Quantum circuits of gates: optimization and entanglement

We study quantum circuits composed of a sequence of \(\texttt {CNOT}\) gates between distant qubits of a n-qubit system. We present some results concerning two important topics related to these circuits: circuit optimization and emergence of entanglement. Regarding the optimization problem, we first describe the group structure underlying quantum circuits generated by \(\texttt {CNOT}\) gates (group isomorphism, group presentation), then we apply these mathematical results to the description of heuristics to reduce the number of gates in these circuits and we also propose an optimization algorithm for circuits of a few qubits. Concerning entanglement, we show how to create some useful entangled states when a circuit of \(\texttt {CNOT}\) gates acts on a fully factorized state. In the case of a 3-qubit system, we prove that a \(\texttt {CNOT}\) circuit acting on a fully factorized state can create all types of entanglement and we propose a method to evaluate the reliability of the implementation of a \(\mathtt {SLOCC}\)-equivalent to the state \(|\mathtt {W}_3\rangle \) in a quantum machine by computing the value of the hyperdeterminant. In the case of a 4-qubit system, we propose a circuit to compute a generic entangled state and a (computer-assisted) proof of the impossibility of creating a \(\mathtt {SLOCC}\)-equivalent to the state \(|\mathtt {W}_4\rangle \) from a circuit of \(\texttt {CNOT}\) gates acting on a factorized state