Quantum codes as robust phases of matter

Abstract: There is a deep connection between quantum error correction and phases of matter for spatially local codes in finite dimensions.  I will show how this analogy extends to more general settings: quantum codes with check soundness are absolutely stable phases of matter.  These codes include constant-rate quantum low-density parity-check codes, which shows that the third law of thermodynamics is false: there exist absolutely stable phases of matter with constant entropy density at zero temperature.

A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures

Abstract: In this talk, I will present a construction of symmetric informationally complete POVMs (SIC-POVMs), a special class of quantum measurements whose existence in all dimensions was conjectured by Zauner in 1999. Equivalently, these are maximal sets of d^2 equiangular lines in ℂ^d. Our approach introduces an explicit mathematical object, the ghost SIC, built from number-theoretic properties of a special modular function, and we show that it is Galois conjugateto an actual SIC.

A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures

In this talk, I will present a construction of symmetric informationally complete POVMs (SIC-POVMs), a special class of quantum measurements whose existence in all dimensions was conjectured by Zauner in 1999. Equivalently, these are maximal sets of d^2 equiangular lines in ℂ^d. Our approach introduces an explicit mathematical object, the ghost SIC, built from number-theoretic properties of a special modular function, and we show that it is Galois conjugateto an actual SIC.

Optimization by Decoded Quantum Interferometry

In this talk I will describe Decoded Quantum Interferometry (DQI), a quantum algorithm for reducing classical optimization problems to classical decoding problems by exploiting structure in the Fourier spectrum of the objective function. (See: https://arxiv.org/abs/2408.08292.) For a regression problem called optimal polynomial intersection, which has been previously studied in the contexts of coding theory and cryptanalysis, we believe DQI achieves an exponential quantum speedup.