1. Scott Aaronson Computer Science, UT Austin AAAS Meeting, Feb. 19, 2017
The Limits of Computation
Quantum Computers and Beyond
Scott Aaronson
Computer Science, UT Austin
AAAS Meeting, Feb. 19, 2017

2. Moore’s Law
Moore’s Law. I’m sure you’ve all seen this before: the number of transistors per computer has doubled pretty much every two years. This is arguably the main thing that’s driven the progress of human civilization since World War II.

3. Extrapolating: Robot uprising?
First and most obvious is the robot uprising. Someday soon Google is going to become sentient, and will instruct all the computers on the Internet to enslave their owners. Don’t believe me? Google it!

4. But even a killer robot would still be “merely” a Turing machine, operating on principles laid down in the 1930s… =
But at a fundamental level, all computers we talk about today (even killer robots!) are still just a Turing machine – this theoretical device that was invented in the 1930s and that we teach you about in our undergraduate courses. Macs, PCs, killer robots: on the inside, they’re all the same stuff. So is there anything else beyond that, that’s more interesting?

5. And it’s conjectured that thousands of interesting problems are inherently intractable for Turing machines…
So if we extrapolate Moore’s Law, what can we look forward to next?
Is there any feasible way to solve these problems, consistent with the laws of physics?

6. 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!

7. 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

8. 21 = 3 × 7 (with high probability)
Quantum Computers
What we’ve learned from quantum computers so far:
21 = 3 × 7 (with high probability)
The first is quantum computers – yes, that’s really what they look like! This happens to be my research area. A quantum computer is a hypothetical machine that would exploit the wave nature of quantum mechanics to solve certain problems, like factoring integers and breaking most of the cryptographic codes used on the Internet, dramatically faster than we know how to solve them with any computer today.
So, what’s been the progress so far in quantum computing? After 16 years, more than a billion of dollars of investment, and the building of ion-trap and nuclear-magnetic resonance devices the size of rooms, we’ve learned that, *with high probability*, 15=3x5.
Alright, so maybe quantum computing still has a ways to go.

9. 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.

10. Any hope for a speedup rides on the magic of quantum interference
Journalists 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 quantum interference

11. 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

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. Some of My Recent Research
“QUANTUM SUPREMACY”: Getting a clear quantum speedup for some task—not necessarily a useful one
BosonSampling (with Alex Arkhipov): A proposal for a simple optical quantum computer to sample a distribution that can’t be sampled efficiently classically (unless P#P=BPPNP)
Experimentally demonstrated with 6 photons by O’Brien group at Bristol
Random Quantum Circuit Sampling: Martinis group at Google is planning a system with high-quality superconducting qubits in the near future; we’re thinking about what to do with it that’s classically hard

14. Complexity of Decoding Hawking Radiation
Hawking famously asked in the 1970s how information can escape from a black hole, as it must if QM is universally valid
His question led to the proposal of black hole complementarity (Susskind, ‘t Hooft 1990s)
But then the “firewall paradox” (AMPS 2012) said that, by doing a suitable measurement on the Hawking radiation, you could destroy the spacetime geometry inside the black hole!
Harlow and Hayden 2013: Yes, but that measurement would probably require performing an exponentially long quantum computation! (For a solar-mass black hole: ~210^67 years)
I’ve improved Harlow and Hayden’s argument to base it on “standard” crypto assumptions (injective OWFs)

15. Summary
From a theoretical standpoint, modern computers are “all the same slop”: polynomial-time Turing machines
Quantum computing is interesting as the first serious proposal that would go beyond thi
Even going a bit beyond the limits of today’s computers (say, with quantum supremacy experiments) is a challenge
Contrary to what you read, even quantum computers would face significant limitations
But those limitations could help protect the geometry of spacetime!