Question

Show that U = 1/√2 [1 1] is a legitimating quantum gate
[i i]

Answer 1

To show that a quantum gate is valid, we need to check if it is unitary and reversible. In this case, the gate U = 1/√2 [1 1] [i i] is unitary and therefore a legitimate quantum gate.

Step-by-step explanation:

To show that a quantum gate is valid, we need to check if it is unitary and reversible. In this case, we have a gate U = 1/√2 [1 1] [i i].

First, we need to check if the gate is unitary. A unitary matrix is one whose conjugate transpose is equal to its inverse. To check this, we calculate the conjugate transpose of the given matrix:

U† = (1/√2 [1 1] [i i])† = (1/√2 [-i -i] [-1 -1])

Next, we multiply U by its conjugate transpose:

UU† = (1/√2 [1 1] [i i])(1/√2 [-i -i] [-1 -1]) = (1/2)([1 1] [-i -i])([-i -i] [-1 -1]) = [1/2*2 1/2*2] = [1 0] [0 1]

Since UU† = I, the identity matrix, the gate U is indeed unitary. Therefore, it is a valid quantum gate.