Optimal Stabilizer Testing and Learning with Limited Quantum Memory

2026-07-02 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Emerging Technologies & Innovation, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

A new study examines the complexity of stabilizer state testing and learning when coherent quantum memory is limited. Traditionally, testing n-qubit stabilizer states requires only 6 copies, independent of dimension, while learning demands Θ(n) copies, demonstrating a clear separation. However, this research reveals that this separation vanishes under memory constraints. Specifically, the sample complexity for testing stabilizer states with k qubits of memory becomes Θ(n-k), linking to the hidden shift problem. For learning stabilizer states in a non-adaptive framework, the complexity is Θ(n^2/k). The findings highlight coherent quantum memory as the critical resource enabling the distinction between testing and learning. For instance, even with k=0.99n qubits of memory, a constant-copy stabilizer tester is impossible, and for k=cn (where 0 < c < 1), both testing and learning require Θ(n) copies.

Key takeaway

For research scientists designing quantum algorithms for stabilizer states, recognize that limited coherent quantum memory fundamentally alters complexity. If your protocol has memory constraints, the traditional 6-copy testing advantage over Θ(n) learning vanishes. You must account for Θ(n-k) testing and Θ(n^2/k) learning complexities. This implies that memory-efficient designs are crucial, as even near-full memory (k=0.99n) prevents constant-copy testing, making testing as hard as learning.

Key insights

Limited coherent quantum memory eliminates the complexity separation between stabilizer state testing and learning.

Principles

• Stabilizer testing complexity is Θ(n-k) with k memory.
• Stabilizer learning complexity is Θ(n^2/k) non-adaptive.
• Coherent memory enables testing-learning separation.

Method

The upper bound for stabilizer testing complexity is derived via a novel connection to the hidden shift problem. Lower bounds use average case likelihood ratios and stochastic orthogonal group combinatorics.

In practice

• Quantum memory constraints impact protocol design.
• Consider memory limits for stabilizer state tasks.
• Purity testing faces exponential lower bounds.

Topics

• Stabilizer States
• Quantum Memory
• Quantum Algorithm Complexity
• Hidden Shift Problem
• Quantum State Testing
• Quantum State Learning
• Purity Testing

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.