



01765nas a2200145   4500008004100000245006400041210006300105260001400168490000700182520131400189100002501503700002401528700002301552856004401575       2005                        eng d00aAsymptotically Optimal Quantum Circuits for d-level Systems0 aAsymptotically Optimal Quantum Circuits for dlevel Systems  c2005/6/140 v943 a  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'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'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.
1 aBullock, Stephen, S.1 aO'Leary, Dianne, P.1 aBrennen, Gavin, K.  uhttp://arxiv.org/abs/quant-ph/0410116v2