The recent developments in condensed matter theory has elucidated the novel ways of the theoretical characterization of matter, it is the era of the topological phases of matter. It can be called the second revolution in condensed matter physics after the discovery of quantum theory of superconductivity. The Landau paradigm to characterize the quantum phases of matter, in the first era, is based on the emergence of an order parameter for a broken local symmtery. The topological phases of matter, in this respect, cannot be categorized by the conventional methods, where one can observe a phase transition even in the absence of a broken local symmetry. Series of seminal works by M. Berry, D. Haldane, R. Laughlin and X. G. Wen have led the development of the topological characterization of matter which in general falls under the umbrella of Projective Symmetry Groups.

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The unsurprising surprise for the physicists is the observation of the topological modes in classical systems. The take-away message is that the quantum system under investigation can be characterized by its spectrum, i.e. Hamiltonian. The corresponding set of coupled differential equations can very well be realized in classical systems such as lumped element electric circuits, sound waves, mechanical systems.

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In these lecture notes, we aim to compile the state of art and give a pedagogical introduction to analog quantum simulation/computation with electric circuits, dubbed as the topo-electric systems.