We study in details the algebraic structure underlying quantum circuits generated by CNOT gates. Our results allow us to propose polynomial heuristics to reduce the number of gates used in a given CNOT gates circuit and we also give algorithms to optimize this type of circuits in some particular cases. Finally we show how to create some usefull entangled states using a CNOT gates circuit acting on a fully factorized state.
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While liberalization in energy markets has been a widely successful process all over the world, incumbents often still hold a dominant position. Thus, electricity wholesale markets are subject to market surveillance. Nevertheless, consolidated findings on abusive practices of market power and their cause and effect in wholesale electricity markets are scarce and non-controversial market monitoring practices fail to exist. Our application of the established measure of market concentration RSI shows that it serves as a decent indicator for the rents that can be gained in the market but also reveals considerable weaknesses of the RSI. Therefore, we propose and apply the "Return on Withholding Capacity Index" (RWC) representing a measure of the firms' incentive of withholding capacity as a complementary index to the RSI.
We have studied the algebraic structure underlying the quantum circuits composed by c − Z and SWAP gates. Our results are applied to optimize the circuits and to understand the emergence of entanglement when a circuit acts on a fully factorized state.
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