# Constructing exact symmetric informationally complete measurements from numerical solutions

## TL;DR

This paper introduces a novel method to convert high-precision numerical solutions of SIC POVMs into exact algebraic solutions using Galois symmetries, significantly expanding the set of known solutions and supporting related number-theoretic conjectures.

## Contribution

The authors develop a new approach combining integer relation algorithms and Galois symmetries to find exact SIC solutions from numerical data, surpassing previous methods in scope.

## Key findings

- Calculated 69 new exact SIC solutions
- Discovered solutions in high-degree number fields up to 12,288
- Confirmed SIC solutions obey number-theoretic conjectures

## Abstract

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work.