On the Role of Entanglement and Statistics in Learning

In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$. 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. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between 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 et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-$2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$. $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.

Supervised learning with quantum enhanced feature spaces

Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here, we propose and experimentally implement two novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1,2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.