1. Scott Aaronson (MIT) QIP08, New Delhi
An Invitation to Quantum Complexity Theory The Study of What We Can’t Do With Computers We Don’t Have
NP-complete
Scott Aaronson (MIT)
QIP08, New Delhi
SZK
BQP

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

3. 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

4. Shor’s factoring algorithm presents us with a choice
Either
the Extended Church-Turing Thesis is false,
textbook quantum mechanics is false, or
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

5. 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

6. 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?

7. Classical Complexity Theory
A 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

8. 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: f x
f(x)
and ignore all other computational steps

9. 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 = minimum number of steps needed to achieve
for all f

10. 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

11. 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

12. 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, Høyer, and Tapp gave a quantum algorithm that needs only ~N1/3 queries
A., Shi: This is the best possible
Quantum algorithms can’t always exploit structure to get exponential speedups!

13. 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

14. Most of us believe (hope
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!

15. 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?)

16. Open Problem #3: Can Quantum Computers Be Simulated In NP?
Most of us don’t believe NPBQP … but what about BQPNP?
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.

17. Related Problems
Is BQPPH (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 prove the answer to a skeptic (who doesn’t even believe quantum mechanics)?
The latter question carries a $25 prize! See

18. 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.

19. 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