Sample-optimal learning of stabilizer states

Abstract: It is well-known that learning a pure $n$-qubit stabilizer state $|ψ\rangle$ both requires, and can be accomplished with, access to a number of copies of $|ψ\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_δ(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<δ<1/8$, satisfies $$n+\lceil\log_2(1/δ)\rceil-3\leq L_δ(n)\leq n+\left\lceil\log_2(1/δ)\right\rceil+4$$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/δ)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $$\mathbb{Z}_4ⁿ \times \mathbb{F}_2<sup>{n(n-1)/2}$$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.