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  "slug": "hybrid-gate-decomposition-and-vector-norm-analysis-for-quantum-pde-solvers",
  "title": "Hybrid Gate Decomposition and Vector-Norm Analysis for Quantum PDE Solvers",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The simulation of high-Reynolds number fluid turbulence remains a grand challenge in computational physics, where the “Curse of Dimensionality” imposes an O(N⁴) complexity scaling on classical solvers (Zylberman, 2025). Quantum algorithms offer a theoretical logarithmic pathway, but their practical implementation is stifled by the limited circuit depth of Noisy Intermediate-Scale Quantum (NISQ) devices. This manuscript introduces a unified framework that primarily leverages vector-norm error analysis to dismantle the “time-step bottleneck” in Hamiltonian simulation. We demonstrate that for transport equations, error bounds based on state smoothness rather than operator norms reduce the required time-steps by a factor of O(4^n), enabling useful simulations on NISQ hardware (Zylberman et al., 2025). Furthermore, we evaluate polylogarithmic-depth gate decompositions as a scalability solution. While these routines offer a “Width-for-Depth” trade-off, our empirical analysis identifies a crossover point at n ≈ 706 qubits; for current NISQ regimes (n < 100), standard linear decompositions remain superior. [...]",
  "abstract_markdown": "The simulation of high-Reynolds number fluid turbulence remains a grand challenge in computational physics, where the “Curse of Dimensionality” imposes an \\(O(N^4)\\) complexity scaling on classical solvers (Zylberman, 2025). Quantum algorithms offer a theoretical logarithmic pathway, but their practical implementation is stifled by the limited circuit depth of Noisy Intermediate-Scale Quantum (NISQ) devices. This manuscript introduces a unified framework that primarily leverages **vector-norm error analysis** to dismantle the “time-step bottleneck” in Hamiltonian simulation. We demonstrate that for transport equations, error bounds based on state smoothness rather than operator norms reduce the required time-steps by a factor of \\(O(4^n)\\), enabling useful simulations on NISQ hardware (Zylberman et al., 2025). Furthermore, we evaluate **polylogarithmic-depth gate decompositions** as a scalability solution. While these routines offer a “Width-for-Depth” trade-off, our empirical analysis identifies a crossover point at \\(n \\approx 706\\) qubits; for current NISQ regimes (\\(n < 100\\)), standard linear decompositions remain superior. [...]",
  "date": "2026-01-01",
  "first_published": "2026-01-01",
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    "titles": [
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        "title": "Hybrid Gate Decomposition and Vector-Norm Analysis for Quantum PDE Solvers"
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        "description": "The simulation of high-Reynolds number fluid turbulence remains a grand challenge in computational physics, where the “Curse of Dimensionality” imposes an O(N⁴) complexity scaling on classical solvers (Zylberman, 2025). Quantum algorithms offer a theoretical logarithmic pathway, but their practical implementation is stifled by the limited circuit depth of Noisy Intermediate-Scale Quantum (NISQ) devices. This manuscript introduces a unified framework that primarily leverages vector-norm error analysis to dismantle the “time-step bottleneck” in Hamiltonian simulation. We demonstrate that for transport equations, error bounds based on state smoothness rather than operator norms reduce the required time-steps by a factor of O(4^n), enabling useful simulations on NISQ hardware (Zylberman et al., 2025). Furthermore, we evaluate polylogarithmic-depth gate decompositions as a scalability solution. While these routines offer a “Width-for-Depth” trade-off, our empirical analysis identifies a crossover point at n ≈ 706 qubits; for current NISQ regimes (n < 100), standard linear decompositions remain superior. [...]",
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