Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.20620

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

Hua-Dong Yao·Sauro Succi
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

physics.comp-phquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hua-Dong Yao, Sauro Succi. 2026-09-08. Representation-split Carleman lattice Boltzmann with near-unit step success probability on quantum computers. https://arxiv.org/abs/2610.20620

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Advanced computational methods for simulating the classical hydrogen atom in stochastic electrodynamics

Stochastic electrodynamics (SED) is a theory of nature seeking to derive key aspects of quantum mechanics and quantum electrodynamics from a purely classical physics perspective. While SED has been successful in many areas, it has had mixed success when addressing the classical hydrogen atom bathed in classical electromagnetic zero-point radiation. Because analytical approaches are largely intractable, advanced numerical simulation methods are required. To address this, we develop a pseudo-wave formulation that integrates the solid angle in k-space utilizing symmetry, resulting in a significantly reduced memory footprint and lower computational overhead for generating the zero-point radiation spectrum. This optimization is paired with a logarithmic frequency grid, a sliding window tracking the instantaneous Keplerian frequency, and a Tukey smoothing function to eliminate discrete force discontinuities. Integrated via an adaptive RK4 algorithm with step-doubling across highly dense grids of up to 100-million bins, these techniques demonstrate the asymptotic convergence of ensemble stability. We show that increasing grid resolution in the high-frequency domain systematically mitigates artificial electron decay, yielding a simulated probability density function that closely aligns with the quantum mechanical ground state P(r). Crucially, the simulations exhibit stable, low-eccentricity orbits without any observed atomic ionization events.

physics.comp-ph↗
arXiv2026

Quantum simulation of dissipative acoustic and elastic waves by Schrödingerisation: absorbing boundaries and explicit polynomial-size circuits

Explicit quantum circuits for the acoustic and elastic wave equations, obtained by Hamiltonian simulation of their first-order forms, conserve energy and therefore describe closed domains: an outgoing wave reflects off the edge of the computational region or, with periodic boundaries, wraps around and re-enters from the opposite side. We open the domain with the simplest physical absorber, a graded sponge layer in which every wave field is damped at a rate that grows towards the outer edge, and implement the resulting dissipative dynamics on a quantum simulator by Schrödingerisation, which turns a non-unitary linear evolution into a family of unitary ones by means of a single auxiliary variable. We derive the whole chain from the partial differential equation to the circuit, in one, two and three dimensions. An independent finite-difference solver confirms that damping all wave fields absorbs far better than the common velocity-only damping mask. For the sponge the quantum analysis is sharp: recovery is exact on the whole non- negative half of the auxiliary axis at every time; that axis can be truncated periodically at a length set by the largest damping rate and the simulated time; and the success probability is a fixed constant, about 0.4, times the fraction of the energy not yet absorbed, with only one or two auxiliary qubits measured. On the circuit side, the damping step, which generic synthesis compiles into a number of controlled-NOT gates exponential in the number of qubits, factorises exactly into polynomially many multi-controlled phase gates. Gate-level simulations on up to 25 qubits agree with the classical solutions of the damped wave equations.

physics.comp-ph↗
arXiv2026

Maximal correlation and optimal transformations of probability distributions in tensor-train format: a variational algorithm

The correlation between two subsystems $A$ and $B$ of a many-body system is usually measured through a pair of observables chosen in advance. The Hirschfeld-Gebelein-Rényi maximal correlation removes the choice: it is the largest correlation coefficient between any function of the configuration of $A$ and any function of the configuration of $B$, and the two functions that reach it, the optimal transformations, are the observables that carry the correlation. Both follow from a singular value decomposition (SVD) of the joint distribution normalized by its marginals, a matrix whose dimension is the number of configurations. We show that this SVD can be computed variationally for a classical distribution of discrete variables, such as the Boltzmann distribution of a spin model, given in the tensor-train (matrix-product) format, without ever forming the matrix. The optimal transformations are sought as tensor trains; each step updates a single core in closed form, and the inverse of a small matrix contracted from the distribution and the remaining cores performs the division by the marginals core by core. Every iterate is a feasible pair, so the objective is a lower bound on the maximal correlation that increases monotonically, and the cost grows linearly with the number of variables. The algorithm reproduces the exact SVD on small systems, recovers the known nonlinear optimal transformations of synthetic distributions of up to ten angular variables ($10^{15}$ configurations), and, for the bipartite antiferromagnetic model with $2^{32}$ configurations, returns an optimal transformation that is a function of the group magnetization, linear above the transition and sign-like below it: the staggered order parameter is read off from the distribution instead of being postulated.

physics.comp-ph↗