An Invitation to Quantum Complexity Theory

The Study of What We Can’t Do With Computers We Don’t Have

Scott Aaronson (MIT)

NP complete

QIP08, New Delhi

SZK BQP

So then why can’t we just ignore quantum computing, and get back to real work?

Because the universe isn’t classical

My picture of reality, as an 11-year-old messing around with BASIC programming: + details (Also some people’s

current

picture of reality) Fancier version:

Extended Church-Turing Thesis

Shor’s factoring algorithm presents us with a choice Either

1. the Extended Church-Turing Thesis is false, 2. textbook quantum mechanics is false, or 3.

there’s an efficient classical factoring algorithm.

All three seem like crackpot speculations.

At least one of them is true!

In my view, this is why everyone should care about quantum computing, whether or not quantum factoring machines are ever built

Outline of Talk

• What is quantum complexity theory?

• The “black-box model” • Three examples of what we know • Five examples of what we don’t

Quantum Complexity Theory

Today, we know fast quantum algorithms to factor integers, compute discrete logarithms, solve certain Diophantine equations, simulate quantum systems … but

not

to solve NP-complete problems.

Quantum complexity theory is the field where we step back and ask:

How much of the classical theory of computation is actually overturned by quantum mechanics? And how much of it can be salvaged (even if in a strange new quantum form)?

But first, what

is

the classical theory of computation?

Classical Complexity Theory

A polytheistic religion with many local gods:

EXP PSPACE IP MIP BPP RP ZPP SL NC AC 0 TC 0 MA AM SZK

But also some gods everyone prays to:

P

: Class of problems solvable efficiently on a deterministic classical computer

NP

: Class of problems for which a “yes” answer has a short, efficiently-checkable proof

Major Goal:

Disprove the heresy that the

P

gods are equal and

NP

The Black-Box Model

In both classical and (especially) quantum complexity theory, much of what we know today can be stated in the

“black-box model”

This is a model where we count only the number of questions to some

black box

or

oracle

f: x

f

f(x) and ignore all other computational steps

Quantum Black-Box Algorithms

Algorithm’s state has the form   A

query

maps each basis state |x,w  to |x,w  f(x)  (f(x) gets “reversibly written to the workspace”) Between two query steps, can apply an arbitrary unitary operation that doesn’t depend on f

Query complexity

needed to achieve = minimum number of steps  , corresponding to right answer  2  2 3 for all f

Example Of Something We Can Prove In The Black-Box Model

Given a function f:[N]  {0,1}, suppose we want to know whether there’s an x such that f(x)=1. How many queries to f are needed?

Classically, it’s obvious the answer is ~N On the other hand, Grover gave a quantum algorithm that needs only ~  N queries Bennett, Bernstein, Brassard, and Vazirani proved that no quantum algorithm can do better

Example #2

Given a periodic function f:[N]  [N], how many queries to f are needed to determine its period?

Classically, one can show ~N queries are needed by any deterministic algorithm, and ~  N by any randomized algorithm On the other hand, Shor (building on Simon) gave a quantum algorithm that needs only O(log N) queries. Indeed, this is the core of his factoring algorithm So quantum query complexity can be exponentially smaller than classical!

Beals, Buhrman, Cleve, Mosca, de Wolf: But only if there’s some “promise” on f, like that it’s periodic

Example #3

Given a function f:[N]  [N], how many queries to f are needed to determine whether f is one-to-one or two-to one? (Promised that it’s one or the other) Classically, ~  N (by the Birthday Paradox) By combining the Birthday Paradox with Grover’s algorithm, Brassard, H øyer, and Tapp gave a quantum algorithm that needs only ~N 1/3 queries A., Shi: This is the best possible Quantum algorithms can’t

always

to get exponential speedups!

exploit structure

Open Problem #1: Are quantum computers more powerful than classical computers?

(In the “real,” non-black-box world?)

More formally, does

BPP

=

BQP

?

BPP

(Bounded-Error Probabilistic Polynomial Time): Class of problems solvable efficiently with use of randomness

Note:

It’s generally believed that

BPP

=

P BQP

(Bounded-Error Quantum Polynomial-Time): Class of problems solvable efficiently by a quantum computer

Most of us believe (hope?) that

BPP



BQP

— among other things, because factoring is in

BQP

!

On the other hand, Bernstein and Vazirani showed that

BPP



BQP



PSPACE

Therefore, you can’t prove

BPP



BQP

without also proving

P



PSPACE

. And that would be almost as spectacular as proving

P



NP

!

Open Problem #2: Can Quantum Computers Solve NP-complete Problems In Polynomial Time?

More formally, is

NP



BQP

?

Contrary to almost every popular article ever written on the subject, most of us think the answer is no For “generic” combinatorial optimization problems, the situation seems similar to that of black-box model —where you only get the quadratic speedup of Grover’s algorithm, not an exponential speedup As for proving this … dude, we can’t even prove

classical

computers can’t solve NP-complete problems in polynomial time!

(Conditional result?)

Open Problem #3: Can Quantum Computers Be Simulated In NP?

Most of us don’t believe

NP



BQP

about

BQP



NP

?

… but what If a quantum computer solves a problem, is there always a short proof of the solution that would convince a skeptic?

(As in the case of factoring?) My own opinion: Not enough evidence even to conjecture either way

Related Problems

Is

BQP



PH

(where

PH

is the Polynomial-Time Hierarchy, a generalization of number of quantifiers)?

NP

to any constant

Gottesman’s Question:

quantum mechanics)?

If a quantum computer solves a problem, can

it itself

interactively prove the answer to a skeptic (who doesn’t even believe

The latter question carries a $25 prize! See www.scottaaronson.com/blog

Open Problem #4: Are Quantum Proofs More Powerful Than Classical Proofs?

That is, does

QMA

=

QCMA

?

QMA

(Quantum Merlin-Arthur): A quantum generalization of

NP

.

Class of problems for which a “yes” answer can be proved by giving a polynomial-size quantum state |  , which is then checked by a

BQP

algorithm.

QCMA

: A “hybrid” between

QMA

and

NP

. The proof is classical, but the algorithm verifying it can be quantum Known:

QMA

-complete problems [Kitaev et al.], “quantum oracle separation” between

QMA

and

QCMA

[A.-Kuperberg]

Open Problem #5: Are Two Quantum Proofs More Powerful Than One?

Does

QMA(2)

=

QMA

?

QMA(2)

: Same as

QMA

, except now the verifier is given

two

quantum proofs |  and |  , which are guaranteed to be unentangled with each other Liu, Christandl, and Verstraete gave a problem called “pure state N-representability,” which is in

QMA(2)

but not known to be in

QMA

Recently A., Beigi, Fefferman, and Shor showed that, if a 3SAT instance of size n is satisfiable, this can be proved using two unentangled proofs of  n polylog n qubits each

www.scottaaronson.com/talks