arxiv: 2604.05577 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Resource Implications of Different Encodings for Quantum Computational Fluid Dynamics

Hans A. K\"osel , Roland Ewert , Jan W. Delfs

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:55 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum computingamplitude encodingcomputational fluid dynamicslattice Boltzmann methodresource estimationstate preparationmeasurement scalingquantum algorithms

0 comments

The pith

The upper bound on executions for amplitude encoding in quantum CFD is too conservative, with empirical evidence suggesting milder scaling that enables a new encoding for the lattice Boltzmann method.

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

The paper examines the initialization and readout costs of amplitude encoding when quantum algorithms must compute entire fields of values, as in computational fluid dynamics. It derives the circuit depth required for state preparation and places an upper bound on the number of executions needed to extract the encoded values to a target accuracy. Although this bound scales quadratically with the number of values times the logarithm, simulations for the equal-probability reference case indicate a gentler linear-log scaling. These observations lead the authors to propose a new encoding approach designed specifically for a quantum algorithm solving the lattice Boltzmann method.

Core claim

The paper claims that standard theoretical bounds on the resources for amplitude encoding in quantum CFD are overly conservative, as empirical study of equal-probability distributions shows the number of executions required for accurate readout follows a scaling of ñ ln(ñ) rather than the derived ñ² ln(ñ), and that the resulting resource insights directly motivate a new encoding tailored to quantum lattice Boltzmann algorithms.

What carries the argument

The empirical scaling law for the number of executions needed to extract encoded values with given accuracy from equal-probability distributions, which informs the proposal of a new encoding approach for quantum lattice Boltzmann methods.

If this is right

Quantum CFD algorithms can operate with lower execution overhead than theoretical upper bounds predict.
Readout of encoded field values becomes feasible with a number of runs that grows only as ñ ln(ñ) in the uniform case.
A specialized encoding can be constructed for lattice Boltzmann methods that exploits the milder empirical scaling.
Resource estimates for quantum algorithms computing entire fluid fields become more realistic and potentially closer to practical implementation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The milder scaling may extend to other quantum algorithms that compute dense fields of values, such as electromagnetic or structural simulations.
Testing the new encoding on small-scale fluid problems could reveal whether the empirical scaling persists when the probability distribution deviates from uniform.
Direct resource comparisons between the new encoding and other proposed quantum CFD approaches would clarify its relative efficiency.

Load-bearing premise

That resource insights from general amplitude encoding and the equal-probability case translate directly into a practical and advantageous new encoding for CFD applications without further validation.

What would settle it

A quantum lattice Boltzmann simulation using the proposed encoding in which the observed number of executions required for target accuracy follows the higher quadratic-log scaling or shows no advantage over standard amplitude encoding.

Figures

Figures reproduced from arXiv: 2604.05577 by Hans A. K\"osel, Jan W. Delfs, Roland Ewert.

**Figure 2.** Figure 2: Quantum circuit schematics of a procedure for disentangling subsequently all qubits of a state [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Quantum circuit schematics for a decomposition [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of the recursive decomposition of a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration of the application of the decomposition in Fig. 3 for the disentangling structure in Fig. 1, where [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Beginning of the gate pattern for the general initialization procedure for amplitude encoding. The blue [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Circuit depth contributions due to the Rj -rotation gates as the native 1-qubit gates and due to the CX-gates for the general initialization procedure for amplitude encoding. CX-gates, it results Xn i=0 (2 · 2 i − 2) = 2Xn i=0 2 i − 2 · (n + 1) = 2 · 1 − 2 n+1 1 − 2 − 2 · (n + 1) = 2n+2 − 2 · (n + 2), (14) which is also the circuit depth contribution of them, because the CX-gates separate the Rj -gates si… view at source ↗

**Figure 8.** Figure 8: Required number of shots N according to equality in formula (21) for the plots (a) and (c) and according to equality in formula (22) for the plots (b) and (d). III. EMPIRICAL STUDIES FOR GENERAL AMPLITUDE ENCODING To verify the claims for the initialization time according to formula (16) and the upper bound for the required number of runs (22), corresponding simulation experiments were conducted, which a… view at source ↗

**Figure 9.** Figure 9: Execution times for the initialization of a general [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 10.** Figure 10: Exemplary results of a single experiment run for studying the read-out accuracy. The number of shots [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗

**Figure 11.** Figure 11: Required number of shots N to read out the probability distribution in the situation that all n˜ = 2n probabilities are equal and shall be read out up to a relative error ϵ = 0.1, respectively, where a success probability of 1 − δ = 0.5 was demanded. The values denoted by ’empirical’ are the found N(n) in [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

**Figure 12.** Figure 12: Illustration of the subproblems in the LBM [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗

**Figure 13.** Figure 13: Example for the general circuit pattern for the implementation of a mapping from input bitstrings to [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗

**Figure 14.** Figure 14: Specific circuit implementation of the mapping [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗

**Figure 15.** Figure 15: Results obtained from the statevector simulation for the circuits of the Figs. 13 and 14, illustrating the approximate realization of the function x 2 according to [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗

**Figure 16.** Figure 16: Circuit schematics of the BV quantum algorithm for the example of the secret 6-bit code ’101000’. The dashed blue box encomprises an implementation of the BV oracle. initial field with the time according to φ(x, t) = φ(x − ct, t0). (A2) A simple numerical treatment of eq. (A1) is the upwind discretization w.r.t. the space and the explicit Euler scheme w.r.t. the time, which reads: φ(xi , tn + ∆t) (A3) = … view at source ↗

**Figure 17.** Figure 17: Minimal example for the implementation of the solving procedure (A4) for the linear advection eq. (A1) [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗

**Figure 18.** Figure 18: Results obtained from the statevector simulation of an extended minimal example according to the schemes [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗

**Figure 19.** Figure 19: Analytical results according to the formula (B12) for the success probability [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗

read the original abstract

For quantum algorithms for problems in which the task is to compute an entire field of values, like e.g. computational fluid dynamics (CFD), it is often proposed amplitude encoding w.r.t. multiple qubits; however, the efforts implied by it for initialization and read-out are not addressed. This work is devoted specifically to this issue: It reviews different encoding schemes in quantum computing, discussing their computational costs for initialization and read-out as well as resulting aspects for their usage via minimal examples. The considerations in previous literature on the required computational resources for amplitude encoding w.r.t. multiple qubits are extended in the presented quantification by explicitly deducing the circuit depth that results for the decomposed initialization procedure of V. V. Shende et al. [1, 2] and deriving an upper bound for the necessary number of executions of a quantum algorithm to extract the encoded values with a specific accuracy. For these two results, an empirical verification via the means provided by IBM's quantum computing simulation framework $\textit{Qiskit}$ [3] is given. In the framework of the study on the required number of runs to achieve a desired accuracy, it is however found that the derived upper bound, scaling like $ {{\tilde{n}}^2} ~ {\ln( {\tilde{n}} )} $ with the number of encoded values $ {\tilde{n}} $, is too conservative to be used for precise estimations. Therefore, a corresponding study of the required runs for the reference distribution of equal probabilities for all basis states is done in particular, which suggests $ {\tilde{n}} ~ { \ln( {\tilde{n}} ) } $ as an empirical scaling law. Since the view regarding CFD applications is taken here, it is presented in particular that the insights from this work lead to a new encoding approach, which is proposed specifically for a quantum algorithm for the lattice Boltzmann method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

read the letter

This paper gives a concrete resource analysis for amplitude encoding in quantum CFD with Qiskit-backed scaling insights, but the new LBM encoding rests on an untested jump from uniform to realistic non-uniform distributions. It reviews several encoding schemes, works out the circuit depth for the Shende initialization procedure, and derives an upper bound on the number of algorithm executions needed to reach a target accuracy. The Qiskit simulations then show that bound is loose, and for the uniform-probability reference case they observe an empirical scaling closer to ñ ln(ñ). Those numbers lead directly to the proposed new encoding for lattice Boltzmann methods. The explicit depth calculation and the simulation check are the parts that hold up best; they turn an abstract cost discussion into something you can actually use for planning. The review with minimal examples is also clear enough to be helpful for readers who are not already deep in the subfield. The soft spot is the move from the uniform-case result to the CFD-specific proposal. Lattice Boltzmann states are not uniform; they carry the structure of the flow field, with peaks and spatial variation. The paper does not report run counts, depth measurements, or accuracy checks on any non-uniform example taken from a simple flow. Without that step the claimed advantage for quantum LBM stays speculative. This is the kind of incremental resource study that people working on quantum algorithms for fluids will want to see. The calculations and simulations are grounded enough to justify sending it to referees, even though the CFD application claims will need more direct validation.

Referee Report

1 major / 0 minor

Summary. The manuscript reviews quantum encoding schemes for applications such as computational fluid dynamics, focusing on the initialization and readout costs of amplitude encoding. It extends prior work by deriving the circuit depth of the Shende et al. initialization procedure and an upper bound scaling as ñ² ln(ñ) on the number of executions required to extract encoded values to a target accuracy. Qiskit simulations empirically verify that this bound is conservative; for the uniform (equal-probability) reference distribution the observed scaling is closer to ñ ln(ñ). These observations are used to motivate and propose a new encoding scheme specifically tailored to a quantum lattice Boltzmann method algorithm.

Significance. If the empirical scaling improvement and the proposed LBM encoding can be shown to persist under the non-uniform probability distributions that arise in actual fluid simulations, the work would supply concrete, actionable resource estimates that could guide the design of practical quantum CFD algorithms. The explicit circuit-depth calculation and the Qiskit verification of the initialization procedure are useful contributions to the literature on quantum state preparation.

major comments (1)

[section proposing new LBM encoding] The proposal of the new encoding approach for the quantum lattice Boltzmann method (final section) is motivated by the ñ ln(ñ) empirical scaling obtained exclusively for the equal-probability reference distribution. Lattice Boltzmann velocity distributions are typically non-uniform and spatially structured; without a corresponding run-count or circuit-depth analysis for at least one representative CFD state (e.g., Poiseuille or lid-driven cavity), the claim that the new encoding yields a practical advantage remains an untested extrapolation and is load-bearing for the central contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important limitation in the motivation for the proposed encoding. We address the comment below and have revised the manuscript to incorporate the feedback.

read point-by-point responses

Referee: [section proposing new LBM encoding] The proposal of the new encoding approach for the quantum lattice Boltzmann method (final section) is motivated by the ñ ln(ñ) empirical scaling obtained exclusively for the equal-probability reference distribution. Lattice Boltzmann velocity distributions are typically non-uniform and spatially structured; without a corresponding run-count or circuit-depth analysis for at least one representative CFD state (e.g., Poiseuille or lid-driven cavity), the claim that the new encoding yields a practical advantage remains an untested extrapolation and is load-bearing for the central contribution.

Authors: We agree that the empirical scaling ñ ln(ñ) was obtained specifically for the uniform reference distribution, while the general upper bound ñ² ln(ñ) derived in the manuscript applies to arbitrary distributions. The new LBM encoding is motivated by the overall resource costs of amplitude encoding (initialization depth from the Shende decomposition plus readout overhead) quantified earlier in the paper, rather than solely by the uniform-case empirical result. Nevertheless, we acknowledge that the practical advantage for the non-uniform, spatially structured distributions that arise in actual lattice Boltzmann simulations has not been verified through explicit run-count or circuit-depth analysis on representative states such as Poiseuille flow or lid-driven cavity. In the revised manuscript we have updated the final section to state explicitly that the proposed encoding constitutes a conceptual direction whose resource savings under realistic CFD distributions remain to be quantified and that further numerical studies on non-uniform cases are required before a definitive practical advantage can be claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations use external references and independent empirical checks

full rationale

The paper derives a theoretical upper bound on the number of executions (scaling as ñ² ln(ñ)) from the Shende et al. initialization procedure and verifies it via Qiskit simulations. A separate empirical study on the equal-probability reference case yields the ñ ln(ñ) scaling as an observed pattern from direct simulation, not by fitting parameters to the paper's own bound or results. The new LBM encoding proposal is motivated by these insights without any self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation. All load-bearing steps reference external sources (Shende procedure, Qiskit) or standalone simulation data, keeping the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The analysis rests on standard quantum circuit decomposition assumptions and statistical sampling for readout; the new encoding is introduced without independent evidence beyond the resource study.

axioms (2)

domain assumption State preparation circuit depth follows the decomposition of Shende et al.
Invoked to deduce explicit circuit depth for amplitude encoding.
standard math Measurement statistics follow standard quantum projective measurement rules for probability extraction.
Used to derive the upper bound on number of executions.

invented entities (1)

New encoding approach for quantum lattice Boltzmann method no independent evidence
purpose: To reduce resource overhead in CFD applications based on the encoding cost analysis
Proposed as a direct outcome of the resource study but lacks independent falsifiable evidence or validation in the abstract.

pith-pipeline@v0.9.0 · 5645 in / 1375 out tokens · 61412 ms · 2026-05-10T19:55:56.799192+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 6 canonical work pages · 2 internal anchors

[1]

Bitstring Encoding For bitstring encoding, it is known that there is either the state|0⟩or|1⟩for every qubit, so that the state cor- responds to a classical configuration of the computation system. For a machine with ideal measurement opera- tions, the accuracy of the solution in bitstring encoding should be determined therefore just by the number of qubi...

1988
[2]

Representing Non-linear Functions On the one hand, the evolution of a quantum system with time is mediated by the time-evolution operator, which is set as a unitary and thus linear operator. On the other hand, measurement processes correspond to Hermi- tian projection operators, which set the system state to an eigenstate of the operator for the measured ...
[3]

Fig. 13 depicts as well that if it is needed that the input ports are the output ports, the qubit states can be exchanged afterwards via SWAP-gates and the an- cilla qubits can then be reset back to the|0⟩-states from |3rd digit⟩ |2nd digit⟩ |1st digit⟩ |0⟩ reset to|0⟩ |0⟩ X reset to|0⟩ Fig. 14: Specific circuit implementation of the mapping of Table 6 ba...
[4]

A Mapping of the Linear Advection Equation to the Bernstein-Vazirani Quantum Algorithm For the potential of QC w.r.t. bitstring encoding, often the conceptual possibility is mentioned that QC allows to feed in all classically possible inputs in parallel and in principle also operate on them simultaneously. A situa- tion that illustrates this relatively we...
[5]

V. V. Shende, S. S. Bullock, and I. L. Markov,Syn- thesis of quantum-logic circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Sys- tems25, 6, pp. 1000-1010 (2006)

2006
[6]

V. V. Shende, S. S. Bullock, and I. L. Markov,Synthesis of quantum logic circuits, Proceedings of the 2005 Asia and South Pacific Design Automation Conference173, pp. 272-275 (2005)

2005
[7]

Quantum computing with Qiskit

A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Na- tion, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta,Quantum computing with Qiskit, arXiv:2405.08810v3 [quant-ph] (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024
[8]

Succi, W

S. Succi, W. Itani, K. Sreenivasan, and R. Steijl,Quan- tum computing for fluids: Where do we stand?, EPL144, 10001 (2023)

2023
[9]

K. P. Griffin, S. S. Jain, T. J. Flint, and W. H. R. Chan,Investigation of quantum algorithms for direct nu- merical simulation of the Navier-Stokes equations, Cen- ter for Turbulence Research Annual Research Briefs, pp. 347–363 (Center for Turbulence Research, Stanford Uni- versity, 2019)

2019
[10]

M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information(10th anniversary edi- tion, Cambridge University Press, 2010)

2010
[11]

Wawrzyniak, J

D. Wawrzyniak, J. Winter, S. Schmidt, T. Indinger, C. F. Janßen, U. Schramm, and N. A. Adams,A quantum algorithm for the lattice-Boltzmann method advection- diffusion equation, Computer Physics Communications 306, 109373 (2025)

2025
[12]

L. Budinski,Quantum algorithm for the Navier Stokes equations by using the streamfunction vortic- ity formulation and the lattice Boltzmann method, arXiv:2103.03804v2 [quant-ph] (2022)

work page arXiv 2022
[13]

Wawrzyniak, J

D. Wawrzyniak, J. Winter, S. Schmidt, T. Indinger, U. Schramm, C. Janßen, and N. A. Adams,Unitary quan- tum algorithm for the lattice-Boltzmann method, 16th World Congress in Computational Mechanics (WCCM), Vancouver, Canada, 2024

2024
[14]

P. Over, S. Bengoechea, P. Brearley, S. Laizet, and T. Rung,Quantum algorithm for the advection-diffusion equation by direct block encoding of the time-marching operator, Phys. Rev. A112, L010401 (2025)

2025
[15]

Quantum time-marching algorithms for solving linear transport problems including boundary conditions

S. Bengoechea, P. Over, and T. Rung,Quantum time- marching algorithms for solving linear transport prob- lems including boundary conditions, arXiv:2511.04271v1 [quant-ph] (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[16]

Nagel and J

A. Nagel and J. Löwe,Quantum lattice boltzmann method for multiple time steps without reinitialization for linear advection-Diffusion problems, Computer Physics Com- munications321, 110040 (2026)

2026
[17]

Kocherla, Z

S. Kocherla, Z. Song, F. Ezahra Chrit, B. Gard, E. F. Du- mitrescu, A. Alexeev, and S. H. Bryngelson,Fully quan- tum algorithm for mesoscale fluid simulations with ap- plication to partial differential equations, AVS Quantum Sci.6, 033806 (2024)

2024
[18]

Krüger, H

T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, E. M. Viggen,The Lattice Boltzmann Method (Springer, 2017)

2017
[19]

Succi,The Lattice Boltzmann Equation – For Com- plex States of Flowing Matter(Oxford University Press, 2022)

S. Succi,The Lattice Boltzmann Equation – For Com- plex States of Flowing Matter(Oxford University Press, 2022)

2022
[20]

Succi,The Lattice Boltzmann Equation – For Fluid Dynamics and Beyond(Oxford University Press, 2013)

S. Succi,The Lattice Boltzmann Equation – For Fluid Dynamics and Beyond(Oxford University Press, 2013)

2013
[21]

M. A. Schalkers and M. Möller,On the importance of data encoding in quantum Boltzmann methods, Quantum Inf. Process.23, 20 (2024)

2024
[22]

A. M. Childs and N. Wiebe,Hamiltonian simulation us- ing linear combinations of unitary operations, Quantum Information and Computation12, No. 11 & 12 (2012)

2012
[23]

Veltheim and E

O. Veltheim and E. Keski-Vakkuri,Optimizing Quantum Measurements by Partitioning Multisets of Observables, Phys. Rev. Lett.134, 030801 (2025)

2025
[24]

Veltheim and E

Supplemental Material of O. Veltheim and E. Keski- Vakkuri,Optimizing Quantum Measurements by Parti- tioning Multisets of Observables, Phys. Rev. Lett.134, 030801 (2025)

2025
[25]

Adedoyin, J

Abhijith J., A. Adedoyin, J. Ambrosiano, P. Anisimov, W. Casper, G. Chennupati, C. Coffrin, H. Djidjev, D. Gunter, S. Karra, N. Lemons, S. Lin, A. Malyzhenkov, 28 D. Mascarenas, S. Mniszewski, B. Nadiga, D. O’malley, D. Oyen, S. Pakin, L. Prasad, R. Roberts, P. Romero, N. Santhi, N. Sinitsyn, P. J. Swart, J. G. Wendelberger, B. Yoon, R. Zamora, W. Zhu, S....

2022
[26]

https://quantum.cloud.ibm.com/docs/en/guides/set- optimization (accessed 12 February 2026, 10:30 a.m. UTC+1)

2026
[27]

Tennie, S

F. Tennie, S. Laizet, S. Lloyd, and L. Magri,Quantum computing for nonlinear differential equations and turbu- lence, Nature Reviews Physics7, pp. 220–230 (2025)

2025
[28]

Kandala, K

A. Kandala, K. X. Wei, S. Srinivasan, E. Magesan, S. Carnevale, G. A. Keefe, D. Klaus, O. Dial, and D. C. McKay,Demonstration of a High-Fidelity CNOT Gate for Fixed-Frequency Transmons with Engineered ZZ Sup- pression, Phys. Rev. Lett.127, 130501 (2021)

2021
[29]

Tutorial on the quantikz package

A. Kay,Tutorial on the Quantikz Package, arXiv:1809.03842v7 [quant-ph] (2023)

work page arXiv 2023
[30]

Jaksch, P

D. Jaksch, P. Givi, A. J. Daley, and T. Rung,Variational Quantum Algorithms for Computational Fluid Dynam- ics, AIAA Journal61, 5, pp. 1885–1894 (2023)

2023
[31]

Quantum algorithm for nonlinear differential equations.arXiv preprint arXiv:2011.06571, 2020

S. Lloyd, G. De Palma, C. Gokler, B. Kiani, Z.- W. Liu, M. Marvian, F. Tennie, and Tim Palmer, Quantum algorithm for nonlinear differential equations, arXiv:2011.06571v2 [quant-ph] (2020)

work page arXiv 2011
[32]

Lăcătuş and M

M. Lăcătuş and M. Möller,Surrogate Quantum Circuit Design for the Lattice Boltzmann Collision Operator, arXiv:2507.12256v2 [quant-ph] (2025)

work page arXiv 2025