Moment-Structured Block Encodings of Periodic Finite-Difference Operators

Quantum linear algebra algorithms, such as the Quantum Singular Value Transformation (QSVT), rely on block encoding to represent non-unitary operators as unitaries. The efficiency of these algorithms is heavily dependent on the subnormalization factor, which dictates the success probability and simulation cost. While explicit block-encoding constructions exist for specific operators, they often lack a general, class-wide optimality guarantee. This paper addresses the challenge of constructing optimal, explicit block encodings for the broad class of translation-invariant finite-difference operators on periodic grids.

The authors bridge the gap between spatial stencil coefficients and spectral properties by introducing a "moment-structured" framework. They define the discrete moments of a stencil and demonstrate that the moment order $m$ acts as a unifying parameter. This parameter simultaneously identifies the continuum differential operator being approximated, the vanishing order of the Fourier symbol, and the costs associated with the block-encoding circuit. By leveraging these relationships, the authors derive an analytic, closed-form optimality criterion that operates directly on stencil coefficients, allowing for the certification of optimality across entire families of operators without needing to compute individual eigenvalues.

The study provides several key contributions to quantum algorithm design:

• Optimality Certification: A closed-form criterion that certifies whether a shift-LCU (Linear Combination of Unitaries) construction achieves the optimal subnormalization, applicable uniformly across grid sizes.
• Unified Parameterization: The moment order $m$ is established as the fundamental descriptor for both the continuum approximation and the quantum implementation cost.
• Success Probability: The authors derive explicit success-probability floors for the advection-diffusion family, a class for which no prior explicit spatial block encoding existed.
• Generalization: The framework recovers known optimal encodings for the Laplacian and extends these results to higher-order operators, such as the biharmonic operator.