SWAP specific phase components of two qubits

Quantum Computing Asked on January 3, 2022

Is it possible to perform an operation on two qubits with initial states as follows:

$$q_1: 1/sqrt(2)(|0rangle + exp(0.a_1a_2a_3)|1rangle)$$
$$q_2: 1/sqrt(2)(|0rangle + |1rangle)$$

To resultant state:-

$$q_1: 1/sqrt(2)(|0rangle + exp(0.a_1a_2)|1rangle)$$
$$q_2: 1/sqrt(2)(|0rangle + exp(0.a_3)|1rangle)$$

Without knowing the value of $a_3$. Where $a_1,a_2,a_3 ∈ [0, 1].$

The idea is to shift the phase of $q_1$ by $exp(-0.00a_3)$ and $q_2$ by $exp(0.a_3)$ with the unitary operation not being aware of the value of $a_3$.

One Answer

No, it's not possible to extract digits of the phase like that. It would violate the Holevo bound. In general there's no way to "amplify" single small phase differences into big phase differences, because of linearity.

Answered by Craig Gidney on January 3, 2022

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