



@article {1373,
	title = {Asymptotically Optimal Quantum Circuits for d-level Systems},
	journal = {Physical Review Letters},
	volume = {94},
	year = {2005},
	month = {2005/6/14},
	abstract = {  As a qubit is a two-level quantum system whose state space is spanned by |0>,
|1>, so a qudit is a d-level quantum system whose state space is spanned by
|0>,...,|d-1>. Quantum computation has stimulated much recent interest in
algorithms factoring unitary evolutions of an n-qubit state space into
component two-particle unitary evolutions. In the absence of symmetry, Shende,
Markov and Bullock use Sard{\textquoteright}s theorem to prove that at least C 4^n two-qubit
unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa
(VMS) use the QR matrix factorization and Gray codes in an optimal order
construction involving two-particle evolutions. In this work, we note that
Sard{\textquoteright}s theorem demands C d^{2n} two-qudit unitary evolutions to construct a
generic (symmetry-less) n-qudit evolution. However, the VMS result applied to
virtual-qubits only recovers optimal order in the case that d is a power of
two. We further construct a QR decomposition for d-multi-level quantum logics,
proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing
the complexity question for all d-level systems (d finite.) Gray codes are not
required, and the optimal Theta(d^{2n}) asymptotic also applies to gate
libraries where two-qudit interactions are restricted by a choice of certain
architectures.
},
	doi = {10.1103/PhysRevLett.94.230502},
	url = {http://arxiv.org/abs/quant-ph/0410116v2},
	author = {Stephen S. Bullock and Dianne P. O{\textquoteright}Leary and Gavin K. Brennen}
}
