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REVIEW 4 major objections 5 minor 54 references

Adaptive Lattice Gas Algorithm: Classical and Quantum implementations

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A deterministic lattice gas with locally adapted collision fractions reproduces LBM equilibrium and low-speed dynamics, and admits a quantum version with a linear collision operator on log(N)+3 qubits.

desk verdict A genuinely new deterministic lattice-gas construction with a clean quantum encoding, but the LBM-equivalence claim rests on equilibrium matching and one benchmark because the adaptive hydrodynamic limit is never derived. read the letter →

arxiv 2504.13549 v2 pith:AE2WGSYY submitted 2025-04-18 quant-ph

Niccolò Fonio , Ljubomir Budinski , Valtteri Lahtinen , Pierre Sagaut This is my paper · ORCID
classification quant-ph MSC 76M2881P68 PACS 47.11.-j03.67.Ax
keywords latticegasautomataintegerBoltzmannmethodadaptivecollisionratequantumD1Q3modellinearcombinationofunitaries
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

Starting from a one-dimensional integer lattice gas (ILGA), this paper replaces the random collision selection of Monte Carlo lattice gas automata with a fixed fraction of the possible crunches and splittings at each site. The crunch fraction $\lambda_c$ is then adapted locally through Eq. (17), so that the equilibrium populations match those of the lattice Boltzmann method (LBM) in the $|u|\approx 0$ limit. The resulting adaptive lattice gas algorithm (ALGA) is deterministic and, on a cosine-wave benchmark, reproduces the shock formation and density/momentum profiles of LBM. The same collision rule is encoded in quantum amplitudes via Eq. (18), making the collision operator linear; the quantum version uses $\log(N)+3$ qubits and costs $O(\log^2 N)$ per evolution step before measurement and reinitialization. If the claim holds, ALGA offers a quantum-friendly, sampling-free classical scheme that inherits LBM's low-Mach behavior.

What carries the argument

The load-bearing object is the adapted crunch fraction $\lambda_c(x,t) = \lambda_s (2 - 3u^2)/(1 - 3|u| + 3u^2)$, which turns a linear-in-velocity equilibrium into LBM's quadratic equilibrium by making the collision intensity a local function of momentum density. Its quantum counterpart is the amplitude encoding $|\Psi\rangle = (1/M)\sum_x |x\rangle(n_0|00\rangle + n_1|01\rangle + n_2|10\rangle + \lambda_c \min(n_0,n_1)|11\rangle)$, which buries the nonlinear min term in an amplitude so the collision update becomes the linear operator $\hat{C}$ of Eq. (19). That operator is implemented through singular-value decomposition and a linear combination of unitaries, with streaming carried out by a quantum shift circuit at $O(\log^2 N)$ cost. The $\min(n_0,n_1)$ term is the nonlinear engine in both the classical and quantum versions.

What would settle it

Measure the effective viscosity of ALGA from the decay rate of the cosine wave over many time steps, at several $|u|$ values inside the validity range, and compare with the LBM viscosity $(\tau - 0.5)/3$ for the best matching relaxation time; if the fitted viscosity varies with amplitude or the density profiles deviate systematically from LBM at matched times, the claim of identical simulation results is falsified.

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Extended reading notes

Core claim

The paper's central claim is that a D1Q3 (one-dimensional, three-velocity) integer lattice gas in which a fraction $\lambda_s$ of splittings and a locally adapted fraction $\lambda_c(x,t)$ of crunches are performed reproduces the equilibrium distribution functions of LBM for $|u|\approx 0$, and therefore simulates the same macroscopic phenomena. The adaptation rule $\lambda_c = \lambda_s (2 - 3u^2)/(1 - 3|u| + 3u^2)$ is obtained by substituting LBM's Maxwell–Boltzmann equilibrium populations into the ILGA equilibrium condition, and is validated by equilibrium measurements and by a cosine-wave shock simulation matching LBM. The paper further claims that encoding the populations, including the nonlinear min term, in quantum amplitudes makes the collision operator the linear matrix of Eq. (19); using singular-value decomposition and a linear combination of unitaries, the per-step quantum cost is $O(\log^2 N)$ with $\log(N)+3$ qubits, but the present implementation needs measurement and reinitialization every time step, so the full $T$-step cost remains $\Omega(NT)$.

Load-bearing premise

The whole result assumes that matching LBM's equilibrium populations makes the large-scale behavior match too; the paper tests this only on one cosine-wave setup, because the governing equations for the adaptive rule are not derived.

Editorial extensions

If this is right

  • In the $|u|\approx 0$ limit, ALGA inherits LBM's low-Mach behavior: the cosine-wave benchmark matches LBM's mass and momentum profiles, and stability is lost at the same high-velocity threshold as LBM.
  • Because ALGA is deterministic, the ensemble averaging needed by Monte Carlo lattice gas automata is not required; with smooth initial data a single run reproduces the averaged results.
  • The quantum encoding uses $\log(N)+3$ qubits, with an $O(1)$ collision via a linear combination of unitaries and $O(\log^2 N)$ streaming from the quantum shift circuit.
  • In the constant-$\lambda_c$ case, time-step concatenation is possible when the min term is always attained by the same population; with local adaptation it is possible if the local momentum is known analytically and has one sign everywhere, removing measurement and reinitialization.
  • Without such concatenation, measurement and reinitialization dominate: the full algorithm costs $\Omega(NT)$, so a quantum advantage awaits efficient tomography or unitary computation of the nonlinear term.

Reading between the lines

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

  • The paper leaves implicit that a second-order Chapman–Enskog expansion with the adaptive $\lambda_c(x,t)$ would settle whether the effective viscosity is LBM's; the numerical hint $\tau \propto (1+\lambda_s)/\lambda_s$ is a concrete starting point for deriving that formula.
  • The encoding's trick of hiding the nonlinear min term in an amplitude so the collision operator stays linear suggests a general recipe for quantum-friendly lattice gases; its scalability will be tested when the same idea is extended to 2D/3D models with multiple nonlinear terms.
  • If the min term can be computed unitarily or supplied analytically, the paper's concatenation argument implies the $T$-step cost drops to $O(T \log^2 N)$, at the price of one ancilla per step and a $2^{-T}$ success probability; that trade-off is the natural next design problem.
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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

4 major / 5 minor

Summary. The paper introduces an adaptive integer lattice gas algorithm (ALGA) for the one-dimensional D1Q3 model. The collision step applies local fractions λ_s and λ_c of the maximum possible splittings and crunches; λ_c is adjusted at each cell via Eq. (17) so that the equilibrium populations match the LBM equilibrium (Eq. 9) in the low-velocity limit. The authors claim that ALGA 'achieves the same simulation results of LBM' in this limit, and support this with equilibrium-distribution measurements and a cosine-wave shock-formation comparison. They then present a quantum encoding (Eq. 18) in which the collision operator is represented as a fixed linear operator (Eq. 19), implemented via SVD and LCU, with streaming by quantum shifts. The per-step circuit cost is stated as O(log² N) using log(N)+3 qubits, but the full evolution still requires measurement and reinitialization, giving total cost Ω(NT). The paper explicitly leaves several issues open, including an analytical τ–λ_s relation and the possibility of time-step concatenation.

Significance. If the central claim is established, ALGA would be a deterministic, quantum-friendly lattice gas that avoids the random sampling of MCLGA while reproducing low-Mach LBM behavior, with a linear collision operator suitable for quantum circuits. The paper is honest about its current limitations: the quantum implementation does not achieve an end-to-end advantage because of tomography and reinitialization, and the authors concede that no analytical viscosity relation is obtained. The main value lies in proposing a concrete construction that bridges ILGA and LBM and in demonstrating numerically that the local-adaptation idea can match LBM for a benchmark. However, the claim of 'same simulation results as LBM' is not yet backed by a hydrodynamic derivation for the adaptive case, and the only external validation is a single cosine-wave test. The paper would be a useful contribution if the missing analysis is supplied.

major comments (4)
  1. [Appendix A, Eq. A13] The Chapman-Enskog expansion in Appendix A is performed only for constant λ_c and λ_s. It yields Eq. A13, which contains a term λ_c/(λ_c+λ_s) ∂|ρu|/∂x rather than the LBM convective term. In the adaptive algorithm, λ_c(x,t) from Eq. 17 depends on the local momentum, so the relaxation rate varies in space and time and the effective viscosity and higher-order Knudsen terms are uncontrolled. The paper's central claim that ALGA 'achieves the same simulation results of LBM' therefore rests on the assertion in Sec. II B that matching equilibrium distributions suffices, but no derivation or quantitative error bound is given for the adaptive case. This missing derivation is load-bearing and directly affects the abstract's main claim.
  2. [Sec. V C, Eq. 17] The equilibrium-distribution agreement in Sec. V C is partially a verification of the construction rather than an independent prediction: λ_c is obtained by substituting the LBM equilibrium g_i^eq into the equilibrium condition Eq. 16, so agreement with Eq. 9 in the |u|≈0 region is enforced by design. The only genuinely external test is the cosine-wave dynamics in Sec. V D. The paper should state this circularity explicitly and distinguish the constructed equilibrium match from an independent validation of the hydrodynamic behavior.
  3. [Sec. V D] The cosine-wave comparison is the sole evidence for the adaptive case, but it is not quantified: no error metric, convergence study, or variation of Reynolds number and wavelength is reported, and the simulation is performed for a single initial condition and a single relaxation time (τ=1). The paper itself states that the numerical analysis of the spatially averaged absolute difference 'did not prove any confirmation' for the τ=1.5 hypothesis and that no analytical τ–λ_s relation was obtained. As a result, the claim that ALGA reproduces LBM behavior in general is not established beyond the single benchmark shown; this is a limitation of the central claim, not a minor presentation issue.
  4. [Sec. II C, Eq. 9] The D1Q3 weights and the equilibrium expansion in Eq. 9 are not the standard LBM equilibrium. The text states w_- = w_+ = 1/3, whereas with lattice velocities ±1 the standard D1Q3 weights are w_± = 1/6, and Eq. 9 as written lacks the velocity-dependent quadratic term and the −ρu²/(2c_s²) correction of the standard low-Mach expansion. Since Eq. 17 is derived from Eq. 9, the authors should specify the velocity rescaling and weight convention used, and verify that the resulting λ_c formula and the numerical fits are consistent with the LBM implementation they compare against.
minor comments (5)
  1. [Sec. IV, Eq. 18] The notation n1 and n2 is used for the populations appearing in the quantum encoding, but the paper elsewhere denotes the populations as n_-, n0, n_+. Please define n1 and n2 explicitly to avoid ambiguity.
  2. [Sec. V D] The condition '|u(x,t)| < 3' appears to be a typo; the bound for λ_c to remain physical is stated in Sec. III A as |u| < √(2/3) (up to the λ_s dependence noted below). Please correct the threshold.
  3. [Sec. III A] The statement that 0 < λ_c < 1 is 'in general' ensured if |u| < √(2/3) is not a complete condition, because the upper bound λ_c < 1 also depends on λ_s through Eq. 17. Please give the precise inequality for λ_c in the allowed range.
  4. [Sec. II C, Eq. 9] The notation 'v_i u' in Eq. 9 is undefined; v_i is not introduced, and it is unclear whether it denotes the lattice velocity c_i, a rescaled velocity, or something else.
  5. [Sec. IV] There is a typo 'algorihtm' in the paragraph after Eq. 19; the word should be 'algorithm'.

Circularity Check

1 steps flagged · score 4.0 of 10

Adaptive λc is fitted to LBM equilibrium, so the equilibrium 'validation' is a consistency check; the cosine benchmark carries the independent content.

  1. fitted input called prediction [Sec. III A, Eq. 17; Sec. V C, Fig. 5]
    "We substitute neq_i from Eq. 16 with geq_i in Eq. 9, and we obtain λc = λs (2 − 3u^2)/(1 − 3|u| + 3u^2). ... The first validation about local-adapted λc concerns the equilibrium distribution functions. As said in Sec. III, we impose the equilibrium distributions adapting the fraction of crunches according to Eq. 17."

    λc(x,t) is not predicted: it is algebraically solved from the ILGA equilibrium condition Eq. 16 after substituting the target LBM equilibrium geq (Eq. 9). Therefore the local equilibrium populations are forced by construction to equal geq whenever the local u is in the valid range. The subsequent Fig. 5 agreement with LBM equilibrium distributions is a consistency check of that imposed condition, not an independent verification. The paper uses this equilibrium match as the bridge to the macroscopic claim ('if a method reproduces the same equilibrium distributions of LBM it should simulate the same phenomena'), but only the cosine-wave simulation in Sec. V D is a genuinely external benchmark.

full rationale

The central new validation with independent content is the cosine-wave evolution in Sec. V D, which compares QALGA with LBM dynamics and is not used to set λc. However, that benchmark is weakened by the paper's own admission that no analytical τ–λs relation was obtained and that the τ=1.5 viscosity hypothesis 'did not prove any confirmation.' The Chapman-Enskog expansion in Appendix A is performed only for constant λc, λs and yields the term ∂|ρu|/∂x (Eq. A13), not the LBM convective term; for space-time varying λc the hydrodynamic limit is never derived. These are correctness/rigor gaps rather than circularity. The quantum complexity analysis is self-contained: the O(log^2 N) streaming step cites [33], the LCU collision is standard, and the paper explicitly acknowledges that measurement and reinitialization cost Ω(NT), so no exponential speedup is claimed for the full evolution. The only concrete circularity is the equilibrium 'validation' of Sec. V C, which restates the fitting equation Eq. 17. Because the cosine-wave dynamics still provides partial independent evidence, the overall circularity score is 4.

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

The paper introduces no new physical entity, force, or conserved quantity. Its novel elements are algorithmic: fractional collision parameters, the local adaptation rule, and an amplitude encoding where the nonlinear min term is stored in the |11> state.

free parameters (3)
  • lambda_s (splitting fraction) = 0.2 in main simulations
    Fraction of possible splitting collisions that is executed. Chosen by hand; no first-principles value is derived. It controls the collision rate and, implicitly, the numerical viscosity; the tau mapping is only fitted empirically.
  • lambda_c (crunching fraction, constant-parameter case) = 0.2 in Sec. V A/B
    In the constant-parameter ALGA, lambda_c is a second free fraction. In the adaptive case it is replaced by Eq. 17, which depends on local u and lambda_s.
  • Best-fit tau(lambda_s) correspondence = tau_min = 0.75 +/- 0.01 at lambda_s_max = 0.436 +/- 0.025; tau_max = 11.36 +/- 0.01
    The relation between lambda_s and the LBM relaxation time tau is obtained by scanning lambda_s and tau and minimizing the LBM-QALGA absolute difference at t0=450. This is a fitted correspondence, used qualitatively, not a derived prediction.
assumptions (5)
  • domain assumption LBM equilibrium distributions geq_i (Eq. 9) define the target behavior; local adaptation is derived by substituting these into the collision equilibrium condition.
    Section II C introduces geq_i as the Maxwell-Boltzmann expansion to O(u^2); Section III A uses it as the fixed target for adapting lambda_c.
  • domain assumption Matching LBM equilibrium distributions is sufficient for reproducing LBM macroscopic dynamics.
    Section II B states this explicitly: 'if a method reproduces the same equilibrium distributions of LBM it should simulate the same phenomena.' The adaptive hydrodynamic limit is never derived.
  • standard math Chapman-Enskog expansion is a valid asymptotic description of the lattice gas dynamics.
    Appendix A uses the standard Taylor and Chapman-Enskog expansion to derive the constant-parameter PDEs (Eqs. A12, A13).
  • domain assumption The integer-particle constraint can be dropped in simulations without changing the model's behavior.
    Section V explains that integer casting introduces noise, and the main results run without integer casting while still using the particle-based derivation.
  • domain assumption Perfect quantum state tomography and reinitialization after each time step are available.
    Sections IV and V D assume perfect tomography so the classical lattice amplitudes can be re-encoded; the authors explicitly state this assumption.

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Pith. "Pith review of Adaptive Lattice Gas Algorithm: Classical and Quantum implementations." pith.science (2026). https://pith.science/paper/AE2WGSYY

@misc{pith2026250413549,
  author       = {Pith},
  title        = {Pith review of: Adaptive Lattice Gas Algorithm: Classical and Quantum implementations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AE2WGSYY}},
  note         = {Machine review of arXiv:2504.13549}
}
read the original abstract

Lattice gas algorithms (LGA) are a class of algorithms including, in chronological order, binary lattice gas cellular automata (LGCA), integer lattice gas algorithms (ILGA) and lattice Boltzmann method (LBM). They are largely used for simulating non-linear systems. Starting from 1-dimensional ILGA, we design an algorithm where we carry out a fraction of the possible collisions. These fractions are then adapted to reproduce LBM equilibrium distributions, resulting in an adaptive lattice gas algorithm (ALGA) that achieves the same simulation results of LBM. Considering this, we develop a quantum algorithm that involves a linear collision operator and capable of simulating the same phenomena, while still using a measurement and reinitialization procedure. Multi-time-step implementation is possible in some specific cases briefly discussed.

Figures

Figures reproduced from arXiv: 2504.13549 by the authors.

Figure 1
Figure 1. FIG. 1. Lattice gas Cellular Automata example of evolu [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum circuit for executing the algorithm. After [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Equilibrium distribution functions averaged over t [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison between Q-ALGA(circles) with constant c [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Equilibrium distribution functions for D1Q3 with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison between QALGA(circles) with local up [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Works this paper leans on

54 extracted references · 46 canonical work pages

  1. [1]

    Rivet and J

    J.-P. Rivet and J. P. Boon, Lattice Gas Hydrodynamics , Cambridge Nonlinear Science Series (Cambridge Univer- sity Press, 2001)

    work page 2001

  2. [2]

    When λc is constant (not in the local adaptation case), the condition for having time-step concatenation is min( n0, n1) = n0 or min( n0, n1) = n1 in any cell at any time

    This algorithm can achieve time-step concatenation, avoiding measurement and reinitialization, when the amplitude of the state |11⟩ can be calculated unitarily. When λc is constant (not in the local adaptation case), the condition for having time-step concatenation is min( n0, n1) = n0 or min( n0, n1) = n1 in any cell at any time. If the adap- tation is l...

    work page

  3. [3]

    Frisch, D

    U. Frisch, D. d’Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics in two and three dimensions, in Lattice Gas Methods for Partial Differential Equations (CRC Press, 2019) pp. 77– 136

    work page 2019

  4. [4]

    Kruger, Lattice Boltzmann Method-Principles and Practice (Springer International Publish, 2016)

    T. Kruger, Lattice Boltzmann Method-Principles and Practice (Springer International Publish, 2016)

    work page 2016

  5. [5]

    Mohamad, Lattice boltzmann method , Vol

    A. Mohamad, Lattice boltzmann method , Vol. 70 (Springer, 2011)

    work page 2011

  6. [6]

    Succi, The lattice Boltzmann equation: for fluid dy- namics and beyond (Oxford university press, 2001)

    S. Succi, The lattice Boltzmann equation: for fluid dy- namics and beyond (Oxford university press, 2001)

    work page 2001

  7. [7]

    (A4) From here on, we substitute the Chapman-Enskog ex- pansion in Eq

    and [5] we have the following Taylor expansion fi(x +ci∆t, t + ∆t) − fi(x, t) = ∆ t ( ∂fi ∂t + ci ∂fi ∂x ) + + ∆t2 2 ( ∂2fi ∂t2 + 2ci ∂2fi ∂t∂x + c2 i ∂2fi ∂x2 ) + o(∆t3).(A1) The Chapman-Enskog expansion corresponds to fi = f eq i + ǫf (1) i + ǫ2f (2) i + o(ǫ3), (A2) ∂ ∂t − →ǫ ∂(1) ∂t + ǫ2 ∂(2) ∂t + o(ǫ3), (A3) ∂ ∂x − →ǫ ∂(1) ∂x . (A4) From here on, we s...

    work page

  8. [8]

    S. S. Bharadwaj and K. R. Sreenivasan, Quan- tum computation of fluid dynamics, arXiv preprint arXiv:2007.09147 15, 1 (2020)

    arXiv 2020

Show all 54 references
  1. [9]

    D. A. Wolf-Gladrow, Lattice-gas cellular automata and lattice Boltzmann models: an introduction (Springer, 2004)

    2004

  2. [10]

    [001] [101] [100] [011] [101]

  3. [11]

    [100] [101] [001] [110] [001] x 0 1 2 3 4 5 Before collision After collision After streaming FIG. 1. Lattice gas Cellular Automata example of evolu- tion for 1-dimensional model with 3 velocities named D1Q3. Each cell consists of 3 bits [ b2b1b0], representing the pres- ence o...

  4. [12]

    B. M. Boghosian and W. Taylor IV, Quantum lattice-gas model for the many-particle schr¨ odinger equation in d dimensions, Physical Review E 57, 54 (1998)

    1998

  5. [13]

    Yepez, Lattice-gas quantum computation, Interna- tional Journal of Modern Physics C 9, 1587 (1998)

    J. Yepez, Lattice-gas quantum computation, Interna- tional Journal of Modern Physics C 9, 1587 (1998)

    1998

  6. [14]

    Succi, The lattice Boltzmann equation: for complex states of flowing matter (Oxford university press, 2018)

    S. Succi, The lattice Boltzmann equation: for complex states of flowing matter (Oxford university press, 2018)

    2018

  7. [15]

    Yepez, Quantum lattice-gas model for computational fluid dynamics, Physical Review E 63, 046702 (2001)

    J. Yepez, Quantum lattice-gas model for computational fluid dynamics, Physical Review E 63, 046702 (2001)

    2001

  8. [16]

    Succi and R

    S. Succi and R. Benzi, Lattice boltzmann equation for quantum mechanics, Physica D: Nonlinear Phenomena 69, 327 (1993)

    1993

  9. [17]

    D. A. Meyer, From quantum cellular automata to quan- tum lattice gases, Journal of Statistical Physics 85, 551 (1996)

    1996

  10. [18]

    In this simulation we are well within the limits for which the collisions take place in each site

    We must notice that this encoding does not allow to avoid collisions when λc is greater than one or negative, as we do in the classical al- gorithm. In this simulation we are well within the limits for which the collisions take place in each site. However, this is not guarante...

  11. [19]

    D. A. Meyer, Quantum mechanics of lattice gas au- tomata: One-particle plane waves and potentials, Physi- cal Review E 55, 5261 (1997)

    1997

  12. [20]

    Itani and S

    W. Itani and S. Succi, Analysis of carleman linearization of lattice boltzmann, Fluids 7, 24 (2022)

    2022

  13. [21]

    Yepez, Quantum lattice-gas model for the diffusion equation, International Journal of Modern Physics C 12, 1285 (2001)

    J. Yepez, Quantum lattice-gas model for the diffusion equation, International Journal of Modern Physics C 12, 1285 (2001)

    2001

  14. [22]

    Sanavio and S

    C. Sanavio and S. Succi, Quantum lattice boltzmann- carleman algorithm, arXiv preprint arXiv:2310.17973 (2023)

    2023 arXiv

  15. [23]

    Fonio, P

    N. Fonio, P. Sagaut, and G. Di Molfetta, Quantum colli- sion circuit, quantum invariants and quantum phase esti- mation procedure for fluid dynamic lattice gas automata, arXiv preprint arXiv:2310.07362 (2023)

    2023 arXiv

  16. [24]

    A. D. B. Zamora, L. Budinski, O. Niemim¨ aki, and V. Lahtinen, Efficient quantum lattice gas automata, Computers & Fluids 286, 106476 (2025)

    2025

  17. [25]

    Budinski, Quantum algorithm for the advection– diffusion equation simulated with the lattice boltzmann method, Quantum Information Processing 20, 57 (2021)

    Lj. Budinski, Quantum algorithm for the advection– diffusion equation simulated with the lattice boltzmann method, Quantum Information Processing 20, 57 (2021)

    2021

  18. [26]

    Lj. Budinski, Quantum algorithm for the navier–stokes equations by using the streamfunction-vorticity formu- lation and the lattice boltzmann method, International Journal of Quantum Information 20, 2150039 (2022)

    2022

  19. [27]

    B. M. Boghosian, J. Yepez, F. J. Alexander, and N. H. Margolus, Integer lattice gases, Physical Review E 55, 4137 (1997)

    1997

  20. [28]

    Itani, K

    W. Itani, K. R. Sreenivasan, and S. Succi, Quantum algo- rithm for lattice boltzmann (qalb) simulation of incom- pressible fluids with a nonlinear collision term, Physics of Fluids 36 (2024)

    2024

  21. [29]

    Masselot and B

    A. Masselot and B. Chopard, A multiparticle lattice-gas model for hydrodynamics, International Journal of Mod- ern Physics C 9, 1221 (1998)

    1998

  22. [30]

    Sanavio and S

    C. Sanavio and S. Succi, Lattice boltzmann–carleman quantum algorithm and circuit for fluid flows at mod- erate reynolds number, A VS Quantum Science 6 (2024)

    2024

  23. [31]

    Succi, F

    S. Succi, F. Fillion-Gourdeau, and S. Palpacelli, Quan- tum lattice boltzmann is a quantum walk, EPJ Quantum Technology 2, 1 (2015)

    2015

  24. [32]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics 16, 1050 (2020)

    2020

  25. [33]

    M. A. Schalkers and M. M¨ oller, On the importance of data encoding in quantum boltzmann methods, Quan- tum Information Processing 23, 20 (2024)

    2024

  26. [34]

    Seekins and A

    N. Seekins and A. J. Wagner, Integer lattice gas with a sampling collision operator for the fluctuating diffusion equation, Physical Review E 105, 035303 (2022)

    2022

  27. [35]

    Chopard, A

    B. Chopard, A. Masselot, and M. Droz, Multiparticle lat- tice gas model for a fluid: application to ballistic annihi- lation, Physical review letters 81, 1845 (1998)

    1998

  28. [36]

    d’Humi` eres, Multiple–relaxation–time lattice boltz- mann models in three dimensions, Philosophical Trans- actions of the Royal Society of London

    D. d’Humi` eres, Multiple–relaxation–time lattice boltz- mann models in three dimensions, Philosophical Trans- actions of the Royal Society of London. Series A: Math- ematical, Physical and Engineering Sciences 360, 437 (2002)

    2002

  29. [37]

    Blommel and A

    T. Blommel and A. J. Wagner, Integer lattice gas with monte carlo collision operator recovers the lattice boltzmann method with poisson-distributed fluctuations, Physical Review E 97, 023310 (2018)

    2018

  30. [38]

    using an ancilla. The streaming step is executed with the following operator ˆS = ∑ x |x + 1⟩ ⟨x| ⊗ |01⟩ ⟨01| + |x − 1⟩ ⟨x| ⊗ |10⟩ ⟨10| , (20) that can be efficiently carried out using the shift algo- rithm described in [33]. This shift operation, however, does not transport the...

    2000

  31. [39]

    A. J. Wagner and K. Strand, Fluctuating lattice boltz- mann method for the diffusion equation, Physical Review E 94, 033302 (2016)

    2016

  32. [40]

    Strand and A

    K. Strand and A. J. Wagner, Overrelaxation in a diffu- sive integer lattice gas, Physical Review E 105, L063301 (2022)

    2022

  33. [41]

    Shakeel, Efficient and scalable quantum walk algo- rithms via the quantum fourier transform, Quantum In- formation Processing 19, 323 (2020)

    A. Shakeel, Efficient and scalable quantum walk algo- rithms via the quantum fourier transform, Quantum In- formation Processing 19, 323 (2020)

    2020

  34. [42]

    P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems, Physical review 94, 511 (1954)

    1954

  35. [43]

    T. Xin, D. Lu, J. Klassen, N. Yu, Z. Ji, J. Chen, X. Ma, G. Long, B. Zeng, and R. Laflamme, Quantum state to- mography via reduced density matrices, Physical review letters 118, 020401 (2017)

    2017

  36. [44]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010)

    2010

  37. [45]

    A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, arXiv preprint arXiv:1202.5822 (2012)

    2012 arXiv

  38. [46]

    Razzoli, G

    L. Razzoli, G. Cenedese, M. Bondani, and G. Benenti, Efficient implementation of discrete-time quantum walks on quantum computers, Entropy 26, 313 (2024)

    2024

  39. [47]

    Budinski, O

    L. Budinski, O. Niemim¨ aki, R. Zamora-Zamora, and V. Lahtinen, Efficient parallelization of quantum basis state shift, Quantum Science and Technology 8, 045031 12 (2023)

    2023

  40. [48]

    M. G. Paris, Quantum estimation for quantum technol- ogy, International Journal of Quantum Information 7, 125 (2009)

    2009

  41. [49]

    D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Physical Review A 64, 052312 (2001)

    2001

  42. [51]

    Binosi, G

    D. Binosi, G. Garberoglio, D. Maragnano, M. Dapor, and M. Liscidini, A tailor-made quantum state tomography approach, APL Quantum 1 (2024)

    2024

  43. [52]

    Gross, Y.-K

    D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis- ert, Quantum state tomography via compressed sensing, Physical review letters 105, 150401 (2010)

    2010

  44. [53]

    Aaronson, Shadow tomography of quantum states, in Proceedings of the 50th annual ACM SIGACT sympo- sium on theory of computing (2018) pp

    S. Aaronson, Shadow tomography of quantum states, in Proceedings of the 50th annual ACM SIGACT sympo- sium on theory of computing (2018) pp. 325–338

    2018

  45. [54]

    Zhang and X

    X.-M. Zhang and X. Yuan, Circuit complexity of quan- tum access models for encoding classical data, npj Quan- tum Information 10, 42 (2024)

    2024

  46. [101]

    [001] [010] [100] [011] [010]

Pith tools

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