Schrödinger-Navier-Stokes Equation for the Quantum Simulation of Navier-Stokes Flows
Luca Cappelli, Sauro Succi, Monica Lacatus, Alessandro Zecchi, Alessandro Roggero
Abstract
The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers. To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.
AI Impact Assessments
(3 models)Scientific Impact Assessment
1. Core Contribution
This paper addresses the problem of quantum simulation of classical Navier-Stokes (NS) flows by revisiting a largely forgotten 1985 formulation by Dietrich and Vautherin that casts the NS equations into a Schrödinger-like wave equation (SNS). The core contributions are threefold:
First, the authors identify why the SNS dissipator—being non-polynomial and non-local—is fundamentally problematic for quantum implementations, and propose retreating to the Hamilton-Jacobi (HJ) formulation (NSHJ) where the nonlinearities become second-order polynomials amenable to Carleman linearization.
Second, they develop a tensor-network representation of the Carleman embedding that reduces memory from O((4G)^{N_C}) to O((N_t 4G)^{N_C-2}/(N_C-2)!), enabling fourth-order Carleman simulations that would otherwise require ~10^5 GB.
Third, they emulate the quantum algorithm classically and benchmark it against Carleman Navier-Stokes (CNS) and Carleman Lattice Boltzmann (CLB) approaches on Kolmogorov-like flows at moderate Reynolds numbers (Re ≈ 5–41).
The claim of being "the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity" is significant if validated, as previous approaches either omitted vorticity or handled dissipation classically.
2. Methodological Rigor
The mathematical derivation from SNS to NSHJ is clearly presented and self-contained. The Carleman embedding is standard but appropriately applied. The discretization uses first-order Euler in time and centered finite differences in space—acknowledged by the authors as numerically fragile but adequate for proof-of-concept.
However, several methodological concerns arise:
3. Potential Impact
The paper opens a "fourth avenue" (alongside CNS, Carleman-Grad, and CLB) for quantum simulation of fluids. The tensor-network technique for Carleman embedding is genuinely useful and transferable to other formulations, potentially benefiting the broader quantum-classical simulation community. The memory savings demonstrated (from 10^5 GB to 10^{-2} GB for N_C=4) are dramatic and practically enabling.
However, the real-world impact is currently limited by:
4. Timeliness & Relevance
The paper is well-timed. Quantum simulation of fluid dynamics is a rapidly growing field, with multiple recent publications (2024-2025) exploring various approaches. The identification and revival of the Dietrich-Vautherin formulation adds genuine historical and scientific value. The comparison with concurrent approaches (CNS, CLB) positions this work within the active research landscape.
5. Strengths & Limitations
Strengths:
Limitations:
Overall Assessment
This is a competent, well-structured paper that makes a meaningful theoretical contribution by connecting wave formulations of fluid dynamics to quantum algorithms via Carleman linearization and tensor networks. The tensor-network technique is the most practically impactful component. However, the work remains largely theoretical/emulatory, with significant gaps between the proposed quantum algorithm and any demonstrated advantage. The low Reynolds number regime and absence of quantum resource analysis limit the near-term significance. It advances the field incrementally rather than transformatively.
Generated Apr 19, 2026
Comparison History (38)
Paper 1 addresses the highly sought-after goal of quantum simulation of Navier-Stokes equations, combining quantum computing with classical fluid dynamics. It proposes a novel quantum algorithm using Carleman embedding with tensor-network representations, targeting a problem (CFD) with enormous real-world applications in engineering and science. The breadth of potential impact across computational physics, aerospace, climate modeling, and quantum computing is vast. Paper 2 makes important but more specialized contributions to quantum information theory (entanglement distillation), with narrower immediate applicability. Paper 1's timeliness in the quantum computing era gives it an edge.
Paper 1 addresses the fundamental challenge of quantum simulation of Navier-Stokes equations, which has enormous potential applications across engineering and science. It introduces novel techniques combining Schrödinger-Navier-Stokes formulation with Carleman embedding and tensor-network representations, tackling a problem (computational fluid dynamics) with vast real-world impact. Paper 2, while interesting in proposing Rydberg-atom simulation of Motzkin spin chains, addresses a more niche topic in condensed matter/mathematical physics. The breadth of potential applications and the timeliness of quantum algorithms for fluid dynamics give Paper 1 higher impact potential.
Paper 2 addresses the highly sought-after goal of quantum simulation of Navier-Stokes equations, bridging quantum computing, fluid dynamics, and computational physics. It introduces a novel quantum algorithm combining Schrödinger-Navier-Stokes formulation with Carleman embedding and tensor networks—a first of its kind. Its breadth of impact spans quantum computing, CFD, and engineering applications. While Paper 1 makes a solid incremental advance in tensor network methods for open quantum systems (extending TEMPO to non-commuting coupling operators), Paper 2's cross-disciplinary novelty and timeliness in quantum computational fluid dynamics give it higher potential impact.
Paper 1 presents a more concrete, scalable algorithmic advance with theoretical error-control guarantees and polynomial complexity claims, plus demonstrated construction of up to 100-qubit ansätze on a real, strongly correlated molecular system relevant to drug/photomedicine—suggesting clearer near-term utility for quantum chemistry and NISQ-era workflows. Paper 2 is ambitious and potentially broad (quantum algorithms for Navier–Stokes), but it relies on challenging dissipative dynamics, Carleman embeddings, and moderate-Re validation; practical quantum advantage and rigor of scalability are less established. Thus Paper 1 has higher likely impact.
Paper 2 addresses a fundamental and broadly applicable problem in near-term quantum computing—measurement complexity reduction—with rigorous theoretical proofs (maximal variance reduction bounds) and scalable numerical demonstrations up to 44 qubits. Its results directly impact variational quantum eigensolvers and other near-term algorithms, making it immediately relevant to the large quantum chemistry and materials science communities. Paper 1, while innovative in combining Schrödinger-Navier-Stokes with Carleman linearization for quantum CFD, addresses a more niche problem and demonstrates results only at moderate Reynolds numbers, limiting near-term practical impact.
Paper 2 is likely higher impact due to its rigorous, broadly applicable theoretical result linking mutual information to neural quantum state (NQS) capacity, with implications across tomography, ground-state learning, and finite-temperature modeling. The information-theoretic scaling law is novel, architecture-agnostic, and timely given rapid growth of NQS/ML-for-quantum. It also offers analytical results (stabilizer rank formula) and validated numerical evidence. Paper 1 is innovative but appears narrower (specific quantum algorithm for Navier–Stokes) and faces significant practical barriers (dissipation/embedding overhead), potentially limiting near-term uptake.
Paper 2 has higher potential impact due to its direct link to a grand-challenge application area (Navier–Stokes simulation) and its cross-disciplinary reach (fluid dynamics, quantum computing, numerical analysis, tensor networks). It proposes an actionable algorithmic framework (Carleman/tensor-network embedding of Hamilton–Jacobi) and provides emulation evidence on canonical flows, improving methodological credibility. The topic is timely given rapid advances in quantum algorithms and simulation. Paper 1 is rigorous and novel within quantum many-body/quantum information, but its applications and breadth are narrower and more foundational.
Paper 1 addresses the fundamental challenge of quantum simulation of Navier-Stokes equations—a problem with enormous implications across science and engineering (fluid dynamics, climate modeling, aerodynamics). It introduces a novel quantum algorithm combining Schrödinger-Navier-Stokes formulation with Carleman embedding and tensor networks, claiming to be the first such algorithm for genuine Navier-Stokes equations including pressure, dissipation, and vorticity. Paper 2 offers a useful but incremental optimization of QFT truncation for NISQ hardware. While practically valuable, it represents a more bounded contribution compared to Paper 1's potentially transformative approach to one of the most important computational problems in physics.
Paper 1 likely has higher impact: it targets quantum algorithms for simulating full Navier–Stokes dynamics (pressure, dissipation, vorticity), a high-value and broadly relevant problem across CFD, turbulence, and quantum computing. It adds methodological contributions (HJ reformulation, tensor-network Carleman embedding with memory savings) and provides classical emulation with convergence/accuracy analysis, supporting rigor and near-term relevance. Paper 2 is novel in structured Volkov wavepackets and could influence strong-field/QED wavepacket engineering, but its applications and cross-field breadth appear narrower and more exploratory.
Paper 1 is more novel and methodologically grounded: it targets inference of invariant *-algebraic structures (e.g., decoherence-free subalgebras) from restricted-access multi-time measurement data via maximum-likelihood model selection, and demonstrates feasibility on multiple models including waveguide QED. This advances open-quantum-systems characterization and could impact quantum control, error mitigation, and device verification broadly. Paper 2 is timely and ambitious but higher-risk: it revisits an older SNS formulation, relies on Carleman/HJ embeddings with demonstrated results only at moderate Reynolds numbers and classical emulation, making near-term practical impact less certain.
Paper 2 has higher impact potential because it reports a first-in-class experimental digital quantum simulation of a bosonic SU(2) matrix model on real trapped-ion hardware, with careful error budgeting and symmetry-violation post-selection—methodologically rigorous and timely for NISQ-era quantum simulation. Its relevance spans quantum computing, gauge theories, quantum chaos, and holography, offering a concrete platform benchmark and roadmap for scaling challenges. Paper 1 is innovative algorithmically for Navier–Stokes, but is validated only via classical emulation at moderate Reynolds numbers, making near-term real-world impact and rigor less compelling.
Paper 1 offers a highly novel, simple, and empirically strong predictor (mod-w defect count) with clear mechanistic explanation, rigorous validation across multiple codes/noise levels, and direct near-term operational value for quantum LDPC decoding (including IBM-targeted codes). It can immediately reduce decoding cost by skipping futile BP runs and informs failure mechanisms, impacting quantum error correction practice and hardware roadmaps. Paper 2 is ambitious and broad, but hinges on challenging quantum implementation details and is demonstrated only via classical emulation on limited flow regimes, making near-term real-world impact and methodological maturity less certain.
Paper 1 offers a concrete, parameter-free analytical theory validated across multiple codes/noise levels and tied to measurable performance (e.g., 330× latency reduction with unchanged logical error rate, hardware-level reproduction). Its methodological rigor and near-term applicability to quantum error correction decoding make impact more immediate and likely across QEC, hardware control, and real-time decoding. Paper 2 is conceptually novel and potentially broad, but remains earlier-stage: feasibility for practical quantum advantage is unclear, validation is limited to classical emulation at moderate Reynolds numbers, and real-world deployment depends on quantum resources not yet available.
Paper 1 demonstrates a first experimental realization of quantum-light–boosted tunneling ionization, with a striking, quantified efficiency gain (300 nJ squeezed vacuum matching 7.1 µJ coherent) and a clear new control knob (squeezing at fixed energy). This is highly novel, timely in strong-field/attosecond science, and plausibly enabling for real applications (efficient HHG/frequency conversion, controlled chemistry). Paper 2 is innovative but primarily theoretical/algorithmic, with feasibility constraints (dissipation, Carleman scaling) and limited validation (moderate-Re classical emulation), making near-term impact less certain.
Paper 2 presents a genuinely novel quantum algorithm for simulating Navier-Stokes equations—a fundamental problem in computational physics and engineering. It introduces new theoretical contributions (SNS formulation, Hamilton-Jacobi strategy, tensor-network Carleman embedding) with broad applications across fluid dynamics, aerospace, climate science, and beyond. Paper 1 is a systematic but incremental empirical study of noise effects on a variational quantum classifier using a standard dataset (Titanic), offering practical observations but limited novelty. Paper 2's methodological innovation, theoretical depth, and cross-disciplinary relevance give it substantially higher impact potential.
Paper 1 introduces a new, broadly applicable fundamental noise source (atomic granularity noise) and a unified scaling law with a clear optimization crossover and a hard limit on quantum-enhanced sensing—high novelty with immediate implications for many atomic-ensemble platforms (magnetometers, clocks, interferometers). Its impact spans quantum metrology, sensing engineering, and quantum optics. Paper 2 is timely but higher-risk: quantum algorithms for full Navier–Stokes face severe scalability/feasibility constraints, and results are limited to moderate-Re tests via classical emulation, reducing near-term real-world and methodological impact.
Paper 2 likely has higher impact due to broader cross-disciplinary reach (fluid dynamics, applied math, quantum algorithms), timeliness with quantum simulation of PDEs, and a potentially transformative application domain (Navier–Stokes). It contributes new algorithmic ideas (HJ reformulation, tensor-network Carleman embedding) and provides convergence/accuracy studies, suggesting methodological depth. Paper 1 is novel and experimentally grounded but is demonstrated on a niche NMR platform and focuses on single-qubit gate families; its impact may be more incremental within quantum control/compilation rather than opening a wider application frontier.
Paper 1 is more novel and timely for near-term quantum hardware: it proposes CV QEC without requiring hard-to-prepare GKP ancillas, using only DV (even single-qubit) ancillas and offering practical error-suppression plus concatenation with DV codes. This addresses a central bottleneck for scalable bosonic quantum computing with clear experimental pathways and broad relevance across CV platforms (superconducting, photonic, trapped-ion). Paper 2 is ambitious but higher-risk: quantum algorithms for full Navier–Stokes via Carleman/HJ embedding face severe resource scaling and limited demonstrated regimes, making near-term impact less certain.
Paper 1 addresses the fundamental challenge of quantum simulation of Navier-Stokes equations, combining quantum computing with computational fluid dynamics—two major fields. It introduces a novel quantum algorithm using tensor-network Carleman embedding of Hamilton-Jacobi equations, potentially enabling quantum speedups for one of the most important equations in physics/engineering. The breadth of applications (turbulence, aerodynamics, climate modeling) and the novelty of being the first quantum algorithm for genuine Navier-Stokes with pressure, dissipation, and vorticity give it broader impact potential than Paper 2's incremental improvement to variational MBQC for generative modeling.
Paper 2 has higher potential impact due to greater novelty and breadth: it proposes a new quantum algorithmic route to simulating full Navier–Stokes (including dissipation, pressure, vorticity) via a Hamilton–Jacobi reformulation and tensor-network Carleman embedding, with demonstrated emulation and error/convergence analysis. If scalable, applications span computational fluid dynamics, engineering, climate, and turbulence—high-impact domains and timely for quantum algorithms. Paper 1 is solid and experimentally grounded but more incremental and narrow (sampling fairness tuning in QA for a specific constrained problem, small-size scaling), limiting cross-field reach.