Charged particle tracking with quantum annealing-inspired optimization

Abhishek Anand; Alexander Zlokapa; Daniel Lidar; Javier M. Duarte; Jean-Roch Vlimant; Joshua Job; Maria Spiropulu

REVIEW 3 major objections 6 minor 3 cited by

Charged particle tracking with quantum annealing-inspired optimization

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Charged particle tracking can be reformulated as a QUBO optimization problem whose lowest-energy solution selects the true tracks, and both simulated and quantum annealing solve it on TrackML data with stable purity and efficiency as…

desk verdict A genuine proof-of-concept for QUBO-based tracking on real annealing hardware, but the evaluation baseline does not yet separate the solver from the KDE prior. read the letter →

arxiv 1908.04475 v1 pith:OMMZDSUM submitted 2019-08-13 quant-ph cs.LGhep-ph

Alexander Zlokapa , Abhishek Anand , Jean-Roch Vlimant , Javier M. Duarte , Joshua Job , Daniel Lidar , Maria Spiropulu This is my paper · ORCID

classification quant-phcs.LGhep-ph

keywords trackreconstructionQUBOquantumannealingsimulatedMLHL-LHCcombinatorialoptimizationparticletracking

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

At the High-Luminosity LHC, track reconstruction must survive an increase from roughly 40 to 200 simultaneous proton-proton collisions per bunch crossing. This paper claims that the tracking task can be rewritten as a quadratic unconstrained binary optimization (QUBO): each candidate hit-to-hit segment is a binary variable, and the assignment that minimizes a geometric energy function selects the tracks. The authors adapt the original geometric network method to HL-LHC conditions with additional helix, beam-spot, and momentum-bias terms, then solve the resulting QUBO with simulated annealing and with a 1,098-qubit programmable quantum annealer. On TrackML events, purity and efficiency stay roughly constant as multiplicity grows, while random edge selection after identical pre-processing collapses, supporting the claim that the optimization itself does the tracking. The formulation also points toward fast hardware-specific implementations at the LHC, while the question of a genuine quantum speedup remains open.

What carries the argument

The central object is the modified geometric-network QUBO energy (equation (8)). Its variables are candidate hit-to-hit edges $s_{ab}$; the energy combines a helical-geometry reward $(\cos^\lambda\theta_{abc} + \rho\cos^\lambda\varphi_{abc})/(r_{ab}+r_{bc})$ for two consecutive segments, a beam-spot $z$-intercept penalty, a bifurcation penalty that penalizes a hit from being shared by two segments, and a prior-probability bias learned by Gaussian kernel density estimation. Around this energy the paper builds $O(h^2)$ classical pre-processing: 32 overlapping azimuthal sectors, Gaussian-KDE edge selection, flood-fill sub-graphing that keeps at most five edges per node, and a multi-stage sparse-QUBO construction. The energy landscape is the mechanism: its low-energy states coincide with true tracks, and the annealing procedures are what navigate it.

What would settle it

Run the identical QUBO pipeline on TrackML events without ground-truth duplicate removal, replacing it with a geometric duplicate-cleaning step; if purity or efficiency drops substantially as pileup increases toward 200, the claim that this annealing-based method reconstructs HL-LHC tracks is refuted. A complementary check is whether random edge selection after pre-processing matches the QUBO's performance on these harder events, which would show the solver itself is not doing the reconstruction.

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Extended reading notes

Core claim

The central claim is that charged particle tracking is a segment-classification problem with a QUBO objective. For every candidate pair of hits $(a,b)$, a binary variable $s_{ab}$ marks whether both hits belong to the same particle's trajectory, and the energy of equation (8) rewards chains of short, collinear segments in cylindrical coordinates, biases toward high-momentum and origin-pointing tracks, and penalizes shared-hit bifurcations. Minimizing this energy with simulated annealing, and with a programmable quantum annealer after minor-embedding onto 33 fully connected logical qubits, reconstructs tracks from TrackML events with roughly constant purity and efficiency up to the tested multiplicities, whereas random edge selection after the same pre-processing fails. Quantum annealing matches simulated annealing on events up to 500 tracks but does not outperform it, so no speedup is claimed. The paper therefore argues that the QUBO solver, not the polynomial $O(h^2)$ pre-processing heuristics, is responsible for the track-finding performance, and that the post-preprocessing annealing time grows exponentially with sub-graph size with a fitted exponent $c = (8.52 \pm 0.076)\times 10^{-3}$.

Load-bearing premise

The load-bearing premise is that deleting duplicate hits with ground-truth information, rather than by a geometric algorithm, does not change the difficulty of track finding; if that premise is wrong, the reported purity and efficiency numbers do not transfer to realistic HL-LHC events.

Editorial extensions

If this is right

The same QUBO can be re-tuned for high efficiency or high purity, allowing it to act as a first-stage track filter before Kalman-style fitting or as a trigger-level preselection.
Because the formulation is hardware-agnostic, it can run on FPGA simulated-annealing chips and fully connected classical Ising machines with more spins than the annealer used here, which could make trigger-level tracking practical.
Scaling to full HL-LHC conditions would require an annealer with roughly 10,000 fully connected qubits after sub-graphing, about twice the capacity of next-generation annealer topologies, so current hardware is limited to smaller events.
Quantum annealing at the shortest tested time (5 microseconds) already solves the sub-QUBOs as well as longer anneals, so the speedup question cannot be settled until larger problems can be embedded.
The exponential growth of simulated-annealing time with sub-graph size means the practical bottleneck is the size distribution of sub-graphs, not the raw number of hits in the event.

Reading between the lines

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

Removing duplicate hits with ground truth (as done in Section 3.3) is the assumption most likely to change the difficulty of the problem; a direct test would rerun the pipeline with geometric duplicate removal and compare at 200 pileup.
The energy function's visible correlation with track efficiency and purity suggests the same QUBO could serve as a differentiable loss or regularizer for learned track-finding networks, although the paper does not explore this.
Because the 5 microsecond anneal already saturates performance on these small sub-QUBOs, any quantum speedup, if it exists, would appear only when sub-QUBOs grow beyond what today's annealers can embed; that is a hardware test the paper cannot yet perform.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I'll give you the two-minute version. The paper genuinely demonstrates that a Denby-Peterson-derived QUBO, extended with LHC-specific terms, can be solved by both simulated annealing and the D-Wave 2X to reconstruct tracks on TrackML data. That's a real proof of concept, and it's more than the prior quantum tracking work, which used triplets and heavier seeding. The authors also put real thought into decomposing the problem to fit the hardware, and they're refreshingly honest that they find no quantum speedup.

Where it gets soft: the baseline used to attribute the tracking quality to the solver is too weak. Section 3.6 compares annealing against random edge selection after the KDE cut. That only says a random selection from ~1% true edges gives low purity. It doesn't show the QUBO's pairwise terms are doing the work. A greedy baseline that takes edges in descending KDE score, while rejecting bifurcations, would settle that. The stress-test note is right: absent that control, the claim that 'tracking is completed in solving the QUBO rather than pre-processing' is unsupported.

The other two concerns are real but less damaging. Duplicate hits are removed using ground truth, which the authors admit. That does remove some of the combinatorial challenge that matters at HL-LHC, though they note it can be handled in post-processing. It should be disclosed more prominently and, ideally, the pipeline should be tested without it. Also, the QUBO parameters are tuned on 500-track TrackML events that appear to overlap with the evaluation set; there is no explicitly separated held-out validation for the parameter choice. The 1000- and 2000-track results help, but the 500-track numbers could be optimistic.

None of this kills the feasibility result. The paper is best read as a proof of concept with a few controls missing, not as a demonstration that a quantum annealer is competitive. For someone working on quantum annealing or on fast tracking, it's worth a look. I'd send it to peer review, but I'd ask the authors to add the KDE-greedy baseline, redo the evaluation without ground-truth duplicate removal, and document the train/test split for the KDE and QUBO parameters.

Referee Report

3 major / 6 minor

Summary. The paper presents a QUBO-based track reconstruction method for charged particles, adapting the Denby-Peterson Hopfield-network formulation to an Ising/QUBO objective with additional terms for helical geometry, beamspot constraints, and a KDE-based edge prior. The authors preprocess TrackML data by sectoring, KDE-based edge selection, and sub-graphing, then solve the resulting QUBOs with simulated annealing and on the D-Wave 2X. They report purity and efficiency that remain roughly stable with track multiplicity, and they attribute the reconstruction quality primarily to the QUBO solver rather than to the preprocessing heuristics. They also analyze the scaling of simulated annealing and explicitly leave the quantum speedup question open.

Significance. If the central claim were fully supported, this would be a useful proof-of-concept for applying quantum and quantum-inspired optimization to HL-LHC track reconstruction, with a clear mapping of a realistic physics problem to QUBO and a concrete hardware demonstration. The paper is careful to state caveats: it does not claim a quantum speedup, it notes that duplicate hits are removed using ground truth, and it reports limited qubit scalability. The explicit formulation of the track segment classification problem and the complexity analysis of preprocessing are valuable. However, the reported performance numbers are currently not fully convincing because the evaluation uses ground truth in preprocessing and lacks a held-out validation, and the baseline used to attribute the tracking to the QUBO solver is too weak.

major comments (3)

[Section 3.6, Eq. (8)] The comparison to random edge selection cannot support the inference that "the patterns of hits are not found during pre-processing, but rather by solving the QUBO." The Gaussian KDE prior P(sab) enters the QUBO as the linear coefficient (βP(sab)-γ); random selection discards this information, so it is not a meaningful control. Add a greedy baseline that orders candidate edges by P(sab) and rejects edges sharing a hit, and compare its purity and efficiency to the SA and QA results. Without this control, the attribution of the reconstruction quality to the combinatorial optimization is unestablished.
[Section 3.3] The decision to "use ground truth information for ease of processing" to remove duplicate hits changes the problem instance. Duplicate hits are part of the HL-LHC hit combinatorics, and removing them using ground truth means the reported efficiencies and purities are not measured on the actual challenge. Please provide an ablation that includes duplicate hits, or an explicit estimate of the effect, or clearly restrict the claims to the duplicate-free simplified problem. As written, this simplification affects the central feasibility claim for HL-LHC conditions.
[Section 3.2 and Section 3.4.1, Table 1] The QUBO parameters are optimized by Bayesian optimization on 500-particle TrackML events to maximize F1, and the KDE prior is trained on ground-truth edges from the same dataset; no held-out validation is described. The performance reported in Section 4 may therefore reflect tuning to the test data. Please report results on an explicitly held-out validation set (e.g., different events or a k-fold split), and state whether the parameters are re-tuned per event or fixed.

minor comments (6)

[Section 3.2, Eq. (8)] The sum over "a,bc" in the beamspot term should be "a,b,c" to match Eq. (6).
[Figure 11 caption] The caption refers to "random annealing" while the text describes "random edge selection"; please make the terms consistent.
[Introduction, last paragraph] The phrase "We provide a concluding analysis of potential real-world applications in section 5. section 5." contains a duplicated "section 5."; remove the repetition.
[Section 4.1] Purity and Efficiency are defined at the track level, but the paper does not specify how a "reconstructed track" is obtained from the selected edge variables (e.g., connected components or a maximum-segment criterion); please clarify so the metrics are reproducible.
[Section 3.4] The notation "1010" and "103" should be rendered with superscripts to avoid ambiguity.
[Supplementary Material, Section 3] The statement that the convergence analysis seeks a lower bound on runtime should be reconciled with the fitted exponential constant, since the epsilon-close criterion for the 2000-track point may make the lower bound non-uniform across multiplicities; suggest adding a sentence in the main text.

Circularity Check

2 steps flagged · score 5.0 of 10

The central attribution claim is partially circular: the trained KDE prior and F1-tuned parameters are embedded in the QUBO, yet the random baseline used to attribute 'the majority of tracking' to the solver discards that prior information.

fitted input called prediction [Section 3.4.1 (Gaussian kernel density estimation), Eq. (7)-(8), Section 3.6 (Benchmark studies)]
"Random edge selection simply randomly selects edges as true according to the expected fraction of true edge segments in the pre-processed data. Since the edge selection by annealing occurs after our heuristic edge selection with the Gaussian KDE and disjoint sub-graph search, comparison to random edge selection demonstrates that the patterns of hits are not found during pre-processing, but rather by solving the QUBO."

The QUBO's linear term in Eq. (8) is -sum(βP(sab)-γ)sab, where P(sab) is the Gaussian KDE prior trained on ground-truth edges from TrackML data. The random baseline discards P(sab), so it is a degenerate control: any sensible use of the KDE scores, such as greedy acceptance by descending P(sab) with a bifurcation constraint, would outperform random selection. Therefore the comparison cannot show that 'patterns of hits' are found by the solver rather than by the trained prior. The attribution of reconstruction quality to 'solving the QUBO' double-counts the fitted prior that was inserted into the QUBO as an input.
fitted input called prediction [Section 3.2 (Modified QUBO for HL-LHC) and Section 4.1 (Tracking efficiency and purity)]
"The parameters are optimized by Bayesian optimization, sampling regions of the parameter space that are expected to provide the largest improvement in the objective function according to Bayesian inference. ... a total of 100 parameter sets were sampled on TrackML events with 500 particles/event to maximize the F1 score (the harmonic mean between purity and efficiency) using SA."

The headline SA purity/efficiency results include the 500-particle/event configuration, which is exactly the configuration on which the QUBO parameters were selected to maximize F1. The 500-track benchmark is therefore an in-sample optimized quantity, not an independent prediction. Combined with the ground-truth-trained KDE prior, the reported performance at 500 tracks is substantially a report of fitted inputs; the remaining extrapolative content is the scaling to 1000 and 2000 tracks, but the central attribution of 'the majority of tracking' to the solver is not independently supported.

full rationale

The paper is not a pure renaming of a known result, and no load-bearing self-citation chain is used: Refs. [8] and [45] support general annealing methodology and gauge averaging, not the tracking claim. However, the central derivation is not fully self-contained. The Gaussian KDE prior P(sab) is trained on ground-truth segments from TrackML data and then inserted into the QUBO via Eq. (7)-(8); the QUBO parameters are selected by Bayesian optimization on TrackML 500-particle events to maximize F1. The reported 500-track purities/efficiencies are therefore in-sample optimized quantities. More importantly, Section 3.6's random-selection baseline cannot establish that the solver, rather than the KDE prior, is responsible for reconstruction quality, because random selection discards the very edge-score information that the QUBO receives. A greedy KDE-only baseline is needed before the attribution claim is supported. The ground-truth-based removal of duplicate hits (Section 3.3) is a stated simplification and a correctness risk, but it is not itself a circular step. Because some extrapolative content remains (fixed parameters at higher multiplicities, geometric pair/triplet terms, and comparison between SA and QA), the circularity is partial rather than total.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a QUBO energy function whose coefficients (lambda, rho, eta, zeta, alpha, beta, gamma) are fitted to maximize F1 on 500-particle TrackML events, plus a ground-truth-trained Gaussian KDE prior and several hand-picked cutoffs (tau, KDE threshold). These are not derived from first principles; they are tuned on the same type of data used for evaluation, which limits the independent support for the conclusion. No new physical entities are postulated.

free parameters (11)

lambda = 13.17
Track angle separator in the QUBO reward term (Table 1), fitted via Bayesian optimization on TrackML 500-particle events.
rho = 5.00
High-momentum bias weight in the QUBO (Table 1), fitted to maximize F1.
eta = 14.41
Beam spot bias weight for the z-intercept penalty (Table 1), fitted.
zeta = 1.79
Beam spot separator exponent (Table 1), fitted.
alpha = 86.20
Bifurcation penalty weight (Table 1), fitted.
beta = 20.91
Edge alignment penalty weight (Table 1), fitted.
gamma = 9.79
Total edge count penalty weight (Table 1), fitted.
tau = 0.996
Threshold on cos(lambda * theta) for edge affinity, set by hand in Section 3.2; not listed in Table 1. A value of 0.9997 is used in the supplementary for larger events.
KDE cutoff threshold = not specified
The cut on the Gaussian KDE prior that reduces the QUBO size is mentioned in Section 3.4.1 but its value is not reported, making it an implicit free parameter.
KDE bandwidth = not specified
Bandwidth of the Gaussian KDE used for the prior is not stated, affecting the prior distribution and the candidate edge cut.
characteristic lengths for scaling = (1000, pi, 1000)
Hand-chosen standardization scales for r, azimuthal angle, and z in Section 3.2; these choices affect the relative weighting of coordinates.

assumptions (5)

domain assumption Charged particle tracks in the LHC tracker are well approximated by helices in a uniform axial magnetic field.
Used to justify the theta and phi angle rewards in the QUBO (Section 3.2, Eq. 4 and Figure 5).
domain assumption The beam spot is small (about 5.5 mm in z), so true tracks should extrapolate back near the origin in the rz-plane.
Motivates the beamspot penalty term in Eq. 6, based on TrackML beamspot size (Section 3.2).
domain assumption The TrackML simulated dataset is representative of HL-LHC conditions with up to about 200 pileup.
The paper evaluates the method on TrackML events and draws conclusions about HL-LHC performance (Sections 3.3 and 4).
domain assumption The Gaussian KDE prior trained on ground-truth edges from TrackML generalizes to the events used for evaluation.
The prior in Section 3.4.1 is trained on ground truth and then used to construct the QUBO for the target events; the paper does not demonstrate a fully independent training set separation.
domain assumption The sub-graphing step that keeps only the 5 highest-bias edges per node preserves enough true edges for reconstruction.
Section 3.4.2 describes this pruning without quantifying how many true edges are discarded; the subsequent performance depends on this assumption.

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Cite this review

Pith. "Pith review of Charged particle tracking with quantum annealing-inspired optimization." pith.science (2026). https://pith.science/paper/OMMZDSUM

@misc{pith2026190804475,
  author       = {Pith},
  title        = {Pith review of: Charged particle tracking with quantum annealing-inspired optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMMZDSUM}},
  note         = {Machine review of arXiv:1908.04475}
}

read the original abstract

At the High Luminosity Large Hadron Collider (HL-LHC), traditional track reconstruction techniques that are critical for analysis are expected to face challenges due to scaling with track density. Quantum annealing has shown promise in its ability to solve combinatorial optimization problems amidst an ongoing effort to establish evidence of a quantum speedup. As a step towards exploiting such potential speedup, we investigate a track reconstruction approach by adapting the existing geometric Denby-Peterson (Hopfield) network method to the quantum annealing framework and to HL-LHC conditions. Furthermore, we develop additional techniques to embed the problem onto existing and near-term quantum annealing hardware. Results using simulated annealing and quantum annealing with the D-Wave 2X system on the TrackML dataset are presented, demonstrating the successful application of a quantum annealing-inspired algorithm to the track reconstruction challenge. We find that combinatorial optimization problems can effectively reconstruct tracks, suggesting possible applications for fast hardware-specific implementations at the LHC while leaving open the possibility of a quantum speedup for tracking.

Figures

Figures reproduced from arXiv: 1908.04475 by the authors.

**Figure 2.** Timing performance of 2016 and 2017 CMS conventional tracking methods, as well as 2017 tracking with cellular automaton (CA) seeding [28] with respect to average pileup. Note that currently LHC collisions contain an average of 40 pileup interactions corresponding to a multiplicity of 2,000 particles (tracks). HL-LHC is expected to operate at an average of 200 pileup. 2.2 Quantum annealing Quantum annealing is a for… view at source ↗

**Figure 3.** figure 3. We begin with edges between pre-selected pairs [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (17 more)

**Figure 3.** Figure 3: Projection in the transverse plane of 50 tracks [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

**Figure 4.** Figure 4: Representation of three hits (a, b, c), the segments (rab, rbc) and the opening angle in cylindrical coordinate θabc. The angle φabc (not represented) is measured in the transverse plane. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Projection of 50 tracks in the rz-plane. Due to the uniform magnetic field, charged particles travel in straight lines in cylindrical coordinates. Hence, we may bias the QUBO towards perfectly helical tracks through the θabc term. As in the original DP method, we add a…

**Figure 6.** Figure 6: Representation of the segment configuration [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: Summary of the heuristic methodology for re [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: Gaussian kernel density estimation of a prior [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 11.** Figure 11: QA and SA benchmarked against random an [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]

**Figure 13.** Figure 13: Efficiency as a function of η for tracks with pT > 1 GeV. The track distribution is typically constant in η. 4.2 Impact of pre-processing on SA performance With SA, we expect the time scaling after pre-processing to increase exponentially as O PK i=1 exp (cmi) for…

**Figure 12.** Figure 12: Track purity and efficiency for SA results for [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

**Figure 14.** Figure 14: SA convergence time with a fit O PK i exp (cmi) . By assuming the logarithm of convergence time is proportional to the distribution of sub-graph sizes {mi}, we find a reasonable model for pre-processed scaling with a proportionality constant of c = (8.52 ± 0.076) ×…

**Figure 15.** Figure 15: SA convergence time bounded from above by [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]

**Figure 16.** Figure 16: Comparison of the purity and efficiency for [PITH_FULL_IMAGE:figures/full_fig_p010_16.png]

**Figure 17.** Figure 17: Correlation of Ising model energy with track efficiency and purity for a 500-track sector. The Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]

**Figure 18.** Figure 18: Summary of dimension reduction methodology. After solving a partial QUBO with no bifurcation or [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]

**Figure 19.** Figure 19: TrackML detector geometry in the rz-plane [34]. By placing a threshold on the minimum prior probability, the candidate set of edges can be classically reduced in size. A QUBO is then constructed from edge compatibility scores without bifurcations or bias terms: E = − …

**Figure 20.** Figure 20: Purity and efficiency dependence on annealing stage in the sparse QUBO construction. Purity is sig [PITH_FULL_IMAGE:figures/full_fig_p017_20.png]

**Figure 21.** Figure 21: Purity and efficiency for with QUBO construction parameters tuned to process larger events with high [PITH_FULL_IMAGE:figures/full_fig_p018_21.png]

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