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Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.
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Explicit block encodings of boundary value problems for many-body elliptic operators
Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.
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- Award ID(s):
- 2120757
- PAR ID:
- 10678970
- Publisher / Repository:
- Quantum
- Date Published:
- Journal Name:
- Quantum
- Volume:
- 9
- ISSN:
- 2521-327X
- Page Range / eLocation ID:
- 1764
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.22331/q-2025-06-04-1764
