The Exercise

The Grover iteration contains a conditional phase shift. On a system whose state space has the computational basis $|0\rangle, |1\rangle, \dots, |d\rangle$, it leaves $|0\rangle$ as it is and changes the sign of every other basis state.^✦ The exercise is to write this operator through the projector onto $|0\rangle$ and the identity $I$.

The Phase Shift

The library states the exercise on $\mathbb{C}^{d+1}$, the space of column vectors with $d+1$ complex entries indexed $0, 1, \dots, d$, for an arbitrary $d$. The standard basis vector $|i\rangle$ of $\mathbb{C}^{n}$, for a dimension $n$ and an index $i$ among $0, \dots, n-1$, is the vector with components $$(|i\rangle)_j = \delta_{ij}.$$(1)^✦ The vectors $|0\rangle, \dots, |d\rangle$ are the computational basis states of $\mathbb{C}^{d+1}$. The projector onto $|0\rangle$ is the operator $|0\rangle\langle 0|$ on $\mathbb{C}^{d+1}$, the outer product of $|0\rangle$ with its conjugate transpose $\langle 0|$.

The phase shift $P$ on $\mathbb{C}^{d+1}$ is the diagonal matrix $$P = \mathrm{diag}(1, \underbrace{-1, \dots, -1}_{d}).$$(2)^✦

The Claim

What Is Checked

The claim is stated over the phase shift $P$ of (2) and the projector $|0\rangle\langle 0|$ formed from the standard basis vector $|0\rangle$ of (1). What the library proves about them is the matrix identity (3).