1. Scott Aaronson (UT Austin) Lakeway Men’s Breakfast Club April 19, 2017
Quantum Computing
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 (UT Austin)
Lakeway Men’s Breakfast Club
April 19, 2017

2. The field of quantum computing and information arguably started here in Austin in the early 1980s—with David Deutsch and other students, faculty, and postdocs in physics
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.
Together with colleagues, we’re now seeking to build up a new quantum computing and information presence at UT Austin

3. 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?

4. 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!

5. 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

6. 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)

7. THE RULES: If a system can be in two distinguishable states, labeled |0 and |1, it can also be in a superposition, written
|0 + |1
Here  and  are complex numbers called amplitudes, which satisfy ||2+||2=1. A 2-state superposition is called a qubit.
If we observe, we see |0 with probability ||2 and |1 with probability ||2. But if the qubit is isolated, it evolves by rules different from those of classical probability.
In the 1980s, Feynman, Deutsch, and others noticed that a system of n qubits seems to take ~2n steps to simulate on a classical computer, because of the phenomenon of entanglement between the qubits. They had the amazing idea of building a quantum computer to overcome that problem

8. Popularizers Beware: A quantum computer is NOT like a massively-parallel classical computer!
Exponentially many possible answers, but you only get to observe one of them
Any hope for a speedup rides on choreographing an interference pattern that boosts the amplitude of the right answer

9. 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

10. Can QCs Actually Be Built?
Where we are now: A quantum computer can factor 21 into 37, with high probability…
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.
The #1 application of QC, in my mind: disproving those people!
What makes many of us optimistic of eventual success: the Quantum Fault-Tolerance Theorem

11. 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

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. 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 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