# SWAP specific phase components of two qubits

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

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

To resultant state:-

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

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 $$a_3$$.

• What are the normalization factors on $q_1, q_2$? Are $|0 \rangle, |1\rangle$ of equal probability or...? Jul 24 '20 at 22:59
• Yes of equal probability. Have updated the question. Jul 25 '20 at 6:02
• Take two possible different choices to your a coefficients, and assume a unitary exists that implements your desired transformation for those elements. By linearity you now know how it works for all input states. Does this coincide with what you want? (I assume not) Jul 25 '20 at 6:46
• Yes it does, but trying to figure out a unitary that doesn't depend on the value of $a_3$ Jul 25 '20 at 7:31