If I have an $n$ qubit system and a projector $P$ such as $P_0 = \left|0\right>^{\otimes n}\left<0\right|^{\otimes n}$ (as an example) on those qubits, is there an efficient way to implement the unitary $e^{-i\theta P}$?

In the specific example given, there is a simple but inefficient implementation in terms of exponentials of Paulis: $$P_0 = \sum_{j\in\left\lbrace0, 1\right\rbrace^n}\bigotimes_{k=1}^nZ_k^{j_k},$$ where $Z^1_k$ is the Pauli-$Z$ operator on qubit $k$ and $Z^0_k$ is the identity on qubit $k$. That is, the given projector is the sum over all possible permutations of identity and $Z$. As $Z$ and identities all commute, this can then be written as $$U_{P_0} = \prod_{j\in\left\lbrace0, 1\right\rbrace^n}e^{-i\theta\bigotimes_{k=1}^nZ_k^{j_k}}.$$ However, for $n$ qubits, this requires an exponential number of gates. Is there any way to improve upon this?

Funnily enough, using e.g. the Reck or Clements scheme (in photonics), this appears to be trivial, as it's simply a phase on a single mode. However, to my knowledge, this doesn't count as it requires an expontial number of resources to recreate the evolution given by a unitary acting on $n$ qubits.


With auxiliary qubits, you could construct a gate that is controlled on the $n$ qubits being $0$ to apply $X$ to the auxiliary. This can be done with polynomially many gates and auxiliaries.

Once you have that, you can apply contolled $e^{-i \theta}$ on any of the original $n$.

Then do all the uncomputation.

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