A Construction for SIC-POVMs

Abstract: In quantum information, symmetric informationally complete measurements (SIC-POVMs) serve as elegant and efficient tools for numerous tasks like state tomography, QKD, and more. These objects have appeared in other contexts such as frame theory and design theory; for example, they are minimal spherical 2-designs. There are even intriguing connections to open problems in number theory. It has been an open problem for 25 years to prove that SIC-POVMs exist in infinitely many dimensions.

A Construction for SIC-POVMs

In quantum information, symmetric informationally complete measurements (SIC-POVMs) serve as elegant and efficient tools for numerous tasks like state tomography, QKD, and more. These objects have appeared in other contexts such as frame theory and design theory; for example, they are minimal spherical 2-designs. There are even intriguing connections to open problems in number theory. It has been an open problem for 25 years to prove that SIC-POVMs exist in infinitely many dimensions.

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.

Quantum complexity in many-body physics: random circuits and thermodynamics

Quantum complexity is emerging as a key property of many-body systems, including black holes, topological materials, and early quantum computers. A state's complexity quantifies the number of computational gates required to prepare the state from a simple tensor product. I will discuss two approaches to better understand the role of quantum complexity in many-body physics. First, we'll consider random circuits, a model for chaotic dynamics. In such circuits, the quantum complexity grows linearly until it saturates at a value exponential in the system size.

Achieving low circuit depth with few qubits, for arithmetic and the QFT

In this work we present fast constructions for the quantum Fourier transform and quantum integer multiplication, using few ancilla qubits compared to the size of the input. For the approximate QFT we achieve depth O(log n) using only n + O(n / log n) total qubits, by applying a new technique we call "optimistic quantum circuits." To our knowledge this is the first circuit for the AQFT with space-time product O(n log n), matching a known lower bound.