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:2610.20620

Representation-split Carleman lattice Boltzmann with near-unit step success probability on quantum computers

Hua-Dong Yao, Sauro Succi

physics.comp-phquant-phSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
arXivHTMLPDF

Abstract

The Carleman lattice Boltzmann method replaces the quadratic collision of the lattice Boltzmann equation by a linear map on a lifted state. Existing block encodings of this map succeed with probability $10^{-5}$ to $10^{-2}$ per step, so that the probability of a multi-step evolution decays exponentially with the number of steps. This study develops multi-relaxation-time Carleman lattice Boltzmann, a representation split in which the Carleman-lifted collision is performed in a moment basis on the amplitude-encoded state, while streaming remains a permutation in population space. The level-2 hierarchy closes under collision but not under streaming, which generates the work of pressure, advection and stress on the momentum flux. The state therefore remains the full pair array, and the moment representation is an orthogonal change of basis of the population one, with identical dynamics and truncation error. The operator to be block encoded changes with the basis, and with it the success probability of a quantum step. In the weighted Hermite basis with the amplitude encoding $f_i/\sqrt{w_i}$, the relaxation is a diagonal operator of norm one, and the level-2 collision stage becomes this diagonal plus a sparse coupling from products of conserved moments. A structured block encoding of this form raises the single-step success probability on the D2Q9 lattice from $10^{-2}$ for the best population encoding to 0.98. With the pair sector in relative coordinates and scaled, one step requires $O(\log^2 N)$ Toffoli gates and is verified by statevector simulation. The encoding and readout overhead of a $T$-step run is then polynomial in $T$ and $N$ rather than accumulating a factor $p^T$. For resolved flows, the remaining exponential loss in coherent multi-step Carleman lattice Boltzmann evolution is set by the non-equilibrium norm relaxed by dissipation rather than by the block encoding.

Comments
15 pages, 4 figures

Related papers

Conserved and relaxing modes enable polynomial-success coherent Carleman lattice Boltzmann evolution

Hua-Dong Yao, Sauro Succi

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.

quant-pharXiv:2609.32310•Sep 29, 2026•0 citations

Deterministic Realization of Classical Dissipation on Quantum Computers

Muhammad Idrees Khan, Sauro Succi, Hua-Dong Yao

Lattice Boltzmann (LB) on quantum devices must reconcile unitary gate evolution with the dissipative \emph{collision} step. In the multiple-relaxation-time (MRT) class, we work in the common setting of \emph{modewise diagonal} moment relaxation, $δm_r'=λ_r\,δm_r$ with $λ_r\in[-1,1]$ (overrelaxation if $λ_r<0$). Embedding that contraction in a unitary by block encoding or a linear combination of unitaries (LCU) typically yields subunitary success probability that decays multiplicatively across modes, sites, and time, a key bottleneck for quantum LB. \emph{For the dissipative MRT block alone} we give a \emph{block-encoding-free} construction: a signed \emph{two-rail} population encoding, then a completely positive trace-preserving (CPTP) map (per-rail amplitude damping with survival $|λ_r|$ and, if $λ_r<0$, a rail SWAP) so that, after the decode, the map agrees with classical MRT relaxation exactly (expectations of the rail number operators, common encoding--decode scale). Trace preservation gives success probability $1$ for that substage. The main result is the dissipative MRT block; construction of the equilibrium moment vector~$m^{\mathrm{eq}}=Mf^{\mathrm{eq}}$ (prescribed~$f^{\mathrm{eq}}$, host moment matrix~$M$; notation as in Section~\ref{subsec:generic-mrt}), moment transforms, streaming, and boundaries are composed with it as in a standard host pipeline and lie outside the scope of the formal theorem. Hybrid and fully coherent encodings, adaptive scales, Carleman-based context, and a one-rail no-go in the same nonnegative population framework are in the main text. Audits of the open-channel map on a long LBM collide-stream simulation and on stencil-free inputs both match the target to machine precision.

physics.comp-phCross-listedarXiv:2604.25429•Apr 29, 2026•0 citations

Decomposition of nonlinear collision operator in quantum Lattice Boltzmann algorithm

E. Dinesh Kumar, Steven H. Frankel

Abstract We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.

arXiv:2408.00387•Europhysics Letters (EPL)•Oct 28, 2024•3 citations

Carleman-lattice-Boltzmann quantum circuit with matrix access oracles

Claudio Sanavio, William Simon, Alexis Ralli +2 more

We apply Carleman linearization of the Lattice Boltzmann (CLB) representation of fluid flows to quantum emulate the dynamics of a 2D Kolmogorov-like flow. We assess the accuracy of the result and find a relative error of the order of $10^{-3}$ with just two Carleman iterates, for a range of the Reynolds number up to a few hundreds. We first define a gate-based quantum circuit for the implementation of the CLB method and then exploit the sparse nature of the CLB matrix to build a quantum circuit based on block-encoding techniques which makes use of matrix oracles. It is shown that the gate complexity of the algorithm is thereby dramatically reduced, from exponential to quadratic. However, due to the need of employing up to seven ancilla qubits, the probability of success of the corresponding circuit for a single time step is too low to enable multi-step time evolution. Several possible directions to circumvent this problem are briefly outlined.

arXiv:2501.02582•Jan 5, 2025•0 citations

A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation

David Wawrzyniak, Josef M. Winter, Steffen J. Schmidt +4 more

We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.

Computer Physics Communications•Sep 10, 2024•33 citations

Quantum lattice Boltzmann method for several time steps: A local Carleman linearization algorithm

Antonio David Bastida Zamora, Ljubomir Budinski, Valtteri Lahtinen +1 more

This article presents a novel encoding for quantum Lattice Boltzmann method algorithm using Carleman linearization. In contrast to previous articles \cite{Sanavio2024LatticeBC,sanavio2025carleman}, the encoding used allows for local collision rules while keeping a higher probability to obtain the right result, which is of the order of $10^{-2}$. The algorithm scales as $O(log_2^2(N)+Q^3)$ each time step with $N$ the number of lattice sites of the 2D lattice and $Q$ the number of channels with a constant number of qubits when using dynamical circuits.

quant-pharXiv:2511.13072•Physical review. E•Nov 18, 2025•2 citations
The Carleman lattice Boltzmann method replaces the quadratic collision of the lattice Boltzmann equation by a linear map on a lifted state. Existing block encodings of this map succeed with probability 10−510^{-5}10−5 to 10−210^{-2}10−2 per step, so that the probability of a multi-step evolution decays exponentially with the number of steps. This study develops multi-relaxation-time Carleman lattice Boltzmann, a representation split in which the Carleman-lifted collision is performed in a moment basis on the amplitude-encoded state, while streaming remains a permutation in population space. The level-2 hierarchy closes under collision but not under streaming, which generates the work of pressure, advection and stress on the momentum flux. The state therefore remains the full pair array, and the moment representation is an orthogonal change of basis of the population one, with identical dynamics and truncation error. The operator to be block encoded changes with the basis, and with it the success probability of a quantum step. In the weighted Hermite basis with the amplitude encoding fi/wif_i/\sqrt{w_i}fi​/wi​​, the relaxation is a diagonal operator of norm one, and the level-2 collision stage becomes this diagonal plus a sparse coupling from products of conserved moments. A structured block encoding of this form raises the single-step success probability on the D2Q9 lattice from 10−210^{-2}10−2 for the best population encoding to 0.98. With the pair sector in relative coordinates and scaled, one step requires O(log⁡2N)O(\log^2 N)O(log2N) Toffoli gates and is verified by statevector simulation. The encoding and readout overhead of a TTT-step run is then polynomial in TTT and NNN rather than accumulating a factor pTp^TpT. For resolved flows, the remaining exponential loss in coherent multi-step Carleman lattice Boltzmann evolution is set by the non-equilibrium norm relaxed by dissipation rather than by the block encoding.
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.
Lattice Boltzmann (LB) on quantum devices must reconcile unitary gate evolution with the dissipative \emph{collision} step. In the multiple-relaxation-time (MRT) class, we work in the common setting of \emph{modewise diagonal} moment relaxation, δmr′=λr δmrδm_r'=λ_r\,δm_rδmr′​=λr​δmr​ with λr∈[−1,1]λ_r\in[-1,1]λr​∈[−1,1] (overrelaxation if λr<0λ_r<0λr​<0). Embedding that contraction in a unitary by block encoding or a linear combination of unitaries (LCU) typically yields subunitary success probability that decays multiplicatively across modes, sites, and time, a key bottleneck for quantum LB. \emph{For the dissipative MRT block alone} we give a \emph{block-encoding-free} construction: a signed \emph{two-rail} population encoding, then a completely positive trace-preserving (CPTP) map (per-rail amplitude damping with survival ∣λr∣|λ_r|∣λr​∣ and, if λr<0λ_r<0λr​<0, a rail SWAP) so that, after the decode, the map agrees with classical MRT relaxation exactly (expectations of the rail number operators, common encoding--decode scale). Trace preservation gives success probability 111 for that substage. The main result is the dissipative MRT block; construction of the equilibrium moment vector~meq=Mfeqm^{\mathrm{eq}}=Mf^{\mathrm{eq}}meq=Mfeq (prescribed~feqf^{\mathrm{eq}}feq, host moment matrix~MMM; notation as in Section~\ref{subsec:generic-mrt}), moment transforms, streaming, and boundaries are composed with it as in a standard host pipeline and lie outside the scope of the formal theorem. Hybrid and fully coherent encodings, adaptive scales, Carleman-based context, and a one-rail no-go in the same nonnegative population framework are in the main text. Audits of the open-channel map on a long LBM collide-stream simulation and on stencil-free inputs both match the target to machine precision.
Abstract We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.
We apply Carleman linearization of the Lattice Boltzmann (CLB) representation of fluid flows to quantum emulate the dynamics of a 2D Kolmogorov-like flow. We assess the accuracy of the result and find a relative error of the order of 10−310^{-3}10−3 with just two Carleman iterates, for a range of the Reynolds number up to a few hundreds. We first define a gate-based quantum circuit for the implementation of the CLB method and then exploit the sparse nature of the CLB matrix to build a quantum circuit based on block-encoding techniques which makes use of matrix oracles. It is shown that the gate complexity of the algorithm is thereby dramatically reduced, from exponential to quadratic. However, due to the need of employing up to seven ancilla qubits, the probability of success of the corresponding circuit for a single time step is too low to enable multi-step time evolution. Several possible directions to circumvent this problem are briefly outlined.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.
This article presents a novel encoding for quantum Lattice Boltzmann method algorithm using Carleman linearization. In contrast to previous articles \cite{Sanavio2024LatticeBC,sanavio2025carleman}, the encoding used allows for local collision rules while keeping a higher probability to obtain the right result, which is of the order of 10−210^{-2}10−2. The algorithm scales as O(log22(N)+Q3)O(log_2^2(N)+Q^3)O(log22​(N)+Q3) each time step with NNN the number of lattice sites of the 2D lattice and QQQ the number of channels with a constant number of qubits when using dynamical circuits.