Quantum Computing and the Limits of the Efficiently Computable

Scott AaronsonMIT

Things we never see…

Warp drive Perpetuum mobile

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

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?

Problem: “Given a flight map, is every airport reachable from every other in 5 flights or less?”

Any specific map is an instance of the problem

The size of an instance, n, is the number of bits used to specify it

An algorithm is polynomial-time if it uses at most knc steps, for some constants k,c

P is the class of all problems for which there’s a deterministic, polynomial-time algorithm that correctly solves every instance

Complexity Theory 101

NP: Nondeterministic Polynomial Time

37976595177176695379702491479374117272627593301950462688996367493665078453699421776635920409229841590432339850906962896040417072096197880513650802416494821602885927126968629464313047353426395204881920475456129163305093846968119683912232405433688051567862303785337149184281196967743805800830815442679903720933

Does

have a factor ending in 7?

NP-hard: If you can solve it, then you can solve every NP problem

NP-complete: NP-hard and in NP

Is there a tour that visits each city once?

P

NP

NP-complete

NP-hard

Graph connectivityPrimality testingMatrix determinantLinear programming…

Matrix permanentHalting problem…

Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Factoring

Graph isomorphism…

Does P=NP?The (literally) $1,000,000 question

If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude.—Gödel to von Neumann, 1956

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”

An important presupposition underlying P vs. NP is the...

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”

Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease)

DNA computers: Just massively parallel classical computers

Other Approaches

Ah, but what about quantum computing?

(you knew it was coming)

Quantum computing: “The power of 2n complex numbers working for YOU”

In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2n time to simulate—and had the amazing idea of building a “quantum computer” to overcome that problem

Quantum mechanics: “Probability theory with minus signs”(Nature seems to prefer it that way)

Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

Journalists Beware:A quantum computer is NOT like a

massively-parallel classical computer!

Exponentially-many basis states, but you only get to

observe one of them

nxx x

2,,1

Any hope for a speedup rides on the magic of

quantum interference

BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993

Shor 1994: Factoring integers is in BQP

NP

NP-complete

P

FactoringBQP

Interesting

Remember: factoring isn’t thought to be NP-complete!

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 2n possible solutions, then even a quantum computer needs ~2n/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)

HiHamiltonian with easily-prepared ground state

HfGround 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

No solutions1 solution to NP-complete problem

Relativity Computer

DONE

Zeno’s Computer

STEP 1

STEP 2

STEP 3STEP 4

STEP 5

Tim

e (s

econ

ds)

Here’s a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel):

Read an integer x{0,…,2n-1} from the futureIf x encodes a valid solution, then output xOtherwise, output (x+1) mod 2n

Closed Timelike Curves (CTCs)

If valid solutions exist, then the only fixed-points of the above program input and output them

Building on work of Deutsch, [A.-Watrous 2008] defined a formal model of CTC computation, and showed that in both the classical and quantum cases, it has exactly the power of PSPACE (believed to be even larger than NP)

Includes PNP 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, CTCs don’t exist...

“The No-SuperSearch Postulate”There is no physical means to solve NP-complete

problems in polynomial time.

Question: What exactly does it mean to “solve” an NP-complete problem?Example: It’s been known for decades that, if you send n identical photons through a network of beamsplitters, the amplitude for the photons to reach some final state is given by the permanent of an nn matrix of complex numbers

nS

n

iiiaA

1,Per

But the permanent is #P-complete (believed even harder than NP-complete)! So how can Nature do such a thing?

Resolution: The amplitudes aren’t directly observable, and require exponentially-many probabilistic trials to estimate

Lesson: If you can’t observe the answer, it doesn’t count!

Recently, Alex Arkhipov and I gave the first evidence that even the observed output distribution of such a linear-optical network would be hard to simulate on a classical computer—

but the argument was necessarily more subtle

One could imagine worse research agendas than the following:

Prove P≠NP (better yet, prove factoring is classically hard, implying P≠BQP)

Prove NPBQP—i.e., that not even quantum computers can solve NP-complete problems

Build a scalable quantum computer(or even more interesting, show that it’s impossible)

Determine whether all of physics can be simulated by a quantum computer

“Derive” as much physics as one can from No-SuperSearch and other impossibility principles

Conclusion

Papers, talk slides, blog:www.scottaaronson.com