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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: Galton’s Board, on display at (e.g.) the Boston Museum of Science
Using only pegs and non-interacting balls, you probably
can’t build a universal computer—but you can do some interesting
computations, like generating the binomial distribution!
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The Quantum CounterpartLet’s 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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What’s Going On?The amplitude for an n-photon final state in an optical experiment is a permanent:
nS
n
iiiaA
1,Per
02
1
2
1
2
1
2
12
1
2
1
Per
For example, the amplitude of the final state |1,1 in the Hong-Ou-Mandel experiment is
where A=(aij) is an nn matrix of transition amplitudes for the individual
photons
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?
Explanation: To get a reasonable estimate of Per(A), you might need to repeat the optical experiment exponentially many times
Theorem (Gurvits 2005): There’s 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: Gurvits’s algorithm can be generalized to estimate probabilities of arbitrary final states
That sounds way too good to be true—it would let us solve NP-complete problems and more using QC!
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Even so, the fact that amplitudes are permanents does let us
Use Quantum Optics to Solve Hard Sampling Problems[A.-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) Shor’s factoring algorithm, we get stronger evidence that a weaker system can do interesting quantum computations
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The Equivalence of Sampling and Searching
[A., CSR 2011]
[A.-Arkhipov] gave a “sampling problem” solvable using quantum optics that seems hard classically—but 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-Hard
[A., Proc. Roy. Soc. 2011]
Valiant famously showed that the permanent is #P-hard—but his proof required strange, custom-made gadgets
We gave a new, more transparent proof by combining three facts:(1)n-photon amplitudes correspond to nn permanents(2) Postselected quantum optics can simulate universal quantum computation [Knill-Laflamme-Milburn 2001](3) Quantum computations can encode #P-hard quantities in their amplitudes
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SummaryThinking 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 Valiant’s result that the permanent is #P-hard- (Indirectly) A new connection between sampling and searching
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Future DirectionsDo our optics experiment!
We’re in contact with two groups working to do so: Terry Rudolph’s at Imperial College London and Andrew White’s in Brisbane, AustraliaCurrent 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”