Towards a Complexity Theory for the Quantum Age


Henry Yuen
Columbia University


Anand Natarajan
Abstract: How hard is it to compress a quantum state? To fast-forward the evolution of a local Hamiltonian? To unscramble the Hawking radiation of a black hole? Traditional complexity theory -- which is centered around decision problems and tasks with classical inputs and outputs -- appears inadequate for reasoning about the complexity of such tasks involving quantum inputs and outputs.

I'll discuss why we need a "fully quantum" complexity theory, and will describe some facets of such a theory. As a key illustration I'll explain how a "fully quantum" task called the Uhlmann Transformation Problem characterizes the complexity of seemingly unrelated problems, such as decoding noisy quantum channels, performing the Harlow-Hayden black hole radiation decoding task, and breaking the security of quantum bit commitment schemes. I will describe some of the many open problems and directions to explore in the world of fully quantum complexity theory.

Bio: Henry Yuen is an Associate Professor of Computer Science at Columbia University. His research focuses on the interplay between quantum computing, complexity theory, cryptography, and information theory. Yuen received a BA in mathematics from the University of Southern California in 2010, and received his PhD in computer science at MIT in 2016. He is a recipient of the NSF CAREER award and a Sloan Fellowship.