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Noisy quantum learning theory
Jordan Cotler1,2, Weiyuan Gong3, Ishaan Kannan4
1Department of Physics, Harvard University, Cambridge, Massachusetts, USA.
Nature Communications
|May 29, 2026
Summary
Noise can erase quantum learning advantages, but new research introduces "noisy BQP" to model fault-tolerant quantum computers. This work explores how noise impacts quantum speedups in real-world experiments, guiding future research toward robust quantum advantages.
Area of Science:
- Quantum Information Science
- Computational Complexity Theory
Background:
- Quantum learning algorithms often assume idealized, noiseless conditions.
- Real-world quantum systems inevitably interact with noisy, uncharacterized environments.
Purpose of the Study:
- To investigate the impact of noise on quantum learning speedups.
- To introduce and analyze the complexity class "noisy BQP" (NBQP) for fault-tolerant quantum computers.
- To identify conditions for achieving meaningful quantum advantages in noisy experimental settings.
Main Methods:
- Introduction of the NBQP complexity class.
- Analysis of noise effects on purity testing and Pauli tomography.
- Derivation of lower bounds for sample complexity in noisy quantum learning tasks.
- Investigation of noise-dependent limitations on quantum metrology.
Main Results:
- Noise can eliminate exponential quantum advantages for idealized learners.
- NBQP learners can be exponentially weaker than noiseless counterparts, but a superpolynomial gap persists between NISQ and fault-tolerant devices.
- The exponential advantage for purity testing collapses under depolarizing noise.
- Noise-dependent polynomial speedups are identified for Pauli tomography.
- Physical structure can restore quantum speedups in specific scenarios.
Conclusions:
- Quantum learning primitives are fragile to noise.
- Realizing practical quantum advantages requires interfacing noise-robust physics with quantum algorithms.
- Future quantum experiments must account for noise to achieve meaningful speedups.
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