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The projector-controlled not gate $\text{C$_\Pi$NOT}$ is defined as

$$\text{C$_\Pi$NOT} \, \colon= \Pi \otimes X + (\mathbb{I}-\Pi)\otimes\mathbb{I}\,, \tag{1}$$ in András Gilyén et. al. (2018)[arXiv:1806.01838].

Can this gate be implemented efficiently, namely, polynomial size to qubits included by the circuit?

Particularly, how about the case that $\Pi$ is a projector onto $r$-dimensional subspace spanned by computational bases? I.e., $$\Pi = \sum_{i=1}^r |i\rangle\langle i|\,,\tag{2}$$where $\{|i\rangle\}_i$ are computational bases.

Maybe, I think that this problem is reduced to whether a measurement $\{\Pi, \mathbb{I}-\Pi\}$ is efficiently implementable in a coherent manner using multi-Toffoli gates and ancilla qubits.

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I suspect that depends on $\Pi$. There will be easily categorised cases where you can.

For example: $\Pi$ projects onto a subset of basis states for which there exists an efficient classical calculation that recognises membership of the subset).

There will be other $\Pi$ for which it's impossible, e.g. (I suspect) a rank 1 projector onto a (family of) quantum states that cannot be efficiently prepared on a quantum computer.

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  • $\begingroup$ How about in the case that $\Pi$ is a projector onto a $r$-dimensional subspace spanned by computational bases? I.e., $\Pi = \sum_{i=1}^r |i\rangle\langle i|$, where $\{| i\rangle\}_i$ is computational bases. I think that if we use $r$ multi-controlled Toffoli gate (which has $\mathcal{O}(n)$ circuit complexity), we can trivially implement this C$_\Pi$NOT gate in $\mathcal{O}(rn)$. However, if $r$ is exponential, it is inefficient. Then, I don't find that more efficiently implementable... $\endgroup$
    – Takimoto.R
    Nov 28, 2023 at 11:08
  • $\begingroup$ Sorry, $n$ denotes the number of qubits $\endgroup$
    – Takimoto.R
    Nov 28, 2023 at 11:24
  • $\begingroup$ this is the same calculation as introducing an ancilla and evaluating $i\leq r$. There's an efficient classical algorithm for that. $\endgroup$
    – DaftWullie
    Nov 28, 2023 at 12:03
  • $\begingroup$ Is it correct to understand it as efficiently implementable (using ancillae maintaining superposition) since there exists an efficient classical algorithm that judges whether i≤r or not, right? If it's not inconvenient, could you please provide me with references? $\endgroup$
    – Takimoto.R
    Nov 29, 2023 at 2:44
  • $\begingroup$ I've got no idea about a reference (not my strong point!), but it's a very simple procedure: compute a 0/1 value onto an ancilla. Do a controlled-not off that ancilla. Then invert the computation that set the value of the ancilla. Any efficient classical algorithm (such as $i\leq r$) can be done in this way. $\endgroup$
    – DaftWullie
    Nov 29, 2023 at 7:46

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