Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations

arXiv:2609.31372 · quant-ph · Submitted 2026-09-25 · Read on arXiv

quant-ph

Submitted: 2026-09-25

Updated: 2026-09-25

Terminology

• Block Encoding — 28× in this paper · explained in 4 episodes
• Linear Combination of Hamiltonian Simulation — 3× in this paper
• Quantum Signal Processing — 3× in this paper
• Generalization — 2× in this paper · explained in 13 episodes
• Linear Combination of Unitaries — 2× in this paper
• Quantum Fourier Transform — 2× in this paper
• Quantum Singular Value Transformation — 2× in this paper
• Ancilla Qubits — 1× in this paper · explained in 1 episode
• Discrete Cosine Transformation — 1× in this paper
• Discrete Sine Transformation — 1× in this paper
• Quantum Circuits — 1× in this paper · explained in 2 episodes
• Quantum Singular Value Transformation (QSVT) — 1× in this paper · explained in 4 episodes
• Sparsity — 1× in this paper · explained in 4 episodes
• Trace — 1× in this paper · explained in 2 episodes
• Unitary Dynamics — 1× in this paper · explained in 2 episodes

Sources

• Quantum algorithms: A survey of applications and end-to-end complexities
• Quantum simulation of partial differential equations via Schrodingerisation: technical details
• Efficient block-encodings require structure
• Efficient and Explicit Block Encoding of Finite Difference Discretizations of the Laplacian
• A Linear Combination of Unitaries Decomposition for the Laplace Operator
• Heat-kernels on the discrete circle and interval
• Hamiltonian Simulation by Uniform Spectral Amplification
• Optimal quantum simulation of linear non-unitary dynamics
• Discrete Cosine Transforms on Quantum Computers
• QRTlib: A Library for Fast Quantum Real Transforms

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