Feed-Forward Corrections

Each single-qubit measurement in MBQC leaves a random Pauli byproduct on the qubits downstream. This lesson shows precisely how those byproducts are managed — by adapting later measurement angles (feed-forward) and by tracking the leftover Paulis classically (the Pauli frame) — so that the logical output is deterministic despite the randomness of every individual outcome.

Where byproducts come from

From the elementary gadget, an $X$ -basis measurement with outcome $s$ leaves the next wire in

X^{s}\,H\,|\psi\rangle ,

and a tilted measurement at angle $\xi$ leaves $X^{s}\,H R_z(\xi)\,|\psi\rangle$ . The intended gate is always applied; the byproduct $X^{s}$ is a known Pauli — known because we recorded the outcome $s$ . The problem is purely that these byproducts accumulate as we measure along the cluster, and they do not commute with subsequent operations.

The commutation rules that make corrections necessary

Two facts drive everything. First, the byproduct on the qubit about to be measured changes that measurement. Propagating a Pauli through the measurement command gives

M^{\xi} X^{s} = M^{(-1)^{s}\,\xi}, \qquad M^{\xi} Z^{t} = M^{\xi + t\pi}.

So a pending $X$ byproduct (from outcomes in the set with total parity $s^X$ ) negates the angle, and a pending $Z$ byproduct (parity $s^Z$ ) shifts it by $\pi$ . The qubit is therefore measured not at its nominal angle $\xi$ but at the adapted angle

\xi' = (-1)^{s^X}\,\xi + s^Z\,\pi .

Choosing $\xi'$ on the fly, conditioned on earlier outcomes, is feed-forward. It is what forces a temporal order on the measurements.

Second, propagating Paulis past the entangling gate spreads them, because $CZ$ conjugates $X_a \mapsto X_a Z_b$ . This is bookkeeping, not extra quantum work: the running product of byproduct Paulis on every qubit is tracked in a classical register called the Pauli frame.

How a Hadamard chain stays deterministic

Consider the two-qubit Hadamard pattern $X_2^{s_1} M_1^{0} E_{12} N_2$ . After measuring qubit $1$ with outcome $s_1$ , the output qubit $2$ is in $X^{s_1} H|\psi\rangle$ . The correction command $X_2^{s_1}$ applies an $X$ exactly when $s_1 = 1$ :

X^{s_1}\big(X^{s_1} H|\psi\rangle\big) = X^{2 s_1} H|\psi\rangle = H|\psi\rangle,

since $X^2 = I$ . The two possible outcomes are mapped to the same logical state. This is the general mechanism: a recorded outcome $s$ produces a byproduct $P^{s}$ , and applying $P^{s}$ again cancels it, because every Pauli is its own inverse.

Two roles, one principle

It is worth separating the two appearances of the same Pauli-tracking idea:

On a qubit yet to be measured, a byproduct cannot simply be applied later — it would corrupt the basis. It is instead folded into the measurement angle via $\xi' = (-1)^{s^X}\xi + s^Z\pi$ . That is feed-forward proper, and it is why measurements are ordered.
On an output qubit, never measured, the byproduct is harmless to track and is undone by a final Pauli correction $X^{s}Z^{t}$ .

Both are instances of the single rule record the outcome, then apply the inverse Pauli (or absorb it into a later angle).

Try it

The exercise models the output-correction case on one qubit. An input $|\psi\rangle = R_x(1.1)|0\rangle$ has acquired a byproduct $X$ (the measurement that produced it returned $s = 1$ ). Apply the feed-forward correction $X^{s} = X$ to cancel it. The grader checks the final Bloch vector equals that of the clean input, $(x, y, z) = (0,\, -\sin 1.1,\, \cos 1.1)$ — confirming the two $X$ operations have annihilated.

Run your code to see the quantum state.