Generative Krylov Subspace Representations for Scalable Quantum Eigensolvers

TL;DR

The paper tackles the high quantum-resource cost of Krylov-based eigensolvers on NISQ devices by introducing GenKSR, a conditional generative modeling framework that learns the distribution of Krylov measurement outcomes. By evaluating Transformer and Mamba backbones, GenKSR demonstrates generalization to unseen Hamiltonians and extrapolation to larger Krylov subspaces, validated in 1D and 2D Heisenberg models and a 20-qubit XXZ hardware experiment. The approach enables ground-state energy reconstruction entirely classically, significantly reducing the need for repeated quantum experiments. This work paves the way for scalable, resource-efficient quantum eigensolvers that leverage classical surrogates learned from quantum data.

Abstract

Predicting ground state energies of quantum many-body systems is one of the central computational challenges in quantum chemistry, physics, and materials science. Krylov subspace methods, such as Krylov Quantum Diagonalization and Sample-based Krylov Quantum Diagonalization, 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 enable effective modeling of quantum systems, GenKSR leverages a conditional generative model framework. We investigate two representative backbone architectures, the standard Transformer and the Mamba state-space model. 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 through simulations of 15-qubit 1D and 16-qubit 2D Heisenberg models, as well as a hardware experiment on a 20-qubit XXZ chain executed on an IBM quantum processor. Our model successfully learns the 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 unseen Hamiltonians, and significantly reduces the need for further quantum computation.

Figures (11)

• Figure 1: The workflow of GenKSR. The framework consists of a data generation step on a quantum computer (top panel, blue) and a generative modeling step on a classical computer (bottom panel, green). Training (solid black arrows): Krylov subspace for a set of training Hamiltonian ${X_\mathrm{train}}$ are prepared and measured on a QPU to collect measurement outputs ${\vec{a}^{(l)}}$, which are used to train a conditional generative model. Inference (dashed green arrows): For a new Hamiltonian $X_\mathrm{test}$, the trained model generates synthetic measurement samples for a chosen evolution time $t_l$, which are then processed by a classical estimator to reconstruct the ground-state energy.
• Figure 2: Overview of the Mamba based CGM
• Figure 3: KQD energy prediction for a 15-qubit Heisenberg model, averaged over 20 unseen test Hamiltonians. (a) Comparison of the CS with the exact simulation. (b) Comparison of the trained Transformer and Mamba models with the exact simulation. Models were trained on data for $D\leq 5$ and evaluated up to $D=15$ using 10,000 measurement shots..
• Figure 4: Performance of exact SKQD, Transformer GenKSR, and Mamba GenKSR on the $4\times4$$J_1$--$J_2$ model at $J_2 = 0.5$. (a) Ground-state energy versus Krylov dimension $D$. (b) Energy error $\Delta E$ in semi-logarithmic scale.
• Figure 5: Comparison of energy error distributions on the 20-qubit ibm_fez processor. The distributions are shown for samples obtained directly from the ibm_fez, and those generated by Transformer and Mamba. Each panel corresponds to a different number of measurement samples (1k, 5k, 10k). Within each panel, the color of each box plot indicates the sample source, while the hatch pattern distinguishes the Krylov dimension: D=5 (no hatch) versus the extrapolated D=10 (hatched). Each box plot displays the median (red line), interquartile range, and data range over 30 test Hamiltonians.