New Computational Insights from Quantum Optics Scott Aaronson Based on joint work with Alex Arkhipov

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?