DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians

Quantum Physics [Submitted on 2 Apr 2026] DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians Zhengfeng Ji, Tongyang Li, Changpeng Shao, Xinzhao Wang, Yuxin Zhang We study the computational complexity of estimating the normalized trace 2^{-n}Tr[f(A)] for a log-local Hamiltonian A acting on n qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions f(x). We show that if f(x) is a continuous function with approximate degree \Omega({\rm poly}(n)), then estimating 2^{-n}Tr[f(A)] up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of f(x). This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when A is sparse, the classical query complexity of this problem is exponential in the approximate degree, assuming a conjectured lower bound for a trace variant of the k-Forrelation problem in the DQC1 query model. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and, conditionally, the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem. Comments: 22 pages Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2604.01519 [quant-ph]   (or arXiv:2604.01519v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.01519 Focus to learn more Submission history From: Changpeng Shao [view email] [v1] Thu, 2 Apr 2026 01:15:12 UTC (43 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)