Lecture notes that organize quantum PDE algorithms around block encodings of finite-difference and finite-element operators, tracking discretization, preparation, normalization, postselection, and measurement costs.
Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems
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abstract
We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.
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2026 1verdicts
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A Quantum Path to Partial Differential Equations
Lecture notes that organize quantum PDE algorithms around block encodings of finite-difference and finite-element operators, tracking discretization, preparation, normalization, postselection, and measurement costs.