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Question: The operator 1/2 (I + sigma_A middot sigma_B) is called the SWAP operator. Its matrix representation in the basis {|00>, |01>, |10>, |11>} is U_SWAP = (1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1). Check that its squareroot U^1/2_SWAP is given by U^1/2_SWAP = 1/1+ i (1 + i 0 0 0 0 1 i 0 0 i 1 0 0 0 0 1 + i). and that the so-called cZ gate can be constructed from cZ = e^i pi sigma^A_z/4 e^- i pi sigma^B_z/4 U^1/2_SWAP e^i pi sigma^A_z/2 U^1/2_SWAP = (I 0 0 sigma_z). – Free Chegg Question Answer

Transcribed text From Image: The operator 1/2 (I + sigma_A middot sigma_B) is called the SWAP operator. Its matrix representation in the basis {|00>, |01>, |10>, |11>} is U_SWAP = (1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1). Check that its squareroot U^1/2_SWAP is given by U^1/2_SWAP = 1/1+ i (1 + i 0 0 0 0 1 i 0 0 i 1 0 0 0 0 1 + i). and that the so-called cZ gate can be constructed from cZ = e^i pi sigma^A_z/4 e^- i pi sigma^B_z/4 U^1/2_SWAP e^i pi sigma^A_z/2 U^1/2_SWAP = (I 0 0 sigma_z).

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