QuantumCore

Hands-on lab · 1 step

Hadamard in the ± basis

Represent the Hadamard operator in the ordered basis (∣+⟩,∣−⟩)(\lvert+\rangle,\lvert-\rangle) using matrix elements Aab=⟨a∣H∣b⟩A_{ab}=\langle a\rvert H\lvert b\rangle.

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Step 1: Matrix of H with respect to (|+⟩, |−⟩)

Let ∣±⟩=12(∣0⟩±∣1⟩)\lvert\pm\rangle=\tfrac1{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle) and H=12(111−1)H=\tfrac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix} in the computational basis. Enter the 2×22\times2 matrix of HH in the ordered basis (∣+⟩,∣−⟩)(\lvert+\rangle,\lvert-\rangle), whose entry in row aa and column bb is ⟨a∣H∣b⟩\langle a\rvert H\lvert b\rangle for a,b∈{+,−}a,b\in\{+,-\}. Entries are accepted within 10−310^{-3}.

Enter each entry as an expression, e.g. 1/sqrt(2), -i/2, exp(i*pi/4). Rows run top to bottom, columns left to right.

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Fill every entry with a valid expression.