Kavli Affiliate: Birgitta Whaley
| First 5 Authors: Tyler Kharazi, Ahmad M. Alkadri, Jin-Peng Liu, Kranthi K. Mandadapu, K. Birgitta Whaley
| Summary:
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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