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.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.

physics.comp-phCross-listedarXiv:2610.20620Sep 9, 20260 citations

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.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:2609.32310Sep 29, 20260 citations

Stochastic approach and irreversible chemical reactions in small systems

J. P. Badiali

Journal of Electroanalytical ChemistryAug 1, 20112 citations

Reaction-Diffusion Processes and Quantum Chains

Francisco Castilho Alcaraz, V. Rittenberg

NATO ASI series. Series B : PhysicsJan 1, 19932 citations

Tunneling Dynamics Using Classical-like Trajectories with an Effective Quantum Force

Huaqing Li, Jens Aage Poulsen, Gunnar Nyman

We develop an effective quantum force to propagate trajectories in Wigner phase space. In this way, the quantum entanglement between trajectories is represented by the effective quantum force, which is related to the variance of the momentum distribution. Reaction probabilities are obtained for three model potentials. For a cubic potential, the reaction probabilities are found to continuously grow in time long after the classical reaction probabilities have become stationary. This growth is due to tunneling and is in good agreement with numerically exact quantum results. The variance of the momentum distribution is updated at each time step of the trajectories, which is the key to the increase of the reaction probabilities at longer times and allows a nice interpretation of tunneling.We develop an effective quantum force to propagate trajectories in Wigner phase space. In this way, the quantum entanglement between trajectories is represented by the effective quantum force, which is related to the variance of the momentum distribution. Reaction probabilities are obtained for three model potentials. For a cubic potential, the reaction probabilities are found to continuously grow in time long after the classical reaction probabilities have become stationary. This growth is due to tunneling and is in good agreement with numerically exact quantum results. The variance of the momentum distribution is updated at each time step of the trajectories, which is the key to the increase of the reaction probabilities at longer times and allows a nice interpretation of tunneling.

The Journal of Physical Chemistry LettersAug 20, 20138 citations

10^{-5}

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.