BQP

PSPACE

NPP

PostBQP

Quantum Complexity and Fundamental Physics

Scott AaronsonMIT

RESOLVED: That the results of quantum complexity 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 QCMA/qpoly and QMAlog(2), sometimes even I wonder whether it’s all just an irrelevant 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

Eleven of my favorite quantum complexity theorems … and their relevance for physics

PART I. BQP-Infused Quantum Foundations

BQP P#P, BBBV lower bound, collision lower bound, limits of random access codes

PART II. BQP-Encrusted Many-Body Physics

QMA-completeness and the limits of adiabatic computing

PART III. Quantum Gravity With a Side of BQP

Black holes as mirrors, topological QFTs, computational power of nonlinearities, postselection, and CTCs

Rest of the Talk

PART I. BQP-Infused Quantum Foundations

BQP

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

Hdtdi

This fact has many profound implications, such as…

BQP

EXP

P#P

I.e., if you want more than the N Grover speedup for solving an NP-complete problem, then you’ll need to exploit problem structure [Bennett, Bernstein, Brassard, Vazirani 1997]

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

BBBVThe “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?

Computational Power of Hidden Variables

2yx

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:Any n-qubit state can be “PAC-learned” using O(n) sample measurements—exponentially better than quantum state 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 II. BQP-Encrusted Many-Body Physics

BQP

QMA-completeness

Just one of many things we learned from this theory:

In general, finding the ground state of a 1D nearest-neighbor Hamiltonian is just as hard as finding the ground state

of any physical Hamiltonian[Aharonov, Gottesman, Irani, Kempe 2007]

One of the great achievements of quantum complexity theory, initiated by Kitaev

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]

PART III. Quantum Gravity With a Side of BQP

BQP

Black Holes as Mirrors

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

Topological Quantum Field Theories

Free

dman

, Kita

ev, L

arse

n, W

ang

2003

Aharonov, Jones, Landau 2006

Witten 1980’s

TQFTs

Jones PolynomialBQP

Beyond Quantum Computing?If QM were nonlinear, one could exploit that to solve NP-complete problems in polynomial time [Abrams & Lloyd 1998]

Quantum computers with closed timelike curves (i.e. time travel) could solve PSPACE-complete problems—but not more than that [A.-Watrous 2008]

Quantum computers with postselected measurement outcomes could solve not only NP-complete problems, but even counting problems [A. 2005]

R CTC R CR

C

000

Answer

I interpret these results as providing additional evidence that nonlinear QM,

postselection, and closed timelike curves are physically impossible.

Why? Because I’m an optimist.

For Even More Interdisciplinary Excitement, Here’s What You

Should Look ForA 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?)

A bold (but true) hypothesis linking complexity and fundamental physics…

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

There is no physical means to solve

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

Prediction: Someday, this hypothesis will be as canonical as no-superluminal-signalling or the Second Law