1. Computational Complexity and Fundamental Physics
ZOOM THRU ZOOM THRU Thanks so much for inviting me! 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
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. Where is it? The second is perpetual-motion machines – the ultimate solution to the world’s energy problems. 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.
So what about the third one? What are the ultimate physical limits on what can be feasibly computed? And do those limits have any implications for physics?

3. NP Efficiently verifiable
NP-hard All NP problems are efficiently reducible to these
Matrix permanent Halting problem …
Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique …
NP-complete
NP Efficiently verifiable
Factoring Graph isomorphism …
Here’s a rough map of the world. At the bottom is P, which includes everything we know how to solve quickly with today’s computers. Containing it is NP, the class of problems where we could recognize an answer if we saw it, and at the top of NP is this huge family of NP-complete problems. There are plenty of problems that are even harder than NP-complete – one famous example is the halting problem, to determine whether a given computer program will ever stop running. Very interestingly, there are also problems believed to be intermediate between P and NP-complete. One example is factoring. These intermediate problems are extremely important for quantum computing, as we’ll see later, and they’re also important for cryptography.
Graph connectivity Primality testing Matrix determinant Linear programming …
P Efficiently solvable

4. The (literally) $1,000,000 question
Does P=NP?
The (literally) $1,000,000 question
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. The Extended Church-Turing Thesis (ECT)
An important presupposition underlying P vs. NP is the
The Extended Church-Turing Thesis (ECT)
“Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory”
But how sure are we of this thesis? What would a challenge to it look like?

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. 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!

8. 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

9. 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)
In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2n time to simulate—and had the amazing idea of building a “quantum computer” to overcome that problem
Quantum computing: “The power of 2n complex numbers working for YOU”

10. Quantum Mechanics in One Slide
Probability Theory:
Linear transformations that conserve 1-norm of probability vectors: Stochastic matrices
Quantum Mechanics:
Linear transformations that conserve 2-norm of amplitude vectors: Unitary matrices
So, let me first explain quantum mechanics in one slide. See, the physicists somehow convinced everyone that quantum mechanics is complicated and hard. The truth is, QM is unbelievably simple, once you take the physics out.
What is quantum mechanics IS, fundamentally, is a certain generalization of the laws of probability. In probability theory, you always represent your knowledge of a system using a vector of nonnegative real numbers, which sum to 1 and which are called probabilities. As the system changes, you update your knowledge by applying a linear transformation to the vector. The linear transformation has to preserve the 1-norm; matrices that do that are called stochastic matrices.
Quantum mechanics is almost the same, except now you represent your knowledge using a vector of complex numbers, called “amplitudes”. And instead of preserving the 1-norm of the vector, you preserve its 2-norm – which God or Nature seems to prefer over the 1-norm in every situation. Matrices that preserve 2-norm are called unitary matrices.

11. Journalists Beware: A quantum computer is NOT like a massively-parallel classical computer!
Exponentially many possible measurement outcomes, but you only see one, probabilistically!
Any hope for a speedup rides on the magic of interference between positive and negative contributions to an amplitude

12. 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

13. Can QCs Actually Be Built?
Where we are now: A quantum computer has factored 21 into 37, with high probability (Martín-López et al. 2012)
Why is scaling up so hard? Because of decoherence: unwanted interaction between a QC and its external environment, “prematurely measuring” the quantum state
A few skeptics, in CS and physics, even argue that building a QC will be fundamentally impossible
As zany as this sounds, Deutsch’s speculations are part of what gave rise to the modern field of quantum computing. So, what’s the idea of quantum computing?
Well, a general entangled state of n qubits requires 2^n amplitudes to specify, since you need to give an amplitude for every configuration of all n of the bits. That’s a staggering amount of information! It suggests that Nature, off to the side somewhere, needs to write down 2^1000 numbers just to keep track of 1000 particles. And that presents an obvious practical problem when people try to use conventional computers to SIMULATE quantum mechanics – they have all sorts of approximate techniques, but even then, something like 10% of supercomputer cycles today are used, basically, for simulating quantum mechanics.
In 1981, Richard Feynman said, if Nature is going to all this work, then why not turn it around, and build computers that THEMSELVES exploit superposition? What would such computers be useful for? Well, at least one thing: simulating quantum physics! As tautological as that sounds, I predict that if QCs ever become practical, simulating quantum physics will actually be the main thing that they’re used for. That actually has *tremendous* applications to materials science, drug design, understanding high-temperature superconductivity, etc.
But of course, what got everyone excited about this field was Peter Shor’s discovery, in 1994, that a quantum computer would be good for MORE than just simulating quantum physics. It could also be used to factor integers in polynomial time, and thereby break almost all of the public-key cryptography currently used on the Internet. (Interesting!)
Where we are: After 18 years and more than a billion dollars, I’m proud to say that a quantum computer recently factored 21 into 3*7, with high probability. (For a long time, it was only 15.) Scaling up is incredibly hard because of decoherence – the external environment, as it were, constantly trying to measure the QC’s state and collapse it down to classical. With classical computers, it took more than 100 years from Charles Babbage until the invention of the transistor. Who knows how long it will take in this case? But unless quantum mechanics itself is wrong, there doesn’t seem to be any fundamental obstacle to scaling this up. On the contrary, we now know that, IF the decoherence can be kept below some finite but nonzero level, then there are very clever error-correcting codes that can render its remaining effects insignificant. So, I’m optimistic that if civilization lasts long enough, we’ll eventually have practical quantum computers.
I don’t expect them to be right, but I hope they are! If so, it would be a revolution in physics
And for me, putting quantum mechanics to the test is the biggest reason to build QCs—the applications are icing!

14. Key point: factoring is not believed to be NP-complete!
And today, we don’t believe quantum computers can solve NP-complete problems in polynomial time in general (though not surprisingly, we can’t prove it)
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!)
If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow

15. 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

16. Most striking application so far of complexity to fundamental physics?
Hawking 1970s: Black holes radiate
The radiation seems thermal (uncorrelated with whatever fell in)—but if quantum mechanics is true, then it can’t be
Susskind et al. 1990s: “Black-hole complementarity.” In string theory / quantum gravity, the Hawking radiation should just be a scrambled re-encoding of the same quantum states that are also inside the black hole

17. The Firewall Paradox [Almheiri et al. 2012]
If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits (after waiting ~1067 years for enough to come out), then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior?
Entanglement among Hawking photons detected!

18. Harlow-Hayden 2013: Sure, there’s some unitary transformation that Alice could apply to the Hawking radiation, that would generate a “firewall” inside the event horizon. But how long would it take her to apply it?
They gave evidence that this problem would take exponential time, even for a quantum computer—in which case, long before one had made a dent in the problem, the black hole would’ve already evaporated anyway!
Their evidence relied on a lower bound on the number of steps needed by a quantum computer to find collisions (i.e., duplicates in a list of numbers), which I proved in 2002
Read their paper – there’s strings and branes flying all over the place but then when you finally get to why the problem is hard – the collision lower bound!
Recently I improved their argument, so that it no longer needs the collision lower bound, just one-way functions

19. Conclusion My suggested research agenda: Prove P≠NP
Prove that not even quantum computers can solve NP-complete problems
Build a scalable quantum computer (or even more interesting, show that it’s impossible)
Clarify whether all of known physics can be simulated by a quantum computer
Use computational complexity ideas to make progress toward a quantum theory of gravity