1. Quantum Computing and the Limits of the Efficiently Computable
GO FAST GO FAST GO FAST Thanks so much for inviting me to this beautiful place! When I typed “quantum computer” into Google Image Search, that’s the first picture that came up. That’s apparently what they look like. (I should warn you that I’m a theorist rather than an engineer.)
Scott Aaronson
MIT

2. Things we never see… Warp drive Perpetuum mobile Übercomputer
GOLDBACH CONJECTURE: TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively
So what about the third one?
The starting point for this talk is, there are certain technologies we never see that would be REALLY cool if we had them. The first is warp drive. For this crowd especially, I can say: what’s taking you so long? The second is perpetual-motion machines. The third is what I like to call the Ubercomputer. This is a machine where you feed it any well-posed mathematical question and it instantly tells you the answer. Currently, even with the fastest computers today, if you ask them to prove a hard theorem, they could do it eventually, but it might take longer than the age of the universe. That’s why there are still human mathematicians.
In this talk, I want to convince you that the impossibility of ubercomputers is also something physicists should think about, and also something that may have implications for physics.

3. Some would say Alan Turing & friends already answered this question in the 1930s
No computer program can infallibly decide the truth or falsehood (or even the provability) of arbitrary mathematical statements!
The big question is whether P=NP. Literally a million dollar question – if you solve it, you get a million dollars from the Clay Math Institute.
In my opinion, it’s the most important of all 7 Clay problems – since if P=NP, then probably you could not only solve that one problem, but also the other six. For you would simply program your computer to find the proofs for you.
I should mention, because of this blog I write, I get claims to solve the P vs. NP problem in my inbox every other week or so. The most recent *relatively-serious* claim was this summer, when this guy Vinay Deolalikar got all over the news claiming to have proved P!=NP. I was on vacation, but eventually it got to the point where I said, listen, if he’s right, I’ll supplement his million-dollar prize by $200,000. I took a lot of flak for that, but in case you’re wondering, the end result was I didn’t have to pay. This is still an open problem, one of the hardest and most profound open problems in mathematics.
But what about “merely” searching all possible proofs with (say) 109 symbols or fewer? Can that be done in a way that avoids exhaustive search?

4. NP: Nondeterministic Polynomial-Time
This sounds like (literally) a $1,000,000 question:
P=NP?
NP: Nondeterministic Polynomial-Time
P: Polynomial-Time
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 big question is whether P=NP. Literally a million dollar question – if you solve it, you get a million dollars from the Clay Math Institute.
In my opinion, it’s the most important of all 7 Clay problems – since if P=NP, then probably you could not only solve that one problem, but also the other six. For you would simply program your computer to find the proofs for you.
I should mention, because of this blog I write, I get claims to solve the P vs. NP problem in my inbox every other week or so. The most recent *relatively-serious* claim was this summer, when this guy Vinay Deolalikar got all over the news claiming to have proved P!=NP. I was on vacation, but eventually it got to the point where I said, listen, if he’s right, I’ll supplement his million-dollar prize by $200,000. I took a lot of flak for that, but in case you’re wondering, the end result was I didn’t have to pay. This is still an open problem, one of the hardest and most profound open problems in mathematics.

5. Extended Church-Turing Thesis
However, an important presupposition underlying P vs. NP is the...
Extended Church-Turing Thesis
“Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory”
So how sure are we of this thesis?
Have there been serious challenges to it?

6. Old proposal: 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-hard problem “instantaneously”

7. Ah, but what about quantum computing? (you knew it was coming)
Quantum mechanics: “Probability theory with minus signs”
(Nature seems to prefer it that way)

8. Quantum Computing
A quantum state of n qubits takes 2n complex numbers to describe:
Chemists and physicists knew that for decades, as a practical problem!
In the 1980s, Feynman, Deutsch, and others had the amazing idea of building a new type of computer that could overcome the problem, by itself exploiting the exponentiality inherent in QM
Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

9. Any hope for a speedup rides on the magic of interference
Popularizers Beware: A quantum computer is NOT like a massively-parallel classical computer!
Exponentially-many basis states, but you only get to observe one of them
Any hope for a speedup rides on the magic of interference

10. BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993
Interesting
Shor 1994: Factoring integers is in BQP NP
NP-complete P
Factoring
BQP

11. But factoring is not believed to be NP-complete!
And today, we don’t believe BQP contains all of NP (though not surprisingly, we can’t prove that it doesn’t)
Bennett et al. 1997: “Quantum magic” won’t be enough
If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!)
So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

12. Quantum Adiabatic Algorithm (Farhi et al. 2000)Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete problem
Problem: “Eigenvalue gap” can be exponentially small

13. 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
1 solution to NP-complete problem
No solutions

14. Relativity Computer DONE
But while we’re waiting for scalable quantum computers, we can also base computers on that other great theory of the 20th century, relativity! The idea here is simple: you start your computer working on some really hard problem, and leave it on earth. Then you get on a spaceship and accelerate to close to the speed of light. When you get back to earth, billions of years have passed on Earth and all your friends are long dead, but at least you’ve got the answer to your computational problem. I don’t know why more people don’t try it!

15. STEP 1 Zeno’s Computer STEP 2 Time (seconds) STEP 3 STEP 4
Another of my favorites is Zeno’s computer. The idea here is also simple: this is a computer that would execute the first step in one second, the next step in half a second, the next in a quarter second, and so on, so that after two seconds it’s done an infinite amount of computation. Incidentally, do any of you know why that WOULDN’T work?
The problem is that, once you get down to the Planck time of 10^{-43} seconds, you’d need so much energy to run your computer that fast that, according to our best current theories, you’d exceed what’s called the Schwarzschild radius, and your computer would collapse to a black hole. You don’t want that to happen.
STEP 3
STEP 4
STEP 5

16. “The No-SuperSearch Postulate” There is no physical means to solve NP-complete problems in polynomial time.
Includes PNP as a special case, but is stronger
No longer a purely mathematical conjecture, but also a claim about the laws of physics
If true, would “explain” why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, closed timelike curves don’t exist...

17. Some of My Recent Research
BosonSampling (with Alex Arkhipov): A proposal for a rudimentary optical quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers
Computational Complexity of Decoding Hawking Radiation: Building on a striking recent proposal by Harlow and Hayden—that part of the resolution of the black hole information problem might be that reconstructing the infalling information from the Hawking radiation would require an exponentially long computation