The Computational Complexity of Linear Optics

Scott Aaronson and Alex ArkhipovMIT

vs

Shor’s Theorem: QUANTUM SIMULATION has no efficient classical algorithm, unless FACTORING does

also

The Extended Church-Turing Thesis (ECT)

Everything feasibly computable in the physical

world is feasibly computable by a (probabilistic) Turing

machine

So the ECT is false … what more evidence could anyone want?

Building a QC able to factor large numbers is damn hard! After 16 years, no fundamental obstacle has been found (or even seriously proposed), but who knows?

Can’t we “meet the physicists halfway,” and show computational hardness for quantum systems closer to what they actually work with now?

FACTORING might be in BPP! At any rate, it’s an extremely “special” problem

Wouldn’t it be great to show that if BPP=BQP, then (say) the polynomial hierarchy collapses?

We define a model of computation based on linear optics: n identical photons traveling through a network of poly(n) beamsplitters, phase-shifters, etc., then a measurement of where the photons ended up

Crucial point: No entangling interactions between pairs of photons needed!

Today: “A New Attack on the ECT”

Our model is contained in BQP, but seems unlikely to be BQP-complete. We don’t know if it solves any decision problems that are hard classically.

But for sampling and search problems, the situation is completely different…

Theorem 1. Suppose that for every linear-optics network, the probability distribution over measurement outcomes can be sampled in classical polynomial time. Then P#P=BPPNP (so PH collapses)

More generally, let O be any oracle that simulates a linear-optics network A, given a description of A and a random string r. Then

So even if linear optics can be simulated in BPPPH, that still collapses PH! (New evidence that QCs have capabilities beyond PH, complementing [A’10],[FU’10])

ONPP BPPP #

“OK, but isn’t the real question the hardness of approximate sampling? After all, experiments are noisy, and not even the linear-optics network

itself can sample exactly!”

Theorem 2. Suppose two plausible conjectures are true: the permanent of a Gaussian random matrix is

(1) #P-hard to approximate, and

(2) not too concentrated around 0.

Let O be any oracle takes as input a description of a linear-optics network A, a random string r, and 01/, and that samples from a distribution -close to A’s in variation distance. Then ONPP BPPP #

In other words: if our conjectures hold, then even simulating noisy linear-optics experiments is

classically intractable, unless PH collapses

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

Particle Physics In One Slide

Indeed, [Valiant 2002, Terhal-DiVincenzo 2002] showed that noninteracting fermion systems can be simulated in BPP. But, confirming Avi’s joke,

we’ll argue that the analogous problem for bosons (such as photons) is much harder…

Linear Optics for DummiesWe’ll be considering a special kind of quantum computer, which is not based on qubits

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 if there are n photons, then s1,…,sm are nonnegative integers summing to n

Initial state: |I=|1,…,1,0,……,0

For us, m=poly(n)

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

U

Example: The “Hong-Ou-Mandel Dip”

Suppose

0Per2 U

.11

11

2

1

U

Then Pr[the two photons land in different modes] is

Pr[they both land in the first mode] is

2

1

11

11

2

1Per

2!

12

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 linear-optics 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 linear-optics sampling. Then P#P=BPPNP.

IrMr

outputs Pr

222Per: AIUIp n

As I said before, I find this result unsatisfying, since it only talks about the classical hardness of exactly sampling the distribution over photon numbers

Difficulty: The sampler might adversarially refuse to output the one submatrix whose permanent we care about! That changes the output distribution by only exp(-n), so we still have an excellent sampler … but we can no longer use it to estimate |Per(A)|2 in BPPNP

What about sampling a distribution that’s 1/poly(n)-close in variation distance?

To get around this difficulty, it seems we need to “smuggle in” the matrix A that we about as a random submatrix of U

U

Consider applying a Haar-random mm unitary matrix U, to n photons in m=poly(n) modes:

Main Result

Suppose there’s a classical algorithm to sample a distribution -close to DU in poly(n,1/) time. Then for all ,1/poly(n), there’s also a BPPNP algorithm to estimate |Per(X)|2 to within additive error n!, with probability 1- over a Gaussian random matrix

Distribution DU over photon numbers

nnCNX

1,0~

Main technical lemma used in proof:

Let mn6. Then an nn submatrix of an mm Haar unitary matrix is Õ(1/n)-close in variation distance to a matrix of independent Gaussians.

So the question boils down to this: how hard is it to additively estimate |Per(X)|2, with high probability over a Gaussian random matrix ?1,0~

nnCNX

We conjecture that it’s #P-hard—in which case, even approximate classical simulation of our linear-optics experiment would imply P#P=BPPNP

We can decompose this conjecture into two plausible sub-conjectures: that multiplicatively estimating Per(X) is #P-hard for Gaussian X, and that Per(X) is “not too concentrated around 0”

The following problem is #P-hard. Given a matrix XCnn of i.i.d. Gaussian entries, together with 01/ and 01/, output an approximation z such that

The Permanent-of-Gaussians Conjecture (PGC)

.1PerPerPr

1,0~

XXz

nnCNX

We can prove #P-hardness if =0 or =0. So what makes the PGC nontrivial is really the

combination of average-case with approximation

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

The Permanent Anti-Concentration Conjecture (PACC)

D

NXCnnX

nnC

!PerPr1,0~

Empirically true!

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

Experimental ProspectsIt seems well within current technology to do our experiment with (say) n=4 photons and m=20 modes

(Current record: n=2 photons)

If you can scale to n photons and error in variation distance, using “poly(n,1/) experimental effort,” then modulo our complexity conjectures, the ECT is false

What would it take to scale to (say) n=20 photons and m=500 modes?- Reliable single-photon sources (standard laser isn’t good enough!)- Reliable photodetector arrays- Stable apparatus to ensure that w.h.p., all n photons arrive at photodetector arrays at exactly the same time

Physicists we consulted: “Sounds hard! But not as hard as building a universal QC”

Remark: No point in scaling this experiment much beyond 20 or 30 photons, since then a

classical computer can’t even verify the answers!

Open Problems

Similar hardness results for other natural quantum systems (besides linear optics)?

Bremner, Jozsa, Shepherd 2010: Another system for which exact classical simulation would collapse PH

Can our linear-optics model solve classically-intractable decision problems? What about problems for which a classical computer can verify the answers?

Do BPP=BQP or PromiseBPP=PromiseBQP have interesting structural complexity consequences?

Prove the PGC ($200) and PACC ($100)!