Abstract:
In this lecture, we proved that any arbitrary unitary operation can be implemented using single-qubit rotations and entangling operations $C\mathrm{NOT}$. By calculating the real parameters, we verified that an exponential number of entanglement operations is required for the exact synthesis of arbitrary unitary transformation. The accuracy of the approximation of unitary operations degrades no worse than linearly with the increase in the number of qubits, that is why arbitrary quantum operation can also be represented by a discrete set of elementary operations. After discussing quantum measurements and classical control, we examined the deferred measurement principle. Finally, we introduced the concept of quantum circuits – arbitrary and unitary – and defined the complexity class $\mathrm{BQP}$.
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