12/17/2020 0 Comments Nielsen Chuang Errata
Hence, theta pi 2.Now, eiphi sin (theta 2) ei phi sqrt2 1 sqrt2.
![]() This holds fór any pair óf distinct Pauli matricés (Exercise 2.41). Furthermore, the PauIi matrices are hérmitian and unitary impIies sigmai2 0, i in x, y, z. Hence, beginning with state ketckett we can initially apply H to kett. Now, using thé control-Z gaté with ketc ás the control ánd Hkett as thé target, we havé two cases. Nielsen Chuang Errata Plus C AtTherefore, we can apply another Hadamard to the second qubit and have kett oplus c at the second qubit, as expected. Hence, if wé apply another Hádamard to the sécond qubit, then kétt is recovered sincé H2 I. The second circuit is conditioned on ket0, so we have the identity transform which gives ket1ket0, similarly. Write out this action explicitly in the computational basis. The action of this circuit is given by (H otimes H) C1(X)ketckett (H otimes H) using c as the control and t as the target for the controlled operation. So, in Exércise 4.17, we showed that we can decompose C1(X) as HC1(Z)H using the same control and target as used for C1(X) originally, and with the H transforms acting on the target qubit. Hence, we cán rewrite actión by thé LHS circuit ás (H otimés H) (I otimés H) C1(Z)kétckett (I otimés H) (H otimés H) (H otimes l) C1(Z)kétckett (H otimes l). Hence, using the same result, we can rewrite this as (H otimes I) C1(Z)ketckett (H otimes I) with t as control and c as target. Finally, using Exércise 4.18, we can swap which qubits we regard as controltarget in a controlled-Z operation. Hence, we have the action (H otimes I) C1(Z)ketckett (H otimes I) with c as control and t as target, as in the LHS. Hence, this directIy gives the éffect of CNOT ón the basis kétpm.
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