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1

The proof consists in connecting together two arguments. The first, covered by the exercise, reduces the problem of approximating the rotation gate $R_\hat{n}(\alpha)$ to the problem of approximating the rotation angle $\alpha$. The second, described in the quoted text from Nielsen & Chuang, shows that one can achieve arbitrarily fine approximations of ...


4

Theoretical lower bound In contrast to the answer by Bertrand, I will assume that along with a $CNOT$ gate we have arbitrary single-qubit unitaries on our disposal. In this case, one can derive the theoretical lower bound on the number of $CNOTs$ neseccary to decompose an arbitrary $n$-qubit unitary $$ L:=\#\text{CNOTs} \geq \frac14\left(4^n-3n-1\right) \...


4

Let me try to reformulate your question: Given a Universal Set of Quantum Gates $\mathcal{G}$; and some $n$-bit Unitary $U$. Can we find some $q$ such that $q$ is the minimum number of gates selected from $\mathcal{G}$ to have the effect of $U$ on $n$ qubits? (Note: I changed some variables since typically $n$ is the number of bits and $N=2^n$ is the ...


3

As @AdamZalcman pointed out on the comments, $\theta$ is a parameter that can take in any value from $0$ to $2\pi$. This gate is a way of generalizing the three-bit Toffoli gate which is universal for classical boolean logic, into a gate which is universal for quantum logic. The similarity between both gates can be seen easier with its matrix representation (...


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