A recent study investigates how limited coherent quantum memory impacts the complexity of testing and learning $n$-qubit stabilizer states. Traditionally, testing stabilizer states requires a constant number of copies of an unknown state, independent of $n$, whereas full state learning scales with $Θ(n)$. This fundamental separation in quantum state characterization is shown to break down when the available coherent memory is restricted.

The researchers demonstrated that the sample complexity for testing stabilizer states with $k$ qubits of memory is $Θ(n-k)$. This stands in stark contrast to the 6-copy result for unrestricted memory. The upper bound for testing was established through a novel connection to the hidden shift problem, while the lower bound was proven using a combinatorial approach to likelihood ratios over the stochastic orthogonal group. Furthermore, the sample complexity for learning stabilizer states in the non-adaptive framework with $k$ qubits of memory is found to be $Θ(n^2/k)$.

These findings suggest that coherent quantum memory is a critical resource enabling the observed separation between stabilizer state testing and learning. For instance, even with $k=0.99n$ qubits of memory, a constant-copy stabilizer tester no longer exists. For $k=cn$ qubits of memory (where $0 < c < 1$), stabilizer testing becomes as hard as learning, with both requiring $Θ(n)$ copies. This has significant implications for designing quantum state characterization protocols in systems with constrained memory resources.