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REVIEW 3 major objections 4 minor 33 references

Sphere Packing on a Quantum Computer for Chromatography Modeling

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Bounded sphere packing for chromatography can be cast as a maximum independent set and solved with QAOA on quantum hardware.

desk verdict A real hardware demo and reasonable formulations, but the scaling claim skips QAOA depth p and overstates the classical comparison. read the letter →

arxiv 2412.00601 v2 pith:M5XUEWJ7 submitted 2024-11-30 quant-ph

Benjamin Hall , Ian Njoroge , Colin Campbell , Bharath Thotakura , Rich Rines , Victory Omole , Maen Qadan This is my paper · ORCID
classification quant-ph MSC 52C1790C2781P68
keywords QuantumApproximateOptimizationAlgorithmmaximumindependentsetspherepackingcolumnchromatographyheterogeneousresourceestimationparameterconcentrationbiopharmaceuticalmanufacturing
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

Column chromatography, the workhorse of biopharmaceutical protein separation, is usually modeled with abstracted particle-level kinetics. This paper proposes replacing that abstraction with a literal sphere-packing model of the resin: circles in a circle for the two-dimensional cross-section, spheres in a cylinder for three dimensions. The homogeneous circle-packing version is discretized onto a grid and cast as a maximum independent set, then solved with the Quantum Approximate Optimization Algorithm on a 20-qubit quantum processor. Heterogeneous circle packing is formulated as a constrained graph optimization, solved classically, and extended to heterogeneous sphere packing; resource counts suggest quantum resources grow polynomially with problem size while classical effort for these NP-hard packings grows exponentially. The value, if the scaling claim holds, is a mechanistic model that captures particle-level adsorption phenomena and a concrete near-term use for quantum optimization.

What carries the argument

The carrying mechanism is a graph encoding of continuous packing: a grid of candidate centers inside the boundary circle becomes the vertex set $V$, edges connect pairs whose disks would overlap, and a valid packing is exactly a maximum independent set. The objective, minimizing empty nodes, and the non-overlap constraints are combined into a single QAOA cost Hamiltonian $H_C = \frac{1}{2}\sum_{v\in V} Z_v + \frac{\lambda}{4}\sum_{v,w\in E}(Z_v Z_w - Z_v - Z_w)$, with the Lagrange multiplier $\lambda$ treated as a tunable hyperparameter. For heterogeneous radii, the 'first quantization' formulation encodes each node's radius choice in $\lceil \log_2(|R|+1)\rceil$ qubits while the 'second quantization' formulation assigns one qubit per radius-location pair, and the same edge criterion $|v-w| \le r+s$ builds the constraint Hamiltonian.

What would settle it

Run exact or near-exact simulations of the QAOA circuit for the homogeneous circle-packing graphs at increasing grid sizes (for example 18, 40, 80, and 160 nodes) and measure the number of QAOA layers $p$ needed to reach a fixed success probability; if $p$ grows faster than any polynomial, the resource-scaling claim is falsified.

Watch

Extended reading notes

Core claim

The central discovery is a reduction: bounded heterogeneous sphere packing, the geometry of a packed chromatography column, can be discretized and expressed as a graph optimization whose quantum cost Hamiltonian is compiled into QAOA circuits. For the homogeneous case, the reduction to maximum independent set is exact on the grid, and the resulting 18-node instance produced the optimal 12-circle packing on real hardware, reaching a packing density of 0.68 versus the proven optimum of 0.74. The heterogeneous cases are written in two complementary quantum formulations, one with logarithmic qubit compression and one with a qubit per radius-location pair, and both are bounded by low-degree polynomial resource counts in the discretization points per side and the number of allowed radii. The paper therefore claims a pathway to quantum advantage for a real biopharmaceutical problem: classical solvers scale exponentially because the underlying maximum independent set is NP-hard, while the quantum resources for each QAOA layer scale polynomially.

Load-bearing premise

The claim that quantum resources stay polynomial assumes the number of QAOA rounds needed to find a good packing grows only as a polynomial of the problem size; the paper never proves this.

Editorial extensions

If this is right

  • If the polynomial resource scaling holds, a sufficiently large chromatography column modeled with this discretization can be optimized on a quantum computer with a qubit and gate count that grows polynomially in grid resolution and the number of radii, whereas classical approximate solvers face an NP-hard search.
  • Parameter concentration evidence means QAOA parameters trained on a small subgraph of a packing instance can be transferred to the full instance, so large untrainable problems remain accessible by scaling up a small trained circuit.
  • The same integer program extends from circles in a circle to spheres in a cylinder by stacking two-dimensional slices and adding inter-slice edges, so the homogeneous hardware result is already a template for the three-dimensional model.
  • The first- versus second-quantization tradeoff offers a resource tunable: use logarithmic qubits with higher-order terms, or more qubits with lower CNOT-degree, depending on which noise source dominates a given device.

Reading between the lines

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

  • The resource estimate counts qubits and CNOT gates per QAOA layer but never bounds the number of layers $p$ needed to reach a good packing; if $p$ must grow faster than polynomially, the claimed polynomial total resource scaling would fail, and testing $p$-scaling on grid graphs of increasing size would settle this.
  • The physical model assumes a regular grid discretization of space, whereas real resin beds are random packed and typically achieve densities around 0.64 for monodisperse spheres; a direct comparison to experimental packed-bed porosity data would be a natural testable extension.
  • The same graph and maximum-independent-set reduction applies to other bounded geometric packing problems such as bin packing or container loading, so a quantum advantage here would carry over to any such NP-hard instance.
  • The heterogeneous second-quantization Hamiltonian is built from pairwise overlap penalties, so it could also be attacked by classical annealing or other variational circuits; benchmarking QAOA against classical local search at matched instance sizes would clarify where the quantum advantage claim actually stands.
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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 / 4 minor

Summary. The paper models column chromatography as bounded sphere packing and treats three increasing levels of complexity: homogeneous circle packing, heterogeneous circle packing, and heterogeneous sphere packing. Homogeneous circle packing is recast as a maximum independent set problem on a grid graph and solved with QAOA on IQM's Garnet device (18 qubits), with hyperparameter selection, compilation, noisy simulation, hardware execution, and a parameter concentration study. Heterogeneous packing is formulated through two Hamiltonian constructions ("first quantization" and "second quantization"), and Section IV gives resource estimates intended to show that the quantum resources scale polynomially with problem size.

Significance. If the polynomial-scaling claim were established, the paper would be a valuable end-to-end demonstration of a concrete industrial optimization use case for QAOA, combining a falsifiable hardware prediction with transferable techniques such as noise-aware depth selection and subgraph parameter transfer. The hardware experiment is the strongest part: the noisy simulation correctly predicted p=3 as the best depth, and the device returned the optimal independent set as the most probable state for p=1, 3, and 5. The resource-scaling analysis, however, is incomplete in a way that directly affects the paper's central quantum-advantage claim, because it never bounds the QAOA layer count p as a function of problem size.

major comments (3)
  1. [Section IV, Eqs. (14)-(19)] The resource counts in Eqs. (14)-(19) are per QAOA layer, but the total quantum cost is p times these counts. The paper never bounds p(n), the number of layers needed to obtain a good packing. The p to infinity convergence guarantee in Sec. I.B and the hardware results for p=1,3,5 on a single 18-node graph (Figs. 11-12) do not establish any scaling for p on larger instances, and the parameter concentration results (Figs. 13-14) concern parameter reuse, not required circuit depth. The statement in Sec. IV that "the quantum resources for this problem only scale polynomially with increased problem size" is therefore unproven, and this is the load-bearing claim of the paper.
  2. [Sections II.B and IV] The classical-complexity comparison is overbroad. The paper argues that classical computation is exponential because MIS is NP-hard in the worst case, but the graphs constructed in Sec. II are grid graphs with unit-disk-like edges, for which polynomial-time approximation schemes exist. The claim of exponential classical difficulty for this particular instance family and for the heterogeneous packing formulations needs instance-specific evidence (or a more careful statement of the complexity assumption) rather than an appeal to worst-case NP-hardness.
  3. [Eqs. (15) and (19)] The displayed resource bounds contain algebraic errors. In Eq. (15), the chain should give |R|^2 q^d (r_m q / R_b)^d, not |R| q^d (r_m q / R_b)^d; a factor of |R| is missing. In Eq. (19), 2^{2 ceil(log2(|R|+1))} is of order |R|^2, so the coefficient is of order |R|^4, not |R|^3. These errors do not change the polynomial degree in q, but they must be corrected because the resource estimates are the quantitative basis of the paper's scaling claims.
minor comments (4)
  1. [Eq. (3)] The identity operator is defined as I = (|0><0|+|1><1|)/2, but the identity should be |0><0|+|1><1|; as written, the relation x_v = (I-Z_v)/2 is inconsistent.
  2. [Section II.C.7] The text says "we train the parameters of both the subgraph QAOA circuit (Csub) and the full graph QAOA circuit (Csub)"; the second circuit name should be Cfull.
  3. [Figure 11] The y-axis labels appear as 10^3, 10^2, 10^1, but the probabilities are manifestly below 1; the labels should be 10^{-3}, 10^{-2}, 10^{-1}.
  4. [Section IV, final paragraph] The comparison between the two formulations says the second-quantization formulation trades an increase in CNOT scaling for an exponential decrease in qubits, but the qubit counts in Eqs. (14) and (17) show that second quantization uses more qubits (|R| q^d) than first quantization (log|R| q^d), not fewer; the sentence should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found: the paper's derivations are self-contained, and its resource claims are upper bounds derived from construction, not from fitted outputs.

full rationale

The paper's central chain is: (1) map bounded homogeneous circle packing to maximum independent set via an explicit grid construction (Eq. 2); (2) transform the integer program to a cost Hamiltonian (Eq. 4) by the standard substitution x_v = (I_v - Z_v)/2; (3) run QAOA using a physical device. None of these steps defines a quantity in terms of the target prediction. The noisy simulation that selected p = 3 used device calibration data (T1, T2, randomized benchmarking, CZ infidelity, readout rates) and was confirmed on hardware; this is an independent prediction, not a fitted input. The hyperparameter λ = 0.5 is tuned for a single instance and is not used to define the resource scaling or the optimality claim. The resource scaling bounds in Section IV are explicit upper bounds in terms of graph size, radius, and dimension — they are derived from the construction and are not fitted to the hardware results. The paper cites its own Superstaq software for compilation, but that is a tooling citation, not a load-bearing premise for any scientific conclusion. The main weakness — that the total QAOA cost requires multiplying per-layer counts by the number of layers p, and no polynomial bound on p is provided — is a correctness/completeness concern about the scaling claim, not circularity. Similarly, the NP-hardness argument is a worst-case complexity statement and may be overbroad for the specific grid graphs, but that is also a correctness issue, not a circular one. There is no equation that reduces to its own input, no fitted parameter renamed as a prediction, and no self-citation chain invoked to forbid alternatives.

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

The central claim depends on a discretization approximation, on heuristic behavior of QAOA, on parameter concentration, and on an unstated assumption that the number of QAOA layers does not grow exponentially. The classically exponential scaling claim also assumes worst-case NP-hardness transfers to structured grid instances.

free parameters (3)
  • Lagrange multiplier lambda = 0.5
    Penalty weight for the MIS constraint, tuned via a hyperparameter sweep (Figure 9) on the hardware instance.
  • Grid spacing a = sqrt(2) for hardware
    Discretization spacing chosen to match IQM Garnet's qubit connectivity; the paper also uses a=1.4, 1.1, 0.9, 0.75 for classical illustrations.
  • Boundary radius Rb = 4.2
    Chosen so the graph fits 18 qubits and the starting circle at the missing corner; the resulting density 0.68 is compared to the optimal 12-circle density 0.74 for Rb=4.02.
assumptions (5)
  • domain assumption Grid discretization convergence: as the grid spacing a goes to 0, the MIS optimum on the lattice approaches the continuous circle packing optimum.
    Invoked in Section II A to justify the integer program; the paper notes the formulation approaches the continuous case as a goes to 0 but provides no convergence proof or error bound.
  • standard math QAOA performance with finite p: increasing p improves the approximation, and in the limit p goes to infinity QAOA reaches the optimal state (Farhi et al. [17]).
    The p to infinity limit is proven, but the paper uses small finite p (1, 3, 5) and assumes the favorable trend holds for larger instances.
  • domain assumption Parameter concentration: QAOA parameters trained on a small subgraph of a sphere packing instance perform nearly as well on the full graph.
    Section II C 7 asserts this based on one numerical comparison (Figure 14) with no statistics or repeated trials.
  • ad hoc to paper Polynomial layer scaling: the QAOA depth p needed for good solutions grows at most polynomially with instance size.
    This is the load-bearing premise for the resource scaling claim in Section IV; the paper never mentions or justifies it.
  • domain assumption Classical exponential scaling: NP-hardness of MIS implies exponential classical time for these sphere packing instances.
    NP-hardness is a worst-case statement; the paper does not analyze the structure of grid-based MIS instances, for which heuristic solvers may scale well in practice.

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Cite this review

Pith. "Pith review of Sphere Packing on a Quantum Computer for Chromatography Modeling." pith.science (2026). https://pith.science/paper/M5XUEWJ7

@misc{pith2026241200601,
  author       = {Pith},
  title        = {Pith review of: Sphere Packing on a Quantum Computer for Chromatography Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5XUEWJ7}},
  note         = {Machine review of arXiv:2412.00601}
}
read the original abstract

Column chromatography is an important process in downstream biopharmaceutical manufacturing that enables high-selectivity separation of proteins through various modalities, such as affinity, ion exchange, hydrophobic interactions, or a combination of the aforementioned modes. Current mechanistic models of column chromatography typically abstract particle-level phenomena, in particular adsorption kinetics. A mechanistic model capable of incorporating particle-level phenomena would increase the value derived from mechanistic models. To this end, we model column chromatography via sphere packing, formulating three versions, each with increasing complexity. The first, homogeneous circle packing, is recast as maximum independent set and solved by the Quantum Approximate Optimization Algorithm on a quantum computer. The second, heterogeneous circle packing, is formulated as a graphical optimization problem and solved via classical simulations, accompanied by a road map to a quantum solution. An extension to the third, heterogeneous sphere packing, is formulated mathematically in a manner suitable to a quantum solution. Finally, detailed resource scaling is conducted to estimate the quantum resources required to simulate the most realistic model, providing a pathway to quantum advantage.

Figures

Figures reproduced from arXiv: 2412.00601 by the authors.

Figure 1
Figure 1. FIG. 1. Three complexity levels of sphere packing for model [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic of chromatography for protein filtration. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Graphs for discretization spacings [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic showing how the size [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Grid qubit coupling map of IQM’s Garnet. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Graph for quantum hardware experiment with first [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Maximum independent set of the graph for the quan [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The probability of measuring the optimal state ver [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of an ideal and noisy simulation of compiled [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Probabilities of the top 10 most probable states [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Probability of measuring the best state versus num [PITH_FULL_IMAGE:figures/full_fig_p008_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. For the “First Quantization” formulation of the [PITH_FULL_IMAGE:figures/full_fig_p009_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. In the “Second Quantization” formulation of the [PITH_FULL_IMAGE:figures/full_fig_p010_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 25 canonical work pages

  1. [1]

    Hamiltonian Formulation To formulate MIS (Eq. 2) as a Hamiltonian, we make the following transformation from the indicator variable to Pauli spin matrices xv → (Iv − Zv)/2, (3) where I = (|0⟩ ⟨0|+|1⟩ ⟨1|)/2 and Z = (|0⟩ ⟨0|−| 1⟩ ⟨1|)/2, implying xv = |1⟩ ⟨1|. Therefore, xv |1⟩v = 1 while xv |0⟩v = 0, as expected from an indicator variable. Ap- plying this...

    work page

  2. [2]

    His conjec- ture was finally proven, with the aid of computer proof assistants, in 2017 [10]

    ≈ 0.74. His conjec- ture was finally proven, with the aid of computer proof assistants, in 2017 [10]. Much less is known about un- bounded heterogeneous packing except for special ratios of radii for two sphere sizes [11], leaving a general an- alytical solution an open problem within the field. The existence of general or analytical solutions is summariz...

    work page 2017

  3. [3]

    As for sphere packing, in 1611 Johannes Kepler conjectured that cubic close packing and the hexagonal close packing are optimal in their density of π/(3 √

    ≈ 0.91. As for sphere packing, in 1611 Johannes Kepler conjectured that cubic close packing and the hexagonal close packing are optimal in their density of π/(3 √

    work page

  4. [4]

    missing corners

    Experiment Setup The hardware selected for this experiment was IQM’s 20-qubit superconducting device, Garnet [18]. It was cho- sen, in large part, because of the grid connectivity of its FIG. 6. Grid qubit coupling map of IQM’s Garnet. FIG. 7. Graph for quantum hardware experiment with first circle touching the boundary. qubit coupling map (Figure 6). It’...

    work page

  5. [5]

    This leads to a packing density (fraction of the boundary circle’s area filled by packing circles) of 0 .68

    Figure 8 shows that the optimal solu- tion (the maximum independent set) for this discretiza- tion spacing is a placement of 12 spheres. This leads to a packing density (fraction of the boundary circle’s area filled by packing circles) of 0 .68. It has been proven that the optimal packing density for 12 spheres 0 .74 (which has a boundary circle of radius...

    work page

  6. [6]

    sweet spot

    Hyperparameter Optimization The Lagrange multiplier λ in the integer program for MIS (Eq. 4) must be chosen carefully. If λ is too small, the energies associated with invalid packing will not be sufficiently separated from valid packings. This will lead to QAOA including too many infeasible states in its search, thus hampering both its search efficiency a...

    work page

  7. [7]

    4) into the QAOA ansatz (Eq

    Compilation Plugging the MIS Hamiltonian (Eq. 4) into the QAOA ansatz (Eq. 1) results in a set of RZZ , RZ, and RX gates. The RZZ operation can be decomposed into single-qubit gates and exactly two CZ gates (Garnet’s native two- qubit gate). Meanwhile, to reduce total gate count, the RZ and R X gates can be combined (as R Z commutes through R ZZ ) and fur...

    work page

  8. [8]

    Noisy Simulation Before executing our QAOA circuits on hardware, we considered an ideal simulation of the QAOA circuit with λ = 0.5 over increasing circuit depth p and found that, as expected, the probability of QAOA measuring the op- timal solution increases with increasing depth (Figure 11). However, because we are currently in the Noisy Intermediate-Sc...

    work page

Show all 33 references
  1. [9]

    sweet spot

    Experiment Results To execute our circuits on Garnet, we compiled and submitted our jobs through Infleqtion’s compilation soft- ware, Superstaq [21] which, in turn, submitted the job through A WS’s Braket. We ran QAOA for p = 3 (based on the noisy simulations of Subsection II ...

  2. [10]

    smaller problem instance

    Parameter Concentration As noise often scales with the number of qubits/gates, and therefore problem size, it becomes harder to train QAOA as the problem size increases. This roadblock can be alleviated, however, if the problem exhibits pa- rameter concentration [22], a phenom...

  3. [11]

    Meet Willow, our state-of-the-art quantum chip — blog.google, https://blog.google/technology/ research/google-willow-quantum-chip/ , [Accessed 16-12-2024]

    2024

  4. [12]

    W. Wang, Z. Ye, H. Gao, and D. Ouyang, Computational pharmaceutics - a new paradigm of drug delivery, Journal of Controlled Release 338, 119–136 (2021)

    2021

  5. [13]

    A. S. Rathore, D. Kumar, and N. Kateja, Recent devel- opments in chromatographic purification of biopharma- ceuticals, Biotechnology Letters 40, 895 (2018)

    2018

  6. [14]

    Kumar and A

    V. Kumar and A. M. Lenhoff, Mechanistic modeling of preparative column chromatography for biotherapeutics, On: Wed 51, 1 (2020)

    2020

  7. [15]

    L. K. Shekhawat and A. S. Rathore, An overview of mechanistic modeling of liquid chromatography, Prepar- ative Biochemistry & Biotechnology 49, 623 (2019), doi: 10.1080/10826068.2019.1615504

    2019

  8. [16]

    N. Borg, Y. Brodsky, J. Moscariello, S. Vunnum, G. Vedantham, K. Westerberg, and B. Nilsson, Model- ing and robust pooling design of a preparative cation- exchange chromatography step for purification of mon- oclonal antibody monomer from aggregates, Journal of Chromatography ...

    2014

  9. [17]

    Espinoza, S

    D. Espinoza, S. Tallvod, N. Andersson, and B. Nilsson, Automatic procedure for modelling, calibration, and op- timization of a three-component chromatographic sep- aration, Journal of Chromatography A 1720, 464805 (2024)

    2024

  10. [18]

    Campos, M

    M. Campos, M. Jenssen, M. Michelen, and J. Sa- hasrabudhe, A new lower bound for sphere packing (2023), arXiv:2312.10026

    2023 arXiv

  11. [19]

    Chang and L.-C

    H.-C. Chang and L.-C. Wang, A simple proof of thue’s theorem on circle packing (2010), arXiv:1009.4322

    2010 arXiv

  12. [20]

    Hales, M

    T. Hales, M. Adams, G. Bauer, T. D. Dang, J. Har- rison, L. T. Hoang, C. Kaliszyk, S. Margron, Vic- torand McLaughlin, T. Nguyen, Tat, and et al., A formal proof of the kepler conjecture, Forum of Mathematics, Pi 5, e2 (2017)

    2017

  13. [21]

    Kennedy, Compact packings of the plane with two sizes of discs (2004), arXiv:math/0407145 [math.MG]

    T. Kennedy, Compact packings of the plane with two sizes of discs (2004), arXiv:math/0407145 [math.MG]

    2004 arXiv

  14. [22]

    D. B. Ekanayake and D. J. LaFountain, Tight partitions for packing circles in a circle (2024)

    2024

  15. [23]

    Mughal, H

    A. Mughal, H. K. Chan, D. Weaire, and S. Hutzler, Dense packings of spheres in cylinders: Simulations, Physical Review E 85, 10.1103/physreve.85.051305 (2012)

    2012 doi

  16. [24]

    Chan, Densest columnar structures of hard spheres from sequential deposition, Physical Review E 84, 10.1103/physreve.84.050302 (2011)

    H.-K. Chan, Densest columnar structures of hard spheres from sequential deposition, Physical Review E 84, 10.1103/physreve.84.050302 (2011)

    2011 doi

  17. [25]

    L. Fu, W. Steinhardt, H. Zhao, J. E. S. Socolar, and P. Charbonneau, Hard sphere packings within cylinders, Soft Matter 12, 2505–2514 (2016)

    2016

  18. [26]

    Jo´ os, On the packing of fourteen congruent spheres in a cube, Geom

    A. Jo´ os, On the packing of fourteen congruent spheres in a cube, Geom. Dedicata 140, 49 (2009)

    2009

  19. [27]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, A quan- tum approximate optimization algorithm (2014), arXiv:1411.4028 [quant-ph]

    2014 arXiv

  20. [28]

    L. A. et al., Technology and performance bench- marks of iqm’s 20-qubit quantum computer (2024), arXiv:2408.12433

    2024 arXiv

  21. [29]

    Fodor, The densest packing of 12 congruent circles in a circle., Beitr¨ age zur Algebra und Geometrie 41, 401 (2000)

    F. Fodor, The densest packing of 12 congruent circles in a circle., Beitr¨ age zur Algebra und Geometrie 41, 401 (2000)

    2000

  22. [30]

    Preskill, Quantum computing in the nisq era and be- yond, Quantum 2, 79 (2018)

    J. Preskill, Quantum computing in the nisq era and be- yond, Quantum 2, 79 (2018)

    2018

  23. [31]

    Campbell, F

    C. Campbell, F. T. Chong, D. Dahl, P. Frederick, P. Goiporia, P. Gokhale, B. Hall, S. Issa, E. Jones, S. Lee, A. Litteken, V. Omole, D. Owusu-Antwi, M. A. Perlin, R. Rines, K. N. Smith, N. Goss, A. Hashim, R. Naik, E. Younis, D. Lobser, C. G. Yale, B. Huang, and J. Liu, Supers...

    2023 arXiv

  24. [32]

    Wurtz and D

    J. Wurtz and D. Lykov, The fixed angle conjecture for qaoa on regular maxcut graphs (2021), arXiv:2107.00677

    2021 arXiv

  25. [33]

    Glover, G

    F. Glover, G. Kochenberger, and Y. Du, A tuto- rial on formulating and using qubo models (2018), arXiv:1811.11538

    2018 arXiv

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