Quantum Computing and the Limits of the Efficiently Computable

Scott Aaronson MIT

Things we never see…

GOLDBACH CONJECTURE: TRUE NEXT QUESTION

Warp drive Perpetuum mobile Übercomputer The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively So what about the third one?

Some would say Alan Turing & friends already answered this question in the 1930s No computer program can infallibly decide the truth or falsehood (or even the provability) of arbitrary mathematical statements!

But what about “merely” searching all possible proofs with (say) 10 9 symbols or fewer? Can that be done in a way that avoids exhaustive search?

This sounds like (literally) a $1,000,000 question:

P = NP ?

NP: Nondeterministic Polynomial-Time P: Polynomial-Time If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), this would have consequences of the greatest magnitude.

—Gödel to von Neumann, 1956

However, an important presupposition underlying

P

vs.

NP

is the...

Extended Church-Turing Thesis

“Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory” So how sure are we of this thesis?

Have there been serious challenges to it?

Old proposal:

Dip two glass plates with pegs between them into soapy water.

Let the soap bubbles form a minimum Steiner tree connecting the pegs—thereby solving a known

NP

-hard problem “instantaneously”

Ah, but what about quantum computing?

(you knew it was coming) Quantum mechanics:

“Probability theory with minus signs”

(Nature seems to prefer it that way)

Quantum Computing

A quantum state of n qubits takes 2 n numbers to describe: 

x

  

n

 complex

x x

Chemists and physicists knew that for decades, as a practical problem!

In the 1980s, Feynman, Deutsch, and others had the amazing idea of building a new type of computer that could overcome the problem, by

itself

exploiting the exponentiality inherent in QM

Actually building a QC:

Damn hard, because of

decoherence

. (But seems possible in principle!)

Popularizers Beware: A quantum computer is

NOT

like a massively-parallel classical computer!

 

x

   1 ,  , 2 

n



x x

Exponentially-many basis states, but you only get to observe one of them Any hope for a speedup rides on the magic of interference

BQP (Bounded-Error Quantum Polynomial-Time):

The class defined by Bernstein and Vazirani in 1993

Shor 1994:

Factoring integers is in

BQP NP-complete BQP NP P

Factoring

But factoring is not believed to be

NP

-complete!

And today, we

don’t

believe

BQP

contains all of

NP

(though not surprisingly, we can’t prove that it doesn’t)

Bennett et al. 1997:

“Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2 n possible solutions, then even a quantum computer needs ~2 n/2 steps to find the correct one (That bound is actually achievable, using

Grover’s algorithm

!) So, is there any quantum algorithm for

NP

-complete problems that

would

exploit their structure?

Quantum Adiabatic Algorithm

(Farhi et al. 2000)

H i Hamiltonian with easily prepared ground state H f Ground state encodes solution to

NP

-complete problem

Problem:

“Eigenvalue gap” can be exponentially small

Nonlinear variants of the Schrödinger Equation

Abrams & Lloyd 1998:

If quantum mechanics were nonlinear, one could exploit that to solve

NP

complete problems in polynomial time 1 solution to NP-complete problem No solutions

DONE

Relativity Computer

Zeno’s Computer

STEP 1

STEP 2 STEP 3 STEP 4 STEP 5

“The No-SuperSearch Postulate”

There is no physical means to solve

NP

-complete problems in polynomial time.

Includes

P



NP

as a special case, but is stronger No longer a purely mathematical conjecture, but also a claim about the laws of physics If true, would “explain” why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, closed timelike curves don’t exist...

Some of My Recent Research

BosonSampling (with Alex Arkhipov):

A proposal for a rudimentary optical quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers

Computational Complexity of Decoding Hawking Radiation:

Building on a striking recent proposal by Harlow and Hayden—that part of the resolution of the black hole information problem might be that reconstructing the infalling information from the Hawking radiation would require an exponentially long computation