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REVIEW 3 major objections 6 minor 39 references

Quantum compressed sensing tomographic reconstruction algorithm

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read QCSTR reconstructs CT images from about 10 percent of the usual projection count.

desk verdict A transparent, reproducible QUBO total-variation formulation for sparse-view CT, but the headline error-free claims rest on per-image hand-tuned weights and highly idealized test images. read the letter →

arxiv 2505.11286 v1 pith:PZIDJ73E submitted 2025-05-16 quant-ph

Arim Ryou , Kiwoong Kim , Kyungtaek Jun This is my paper · ORCID
classification quant-ph
keywords quantumcompressedsensingcomputedtomographyQUBOtotalvariationtomographicreconstructionX-raymassattenuationcoefficienthybridsolverradiationdosereduction
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 tries to show that CT image reconstruction can be done from far fewer X-ray projections than classical methods require by posing the problem as a single quadratic unconstrained binary optimization (QUBO) model: a data term that matches simulated projections to measured sinogram data plus a total-variation term that penalizes pixel-to-pixel jumps. The authors test the model with a hybrid quantum-classical optimizer on synthetic anatomical phantoms and on a resized body CT image whose pixels are encoded as three discrete X-ray mass-attenuation values. They report error-free reconstructions of 30x30 images from five projection angles and 60x60 images from six projection angles, about 10 to 20 percent of the projection count normally needed, and recognizable reconstructions from noisy data at 20 percent projections. If this holds, the algorithm points toward cutting CT radiation dose by up to 90 percent while keeping image quality, which matters for medical imaging and for any X-ray tomography setting where dose or scan time is limiting.

What carries the argument

The load-bearing object is the QUBO model $Q = aQ_1 + bQ_2$. QUBO is a quadratic unconstrained binary optimization problem, an energy-minimization problem over binary variables. Here $Q_1$ encodes the squared difference between the simulated projection of the binary-encoded image and the measured sinogram, while $Q_2$ sums squared differences of neighboring pixel values, a total-variation regularizer. The paper encodes pixel values not in binary place-value form but through cumulative X-ray mass-attenuation levels, $I_{ij} = \alpha_1 q_1^{ij} + \sum_{k=2}^{m}(\alpha_k-\alpha_{k-1})q_k^{ij}$, which makes the $L^1$ and $L^2$ penalties coincide and keeps the solver from getting stuck at the wrong minimum. A hybrid quantum-classical binary quadratic model optimizer minimizes the combined QUBO; the balance between the data term and the smoothness term is set by small integer weights $a$ and $b$, with $(a,b)=(1,1)$ or $(1,2)$ used in the reported error-free cases.

What would settle it

Apply the same $Q = aQ_1 + bQ_2$ formulation to a $60\times60$ phantom whose pixel attenuation values vary continuously over a range rather than taking only values 1, 2, and 3, generate its sinogram from six projections, and solve to the claimed global minimum; if the reconstructed image has more than a handful of wrong pixels, the 'about 10 percent of projections' claim fails for continuous-attenuation objects.

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Extended reading notes

Core claim

The paper's central claim is that CT reconstruction can be written as a single quadratic unconstrained binary optimization problem $Q = aQ_1 + bQ_2$, where $Q_1$ is the tomographic data-fidelity term and $Q_2$ is a total-variation smoothness term over adjacent pixels, and that solving this QUBO on a hybrid quantum-classical optimizer returns the global minimum energy associated with the true image. With pixels encoded in the X-ray mass-attenuation-coefficient representation $I_{ij} = \alpha_1 q_1^{ij} + \sum_{k=2}^{m}(\alpha_k-\alpha_{k-1})q_k^{ij}$, the authors report error-free reconstruction of a $30\times30$ image from 5 projection angles and a $60\times60$ image from 6 projection angles, about 10 to 20 percent of the projection count classical reconstruction uses, and recognizable reconstruction from sinograms with about 5 percent Gaussian noise using 20 percent of the projections. The authors interpret the result as a path to cutting CT radiation dose by up to 90 percent once quantum hardware can handle larger QUBO instances.

Load-bearing premise

The demonstration assumes each pixel belongs to a small set of known materials with known X-ray attenuation values, and that the image has been smoothed so adjacent pixels change continuously; if real anatomy has many materials, unknown coefficients, or sharp edges, far more projections may be needed.

Editorial extensions

If this is right

  • A 1000x1000 clinical CT image that normally needs about 1000 projections could in principle be reconstructed from about 100 projections, cutting radiation dose by up to 90 percent.
  • The same QUBO construction transfers to fan-beam, cone-beam, and parallel-beam geometries and to 3D, because the data-fidelity term only needs the line-integral projection geometry.
  • With 5 percent Gaussian noise in the sinogram, the algorithm still identifies internal structure from 20 percent of the data, so the dose reduction is not limited to noiseless idealized data.
  • The optimal weights $a$ and $b$ are not universal: the paper finds different small integer values for different images, so a practical scanner would need an automated rule for choosing them.
  • If future quantum hardware provides enough qubits for QPU-only solving, the $O(\log^2 n)$ scaling of the linear-system formulation makes real-time low-dose CT during procedures a plausible target.

Reading between the lines

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

  • The setup is arguably more a known-material image segmentation or labeling problem than a general CT reconstruction: with pixel values restricted to 1, 2, and 3, the solver is choosing a three-color labeling, and the same QUBO could apply to spectral CT material decomposition where the material classes are known.
  • The 10-percent projection regime likely depends on the smoothness of the test samples; a stress test with textured or edge-rich phantoms would show whether the ratio degrades, and if so, the projection-count claim should be read as a property of smooth, low-cardinality objects rather than of CT in general.
  • A natural next experiment is to replace the squared total-variation term with a direct $L^1$-style penalty via auxiliary binary variables; if QUBO size permits, this would test whether the paper's $L^1$/$L^2$ alignment trick is necessary or whether a direct $L^1$ penalty improves robustness on noisy data.
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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 / 6 minor

Summary. The paper presents a QUBO formulation for sparse-view CT reconstruction that linearly combines a data-fidelity term (Q1) and a total-variation term (Q2), with Q = aQ1 + bQ2, solved by D-Wave's hybrid BQM solver. Experiments on 30x30 and 60x60 Shepp-Logan and body CT samples, restricted to 1-3 known X-ray MAC values and Gaussian-blurred to be continuous, report error-free reconstructions with 5-6 projections (10-20% of the conventional count). The paper extrapolates to a potential 90% radiation dose reduction in clinical CT.

Significance. The QUBO algebra in Eqs. (1)-(9) is explicit and the energy values reported are consistent with the known samples, and the supplementary code supports reproducibility. If the hyperparameter issue were resolved, the demonstration would establish a useful proof-of-concept for quantum annealing in sparse-view CT for discrete multi-material objects. However, the significance is currently limited because the central claim is tied to per-image tuned coefficients, a highly idealized sample model, and no classical compressed-sensing baseline, so the practical advantage over existing algorithms is not established.

major comments (3)
  1. [Result and implementation / Discussion] The central claim of error-free reconstruction with 5-6 projections is not a property of the algorithm as stated, because the coefficients (a,b) in Eq. (9) are selected per image. The Results state 'We varied b to obtain optimal CT images' for the 60x60 experiments (Figs. 3 and 6), and the Discussion concedes 'the new algorithm requires different a and b to be combined depending on the type of image, and has not yet found the optimal values for the coefficients.' Since (a,b)=(1,3) already gives a one-pixel error for the 60x60 body sample, the reported error-free results are outcomes of a small grid search rather than of a pre-specified algorithm. Please provide a fixed rule for choosing (a,b), or explicitly scope the claim to the selected coefficients.
  2. [Result and implementation / Discussion] The experimental setting is far more idealized than the abstract suggests: the objects have 1-3 known discrete MAC values (1-1.6 bits per pixel), the samples are Gaussian-blurred to make interior changes as continuous as possible, and the sinograms are ideal. The Discussion acknowledges that 'if the internal structure of the sample has many discontinuous changes or many fast-changing continuous pixels, more projection images are required,' which directly undermines the abstract's unconditional statement that 'the new algorithm was able to obtain a solution within 5 projection images... reconstructing error-free CT images.' The extrapolation to 1000x1000 clinical CT with 90% dose reduction is unsupported by the present experiments.
  3. [Comparison of classical and quantum CT images] The comparison with classical algorithms is incomplete: SART and FBP are standard baselines, but the paper does not include a classical compressed-sensing reconstruction (e.g., TV-regularized SART or L1 minimization) that would exploit the same known discrete label structure. Without such a baseline, the claim that the QCSTR algorithm 'is less affected by artifacts or noise than classical algorithms' is not demonstrated; the observed success may be due to the strong prior (known labels, smooth image) rather than to the quantum solver.
minor comments (6)
  1. [Result and implementation B] The text contains a typo: 'exactly the same evergy' should be 'exactly the same energy'.
  2. [Result and implementation (noise experiment)] The noise model in the paragraph beginning 'We have investigated...' is labeled Eq. (10) in the text but appears as Eq. (9) in the manuscript; renumber the equations accordingly.
  3. [Comparison of classical and quantum CT images] In Figure 7, the early QTR algorithm is cited as reference [17], but the quantum tomographic reconstruction algorithm is reference [19]; verify the citation.
  4. [Introduction] The phrase 'quantum supremacy' is used loosely to describe image quality improvements and is misleading in this context; please replace it with 'quantum advantage' or a more specific statement.
  5. [Discussion] The statement that 'quantum compressed sensing algorithms can be calculated within a few flops through parallel operations' is vague and should be replaced with a precise computational complexity statement or omitted.
  6. [Comparison of classical and quantum CT images] The spelling 'Helgason-Ludwing' should be 'Helgason-Ludwig'.

Circularity Check

2 steps flagged · score 6.0 of 10

Central 5-6 projection error-free claim relies on per-image tuned weights (a,b) and a self-referential energy target; no fixed coefficient rule is given.

  1. fitted input called prediction [Results, Sections A and B (Q = aQ1 + bQ2, Eq. 9); Discussion, final paragraph]
    "We varied 𝑏 to obtain optimal CT images. ... The new algorithm requires different 𝑎 and b to be combined depending on the type of image, and has not yet found the optimal values for the coefficients."

    For the 60×60 experiments, the reported error-free reconstructions are obtained by scanning b (and sometimes a) over a small integer grid and selecting the value that yields zero error; (a,b)=(1,2) is chosen for Figs. 3 and 6, while (1,3) gives a one-pixel error for the body sample. Since Q=aQ1+bQ2 and the data/TV balance is set per image after seeing the result, the 'new algorithm was able to reconstruct error-free' statement is a retrospective selection, not the output of a fixed algorithm. The paper concedes optimal coefficients are unknown, so the abstract's unconditional claim is not supported by a pre-specified procedure.

  2. self definitional [Method, Eq. (2); Results, e.g. 30×30 Shepp-Logan experiment]
    "The global minimum energy for reconstructing error-free CT images is − ∑ ∑ (𝑃(𝜃, 𝑠))^2. ... The global minimum energy of the QUBO model desired by the new algorithm is −14460.70, and we obtain exactly the same energy ... At this time, the TV is 150 and the reconstructed CT image is error-free."

    The target energy is, by the definition of F1 = Σ(I_P−P)^2 − ΣP^2, the lower bound attained exactly when the solution's projections equal the measured sinogram. With only 3–6 projection angles the data term is heavily underdetermined, so reaching −ΣP^2 merely confirms the solver can fit the input sinogram; it does not independently certify the reconstructed image. Presenting this same-energy match as the 'desired' success value makes the validation loop back to the data and model construction rather than to an external ground truth.

full rationale

The main derivation—QUBO encoding of tomographic data fidelity and total variation—is internally consistent: F1 and F2 are explicit quadratic forms, and Eq. 9 is a linear combination. The use of the X-ray MAC representation follows the authors' prior work [29], and there is no imported uniqueness theorem forcing the ansatz, so I do not flag self-citation as circular. However, the central experimental claim is not self-contained as a predictive algorithm. The coefficients (a,b) are tuned per test image ('We varied b to obtain optimal CT images'; 'requires different a and b ... has not yet found the optimal values'), so the 5-6 projection error-free results are selected outcomes of a small grid search rather than outputs of a fixed rule. Additionally, the 'global minimum energy desired' is defined as the data-fit lower bound −ΣP^2, so matching it is a consistency check with the input sinogram, not an independent prediction. These two issues make the headline claim partially circular: fitted regularization weights and a self-referential energy target are presented as an algorithm capability. The assumptions of known X-ray MACs, Gaussian blurring, and low-cardinality labels further narrow the scope but are acknowledged limitations rather than circularity. Overall score 6.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on two cheap inputs: known discrete material coefficients (alpha_k) and smoothing of the test images, plus per-image tuning of the combination weights (a, b). The supporting math is otherwise standard least-squares plus total variation. The solver optimality assumption is unverified. No new physical entities are invented.

free parameters (3)
  • regularization weights (a, b) = (1,1), (1,2), (1,3) per experiment
    Chosen by grid search over integers 1 to 3 per image to hit the target energy; 'we varied b to obtain optimal CT images' (Results B); the authors state optimal values have not yet been found (Discussion).
  • number of projection images per experiment = 3, 5, or 6 depending on sample
    The headline figure (5 for 30 by 30, 6 for 60 by 60) is the successful endpoint of a manual scan over projection counts, not a predicted or derived requirement.
  • per-sample X-ray MAC set (alpha values) = 1 (single material); 1, 2, 3 (three materials)
    Sample intensities are quantized to consecutive integers so that L1 and L2 norms coincide (Discussion); this limits the demonstration to images with 1 to 3 known attenuation levels, making the reconstruction a low-cardinality labeling problem.
assumptions (5)
  • domain assumption An ideal, noiseless sinogram is obtainable from the CCD; all error-free headline results use this ideal data.
    Stated in Method: 'we proceed under the condition that an ideal sinogram can be obtained from a CCD'. The headline claim does not address real detector noise.
  • domain assumption The sample's X-ray mass attenuation coefficients are known in advance and the image is exactly composed of those discrete values.
    Stated in Results: 'We assumed that the X-ray MAC of the samples used in each experiment was already known.' These values are the coefficients alpha_k of the QUBO encoding (Eq. 5), so the whole formulation presupposes known materials.
  • ad hoc to paper Test samples are Gaussian blurred so the interiors change as continuously as possible.
    Stated in Discussion; the smoothing is what makes the L2 total-variation term effective at 10 to 20 percent projections, and it is a preprocessing choice the authors control, not a property of real anatomy.
  • ad hoc to paper The D-Wave BQM hybrid solver returns the global minimum energy of the QUBO model.
    The paper equates solver output with 'the global minimum energy' and verifies only empirically to two decimal places; hybrid solvers are heuristics and provide no optimality certificate, so the premise is assumed.
  • standard math About nx projections are needed classically to satisfy the Helgason-Ludwig consistency condition for an nx by nx image.
    Invoked in Results C to set the 60-projection classical baseline for a 60 by 60 image; standard tomographic background.

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Cite this review

Pith. "Pith review of Quantum compressed sensing tomographic reconstruction algorithm." pith.science (2026). https://pith.science/paper/PZIDJ73E

@misc{pith2026250511286,
  author       = {Pith},
  title        = {Pith review of: Quantum compressed sensing tomographic reconstruction algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZIDJ73E}},
  note         = {Machine review of arXiv:2505.11286}
}
read the original abstract

Computed tomography (CT) is a non-destructive technique for observing internal images and has proven highly valuable in medical diagnostics. Recent advances in quantum computing have begun to influence tomographic reconstruction techniques. The quantum tomographic reconstruction algorithm is less affected by artifacts or noise than classical algorithms by using the square function of the difference between pixels obtained by projecting CT images in quantum superposition states and pixels obtained from experimental data. In particular, by using quantum linear systems, a fast quadratic unconstrained binary optimization (QUBO) model formulation for quantum tomographic reconstruction is possible. In this paper, we formulate the QUBO model for quantum compressed sensing tomographic reconstruction, which is a linear combination of the QUBO model for quantum tomographic reconstruction and the QUBO model for total variation in quantum superposition-state CT images. In our experiments, we used sinograms obtained by using the Radon transform of Shepp-Logan images and body CT images. We evaluate the performance of the new algorithm by reconstructing CT images using a hybrid solver with the QUBO model computed from each sinogram. The new algorithm was able to obtain a solution within 5 projection images for 30 by 30 image samples and within 6 projection images for 60 by 60 image samples, reconstructing error-free CT images. We anticipate that quantum compressed sensing tomographic reconstruction algorithms could significantly reduce the total radiation dose when quantum computing performance advances.

Figures

Figures reproduced from arXiv: 2505.11286 by the authors.

Figure 1
Figure 1. 𝟑𝟎 × 𝟑𝟎 CT images for a Shepp-Logan image sample with one X-ray MAC. In this experiment, three projection images with projection angles varying by 60 degrees were used. (a) An image sample and a CT image reconstructed by the QCSTR algorithm for the QUBO model, (b, c) CT images reconstructed by the QCSTR algorithm for each QUBO model, (d) A CT image reconstructed by the QTR algorithm. The sample used in the second ex… view at source ↗
Figure 3
Figure 3. 𝟔𝟎 × 𝟔𝟎 CT images for a Shepp-Logan image sample with three X-ray MAC: 1, 2, and 3. In this experiment, six projection images were used in the QUBO model formulation. (a, c) CT images reconstructed by the QCSTR algorithm for the QUBO model, (b) An image sample and a CT image reconstructed by the QCSTR algorithm for the QUBO model, (d) A CT image reconstructed by the QTR algorithm. B. Body CT image from Kaggle A body… view at source ↗

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