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The Learning Stabilizers with Noise Problem

Authors Alexander Poremba, Yihui Quek, Peter Shor

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LIPIcs.ITCS.2026.108.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Random quantum stabilizer codes
  • average-case hardness

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Abstract

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory.
Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove worst-case to average-case reductions for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.

Cite As Get BibTex

Alexander Poremba, Yihui Quek, and Peter Shor. The Learning Stabilizers with Noise Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 108:1-108:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.108

Author Details

Alexander Poremba
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
  • Department of Computer Science and Department of Physics, Boston University, MA, USA
Yihui Quek
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
  • School of Computer and Communication Sciences & School of Basic Sciences, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
Peter Shor
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

Supported by the Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator, under Grant number DOE DE-SC0012704, and Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704.

Acknowledgements

The authors would like to thank Alexandru Gheorghiu, Anand Natarajan, Aram Harrow, Henry Yuen, John Bostanci, Jonas Haferkamp, Jonas Helsen, Jonathan Conrad, Soonwon Choi, Thomas Vidick and Vinod Vaikuntanathan for useful discussions.

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