Speaker
Description
Quantum algorithms for partial differential equations (PDEs) are a promising route to computational advantage in fluid dynamics, yet advantage is typically assumed or shown case-by-case rather than derived. We address this gap directly, investigating two architectures within a unified framework in which PDEs are converted to tensor networks and then mapped to quantum circuits: Hamiltonian simulation (with Schrödingerization where the spatial operator is non-unitary) and linearization via a quantum linear systems algorithm. For each, we derive the structural conditions under which genuine quantum advantage is recovered, using the one-dimensional advection and diffusion equations as qualitatively distinct model problems.
We validate the Hamiltonian simulation pipeline end-to-end for advection, achieving clean MPS/MPO compression (bond dimension χ = 2), shallow Trotterized circuits, and predicted spatial and Trotter error scaling. In parallel, we derive the full linearization complexity landscape: advection admits an exponential speedup, whereas naive tensorization of diffusion inverts advantage structurally. We prove that tensorization alone conserves the conditioning and LCU 1-norm exactly, so improvement must come from truncation. Spectral truncation then yields a master feasibility condition which is an explicit, non-empty window in truncation rank governing when the linearization route recovers advantage for diffusion.
Beyond these two model PDEs, we aim to broaden operator coverage across both directions into a consolidated body of results, surveying candidate mappings of common CFD operators onto quantum circuits. The outcome is a systematic, principled basis for selecting between Hamiltonian simulation and linearization for a given spatial operator.
| Topic | Tensor methods and quantum-inspired machine learning |
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