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 non-interacting balls, you probably

can’t build a universal computer—but you can do some interesting

computations, like generating the binomial distribution!

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!

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!

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!

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

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 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

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

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”