Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Quantum Circuit Complexity Equivalence   Nielsen \cite{Nielsen05} recently asked the following question: "What is the
minimal size quantum circuit required to exactly implement a specified $%
\mathit{n}$-qubit unitary operation $U$, without the use of ancilla qubits?"
Nielsen was able to prove that a lower bound on the minimal size circuit is
provided by the length of the geodesic between the identity $I$ and $U$, where
the length is defined by a suitable Finsler metric on $SU(2^{n})$. We prove
that the minimum circuit size that simulates $U$ is in linear relation with the
geodesic length and simulation parameters, for the given Finsler structure $F$.
As a corollary we prove the highest lower bound of $O(\frac{%
n^{4}}{p}d_{F_{p}}^{2}(I,U)L_{F_{p}}(I,\tilde{U})) $and the lowest upper bound
of $\Omega (n^{4}d_{F_{p}}^{3}(I,U))$, for the standard simulation technique.
Therefore, our results show that by standard simulation one can not expect a
better then $n^{2}$ times improvement in the upper bound over the result from
Nielsen, Dowling, Gu and Doherty \cite{Nielsen06}. Moreover, our equivalence
result can be applied to the arbitrary path on the manifold including the one
that is generated adiabatically.

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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom