Quantum money and scalable 21-cm cosmology

Abstract

This thesis covers two unrelated topics. The first part of my thesis is about quantum money, a cryptographic protocol in which a mint can generate a quantum state that no one can copy. In public-key quantum money, anyone can verify that a given quantum state came from the mint, and in collision-free quantum money, even the mint cannot generate two valid quantum bills with the same serial number. I present quantum state restoration, a new quantum computing technique that can be used to counterfeit several designs for quantum money. I describe a few other approaches to quantum money, one of which is published, that do not work. I then present a technique that seems to be secure based on a new mathematical object called a component mixer, and I give evidence money using this technique is hard to counterfeit. I describe a way to implement a component mixer and the corresponding quantum money using techniques from knot theory. The second part of my thesis is about 21-cm cosmology and the Fast Fourier transform telescope. With the FFT telescope group at MIT, I worked on a design for a radio telescope that operates between 120 and 200 MHz and will scale to an extremely large number of antennas N. We use an aperture synthesis technique based on Fast Fourier transforms with computational costs proportional toN logN instead of N2. This eliminates the cost of computers as the main limit on the size of a radio interferometer. In this type of telescope, the cost of each antenna matters regardless of how large the telescope becomes, so we focus on reducing the cost of each antenna as much as possible. I discuss the FFT aperture synthesis technique and its equivalence to standard techniques on an evenly spaced grid. I describe analog designs that can reduce the cost per antenna. I give algorithms to analyze raw data from our telescope to help debug and calibrate its components, with particular emphasis on cross-talk between channels and I/Q imbalance. Finally, I present a scalable design for a computer network that can solve the corner-turning problem.