Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.20183

Dequantization and Hardness of Spectral Sum Estimation

Roman Edenhofer·Atsuya Hasegawa·François Le Gall
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $\poly(\log(N), s, \kappa, 1/\varepsilon)$, where $s$ is the sparsity and $\kappa$ 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 for the log-determinant runs in time $\polylog(N)\cdot s^{O(\sqrt{\kappa}\log(\kappa/\varepsilon))}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. We complement our classical upper bounds with complexity-theoretic limitations. We prove that estimating normalized traces of polynomial powers and inverses of log-local Hamiltonians to inverse-polynomial additive accuracy is DQC1-complete, resolving an open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. Finally, we prove a general PP-completeness result for unnormalized spectral sums: under mild polynomial-approximability and nondegeneracy assumptions on $f$, estimating $\mathrm{tr}[f(A)]$ to constant additive accuracy is PP-complete.

Explore related subjects

quant-phcs.CCcs.DS

Keep this discovery

BibTeXRIS

Roman Edenhofer, Atsuya Hasegawa, François Le Gall. 2025-09-24. Dequantization and Hardness of Spectral Sum Estimation. https://arxiv.org/abs/2509.20183

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Probing the Error-Mitigation Threshold with Matrix Product States

Quantum error mitigation relies on accurate noise characterization, but mismatches between the actual and characterized noise can be amplified and drive a sharp threshold between successful and failed mitigation. In random circuits, this threshold maps onto a random-field Ising transition, but previous exact numerics were limited to small one-dimensional and all-to-all systems, leaving explicit two-dimensional architectures unresolved. We develop a fixed-bond-dimension matrix-product-state method for the replicated transfer dynamics that extends threshold calculations beyond exact propagation while retaining the finite-size signatures of the transition. At system sizes beyond previous exact studies, we recover the predicted absence of a threshold for quenched disorder in 1D, obtain a sharper annealed all-to-all critical point, and resolve architecture-dependent finite-depth thresholds in 2D square and heavy-hex circuits. These results establish replicated tensor-network dynamics as a practical tool for probing error-mitigation thresholds in large and higher-dimensional noisy circuits.

quant-ph↗
arXiv2026

Low-cost algorithm-to-execution framework for surface-code quantum computing

The execution of useful quantum algorithms on fault-tolerant processors requires more than a mapping from logical gates to encoded operations: the spatial organization, non-Clifford resource supply, and execution schedule must also be determined while keeping physical overhead within practical limits. Although the theoretical hierarchy from logical circuits to fault-tolerant operations is well established, these implementation choices are often specified and optimized separately. Here we develop a low-cost algorithm-to-execution framework for surface-code quantum computing. From hierarchical algorithm descriptions, it constructs dependency-preserving logical schedules and an executable workload capturing logical interactions, operation parallelism, and time-resolved non-Clifford demand, thereby linking logical computation to surface-code organization, resource-state preparation, and fault-tolerant execution in a traceable workflow. We apply the framework to twenty benchmark circuits across seven algorithm families and a hierarchically composed application-scale elliptic-curve discrete-logarithm workload. Physical costs vary substantially even for circuits with similar logical resource counts. Under our direct-rotation calibration, non-Clifford implementation selection reduces space-time volume by up to 241.5 times versus an all-synthesis baseline for the QAOA amplitude-amplification workload. Circuit-specific surface-code layouts reduce routed-latency estimates for all twenty benchmarks; thirteen also reduce space-time volume because communication savings outweigh added spatial overhead. These results show that low-cost fault-tolerant execution depends on computation scheduling and organization, not aggregate logical resource counts alone.

quant-ph↗
arXiv2026

Sample-optimal learning of stabilizer states

It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<\delta<1/8$, satisfies $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.

quant-ph↗