Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.00302

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

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

quant-phSubmitted May 1, 2026• Announced May 4, 2026• 0 citations• Revised Jun 4, 2026 (v4)
v1 · May 1, 2026v2 · May 11, 2026v3 · May 15, 2026v4 · Jun 4, 2026
arXivHTMLPDF

Abstract

Nonlinear ordinary and partial differential equations are ubiquitous in science and engineering, yet finding their solutions is often computationally intractable for classical hardware. To determine if quantum computers can offer a practical advantage, one critical challenge that must be solved is determining how to efficiently load exponentially sized matrices onto quantum hardware. In this article, we introduce an alternative linear combination of unitaries (LCU) strategy which relies on an intermediate linear combination of non-unitaries (LCNU) and a systematic embedding procedure. One advantage of this LCU strategy is that it maintains the exact number of terms as in the LCNU. Therefore, this approach offers a data loading framework for matrices that lack an efficient decomposition using the standard LCU alone. Using this approach, we construct a generalized LCNU framework for any Carleman linearized autonomous dynamical system having a polynomial nonlinearity. To demonstrate the effectiveness of our approach, we construct an LCNU for the 3D Carleman linearized lattice Boltzmann equation (LBE). Here, we find that the number of terms in the decomposition scales like $N_s\sim\mathcal{O}(α^2Q^2)$, where $α$ is the Carleman truncation order and $Q$ is the number of discrete velocities. Importantly, $N_s$ is independent of the number of spatial and temporal discretization points. We then perform a resource estimation of our LCNU's T gate cost when combined with the (1) PREP and SELECT block encoding oracles, and (2) variational quantum linear solver. In the former, the T cost scales like $\mathcal{O}(α^3Q^2(\log_2n)^2)$, where $n$ is the total number of spatial grid points. The latter requires exactly $N_s^2(\log_2 (2n_tn^α)+1)$ circuits per iteration for $n_t$ time steps, with a worst case T gate cost of $\mathcal{O}(α(\log_2Qn)^2)$ among them.

Comments
Corrected minor mistakes

Related papers

Efficient decomposition of the Carleman linearized Burgers' equation

Reuben Demirdjian, Thomas Hogancamp, Daniel Gunlycke

Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order $α$. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of $\mathcal{O}(\log n_t + α^2\log n_x)$ terms for $n_t$ time steps and $n_x$ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like $\mathcal{O}(α(\log n_x)^2)$ and $\mathcal{O}((\log n_x)^2)$, respectively. This is therefore the first explicit polylogarithmic data loading method with respect to $n_x$ and $n_t$ for a Carleman linearized system.

arXiv:2505.00285•Physical Review A•Jan 23, 2026•2 citations

A Scalable Approach to Solve the Carleman Linearized Burgers' Equation on a Quantum Computer

Reuben Demirdjian, Yvan Quinn, Vincent P. Su +2 more

Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to $2^{80} \approx 10^{24}$ are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.

quant-pharXiv:2607.08976•Jul 13, 2026•0 citations

Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

Tayyab Ali

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order $N$, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component $\mathcal{H}_{j}$ of the decomposition.

quant-pharXiv:2602.07097•Feb 9, 2026•0 citations

Explicit Quantum Circuit Simulation of Nonlinear 1-Dimensional Fluid with Carleman-linearized Boltzmann Method

Keita Kanno, Kazumasa Ueno, Hayato Higuchi +5 more

Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.

quant-pharXiv:2606.12770•Jun 12, 2026•0 citations

Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization

Koichi Miyamoto

Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) $\mathrm{d} \mathbf{X}_t = \left(A(t) \mathbf{X}_t + F(t) \mathbf{X}_t^{\otimes s}\right) \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$ for an $\mathbb{R}^N$-valued random process $\mathbf{X}_t$, aiming at quantum speed-up with respect to the dimension $N$. We introduce auxiliary processes $\mathbf{Y}^j_t$ that approximate $\mathbf{X}_t^{\otimes j}$, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of $\mathbf{X}_t$ by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order $J$ in CL, and we can estimate the target expectation with a ${\rm polylog}(N)$ number of queries to unitaries encoding coefficient matrices $A$, $F$, and $B$. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as $J$ increases.

quant-pharXiv:2610.27416•Oct 21, 2026•0 citations

An end-to-end quantum algorithm for weakly nonlinear plasma physics with superquadratic speedup

Bjorn K. Berntson, David Jennings, Matteo Lostaglio +1 more

Nonlinear kinetic plasma simulation is high-dimensional and classically demanding, while quantum algorithms face different bottlenecks: embedding nonlinear dynamics into a linear computation, loading dense field-interaction data, and efficiently extracting information. We present an end-to-end quantum algorithm, with rigorous convergence guarantees, for a weakly nonlinear kinetic plasma model. The system describes a 3D electron-ion plasma with adiabatic electrons, kinetic ions, Debye screening, and Krook relaxation. After Fourier-Hermite truncation, the dynamics reduces to a high-dimensional quadratic ordinary differential equation. To tackle quantum bottlenecks we combine three key ingredients. First, we use a plasma free energy to identify a Lyapunov transform under which a Carleman linear embedding converges exponentially in the truncation order within a certified weakly nonlinear regime. Second, we develop a hierarchical block-encoding protocol for dense matrices, exploiting the spatial decay of the field to avoid polynomial overhead from sparse access encodings. Third, we introduce a subroutine for information extraction that exploits nonlinear components encoded in the full Carleman history state to improve the estimation of linear observables. We construct a quantum algorithm to estimate the spacetime-averaged kinetic energy using $\widetilde{O}\!\left( N_F N_H^{1/2} \operatorname{polylog}\!\left(\frac{T}ε\right)\frac{1}ε\right)$ gates and $\widetilde{O}\!\left(\log\!\left(N_F N_H^{1/2}T\right)\log\!\left(\frac{1}ε\right)\right)$ qubits, where $N_F$ and $N_H$ are the Fourier and Hermite cutoffs. Relative to a Fourier-Hermite spectral solver, this yields exponential memory savings and superquadratic improvements in time. Together, these results establish a controlled nonlinear plasma benchmark for quantum simulation.

quant-pharXiv:2607.14308•Jul 17, 2026•1 citation
Nonlinear ordinary and partial differential equations are ubiquitous in science and engineering, yet finding their solutions is often computationally intractable for classical hardware. To determine if quantum computers can offer a practical advantage, one critical challenge that must be solved is determining how to efficiently load exponentially sized matrices onto quantum hardware. In this article, we introduce an alternative linear combination of unitaries (LCU) strategy which relies on an intermediate linear combination of non-unitaries (LCNU) and a systematic embedding procedure. One advantage of this LCU strategy is that it maintains the exact number of terms as in the LCNU. Therefore, this approach offers a data loading framework for matrices that lack an efficient decomposition using the standard LCU alone. Using this approach, we construct a generalized LCNU framework for any Carleman linearized autonomous dynamical system having a polynomial nonlinearity. To demonstrate the effectiveness of our approach, we construct an LCNU for the 3D Carleman linearized lattice Boltzmann equation (LBE). Here, we find that the number of terms in the decomposition scales like Ns∼O(α2Q2)N_s\sim\mathcal{O}(α^2Q^2)Ns​∼O(α2Q2), where ααα is the Carleman truncation order and QQQ is the number of discrete velocities. Importantly, NsN_sNs​ is independent of the number of spatial and temporal discretization points. We then perform a resource estimation of our LCNU's T gate cost when combined with the (1) PREP and SELECT block encoding oracles, and (2) variational quantum linear solver. In the former, the T cost scales like O(α3Q2(log⁡2n)2)\mathcal{O}(α^3Q^2(\log_2n)^2)O(α3Q2(log2​n)2), where nnn is the total number of spatial grid points. The latter requires exactly Ns2(log⁡2(2ntnα)+1)N_s^2(\log_2 (2n_tn^α)+1)Ns2​(log2​(2nt​nα)+1) circuits per iteration for ntn_tnt​ time steps, with a worst case T gate cost of O(α(log⁡2Qn)2)\mathcal{O}(α(\log_2Qn)^2)O(α(log2​Qn)2) among them.
Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order ααα. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of O(log⁡nt+α2log⁡nx)\mathcal{O}(\log n_t + α^2\log n_x)O(lognt​+α2lognx​) terms for ntn_tnt​ time steps and nxn_xnx​ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like O(α(log⁡nx)2)\mathcal{O}(α(\log n_x)^2)O(α(lognx​)2) and O((log⁡nx)2)\mathcal{O}((\log n_x)^2)O((lognx​)2), respectively. This is therefore the first explicit polylogarithmic data loading method with respect to nxn_xnx​ and ntn_tnt​ for a Carleman linearized system.
Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to 280≈10242^{80} \approx 10^{24}280≈1024 are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.
The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order NNN, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component Hj\mathcal{H}_{j}Hj​ of the decomposition.
Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) dXt=(A(t)Xt+F(t)Xt⊗s)dt+B(t)dWt\mathrm{d} \mathbf{X}_t = \left(A(t) \mathbf{X}_t + F(t) \mathbf{X}_t^{\otimes s}\right) \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_tdXt​=(A(t)Xt​+F(t)Xt⊗s​)dt+B(t)dWt​ for an RN\mathbb{R}^NRN-valued random process Xt\mathbf{X}_tXt​, aiming at quantum speed-up with respect to the dimension NNN. We introduce auxiliary processes Ytj\mathbf{Y}^j_tYtj​ that approximate Xt⊗j\mathbf{X}_t^{\otimes j}Xt⊗j​, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of Xt\mathbf{X}_tXt​ by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order JJJ in CL, and we can estimate the target expectation with a polylog(N){\rm polylog}(N)polylog(N) number of queries to unitaries encoding coefficient matrices AAA, FFF, and BBB. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as JJJ increases.
Nonlinear kinetic plasma simulation is high-dimensional and classically demanding, while quantum algorithms face different bottlenecks: embedding nonlinear dynamics into a linear computation, loading dense field-interaction data, and efficiently extracting information. We present an end-to-end quantum algorithm, with rigorous convergence guarantees, for a weakly nonlinear kinetic plasma model. The system describes a 3D electron-ion plasma with adiabatic electrons, kinetic ions, Debye screening, and Krook relaxation. After Fourier-Hermite truncation, the dynamics reduces to a high-dimensional quadratic ordinary differential equation. To tackle quantum bottlenecks we combine three key ingredients. First, we use a plasma free energy to identify a Lyapunov transform under which a Carleman linear embedding converges exponentially in the truncation order within a certified weakly nonlinear regime. Second, we develop a hierarchical block-encoding protocol for dense matrices, exploiting the spatial decay of the field to avoid polynomial overhead from sparse access encodings. Third, we introduce a subroutine for information extraction that exploits nonlinear components encoded in the full Carleman history state to improve the estimation of linear observables. We construct a quantum algorithm to estimate the spacetime-averaged kinetic energy using O~ ⁣(NFNH1/2polylog⁡ ⁣(Tε)1ε)\widetilde{O}\!\left( N_F N_H^{1/2} \operatorname{polylog}\!\left(\frac{T}ε\right)\frac{1}ε\right)O(NF​NH1/2​polylog(εT​)ε1​) gates and O~ ⁣(log⁡ ⁣(NFNH1/2T)log⁡ ⁣(1ε))\widetilde{O}\!\left(\log\!\left(N_F N_H^{1/2}T\right)\log\!\left(\frac{1}ε\right)\right)O(log(NF​NH1/2​T)log(ε1​)) qubits, where NFN_FNF​ and NHN_HNH​ are the Fourier and Hermite cutoffs. Relative to a Fourier-Hermite spectral solver, this yields exponential memory savings and superquadratic improvements in time. Together, these results establish a controlled nonlinear plasma benchmark for quantum simulation.