1. Quantum Complexity and Fundamental Physics
BQP
PSPACE NP P
PostBQP
Scott Aaronson
MIT

2. RESOLVED: That the results of quantum computing research can deepen our understanding of physics.
That this represents an intellectual payoff from quantum computing, whether or not scalable QCs are ever built.
A Personal Confession
When proving theorems about obscure quantum complexity classes, sometimes even I wonder whether it’s all just a mathematical game…

3. But then I meet distinguished physicists who say things like:
“A quantum computer is obviously just a souped-up analog computer: continuous voltages, continuous amplitudes, what’s the difference?”
“A quantum computer with 400 qubits would have ~2400 classical bits, so it would violate a cosmological entropy bound”
“My classical cellular automaton model can explain everything about quantum mechanics! (How to account for, e.g., Schor’s algorithm for factoring prime numbers is a detail left for specialists)”
“Who cares if my theory requires Nature to solve the Traveling Salesman Problem in an instant? Nature solves hard problems all the time—like the Schrödinger equation!”

4. That’s why YOU should care about quantum computing
The biggest implication of QC for fundamental physics is obvious: “Shor’s Trilemma”
Because of Shor’s factoring algorithm, either
the Extended Church-Turing Thesis—the foundation of theoretical CS for decades—is wrong,
textbook quantum mechanics is wrong, or
there’s a fast classical factoring algorithm. 
That’s why YOU should care about quantum computing
All three seem like crackpot speculations.
At least one of them is true!

5. Rest of the Talk Why computer scientists won’t shut up about P vs. NP
PART I. Classical Complexity Background
Why computer scientists won’t shut up about P vs. NP
PART II. How QC Changes the Picture
Physics invades Platonic heaven
PART III. The NP Hardness Hypothesis
A falsifiable prediction about complexity and physics

6. PART I. Classical Complexity Background

7. 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

8. NP: Nondeterministic Polynomial Time
Does
have a prime factor ending in 7?

9. 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)?

10. NP P NP-hard NP-complete
Matrix permanent Halting problem …
Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique …
NP-complete NP
Factoring Graph isomorphism …
Graph connectivity Primality testing Matrix determinant Linear programming … P

11. The (literally) $1,000,000 question
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!

12. 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

13. PART II. How QC Changes the Picture

14. BQP: Bounded-Error Quantum Polynomial-Time
BQP contains integer factoring [Shor 1994]
But factoring isn’t believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? (Is NPBQP?)
Obviously we don’t have a proof that they can’t…
But “quantum magic” won’t be enough [BBBV 1997]
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

15. QCs Don’t Provide Exponential Speedups for Black-Box Search
BBBV
The “BBBV No SuperSearch Principle” can even be applied in physics (e.g., to lower-bound tunneling times)
Is it a historical accident that quantum mechanics courses teach the Uncertainty Principle but not the No SuperSearch Principle?

16. The Quantum Adiabatic Algorithm
An amazing quantum analogue of simulated annealing [Farhi, Goldstone, Gutmann et al. 2000]
This algorithm seems to come tantalizingly close to solving NP-complete problems in polynomial time! But…
Why do these two energy levels almost “kiss”?
Answer: Because otherwise we’d be solving an NP-complete problem!
[Van Dam, Mosca, Vazirani 2001; Reichardt 2004]

17. Quantum Computing Is Not Analog
is a linear equation, governing quantities (amplitudes) that are not directly observable
This fact has many profound implications, such as…
BQP
EXP
P#P
The Fault-Tolerance Theorem
Absurd precision in amplitudes is not necessary for scalable quantum computing

18. Computational Power of Hidden Variables
Consider the problem of breaking a cryptographic hash function: given a black box that computes a 2-to-1 function f, find any x,y pair such that f(x)=f(y)
Conclusion [A. 2005]:
If, in a hidden-variable theory like Bohmian mechanics, your whole life trajectory flashed before you at the moment of your death, then you could solve problems that are presumably hard even for quantum computers
(Probably not NP-complete problems though)
Can also reduce graph isomorphism to this problem 
QCs can “almost” find collisions with just one query to f!
Measure 2nd register
Nevertheless, any quantum algorithm needs (N1/3) queries to find a collision [A.-Shi 2002]

19. The Absent-Minded Advisor Problem
Can you give your graduate student a state | with poly(n) qubits—such that by measuring | in an appropriate basis, the student can learn your answer to any yes-or-no question of size n?
NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999]
Some consequences:
Not even quantum computers with “magic initial states” can do everything: BQP/qpoly  PostBQP/poly
An n-qubit state  can be “PAC-learned” using only O(n) measurements—exponentially better than tomography [A. 2006]
One can give a local Hamiltonian H on poly(n) qubits, such that any ground state of H can be used to simulate  on all yes/no measurements with small circuits [A.-Drucker 2009]

20. PART III. The NP Hardness Hypothesis

21. But does the absence of these devices have any scientific importance?
Things we never see…
GOLDBACH CONJECTURE: TRUE
NEXT QUESTION
YES
YES
Warp drive
Perpetuum mobile
Übercomputer
But does the absence of these devices have any scientific importance?

22. A falsifiable hypothesis linking complexity and physics…
There is no physical means to solve NP-complete problems in polynomial time.
Encompasses NPP, NPBQP, NPLHC…
Does this hypothesis deserve a similar status as (say) no-superluminal-signalling or the Second Law?

23. Some alleged ways to solve NP-complete problems…
Protein folding
DNA computing
A proposal for massively parallel classical computing
Can get stuck at local optima (e.g., Mad Cow Disease)

24. 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)

25. “Relativity Computing”
DONE

26. Topological Quantum Field Theories
TQFTs
Witten 1980’s
Freedman, Kitaev, Larsen, Wang 2003
Jones Polynomial
BQP
Aharonov, Jones, Landau 2006

27. Quantum Gravity Computing?
We know almost nothing—but there are hints of a nontrivial connection between complexity and QG
Example: Against many physicists’ intuition, information dropped into a black hole seems to come out as Hawking radiation almost immediately—provided you know the black hole’s state before the information went in [Hayden & Preskill 2007]
Their argument uses explicit constructions of approximate unitary 2-designs

28. “Zeno Computing”
Do the first step of a computation in 1 second, the next in ½ second, the next in ¼ second, etc.
Problem: “Quantum foaminess”
Below the Planck scale (10-33 cm or sec), our usual picture of space and time breaks down in not-yet-understood ways

29. 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 additional argument for why QM is linear
1 solution to NP-complete problem
No solutions

30. Closed Timelike Curve Computing
Answer
Polynomial Size Circuit C
“CTC Register”
“Causality-Respecting Register”
R CTC
R CR
Quantum computers with closed timelike curves could solve PSPACE-complete problems—though not more than that [A.-Watrous 2008]

31. Why? Because I’m an optimist.
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
Again, I interpret these results as providing additional evidence that nonlinear QM, closed timelike curves, postselection, etc. aren’t possible.
Why? Because I’m an optimist.
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

32. For Even More Interdisciplinary Excitement, Here’s What You Should Look For
A plausible complexity-theoretic story for how quantum computing could fail (see A. 2004)
Intermediate models of computation between P and BQP (highly mixed states? restricted sets of gates?)
Foil theories that lead to complexity classes slightly larger than BQP (only example I know of: hidden variables)
A sane notion of “quantum gravity polynomial-time” (first step: a sane notion of time in quantum gravity?)

33. Scientific American, March 2008: