[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86108-en":3,"doc-seo-86108-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86108,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers via Spectral Homotopy","Data re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their high expressivity leads to oscillatory, non-convex loss landscapes that hinder gradient optimization. The work identifies a structural failure mode, Fourier locking (FL), where nonlinear coupling between encoding weights and entangling layers causes high-frequency initialization to collapse into spurious local minima. Two Fisher diagnostics quantify FL, and a frequency-staged homotopy protocol paces target frequencies to convexify early loss, enabling escape and improving training dynamics.","Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers  \nvia Spectral Homotopy  \narXiv :2607 . 11013v1 [ quant-ph] 13 Jul 2026  \nSpencer Topel∗  \nMoth, Brooklyn, New York  \n(Dated: July 14, 2026)  \nData re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their expressivity produces oscillatory, non-convex loss landscapes that resist gradientbased optimization. We show that the primary optimization bottleneck in DRU-PQCs is not insufficient capacity but a structural failure mode we term Fourier locking (FL): because encoding weightsand entangling layers are nonlinearly coupled, random initialization on high-frequency targets collapses the encoding parameters into spurious local minima. Two Fisher diagnostics characterize FL. The input-space quantum Fisher information Fx measures the effective frequency content of the encoded state; the Fisher discriminant ratio of the measured features measures their alignment with the class labels. In two independent 50-seed experiments, the locking is literal: trapped circuits hold Fx frozen for the entire run, while escaping circuits migrate their frequency content (direct training: rpb = −0 .48; curriculum: d = 1 .34; both p \u003C 0.001) . The replicated signature is this spectral mobility, not any endpoint value of Fx , and trapped circuits retain a fully non-degenerate parameter-space QFIM (rpb ≈ 0): the failure is spectral misalignment of a responsive state, not a loss of geometric sensitivity. A frequency-staged homotopy protocol that paces the target frequency (f : 1 .0 → 3.0) convexifies the early loss landscape; escaping circuits raise Fx in step with the curriculum, and the escape rate triples (18% vs. 6%) . Fourier locking is a frequency-alignment problem, and its remedy is frequency pacing.  \nI. INTRODUCTION  \nIn current literature, parameterized quantum circuits utilizing data re-uploading (or DRU-PQCs) are often assumed to be advantageous due to their universal expressivity. However, as noted in recent theoretical work, high expressivity degrades trainability, generating highly oscillatory, non-convex loss landscapes [1] .  \nIn this paper we demonstrate that the primary optimization bottleneck in DRU-PQCs is not a deficit in expressive capacity. Instead, we argue it is a spectral failure mode we term Fourier locking (FL) . Because encoding weights (frequency selectors) and entangling layers (signal routers) are non-linearly coupled, random initialization on high-frequency targets causes the encoding parameters to collapse into isolated, spurious local minima.  \nA contribution of this paper is demonstrating that standard gradient magnitudes fail to diagnose FL traps, and that the natural quantum-geometric suspects fail as well. We measure the parameter-space Quantum Fisher Information Matrix (QFIM) throughout training and find it does not collapse in trapped circuits: the locked model remains fully sensitive to its parameters. Instead, two quantities carry the diagnostic signal. The Fisher discriminant ratio (FDR) of the measured features collapses when a circuit locks, identifying the trap as a failure of label alignment at the readout. And the input-space quantum Fisher information Fx—the Fubini–Study susceptibility of the state to the encoded data—identifies its dynamical signature: the ef-  \n∗ [spencer@mothquantum.com](spencer@mothquantum.com)  \nfective frequency content of a trapped circuit is frozen at a misaligned value from initialization onward, while escaping circuits migrate theirs over training.  \nTo test this mechanism we deploy frequency-staged homotopy, sequentially pacing the target frequencies (f = 1.0 → 3.0) to smooth the initial loss landscape. Rather than presenting this pacing as a definitive training solution, we use it as a probe of the frequency-alignment picture. Tracking Fx across the curriculum reveals a distinct latent scaffold mechanism: circuits that escape grow their effective frequency content i","cbCaibR5lpyTcOC0","https://ap.wps.com/l/cbCaibR5lpyTcOC0","pdf",478716,6,1,14,"English","en",105,"# Introduction\n## Quantum data re-uploading and universal expressivity\n## Fourier representation of quantum models\n## Expressivity trap and barren plateaus","[{\"question\":\"What is Fourier locking (FL) in DRU-PQCs?\",\"answer\":\"FL is a spectral failure mode where nonlinear coupling between encoding weights (frequency selectors) and entangling layers causes random initialization on high-frequency targets to collapse into isolated, spurious local minima.\"},{\"question\":\"How is FL diagnosed in the paper?\",\"answer\":\"Two Fisher diagnostics are used: input-space quantum Fisher information (Fx) tracks effective frequency content of the encoded state, and a Fisher discriminant ratio measures alignment of measured features with class labels.\"},{\"question\":\"How does spectral homotopy help overcome FL?\",\"answer\":\"A frequency-staged homotopy protocol sequentially paces target frequencies (e.g., f=1.0→3.0), convexifying the early loss landscape so escaping circuits increase Fx with the curriculum and the escape rate rises substantially.\"}]",1784208563,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"overcoming-fourier-locking-in-quantum-data-re-uploading-classifiers-via-spectral-homotopy","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/overcoming-fourier-locking-in-quantum-data-re-uploading-classifiers-via-spectral-homotopy/86108/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is Fourier locking (FL) in DRU-PQCs?","Question",{"text":76,"@type":77},"FL is a spectral failure mode where nonlinear coupling between encoding weights (frequency selectors) and entangling layers causes random initialization on high-frequency targets to collapse into isolated, spurious local minima.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is FL diagnosed in the paper?",{"text":81,"@type":77},"Two Fisher diagnostics are used: input-space quantum Fisher information (Fx) tracks effective frequency content of the encoded state, and a Fisher discriminant ratio measures alignment of measured features with class labels.",{"name":83,"@type":74,"acceptedAnswer":84},"How does spectral homotopy help overcome FL?",{"text":85,"@type":77},"A frequency-staged homotopy protocol sequentially paces target frequencies (e.g., f=1.0→3.0), convexifying the early loss landscape so escaping circuits increase Fx with the curriculum and the escape rate rises 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