We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.<\/p>\n","protected":false},"excerpt":{"rendered":"
We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"arXiv","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2026-04-21","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":false,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[{"provider":"s2","id":"0f5fc8741584f391cd53471977fd803d324e015d"},{"provider":"arxiv","id":"2604.20033"}],"msr_hide_image_in_river":null,"footnotes":""},"msr-research-highlight":[],"research-area":[243138],"msr-publication-type":[193724],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[249337,250069],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-1169851","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-quantum","msr-locale-en_us","msr-field-of-study-physics","msr-field-of-study-quantum"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2026-04-21","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"arXiv","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":0,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/arxiv.org\/abs\/2604.20033","label_id":"243109","label":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"Vadym Kliuchnikov","user_id":34468,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Vadym Kliuchnikov"},{"type":"name","value":"Jendrik Brachter","user_id":0,"rest_url":false},{"type":"name","value":"M. 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