On the Role of Entanglement and Statistics in Learning

Srinivasan A, Vojtěch Havlíček, Louis Schatzki

Keywords: Quantum Computing, Statistical Learning, Quantum learning theory, Entanglement

TL;DR: We show an equivalence between entangled and separable exact learning, exponential separation between quantum statistical queries and noisy measurements, and generalize known limitations on quantum error mitigation and learning distributions.

Abstract:

In this work we make progress in understanding the relationship between learning models when given access to entangled measurements, separable measurements and statistical measurements in the quantum statistical query ($\mathsf{QSQ}$) model. To this end, we show the following results.

$\textbf{Entanglement versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $\mathcal{C} \subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \ket{x,f(x)}$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. Additionally, we exhibit a concept class $\mathcal{C}$ for which, in order to learn some \emph{property} of $f$, the sample complexity of learning using entangled measurements is exponentially smaller than separable measurements.

$\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in \mathcal{C}$ given access to separable measurements and statistical measurements. We exhibit a concept class $\mathcal{C}$ based on degree-$2$ functions that gives an exponential separation between $\mathsf{QSQ}$ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of (Blum, 2003) that separates classical $\mathsf{SQ}$ learning from classical $\mathsf{PAC}$ learning with classification~noise.

$\textbf{$\mathsf{QSQ}$ lower bounds for learning states.}$ The main technical contribution is to introduce a quantum statistical query dimension ($\mathsf{QSDA}$), which we use to give lower bounds on the $\mathsf{QSQ}$ complexity of learning. Using this, we prove exponential $\mathsf{QSQ}$ lower bounds for testing purity of quantum states, learning CCHL states, coset states of Abelian groups, degree-$2$ functions, planted bi-clique states and learning output states of Clifford circuits of depth polylog($n$).

$\textbf{Further applications.}$ Using our $\mathsf{QSQ}$ lower bounds give an $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the $\mathsf{QSQ}$ model. Prior works by (Quek et al., 2022), (Hinsche et al., 2022), and (Neitner et al., 23) proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.

Submission Number: 13296