Sherrington-Kirkpatrick Hamiltonian model of meta-surfaces in complex environments
Gabriele Gradoni, Sergio Terranova, Emanuel Colella, Qi Jian Lim, Charles Ross, Zhen Peng · 2023
The marriage of quantum computing (QC) and computational electromagnetics is envisaging fast algorithms for the solution of large design problems with a high number of degrees-of-freedom. We have recently adopted this paradigm and performed a lattice based simulation of reconfigurable meta-surfaces, for both beam-and nullforming in free-space. In particular, we have demonstrated a quantum advantage through the radar cross section of a binary reconfigurable intelligent surface (RIS) under oblique plane-wave incidence The RIS is a meta-surface that dynamically controls waves within complex electromagnetic environments (EMEs) and is being considered for integration in deployed 5G networks A good wealth of scientific literature exist on the evaluation of the communication performance of RIS-assisted wireless links, as well as related physics-based modelling [3]. However, while researchers have devoted substantial efforts in RIS optimisation methods that achieve a specific scattering profile in free-space, a question arises on how to select the phase configuration in presence of multiple propagation paths generated within complex EMEs. In this work, we formulate a random Ising Hamiltonian whose couplings fluctuate on account of the interaction between multiple propagation paths within EMEs, i.e., multi-path fading. The Hamiltonian is constructed from a communications model based on mutual impedance matrices that describe the RIS-assisted wireless link The fading originating from complex EMEs is formulated and cast into impedance formalism via a statistical model grounded on wave chaos theory: The random coupling model (RCM) We show that the so obtained channel transfer matrix configures as a Sherrington-Kirkpatrick (SH) Hamiltonian with generalised complex-valued Hopfield couplings. For both SISO and MIMO systems, we perform numerical Montecarlo simulations that elucidate the transition to Gaussianity of couplings, as parameterised by the average environment losses and RIS line-of-sight channel strengths. Finally, we briefly discuss a research roadmap that includes: i) Retrieve the transmission matrix of complex environments from random coupling fluctuation; ii) Study the relation between channel hardening and onset of a spin-glass phase in RIS; iii) Implement the SH Hamiltonian via adiabatic QC adopting quantum annealing. Achieved results are important for smart surface environment design.