BQP

PSPACE

NP

P

PostBQP

Limits on Efficient Computation in the Physical World

Scott Aaronson

MIT

Things we never see…

Warp drive Perpetuum mobile

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

Übercomputer

But does the absence of these devices have any scientific importance?

YES YES

Goal of talk: Explain why the impossibility of übercomputers is a great question for 21st-century science

$3 billion

Problem: “Given a graph, is it connected?”

Each particular graph is an instance

The size of the instance, n, is the number of bits needed 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 that have polynomial-time algorithms

CS Theory 101

NP: Nondeterministic Polynomial Time

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Does

have a prime factor ending in 7?

NP-hard: If you can solve it, you can solve everything in NP

NP-complete: NP-hard and in NP

Is there a Hamilton cycle (tour that visits each vertex exactly 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

Q: What if P=NP, and the algorithm takes n10000 steps?

A: Then we’d just change the question!

Q: Why is it so hard to prove PNP?

A: Because polynomial-time algorithms are so rich

What about quantum computers?

Shor 1994: BQP contains integer factoring

But factoring isn’t believed to be NP-complete.So the question remains: can quantum computers solve NP-complete problems efficiently?

Bennett et al. 1997: “Quantum magic” won’t be enough

If we throw away the problem structure, and just consider a “landscape” of 2n possible solutions, even a quantum computer needs ~2n/2 steps to find a correct solution

BQP: Bounded-Error Quantum Polynomial-Time

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

Other Alleged Ways to Solve NP-complete Problems

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

DNA computers: A proposal for massively parallel classical computing

The cognitive science approach: Think about it really hard

My Personal FavoriteDip 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-complete problem)

What would the world actually be like if we could solve NP-complete

problems efficiently?

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

Proof of Riemann hypothesis with

10,000,000 symbols?Shortest efficient

description of stock market data?

• Implies, but is stronger than, PNP

• As falsifiable as it gets

• Consistent with currently-known physical theory

• Scientifically fruitful?

Alright, what can we say about this assumption?

The NP Hardness AssumptionThere is no physical means to solve NP complete problems in polynomial time.

Rest of talk: Try to give indications that it is

1. “Relativity Computing”

DONE

2. Topological Quantum Field Theories (TQFT’s)

Freedman, Kitaev, Wang 2000: Equivalent to ordinary quantum computers

3. 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

Can take as an additional

argument for why QM is linear

4. Anthropic PrincipleFoolproof way to solve NP-complete problems in polynomial time (at least in the Many-Worlds Interpretation):

First guess a random solution. Then, if it’s wrong, kill yourself

Technicality: If there are no solutions, you’d seem to be out of luck!Solution: With tiny probability don’t do anything. Then, if you find yourself in a universe where you didn’t do anything, there probably were no solutions, since otherwise you would’ve found one

What if we combine quantum computing with the Anthropic Principle?

I.e. perform a polynomial-time quantum computation, but where we can measure a qubit and assume the outcome will be |1

Leads to a new complexity class:PostBQP (Postselected BQP)

A. 2005: PostBQP=PP—and this yields a 1-page proof of the Beigel-Reingold-Spielman theorem, that PP is closed under intersection

Everyone’s first idea for a time travel computer: Do an arbitrarily long computation, then send the answer back in time to before you started

THIS DOES NOT WORK

Why not?

• Ignores the Grandfather Paradox

• Doesn’t take into account the computation you’ll have to do after getting the answer

5. Time Travel

Deutsch’s ModelA closed timelike curve (CTC) is a computational resource that, given an efficiently computable function f:{0,1}n{0,1}n, immediately finds a fixed point of f—that is, an x such that f(x)=x

Admittedly, not every f has a fixed point

But there’s always a distribution D such that f(D)=D

Probabilistic Resolution of the Grandfather Paradox- You’re born with ½ probability- If you’re born, you back and kill your grandfather- Hence you’re born with ½ probability

Let PCTC be the class of problems solvable in polynomial

time, if for any function f:{0,1}n{0,1}n described by a poly-size circuit, we can immediately get an x{0,1}n such that f(m)(x)=x for some m

Theorem: PCTC = PSPACE

What if we perform a quantum computation around a CTC?

Let BQPCTC be the class of problems solvable in

quantum polynomial time, if for any superoperator E described by a quantum circuit, we can immediately get a mixed state such that E() =

Clearly PSPACE = PCTC BQPCTC

A., Watrous 2008: BQPCTC = PSPACE

If closed timelike curves exist, then quantum computers are no more powerful than classical ones

Concluding RemarksAre NP-complete problems intractable in the physical universe? I conjecture that they are, but fully understanding why will bring in:

• Math and computer science (duh): The P vs. NP question

• Quantum mechanics: The NP vs. BQP question

• Other physics: Quantum field theory, quantum gravity, closed timelike curves…

• Biology, cognitive science, economics?

Prediction: The “NP Hardness Assumption” will eventually be seen as analogous to Second Law

of Thermodynamics or the impossibility of superluminal signaling

Open Question: What is “polynomial time” in quantum gravity?

Scientific American, March 2008:

www.scottaaronson.com