Assuming a cloning oracle, satisfiability, which is an NP complete problem, is shown to belong to $BPP^C$ and $BQP^C$ (depending on the ability of the oracle C to clone either a binary random variable or a qubit). The same result is extended in the case of an approximate cloning oracle, thus establishing that $NP \subseteq BPP^C \subseteq BQP^C$ and $NP \subseteq BPP^{AC} \subseteq BQP^{AC}$, where C and AC are the exact and approximate cloning oracles, respectively. Although exact cloning is impossible in quantum systems, approximate cloning remains a possibility. However, the best known methods for approximate cloning (based on unitary evolution) do not currently achieve the desired precision levels. And it remains an open question whether they could be improved when non-linear (or non-unitary) operators are used. Finally, a straightforward attempt to dispense with cloning, replacing it by unitary evolution, is proved to be impossible.
Teleportation scheme implementing contextually the Universal Optimal Quantum Cloning Machine and the Universal Not Gate. Complete experimental realization
Beyond Maximal Entanglement: Exact Resources for Multiparty Encrypted Quantum Cloning
S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）；Centre for Interdisciplinary Areas in Quantum Computing, Indian Institute of Technology Indore（印度印多尔理工学院量子计算跨学科领域中心）；Department of Physics, Indian Institute of Technology Indore（印度印多尔理工学院物理系）
University of Colorado, Boulder（科罗拉多大学博尔德分校）；University of Maryland, College Park（马里兰大学帕克分校）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）
