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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
SZK
BQP
NPcomplete
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
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
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

x ,w
x, w
x ,w
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
2
needed to achieve
2
for all f
 x ,w 

x ,w
corresponding to
right answer
3
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, 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!
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 PolynomialTime): 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 … 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.
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
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.
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