Verification of BosonSampling Devices

Scott Aaronson (MIT)Talk at Simons Institute, February 28, 2014

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, 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 have a fast classical algorithm! At any rate, it’s an extremely “special” problem

Wouldn’t it be great to show that if, quantum computers can be simulated classically, then (say) P=NP?

BosonSampling (A.-Arkhipov 2011)

A rudimentary type of quantum computing, involving only non-interacting photons

Classical counterpart: Galton’s Board

Replacing the balls by photons leads to famously counterintuitive phenomena,

like the Hong-Ou-Mandel dip

In general, we consider a network of beamsplitters, with n input “modes” (locations) and m>>n output modesn identical photons enter, one per input modeAssume for simplicity they all leave in different modes—there are possibilities

The beamsplitter network defines a column-orthonormal matrix ACmn, such that

nS

n

iiixX

1,Per

n

m

2PeroutcomePr SAS

where

is the matrix permanent

nn submatrix of A corresponding to S

So, Can We Use Quantum Optics to Solve a #P-Complete Problem?

Explanation: If X is sub-unitary, then |Per(X)|2 will usually be exponentially small. So to get a reasonable estimate of |Per(X)|2 for a given X, we’d generally need to repeat the optical experiment exponentially many times

That sounds way too good to be true…

Better idea: Given ACmn as input, let BosonSampling be the problem of merely sampling from the same distribution DA that the beamsplitter network samples from—the one defined by Pr[S]=|Per(AS)|2

Theorem (A.-Arkhipov 2011): Suppose BosonSampling is solvable in classical polynomial time. Then P#P=BPPNP

Better Theorem: Suppose we can sample DA even approximately in classical polynomial time. Then in BPPNP, it’s possible to estimate Per(X), with high probability over a Gaussian random matrix nn

CΝX 1,0~

Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence that a

weaker system can do something classically hard

We conjecture that the above problem is already #P-complete. If it is, then a fast classical algorithm for approximate BosonSampling would already have the consequence that P#P=BPPNP

BosonSampling Experiments

# of experiments ≥ # of photons!

Last year, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3- and 4-photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 and 4x4 permanents

Obvious challenge for scaling up: Need n-photon coincidences (requires either postselection or deterministic single-photon sources)Recent idea: Scattershot BosonSampling

Verifying BosonSampling DevicesCrucial difference from factoring: Even verifying the output of a claimed BosonSampling device would presumably take exp(n) time, in general!

Recently underscored by [Gogolin et al. 2013] (alongside specious claims…)

Our responses:(1)Who cares? Take n=30(2) If you do care, we can show how to distinguish the output of a BosonSampling device from all sorts of specific “null hypotheses”

Is a BosonSampling device’s output just uniform noise?

No way, not even close (A.-Arkhipov, arXiv:1309.7460)

Under the uniform distribution

Histogram of (normalized) probabilities under a Haar-random BosonSampling distribution

Theorem (A. 2013): Let ACmn be Haar-random, where m>>n. Then there’s a classical polytime algorithm C(A) that distinguishes the BosonSampling distribution DA from the uniform distribution U (whp over A, and using only O(1) samples)Strategy: Let AS be the nn submatrix of A corresponding to output S. Let P be the product of squared 2-norms of AS’s rows. If P>E[P], then guess S was drawn from DA; otherwise guess S was drawn from U

P under uniform distribution (a lognormal random variable)

P under a BosonSampling distributionA

AS

?22

1

n

n m

nvvP

Recent realization: You can use the number of multi-photon collisions to efficiently distinguish DA from EA

Given a matrix, A, let EA be like the BosonSampling distribution DA, but with distinguishable particles:

Observe that the row-norm estimator, P, fails completely to distinguish DA from EA! (Why?)

2### ,PeroutcomePrijSijSS

EaaAS

A

Conjecture: Could also distinguish without looking at collisions

The Classical Mockup ChallengeGiven a matrix ACmn, is there some classically efficiently-samplable distribution CA, which is indistinguishable from the BosonSampling distribution DA by any polynomial-time algorithm?

Observation: If we just wanted an efficiently-samplable distribution that’s indistinguishable from DA by any (say) n2-time algorithm, that’s trivial to get!

Brandao: We can even get such a mockup distribution with a large min-entropy, using Trevisan-Tulsiani-Vadhan

The NP ChallengeCan our linear-optics model solve a classically-intractable problem (say, a search or decision problem) for which a classical computer can efficiently verify the answer?

Given an nn matrix with large (1/poly(n)) permanent, can one “smuggle” it as a submatrix of a unitary matrix?

What kinds of (sub)unitary matrices can have ≥1/poly(n) permanents? Must every such matrix be “close to the identity,” in some sense?

Arkhipov: Every unitary with permanent ≥1-1/e has a “large” diagonal

The Interactive Protocol ChallengeCan a BosonSampling device convince a classical skeptic of its post-classical powers via an interactive protocol?

Arora et al. 2012: An oracle for Gaussian permanent estimation would be self-checkable. (But alas, a BosonSampling device is not such an oracle!)