# Quantum Learning Boolean Linear Functions w.r.t. Product Distributions

## TL;DR

This paper develops efficient quantum algorithms for learning Boolean linear functions under biased product distributions, extending previous uniform-distribution methods and analyzing their sample complexity and robustness to noise.

## Contribution

It introduces new quantum algorithms using biased quantum Fourier transforms for learning under biased distributions, with proven efficiency and noise stability, and compares classical and quantum sample complexities.

## Key findings

- First algorithm requires O(log n) quantum examples for any bias except full bias.
- Second algorithm uses a constant number of quantum examples, effective for small bias.
- Quantum sample complexity lower bound is Omega(log n) under large bias.

## Abstract

The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein-Vazirani algorithm. A similar strategy can be applied in the case of noisy quantum training data, as was observed in arXiv:1702.08255v2 [quant-ph]. However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in arXiv:1802.05690v2 [quant-ph] to develop efficient quantum algorithms for learning Boolean linear functions on $n$ bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires $\mathcal{O}(\ln (n))$ quantum examples. The number of quantum examples used by our second algorithm is independent of $n$, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, $\Omega (n)$ examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of $\Omega (\ln (n))$ only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.