Dequantization and Hardness of Spectral Sum Estimation
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866918147300261888 |
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| author | Edenhofer, Roman Hasegawa, Atsuya Gall, François Le |
| author_facet | Edenhofer, Roman Hasegawa, Atsuya Gall, François Le |
| contents | We give new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $\varepsilon$-relative accuracy in time polylogarithmic in the dimension $N$, specifically in time $\mathrm{poly}(\mathrm{log}(N), s, κ, 1/\varepsilon)$, where $s$ is the sparsity and $κ$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension. Our classical algorithm runs in time $\mathrm{polylog}(N)\cdot s^{O(\sqrtκ\log κ/\varepsilon)}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes.
We complement our classical upper bound with $\mathsf{DQC1}$-completeness results for estimating specific spectral sums such as the trace of the inverse and the trace of matrix powers for log-local Hamiltonians, with parameter scalings analogous to those of known quantum algorithms. Assuming $\mathsf{BPP}\subsetneq\mathsf{DQC1}$, this rules out classical algorithms with the same scalings. It also resolves a main open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. We further analyze a block-encoding input model, where instead of a classical description of a sparse matrix, we are given a block-encoding of it. We show $\mathsf{DQC}1$-completeness in a very general way in this model for estimating $\mathrm{tr}[f(A)]$ whenever $f$ and $f^{-1}$ are sufficiently smooth.
We conclude our work with $\mathsf{BQP}$-hardness and $\mathsf{PP}$-completeness results for high-accuracy log-determinant estimation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dequantization and Hardness of Spectral Sum Estimation Edenhofer, Roman Hasegawa, Atsuya Gall, François Le Quantum Physics Computational Complexity Data Structures and Algorithms We give new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $\varepsilon$-relative accuracy in time polylogarithmic in the dimension $N$, specifically in time $\mathrm{poly}(\mathrm{log}(N), s, κ, 1/\varepsilon)$, where $s$ is the sparsity and $κ$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension. Our classical algorithm runs in time $\mathrm{polylog}(N)\cdot s^{O(\sqrtκ\log κ/\varepsilon)}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. We complement our classical upper bound with $\mathsf{DQC1}$-completeness results for estimating specific spectral sums such as the trace of the inverse and the trace of matrix powers for log-local Hamiltonians, with parameter scalings analogous to those of known quantum algorithms. Assuming $\mathsf{BPP}\subsetneq\mathsf{DQC1}$, this rules out classical algorithms with the same scalings. It also resolves a main open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. We further analyze a block-encoding input model, where instead of a classical description of a sparse matrix, we are given a block-encoding of it. We show $\mathsf{DQC}1$-completeness in a very general way in this model for estimating $\mathrm{tr}[f(A)]$ whenever $f$ and $f^{-1}$ are sufficiently smooth. We conclude our work with $\mathsf{BQP}$-hardness and $\mathsf{PP}$-completeness results for high-accuracy log-determinant estimation. |
| title | Dequantization and Hardness of Spectral Sum Estimation |
| topic | Quantum Physics Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.20183 |