Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)

TL;DR

This paper develops a geometric framework using Bloch sphere analogs for higher-dimensional SU(N) groups, especially SU(4), to describe quantum evolution as rotations of real vectors, aiding quantum computation understanding.

Contribution

It introduces a systematic method to construct Bloch-like representations for SU(N), focusing on SU(4), and describes the associated geometric manifolds for quantum evolution.

Findings

Derived Bloch-like rotations for SU(4) Hamiltonians

Identified geometric manifolds for SU(4) and its subgroups

Expressed two-qubit evolution as rotations of real vectors

Abstract

A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) sub-group and an eight-dimensional Grassmannian manifold for the general SU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint.