New Computational Insights from Quantum Optics

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New Computational Insights from Quantum Optics

• Scott Aaronson

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What Is Quantum Optics?

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

Classical counterpart Galtons Board, on display
at (e.g.) the Boston Museum of Science
Using only pegs and non-interacting balls, you
probably cant build a universal computerbut you
can do some interesting computations, like
generating the binomial distribution!
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The Quantum Counterpart

• Lets replace the balls by identical single
  photons, and the pegs by beamsplitters

Then the fact that photons obey Bose statistics
leads to strange phenomena, like the
Hong-Ou-Mandel dip
The two photons are now correlated, even though
they never interacted!
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Whats Going On?

• The amplitude for an n-photon final state in an
  optical experiment is a permanent

where A(aij) is an n?n matrix of transition
amplitudes for the individual photons
For example, the amplitude of the final state
1,1? in the Hong-Ou-Mandel experiment is
The two contributions to the amplitude interfere
destructively, cancelling each other out!
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So, Can We Use Quantum Optics to Calculate the
Permanent?
That sounds way too good to be trueit would let
us solve NP-complete problems and more using QC!
Explanation To get a reasonable estimate of
Per(A), you might need to repeat the optical
experiment exponentially many times
Theorem (Gurvits 2005) Theres an O(n2/?2)
classical randomized algorithm to estimate the
probability that there will be one photon in each
of n slots, to ?? accuracy
A. 2011 Gurvitss algorithm can be generalized
to estimate probabilities of arbitrary final
states
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Even so, the fact that amplitudes are permanents
does let usUse Quantum Optics to Solve Hard
Sampling ProblemsA.-Arkhipov, STOC 2011
Our Basic Result Suppose there were a
polynomial-time classical randomized algorithm
that took as input a description of a quantum
optics experiment, and output a sample from the
correct final distribution over n-photon
states. Then the polynomial hierarchy would
collapse.
Motivation Compared to (say) Shors factoring
algorithm, we get stronger evidence that a weaker
system can do interesting quantum computations
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The Equivalence of Sampling and SearchingA.,
CSR 2011
A.-Arkhipov gave a sampling problem solvable
using quantum optics that seems hard
classicallybut 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
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Using Quantum Optics to Prove that the Permanent
is P-HardA., Proc. Roy. Soc. 2011
Valiant famously showed that the permanent is
P-hardbut his proof required strange,
custom-made gadgets

• We gave a new, more transparent proof by
  combining three facts
• n-photon amplitudes correspond to n?n permanents
• (2) Postselected quantum optics can simulate
  universal quantum computation Knill-Laflamme-Milb
  urn 2001
• (3) Quantum computations can encode P-hard
  quantities in their amplitudes

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Summary

• Thinking 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 Valiants result that the
  permanent is P-hard
• (Indirectly) A new connection between sampling
  and searching

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Future Directions
Do our optics experiment! Were in contact with
two groups working to do so Terry Rudolphs at
Imperial College London and Andrew Whites in
Brisbane, Australia Current focus 3-4 photons
Prove that even approximate classical simulation
of our experiment is infeasible assuming PH is
infinite Most of A.-Arkhipov 2011 is devoted to
a program for proving this, but big pieces remain
Find more ways for quantum complexity theory to
meet the experimentalists halfway