We introduce a new class of fully-quantum Metropolis walks in which both the proposal and acceptance steps are intrinsically quantum. Unlike standard quantum walks obtained by quantizing classically efficient Markov chains, our algorithm employs Hamiltonian simulation as a quantum-native proposal mechanism, enlarging the class of quantum walks beyond classical counterparts. We target the problem of sampling from the low-temperature Gibbs distribution of classical dense Ising models, within a fixed error in total variation distance. This approach achieves about a cubic polynomial asymptotic advantage over previous quantum-walks, resulting in a total sixth-degree polynomial queries speedup compared to the best classical walk. This shows that speedups beyond the widely assumed quadratic limit are possible within the quantum walk formalism. We perform a complete fault-tolerant compilation of all algorithmic primitives and benchmark against CPU, GPU, and FPGA implementations of the best classical Markov chain. Under identical hardware assumptions, the resulting advantage runtime crossover is reduced from approximately $10^3$ years for conventional quantum walks to less than one day. These results identify fully-quantum Markov chains as a promising route toward practical quantum advantage.
Given that quantum computers are naturally suited to simulate the behavior of quantum many-body systems, an immediate question arises: can one formulate physically motivated quantum machine learning (QML) tasks that exhibit learning separations? We address this problem by studying the learnability of quantum many-body dynamics from the perspective of probably approximately correct (PAC)-learning. Concretely, we devise a supervised learning problem where the training set consists of specifications of randomized stabilizer probe states, evolution times sampled uniformly from a polynomially large time interval $[0,T]$, coupled with expectation values of certain observables evaluated on the resulting time-evolved state under an unknown Hamiltonian. For this learning task, we provide an efficient quantum procedure whose training phase learns the underlying Hamiltonian from short-time training samples, and whose deployment phase combines Hamiltonian simulation with the classical shadows protocol to perform inference on a newly given data point. By contrast, the existence of $O(\mathsf{poly}(n))$-time instances ensures classical hardness: by embedding a $\mathsf{BQP}$-complete computation into the polynomially long time-dynamics of a low-intersection variant of the Feynman-Kitaev clock Hamiltonian construction, we show that, for a certain family of input distributions, no randomized classical polynomial-time algorithm can fulfill our learning condition, unless $\mathsf{BQP}\subseteq\mathsf{P/poly}$. Furthermore, we show that the classically hard instance maintains quantum learnability. We also give an interpretation of our results in learning-assisted certified quantum simulation. Taken together, our results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.