Hardness and Complexity Transition of Noisy Random Circuit Sampling

Quantum Physics arXiv:2607.20804 (quant-ph) [Submitted on 23 Jul 2026] Title:Hardness and Complexity Transition of Noisy Random Circuit Sampling Authors:Byeongseon Go, Changhun Oh, Hyunseok Jeong View a PDF of the paper titled Hardness and Complexity Transition of Noisy Random Circuit Sampling, by Byeongseon Go and 2 other authors View PDF HTML (experimental) Abstract:Random circuit sampling (RCS) is a leading candidate for demonstrating quantum advantage, supported by strong complexity-theoretic evidence of hardness in the ideal setting and by rapid experimental progress to date. In practice, however, noise is unavoidable, and a central problem is to identify the noise-strength boundary between classically simulable and classically hard regimes. In this work, we establish an architecture-general hardness bound for this boundary for the standard local depolarizing noise of strength $\gamma$. Assuming the standard average-case #P-hardness conjecture for ideal RCS, we show that, for any circuit architecture satisfying this conjecture, noisy RCS on the same architecture remains hard to simulate classically within any inverse-polynomial total variation distance whenever $\gamma=O(\log n/(nd))$ for $n$-qubit circuits of depth $d$, unless the polynomial hierarchy collapses. Crucially, noisy-RCS hardness follows without any additional conjectural or architecture-specific assumption beyond those already entering the ideal-RCS hardness framework. Our proof combines a low-degree polynomial extrapolation with a monotonicity reduction showing that efficient classical simulation at one depolarizing noise strength implies efficient simulation at every larger strength. Together, these ingredients transfer the standard ideal-RCS hardness conjecture to sampling hardness at a prespecified noise strength. Finally, combining the convergence-to-uniformity result of Dalzell et al. [Commun. Math. Phys. 405, 78 (2024)] with our monotonicity reduction yields efficient classical simulation for $\gamma=\omega(\log n/(nd))$ on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies $\gamma=\Theta(\log n/(nd))$ as the asymptotic complexity-transition scale. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2607.20804 [quant-ph] (or arXiv:2607.20804v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2607.20804 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Byeongseon Go [view email] [v1] Thu, 23 Jul 2026 00:23:18 UTC (303 KB) Full-text links: Access Paper: View a PDF of the paper titled Hardness and Complexity Transition of Noisy Random Circuit Sampling, by Byeongseon Go and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-07 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?)