### Paul Bernays Lectures

Monday, September 9th, 2019Last week, I had the honor of giving the annual Paul Bernays Lectures at ETH Zürich. My opening line: “as I look at the list of previous Bernays Lecturers—many of them Nobel physics laureates, Fields Medalists, etc.—I think to myself, how badly did you have to screw up this year in order to end up with me?”

Paul Bernays was the primary assistant to David Hilbert, before Bernays (being Jewish by birth) was forced out of Göttingen by the Nazis in 1933. He spent most of the rest of his career at ETH. He’s perhaps best known for the von Neumann-Bernays-Gödel set theory, and for writing (in a volume by “Hilbert and Bernays,” but actually just Bernays) arguably the first full proof of Gödel’s Second Incompleteness Theorem.

Anyway, the idea of the Paul Bernays Lectures is to rotate between Bernays’s different interests in his long, distinguished career—interests that included math, philosophy, logic, and the foundations of physics. I mentioned that, if there’s any benefit to carting me out to Switzerland for these lectures, it’s that quantum computing theory combines *all* of these interests. And this happens to be the moment in history right before we start finding out, directly from experiments, whether quantum computers can indeed solve certain special problems much faster.

The general theme for my three lectures was “Quantum Computing and the Fundamental Limits of Computation.” The attendance was a few hundred. My idea was to take the audience from Church and Turing in the 1930s, all the way to the quantum computational supremacy experiments that Google and others are doing now—as part of a single narrative.

If you’re interested, streaming video of the lectures is available as of today (though I haven’t watched it—let me know if the quality is OK!), as well as of course my slides. Here you go:

**Lecture 1: The Church-Turing Thesis and Physics (watch streaming / PowerPoint slides)** **(with an intro in German by Giovanni Sommaruga, who knew Bernays, and a second intro in English by Renato Renner, who appeared on this blog here)**

Abstract:Is nature computable? What should we even mean in formulating such a question? For generations, the identification of “computable” with “computable by a Turing machine” has been seen as either an arbitrary mathematical definition, or a philosophical or psychological claim. The rise of quantum computing and information, however, has brought a fruitful new way to look at the Church-Turing Thesis: namely, as a falsifiable empirical claim about the physical universe. This talk seeks to examine the computability of the laws of physics from a modern standpoint—one that fully incorporates the insights of quantum mechanics, quantum field theory, quantum gravity, and cosmology. We’ll critically assess ‘hypercomputing’ proposals involving (for example) relativistic time dilation, black holes, closed timelike curves, and exotic cosmologies, and will make a 21st-century case for the physical Church-Turing Thesis.

**Lecture 2: The Limits of Efficient Computation (watch streaming / PowerPoint slides)**

Abstract:Computer scientists care about what’s computable not only in principle, but within the resource constraints of the physical universe. Closely related, which types of problems are solvable using a number of steps that scales reasonably (say, polynomially) with the problem size? This lecture will examine whether the notorious NP-complete problems, like the Traveling Salesman Problem, are efficiently solvable using the resources of the physical world. We’ll start with P=?NP problem of classical computer science—its meaning, history, and current status. We’ll then discuss quantum computers: how they work, how they can sometimes yield exponential speedups over classical computers, and why many believe that not even they will do so for the NP-complete problems. Finally, we’ll critically assess proposals that would use exotic physics to go even beyond quantum computers, in terms of what they would render computable in polynomial time.

**Lecture 3: The Quest for Quantum Computational Supremacy (watch streaming / PowerPoint slides)**

Abstract:Can useful quantum computers be built in our world? This talk will discuss the current status of the large efforts currently underway at Google, IBM, and many other places to build noisy quantum devices, with 50-100 qubits, that can clearly outperform classical computers at least on some specialized tasks — a milestone that’s been given the unfortunate name of “quantum supremacy.” We’ll survey recent theoretical work (on BosonSampling, random circuit sampling, and more) that aims to tell us: which problems should we give these devices, that we’re as confident as possible are hard for classical computers? And how should we check whether the devices indeed solved them? We’ll end by discussing a new protocol, for generating certified random bits, that can be implemented almost as soon as quantum supremacy itself is achieved, and which might therefore become the first application of quantum computing to be realized.

Finally, thanks so much to Giovanni Sommaruga and everyone else at ETH for arranging a fantastic visit.