Curator's Take
This article shows that the dramatic gap between constant‑copy stabilizer testing and Θ(n) learning vanishes once a quantum processor can only retain a sublinear number of qubits in coherent memory, with the test sample complexity rising to Θ(n − k). By linking the problem to hidden‑shift algorithms and developing new combinatorial lower‑bound techniques, the authors pinpoint coherent memory itself as the key resource that enables efficient verification—a finding especially relevant for near‑term devices where long‑lived qubits are scarce. The work therefore reshapes expectations for practical state certification and purity testing on hardware with limited quantum RAM, warning that even modest memory constraints can erase the theoretical speedups previously celebrated in the literature.
— Mark Eatherly
Summary
We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown $n$-qubit state, but may keep only $k$ qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test $n$-qubit stabilizer states using $6$ copies, which is dimension independent, unlike the learning complexity of $Θ(n)$. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the $k$-qubit memory framework is $Θ(n-k)$. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with $k$ qubits of memory, in the non-adaptive framework, is $Θ(n^2/k)$. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with $k=0.99n$ qubits of memory, there is no constant-copy stabilizer tester; furthermore for $k=cn$ qubits of memory (for $0< c < 1$), stabilizer testing is as hard as learning, with both requiring $Θ(n)$ copies.