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REVIEW 3 major objections 5 minor 59 references

Quantum Higher Order Singular Value Decomposition

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Pith's one-line read The paper presents two quantum algorithms for higher-order singular value decomposition (HOSVD), claiming polylogarithmic runtime and exponential speedup over the classical method.

desk verdict Algorithm 1's complexity claim is internally inconsistent, but Algorithm 2 and the recommendation scheme are worth a major-revision review. read the letter →

arxiv 1908.00719 v2 pith:OIKNYU2M submitted 2019-08-02 quant-ph

Lejia Gu , Xiaoqiang Wang , H. W. Joseph Lee , Guofeng Zhang This is my paper · ORCID
classification quant-ph MSC 81P6868Q1215A69 PACS 03.67.Ac03.67.Lx
keywords higher-ordersingularvaluedecompositionquantumalgorithmtensorprincipalcomponentanalysisestimationqRAMrecommendationsystemsexponentialspeedup
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to bring higher-order singular value decomposition (HOSVD), a standard tool for analyzing multidimensional data, into the quantum setting. It presents two quantum algorithms that prepare a superposition state encoding a tensor's singular values and singular vectors. For an $m$th-order $n$-dimensional tensor, the first runs in $O(m^2 \mathrm{polylog}\, n)$ and the second in $O(m^3 \mathrm{polylog}\, n)$, compared with $O(m n^{m+1})$ for classical HOSVD; the first assumes low-rank unfolding matrices, while the second removes that restriction. If the speedup holds, tensor-based machine learning, recommendation systems, and quantum-chemistry computations could handle far larger datasets than classical algorithms permit. The authors also propose a hybrid quantum-classical HOSVD-based recommendation algorithm.

What carries the argument

The load-bearing object is the Hermitian extension of the mode-$k$ unfolding, $\tilde A^{(k)}$ (the block matrix with $A^{(k)}$ and $A^{(k)\dagger}$ off-diagonal), together with a SWAP-like operator $S^{(k)}_{\tilde A}$ built from the entries of the original tensor. Because $S^{(k)}_{\tilde A}$ is one-sparse in a larger space, its Hamiltonian simulation is efficient, and qPCA on it reveals the singular values and singular vectors of each unfolding. The alternative algorithm replaces this with the quantum singular value estimation operator $W=(2PP^\dagger-I)(2QQ^\dagger-I)$, constructed from the row and column state-preparation isometries $P$ and $Q$; phase estimation on $W$ yields the singular values directly. Both algorithms rely on a quantum-accessible tree data structure (with qRAM access) that prepares the tensor state and row states in polylogarithmic time.

What would settle it

Compare end-to-end runtimes including data-structure construction: if any faithful preparation of the input requires reading all $n^m$ entries, the total time is $\Omega(n^m)$, so the claimed exponential advantage over $O(m n^{m+1})$ classical HOSVD would not hold as an end-to-end statement. A concrete check would be to instantiate the qRAM data structure for a dense $n\times n\times n$ tensor and count the entry-loading operations, or to run Algorithm 2 with the data-structure build cost included and show it saturates the classical time.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that HOSVD, previously treated as a classical multilinear-algebra routine, can be implemented quantumly with polylogarithmic scaling in the tensor dimension. Algorithm 1 uses quantum principal component analysis on a Hermitian extension of each mode-$k$ unfolding to estimate tensor singular values and vectors; Algorithm 2 uses quantum singular value estimation on the unfolding matrices directly and works for arbitrary input structure. The claimed complexities for preparing the superposition state are $O(m^2 \mathrm{polylog}\, n)$ and $O(m^3 \mathrm{polylog}\, n)$, respectively, versus $O(m n^{m+1})$ classically. Reconstructing the singular matrices and core tensor explicitly costs $O(m^3 n^2 \mathrm{polylog}\, n)$ and $O(m^4 n^2 \mathrm{polylog}\, n)$. In addition, the paper applies the HOSVD model to a hybrid quantum-classical recommendation system, where gradient computations are accelerated by quantum inner-product estimation.

Load-bearing premise

The exponential speedup assumes a pre-existing quantum-accessible data structure that can supply the $n^m$ tensor entries (or every mode-$k$ row state) in polylogarithmic time; building or updating that structure costs at least $O(n^m)$, and that cost is excluded from the complexity claims.

Editorial extensions

If this is right

  • If correct, HOSVD-based tensor analysis in machine learning, signal processing, and quantum chemistry can be run on large tensors whose classical decomposition cost $O(m n^{m+1})$ would be prohibitive.
  • Algorithm 2 removes the low-rank requirement on the unfolding matrices, so the method applies to dense, high-rank input tensors once the quantum data structure is available.
  • The hybrid recommendation algorithm shows a concrete downstream use: gradient updates in tensor completion can be evaluated with polylogarithmic quantum subroutines instead of classical $O(K m d^m)$ computations.
  • The explicit-output versions of both algorithms, with $O(m^3 n^2 \mathrm{polylog}\, n)$ and $O(m^4 n^2 \mathrm{polylog}\, n)$ costs, set a benchmark for what full HOSVD reconstruction would cost on a quantum computer.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The end-to-end speedup is conditional on the quantum-accessible data structure already existing: loading an $n^m$-entry tensor classically costs at least $O(n^m)$ time and memory, which is not counted in the polylogarithmic bounds; if that loading cost is included, the exponential separation over classical HOSVD disappears.
  • Because the same data-structure oracle underlies most quantum machine-learning speedup claims, the HOSVD result inherits the general open question of whether qRAM can be built and maintained cheaply in practice.
  • A natural extension, not pursued in the paper, is to adapt the two algorithms to other tensor decompositions such as CP or tensor-train, where the same mode-wise singular-value ideas could apply.
  • The recommendation-system application suggests a testable small-scale experiment: simulate Algorithm 3 with quantum gradient estimates and compare convergence against classical SGD; the paper does not report numerical results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes two quantum algorithms for higher-order singular value decomposition (HOSVD) of an mth-order n-dimensional tensor. Algorithm 1 builds a Hermitian extension of each mode-k unfolding, applies a qPCA-style simulation via a sparse SWAP-like operator, and uses phase estimation to prepare a superposition of tensor singular values and singular vectors. Algorithm 2 instead uses the quantum singular value estimation (QSVE) subroutine of Kerenidis and Prakash on each unfolding matrix. The paper claims O(m^2 polylog n) and O(m^3 polylog n) query complexity for the two algorithms, respectively, and hence an exponential speedup over classical HOSVD's O(m n^{m+1}); it also presents a hybrid quantum-classical recommendation-system algorithm based on HOSVD and SGD. The central claims are the two quantum HOSVD algorithms and their complexity analysis in Sections III-V and the summary in Section VII.

Significance. Quantum tensor algorithms are a timely and underexplored topic, and the paper is one of the first to formulate quantum HOSVD. Algorithm 2's structure around QSVE is natural and, under the standard quantum-accessible data structure assumption, could plausibly yield polylogarithmic query complexity for preparing singular-value/vector superpositions of tensor unfoldings. The hybrid recommendation application is also a reasonable extension of existing quantum recommendation ideas. However, the paper's headline exponential speedup is not established: Algorithm 1's complexity argument contains an internal inconsistency in the eigenvalue-resolvability analysis, and the cost of producing explicit singular matrices/core tensors as well as the preprocessing cost of the data structure are not accounted for. These issues are load-bearing for the main claims, so the contribution as it stands is not sound enough for publication.

major comments (3)
  1. [Section V, Eq. (45)] The eigenvalue-resolvability argument is internally inconsistent. With the normalization ||A||_F = 1, the Hermitian extension ~A^(k) in Eq. (16) has Frobenius norm sqrt(2), so the simulated Hamiltonian H = ~A^(k)/N has eigenvalues mu_j satisfying sum_j mu_j^2 = ||H||_F^2 = 2/N^2; in particular |mu_j| <= sqrt(2)/N. The condition |mu_j| = Omega(1/t) that the text imposes for phase estimation at simulation time t therefore forces t = Omega(N). Substituting this spectral bound into the lower bound in Eq. (45) gives Omega(r/t^2) <= tr(H_r^2) <= 2/N^2, hence t = Omega(N) unless r = 0. This contradicts the subsequent choice t = O(polylog n) and invalidates the derived rank bound r = O(||A||_max^2 t^2). Consequently the advertised O(m^2 polylog n) complexity of Algorithm 1 does not follow from the stated simulation.
  2. [Section V (final paragraph) and Sections III.D-III.E] The claimed complexities O(m^3 n^2 polylog n) and O(m^4 n^2 polylog n) for obtaining the singular matrices and core tensor are not derived and appear inconsistent with the algorithm as written. Step 4 obtains each U^(k) by measurement and amplitude amplification in T_U = O(n^{3/2}) per mode, giving O(m n^{3/2}) in total, while Step 5 costs O(m sqrt(n)/epsilon) by Eq. (35). No m^3 n^2 or m^4 n^2 factor appears in these ingredients. If the intention is to output the complete classical matrices and tensor, a readout lower bound of Omega(n^2) or Omega(n^m) applies and is not discussed. The summary in Section VII therefore overstates what the algorithm is shown to deliver.
  3. [Section VII vs. Theorem 1 and Lemma 3] The exponential-speedup comparison excludes the cost of constructing the quantum-accessible data structure. Theorem 1 and Lemma 3 provide O(polylog(n^m)) state-preparation time only after the tree structure has been built, and building it from the n^m entries requires Omega(n^m) time and memory. Algorithm 1's Step 1 says 'Load A into qRAM', but this preprocessing is never included in the runtime. Since the abstract and Section VII compare against the full classical cost O(m n^{m+1}), the comparison is between a query complexity under a strong input model and an unconditional classical cost. This distinction should be stated explicitly and the speedup claims adjusted accordingly.
minor comments (5)
  1. [Section III.D, after Eq. (29)] The text says that projecting onto the u_j part succeeds 'with probability <~u_j|u_j,0> = 1/2'; since |~u_j> = (u_j; +/- v_j)/sqrt(2), the inner product is 1/sqrt(2) and the probability is |<~u_j|u_j,0>|^2 = 1/2. Please correct the displayed value.
  2. [Section V, Algorithm 2 complexity] The claimed O(m^3 polylog n) complexity for Algorithm 2 lacks a derivation. The stated ingredients give O(m polylog n) per mode, so a sequential implementation would cost O(m^2 polylog n); either way the extra factor of m needs explanation.
  3. [Section VI, Algorithm 3] In the recommendation application, the cost of preparing the subtensor states |s> after each SGD update is not counted. The factor matrices and core tensor change at every iteration, so the underlying data structure must be updated, but the stated complexity O(K m^2 d polylog d) does not include this.
  4. [Section III.E, Eq. (31)] The notation for the column norms of U^(k) is used inconsistently: the state in Eq. (31) includes a factor ||U^(k)_{bullet j_k}||_2 while the surrounding text sometimes treats the columns as unnormalized; please clarify the normalization convention.
  5. [General] There are several typographical and grammatical errors, including 'Base on this algorithm' in Section I and inconsistent spacing in displayed equations; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Q-HOSVD algorithms are compositions of external qPCA and QSVE subroutines; self-citations are comparative and not load-bearing.

full rationale

The derivation chain starts from the standard HOSVD definition (De Lathauwer et al. [10]), then builds Algorithm 1 on external qPCA [24] and phase estimation [22], and Algorithm 2 on the external QSVE data-structure and estimation lemmas [20,29]. No parameter is fitted to a subset of data and then repackaged as a prediction. The outputs (singular values, singular matrices, core tensor) are defined by the standard HOSVD equations (6)-(8), not by the quantum circuits. The self-citations [14] and [44] are used only in comparative remarks: 'Algorithm 1 is similar to that in [14]' and 'We have carried out the similar SVD and truncation operations in [44]', so they do not carry the correctness or speedup claims. The quantum-accessible data structure (Theorem 1 and Lemma 3) is an explicit oracle assumption; its construction cost is not counted, but that is a modeling caveat and an unfair-comparison concern, not a reduction of the output to the input by construction. The low-rank restriction for Algorithm 1 is stated openly in Sections IV and VII, and the eigenvalue-resolvability issue in Eq. (45) identified by the skeptic is a potential complexity/correctness gap, not a circularity, because it is not a case of a fitted parameter being renamed a prediction or a definition being used as its own proof.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No data-fitted parameters appear, but Algorithm 1 uses a hand-chosen simulation time that is incompatible with the eigenvalue scale. The main assumptions are quantum-access oracles and the low-rank condition for Algorithm 1. No new physical entities are introduced.

free parameters (1)
  • simulation time t = O(polylog n)
    Section V chooses t=O(polylog n) to force the effective rank r=O(t^2) to be polylogarithmic. This choice is load-bearing and is inconsistent with the eigenvalue scale of ~A/N, which is at most about 1/N.
assumptions (4)
  • domain assumption A pre-built qRAM or tensor data structure provides quantum access to all tensor entries and to the row states of every mode-k unfolding in O(polylog n) time.
    Invoked in Theorem 1 and Lemma 3 and used in both algorithms. The paper does not count the O(n^m) cost of building or updating this structure.
  • standard math The qPCA identity in Lemma 1, cited from [24], correctly simulates the evolution of ~A/N in the form used by the paper.
    Used at Eq. (18) and throughout Algorithm 1. The paper's use introduces the 1/N factor that causes the scaling problem.
  • domain assumption The mode-k unfolding matrices are low-rank with rank r=O(polylog n) for Algorithm 1.
    Section V explicitly concludes that t=O(polylog n) forces r=O(polylog n), so Algorithm 1 only runs in polylogarithmic time under this low-rank condition.
  • standard math The quantum singular value estimation result of Lemma 4 from [20] is correct.
    Algorithm 2 is built entirely on QSVE, and the paper does not re-derive or verify this external theorem.

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Cite this review

Pith. "Pith review of Quantum Higher Order Singular Value Decomposition." pith.science (2026). https://pith.science/paper/OIKNYU2M

@misc{pith2026190800719,
  author       = {Pith},
  title        = {Pith review of: Quantum Higher Order Singular Value Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIKNYU2M}},
  note         = {Machine review of arXiv:1908.00719}
}
read the original abstract

Higher order singular value decomposition (HOSVD) is an important tool for analyzing big data in multilinear algebra and machine learning. In this paper, we present two quantum algorithms for HOSVD. Our methods allow one to decompose a tensor into a core tensor containing tensor singular values and some unitary matrices by quantum computers. Compared to the classical HOSVD algorithm, our quantum algorithms provide an exponential speedup. Furthermore, we introduce a hybrid quantum-classical algorithm of HOSVD model applied in recommendation systems.

Figures

Figures reproduced from arXiv: 1908.00719 by the authors.

Figure 1
Figure 1. FIG. 1. Block diagram of the HOSVD for a third-order tensor. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The tensor network notation of equation (6). [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The data structure for a real 2 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The data structure for a complex 2 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The quantum circuit to perform mode-3 matrix un [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Note that QSVE obtains the superposition state [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Circuit of QSVE on matrix [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Reference graph

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