Randomized product formulas beyond optimal deterministic scaling

Quantum Physics arXiv:2608.07720 (quant-ph) [Submitted on 7 Aug 2026] Title:Randomized product formulas beyond optimal deterministic scaling Authors:Leeseok Kim, Luis Pedro García-Pintos View a PDF of the paper titled Randomized product formulas beyond optimal deterministic scaling, by Leeseok Kim and 1 other authors View PDF HTML (experimental) Abstract:Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, $H=A+\alpha B$, where $\alpha$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal O(\alpha^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $\Omega(\alpha)$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+\alpha B_\ell$ for $B = \sum_{\ell}B_\ell$, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves $\mathcal O(\alpha^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $\Omega(\alpha^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2608.07720 [quant-ph] (or arXiv:2608.07720v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.07720 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Leeseok Kim [view email] [v1] Fri, 7 Aug 2026 19:13:09 UTC (409 KB) Full-text links: Access Paper: View a PDF of the paper titled Randomized product formulas beyond optimal deterministic scaling, by Leeseok Kim and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-08 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?)