Paper Citation Record · LEDGER

Unitary designs in nearly optimal depth

As of 20 August 2026, this Paper Citation Record lists 18 of 18 outbound references and 19 inbound Pith citation observations for arXiv:2507.06216.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2507.06216 v2

Coverage vector

measured 18 of 18 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T19:27:42.524431Z

measured 37 of 37 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-20T06:33:59.587034+00:00

measured 19 of 19 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-15T18:20:11.988425Z

measured 1 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Reference resolution

18 of 18 outbound references displayed

verified exact0
verified fuzzy7
unresolved11
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

0
arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Outbound references

Observation f550df09-e16d-4ea3-bf60-f344df841b2f · outbound

This paper cites Since p(x) is fixed, µ(x) can be precomputed once (on a classical computer) and reused.

Unitary designs in nearly optimal depth Since p(x) is fixed, µ(x) can be precomputed once (on a classical computer) and reused

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.819479Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation ea3d2967-24af-4ae3-b40b-c1fdcc8c1bbe · outbound

This paper cites This multiplication can be implemented using the Sch¨ onhage-Strassen algorithm with circuit depth O(log n) and size O(n log n log logn).

Unitary designs in nearly optimal depth This multiplication can be implemented using the Sch¨ onhage-Strassen algorithm with circuit depth O(log n) and size O(n log n log logn)

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.803852Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation f0c37677-e8b9-4756-be80-249c4b0c1faf · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.787996Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 29ff99f8-1b2d-462b-af00-6e1e992bcf62 · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.776514Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation e702a00b-a562-4fb8-b8d3-8f40664542f2 · outbound

This paper cites This requires another polynomial multiplication q(x) · p(x), which uses a circuit depth of O(log n) and a size of O(n log n log logn).

Unitary designs in nearly optimal depth This requires another polynomial multiplication q(x) · p(x), which uses a circuit depth of O(log n) and a size of O(n log n log logn)

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.761207Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 27577c8e-9b27-400c-b904-55f7e9fd0dbc · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 6

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.747283Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 76cde0ed-92f0-4af0-82cf-8e7fcdd8734b · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.731495Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 75f7ab47-18e9-4772-bae9-7449cdd120fd · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.711941Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 53d18c2e-d55f-4436-aa1f-88b1f67ac20d · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.697852Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 51c9175e-a68a-4f83-bb13-2c453daa3e9d · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.685144Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 4cf7f142-ce4b-4f52-8677-739066667fb2 · outbound

This paper cites ket” and “bra.

Unitary designs in nearly optimal depth ket” and “bra

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.672662Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 88d80d16-c2c7-4a0c-a087-113109a7e293 · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.656444Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation fcc2c411-7b37-4d84-888d-399e7a37ea15 · outbound

This paper cites glued” together by a random function acting on small subsets of both systems. In order to analyze the moments of this ensemble, we formulate a procedure to “glue.

Unitary designs in nearly optimal depth glued” together by a random function acting on small subsets of both systems. In order to analyze the moments of this ensemble, we formulate a procedure to “glue

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.639516Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 1672b787-12f9-4707-af83-d048e08f36e1 · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.624054Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 05ee3af7-4e1d-4fbf-bb55-269cc08abb83 · outbound

This paper cites Both implementations use 3nk + (5/2)n bits of randomness.

Unitary designs in nearly optimal depth Both implementations use 3nk + (5/2)n bits of randomness

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.610990Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation fa4fb1b2-5416-49cd-8974-fc47890eba58 · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.596223Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 636bfacc-fcb4-4b75-b456-89e1166ab73a · outbound

This paper cites an unresolved cited work.

Unitary designs in nearly optimal depth Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-06T19:27:42.583194Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 37bd2c85-33c6-446b-b5f6-db04f46dbbf9 · outbound

This paper cites amplification.

Unitary designs in nearly optimal depth amplification

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T19:27:42.568021Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Pith citing papers

Observation e9ccf646-7c8b-47de-b187-0786cdcd642a · inbound

Shallow quantum circuit for generating extremely low-entangled approximate state designs cites this paper.

Shallow quantum circuit for generating extremely low-entangled approximate state designs Unitary designs in nearly optimal depth

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-06T14:54:42.334391Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8da9e755-5e16-4702-b62e-95726c33d830 · inbound

Unconditional Pseudorandomness against Shallow Quantum Circuits cites this paper.

Unconditional Pseudorandomness against Shallow Quantum Circuits Unitary designs in nearly optimal depth

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-15T18:20:11.988425Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fb93dec9-d3e5-460d-8e5b-4ecd1aec5769 · inbound

Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit cites this paper.

Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit Unitary designs in nearly optimal depth

Reference 40

Resolution: verified exact; arxiv_id, observed 2026-05-19T01:06:57.481547Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation e1f1eb75-ebb9-4d34-9ce2-e2d3a6f0bb4f · inbound

Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit cites this paper.

Near-Heisenberg-limited parallel amplitude estimation with logarithmic depth circuit Unitary designs in nearly optimal depth

Reference 40

Resolution: unresolved; no resolver link, observed 2026-08-05T23:07:25.863452Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a1edd1eb-fdd2-42fe-a660-c617300d6624 · inbound

Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms cites this paper.

Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms Unitary designs in nearly optimal depth

Reference 68

Resolution: unresolved; no resolver link, observed 2026-08-04T13:37:04.450743Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 53882597-dbdb-499f-8a3e-a28c7807d379 · inbound

Growth and collapse of subsystem complexity under random unitary circuits cites this paper.

Growth and collapse of subsystem complexity under random unitary circuits Unitary designs in nearly optimal depth

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-04T08:55:58.222770Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f0bed99e-6e3d-415d-bc98-7f097a760e30 · inbound

Apparent Universal Behavior in Second Moments of Random Quantum Circuits cites this paper.

Apparent Universal Behavior in Second Moments of Random Quantum Circuits Unitary designs in nearly optimal depth

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-04T07:56:55.947030Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2d49a7da-0fb9-49eb-a167-4e2b469422c1 · inbound

Noisy Monitored Quantum Circuits cites this paper.

Noisy Monitored Quantum Circuits Unitary designs in nearly optimal depth

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-03T14:57:25.706343Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4860bc47-9699-4816-89ea-4df4bb71e31e · inbound

The most discriminable quantum states in the multicopy regime cites this paper.

The most discriminable quantum states in the multicopy regime Unitary designs in nearly optimal depth

Reference 35

Resolution: verified exact; arxiv_id, observed 2026-05-12T09:41:25.779736Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation bbda72a2-0930-4499-874d-c21e53553416 · inbound

The most discriminable quantum states in the multicopy regime cites this paper.

The most discriminable quantum states in the multicopy regime Unitary designs in nearly optimal depth

Reference 35

Resolution: verified exact; arxiv_id, observed 2026-07-01T08:35:33.577237Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation ec5d6f48-0b6d-40f3-beb6-f81c59e32626 · inbound

Arts & crafts: Strong random unitaries and geometric locality cites this paper.

Arts & crafts: Strong random unitaries and geometric locality Unitary designs in nearly optimal depth

Reference 31

Resolution: metadata mismatch; arxiv_id, observed 2026-05-09T05:55:31.085631Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation f199b821-899b-4fc4-99eb-ccec2bc999b2 · inbound

Typical Mixing and Rare-State Bottlenecks in Open Quantum Systems cites this paper.

Typical Mixing and Rare-State Bottlenecks in Open Quantum Systems Unitary designs in nearly optimal depth

Reference 73

Resolution: verified exact; arxiv_id, observed 2026-05-11T02:30:55.135163Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 35659634-f704-49ef-bdb4-cbf111334c76 · inbound

Quantum Nonlinear Properties from a Single Measurement Setting cites this paper.

Quantum Nonlinear Properties from a Single Measurement Setting Unitary designs in nearly optimal depth

Reference 60

Resolution: verified exact; arxiv_id, observed 2026-05-12T06:31:27.576393Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 04cedd6c-9b67-485b-97ed-c52392a9232e · inbound

Random Projections for Multi-Copy Quantum Algorithms cites this paper.

Random Projections for Multi-Copy Quantum Algorithms Unitary designs in nearly optimal depth

Reference 49

Resolution: verified exact; arxiv_id, observed 2026-07-04T04:29:35.165277Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 3f82abf7-078e-48cb-8fef-f501eca0f642 · inbound

Particle-preserving fermionic shadows with mode-independent sample complexity cites this paper.

Particle-preserving fermionic shadows with mode-independent sample complexity Unitary designs in nearly optimal depth

Reference 34

Resolution: metadata mismatch; arxiv_id, observed 2026-06-26T05:39:00.080931Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation d02bf908-d761-488a-a850-1e0f9a508d7e · inbound

Exact Hilbert-space ergodicity from continuous monitoring cites this paper.

Exact Hilbert-space ergodicity from continuous monitoring Unitary designs in nearly optimal depth

Reference 21

Resolution: verified exact; arxiv_id, observed 2026-06-30T13:24:40.736845Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 9f07b4e9-6d95-414a-9103-4f4081c82193 · inbound

Exact Hilbert-space ergodicity from continuous monitoring cites this paper.

Exact Hilbert-space ergodicity from continuous monitoring Unitary designs in nearly optimal depth

Reference 21

Resolution: verified exact; arxiv_id, observed 2026-07-01T09:45:39.466385Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation fac093e9-d52a-4258-a8f5-262000b9f5a4 · inbound

Strong unitary designs in optimal depth and space cites this paper.

Strong unitary designs in optimal depth and space Unitary designs in nearly optimal depth

Reference 96

Resolution: unresolved; no resolver link, observed 2026-08-14T06:03:59.950516Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cee41bfc-3d32-44df-89fc-7ca5ca3afa24 · inbound

Ambient unitaries don't enable shallow group designs cites this paper.

Ambient unitaries don't enable shallow group designs Unitary designs in nearly optimal depth

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-14T05:14:10.174197Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.