast classical randomized algorithm to approximate Per(A) for nonnegative A

The Quantum Optics ModelA 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!

2

1

2

12

1

2

1

The Quantum VersionLet’s replace the balls by identical single photons,

and the pegs by beamsplitters

Then we see strange things like the Hong-Ou-Mandel dip

The two photons are now correlated, even

though they never interacted!

Explanation involves destructive interference of amplitudes:Final amplitude of non-collision is

02

1

2

1

2

1

2

1

Getting Formal The basis states have the form |S=|s1,…,sm, where si is the number of photons in the ith “mode”

We’ll never create or destroy photons. So s1+…+sm=n is constant.

Initial state: |I=|1,…,1,0,……,0

For us, m=poly(n)

U

You get to apply any mm 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

n

nmM

1:

In general, there are ways to distribute n identical photons into m modes

U induces an MM unitary (U) on the n-photon states as follows:

!!!!

PerU

11

,,

mm

TSTS

ttss

U

Here US,T is an nn submatrix of U (possibly with repeated

rows and columns), obtained by taking si copies of the ith row of U and tj copies of the jth column for all i,j

Beautiful Alternate PerspectiveThe “state” of our computer, at any time, is a degree-n polynomial over the variables x=(x1,…,xm) (n<<m)

Initial state: p(x) := x1xn

We can apply any mm unitary transformation U to x, to obtain a new degree-n polynomial

m

m

ssS

sm

sS xxUxpxp

,,1

1

1'

Then on “measuring,” we see the monomialwith probability !!1

2

mS ss

msm

s xx 11

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

222Per: AIUIp n

Given any matrix ACnn, we can construct an mm unitary U (where m2n) as follows:

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

DC

BAU

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

Idea: Valiant proved that the PERMANENT is #P-complete.

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

Claim 2: Suppose we had a fast classical algorithm for boson sampling. Then we could estimate p in BPPNP

Idea: Let M be our classical sampling algorithm, and let r be its randomness. Use approximate counting to estimate

Conclusion: Suppose we had a fast classical algorithm for boson sampling. Then P#P=BPPNP.

IrMr

outputs Pr

222Per: AIUIp n

The previous result hinged on the difficulty of estimating a single, exponentially-small probability p—but what about noise and error?

The Elephant in the Room

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

CNA

1,0~

Estimating |Per(A)|2, for most Gaussian matrices A, is a #P-hard problem

Our Main Conjecture

We can prove it if you replace “estimating” by “calculating,” or “most” by “all”

If the conjecture holds, then even a noisy n-photon experiment could falsify the Extended Church Thesis, assuming P#PBPPNP!

Most of our paper is devoted to giving evidence for this conjecture

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

First step: understand the distribution of |Per(A)|2 for random A

D

NXCnnX

nnC

!PerPr1,0~

Empirically true!

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

Conjecture:

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 nn 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 IssuesBasically, 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 (Imperial College, U. of

Queensland) are currently working to do our experiment with 3 photons. Largest challenge they

face: reliable single-photon sources

SummaryThinking 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

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

Bremner, Jozsa, Shepherd 2011: QC with commuting Hamiltonians can sample hard distributions

Can our model solve classically-intractable decision problems?

Prove our main conjecture: that Per(A) is #P-hard to approximate for a matrix A of i.i.d. Gaussians ($1,000)!

Fault-tolerance in the quantum optics model?