#### Transcript Slide 1

```Limits on Efficient Computation
in the Physical World
Scott Aaronson
MIT
Things we never see…
GOLDBACH
CONJECTURE:
TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
\$3 billion
But does the absence of these devices
have any scientific importance?
Goal of talk: Explain why the impossibility of
übercomputers is a great question for 21st-century
science
CS Theory 101
Problem: “Given a graph, is it connected?”
Each particular graph is an instance
The size of the instance, n, is the number of
bits needed to specify it
An algorithm is polynomial-time if it uses at
most knc steps, for some constants k,c
P is the class of all problems that have
polynomial-time algorithms
NP: Nondeterministic
Polynomial Time
Does
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have a prime factor ending in 7?
NP-hard: If you can solve it, you
can solve everything in NP
NP-complete: NP-hard and in NP
Is there a
Hamilton cycle
(tour that visits
each vertex
exactly once)?
NP-hard
Hamilton cycle
Steiner tree
Graph 3-coloring
Satisfiability
Maximum clique
…
NPcomplete
NP
Graph connectivity
Primality testing
Matrix determinant
Linear programming
…
P
Matrix permanent
Halting problem
…
Factoring
Graph isomorphism
…
Does P=NP?
The (literally) \$1,000,000 question
Q: What if P=NP, and the algorithm takes n10000 steps?
A: Then we’d just change the question!
Q: Why is it so hard to prove PNP?
A: Because polynomial-time algorithms are so rich
computers?
BQP: Bounded-Error Quantum Polynomial-Time
Shor 1994: BQP contains integer factoring
But factoring isn’t believed to be NP-complete.
So the question remains: can quantum computers solve
NP-complete problems efficiently?
Bennett et al. 1997: “Quantum magic” won’t be enough
If we throw away the problem structure, and just consider
a “landscape” of 2n possible solutions, even a quantum
computer needs ~2n/2 steps to find a correct solution
(Farhi et al. 2000)
Hi
Hamiltonian with
easily-prepared
ground state
Hf
Ground state encodes
solution to NPcomplete problem
Problem: Eigenvalue gap can be
exponentially small
Other Alleged Ways to Solve
NP-complete Problems
Protein folding: Can also get stuck at local optima
DNA computers: A proposal for massively parallel
classical computing
The cognitive science approach: Think about it
really hard
My Personal Favorite
Dip two glass plates with pegs between them into
soapy water; let the soap bubbles form a minimum
“Steiner tree” connecting the pegs (thereby solving a
known NP-complete problem)
What would the world actually be like
if we could solve NP-complete
problems efficiently?
Proof of Riemann hypothesis
with 10,000,000 symbols?
Shortest efficient description
of stock market data?
If there actually were a machine with
[running time] ~Kn (or even only with
~Kn2), this would have consequences
of the greatest magnitude.
—Gödel to von Neumann, 1956
The NP Hardness Assumption
There is no physical means to solve NP
complete problems in polynomial time.
• Implies, but
is
stronger
than,
PNP
Rest of talk: Try to give
indications
• As falsifiable
as it getsthat it is
• Consistent with currently-known physical theory
• Scientifically fruitful?
1. “Relativity Computing”
DONE
2. Topological Quantum Field
Theories (TQFT’s)
Freedman, Kitaev, Wang 2000:
Equivalent to ordinary quantum computers
3. Nonlinear variants of the
Schrödinger Equation
Abrams & Lloyd 1998: If quantum mechanics
were nonlinear, one could exploit that to solve
NP-complete problems in polynomial time
Can take as an
argument for why
QM is linear
1 solution to NP-complete problem
No solutions
4. Anthropic Principle
Foolproof way to solve NP-complete problems in
polynomial time (at least in the Many-Worlds Interpretation):
First guess a random solution. Then, if it’s wrong,
kill yourself
Technicality: If there are no solutions, you’d seem
to be out of luck!
Solution: With tiny probability don’t do anything. Then, if you find
yourself in a universe where you didn’t do anything, there probably were
no solutions, since otherwise you would’ve found one
What if we combine quantum computing
with the Anthropic Principle?
I.e. perform a polynomial-time quantum
computation, but where we can measure a
qubit and assume the outcome will be |1
Leads to a new complexity class:
PostBQP (Postselected BQP)
A. 2005: PostBQP=PP—and this yields a 1page proof of the Beigel-Reingold-Spielman
theorem, that PP is closed under intersection
5. Time Travel
Everyone’s first idea for a time travel computer:
Do an arbitrarily long computation, then send the
answer back in time to before you started
THIS DOES NOT WORK
Why not?
• Doesn’t take into account the computation you’ll have
to do after getting the answer
Deutsch’s Model
A closed timelike curve (CTC) is a computational
resource that, given an efficiently computable function
f:{0,1}n{0,1}n, immediately finds a fixed point of f—
that is, an x such that f(x)=x
Admittedly, not every f has a fixed point
But there’s always a distribution D such that f(D)=D
Probabilistic Resolution of the Grandfather Paradox
- You’re born with ½ probability
- If you’re born, you back and kill your grandfather
- Hence you’re born with ½ probability
Let PCTC be the class of problems solvable in
polynomial time, if for any function f:{0,1}n{0,1}n
described by a poly-size circuit, we can immediately get
an x{0,1}n such that f(m)(x)=x for some m
Theorem: PCTC = PSPACE
What if we perform a quantum
computation around a CTC?
Let BQPCTC be the class of problems solvable in
quantum polynomial time, if for any superoperator E
described by a quantum circuit, we can immediately get
a mixed state  such that E() = 
Clearly PSPACE = PCTC  BQPCTC
A., Watrous 2008:
BQPCTC = PSPACE
If closed timelike curves exist, then quantum
computers are no more powerful than classical ones
Concluding Remarks
Are NP-complete problems intractable in the physical
universe? I conjecture that they are, but fully
understanding why will bring in:
• Math and computer science (duh): The P vs. NP
question
Prediction: The “NP Hardness Assumption” will
eventually
be seen as
analogous
to Second
Law
• Quantum
mechanics:
The
NP vs. BQP
question
of Thermodynamics or the impossibility of
• Other physics: superluminal
Quantum fieldsignaling
theory, quantum gravity,
closed timelike curves…
• Biology, cognitive science, economics?
Open Question: What is “polynomial
time” in quantum gravity?
Scientific American, March 2008:
www.scottaaronson.com
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