An Invitation to Quantum Complexity Theory The Study of What We Cant Do With Computers We Dont Have Scott Aaronson (MIT) QIP08, New Delhi BQP NP- complete.

An Invitation to Quantum Complexity Theory

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

Scott Aaronson (MIT)

QIP08, New Delhi

BQP

NP-complete

SZK

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:

+ detailsFancier version: Extended Church-Turing Thesis

(Also some people’s current picture of reality)

Shor’s factoring algorithm presents us with a choice

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!

Either

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 TheoryToday, 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 TheoryA polytheistic religion with many local gods:

EXP PSPACE IP MIP BPP RP ZPP SL NC AC0 TC0 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 and NP gods are equal

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:

fx f(x)

and ignore all other computational steps

The Black-Box Model

Algorithm’s state has the form ,,

,x wx w

x wA query maps each basis state |x,w to |x,wf(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 = minimum number of steps needed to achieve 2

,,

corresponding toright answer

2

3x wx w

for all f

Quantum Black-Box Algorithms

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?

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

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

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

Example #2

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

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)

Example #3

Classically, ~N (by the Birthday Paradox)

A., Shi: This is the best possible

Quantum algorithms can’t always exploit structure to get exponential speedups!

By combining the Birthday Paradox with Grover’s algorithm, Brassard, Høyer, and Tapp gave a quantum algorithm that needs only ~N1/3 queries

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

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

(In the “real,” non-black-box world?)

Most of us believe (hope?) that BPPBQP—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 BPPBQP without also proving PPSPACE. And that would be almost as spectacular as proving PNP!

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

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

As for proving this … dude, we can’t even prove classical computers can’t solve NP-complete problems in polynomial time!

More formally, is NP BQP?

(Conditional result?)

Most of us don’t believe NPBQP … but what about BQPNP?

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

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

Is BQPPH (where PH is the Polynomial-Time Hierarchy, a generalization of NP to any constant number of quantifiers)?

Gottesman’s Question: If a quantum computer solves a problem, can it itself interactively prove the answer to a skeptic (who doesn’t even believe quantum mechanics)?

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

Related Problems

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

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

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

www.scottaaronson.com/talks