A Study of Linear Joint and Tool Models in Spindle-Holder-Tool Receptance Coupling
Timothy J. Burns, T L. Schmitz
As a qubit is a two-level system whose state space is spanned by and , so a qudit is a -level system whose state space is spanned by ,..., . Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an -qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov, and Bullock use Sard's theorem to prove that at least two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands two-qudit unitary evolutions to construct a generic (symmetry-less) -qudit evolution. However, the VMS result applied to virtual qubits only recovers optimal order in the case that is a power of two. We further construct a decomposition for multi-level quantum logics, proving a sharp asymptotic of two-qudit gates and thus closing the complexity question for all -level systems ( finite). Gray codes are not required.
Proceedings of the Fifth ASME International Conference on Multibody Systems Nonlinear Dynamics and Control
Long Beach, CA
September 24-28, 2005
ASME 2005 International Design Engineering Technical Conferences
and Schmitz, T.
A Study of Linear Joint and Tool Models in Spindle-Holder-Tool Receptance Coupling, Proceedings of the Fifth ASME International Conference on Multibody Systems Nonlinear Dynamics and Control
(Accessed June 8, 2023)