DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians
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Quantum Physics arXiv:2604.01519 (quant-ph) [Submitted on 2 Apr 2026] Title:DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians Authors:Zhengfeng Ji, Tongyang Li, Changpeng Shao, Xinzhao Wang, Yuxin Zhang View a PDF of the paper titled DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians, by Zhengfeng Ji and 4 other authors View PDF HTML (experimental) Abstract: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: 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 arXiv-issued DOI via DataCite (pending registration) Submission history From: Changpeng Shao [view email] [v1] Thu, 2 Apr 2026 01:15:12 UTC (43 KB) Full-text links: Access Paper: View a PDF of the paper titled DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians, by Zhengfeng Ji and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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 Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)