BQP

PSPACE

NP

P

PostBQP

Quantum Complexity and Fundamental Physics

Scott Aaronson

MIT

RESOLVED: That the results of quantum computing research can deepen our understanding of physics.

That this represents an intellectual payoff from quantum computing, whether or not scalable QCs are ever built.

A Personal ConfessionWhen proving theorems about obscure quantum complexity classes, sometimes even I wonder whether it’s all just a mathematical game…

“A quantum computer is obviously just a souped-up analog computer: continuous voltages, continuous amplitudes, what’s the difference?”

“A quantum computer with 400 qubits would have ~2400 classical bits, so it would violate a cosmological entropy bound”

“My classical cellular automaton model can explain everything about quantum mechanics!(How to account for, e.g., Schor’s algorithm for factoring prime numbers is a detail left for specialists)”

“Who cares if my theory requires Nature to solve the Traveling Salesman Problem in an instant? Nature solves hard problems all the time—like the Schrödinger equation!”

But then I meet distinguished physicists who say things like:

The biggest implication of QC for fundamental physics is obvious:

“Shor’s Trilemma”

1. the Extended Church-Turing Thesis—the foundation of theoretical CS for decades—is wrong,

2. textbook quantum mechanics is wrong, or

3. there’s a fast classical factoring algorithm.

All three seem like crackpot speculations.

At least one of them is true!

That’s why YOU

should care about quantum

computing

Because of Shor’s factoring algorithm, either

PART I. Classical Complexity Background

Why computer scientists won’t shut up about P vs. NP

PART II. How QC Changes the Picture

Physics invades Platonic heaven

PART III. The NP Hardness Hypothesis

A falsifiable prediction about complexity and physics

Rest of the Talk

PART I. Classical Complexity Background

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

37976595177176695379702491479374117272627593301950462688996367493665078453699421776635920409229841590432339850906962896040417072096197880513650802416494821602885927126968629464313047353426395204881920475456129163305093846968119683912232405433688051567862303785337149184281196967743805800830815442679903720933

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!

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?

PART II. How QC Changes the Picture

BQP contains integer factoring [Shor 1994]

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

But “quantum magic” won’t be enough [BBBV 1997]

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

Obviously we don’t have a proof that they can’t…

QCs Don’t Provide Exponential Speedups for Black-Box Search

BBBV

The “BBBV No SuperSearch Principle” can even be applied in physics (e.g., to lower-bound tunneling times)

Is it a historical accident that quantum mechanics courses teach the Uncertainty Principle but not the No SuperSearch Principle?

The Quantum Adiabatic Algorithm

Why do these two energy levels almost “kiss”?

An amazing quantum analogue of simulated annealing [Farhi, Goldstone, Gutmann et al. 2000]

This algorithm seems to come tantalizingly close to solving NP-complete problems in polynomial time! But…

Answer: Because otherwise we’d be solving an NP-complete problem!

[Van Dam, Mosca, Vazirani 2001; Reichardt 2004]

Quantum Computing Is Not Analog

The Fault-Tolerance Theorem

Absurd precision in amplitudes is not

necessary for scalable quantum

computing

is a linear equation, governing quantities (amplitudes) that are not directly observable

Hdt

di

This fact has many profound implications, such as…

BQP

EXP

P#P

Computational Power of Hidden Variables

2

yx

N

x

xfxN 1

1Measure 2nd

register

xf

Consider the problem of breaking a cryptographic hash function: given a black box that computes a 2-to-1 function f, find any x,y pair such that f(x)=f(y)

Can also reduce graph isomorphism to this problem

QCs can “almost” find collisions with just one query to f!

Nevertheless, any quantum algorithm needs (N1/3) queries to find a collision [A.-Shi 2002]

Conclusion [A. 2005]:If, in a hidden-variable theory like Bohmian mechanics, your whole life trajectory flashed before you at the moment of your death, then you could solve problems that are presumably hard even for quantum computers

(Probably not NP-complete problems though)

The Absent-Minded Advisor Problem

Some consequences:Not even quantum computers with “magic initial states” can do everything: BQP/qpoly PostBQP/poly

An n-qubit state can be “PAC-learned” using only O(n) measurements—exponentially better than tomography [A. 2006]

One can give a local Hamiltonian H on poly(n) qubits, such that any ground state of H can be used to simulate on all yes/no measurements with small circuits [A.-Drucker 2009]

Can you give your graduate

student a state | with poly(n) qubits—such that by measuring | in an appropriate basis, the

student can learn your answer to any yes-or-no question of size n?

NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999]

PART III. The NP Hardness Hypothesis

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

A falsifiable hypothesis linking complexity and physics…

There is no physical means to solve

NP-complete problems in polynomial time.Encompasses NPP, NPBQP, NPLHC…

Does this hypothesis deserve a similar status as (say) no-superluminal-signalling or the Second Law?

Some alleged ways to solve NP-complete problems…

Protein folding DNA computing

Can get stuck at local optima (e.g., Mad Cow Disease)

A proposal for massively parallel classical computing

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)

“Relativity Computing”

DONE

Topological Quantum Field Theories

Free

dman

, Kita

ev, L

arse

n, W

ang

2003

Aharonov, Jones, Landau 2006

Witten 1980’s

TQFTs

Jones PolynomialBQP

Quantum Gravity Computing?

Example: Against many physicists’ intuition, information dropped into a black hole seems to come out as Hawking radiation almost immediately—provided you know the black hole’s state before the information went in [Hayden & Preskill 2007]

Their argument uses explicit constructions of approximate unitary 2-designs

We know almost nothing—but there are hints of a nontrivial connection between complexity and QG

Do the first step of a computation in 1 second, the next in ½ second, the next in ¼ second, etc.

Problem: “Quantum foaminess”

“Zeno Computing”

Below the Planck scale (10-33 cm or 10-43 sec), our usual picture of space and time breaks down in not-yet-understood ways

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

Closed Timelike Curve Computing

Quantum computers with closed timelike curves could solve PSPACE-complete problems—though not more than that[A.-Watrous 2008]

R CTC R CR

C

0 0 0

Answer

“Causality-Respecting Register”

“CTC Register”

Polynomial Size Circuit

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

Again, I interpret these results as providing additional evidence that

nonlinear QM, closed timelike curves, postselection, etc. aren’t

possible.

Why? Because I’m an optimist.

For Even More Interdisciplinary Excitement, Here’s What You

Should Look For

A plausible complexity-theoretic story for how quantum computing could fail (see A. 2004)

Intermediate models of computation between P and BQP (highly mixed states? restricted sets of gates?)

Foil theories that lead to complexity classes slightly larger than BQP (only example I know of: hidden variables)

A sane notion of “quantum gravity polynomial-time” (first step: a sane notion of time in quantum gravity?)

Scientific American, March 2008:

www.scottaaronson.com