Quantum Algorithm for Estimating Intrinsic Geometry | Paper Citation Record

Observation 679f28e8-ead8-4957-a1c3-9299ef4c8589 · outbound

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Reference 1

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This paper cites It can be seen that the first column of this unitary is P j,xj ∈Ni |j⟩ xi.

Quantum Algorithm for Estimating Intrinsic Geometry It can be seen that the first column of this unitary is P j,xj ∈Ni |j⟩ xi

Reference 2

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Observation caf300e1-e1ca-4d61-9d6d-57e31127a2d7 · outbound

This paper cites • Perform singular decomposition on Ci and obtain a series of singular values σ1 ≥ σ2 ≥.

Quantum Algorithm for Estimating Intrinsic Geometry • Perform singular decomposition on Ci and obtain a series of singular values σ1 ≥ σ2 ≥

Reference 3

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Observation 285bf9ee-97e0-4ad7-b655-869a20dc0d41 · outbound

This paper cites In this step, we first need to use Lemma C.4 to obtain the unitary Uexi , which has complexity O (log |Ni|).

Quantum Algorithm for Estimating Intrinsic Geometry In this step, we first need to use Lemma C.4 to obtain the unitary Uexi , which has complexity O (log |Ni|)

Reference 4

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Observation a07ba53f-33a3-4e77-bba8-c9735e2131f5 · outbound

This paper cites In this step, we need to perform principal component analysis on∝ C † i Ci to find the local dimension di.

Quantum Algorithm for Estimating Intrinsic Geometry In this step, we need to perform principal component analysis on∝ C † i Ci to find the local dimension di

Reference 5

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Observation d6d07cd1-002a-4176-9afc-2ba406c3c5e8 · outbound

This paper cites Before fitting the quadratic curve, we need to use classical computer to compute the sampling density ρ(xi), which is efficient.

Quantum Algorithm for Estimating Intrinsic Geometry Before fitting the quadratic curve, we need to use classical computer to compute the sampling density ρ(xi), which is efficient

Reference 6

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Quantum Algorithm for Estimating Intrinsic Geometry Unresolved cited work

Reference 7

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Observation 009e55fc-ab9b-489e-804d-68918ad204f5 · outbound

This paper cites Laplacian eigen- maps for dimensionality reduction and data representa- tion.

Quantum Algorithm for Estimating Intrinsic Geometry Laplacian eigen- maps for dimensionality reduction and data representa- tion

Reference 19

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Observation ca476147-7968-48a0-81f7-c247791ef8cc · outbound

This paper cites Quantum diffu- sion map for nonlinear dimensionality reduction.Physical Review A, 104(5):052410, 2021.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum diffu- sion map for nonlinear dimensionality reduction.Physical Review A, 104(5):052410, 2021

Reference 20

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Observation 76426456-f6b1-4d08-80ff-a58ed5fdfc95 · outbound

This paper cites Riemannian geometry as determined by the volumes of small geodesic balls.

Quantum Algorithm for Estimating Intrinsic Geometry Riemannian geometry as determined by the volumes of small geodesic balls

Reference 21

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Observation f761112a-bafe-4b62-b5c4-4b4af182ec17 · outbound

This paper cites Quantum singular value transformation and be- yond: exponential improvements for quantum matrix arithmetics.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum singular value transformation and be- yond: exponential improvements for quantum matrix arithmetics

Reference 22

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Observation f908e433-0a05-4c08-bfcf-f5edb16769fc · outbound

This paper cites Optimal hamilto- nian simulation by quantum signal processing.

Quantum Algorithm for Estimating Intrinsic Geometry Optimal hamilto- nian simulation by quantum signal processing

Reference 23

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Observation bafde34a-3a1d-4e3e-8e8c-f6ef218df196 · outbound

This paper cites Hamiltonian sim- ulation by qubitization.

Quantum Algorithm for Estimating Intrinsic Geometry Hamiltonian sim- ulation by qubitization

Reference 24

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Observation 6f5ba7c6-1c2d-408a-89b3-d6b0da82ed5c · outbound

This paper cites Synthesis of quantum superpositions by quantum computation.

Quantum Algorithm for Estimating Intrinsic Geometry Synthesis of quantum superpositions by quantum computation

Reference 25

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Observation 1b61a76b-7649-490e-a21a-f0c92efb2356 · outbound

This paper cites Creating superpositions that correspond to efficiently integrable probability distributions.

Quantum Algorithm for Estimating Intrinsic Geometry Creating superpositions that correspond to efficiently integrable probability distributions

Reference 26

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Observation 5317e358-f234-4fb9-ab06-3ae32c28dc20 · outbound

This paper cites Quantum-state preparation with universal gate decompositions.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum-state preparation with universal gate decompositions

Reference 27

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Observation a6d6664f-035f-4b5f-8826-6dca7adf83e5 · outbound

This paper cites Supervised learning with quantum computers , volume 17.

Quantum Algorithm for Estimating Intrinsic Geometry Supervised learning with quantum computers , volume 17

Reference 28

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Observation 38b6902b-257d-4577-be97-bfce716301b3 · outbound

This paper cites Approxi- mate amplitude encoding in shallow parameterized quan- tum circuits and its application to financial market indi- cators.

Quantum Algorithm for Estimating Intrinsic Geometry Approxi- mate amplitude encoding in shallow parameterized quan- tum circuits and its application to financial market indi- cators

Reference 29

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Observation 57655f59-2057-4089-a2bc-a3eca9e75cd7 · outbound

This paper cites Quantum algorithms for approximate func- tion loading.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum algorithms for approximate func- tion loading

Reference 30

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Observation 5c9befbd-ec8b-44a7-9f6d-850d56ce09af · outbound

This paper cites Quantum generative adversarial networks for learning and loading random distributions.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum generative adversarial networks for learning and loading random distributions

Reference 31

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Observation 9f8dd336-d355-4bd2-9e24-b2cfd032de76 · outbound

This paper cites Quan- tum state preparation with optimal circuit depth: Im- plementations and applications.

Quantum Algorithm for Estimating Intrinsic Geometry Quan- tum state preparation with optimal circuit depth: Im- plementations and applications

Reference 32

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Observation a5eb5e14-f1e2-48a6-a138-65293785f5a0 · outbound

This paper cites Approximate quantum circuit synthesis using block encodings.

Quantum Algorithm for Estimating Intrinsic Geometry Approximate quantum circuit synthesis using block encodings

Reference 33

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Observation 44e10934-8a68-44f1-af11-5e5ee6b369aa · outbound

This paper cites Refined Quantum Algorithms for Principal Component Analysis and Solving Linear System.

Quantum Algorithm for Estimating Intrinsic Geometry Refined Quantum Algorithms for Principal Component Analysis and Solving Linear System

Reference 34

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Observation daaf0a70-c577-41fb-a9fc-e4e3095f4472 · outbound

This paper cites Quantum principal component analysis.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum principal component analysis

Reference 35

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Observation 67d3fbad-22c8-47a9-8197-4854156acd4e · outbound

This paper cites Geometric analysis of nonlinear manifold clustering.

Quantum Algorithm for Estimating Intrinsic Geometry Geometric analysis of nonlinear manifold clustering

Reference 36

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Observation cc625d39-e57d-4c05-a811-ebc45a6d05e1 · outbound

This paper cites A Comprehensive Introduction to Dif- ferential Geometry, Volume 2.

Quantum Algorithm for Estimating Intrinsic Geometry A Comprehensive Introduction to Dif- ferential Geometry, Volume 2

Reference 37

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Observation 0c777578-3cee-4eb4-9d4a-bb8a9e8082ad · outbound

This paper cites Curvature in mathematics and physics.

Quantum Algorithm for Estimating Intrinsic Geometry Curvature in mathematics and physics

Reference 38

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Observation 698970e4-825f-42b5-906b-eb24bc74d76b · outbound

This paper cites A note on Riemann normal coordinates.

Quantum Algorithm for Estimating Intrinsic Geometry A note on Riemann normal coordinates

Reference 39

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Observation 9c3a8b02-4d67-4f98-947a-f2d50a419e1c · outbound

This paper cites Asymptotic probabilities and differential equations.

Quantum Algorithm for Estimating Intrinsic Geometry Asymptotic probabilities and differential equations

Reference 40

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Observation fa4c2ad7-e2ec-4a5e-b02d-68d002d291f6 · outbound

This paper cites Ele- mentary gates for quantum computation.

Quantum Algorithm for Estimating Intrinsic Geometry Ele- mentary gates for quantum computation

Reference 41

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Observation 917f5c35-6781-40d3-aa27-bf656c6c15b5 · outbound

This paper cites Synthesis of quantum logic circuits.

Quantum Algorithm for Estimating Intrinsic Geometry Synthesis of quantum logic circuits

Reference 42

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Observation 50990be2-2ba3-4aca-97c6-880d04f3ecb6 · outbound

This paper cites Approximation theory and approxi- mation practice, extended edition.

Quantum Algorithm for Estimating Intrinsic Geometry Approximation theory and approxi- mation practice, extended edition

Reference 43

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Observation 46aa5b26-229f-47ab-a5da-6a386f44f23d · outbound

This paper cites The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation.

Quantum Algorithm for Estimating Intrinsic Geometry The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation

Reference 44

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Observation ac8885f4-88b3-40b0-9448-aa646045d5a5 · outbound

This paper cites Quantum algorithms for estimating physi- cal quantities using block encodings.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum algorithms for estimating physi- cal quantities using block encodings

Reference 45

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Observation b5e35f6e-3cae-4e2c-8938-8df80cb79cf7 · outbound

This paper cites Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications.

Quantum Algorithm for Estimating Intrinsic Geometry Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications

Reference 46

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Observation 84d40f41-c8f8-4077-837c-53d94927666b · outbound

This paper cites Nonlin- ear transformation of complex amplitudes via quantum singular value transformation.

Quantum Algorithm for Estimating Intrinsic Geometry Nonlin- ear transformation of complex amplitudes via quantum singular value transformation

Reference 47

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Observation b9b56b14-672e-46cb-915a-b05e62f01ec1 · outbound

This paper cites Improved quantum power method and nu- merical integration using a quantum singular-value trans- formation.

Quantum Algorithm for Estimating Intrinsic Geometry Improved quantum power method and nu- merical integration using a quantum singular-value trans- formation

Reference 48

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Observation c07abbbc-7c55-45eb-88b7-4eb0a031a33f · outbound

This paper cites Improved Quantum Algorithms for Eigenvalues Finding and Gradient Descent.

Quantum Algorithm for Estimating Intrinsic Geometry Improved Quantum Algorithms for Eigenvalues Finding and Gradient Descent

Reference 49

Resolution: verified exact; local_arxiv, observed 2026-08-05T22:53:55.043652Z

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Observation 34048a89-d640-4758-89cf-e34e3df4d8ca · outbound

This paper cites A quantum speed-up for approximating the top eigen- vectors of a matrix.

Quantum Algorithm for Estimating Intrinsic Geometry A quantum speed-up for approximating the top eigen- vectors of a matrix

Reference 50

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Observation c1e47887-ad30-4717-a56b-76902b83f31b · outbound

This paper cites Error bounds on the power method for determining the largest eigenvalue of a symmetric, posi- tive definite matrix.

Quantum Algorithm for Estimating Intrinsic Geometry Error bounds on the power method for determining the largest eigenvalue of a symmetric, posi- tive definite matrix

Reference 51

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Observation 556e9e50-59d0-46ce-9d0c-767865f6d617 · outbound

This paper cites Matrix compu- tations.

Quantum Algorithm for Estimating Intrinsic Geometry Matrix compu- tations

Reference 52

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Observation e69a0200-b748-437b-bd8e-6c5e76dbb6e0 · outbound

This paper cites Quantum algorithm for systems of linear equa- tions with exponentially improved dependence on pre- cision.

Quantum Algorithm for Estimating Intrinsic Geometry Quantum algorithm for systems of linear equa- tions with exponentially improved dependence on pre- cision

Reference 53

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Observation e7240e47-25e3-4240-8708-7c83c9d00713 · outbound

This paper cites Geometry, topology and physics.

Quantum Algorithm for Estimating Intrinsic Geometry Geometry, topology and physics

Reference 54

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Observation a72699ee-4184-4f5d-a07e-11afe65cf846 · outbound

This paper cites − dG(xi, xj) h 2# (B.11) is exactly the heat kernel (i.e. the fundamental solution of the diffusion equation at “time.

Quantum Algorithm for Estimating Intrinsic Geometry − dG(xi, xj) h 2# (B.11) is exactly the heat kernel (i.e. the fundamental solution of the diffusion equation at “time

Reference 55

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Observation d721d5bc-98cf-4ec4-bf98-533d4bb15151 · outbound

This paper cites In this part, we first need to use Lemma C.4 to prepare a N -dimensional state |ϕ⟩, which incurs complexity O (log N ).

Quantum Algorithm for Estimating Intrinsic Geometry In this part, we first need to use Lemma C.4 to prepare a N -dimensional state |ϕ⟩, which incurs complexity O (log N )

Reference 56

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Observation 07390a63-dc15-4876-bb65-ec330f1f3ec4 · outbound

This paper cites First, twe use Lemma D.2 to obtain the block-encoding of the matrix E = PN i=1|i−1⟩⟨i−1|⊗Ei ||E||F.

Quantum Algorithm for Estimating Intrinsic Geometry First, twe use Lemma D.2 to obtain the block-encoding of the matrix E = PN i=1|i−1⟩⟨i−1|⊗Ei ||E||F

Reference 57

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.259909Z

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Observation e5b15c03-12df-4a58-95f9-e2e7955a371f · outbound

This paper cites The neighborhood of xi is defined as |Ni| closest points to xi, in terms of geodesic distance.

Quantum Algorithm for Estimating Intrinsic Geometry The neighborhood of xi is defined as |Ni| closest points to xi, in terms of geodesic distance

Reference 58

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Observation 749b8a6f-7fa7-4276-b7e8-3c364066c54a · outbound

This paper cites Algorithm 2 (Diffusion Map).

Quantum Algorithm for Estimating Intrinsic Geometry Algorithm 2 (Diffusion Map)

Reference 59

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Observation b486810f-2d91-46d8-a1d2-7db03f65aae7 · outbound

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Quantum Algorithm for Estimating Intrinsic Geometry Unresolved cited work

Reference 60

Resolution: unresolved; raw_fallback, observed 2026-08-05T22:53:55.215786Z

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Observation 140671e6-cbc8-4175-978b-3052902fb80b · outbound

This paper cites Let |x0⟩ denote some initial state, generated by some known circuit U0 (assuming to have O(1) depth).

Quantum Algorithm for Estimating Intrinsic Geometry Let |x0⟩ denote some initial state, generated by some known circuit U0 (assuming to have O(1) depth)

Reference 61

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Observation 48d4761b-0a38-4f5f-92a2-998a080f5bfb · outbound

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Quantum Algorithm for Estimating Intrinsic Geometry Unresolved cited work

Reference 62

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Quantum Algorithm for Estimating Intrinsic Geometry Unresolved cited work

Reference 63

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Observation 7533a065-bf41-46f0-b5a7-0bbc63f68f41 · outbound

This paper cites We use Lemma J.2 and J.1 to transform the block-encoded operator: γ |xk⟩ ⟨xk| − →e−β(1−γ) |xk⟩ ⟨xk| (J.2).

Quantum Algorithm for Estimating Intrinsic Geometry We use Lemma J.2 and J.1 to transform the block-encoded operator: γ |xk⟩ ⟨xk| − →e−β(1−γ) |xk⟩ ⟨xk| (J.2)

Reference 64

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Observation fb6a142a-abbe-4617-ad4f-985e61249835 · outbound

This paper cites Now we take the above block encoding and apply it to |x0⟩, and according to K.1, we obtain the following state: |0⟩ ⟨xk, x0⟩ e−β(1−γ) |xk⟩ + |Garbage⟩ (J.3).

Quantum Algorithm for Estimating Intrinsic Geometry Now we take the above block encoding and apply it to |x0⟩, and according to K.1, we obtain the following state: |0⟩ ⟨xk, x0⟩ e−β(1−γ) |xk⟩ + |Garbage⟩ (J.3)

Reference 65

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.181028Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 65c9bc4a-313b-4f57-a7f4-6d18a60b99d0 · outbound

This paper cites The suc- cess probability of this measurement is | ⟨xk, x0⟩ |2e−2β(1−γ), which can be improved quadratically better using amplitude amplification.

Quantum Algorithm for Estimating Intrinsic Geometry The suc- cess probability of this measurement is | ⟨xk, x0⟩ |2e−2β(1−γ), which can be improved quadratically better using amplitude amplification

Reference 66

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.173278Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 3110f8e1-f2d9-40c0-90b3-0da3befab9d3 · outbound

This paper cites Tn i=1 Si ∈ Cwhere Si ∈ Cand n ∈ Z≥1.

Quantum Algorithm for Estimating Intrinsic Geometry Tn i=1 Si ∈ Cwhere Si ∈ Cand n ∈ Z≥1

Reference 67

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.158609Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 39767d64-d20a-4b9b-8e32-5736f1d860d8 · outbound

This paper cites S i∈T Si ∈ Cwhere Si ∈ Cand T is any index set (finite or infinite).

Quantum Algorithm for Estimating Intrinsic Geometry S i∈T Si ∈ Cwhere Si ∈ Cand T is any index set (finite or infinite)

Reference 68

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.151170Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 1e0f0504-fac5-49f9-bd1e-7feb8789964e · outbound

This paper cites (Some technical conditions allow the space to behave nicely; a first-time reader can ignore those conditions, which we include here for completeness).

Quantum Algorithm for Estimating Intrinsic Geometry (Some technical conditions allow the space to behave nicely; a first-time reader can ignore those conditions, which we include here for completeness)

Reference 69

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.143258Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation fda6a2b9-605a-4eed-97f5-54065e131a7e · outbound

This paper cites The pair (Ui, φi) is called a chart.

Quantum Algorithm for Estimating Intrinsic Geometry The pair (Ui, φi) is called a chart

Reference 70

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.135071Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 4030fae4-9a61-4ae5-ba52-3ac5ef0af026 · outbound

This paper cites The collection of such charts is called an atlas.

Quantum Algorithm for Estimating Intrinsic Geometry The collection of such charts is called an atlas

Reference 71

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.127156Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 076d1ead-fb5d-4d16-8ea9-350f6bdf9d44 · outbound

This paper cites Note that this forces dim M ≤ dim N.

Quantum Algorithm for Estimating Intrinsic Geometry Note that this forces dim M ≤ dim N

Reference 72

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.119530Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation dc5542ce-e2f0-46e9-a73c-4de2cde639bd · outbound

This paper cites The image f (M ) is called a submanifold of N.

Quantum Algorithm for Estimating Intrinsic Geometry The image f (M ) is called a submanifold of N

Reference 73

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.111709Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 5ac3f922-ac4a-4469-8529-b41dfaee7edf · outbound

This paper cites an unresolved cited work.

Quantum Algorithm for Estimating Intrinsic Geometry Unresolved cited work

Reference 74

Resolution: unresolved; raw_fallback, observed 2026-08-05T22:53:55.104223Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 5ce22af0-c925-4728-9fd3-b663711d71a6 · outbound

This paper cites 35 In the local coordinate {xµ}, we can write the inner product between vectors U and V as: U = U µ(x) ∂ ∂xµ , V = V ν(x) ∂ ∂xν =⇒ gp(U, V) = gµν(x)U µ(x)V ν(x).

Quantum Algorithm for Estimating Intrinsic Geometry 35 In the local coordinate {xµ}, we can write the inner product between vectors U and V as: U = U µ(x) ∂ ∂xµ , V = V ν(x) ∂ ∂xν =⇒ gp(U, V) = gµν(x)U µ(x)V ν(x)

Reference 75

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T22:53:55.095622Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

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Reference resolution

Outbound references

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