Overcoming the su(2^n) sufficient condition for the coherent control of n-qubit systems

Abstract

We study quantum systems with even numbers N of levels that are completely state-controlled by unitary transformations generated by Lie algebras isomorphic to sp(N) of dimension N(N+1)/2. These Lie algebras are smaller than the respective su(N) with dimension N^2-1. We show that this reduction constrains the Hamiltonian to have symmetric energy levels. An example of such a system is an n-qubit system. Using a geometric representation for the quantum wave function of a finite system, we present an explicit example that shows a two-qubit system can be controlled by the elements of the Lie algebra sp(4) (isomorphic to spin(5) and so(5)) with dimension ten rather than su(4) with dimension fifteen. These results enable one to envision more efficient algorithms for the design of fields for quantum-state engineering, and they provide more insight into the fundamental structure of quantum control.

arXiv comment13 pp., 2 figures

References in corpus (6)

• Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance
• Quantum computation over continuous variables
• Time Optimal Control in Spin Systems
• Control of trapped-ion quantum states with optical pulses
• Four-level and two-qubit systems, sub-algebras, and unitary integration
• Geometric phase for SU(N) through fiber bundles and unitary integration

Cited by in corpus (4)

• Lattice QCD and heavy ion collisions: a review of recent progress
• Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)
• Complete controllability of finite quantum systems with two-fold energy level degeneracy
• A general formulation based on algebraic spinors for the quantum computation