Quantum eigenpair solver with minimal sampling overhead

TL;DR

This work tackles the sampling overhead that arises when extracting eigenpairs encoded in quantum states by proposing an amplitude-amplification-based post-filtering, enabling computation of a targeted subset of eigenpairs. Building on a previously introduced complete eigenpair solver, the authors integrate an amplitude-encoded eigenvalue filter to boost the success probability for eigenpairs within a chosen interval, achieving a runtime advantage that scales as $\sqrt{N/k}$ over the complete solver. They demonstrate exponential memory savings relative to classical methods and provide a comparative analysis against classical partial solvers such as the power method and Lanczos, showing favorable scaling in worst-case scenarios. The results indicate a practical, end-to-end quantum algorithm with real-world impact in science and engineering, with future directions including quantum amplitude estimation and hardware benchmarking.

Abstract

The advantage that many quantum algorithms have over their classical counterparts may be lost when the results are extracted as classical data (output problem). One example are eigenpair solvers, which encode the eigenpairs in a quantum state. Extracting these states results in significant sampling overheads. We propose an amplitude-amplification-based post-filtering process that reduces the number of eigenpairs encoded in the final state to a feasible amount. Often for practical applications, computing a subset of all eigenpairs is sufficient, which drastically reduces the sampling overhead. We show, that our adapted eigenpair solver does not only compete with classical alternatives but outperforms them in terms of memory requirements, runtime, and versatility. This makes it an efficient end-to-end quantum algorithm with real-world application in science and engineering.

Figures (3)

• Figure 1: pes based on qpe danz2024response. The original ces consists only of the encoding of a computational basis state via NOT gates followed by the . This computes all eigenpairs in parallel. We extend the routine by an aa reducing the number of solutions.
• Figure 2: Reflection $R_G$ around the good states $\ket{v_j},\forall j\in\mathcal{J}$. We use the quantum adder $\mathrm{qcADD}_l$ and subtractor $\mathrm{qcSUB}_l$ to compare $\varphi_j$ with the left and right limits defining the good states $[\varphi_\mathrm{l},\varphi_\mathrm{r})$ before combining the results with a controlled $Z$ gate. At last, all qubits are returned into their original bases with the Hermitian conjugate of the first three steps leaving only the global phase.
• Figure 3: Runtime $t$ comparison for the pes, the power method, the Lanczos method, the ces, and the QR algorithm under worst conditions (i.e. $\varepsilon, \Delta_\lambda^{(u,\mathcal{J})},\delta,(1-\lambda_2/\lambda_1)\propto N^{-1}$ with $N$ the matrix size). All other parameters are set to 1. The plot on the right shows the regime illustrated in gray in the left plot.