arXiv quant-ph/07030822007-03-09quant-ph

关于量子线路的复杂性等价

On the Quantum Circuit Complexity Equivalence

Milosh Drezgich, Shankar Sastry

备注：: QIP 2007

摘要：

Nielsen \cite{Nielsen05}最近提出了以下问题：“在不使用辅助量子比特的情况下，精确实现指定的$% \mathit{n}$ -量子比特酉操作$U$所需的最小量子线路是多少？“尼尔森能够证明，最小尺寸电路的下界是由恒等式$I$和$U$之间的测地线的长度提供的，其中长度由$SU(2^{n})$上的适当Finsler度量定义。对于给定的Finsler结构$F$，我们证明了模拟$U$的最小电路尺寸与测地线长度和模拟参数成线性关系.作为一个推论，我们证明了$O(\frac{% n^{4}}{p}d_{F_{p}}^{2}(I,U)L_{F_{p}}(I,\tilde{U})) $的最高下限和最低上限的$Ω(n^{4}d_{F_{p}}^{3}(I,U))$，为标准的模拟技术。因此，我们的研究结果表明，通过标准模拟，人们不能期望一个更好的$n^{2}$倍的改善，从尼尔森，道林，顾和多尔蒂\cite{Nielsen06}的结果的上限。此外，我们的等价结果可以应用到流形上的任意路径，包括一个生成的几何。

英文摘要：

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 $Ω(n^{4}d_{F_{p}}^{3}(I,U))$, 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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英文摘要：

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