{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fast-estimation-of-outcome-probabilities-for","title":"Fast estimation of outcome probabilities for quantum circuits","arxiv_id":"2101.12223","date":"2021-01-28","proceeding":null,"authors":["Hakop Pashayan","Oliver Reardon-Smith","Kamil Korzekwa","Stephen D. Bartlett"],"abstract":"We present two classical algorithms for the simulation of universal quantum circuits on $n$ qubits constructed from $c$ instances of Clifford gates and $t$ arbitrary-angle $Z$-rotation gates such as $T$ gates. Our algorithms complement each other by performing best in different parameter regimes. The $\\tt{Estimate}$ algorithm produces an additive precision estimate of the Born rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (which scales like $\\approx 1.17^t$ for $T$ gates). Our algorithm is state-of-the-art for this task: as an example, in approximately $13$ hours (on a standard desktop computer), we estimated the Born rule probability to within an additive error of $0.03$, for a $50$-qubit, $60$ non-Clifford gate quantum circuit with more than $2000$ Clifford gates. Our second algorithm, $\\tt{Compute}$, calculates the probability of a chosen measurement outcome to machine precision with run-time $O(2^{t-r} t)$ where $r$ is an efficiently computable, circuit-specific quantity. With high probability, $r$ is very close to $\\min \\{t, n-w\\}$ for random circuits with many Clifford gates, where $w$ is the number of measured qubits. $\\tt{Compute}$ can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+$T$ circuits with $n=55$, $w=5$, $c=10^5$ and $t=80$ $T$ gates, and then compute the Born rule probability with a run-time consistently less than $10$ minutes using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms and benchmark them using random circuits, the hidden shift algorithm and the quantum approximate optimization algorithm (QAOA).","url_abs":"https://arxiv.org/abs/2101.12223v3","url_pdf":"https://arxiv.org/pdf/2101.12223v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fast-estimation-of-outcome-probabilities-for","repo_url":"https://github.com/or1426/Clifford-T-estimator","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}