---
title: "Optimal Stabilizer Testing and Learning with Limited Quantum Memory"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.02444v1"
permalink: "https://feed.astralor.com/items/itm_150efbfb2edc6e9e7dc3a6fc"
published: "2026-07-02T17:11:38.000Z"
score: 59
channel: "tech"
tags: ["Sample Complexity","stabilizer states","quantum learning","quantum testing","quantum memory"]
---

# Optimal Stabilizer Testing and Learning with Limited Quantum Memory

## At a glance

This paper investigates testing and learning of n-qubit stabilizer states under limited coherent quantum memory. The authors find that the usual separation between stabilizer testing and learning vanishes under memory constraints. Specifically, in the k-qubit memory framework, the sample complexity of testing is Θ(n-k), while that of non-adaptive learning is Θ(n²/k). Additionally, an exponential lower bound for purity testing is proved even when memory remains coherent throughout the protocol. The results identify coherent quantum memory as the key resource enabling the separation between stabilizer testing and learning.

## Key points

- Sample complexity for testing stabilizer states scales linearly with available quantum memory k: Θ(n-k).
- Sample complexity for non-adaptive learning stabilizer states is Θ(n²/k), significantly higher than testing.
- Purity testing requires exponential sample complexity even with fully coherent memory.
- Coherent quantum memory is the fundamental resource enabling the separation between stabilizer testing and learning.

## Full text

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.

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