Verification of BosonSampling
Devices
Scott Aaronson (MIT)
Talk at Simons Institute, February 28, 2014
The Extended ChurchTuring Thesis (ECT)
Everything feasibly
computable in the physical
world is feasibly computable
by a (probabilistic) Turing
machine
Shor’s Theorem: QUANTUM SIMULATION has no efficient
classical algorithm, unless FACTORING does also
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 modes
n identical photons enter, one per input mode
Assume for simplicity they all leave in
different modes—there are  m  possibilities
n
 
The beamsplitter network defines a column-orthonormal
matrix ACmn, such that Pr outcome S  Per  A  2

where
Per X  

n
x  


S n i 1
is the matrix permanent
i,
i
S
nn submatrix of A
corresponding to S
So, Can We Use Quantum Optics to
Solve a #P-Complete Problem?
That sounds way too good to be true…
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
Better idea: Given ACmn 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
Upshot: Compared to (say) Shor’s factoring
Better Theorem: Suppose we can sample DA even
algorithm, we get different/stronger evidence that a
approximately in classical polynomial time. Then in
weaker system can do something classically hard
NP
BPP , it’s possible to estimate Per(X), with high
nn
probability over a Gaussian random matrix X ~ Ν 0,1C
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
Last year, groups in Brisbane,
Oxford, Rome, and Vienna
reported the first 3- and 4photon BosonSampling
experiments, confirming that
the amplitudes were given by
3x3 and 4x4 permanents
# of experiments ≥ # of photons!
Obvious challenge for scaling up: Need n-photon
coincidences (requires either postselection or
deterministic single-photon sources)
Recent idea: Scattershot BosonSampling
Verifying BosonSampling Devices
Crucial 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)
Histogram of (normalized)
probabilities under a Haarrandom BosonSampling
distribution
Under the uniform
distribution
Theorem (A. 2013): Let ACmn 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 nn 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)
AS
P under a BosonSampling
distribution
A
n
P  v1  vn
2
2
n
  ?
m
Given a matrix, A, let EA be like the BosonSampling
distribution DA, but with distinguishable particles:
Proutcome S   Per A ,
EA
#
S
a 
#
S ij
a
# 2
S ij
Observe that the row-norm estimator, P, fails
completely to distinguish DA from EA! (Why?)
Recent realization: You can use the number of multiphoton collisions to efficiently distinguish DA from EA
Conjecture: Could also distinguish without
looking at collisions
The Classical Mockup Challenge
Given a matrix ACmn, 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 Challenge
Can 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 nn 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 Challenge
Can 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!)