The Norton Theorem for Quantum Circuits and Systems

Abstract

It is well known that classical analog circuits and systems benefit from the powerful Thevenin and Norton theorems. These theorems enable rigorous and exact mathematical simplification and representation of the effect of the rest of a system on the part we want to focus on. We show that there are corresponding versions of a "Quantum Norton Theorem" that enable exact simplification and reduction for quantum circuits and systems, not just for one port, but also for multiport and even open quantum systems. We demonstrate the method on two level systems, chains, system environment partitions, Lindblad examples, and Grover search. The Grover examples show how the reduced network isolates the bright pole governing ideal search and exposes how diagonal disorder transfers spectral weight into dark poles that degrade performance. By partitioning finite dimensional Schrodinger dynamics into retained and eliminated sectors, we show that Gaussian elimination gives an exact reduced equation in which the eliminated subsystem appears as a dynamical self energy and a source term. In the circuit representation, these terms become the Norton admittance and Norton current source seen by the retained quantum port. The same Schur complement construction extends to multiport reductions, composite system environment partitions, density matrix dynamics, and Lindblad evolution in Liouville space.