{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:DC6UQ23T76GPPFDVDRWRQP35YK","short_pith_number":"pith:DC6UQ23T","canonical_record":{"source":{"id":"2306.16369","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2023-06-28T17:02:24Z","cross_cats_sorted":[],"title_canon_sha256":"dd77a5595a9732f70bbcff0eeeeb274f9340d4f6fd652b7f9da471382fd59dc1","abstract_canon_sha256":"368d557b05434e5005fea38e6fd9da2bb6dfb27e81a96c0ba2644907604786ee"},"schema_version":"1.0"},"canonical_sha256":"18bd486b73ff8cf794751c6d183f7dc2b6561379d60465d4cea5d980e5f7172f","source":{"kind":"arxiv","id":"2306.16369","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.16369","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"arxiv_version","alias_value":"2306.16369v2","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.16369","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_12","alias_value":"DC6UQ23T76GP","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_16","alias_value":"DC6UQ23T76GPPFDV","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_8","alias_value":"DC6UQ23T","created_at":"2026-07-05T06:46:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:DC6UQ23T76GPPFDVDRWRQP35YK","target":"record","payload":{"canonical_record":{"source":{"id":"2306.16369","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2023-06-28T17:02:24Z","cross_cats_sorted":[],"title_canon_sha256":"dd77a5595a9732f70bbcff0eeeeb274f9340d4f6fd652b7f9da471382fd59dc1","abstract_canon_sha256":"368d557b05434e5005fea38e6fd9da2bb6dfb27e81a96c0ba2644907604786ee"},"schema_version":"1.0"},"canonical_sha256":"18bd486b73ff8cf794751c6d183f7dc2b6561379d60465d4cea5d980e5f7172f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:46:22.981585Z","signature_b64":"gy2Kfc/2SBHukADcON7a6UVtyeAxzpxiJG5zABOAxEI2x5PRCnH6D1Ssdq09jQXP/0hwYK4CSO3r32ctlqYQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18bd486b73ff8cf794751c6d183f7dc2b6561379d60465d4cea5d980e5f7172f","last_reissued_at":"2026-07-05T06:46:22.981023Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:46:22.981023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2306.16369","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:46:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lQnmb78z3GDhgv0c0jy8Dc7JMYbcUEZYms9h0y8dynjFJoamOegGiDYIAMTFQyN0aop9DE1urJULhzIRLtDwAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T07:12:40.981414Z"},"content_sha256":"80d2521e8dbc0522edc0f0623434c89631cc931594a8d32adf8b0805419a586d","schema_version":"1.0","event_id":"sha256:80d2521e8dbc0522edc0f0623434c89631cc931594a8d32adf8b0805419a586d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:DC6UQ23T76GPPFDVDRWRQP35YK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complete Equational Theories for the Sum-Over-Paths with Unbalanced Amplitudes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Matthew Amy","submitted_at":"2023-06-28T17:02:24Z","abstract_excerpt":"Vilmart recently gave a complete equational theory for the balanced sum-over-paths over Toffoli-Hadamard circuits, and by extension Clifford+Rz(2pi/2^k) circuits. Their theory is based on the phase-free ZH-calculus which crucially omits the average rule of the full ZH-calculus, dis-allowing the local summation of amplitudes. Here we study the question of completeness in unbalanced path sums which naturally support local summation. We give a concrete syntax for the unbalanced sum-over-paths and show that, together with symbolic multilinear algebra and the interference rule, various formulations"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.16369","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2306.16369/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:46:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hxUXPsM9ZcJgL8Up6R3p2/pRtKG1GbHqesNfxl6T0IBvz0Z1oPeJCE8+AjlBxgFy990I+748vIZJkTLMSrTtCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T07:12:40.981924Z"},"content_sha256":"761874fb7c829ef0596ffeb75ecd4451b19f53cf0053ede32e312593b0381d83","schema_version":"1.0","event_id":"sha256:761874fb7c829ef0596ffeb75ecd4451b19f53cf0053ede32e312593b0381d83"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DC6UQ23T76GPPFDVDRWRQP35YK/bundle.json","state_url":"https://pith.science/pith/DC6UQ23T76GPPFDVDRWRQP35YK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DC6UQ23T76GPPFDVDRWRQP35YK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-03T07:12:40Z","links":{"resolver":"https://pith.science/pith/DC6UQ23T76GPPFDVDRWRQP35YK","bundle":"https://pith.science/pith/DC6UQ23T76GPPFDVDRWRQP35YK/bundle.json","state":"https://pith.science/pith/DC6UQ23T76GPPFDVDRWRQP35YK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DC6UQ23T76GPPFDVDRWRQP35YK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:DC6UQ23T76GPPFDVDRWRQP35YK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"368d557b05434e5005fea38e6fd9da2bb6dfb27e81a96c0ba2644907604786ee","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2023-06-28T17:02:24Z","title_canon_sha256":"dd77a5595a9732f70bbcff0eeeeb274f9340d4f6fd652b7f9da471382fd59dc1"},"schema_version":"1.0","source":{"id":"2306.16369","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.16369","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"arxiv_version","alias_value":"2306.16369v2","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.16369","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_12","alias_value":"DC6UQ23T76GP","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_16","alias_value":"DC6UQ23T76GPPFDV","created_at":"2026-07-05T06:46:22Z"},{"alias_kind":"pith_short_8","alias_value":"DC6UQ23T","created_at":"2026-07-05T06:46:22Z"}],"graph_snapshots":[{"event_id":"sha256:761874fb7c829ef0596ffeb75ecd4451b19f53cf0053ede32e312593b0381d83","target":"graph","created_at":"2026-07-05T06:46:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2306.16369/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Vilmart recently gave a complete equational theory for the balanced sum-over-paths over Toffoli-Hadamard circuits, and by extension Clifford+Rz(2pi/2^k) circuits. Their theory is based on the phase-free ZH-calculus which crucially omits the average rule of the full ZH-calculus, dis-allowing the local summation of amplitudes. Here we study the question of completeness in unbalanced path sums which naturally support local summation. We give a concrete syntax for the unbalanced sum-over-paths and show that, together with symbolic multilinear algebra and the interference rule, various formulations","authors_text":"Matthew Amy","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2023-06-28T17:02:24Z","title":"Complete Equational Theories for the Sum-Over-Paths with Unbalanced Amplitudes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.16369","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:80d2521e8dbc0522edc0f0623434c89631cc931594a8d32adf8b0805419a586d","target":"record","created_at":"2026-07-05T06:46:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"368d557b05434e5005fea38e6fd9da2bb6dfb27e81a96c0ba2644907604786ee","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2023-06-28T17:02:24Z","title_canon_sha256":"dd77a5595a9732f70bbcff0eeeeb274f9340d4f6fd652b7f9da471382fd59dc1"},"schema_version":"1.0","source":{"id":"2306.16369","kind":"arxiv","version":2}},"canonical_sha256":"18bd486b73ff8cf794751c6d183f7dc2b6561379d60465d4cea5d980e5f7172f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18bd486b73ff8cf794751c6d183f7dc2b6561379d60465d4cea5d980e5f7172f","first_computed_at":"2026-07-05T06:46:22.981023Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:46:22.981023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gy2Kfc/2SBHukADcON7a6UVtyeAxzpxiJG5zABOAxEI2x5PRCnH6D1Ssdq09jQXP/0hwYK4CSO3r32ctlqYQBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:46:22.981585Z","signed_message":"canonical_sha256_bytes"},"source_id":"2306.16369","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80d2521e8dbc0522edc0f0623434c89631cc931594a8d32adf8b0805419a586d","sha256:761874fb7c829ef0596ffeb75ecd4451b19f53cf0053ede32e312593b0381d83"],"state_sha256":"eb4a8bc706a7806efecfa16a082ba693f80977940ff9c49594211dfd629e9f60"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ozWDhZohzGtD+d1zwvLnRvEoHcDHb4iG82cD2PelHowkbZ58Lc3sL66nMFUjaVNRhhJjrfC95SuSQN7grtKyDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T07:12:40.988004Z","bundle_sha256":"b534446ff1bb28b819075408861b07f34f3d5661633f1bcd2ab4085b2312b454"}}