Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids
Summarize this article with:
Quantum Physics arXiv:2606.17589 (quant-ph) [Submitted on 16 Jun 2026] Title:Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids Authors:Chengzhuo Xu, Xiao Chen, Zhihao Liu, Zhigang Li View a PDF of the paper titled Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids, by Chengzhuo Xu and 2 other authors View PDF HTML (experimental) Abstract:Diagonal unitaries are a fundamental but resource-intensive class of quantum operations, arising as the phase separators of QAOA and the time-evolution blocks of Hamiltonian simulation. Under all-to-all connectivity their optimal depth is established, but on nearest-neighbor hardware general-purpose compilers fall back on heuristic search, which yields no analyzable cost bound and becomes intractable at the very sizes where depth is the bottleneck. We address synthesis and compilation jointly. On the synthesis side, we develop a Gray-Path Framework (GPF) that realizes any $n$-qubit diagonal unitary in asymptotically optimal $R_z$ and CNOT depth $O(2^n/n)$ without ancillas. Our main result is that compiling GPF onto a two-dimensional nearest-neighbor grid preserves this optimality: routing adds depth $\Theta(2^n/n)$ and gate count $\Theta(2^n)$. Because GPF fixes its entire interaction structure in advance, routing reduces to scheduling a known sequence, with no heuristic search. We give the construction both with and without ancillas: the ancilla-free, cost-optimized layout is a two-row grid, and a $2k$-row layout introduces a space--time tradeoff that cuts depth by $1/k$ while remaining asymptotically optimal for the enlarged register; both are deterministic and analyzed in closed form. The same complexity is also attained on a linear nearest-neighbor chain, so the preservation is topology-independent, holding on any architecture that contains such a chain. All routing bounds are closed-form, giving the concrete resource estimates that heuristic compilers cannot provide at scale. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2606.17589 [quant-ph] (or arXiv:2606.17589v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2606.17589 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Chengzhuo Xu [view email] [v1] Tue, 16 Jun 2026 06:51:26 UTC (2,927 KB) Full-text links: Access Paper: View a PDF of the paper titled Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids, by Chengzhuo Xu and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-06 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?)