Controlled-U Powers

Quantum phase estimation (QPE) extracts the eigenphase of a unitary $U$ by entangling a register of control qubits with repeated applications of $U$. Each control qubit $k$ (indexed $0, 1, \ldots, n-1$) is coupled to the target through the powered unitary $U^{2^k}$. Understanding how to build and apply these controlled powers is the central programming skill of QPE.

Eigenphases and powers

Suppose $U|u\rangle = e^{2\pi i \varphi}|u\rangle$ where $\varphi \in [0,1)$ is the unknown phase. Applying $U$ $m$ times accumulates the phase multiplicatively:

$U^m |u\rangle = e^{2\pi i m \varphi}|u\rangle.$

This is why QPE uses $U^{2^k}$: control qubit $k$ picks up exactly the bit $\varphi_k$ of the binary expansion $\varphi = 0.\varphi_0\varphi_1\cdots\varphi_{n-1}$ through the phase kickback mechanism.

Building $U^m$ in a circuit

The most direct way to realize $U^m$ is to apply $U$ exactly $m$ times in sequence. Each application is an independent gate invocation on the same wires. For the controlled version, every one of those $m$ applications is conditioned on the same control qubit:

$\text{controlled-}U^m = (\text{controlled-}U)^m.$

Concretely, if your controlled gate is c.cp(lambda, ctrl, target), then controlled-U^m is obtained by calling that line $m$ times with the same arguments.

Example: $S^2 = Z$

The $S$ gate is the quarter-turn phase gate $S = P(\pi/2)$, with matrix

$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}.$

Squaring it:

$S^2 = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}^2 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = Z.$

So $\text{controlled-}S^2 = CZ$. You can verify this by applying the controlled-$S$ gate twice — the phase angles add: $P(\pi/2)$ followed by $P(\pi/2)$ gives $P(\pi) = Z$.

In circuit notation with qubit 0 as the control and qubit 1 as the target, c.cp(pi/2, 0, 1) is the controlled-$S$ gate, and calling it twice implements controlled-$S^2 = CZ$.

Try it

Build the controlled-$S^2$ operator on 2 qubits by applying the controlled-$S$ gate twice. The grader checks the full unitary matrix, so an empty circuit (or a single application) will fail.

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