Transcript The Learnability of Quantum States

New Computational Insights from
Quantum Optics
Scott Aaronson
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 noninteracting balls, you probably
can’t build a universal computer—
but you can do some interesting
computations, like generating the
binomial distribution!
The Quantum Counterpart
Let’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!
What’s Going On?
The amplitude for an n-photon final state
in an optical experiment is a permanent:
Per  A 
n
a  


S n i 1
i,
i
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


Per



1
2
1
2
1 

2  11 0
1  2 2


2
The two contributions to
the amplitude interfere
destructively, cancelling
each other out!
So, Can We Use Quantum Optics to
Calculate the Permanent?
That sounds way too good to be true—it 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): 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
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 polynomialtime 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
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
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 nn 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
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 Valiant’s result that the permanent
is #P-hard
- (Indirectly) A new connection between sampling
and searching
Future Directions
Do 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, 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”