Quantum Data Fitting Algorithm for Non-sparse Matrices

Guangxi Li; Youle Wang; Yuan Feng; Yu Luo

arxiv: 1907.06949 · v1 · pith:WV2V3M5Knew · submitted 2019-07-16 · 🪐 quant-ph · cs.LG

Quantum Data Fitting Algorithm for Non-sparse Matrices

Guangxi Li , Youle Wang , Yu Luo , Yuan Feng This is my paper

Pith reviewed 2026-05-24 21:14 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG

keywords quantum data fittingnon-sparse matricesquantum singular value estimationeigenvalue sign recoveryregularizationcondition numberquantum machine learningruntime complexity

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The pith

A quantum algorithm fits data to non-sparse Hermitian matrices with runtime O(κ²√N polylog(N)/(ε log κ)) that does not depend on sparsity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends an existing quantum data fitting method from sparse matrices to the non-sparse case. It does so by adding a regularization term to control overfitting and by introducing a subroutine that recovers the signs of eigenvalues after quantum singular value estimation. The resulting procedure runs in time polynomial in the square root of the matrix dimension N while keeping a strictly sub-quadratic dependence on the condition number κ. A reader would care because this removes a major practical restriction on which matrices can be handled by quantum linear-algebra routines.

Core claim

By combining quantum singular value estimation with a new efficient sign-recovery procedure for eigenvalues and by inserting a regularization term, the algorithm generalizes the sparse-matrix data-fitting routine of Wiebe, Braun and Lloyd to arbitrary Hermitian matrices F of size N×N while preserving the sparsity-independent runtime bound O(κ²√N polylog(N)/(ε log κ)).

What carries the argument

Quantum singular value estimation together with an efficient eigenvalue sign-recovery subroutine that works for non-sparse matrices.

If this is right

Data fitting becomes possible on dense matrices without an exponential penalty in sparsity.
The algorithm supplies a polynomial speedup in matrix dimension relative to classical methods.
The dependence on condition number remains strictly less than quadratic.
Regularization prevents overfitting while the quantum runtime advantage is retained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sign-recovery technique could be reused in other quantum linear-algebra tasks that require full spectral information on non-sparse operators.
If the method extends beyond data fitting, it would enlarge the set of machine-learning problems that fit inside the quantum linear-algebra framework.
Numerical checks on small non-sparse test matrices would directly test whether the claimed overhead of sign recovery is realized.

Load-bearing premise

The new sign-recovery subroutine works correctly and with only logarithmic overhead on non-sparse matrices.

What would settle it

An explicit non-sparse matrix for which the sign-recovery step either fails or takes time that grows with the number of non-zero entries would falsify the claimed runtime bound.

Figures

Figures reproduced from arXiv: 1907.06949 by Guangxi Li, Youle Wang, Yuan Feng, Yu Luo.

read the original abstract

We propose a quantum data fitting algorithm for non-sparse matrices, which is based on the Quantum Singular Value Estimation (QSVE) subroutine and a novel efficient method for recovering the signs of eigenvalues. Our algorithm generalizes the quantum data fitting algorithm of Wiebe, Braun, and Lloyd for sparse and well-conditioned matrices by adding a regularization term to avoid the over-fitting problem, which is a very important problem in machine learning. As a result, the algorithm achieves a sparsity-independent runtime of $O(\kappa^2\sqrt{N}\mathrm{polylog}(N)/(\epsilon\log\kappa))$ for an $N\times N$ dimensional Hermitian matrix $\bm{F}$, where $\kappa$ denotes the condition number of $\bm{F}$ and $\epsilon$ is the precision parameter. This amounts to a polynomial speedup on the dimension of matrices when compared with the classical data fitting algorithms, and a strictly less than quadratic dependence on $\kappa$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract claims a sparsity-independent runtime for quantum data fitting but provides no derivation for the key sign recovery method.

read the letter

The one thing to know is that this paper claims to have found a way to do quantum data fitting on dense matrices without the sparsity requirement that limited earlier work, by adding regularization and a sign recovery step, but the abstract does not include any of the supporting math or complexity analysis. The work generalizes the Wiebe, Braun, and Lloyd algorithm. That earlier paper handled sparse, well-conditioned matrices. Here they add a regularization term, which is useful because overfitting is a real issue in data fitting tasks from machine learning. They also introduce a method to recover the signs of eigenvalues so that the quantum singular value estimation can be applied even when the matrix is not sparse. The claimed runtime is O(κ² √N polylog(N) / (ε log κ)), which removes the dependence on the number of non-zero entries and gives a polynomial improvement over classical methods in the dimension N. This is a reasonable direction to explore. Removing the sparsity assumption opens the algorithm to a wider class of problems. The regularization is a practical addition that aligns the quantum method more closely with how classical fitting is done. The soft spot is the lack of detail on the sign recovery method. The runtime bound only holds if that subroutine runs efficiently and does not introduce factors that grow with the matrix density. The abstract asserts that the method is efficient, but there are no steps shown for how it works, no error bounds, and no gate complexity. Without those, it is not possible to verify that the claimed scaling is preserved for a matrix with Θ(N²) entries. This is the load-bearing part of the argument, and it is not supported in the text. The paper cites the relevant prior work on QSVE and the sparse case, so the citation pattern is appropriate. There is no obvious circularity. This paper is for specialists in quantum machine learning who follow developments in quantum linear algebra algorithms. A reader can get the high-level idea of how one might try to relax the sparsity constraint, but the absence of the technical justification means there is not much to engage with in depth. It does not rise to the level where a serious editor would send it to peer review, because the central claim cannot be evaluated from what is presented. I would not bring it to a reading group or cite it based on this version.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a quantum data fitting algorithm for non-sparse Hermitian matrices. It extends the QSVE-based approach of Wiebe, Braun, and Lloyd by incorporating a regularization term to mitigate overfitting and introducing a novel subroutine for recovering the signs of eigenvalues. The central claim is a sparsity-independent runtime of O(κ² √N polylog(N) / (ε log κ)) for an N×N matrix F.

Significance. If the sign-recovery subroutine is shown to incur no density-dependent overhead that invalidates the √N or 1/log κ factors, the result would constitute a polynomial improvement over classical dense-matrix fitting algorithms together with a strictly sub-quadratic κ dependence. The regularization step is a standard and useful addition for machine-learning applicability.

major comments (2)

[Sign-recovery subroutine (main algorithm section)] The section describing the novel eigenvalue sign-recovery method (the step added to QSVE): the manuscript asserts that this subroutine enables the sparsity-independent bound, yet supplies no explicit query or gate complexity that demonstrates preservation of the √N scaling when the input matrix has Θ(N²) non-zeros. Any hidden linear or quadratic factor in N would invalidate the headline runtime.
[Abstract / runtime statement] Abstract and runtime claim: the stated complexity O(κ² √N polylog(N)/(ε log κ)) is presented without an accompanying derivation or error-propagation analysis that accounts for the cost of sign recovery on dense matrices; the bound therefore rests on an unverified assumption about the subroutine's overhead.

minor comments (1)

[Introduction / preliminaries] Notation for the matrix F and the regularization parameter should be introduced with explicit definitions before the complexity statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. The two major comments both concern the missing explicit complexity analysis for the eigenvalue sign-recovery subroutine on dense matrices. We agree that this analysis is required to substantiate the claimed runtime and will add it in revision.

read point-by-point responses

Referee: [Sign-recovery subroutine (main algorithm section)] The section describing the novel eigenvalue sign-recovery method (the step added to QSVE): the manuscript asserts that this subroutine enables the sparsity-independent bound, yet supplies no explicit query or gate complexity that demonstrates preservation of the √N scaling when the input matrix has Θ(N²) non-zeros. Any hidden linear or quadratic factor in N would invalidate the headline runtime.

Authors: We acknowledge that the manuscript does not supply an explicit gate or query complexity for the sign-recovery subroutine on dense (Θ(N²)-nonzero) matrices. The current text only states that the subroutine is efficient without a full circuit or oracle analysis. We will add a dedicated subsection (or appendix) deriving the gate complexity of sign recovery, showing that it incurs only polylog(N) overhead per singular vector when matrix elements are accessed via a dense oracle, thereby preserving the overall O(√N) scaling. This will include the number of controlled rotations and the error-propagation bounds from the QSVE step. revision: yes
Referee: [Abstract / runtime statement] Abstract and runtime claim: the stated complexity O(κ² √N polylog(N)/(ε log κ)) is presented without an accompanying derivation or error-propagation analysis that accounts for the cost of sign recovery on dense matrices; the bound therefore rests on an unverified assumption about the subroutine's overhead.

Authors: We agree that the abstract and main runtime claim lack a consolidated derivation that folds in the sign-recovery cost. In the revised manuscript we will (i) move the runtime statement to a theorem with a proof sketch, (ii) explicitly bound the additional error introduced by sign recovery, and (iii) show that the total complexity remains O(κ² √N polylog(N)/(ε log κ)) with no extra polynomial factors in N. The regularization term will also be folded into the error analysis. revision: yes

Circularity Check

0 steps flagged

No circularity; runtime bound derives from external QSVE plus independent sign-recovery subroutine.

full rationale

The paper explicitly builds the algorithm on the external QSVE subroutine from prior literature and introduces a novel sign-recovery method plus regularization term to generalize Wiebe et al. The claimed O(κ²√N polylog(N)/(ε log κ)) bound is stated as following from these additions without any reduction of the new subroutine's complexity to a fitted parameter, self-definition, or self-citation chain. No load-bearing step equates a prediction to its input by construction; the central claim retains independent content in the proposed method for non-sparse matrices.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the matrix being Hermitian and on the unelaborated sign-recovery subroutine working as stated.

axioms (1)

domain assumption The input matrix F is Hermitian and N x N dimensional
Required for application of QSVE as stated in the abstract.

pith-pipeline@v0.9.0 · 5687 in / 1158 out tokens · 20845 ms · 2026-05-24T21:14:23.365193+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 7 internal anchors

[1]

Note that we can assume without loss of generality that the matrix F is Hermitian

is given by w∗ = (F †F +γIn)− 1F †y, (3) where In denotes the n-by-n identity matrix. Note that we can assume without loss of generality that the matrix F is Hermitian. Otherwise, deﬁne ˜F = ˜F † := [ 0 F F † 0 ] and ˜y := [ y 0 ] ∈ Cm+n. Then it is easy to check that w∗ satisﬁes Eq. ( 3) if and only if ˜w∗ = ( ˜F † ˜F +γIm+n)− 1 ˜F † ˜y, (4) 2 Here, we, ...

work page
[2]

by expanding the vector’s dimension [ 3]. III. QUANTUM SINGULAR V ALUE ESTIMATION Quantum Singular Value Estimation (QSVE) can be viewed as extending Phase Estimation [ 24] from unitary to nonunitary matrices, which is also the primary algo- rithm subroutine for our quantum data ﬁtting algorithm. We brieﬂy state it in the following: Given a matrix A ∈ Rm×...

work page
[3]

Generate a value of hyper-parameter γ ∈ [ ∥F ∥2 ∗ κ2 , ∥F ∥2 ∗ ] according to the log-uniform distribution

work page
[4]

Create the quantum state |y⟩ = ∑ i β i |vi⟩ which is proportional to y, with vi’s being the eigenvectors of F

work page
[5]

Perform the QSVE subroutine for matrix ˆF with precision δ = ∥F ∥∗ 4κ ǫ to obtain the state ∑ i β i |vi⟩ ⏐ ⏐ ⏐ ˆλ i ⟩

work page
[6]

Add an auxiliary register and apply a rotation conditioned on the second register, and uncompute the QSVE subroutine to erase the second register, obtaining ∑ i β i |vi⟩    C λ i λ i 2 + γ |0⟩ +    √ 1 − ( Cλ i λ i 2 + γ ) 2 |1⟩    , (8) where λ i = ˆλ i − ∥ F ∥∗ is the estimation of the eigenvalue λ i of F and C = C0 √ γ (0 < C 0 < 2) is a constant

work page
[7]

AI meets Quantum: Quantum algorithms for knowledge represen- tation and learning

Post-select on the auxiliary register being in state |0⟩. where p := ∑ i |βi|2 ( h(λ i) ) 2 and h is deﬁned as follows: h(λ) := Cλ λ 2 +γ = √ γλ λ 2 +γ. (10) Here, we take C = C0 √ γ = √ γ as an example (Other values of C0 ∈ (0, 2) are similar). The ideal state |w∗ ⟩ should be |w∗ ⟩ = ∑ iβih(λ i) |vi⟩ |0⟩√∑ i |βi|2 (h(λ i))2 = ∑ iβih(λ i) |vi⟩ |0⟩√ p , (1...

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Biamonte, P

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017)

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Rebentrost, M

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Quantum Recommendation Systems

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Quantum Deep Learning

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S. Lloyd, M. Mohseni, and P. Rebentrost, Nature Physics 10, 631 (2014)

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G. H. Low, T. J. Yoder, and I. L. Chuang, Physical Review A 89, 062315 (2014)

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Quantum gradient descent and Newton's method for constrained polynomial optimization

P. Rebentrost, M. Schuld, L. Wossnig, F. Petruccione, and S. Lloyd, arXiv preprint arXiv:1612.01789 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
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Ciliberto, M

C. Ciliberto, M. Herbster, A. D. Ialongo, M. Pontil, A. Rocchetto, S. Severini, and L. Wossnig, Proceedings Of The Royal Society A: Mathematical, Physical and En- gineering Sciences 474, 20170551 (2018)

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Wiebe, D

N. Wiebe, D. Braun, and S. Lloyd, Physical review let- ters 109, 050505 (2012)

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A. M. Childs, Communications in Mathematical Physics 294, 581 (2010)

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work page internal anchor Pith review Pith/arXiv arXiv 2009
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Y. Liu and S. Zhang, in International Workshop on Fron- tiers in Algorithmics (Springer, 2015) pp. 204–216

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Wossnig, Z

L. Wossnig, Z. Zhao, and A. Prakash, Physical review letters 120, 050502 (2018)

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Meng, X.-T

F.-X. Meng, X.-T. Yu, R.-Q. Xiang, and Z.-C. Zhang, IEEE Access (2018)

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[1] [1]

Note that we can assume without loss of generality that the matrix F is Hermitian

is given by w∗ = (F †F +γIn)− 1F †y, (3) where In denotes the n-by-n identity matrix. Note that we can assume without loss of generality that the matrix F is Hermitian. Otherwise, deﬁne ˜F = ˜F † := [ 0 F F † 0 ] and ˜y := [ y 0 ] ∈ Cm+n. Then it is easy to check that w∗ satisﬁes Eq. ( 3) if and only if ˜w∗ = ( ˜F † ˜F +γIm+n)− 1 ˜F † ˜y, (4) 2 Here, we, ...

work page

[2] [2]

by expanding the vector’s dimension [ 3]. III. QUANTUM SINGULAR V ALUE ESTIMATION Quantum Singular Value Estimation (QSVE) can be viewed as extending Phase Estimation [ 24] from unitary to nonunitary matrices, which is also the primary algo- rithm subroutine for our quantum data ﬁtting algorithm. We brieﬂy state it in the following: Given a matrix A ∈ Rm×...

work page

[3] [3]

Generate a value of hyper-parameter γ ∈ [ ∥F ∥2 ∗ κ2 , ∥F ∥2 ∗ ] according to the log-uniform distribution

work page

[4] [4]

Create the quantum state |y⟩ = ∑ i β i |vi⟩ which is proportional to y, with vi’s being the eigenvectors of F

work page

[5] [5]

Perform the QSVE subroutine for matrix ˆF with precision δ = ∥F ∥∗ 4κ ǫ to obtain the state ∑ i β i |vi⟩ ⏐ ⏐ ⏐ ˆλ i ⟩

work page

[6] [6]

Add an auxiliary register and apply a rotation conditioned on the second register, and uncompute the QSVE subroutine to erase the second register, obtaining ∑ i β i |vi⟩    C λ i λ i 2 + γ |0⟩ +    √ 1 − ( Cλ i λ i 2 + γ ) 2 |1⟩    , (8) where λ i = ˆλ i − ∥ F ∥∗ is the estimation of the eigenvalue λ i of F and C = C0 √ γ (0 < C 0 < 2) is a constant

work page

[7] [7]

AI meets Quantum: Quantum algorithms for knowledge represen- tation and learning

Post-select on the auxiliary register being in state |0⟩. where p := ∑ i |βi|2 ( h(λ i) ) 2 and h is deﬁned as follows: h(λ) := Cλ λ 2 +γ = √ γλ λ 2 +γ. (10) Here, we take C = C0 √ γ = √ γ as an example (Other values of C0 ∈ (0, 2) are similar). The ideal state |w∗ ⟩ should be |w∗ ⟩ = ∑ iβih(λ i) |vi⟩ |0⟩√∑ i |βi|2 (h(λ i))2 = ∑ iβih(λ i) |vi⟩ |0⟩√ p , (1...

work page

[8] [8]

Biamonte, P

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017)

work page 2017

[9] [9]

Wittek, Quantum machine learning: what quantum computing means to data mining (Academic Press, 2014)

P. Wittek, Quantum machine learning: what quantum computing means to data mining (Academic Press, 2014)

work page 2014

[10] [10]

A. W. Harrow, A. Hassidim, and S. Lloyd, Physical re- view letters 103, 150502 (2009)

work page 2009

[11] [11]

Rebentrost, M

P. Rebentrost, M. Mohseni, and S. Lloyd, Physical re- view letters 113, 130503 (2014)

work page 2014

[12] [12]

Quantum Recommendation Systems

I. Kerenidis and A. Prakash, arXiv preprint arXiv:1603.08675 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

Schuld, I

M. Schuld, I. Sinayskiy, and F. Petruccione, Physical Review A 94, 022342 (2016)

work page 2016

[14] [14]

Quantum Algorithms for Nearest-Neighbor Methods for Supervised and Unsupervised Learning

N. Wiebe, A. Kapoor, and K. Svore, arXiv preprint arXiv:1401.2142 (2014)

work page internal anchor Pith review Pith/arXiv arXiv 2014

[15] [15]

Quantum Deep Learning

N. Wiebe, A. Kapoor, and K. M. Svore, arXiv preprint arXiv:1412.3489 (2014)

work page internal anchor Pith review Pith/arXiv arXiv 2014

[16] [16]

Kapoor, N

A. Kapoor, N. Wiebe, and K. Svore, in Advances in Neural Information Processing Systems (2016) pp. 3999– 4007

work page 2016

[17] [17]

Z. Zhao, J. K. Fitzsimons, and J. F. Fitzsimons, arXiv preprint arXiv:1512.03929 (2015)

work page internal anchor Pith review Pith/arXiv arXiv 2015

[18] [18]

Lloyd, M

S. Lloyd, M. Mohseni, and P. Rebentrost, Nature Physics 10, 631 (2014)

work page 2014

[19] [19]

G. H. Low, T. J. Yoder, and I. L. Chuang, Physical Review A 89, 062315 (2014)

work page 2014

[20] [20]

Quantum gradient descent and Newton's method for constrained polynomial optimization

P. Rebentrost, M. Schuld, L. Wossnig, F. Petruccione, and S. Lloyd, arXiv preprint arXiv:1612.01789 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[21] [21]

Ciliberto, M

C. Ciliberto, M. Herbster, A. D. Ialongo, M. Pontil, A. Rocchetto, S. Severini, and L. Wossnig, Proceedings Of The Royal Society A: Mathematical, Physical and En- gineering Sciences 474, 20170551 (2018)

work page 2018

[22] [22]

Wiebe, D

N. Wiebe, D. Braun, and S. Lloyd, Physical review let- ters 109, 050505 (2012)

work page 2012

[23] [23]

A. M. Childs, Communications in Mathematical Physics 294, 581 (2010)

work page 2010

[24] [24]

D. W. Berry and A. M. Childs, arXiv preprint arXiv:0910.4157 (2009)

work page internal anchor Pith review Pith/arXiv arXiv 2009

[25] [25]

Liu and S

Y. Liu and S. Zhang, in International Workshop on Fron- tiers in Algorithmics (Springer, 2015) pp. 204–216

work page 2015

[26] [26]

D. M. Hawkins, Journal of chemical information and computer sciences 44, 1 (2004)

work page 2004

[27] [27]

A. E. Hoerl and R. W. Kennard, Technometrics 12, 55 (1970)

work page 1970

[28] [28]

Wossnig, Z

L. Wossnig, Z. Zhao, and A. Prakash, Physical review letters 120, 050502 (2018)

work page 2018

[29] [29]

Meng, X.-T

F.-X. Meng, X.-T. Yu, R.-Q. Xiang, and Z.-C. Zhang, IEEE Access (2018)

work page 2018

[30] [30]

C.-H. Yu, F. Gao, and Q.-Y. Wen, arXiv preprint arXiv:1707.09524 (2017)

work page arXiv 2017

[31] [31]

A. Y. Kitaev, arXiv preprint quant-ph/9511026 (1995)

work page internal anchor Pith review Pith/arXiv arXiv 1995

[32] [32]

Svergun, Journal of applied crystallography 25, 495 (1992)

D. Svergun, Journal of applied crystallography 25, 495 (1992)

work page 1992

[33] [33]

Brassard, P

G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Con- temporary Mathematics 305, 53 (2002)

work page 2002

[34] [34]

Montavon, G

G. Montavon, G. B. Orr, and K.-R. M¨ uller, (1998)

work page 1998