Evaluating QAOA expectation values can be as hard as counting optimal solutions

Quantum Physics arXiv:2608.11385 (quant-ph) [Submitted on 11 Aug 2026] Title:Evaluating QAOA expectation values can be as hard as counting optimal solutions Authors:Stuart Hadfield View a PDF of the paper titled Evaluating QAOA expectation values can be as hard as counting optimal solutions, by Stuart Hadfield View PDF HTML (experimental) Abstract:Evaluating expectation values is a critical task for variational quantum eigensolvers, and for parameterized quantum circuits and other quantum algorithms more generally. We consider the well-studied case of the Quantum Approximate Optimization Algorithm (QAOA) for the MaxCut problem. Recent work of Wang et al. [arXiv:2511.20212] showed this task to be NP-hard in general for any QAOA depth $p\geq 2$, complementing past results showing efficiently computable formulas for $p=1$ with arbitrary problem graphs. We sharpen this dichotomy showing that for $p\geq 2$ exact or exponentially precise cost expectation value evaluation is #P-hard under deterministic polynomial-time Turing reductions. Hardness at $p\geq 2$ is shown to remain even for evaluating single pairwise correlators $\langle Z\otimes Z\rangle $, as well as for highly restricted sets of algorithm parameters. Our proof refines the NP-hardness construction of Wang et al. that recovers the maximum cut value from the largest exponent of a QAOA Laurent polynomial, utilizing a distinct and simpler construction that extracts a value proportional to the total number of maximum cuts, in addition to the optimal cut value. Thus we show that the QAOA expectation value hardness transition from $p=1$ to $p=2$ is not only from tractability to optimization hardness, but to that of counting optimal solutions. As an application we show our results imply analogous hardness results for computing gradients and Hessians of QAOA circuits. Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2608.11385 [quant-ph] (or arXiv:2608.11385v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2608.11385 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Stuart Hadfield [view email] [v1] Tue, 11 Aug 2026 19:47:04 UTC (1,216 KB) Full-text links: Access Paper: View a PDF of the paper titled Evaluating QAOA expectation values can be as hard as counting optimal solutions, by Stuart HadfieldView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-08 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?)