Adaptivity is all you need: Optimal stabilizer learning using just single-copy measurements
Abstract
Stabilizer states are central to quantum computing, underlying quantum error correction, benchmarking, and efficient classical simulation. Yet their learnability exhibits a striking gap: an n-qubit stabilizer state can be learned from Θ(n) copies using two-copy Bell measurements, whereas non-adaptive single-copy measurements require Ω(n2) copies. Here we show that adaptivity completely closes this gap. We give a polynomial-time adaptive algorithm that learns an arbitrary n-qubit stabilizer state from Θ(n) single-copy Clifford measurements, matching the optimal sample complexity of Bell sampling without any multi-copy measurements. The same ideas yield a sample-optimal single-copy tolerant tester and, with k qubits of quantum memory, the optimal testing tradeoff Θ(n−k+1/ε) at infidelity ε. Finally, we show that this adaptive mechanism extends beyond exact stabilizer states: states of stabilizer nullity at most r, including states prepared by Clifford circuits with a bounded number of T gates, can be learned using O(n2r) single-copy measurements.
Related
- Learning unknown stabilizer codes using product measurements
- Sample-optimal learning of stabilizer states
- Optimal Stabilizer Testing and Learning with Limited Quantum Memory
- Bounding Fidelity of Non-Stabilizer States from Local Pauli Measurements
- Learning sparse quantum states from single-qubit measurements
BibTeX
@misc{bittel2026adaptivity,
title = {{Adaptivity is all you need: Optimal stabilizer learning using just single-copy measurements}},
author = {L. Bittel and J. Eisert and W. Gong and A. A. Mele and L. Schatzki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02031},
eprint = {2610.02031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adaptivity is all you need: Optimal stabilizer learning using just single-copy measurements
AU - L. Bittel
AU - J. Eisert
AU - W. Gong
AU - A. A. Mele
AU - L. Schatzki
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02031
UR - https://arxiv.org/abs/2610.02031
AB - Stabilizer states are central to quantum computing, underlying quantum error correction, benchmarking, and efficient classical simulation. Yet their learnability exhibits a striking gap: an n-qubit stabilizer state can be learned from Θ(n) copies using two-copy Bell measurements, whereas non-adaptive single-copy measurements require Ω(n^2) copies. Here we show that adaptivity completely closes this gap. We give a polynomial-time adaptive algorithm that learns an arbitrary n-qubit stabilizer state from Θ(n) single-copy Clifford measurements, matching the optimal sample complexity of Bell sampling without any multi-copy measurements. The same ideas yield a sample-optimal single-copy tolerant tester and, with k qubits of quantum memory, the optimal testing tradeoff Θ(n−k+1/ε) at infidelity ε. Finally, we show that this adaptive mechanism extends beyond exact stabilizer states: states of stabilizer nullity at most r, including states prepared by Clifford circuits with a bounded number of T gates, can be learned using O(n2^r) single-copy measurements.
ER -