BosonSampling

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BosonSampling

• Scott Aaronson (MIT)
• Talk at SITP, February 21, 2014

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The Extended Church-Turing Thesis (ECT)
Everything feasibly computable in the physical
world is feasibly computable by a (probabilistic)
Turing machine
Shors Theorem Quantum Simulation has no
efficient classical algorithm, unless Factoring
does also
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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? Cant 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, its an extremely
special problem Wouldnt it be great to show
that if, quantum computers can be simulated
classically, then (say) PNP?
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BosonSampling (A.-Arkhipov 2011)

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

Classical counterpart Galtons Board
Replacing the balls by photons leads to famously
counterintuitive phenomena, like the
Hong-Ou-Mandel dip
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• In general, we consider a network of
  beamsplitters, with n input modes (locations)
  and mgtgtn output modes

n identical photons enter, one per input
mode Assume for simplicity they all leave in
different modesthere are possibilities
The beamsplitter network defines a
column-orthonormal matrix A?Cm?n, such that
where
is the matrix permanent
For simplicity, Im ignoring outputs with 2
photons per mode
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Example
For Hong-Ou-Mandel experiment,
In general, an n?n complex permanent is a sum of
n! terms, almost all of which cancel How hard is
it to estimate the tiny residue left
over? Answer P-complete, even for
constant-factor approx (Contrast with
nonnegative permanents!)
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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,
wed generally need to repeat the optical
experiment exponentially many times
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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 fromthe one defined
by PrSPer(AS)2
Theorem (A.-Arkhipov 2011) Suppose BosonSampling
is solvable in classical polynomial time. Then
PPBPPNP
Upshot Compared to (say) Shors factoring
algorithm, we get different/stronger evidence
that a weaker system can do something classically
hard
Better Theorem Suppose we can sample DA even
approximately in classical polynomial time. Then
in BPPNP, its possible to estimate Per(X), with
high probability over a Gaussian random matrix
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 PPBPPNP
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Related Work
Valiant 2001, Terhal-DiVincenzo 2002, folklore
A QC built of noninteracting fermions can be
efficiently simulated by a classical computer
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
Gurvits 2002 Fast classical randomized algorithm
to approximate n-photon amplitudes to ? additive
error
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BosonSampling Experiments
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
of experiments of photons!
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• Obvious Challenges for Scaling Up
• Reliable single-photon sources (optical
  multiplexing?)
• Minimizing losses
• Getting high probability of n-photon coincidence

• Goal (in our view) Scale to 10-30 photons
• Dont want to scale much beyond thatboth because
• you probably cant without fault-tolerance, and
• a classical computer probably couldnt even
  verify the results!

Theoretical Challenge Argue that, even with
photon losses and messier initial states, youre
still solving a classically-intractable sampling
problem
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Scattershot BosonSampling
Wonderful new idea, proposed by several
experimental groups, for sampling a hard
distribution even with highly unreliable (but
heralded) photon sources, like Spontaneous
Parametric Downconversion (SPDC) crystals The
idea Say you have 100 sources, of which only 10
(on average) generate a photon. Then just detect
which sources succeeded, and use those to define
your BosonSampling instance! Complexity analysis
goes through essentially without change
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Verifying BosonSampling Devices

• As mentioned before, 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
• Who cares? Take n30
• If you do care, we can show how to distinguish
  the output of a BosonSampling device from all
  sorts of specific null hypotheses

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Theorem (A. 2013) Let A?Cm?n be Haar-random,
where mgtgtn. Then theres 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 ASs rows. If PgtEP, then
guess S was drawn from DA otherwise guess S was
drawn from U
Recent realization You can also use the number
of multi-photon collisions to distinguish DA from
DA, the same distribution but with classical
distinguishable particles
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Turning the Logic AroundA., Proc. Roy. Soc.
2011
Arkhipov and I used the P-completeness of the
permanenta great discovery of CS theory from the
1970sto argue that bosonic sampling is hard for
a classical computer Later, I realized that one
can also go in the reverse direction! Using the
power of postselected linear-optical quantum
computingshown by Knill-Laflamme-Milburn
2001and the connection between LOQC and the
permanent, I gave a new, arguably-simpler proof
that the permanent is P-complete
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Open Problems
Prove that approximating the permanent of an
i.i.d. Gaussian matrix is P-hard!
Can our linear-optics model solve a
classically-intractable problem for which a
classical computer can efficiently verify the
answer?
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