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This paper investigates the complexities of stabilizer state testing and learning under constraints of coherent quantum memory, revealing that the previously established testing-learning separation collapses when memory is limited. Specifically, it establishes that the sample complexity for testing stabilizer states with $k$ qubits of memory is $\Theta(n-k)$, and for learning, it is $\Theta(n^2/k)$, highlighting the critical role of coherent quantum memory in these processes. The findings also include an exponential lower bound for purity testing, emphasizing the challenges posed by memory limitations in quantum state analysis.
Memory constraints fundamentally alter the landscape of stabilizer state testing and learning, revealing a surprising collapse of the established complexity separation.
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 $\Theta(n)$. 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 $\Theta(n-k)$. 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 $\Theta(n^2/k)$. 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
