is it true (H⊗H)CZ(H⊗H) = CNOT?

is it true (H⊗H)CZ(H⊗H) = CNOT? Ask Question Asked 1 year ago Modified today Viewed 132 times 0 is it true (H⊗H)CZ(H⊗H) = CNOT?, I have calculated it but did not get the right result. But people said that the result of the operation is the CNOT gate. quantum-gate Share Improve this question Follow asked Mar 10, 2025 at 9:48 duest 1 Add a comment 3 Answers Sorted by: Highest score (default) Date modified (newest first) Date created (oldest first) 2 Recall that HXH=Z 𝐻 𝑋 𝐻 = 𝑍 (Hadamard interchanges between the X 𝑋 and Z 𝑍 bases), and that the CZ 𝐶 𝑍 gate is invariant under swapping the qubits (you can think of either qubit as the target/control). Then (I⊗H) CZ (I⊗H)=C X 12 ( 𝐼 ⊗ 𝐻 )   𝐶 𝑍   ( 𝐼 ⊗ 𝐻 ) = 𝐶 𝑋 12 with control on qubit 1 and target on qubit 2, and (H⊗I) CZ (H⊗I)=C X 21 ( 𝐻 ⊗ 𝐼 )   𝐶 𝑍   ( 𝐻 ⊗ 𝐼 ) = 𝐶 𝑋 21 with control on qubit 2 and target on qubit 1. Share Improve this answer Follow answered Mar 10, 2025 at 10:50 JenBones 211 1 bronze badge Nice. H⊗H 𝐻 ⊗ 𝐻 and CZ 𝐶 𝑍 are invariant under permutation of the qubits but CX 𝐶 𝑋 is not. Your explanation gets to why the symmetry is broken. –  Mark Spinelli Commented Mar 10, 2025 at 19:11 Add a comment 1 No, it is (I⊗H)CZ(I⊗H)=CNOT ( 𝐼 ⊗ 𝐻 ) 𝐶 𝑍 ( 𝐼 ⊗ 𝐻 ) = 𝐶 𝑁 𝑂 𝑇 , with the target of the CNOT on the second qubit (the one with the Hadamard gate). Share Improve this answer Follow answered Mar 10, 2025 at 9:53 nippon 1,60910 10 silver badges 24 24 bronze badges Add a comment 0 To complement the previous answers, the expression you are asking about doesn't give any single "known gate", but rather: ( H 1 ⊗ H 2 )CZ( H 1 ⊗ H 2 )= H 1 CNOT 12 H 1 = H 2 CNOT 21 H 2 ( 𝐻 1 ⊗ 𝐻 2 ) 𝐶 𝑍 ( 𝐻 1 ⊗ 𝐻 2 ) = 𝐻 1 C N O T 12 𝐻 1 = 𝐻 2 C N O T 21 𝐻 2 The fact that we can express it in those two different forms is due to the symmetry in CZ for its control and target qubits, as pointed out by @JenBones. Actually, this gate is a CZ 𝐶 𝑍 in the X 𝑋 basis (as can be understood by changing the basis with the Hadamard gates). Compare this against the helpful identity of a similar form for the CNOT: ( H 1 ⊗ H 2 ) CNOT 12 ( H 1 ⊗ H 2 )= CNOT 21 ( 𝐻 1 ⊗ 𝐻 2 ) C N O T 12 ( 𝐻 1 ⊗ 𝐻 2 ) = C N O T 21 Share Improve this answer Follow answered 35 mins ago Cristian Emiliano Godinez 234 4 bronze badges Add a comment Your Answer Sign up or log in Sign up using Google Sign up using Email and Password Post as a guest Name Email Required, but never shown Post Your Answer By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions quantum-gate See similar questions with these tags. 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