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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2606.01270 |
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| _version_ | 1866916071929282560 |
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| author | Li, Xiantao |
| author_facet | Li, Xiantao |
| contents | A central test case for quantum linear system algorithms (QLSA) is elliptic PDEs after a finite element discretization. Most existing analyses focus on preparing a normalized solution state. But an end-to-end quantum PDE solver must also extract physical quantities of interest, such as fluxes, currents, tractions, and energy. These outputs require quantum measurement, and their observable norms may grow like $h^{-χ} $ with mesh size $h $, creating a readout bottleneck even when a quantum preconditioner reduces the condition-number dependence on $h$.
We present a multilevel framework for this readout problem, motivated by the variance-reduction mechanism of multilevel Monte Carlo (MLMC), which is naturally compatible with a multi-level finite element discretization. Instead of estimating the full fine-grid observable directly, the method estimates a telescoping sum of interlevel corrections, so that the fine-coarse cancellation is exposed before quantum measurement. Our algorithm is based on Schur-complement factorization of the corrected Green's operator through a Ritz-complement map. For quantities of interest with readout order $χ\leq 2$, the multilevel estimator removes the polynomial $h$-dependent readout overhead. With amplitude estimation, the remaining statistical dependence is $ \widetilde{O}(1/\varepsilon)$, i.e., Heisenberg scaling in the inference precision up to logarithmic factors and with direct sampling, the complexity is reduced to standard Monte Carlo scaling $\widetilde{O}(1/ε^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01270 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation Li, Xiantao Quantum Physics A central test case for quantum linear system algorithms (QLSA) is elliptic PDEs after a finite element discretization. Most existing analyses focus on preparing a normalized solution state. But an end-to-end quantum PDE solver must also extract physical quantities of interest, such as fluxes, currents, tractions, and energy. These outputs require quantum measurement, and their observable norms may grow like $h^{-χ} $ with mesh size $h $, creating a readout bottleneck even when a quantum preconditioner reduces the condition-number dependence on $h$. We present a multilevel framework for this readout problem, motivated by the variance-reduction mechanism of multilevel Monte Carlo (MLMC), which is naturally compatible with a multi-level finite element discretization. Instead of estimating the full fine-grid observable directly, the method estimates a telescoping sum of interlevel corrections, so that the fine-coarse cancellation is exposed before quantum measurement. Our algorithm is based on Schur-complement factorization of the corrected Green's operator through a Ritz-complement map. For quantities of interest with readout order $χ\leq 2$, the multilevel estimator removes the polynomial $h$-dependent readout overhead. With amplitude estimation, the remaining statistical dependence is $ \widetilde{O}(1/\varepsilon)$, i.e., Heisenberg scaling in the inference precision up to logarithmic factors and with direct sampling, the complexity is reduced to standard Monte Carlo scaling $\widetilde{O}(1/ε^2)$. |
| title | Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2606.01270 |