Optimal Stabilizer Testing and Learning with Limited Quantum Memory

Q: What problem does the document address?

It analyzes how stabilizer-state testing and learning change when the algorithm is restricted to storing only k qubits coherently between measurements.

Q: How does limited coherent quantum memory affect sample complexity?

Testing stabilizer states requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies, showing a loss of the usual testing-vs-learning separation.

Q: What resource enables the separation between stabilizer testing and learning?

Coherent quantum memory: the results indicate that without enough coherent memory, testing becomes as hard as learning, and even near-linear memory does not yield a constant-copy stabilizer tester.

Jul 16, 2026

Upload by Maeve

Studies stabilizer state testing and learning when an algorithm has only limited coherent quantum memory between measurements. A sequential procedure receives copies of an unknown n-qubit state while retaining at most k coherent qubits. With unlimited memory, testing uses a constant number of copies independent of dimension, unlike learning. Under memory constraints, the testing-vs-learning separation collapses: testing requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies. An application yields exponential lower bounds for purity testing and identifies coherent quantum memory as the key resource enabling the separation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages66

Reading Time2 h 46 min

AccessFree to download

Introduction

Main results

Proof sketch of testing upper bound

Proof sketch of testing lower bound

Proof sketch of learning bounds

Discussion and open questions

Preliminaries

Paulis and Cliffords

Fourier analysis

Stochastic orthogonal group

Technical toolkit

Partial bell sampling

Ensembles corresponding to deg-2 phase states

Structure of protocols with quantum memory

Testing upper bound

Partial Bell sampling

The hidden-shift subroutine

The final tester

Conditional soundness of the graph test

Global soundness: finding a bad prefix

Testing lower bounds

Information Theoretic Lower Bounds

Construction of hard ensemble

Rewriting our phase state ensemble

Lower bound for testing

Proof of Lemma 5.8

Non-adaptive learning stabilizer states

Upper bounds

Lower bound

Purity Testing Lower Bound

Optimal stabilizer testing and learning with limited quantum memory arXiv :2607 .02444v1 [ quant-ph] 2 Jul 2026 Srinivasan Arunachalam IBM Research Silicon Valley [Srinivasan.Arunachalam@ibm.com](Srinivasan.Arunachalam@ibm.com) Louis Schatzki Dahlem Center for Complex Quantum Systems Freie Universit¨at Berlin [real-louismares98@zedat.fu-berlin.de](real-louismares98@zedat.fu-berlin.de) July 3, 2026 Abstract 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 [GNW21] 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 nonadaptive framework, is Θ(n2 /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 constantcopy 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. Contents 1 Introduction 3 1.1 Main results ................................................... 4 1.2 Proof sketch of testing upper bound ..................................... 6 1.3 Proof sketch of testing lower bound ...................................... 8 1.4 Proof sketch of learning bounds ........................................ 10 1.5 Discussion and open questions ......................................... 11 2 Preliminaries 13 2.1 Paulis and Cliffords ............................................... 13 2.2 Fourier analysis ................................................. 15 2.3 Stochastic orthogonal group ......................................... 16 3 Technical toolkit 17 3.1 Partial bell sampling .............................................. 17 3.2 Ensembles corresponding to deg-2 phase states ............................... 20 3.3 Structure of protocols with quantum memory ................................ 21 4 Testing upper bound 24 4.1 Partial Bell sampling .............................................. 24 4.2 The hidden-shift subroutine .......................................... 24 4.3 The final tester ................................................. 29 4.4 Conditional soundness of the graph test ................................... 30 4.5 Global soundness: finding a bad prefix .................................... 32 5 Testing lower bounds 36 5.1 Information Theoretic Lower Bounds ..................................... 36 5.2 Construction of hard ensemble ........................................ 38 5.3 Rewriting our phase state ensemble ...................................... 39 5.4 Lower bound for testing ............................................ 41 5.5 Proof of Lemma 5.8 .............................................. 44 6 Non-adaptive learning stabilizer states 52 6.1 Upper bounds .................................................. 52 6.2 Lower bound .................................................. 56 A Total variatio

FAQs

What problem does the document address?

It analyzes how stabilizer-state testing and learning change when the algorithm is restricted to storing only k qubits coherently between measurements.

How does limited coherent quantum memory affect sample complexity?

Testing stabilizer states requires Θ(n−k) copies, and nonadaptive learning requires Θ(n²/k) copies, showing a loss of the usual testing-vs-learning separation.

What resource enables the separation between stabilizer testing and learning?

Coherent quantum memory: the results indicate that without enough coherent memory, testing becomes as hard as learning, and even near-linear memory does not yield a constant-copy stabilizer tester.

Jul 16, 2026

Upload by Maeve