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arXiv:2604.25429

Deterministic Realization of Classical Dissipation on Quantum Computers

Muhammad Idrees Khan, Sauro Succi, Hua-Dong Yao

physics.comp-phquant-phSubmitted Apr 28, 2026• Announced Apr 29, 2026• 0 citations
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Abstract

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.

Comments
22 pages, 4 figures

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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.
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.
Turbulent fluid simulation is computationally demanding because nonlinear interactions couple a wide range of scales. Quantum computing offers another computational model, but fluid evolution is nonlinear and dissipative whereas closed gate-based dynamics is linear and reversible. The lattice Boltzmann method (LBM) is an attractive discrete target because streaming is a local permutation, yet conventional collision reconstructs a nonlinear equilibrium and applies dissipative relaxation. We present a digital quantum lattice Boltzmann (QLBM) algorithm that keeps that nonlinear timestep in computational-basis registers. Each step writes the updated populations into a clean destination bank, uncomputes the workspace, and resets the obsolete source. The resulting channel is completely positive and trace preserving (CPTP), composes without intermediate measurement, and applies to any lattice Boltzmann stencil that can be evaluated reversibly into a clean bank. Forced D3Q19 homogeneous isotropic turbulence, in direct numerical simulation (DNS) and Smagorinsky large-eddy simulation (LES) at Reynolds number 15000 over one million lattice steps, shows statistical agreement in mass, energy budgets, spectra, and intermittency. The three-dimensional calculations use a digital-register emulator of the circuit. A separate D2Q9 implementation compiles the same contract to reversible gates and agrees bit-exactly with its integer twin. Two-bank storage and serial lattice depth limit near-term feasibility.
The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its translation to quantum algorithms is impeded by the collision process, which is intrinsically nonlinear and irreversible. Several existing quantum formulations implement this process through repeated quantum tomography and state preparation at every timestep, leading to significant overheads. We introduce a quantum LBM based on a denoising-type collision operator that avoids tomography-based updates. The collision dynamics are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric approach filters non-equilibrium components while preserving lattice symmetries and approximating nonlinear terms needed to recover hydrodynamic behavior. A complete pipeline is presented with efficient gate-level realizations, incorporating encoding of distributions, collision, streaming, boundary conditions, and measurement of physical quantities such as hydrodynamic forces. In addition, we outline an approach for implementing projector-based operators deterministically without postselection, paving the way to fully coherent multi-timestep LBM simulations. Numerical experiments for advection-diffusion and flow problems demonstrate that the method reproduces macroscopic behaviors with high accuracy, with performance depending on the choice of reference state.
Reliable multistep quantum lattice-Boltzmann evolution requires controlling computational faults. When ideal collision preserves encoded mass and momentum exactly, changes in those charges can provide syndromes for selected faults at the post-collision checkpoint. A fault that shifts one population then leaves a unique mass-momentum residual labeled by the lattice velocity. A coherent reference register stores the expected charges, so the same test remains valid while the physical fluid charges vary across the lattice. Single bit flips on population number registers occupy disjoint syndrome sectors and satisfy the Knill-Laflamme condition. A state-vector demonstration on eighteen data qubits of the two-dimensional nine-velocity lattice (D2Q9) recovers that single-bit family to double-precision roundoff. Conservation still leaves a fifteen-dimensional kinetic nullspace on the three-dimensional nineteen-velocity lattice (D3Q19). Exact multiple-relaxation-time (MRT) streaming analysis ranks those charge-preserving modes by the order at which they return to density or momentum. A classical D3Q19 decaying-flow simulation at moderate Reynolds number then injects about 1.75×1041.75\times10^41.75×104 selected single-population shifts per realization. Recovery of that alphabet returns the trajectory to floating-point roundoff. The quantum statements assume an ideal charge-preserving collision and a reliable reference. The result is an inner recovery map for a stated charge-changing alphabet together with a kinetic classification of the unresolved sector.
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 lifting converts nonlinear polynomial dynamics into finite linear systems, but the resulting truncations are generally nonunitary and need not correspond to physical quantum evolution. We prove that a finite linear endpoint admits an autonomous Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) realization on vacuum coherences if and only if it is invertible and power bounded. The construction is explicit and realizes the nonunitary map directly as open-system dynamics, with no endpoint postselection and with one encoding and one decoding over repeated timesteps. We apply the result to the complete D2Q9 multiple-relaxation-time lattice Boltzmann (LB) timestep by compiling collision and periodic streaming into a single Carleman endpoint. The resulting GKSL evolution reproduces the classical Carleman trajectory over multiple timesteps, while the remaining discrepancy from nonlinear LB dynamics is the expected Carleman truncation error. The result establishes a general criterion for autonomous open-quantum realization of finite nonunitary dynamics, with Carleman--LB dynamics as a concrete example.