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For a single qubit and a quantum operation, the fidelity $$ F_{|\psi\rangle\!\langle \psi|} = \mathrm{Tr}\bigg(U | \psi \rangle\!\langle \psi | U^\dagger \mathcal M\big(| \psi \rangle\!\langle \psi |\big)\bigg), $$

averaged over all possible initial states $|\psi\rangle$ (i.e., over the Bloch sphere) can be calculated by the fidelity averaged over the six states $|\pm\rangle,|\pm\rangle_x,|\pm \rangle_y$ (see this paper).

Is there a similar result for two qubits or more?

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    $\begingroup$ Could you add more details to your question? Please make your question independent of references. $\endgroup$
    – MonteNero
    Commented Oct 27, 2023 at 3:59
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    $\begingroup$ have you come across the concept of a quantum $t$-design? I'm not overly familiar with the details, but the broad concept is that a $t$-design gives a set of states on up to $t$ qubits that, for calculating averages, is as good as averaging over all possible states. IIRC the Clifford group (i.e. stabilizer states) serves your purpose. $\endgroup$
    – DaftWullie
    Commented Oct 27, 2023 at 7:48
  • $\begingroup$ @DaftWullie No, do you have a good reference for it that is not too mathematical? $\endgroup$
    – Omar Nagib
    Commented Oct 27, 2023 at 13:42
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    $\begingroup$ These things have a habit of getting very technical very quickly! $\endgroup$
    – DaftWullie
    Commented Oct 27, 2023 at 14:50

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You can, as long as the states form a complex-projective 2-design.

To see it, start from expanding the given expression and observing that it is linear in $\mathbb{P}_\psi\otimes\mathbb{P}_\psi$, where $\mathbb{P}_\psi\equiv|\psi\rangle\!\langle \psi|$. More explicitly, this means that it can be reframed as an expression of the form $$F_{\psi}= \langle A, \mathbb{P}_\psi\otimes\mathbb{P}_\psi\rangle \equiv \operatorname{tr}(A^\dagger(\mathbb{P}_\psi\otimes\mathbb{P}_\psi)),$$ for some operator $A$ that you can find explicitly expanding all terms in your definition.

Therefore when averaging the fidelity over all states, you are effectively averaging over $\mathbb{P}_\psi\otimes\mathbb{P}_\psi$ over all (uniformly random) states $\psi$. A complex-projective 2-design is defined as a finite set of vectors such that averaging this quantity over these vectors equals the average over all states, hence the result. See e.g. https://en.wikipedia.org/wiki/Quantum_t-design for more info.

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