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Prove that sets of gates $\{CNOT^a, H^b, T^c\}$ are not universal

First gate set: only has single qubit gates. The group of single qubit gates is closed under composition. E.g. you can't get to CNOT because it's not a single qubit gate. Second gate set: only has ...
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Can anybody explain the following circuit?

The notation of $\oplus$ does not simply denote addition. It denotes bitwise addition modulo 2, i.e. $$ 0\oplus 0=0,\quad 0\oplus1=1\oplus 0=1,\quad 1\oplus 1=0 $$ which is exactly the same as the xor ...
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3 votes
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Can anybody explain the following circuit?

Note that the first CNOT gives you $$ CNOT |a\rangle |0\rangle = |a\rangle |a \oplus 0 \rangle = |a\rangle |a \rangle$$ and then the second CNOT gives you $$ CNOT |b\rangle |a\rangle = |b\rangle |b ...
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5 votes
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How can a finite set of gates be universal?

$\{H,CCZ\}$ is just the generating set. The set of gates it generates, i.e. $\langle H, CCZ \rangle$, has an infinite number of gates so it could be universal. The argument in Eastin-Knill says the ...
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