Solving Hard Problems With Light Scott Aaronson (Assoc. Prof., EECS) Joint work with Alex Arkhipov vs.

Solving Hard Problems With Light

Scott Aaronson (Assoc. Prof., EECS)Joint work with Alex Arkhipov

vs

In 1994, something big happened in the foundations of computer science, whose meaning

is still debated today…

Why exactly was Shor’s algorithm important?

Boosters: Because it means we’ll build QCs!

Skeptics: Because it means we won’t build QCs!

Me: For reasons having nothing to do with building QCs!

Shor’s algorithm was a hardness result for one of the central computational problems

of modern science: QUANTUM SIMULATION

Shor’s Theorem:

QUANTUM SIMULATION is not

solvable efficiently (in polynomial time),

unless FACTORING is also

Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

Advantages:

Based on more “generic” complexity assumptions than the hardness of FACTORING

Gives evidence that QCs have capabilities outside the entire

“polynomial hierarchy”

Requires only a very simple kind of quantum computation:

nonadaptive linear optics (testable before I’m dead?)

Today, a different kind of hardness result for simulating quantum mechanics

Disadvantages:

Applies to relational problems (problems with many possible outputs) or sampling

problems, not decision problems

Harder to convince a skeptic that your computer is indeed

solving the relevant hard problem

Less relevant for the NSA

Bestiary of Complexity Classes

BQP

P#P

BPP

P

NP

PH

FACTORIN

G

PERMANENT

COUNTING

3SAT

XYZ…

How complexity theorists say “such-and-such is damn unlikely”:

“If such-and-such is true, then PH collapses to a finite level”

Suppose the output distribution of any linear-optics circuit can be efficiently sampled by a classical algorithm. Then the polynomial hierarchy collapses.

Indeed, even if such a distribution can be sampled by a classical computer with an oracle for the polynomial hierarchy, still the polynomial hierarchy collapses.

Suppose two plausible conjectures are true: the permanent of a Gaussian random matrix is(1) #P-hard to approximate, and(2) not too concentrated around 0.Then the output distribution of a linear-optics circuit can’t even be approximately sampled efficiently classically, unless the polynomial hierarchy collapses.

Our Results

If our conjectures hold, then even a noisy linear-optics experiment can

sample from a probability distribution that no classical

computer can feasibly sample from

nS

n

iiiaA

1,Per

BOSONS

nS

n

iiiaA

1,

sgn1Det

FERMIONS

There are two basic types of particle in the universe…

Their transition amplitudes are given respectively by…

All I can say is, the bosons got the harder job

Particle Physics In One Slide

High-Level IdeaEstimating a sum of exponentially many positive or

negative numbers: #P-hard

Estimating a sum of exponentially many nonnegative numbers: Still hard, but known to be in PH

If quantum mechanics could be efficiently simulated classically, then these two problems would become

equivalent—thereby placing #P in PH, and collapsing PH

So why aren’t we done?

Because real quantum experiments are subject to noise

Would an efficient classical algorithm that simulated a noisy optics experiment still collapse the polynomial hierarchy?

Main Result: Yes, assuming two plausible conjectures about permanents of random matrices (the “PCC” and the

“PGC”)

U

Particular experiment we have in mind: Take a system of n identical photons with m=O(n2) modes. Put each photon in a known mode, then apply a Haar-random mm unitary transformation U:

Then measure which modes have 1 or more photon in them

There exists a polynomial p such that for all n,

The Permanent Concentration Conjecture (PCC)

2

1,0~

1!PerPr

nnp

nX

nnCNX

Empirically true!

Also, we can prove it with determinant in place of

permanent

Let X be an nn matrix of independent, N(0,1) complex Gaussian entries. Then approximating Per(X) to within a 1/poly(n) multiplicative error, for a 1-1/poly(n) fraction of X, is a #P-hard problem.

The Permanent-of-Gaussians Conjecture (PGC)

Experimental ProspectsWhat would it take to implement the requisite experiment?• Reliable phase-shifters and beamsplitters, to implement an arbitrary unitary on m photon modes• Reliable single-photon sources• Photodetector arrays that can reliably distinguish 0 vs. 1 photonBut crucially, no nonlinear optics or postselected measurements!

Our Proposal: Concentrate on (say)

n=20 photons and m=400 modes, so that classical simulation is

nontrivial but not impossible

SummaryI often say that Shor’s algorithm presented us with three choices. Either

(1)The laws of physics are exponentially hard to simulate on any computer today,

(2)Textbook quantum mechanics is false, or

(3)Quantum computers are easy to simulate classically.

For all intents and purposes?