Title: Generative Krylov Subspace Representations for Scalable Quantum Eigensolvers
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Quantum computing, generative models, Krylov subspace

Abstract: Predicting ground state energies of quantum many-body systems is one of the most central computational challenges in quantum chemistry, physics, and materials science. Krylov subspace methods, such as Krylov Quantum Diagonalization (KQD) and Sample-based Krylov Quantum Diagonalization (SKQD), are promising approaches for this task on near-term quantum computers. However, both require repeated quantum circuit executions for each Krylov subspace and for every new Hamiltonian, posing a major bottleneck under noisy hardware constraints. We introduce Generative Krylov Subspace Representations (GenKSR), a framework that learns a classical, generative representation of the entire Krylov diagonalization process. To overcome the scalability limitations of Transformer-based models, GenKSR leverages a conditional generative model (CGM) built on the Mamba state-space architecture, enabling efficient modeling of large-scale quantum systems. By learning the distribution of measurement outcomes conditioned on Hamiltonian parameters and evolution time, GenKSR generates Krylov subspace samples for unseen Hamiltonians and for larger subspace dimensions than those used in training. This enables full energy reconstruction purely from the classical model, without additional quantum experiments. We validate our approach on simulations of Heisenberg models up to 15 qubits and on a 20-qubit XXZ chain using an IBM quantum processor. Our model successfully learns the noisy distribution from experimental data and generates a high-fidelity representation of the quantum process. This representation enables classical reproduction of experimental outcomes, supports reliable energy estimates for new, unseen Hamiltonians, and significantly reduces the need for further quantum computation.

TL;DR: We propose GenKSR, a generative Krylov subspace representation based on Mamba that reduces quantum experiments for scalable eigensolvers.