Geometric phase for SU(N) through fiber bundles and unitary integration

Abstract

Geometric phases are important in quantum physics and now central to fault tolerant quantum computation. For spin-1/2 and SU(2), the Bloch sphere $S^2$, together with a U(1) phase, provides a complete SU(2) description. We generalize to $N$-level systems and SU($N$) in terms of a $2(N-1)$-dimensional base space and reduction to a ($N$-1)-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an $N$-level system and gives ($N$-1) geometric phases explicitly. A complete analytical construction of a S^4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).

4 pages. Following referee reports in April 2006, stylistic changes made throughout and a specific application added at the end

References in corpus (6)

• Demonstration of conditional gate operation using superconducting charge qubits
• Nonadiabatic Geometric Quantum Computation Using A Single-loop Scenario
• Four-level and two-qubit systems, sub-algebras, and unitary integration
• A note on the geometric phase in adiabatic approximation
• Complete time-dependent treatment of a three-level system
• Geometric Phase in Eigenspace Evolution of Invariant and Adiabatic Action Operators

Cited by in corpus (10)

• Algebraic characterization of X-states in quantum information
• Dynamical invariants for quantum control of four-level systems
• General linear-optical quantum state generation scheme: Applications to maximally path-entangled states
• Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)
• Bloch sphere like construction of SU(3) Hamiltonians using unitary integration
• Overcoming the su(2^n) sufficient condition for the coherent control of n-qubit systems
• Quantum Hall Effect on the Grassmannians ${\mathbf{Gr}}_2({\mathbb{C}}^N)$
• Reduced Dynamics from the Unitary Group to Some Flag Manifolds : Interacting Matrix Riccati Equations
• Vector Properties of Entanglement in a Three-Qubit System
• Hilbert-Schmidt measure of pairwise quantum discord for three-qubit $X$ states