Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotically Optimal Quantum Circuits for d-level Systems   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.

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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
1
The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: math
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The categories of this paper are: chem-ph