Optimal algorithms for materializing stabilizer states and Clifford gates from compact descriptions

Stabilizer states admit compact classical descriptions, but many downstream tasks still require their full amplitude vectors. Since the output itself has size $2^n$, the main algorithmic question is whether one can materialize an $n$-qubit stabilizer state vector in optimal $O(2^n)$ time, rather than paying an additional polynomial overhead. We answer this question in the affirmative. Starting from the standard quadratic-form representation of stabilizer states, we give an algorithm that runs in $O(2^n)$ time and $O(2^n)$ space. The idea is to maintain a cached parity word that records all future off-diagonal quadratic phase increments simultaneously. As consequences, we obtain an optimal procedure for materializing a stabilizer state vector from a standard check-matrix description, and an optimal algorithm for expanding a Clifford tableau into its full dense matrix. These results close the asymptotic gap for dense stabilizer and Clifford materialization.