pith. machine review for the scientific record. sign in

arxiv: 2605.00302 · v2 · submitted 2026-05-01 · 🪐 quant-ph

Recognition: no theorem link

Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation

Abeynaya Gnanasekaran, Amit Surana, Daniel Gunlycke, Reuben Demirdjian, Thomas Hogancamp

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum linear combination of unitariesCarleman linearizationlattice Boltzmann equationquantum data loadingnon-unitary decompositionvariational quantum linear solverquantum simulation of fluids
0
0 comments X

The pith

A decomposition of any square matrix into non-unitaries yields an LCU for Carleman-linearized systems whose term count depends only on truncation order and velocity count.

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

The paper introduces a general method to express an arbitrary square matrix as a linear combination of non-unitaries, each of which is then embedded into a unitary, producing an LCU with the same number of terms. This construction supplies a framework that works for any autonomous dynamical system that has been linearized via Carleman truncation of a polynomial nonlinearity. When the framework is specialized to the three-dimensional lattice-Boltzmann equation, the resulting LCU contains Ns terms that scale only as O(α² Q²) and remain completely independent of the number of spatial grid points and time steps. Because the term count does not grow with discretization size, the approach keeps the resources needed for quantum data loading from exploding as simulations are refined. The authors also supply explicit T-gate cost estimates when the LCU is used with both block-encoding oracles and the variational quantum linear solver.

Core claim

An arbitrary square matrix can be written as a linear combination of non-unitaries (LCNU) in which each non-unitary is embedded into a unitary, thereby producing an LCU with exactly the same number of terms. The resulting LCU construction applies to any Carleman-linearized autonomous system whose nonlinearity is polynomial. For the three-dimensional Carleman-linearized lattice-Boltzmann equation the number of terms satisfies Ns ∼ O(α² Q²), where α is the truncation order and Q is the number of discrete velocities; this count is independent of both the spatial grid size n and the number of time steps nt.

What carries the argument

The linear combination of non-unitaries (LCNU) decomposition, each term of which is embedded into a unitary to form an LCU whose size is fixed by the Carleman order α and the velocity set size Q rather than by the discretization parameters.

If this is right

  • The quantum resources required to load the LBE system into a quantum computer remain bounded even when the spatial and temporal grids are made arbitrarily fine.
  • The same LCNU-to-LCU construction supplies an efficient data-loading method for any other Carleman-linearized polynomial dynamical system.
  • When combined with PREP and SELECT oracles the T-gate cost scales as O(α³ Q² (log₂ n)²).
  • When used inside the variational quantum linear solver the method requires Ns² circuits per iteration, each with worst-case T cost O(α (log₂ Q n)²).

Where Pith is reading between the lines

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

  • The independence of Ns from discretization size suggests that the method could be applied to other high-resolution fluid or plasma simulations that admit a Carleman linearization.
  • If the LCNU construction generalizes beyond the lattice-Boltzmann case, it may reduce the qubit and gate overhead for quantum simulation of any nonlinear PDE that is first linearized by polynomial truncation.
  • The approach separates the cost of data loading from the cost of time evolution, which could allow hybrid classical-quantum workflows in which only the loading step uses the new LCU.

Load-bearing premise

That the Carleman-linearized system matrix can always be decomposed into a linear combination of O(α² Q²) non-unitary operators, each of which can be embedded into a unitary without introducing additional factors that depend on the number of grid points or time steps.

What would settle it

An explicit matrix decomposition for the three-dimensional Carleman-linearized LBE showing that the number of non-unitary terms grows with the spatial grid size n or the number of time steps nt.

Figures

Figures reproduced from arXiv: 2605.00302 by Abeynaya Gnanasekaran, Amit Surana, Daniel Gunlycke, Reuben Demirdjian, Thomas Hogancamp.

Figure 4.1
Figure 4.1. Figure 4.1: Embeddings for the two nontrivial L (e) 1 terms of (62): the emebedded L1,2 term (left) and the embedded L1,3 term (right). The |a⟩ wire is a single ancillary qubit required to embed each term into a unitary operation. A control operation or a single qubit gate on a multi-qubit register should be interpreted as the respective operation being applied to every wire within that register. The vertical dashed… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Embedding for L lin,1 λ as defined in (65b) using λ = (2, 1, 0, x, +1, m) for any m ∈ {1, . . . , NEη }, which is the most expensive circuit among any λ ∈ Λ1. The σf(η,m) block is the mth term in the Pauli decomposition of x-component of (52), and is therefore a tensor product of log Q Pauli gates. The S n +1 block is the incrementer circuit from (24). Then, by combining (25) with the nonlinear component… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Embedding for L lin,2 λ as defined in (68b) using λ = (2, 1, 0, m) for any m ∈ {1, . . . , NR}, which is the most expensive circuit among any λ ∈ Λ2. The σg(m) gate is the mth term in the Pauli decomposition used in (55), and is therefore a tensor product of log Q Pauli gates. The VR and WR circuits come from the SVD in (55) and depend on the specific lattice structure used. where ˜Ii is defined in Secti… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Embedding for L nlin λ as defined in (71b) using λ = (3, 2, α−2, α−3, q, m) for any m ∈ {1, . . . , NΓq } and q ∈ {1, . . . , Q}, which is the most expensive circuit among any λ ∈ Λ3. The circuits for P3, B3,q and the commutation matrix K(a,b) for integers a and b are provided in Appendices E.1, E.4 and E.5, respectively. The σh(q,m) gate is the mth term in the Pauli decomposition used in (59) and is the… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: T gate count to encode the Carleman linearized LBE matrix using the PREP and SELECT view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: The number of circuits (NoC) and maximum T cost per circuit to encode the Carleman linearized view at source ↗
read the original abstract

Herein, we introduce a strategy to decompose an arbitrary square matrix into a linear combination of non-unitaries (LCNU) where each non-unitary term is embedded into a unitary matrix. The result is a linear combination of unitaries (LCU) with an equal number of terms as the LCNU. Using this approach, we construct a generalized LCU framework for any Carleman linearized autonomous dynamical system with a polynomial nonlinearity. This framework is then used to construct an LCU for the 3-dimensional Carleman linearized lattice Boltzmann equation (LBE) in which the number of terms scales like $N_s \sim \mathcal{O}(\alpha^2 Q^2)$, where $\alpha$ is the Carleman truncation order and $Q$ is the number of discrete velocities from the LBE. Importantly, $N_s$ is completely independent of both the number of temporal and spatial discretization points. Lastly, we provide an estimate of our LCNU strategy's T gate cost in conjunction with (1) PREP and SELECT block encoding oracles, and (2) the variational quantum linear solver. In the former, the T cost scales like $\mathcal{O}(\alpha^3 Q^2 (\log_2 n)^2)$, where $n$ is the total number of spatial grid points across all dimensions. Next, the latter requires $N_s^2(\log_2 (2n_tn^\alpha)+1)$ circuits per iteration, with a worst case T gate cost of $\mathcal{O}(\alpha (\log_2 Qn)^2)$ among them. We, therefore, provide an efficient decomposition strategy useful for both fault-tolerant and variational approaches.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a decomposition of an arbitrary square matrix into a linear combination of non-unitaries (LCNU) that yields a linear combination of unitaries (LCU) with the same term count. This is used to build a generalized LCU framework for any Carleman-linearized autonomous dynamical system with polynomial nonlinearity, then specialized to the 3D Carleman-linearized lattice Boltzmann equation (LBE). The central claim is that the LCU term count scales as Ns ∼ O(α² Q²) and is independent of both spatial discretization points n and temporal points nt. T-gate cost estimates are supplied for PREP/SELECT block-encoding oracles (O(α³ Q² (log₂ n)²)) and for the variational quantum linear solver (Ns² circuits per iteration with per-circuit T-cost O(α (log₂ Q n)²)).

Significance. If the claimed independence of Ns from n and nt holds without hidden discretization-dependent overheads in the unitary embeddings, the work would supply a useful data-loading primitive for quantum simulation of nonlinear fluid dynamics via Carleman linearization. The generalization to arbitrary polynomial nonlinearities and the explicit scaling for the LBE, together with concrete resource estimates that span both fault-tolerant and variational regimes, are constructive contributions.

major comments (1)
  1. [Abstract] Abstract: the load-bearing claim that 'Ns is completely independent of both the number of temporal and spatial discretization points' is asserted without any derivation or explicit construction. The LCNU decomposition into O(α² Q²) non-unitaries for the Carleman operator (whose dimension scales as n^α) must be shown to admit block-encoding unitaries whose SELECT/PREP oracles introduce neither additional terms nor ancilla scaling with n; the manuscript supplies no such verification.
minor comments (1)
  1. [Abstract] Abstract: the symbols n (total spatial grid points) and nt (temporal points) appear without prior definition, and the relation between the Carleman truncation order α and the full system dimension is not stated explicitly before the cost expressions are given.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the central scaling claim. We address the major comment below and will incorporate the requested details into the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the load-bearing claim that 'Ns is completely independent of both the number of temporal and spatial discretization points' is asserted without any derivation or explicit construction. The LCNU decomposition into O(α² Q²) non-unitaries for the Carleman operator (whose dimension scales as n^α) must be shown to admit block-encoding unitaries whose SELECT/PREP oracles introduce neither additional terms nor ancilla scaling with n; the manuscript supplies no such verification.

    Authors: We agree that the abstract asserts the independence of Ns without a self-contained derivation and that the manuscript would benefit from an explicit construction showing that the block-encoding oracles do not introduce n-dependent overhead in the number of terms or ancilla count. In the revised manuscript we will add a dedicated subsection in Section 3 that derives the LCNU decomposition for the 3D Carleman-linearized LBE. The derivation proceeds by expressing the Carleman operator as a sum of terms indexed by pairs of velocity labels (arising from the quadratic collision operator and the finite set of Q discrete velocities) together with the appropriate spatial shift or identity operator at each Carleman order. Because the velocity space is finite and of size Q, and because the polynomial degree is bounded by α, the distinct velocity-index combinations produce at most O(α² Q²) non-unitary matrices; each such matrix is then block-encoded into a unitary by the addition of a single ancilla qubit whose control acts on the position register. The resulting LCU therefore contains exactly Ns = O(α² Q²) unitaries. The PREP oracle prepares a state over these Ns coefficients and requires only O(log Ns) ancilla qubits, independent of n. The SELECT oracle is a controlled application of the appropriate unitary indexed by the term label; because the index set size is independent of n, no additional terms are introduced and the ancilla overhead for the index register remains O(log(α² Q²)). The spatial shift operators themselves act on the log n position qubits but do not multiply the number of LCU terms. We will include this explicit construction, together with a small illustrative example for α=2, to substantiate the claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; LCU construction and term scaling derive from explicit decomposition and LBE operator structure.

full rationale

The paper introduces a novel LCNU-to-LCU embedding strategy for arbitrary square matrices, then applies it to the known polynomial form of any Carleman-linearized autonomous system. For the specific 3D LBE, the O(α² Q²) term count follows directly from indexing over Carleman orders (α) and discrete velocities (Q) in the streaming and collision operators; this counting is independent of grid size n by the locality of those operators. T-gate estimates use standard PREP/SELECT oracles and VQLS iteration formulas without any fitted parameters or self-referential definitions. No load-bearing step reduces by construction to its own inputs, and no self-citation chain is invoked for the central claims.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the new LCNU decomposition for arbitrary matrices and the polynomial structure of the Carleman-linearized LBE. Alpha and Q are input model parameters rather than fitted values. No invented entities are introduced.

free parameters (2)
  • alpha
    Carleman truncation order chosen for accuracy; treated as an input parameter in the scaling statements.
  • Q
    Number of discrete velocities in the LBE model; treated as an input parameter in the scaling statements.
axioms (2)
  • domain assumption Any square matrix can be decomposed into a linear combination of non-unitaries that can each be embedded into a unitary while preserving the overall LCU structure.
    Invoked at the start of the strategy to convert LCNU into LCU with equal number of terms.
  • domain assumption The Carleman-linearized LBE matrix has a polynomial nonlinearity whose structure permits an LCU with term count O(α² Q²) independent of discretization.
    Central to the application section and the independence claim.

pith-pipeline@v0.9.0 · 5628 in / 1793 out tokens · 78838 ms · 2026-05-12T05:18:41.352492+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages · 2 internal anchors

  1. [1]

    Efficient quantum algorithm for dissipative nonlinear differential equations

    Jin-Peng Liu et al. “Efficient quantum algorithm for dissipative nonlinear differential equations”. In: Proceedings of the National Academy of Sciences118.35 (Aug. 2021).issn: 1091-6490.doi:10.1073/ pnas.2026805118

    work page 2021

  2. [2]

    World Scientific, 1991

    Krzysztof Kowalski and Willi-hans Steeb.Nonlinear dynamical systems and Carleman linearization. World Scientific, 1991

    work page 1991

  3. [3]

    Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision

    Andrew M. Childs, Robin Kothari, and Rolando D. Somma. “Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision”. In:SIAM Journal on Computing 46.6 (Jan. 2017), pp. 1920–1950.issn: 1095-7111.doi:10.1137/16m1087072

    work page doi:10.1137/16m1087072 2017

  4. [4]

    Quantum Algorithm for Linear Systems of Equations

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum Algorithm for Linear Systems of Equations”. In:Physical Review Letters103.15 (Oct. 2009).issn: 1079-7114.doi:10 . 1103 / physrevlett.103.150502

    work page 2009

  5. [5]

    Variational quantum linear solver

    Carlos Bravo-Prieto et al. “Variational quantum linear solver”. In:Quantum7 (2023), p. 1188

    work page 2023

  6. [6]

    Quantum Linear System Solvers: A Survey of Algorithms and Applications

    Mauro ES Morales et al. “Quantum linear system solvers: A survey of algorithms and applications”. In:arXiv preprint arXiv:2411.02522(2024)

    work page arXiv 2024

  7. [7]

    A shortcut to an optimal quantum linear system solver

    Alexander M Dalzell. “A shortcut to an optimal quantum linear system solver”. In:arXiv preprint arXiv:2406.12086(2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  8. [8]

    Read the fine print

    Scott Aaronson. “Read the fine print”. In:Nature Physics11.4 (2015), pp. 291–293

    work page 2015

  9. [9]

    Tensorized Pauli decomposition algorithm

    Lukas Hantzko, Lennart Binkowski, and Sabhyata Gupta. “Tensorized Pauli decomposition algorithm”. In:Physica Scripta99.8 (2024), p. 085128

    work page 2024

  10. [10]

    Seitz, M

    Abeynaya Gnanasekaran and Amit Surana. “Efficient Variational Quantum Linear Solver for Struc- tured Sparse Matrices”. In:2024 IEEE International Conference on Quantum Computing and Engi- neering (QCE). IEEE, Sept. 2024, pp. 199–210.doi:10.1109/qce60285.2024.00033.url:http: //dx.doi.org/10.1109/QCE60285.2024.00033

    work page doi:10.1109/qce60285.2024.00033.url:http: 2024

  11. [11]

    Efficient quantum access model for sparse structured matrices using linear combination of “things

    Abeynaya Gnanasekaran and Amit Surana. “Efficient quantum access model for sparse structured matrices using linear combination of “things””. In:Physical Review A113.2 (2026), p. 022437

    work page 2026

  12. [12]

    Variational quantum framework for partial differential equation constrained optimization

    Amit Surana and Abeynaya Gnanasekaran. “Variational quantum framework for partial differential equation constrained optimization”. In:ACM Transactions on Quantum Computing7.1 (2025), pp. 1– 36. 31

    work page 2025

  13. [13]

    Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians

    Hui-Min Li, Zhi-Xi Wang, and Shao-Ming Fei. “Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians”. In:Physical Review A108.3 (2023), p. 032418

    work page 2023

  14. [14]

    Hardware efficient decomposition of the Laplace operator and its application to the Helmholtz and the Poisson equation on quantum computer

    Jaehyun Bae et al. “Hardware efficient decomposition of the Laplace operator and its application to the Helmholtz and the Poisson equation on quantum computer”. In:Quantum Information Processing 23.7 (July 2024).issn: 1573-1332.doi:10.1007/s11128-024-04458-y

    work page doi:10.1007/s11128-024-04458-y 2024

  15. [15]

    Variational quantum algorithm for the Poisson equation

    Hai-Ling Liu et al. “Variational quantum algorithm for the Poisson equation”. In:Phys. Rev. A104 (2 Aug. 2021), p. 022418.doi:10.1103/PhysRevA.104.022418.url:https://link.aps.org/doi/ 10.1103/PhysRevA.104.022418

    work page doi:10.1103/physreva.104.022418.url:https://link.aps.org/doi/ 2021

  16. [16]

    Computationally efficient quantum expectation with extended bell measurements

    Ruho Kondo et al. “Computationally efficient quantum expectation with extended bell measurements”. In:arXiv preprint arXiv:2110.09735(2021)

    work page arXiv 2021

  17. [17]

    Efficient decomposition of the Car- leman linearized Burgers’ equation

    Reuben Demirdjian, Thomas Hogancamp, and Daniel Gunlycke. “Efficient decomposition of the Car- leman linearized Burgers’ equation”. In:Phys. Rev. A(Jan. 2026).doi:10.1103/g27q- r2gk.url: https://link.aps.org/doi/10.1103/g27q-r2gk

    work page doi:10.1103/g27q- 2026

  18. [18]

    Variational quantum solutions to the advection–diffusion equation for ap- plications in fluid dynamics

    Reuben Demirdjian et al. “Variational quantum solutions to the advection–diffusion equation for ap- plications in fluid dynamics”. In:Quantum Information Processing21.9 (Sept. 2022).issn: 1573-1332. doi:10.1007/s11128-022-03667-7.url:http://dx.doi.org/10.1007/s11128-022-03667-7

    work page doi:10.1007/s11128-022-03667-7.url:http://dx.doi.org/10.1007/s11128-022-03667-7 2022

  19. [19]

    Real and Fourier space readout methods: Comparison of complexity and appli- cations to CFD problems

    Xinchi Huang et al. “Real and Fourier space readout methods: Comparison of complexity and appli- cations to CFD problems”. In:arXiv preprint arXiv:2511.20017(2025)

    work page arXiv 2025

  20. [20]

    Quantum Algorithms for Partial Differential Equations: A Performance Review and Future Trajectories

    Thanh Nguyen. “Quantum Algorithms for Partial Differential Equations: A Performance Review and Future Trajectories”. In:Intelligent Sustainable Systems. Ed. by Nagar Atulya K. et al. Cham: Springer Nature Switzerland, 2025, pp. 18–37.isbn: 978-3-032-11524-9

    work page 2025

  21. [21]

    Compact quantum algorithms for time-dependent differential equations

    Sachin S Bharadwaj and Katepalli R Sreenivasan. “Compact quantum algorithms for time-dependent differential equations”. In:Physical Review Research7.2 (2025), p. 023262

    work page 2025

  22. [22]

    Simulating non-trivial incompressible flows with a quantum lattice Boltzmann algorithm

    David Jennings et al. “Simulating non-trivial incompressible flows with a quantum lattice Boltzmann algorithm”. In:AIAA SCITECH 2026 Forum. 2026, p. 1936

    work page 2026

  23. [23]

    Limitations for quantum algorithms to solve turbulent and chaotic systems

    Dylan Lewis et al. “Limitations for quantum algorithms to solve turbulent and chaotic systems”. In: Quantum8 (2024), p. 1509

    work page 2024

  24. [25]

    Quantum variational solving of nonlinear and multidimensional partial differen- tial equations

    Abhijat Sarma et al. “Quantum variational solving of nonlinear and multidimensional partial differen- tial equations”. In:Physical Review A109.6 (2024), p. 062616

    work page 2024

  25. [26]

    Incompressible Navier–Stokes solve on noisy quantum hardware via a hybrid quantum–classical scheme

    Zhixin Song et al. “Incompressible Navier–Stokes solve on noisy quantum hardware via a hybrid quantum–classical scheme”. In:Computers & Fluids288 (2025), p. 106507

    work page 2025

  26. [27]

    Quantum algorithms for nonlinear dynamics: Revis- iting carleman linearization with no dissipative conditions

    Hsuan-Cheng Wu, Jingyao Wang, and Xiantao Li. “Quantum algorithms for nonlinear dynamics: Revis- iting carleman linearization with no dissipative conditions”. In:SIAM Journal on Scientific Computing 47.2 (2025), A943–A970

    work page 2025

  27. [28]

    An efficient quantum algorithm for simu- lating polynomial dynamical systems

    Amit Surana, Abeynaya Gnanasekaran, and Tuhin Sahai. “An efficient quantum algorithm for simu- lating polynomial dynamical systems”. In:Quantum Information Processing23.3 (Mar. 2024).issn: 1573-1332.doi:10.1007/s11128-024-04311-2.url:http://dx.doi.org/10.1007/s11128-024- 04311-2. 32

    work page doi:10.1007/s11128-024-04311-2.url:http://dx.doi.org/10.1007/s11128-024- 2024

  28. [29]

    Tensor networks enable the calculation of turbulence probability distribu- tions

    Nikita Gourianov et al. “Tensor networks enable the calculation of turbulence probability distribu- tions”. In:Science Advances11.5 (2025), eads5990.doi:10.1126/sciadv.ads5990. eprint:https: //www.science.org/doi/pdf/10.1126/sciadv.ads5990.url:https://www.science.org/doi/ abs/10.1126/sciadv.ads5990

    work page doi:10.1126/sciadv.ads5990 2025

  29. [30]

    Feasibility of Accelerating Incompressible Com- putational Fluid Dynamics Simulations With Fault-Tolerant Quantum Computers

    Michael C Garrett, Dmitry Ponkratov, and Wei Qiu. “Feasibility of Accelerating Incompressible Com- putational Fluid Dynamics Simulations With Fault-Tolerant Quantum Computers”. In:Marine Tech- nology Society Journal59.1 (2025), pp. 62–65

    work page 2025

  30. [31]

    The cost of solving linear differential equations on a quantum computer: fast- forwarding to explicit resource counts

    David Jennings et al. “The cost of solving linear differential equations on a quantum computer: fast- forwarding to explicit resource counts”. In:Quantum8 (Dec. 2024), p. 1553.issn: 2521-327X.doi: 10.22331/q-2024-12-10-1553

    work page doi:10.22331/q-2024-12-10-1553 2024

  31. [32]

    Quantum Computers for Weather and Climate Prediction: The Good, the Bad, and the Noisy

    F. Tennie and T. N. Palmer. “Quantum Computers for Weather and Climate Prediction: The Good, the Bad, and the Noisy”. In:Bulletin of the American Meteorological Society104.2 (Feb. 2023), E488– E500.issn: 1520-0477.doi:10.1175/bams-d-22-0031.1

    work page doi:10.1175/bams-d-22-0031.1 2023

  32. [33]

    npj Quan- tum Information6, 61 (2020) https://doi

    Frank Gaitan. “Finding flows of a Navier–Stokes fluid through quantum computing”. In:npj Quantum Information6.1 (July 2020).issn: 2056-6387.doi:10.1038/s41534-020-00291-0

    work page doi:10.1038/s41534-020-00291-0 2020

  33. [34]

    Quantum iterative methods for solving differential equations with applica- tion to computational fluid dynamics

    Chelsea A Williams et al. “Quantum iterative methods for solving differential equations with applica- tion to computational fluid dynamics”. In:Advanced Quantum Technologies(2025), e00618

    work page 2025

  34. [35]

    Solving Burgers’ equation with quantum computing

    Furkan Oz et al. “Solving Burgers’ equation with quantum computing”. In:Quantum Information Processing21.1 (Dec. 2021).issn: 1573-1332.doi:10.1007/s11128-021-03391-8

    work page doi:10.1007/s11128-021-03391-8 2021

  35. [36]

    Quantum algorithms for nonlinear partial differential equations

    Shi Jin and Nana Liu. “Quantum algorithms for nonlinear partial differential equations”. In:Bulletin des Sciences Math´ ematiques194 (Sept. 2024), p. 103457.issn: 0007-4497.doi:10.1016/j.bulsci. 2024.103457

    work page doi:10.1016/j.bulsci 2024

  36. [37]

    A hybrid quantum-classical CFD methodology with benchmark HHL solutions

    Leigh Lapworth. “A hybrid quantum-classical CFD methodology with benchmark HHL solutions”. In: arXiv preprint arXiv:2206.00419(2022)

    work page arXiv 2022

  37. [38]

    Quantum algorithm for the Navier–Stokes equations by using the streamfunction- vorticity formulation and the lattice Boltzmann method

    Budinski Ljubomir. “Quantum algorithm for the Navier–Stokes equations by using the streamfunction- vorticity formulation and the lattice Boltzmann method”. In:International Journal of Quantum In- formation20.02 (Feb. 2022).issn: 1793-6918.doi:10.1142/s0219749921500398

    work page doi:10.1142/s0219749921500398 2022

  38. [39]

    Variational quantum algorithms for nonlinear problems

    Michael Lubasch et al. “Variational quantum algorithms for nonlinear problems”. In:Physical Review A101.1 (Jan. 2020).issn: 2469-9934.doi:10.1103/physreva.101.010301

    work page doi:10.1103/physreva.101.010301 2020

  39. [40]

    A Linear Combination of Unitaries Decomposition for the Laplace Operator

    Thomas Hogancamp, Reuben Demirdjian, and Daniel Gunlycke. “A Linear Combination of Unitaries Decomposition for the Laplace Operator”. In:arXiv preprint arXiv:2601.06370(2026)

    work page arXiv 2026

  40. [41]

    Quantum Carleman linearization efficiency in nonlinear fluid dynamics

    Javier Gonzalez-Conde et al. “Quantum Carleman linearization efficiency in nonlinear fluid dynamics”. In:Physical Review Research7.2 (2025), p. 023254

    work page 2025

  41. [42]

    Challenges for quantum computation of nonlinear dynamical systems using linear representations

    Yen Ting Lin et al. “Challenges for quantum computation of nonlinear dynamical systems using linear representations”. In:arXiv preprint arXiv:2202.02188(2022)

    work page arXiv 2022

  42. [43]

    Quantum algorithms for general nonlinear dynamics based on the Carleman embedding

    David Jennings et al. “Quantum algorithms for general nonlinear dynamics based on the Carleman embedding”. In:arXiv preprint arXiv:2509.07155(2025)

    work page arXiv 2025

  43. [45]

    An end-to-end quantum algorithm for nonlinear fluid dynamics with bounded quantum advantage.arXiv preprint arXiv:2512.03758,

    David Jennings et al. “An end-to-end quantum algorithm for nonlinear fluid dynamics with bounded quantum advantage”. In:arXiv preprint arXiv:2512.03758(2025). 33

    work page arXiv 2025

  44. [46]

    Discovering polynomial and quadratic structure in nonlinear ordi- nary differential equations

    Boris Kramer and Gleb Pogudin. “Discovering polynomial and quadratic structure in nonlinear ordi- nary differential equations”. In:arXiv preprint arXiv:2502.10005(2025)

    work page arXiv 2025

  45. [47]

    Commutation matrices and commutation tensors

    Changqing Xu, Lingling He, and Zerong Lin. “Commutation matrices and commutation tensors”. In: Linear and Multilinear Algebra68.9 (2018), pp. 1721–1742.doi:10.1080/03081087.2018.1556242

    work page doi:10.1080/03081087.2018.1556242 2018

  46. [48]

    Timm Kr¨ uger et al.The lattice Boltzmann method. Vol. 10. 978-3. Springer, 2017

    work page 2017

  47. [49]

    A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems

    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”. In:Phys. Rev.94 (3 May 1954), pp. 511–525.doi:10.1103/PhysRev.94.511.url:https://link.aps.org/doi/10.1103/ PhysRev.94.511

    work page doi:10.1103/physrev.94.511.url:https://link.aps.org/doi/10.1103/ 1954

  48. [50]

    Potential quantum advantage for simulation of fluid dynamics

    Xiangyu Li et al. “Potential quantum advantage for simulation of fluid dynamics”. In:Physical Review Research7.1 (2025), p. 013036

    work page 2025

  49. [51]

    Tutorial on the quantikz package

    Alastair Kay. “Tutorial on the quantikz package”. In:arXiv preprint arXiv:1809.03842(2018)

    work page arXiv 2018

  50. [52]

    Hamiltonian Simulation Using Linear Com- binations of Unitary Operations

    Andrew M Childs and Nathan Wiebe. “Hamiltonian simulation using linear combinations of unitary operations”. In:arXiv preprint arXiv:1202.5822(2012)

    work page arXiv 2012

  51. [53]

    Strategies for simulating the time evolution of Hamiltonian lattice field theories

    Siddharth Hariprakash et al. “Strategies for simulating the time evolution of Hamiltonian lattice field theories”. In:Physical Review A111.2 (2025), p. 022419

    work page 2025

  52. [54]

    Quantum circuits ofT-depth one

    Peter Selinger. “Quantum circuits ofT-depth one”. In:Phys. Rev. A87 (4 Apr. 2013), p. 042302.doi: 10.1103/PhysRevA.87.042302.url:https://link.aps.org/doi/10.1103/PhysRevA.87.042302

    work page doi:10.1103/physreva.87.042302.url:https://link.aps.org/doi/10.1103/physreva.87.042302 2013

  53. [55]

    Rise of conditionally clean ancillae for efficient quantum circuit constructions

    Tanuj Khattar and Craig Gidney. “Rise of conditionally clean ancillae for efficient quantum circuit constructions”. In:Quantum9 (2025), p. 1752. [56]Constructing Large Increment Gates. June 2015.url:https://algassert.com/circuits/2015/06/ 12/Constructing-Large-Increment-Gates.html

    work page 2025

  54. [56]

    Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging

    Pedro MQ Cruz and Bruno Murta. “Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging”. In:APL Quantum1.1 (2024)

    work page 2024

  55. [57]

    Variational Quantum Framework for Non- linear PDE Constrained Optimization Using Carleman Linearization

    Abeynaya Gnanasekaran, Amit Surana, and Hongyu Zhu. “Variational Quantum Framework for Non- linear PDE Constrained Optimization Using Carleman Linearization”. In:Quantum Information & Computation25.3 (2025), pp. 260–289.doi:10.2478/qic-2025-0014.url:https://doi.org/10. 2478/qic-2025-0014

    work page doi:10.2478/qic-2025-0014.url:https://doi.org/10 2025

  56. [58]

    Quantum computing with Qiskit

    Ali Javadi-Abhari et al. “Quantum computing with Qiskit”. In:arXiv preprint arXiv:2405.08810 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  57. [59]

    Preconditioned block encodings for quantum linear sys- tems

    Leigh Lapworth and Christoph S¨ underhauf. “Preconditioned block encodings for quantum linear sys- tems”. In:Quantum Science and Technology10.4 (2025), p. 045064

    work page 2025

  58. [60]

    Quantum preconditioning method for linear systems problems via Schrodingerization

    Shi Jin et al. “Quantum preconditioning method for linear systems problems via Schrodingerization”. In:arXiv preprint arXiv:2505.06866(2025)

    work page arXiv 2025

  59. [61]

    Preconditioning for a variational quantum linear solver

    Aruto Hosaka et al. “Preconditioning for a variational quantum linear solver”. In:arXiv preprint arXiv:2312.15657(2023)

    work page arXiv 2023

  60. [62]

    Quantum computing and preconditioners for hydrological linear systems

    John Golden, Daniel O’Malley, and Hari Viswanathan. “Quantum computing and preconditioners for hydrological linear systems”. In:Scientific Reports12.1 (2022), p. 22285

    work page 2022

  61. [63]

    Barren plateaus in variational quantum computing

    Martin Larocca et al. “Barren plateaus in variational quantum computing”. In:Nature Reviews Physics (2025), pp. 1–16. 34

    work page 2025

  62. [64]

    Nonlinear dynamics as a ground-state solution on quantum computers

    Albert J Pool et al. “Nonlinear dynamics as a ground-state solution on quantum computers”. In: Physical Review Research6.3 (2024), p. 033257

    work page 2024

  63. [65]

    Accelerating MPI allreduce communication with efficient gpu-based, compression schemes on modern GPU clusters,

    Christo Meriwether Keller et al. “Hierarchical Multigrid Ansatz for Variational Quantum Algorithms”. In:ISC High Performance 2024 Research Paper Proceedings (39th International Conference). 2024, pp. 1–11.doi:10.23919/ISC.2024.10528934

    work page doi:10.23919/isc.2024.10528934 2024

  64. [66]

    Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator

    Daniel Gunlycke et al. “Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator”. In:arXiv preprint arXiv:2011.08942(2020)

    work page arXiv 2011

  65. [67]

    Hybrid numerical methods based on the lattice Boltzmann approach with appli- cation to non-uniform grids

    Tobias Horstmann. “Hybrid numerical methods based on the lattice Boltzmann approach with appli- cation to non-uniform grids”. PhD thesis. Universit´ e de Lyon, 2018

    work page 2018

  66. [68]

    The logarithmic norm. History and modern theory

    Gustaf S¨ oderlind. “The logarithmic norm. History and modern theory”. In:BIT Numerical Mathemat- ics46.3 (2006), pp. 631–652

    work page 2006

  67. [69]

    2017.url:https://algassert

    Craig Gidney.Simple Algorithm for Multiplicative Inverses mod2 n. 2017.url:https://algassert. com/post/1709. (accessed: 11.26.2024). 35 A Notation Carleman Linearization L Matrix from the linear system resulting from Carleman linearization L(e) Matrix from the linear system resulting from Carleman linearization and zero padding α Truncation order ∆ =Pα j=...

    work page 2017

  68. [70]

    It’s clear from (98) that ˜W2 ={0}impliesQ 1Q2 = 0

    +Q 1Q2(w⊥ 2 ) =Q 1Q2( ˜w2) +Q 1 Q2( ˜w ′ 2) =Q 1Q2( ˜w2) (98) where the second inequality follows fromQ 2(w⊥ 2 ) = 0, and the third is true sinceQ 2( ˜w ′ 2)∈W ⊥ 1 . It’s clear from (98) that ˜W2 ={0}impliesQ 1Q2 = 0. Next, for the forward direction, assumeQ 1Q2 = 0. Then, it follows from our hypothesis that 0 =∥Q 1Q2( ˜w2)∥=∥Q 2( ˜w2)∥=∥˜w2∥,(99) for all...

    work page

  69. [71]

    Suppose also that ˜W2 :={w∈W 2 |Q 2w∈ W1} ̸={0}and considerQ 1Q2 : ˜W2 →V

    LetW i be the rowspace ofQ i, and letQ i be unitary onW i. Suppose also that ˜W2 :={w∈W 2 |Q 2w∈ W1} ̸={0}and considerQ 1Q2 : ˜W2 →V. If Qi is a valid unitary completion ofQ i, then Q1Q2 is a valid unitary completion ofQ 1Q2. Proof.To prove that Q1Q2 is a unitary completion toQ 1Q2, we must show that (1) Q1Q2w=Q 1Q2wwhere w∈ ˜W2, and (2) Q1Q2 in unitary o...

    work page

  70. [72]

    The first condition follows from the fact that each factor on the right hand side of (10) is unitary

    That Lis unitary, and 2) that Lw=Lwfor allw∈W 1. The first condition follows from the fact that each factor on the right hand side of (10) is unitary. The second is verified by Lw=U   mY i=1 niO j=1 P i,j   (V w) =U   mY i=1 niO j=1 Pi,j   (V w) =Lw , where the third equality follows from Definition 1 and the hypothesis thatV w∈W ′ 1. B.8 Proof of...

    work page

  71. [73]

    We claim thatE n i,j =Nlogn l=1 rl for somer l ∈P ρ. As an example, consider E4 1,2 =   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0   =ρ 1 ⊗ρ 2 .(104) The goal of this preamble is to determine how to construct the Kronecker product given (i, j) andn(see Theorem 3 of [11] for a similar discussion). First, define the functionB β(d), which maps the base-ten numb...

    work page

  72. [74]

    1| {z } logN . . .01. . .1| {z } logN 1 | {z } β1 , where we have usedB β1(N α−j)− B β1(1) = 01. . .1 in the third equality and successive applications of (109) for the forth. Sincer 1 =c 1, we have F(B β1(r1),B β1(c1)) = 03. . .3| {z } logN

    work page

  73. [75]

    3| {z } logN . . .03. . .3| {z } logN 3 | {z } β1 , and therefore E2N α−j r1,c1 =ρ F(B β1(r1),Bβ1(c1)) = ρ0 ⊗ρ ⊗logN−1 3 ⊗α−j ⊗ρ 3 .(110) Next, we apply a similar procedure toE 2N α−j−k+1 r2,c2 . Letβ 2 = log 2Nα−j−k+1, then Bβ2(r2) =B β2(N α−j−k+1 −2−N− · · · −N α−j−k) =B β2(N α−j−k+1)− B β2(2)− B β2(N)− · · · − B β2(N α−j−k) = 01. . .10| {z } β2 −Bβ2(N)...

    work page

  74. [76]

    1| {z } logN . . .01. . .1| {z } logN 0 | {z } β2 . Similarly, the column index in binary is Bβ2(c2) =B β2(N α−j−k+1 −1−N− · · · −N α−j−k) =B β2(N α−j−k+1)− B β2(1)− B β2(N)− · · · − B β2(N α−j−k) = 01. . .1| {z } β2 −Bβ2(N)− · · · − B β2(N α−j−k) = 01. . .1| {z } logN

    work page

  75. [77]

    1| {z } logN . . .01. . .1| {z } logN 1 | {z } β2 . This gives F(B β2(r2),B β2(c2)) = 03. . .3| {z } logN

    work page

  76. [78]

    3| {z } logN . . .03. . .3| {z } logN 1 | {z } β2 , 45 and therefore E2N α−j−k+1 r2,c2 =ρ F(B β2(r2),Bβ2(c2)) = ρ0 ⊗ρ ⊗logN−1 3 ⊗α−j−k+1 ⊗ρ 1 .(111) Next, we use the following relation   0b×a Aj j+k−1 0c×a   = (Pk ⊗I N j)A(e),j j+k−1 ,(112) where we recall thatP k ∈C N k−1×N k−1 is defined in Appendix E.1 and we have used the definition ofA (e),j j+k−...

    work page

  77. [79]

    spaced everyQn+ 1 elements apart,

    work page

  78. [80]

    Putting these requirements together, it follows thatB 2,q(i, j) = 1 where (i, j)∈ {(0, b),(1, a+b),(2,2a+b),

    shifted forward by (q−1)nelements. Putting these requirements together, it follows thatB 2,q(i, j) = 1 where (i, j)∈ {(0, b),(1, a+b),(2,2a+b), . . . ,(n−1,(n−1)a+b)},(142) wherea=Qn+ 1,b= (q−1)n,q∈ {1, . . . , Q},Q= 2 qQ andn= 2 qn. We must therefore find an operator to perform the mapping |ia+b⟩ 2qn+qQ → |i⟩2qn+qQ ,(143) wherei∈ {0, . . . , n−1}. Using ...

    work page

  79. [81]

    in each row the 1’s are spaced byQn 2,

    work page

  80. [82]

    in adjacent rows the pattern of 1’s are spaced by (Qn) 2 +Qn+ 1,

    work page

Showing first 80 references.