Logical depth of a measurement pattern

Logical measurement depth is the number of sequential measurement layers required by outcome-dependent basis choices. Depth one means all bases are fixed in advance, so the quantum measurements on distinct qubits can be simultaneous. Classical output corrections may still depend on all outcomes. This does not assert depth-one graph-state preparation or constant-depth classical processing.

A path graph state simulates a sequence of Clifford gates $J({\alpha }_j)$, ${\alpha }_j\in \{0,\pi /2\}$, using fixed equatorial qubit measurement bases on successive path vertices. Let $s_j$ be raw outcomes, $t_j=2{\alpha }_j/\pi \in \{0,1\}$, and start the Pauli frame at $a_0=b_0=0$. Matrix commutation and $J(-\pi /2)=XJ(\pi /2)$ giveInduction shows the final unmeasured state differs from the ideal circuit output by $X^{a_m}{Z}^{b_m}$, up to global phase. A final computational basis result $d$ is corrected to $d\oplus a_m$. All bases are fixed, and projectors on distinct vertices commute, so all measurements, including the final one, can occur in a single layer. The frame recurrence is classical outcome processing, not quantum feed-forward.