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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 May 15, 2026 (v3)
v1 · May 1, 2026v2 · May 11, 2026v3 · May 15, 2026v4 · Jun 4, 2026
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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 $N_s \sim \mathcal{O}(α^2 Q^2)$, where $α$ 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 $\mathcal{O}(α^3 Q^2 (\log_2 n)^2)$, where $n$ is the total number of spatial grid points across all dimensions. Next, the latter requires $N_s^2(\log_2 (2n_tn^α)+1)$ circuits per iteration, with a worst case T gate cost of $\mathcal{O}(α(\log_2 Qn)^2)$ among them. We, therefore, provide an efficient decomposition strategy useful for both fault-tolerant and variational approaches.

Comments
Corrected minor mistakes

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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 Ns∼O(α2Q2)N_s \sim \mathcal{O}(α^2 Q^2)Ns​∼O(α2Q2), where ααα is the Carleman truncation order and QQQ is the number of discrete velocities from the LBE. Importantly, NsN_sNs​ 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 O(α3Q2(log⁡2n)2)\mathcal{O}(α^3 Q^2 (\log_2 n)^2)O(α3Q2(log2​n)2), where nnn is the total number of spatial grid points across all dimensions. Next, the latter requires Ns2(log⁡2(2ntnα)+1)N_s^2(\log_2 (2n_tn^α)+1)Ns2​(log2​(2nt​nα)+1) circuits per iteration, with a worst case T gate cost of O(α(log⁡2Qn)2)\mathcal{O}(α(\log_2 Qn)^2)O(α(log2​Qn)2) among them. We, therefore, provide an efficient decomposition strategy useful for both fault-tolerant and variational approaches.
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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.
Carleman linearization represents nonlinear lattice Boltzmann collision dynamics in a form amenable to quantum block encoding, but low postselection probabilities in existing population-space encodings make coherent multi-step evolution exponentially unlikely to succeed. We present a quantum Carleman lattice Boltzmann scheme based on the multi-relaxation-time collision operator in a weighted Hermite moment basis. In this basis, the lifted collision operator separates into small coupled blocks whose norm exceeds unity because of the coupling between conserved and relaxing modes. This block structure enables a norm-attaining block encoding with a single-step success probability close to unity. Crucially, scaling the pair sector for a prescribed evolution horizon makes the combined encoding and sector-selection overhead grow polynomially rather than exponentially with the number of steps. For a hundred-step evolution, the resulting complete-run success probability exceeds that of population-space encodings by orders of magnitude. Over horizons of several tens of steps, the truncated dynamics reproduces the nonlinearly generated modes of a benchmark flow to within a few percent. We verify the combined preparation, evolution and readout protocol as a single unitary circuit and give a complete gate-level resource estimate for extracting a nonlinear observable of the benchmark flow from a coherent Carleman evolution. The block structure of conserved and relaxing modes holds equally on three-dimensional lattices and provides a constructive route to norm-attaining encodings for lattice Boltzmann models with quadratic equilibria.