Noisy Quantum Learning Theory

TL;DR

This paper develops a framework for understanding the limits of quantum learning in noisy environments, showing how noise impacts quantum advantages and proposing methods to analyze and mitigate these effects.

Contribution

It introduces the complexity class NBQP for noisy quantum learning, analyzes the impact of noise on quantum advantages, and provides bounds and algorithms for noisy quantum tasks.

Findings

Noise eliminates exponential quantum advantages in unphysical, noiseless models.

A superpolynomial gap persists between NISQ devices and fault-tolerant quantum computers.

Noise-resilient structures can restore quantum advantages in certain settings.

Abstract

We develop a framework for learning from noisy quantum experiments in which fault-tolerant devices access uncharacterized systems through noisy couplings. Introducing the complexity class $\textsf{NBQP}$ ("noisy BQP''), we model noisy fault-tolerant quantum computers that cannot generally error-correct the oracle systems they query. Using this class, we prove that while noise can eliminate the exponential quantum learning advantages of unphysical, noiseless learners, a superpolynomial gap remains between $\textsf{NISQ}$ and fault-tolerant devices. Turning to canonical learning tasks in noisy settings, we find that the exponential two-copy advantage for purity testing collapses under local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient physical structure restores this quantum learning advantage. We then analyze noisy Pauli shadow tomography, deriving lower bounds characterizing how instance size, quantum memory and noise jointly control sample complexity, and design algorithms with parametrically matching scalings. We study similar tradeoffs in quantum metrology, and show that the Heisenberg-limited sensitivity of existing error-correction-based protocols persists only up to a timescale inverse-polynomial in the error rate per probe qubit. Together, our results demonstrate that the primitives underlying quantum-enhanced experiments are fundamentally fragile to noise, and that realizing meaningful quantum advantages in future experiments will require interfacing noise-robust physical properties with available algorithmic techniques.