Emulation of Density Matrix Dynamics With Classical Analog Circuits

Analog circuits have emerged as a valuable quantum emulation and simulation platform. Specifically, they have been experimentally shown to excel in emulating coherent state vector dynamics and motifs of quantum circuits, such as the quantum Fourier transform, tensor product superpositions, two-level systems such as Josephson junctions, and nuclear magnetic resonance state dynamics, all on a very large scale integration chip at room temperature (Cressman et al., 2022; Sarpeshkar, 2019a, 2019b, 2019c; Sarpeshkar, 2020). However, the ability to model simple state vectors is insufficient for modeling open quantum systems, i.e., systems with environmental noise. Noisy quantum systems are essential in practical implementations and applications that exploit noise. The density matrix formalism enables us to model such states, including finite reservoir state systems, and all states that can be represented as state vectors. To our knowledge, no one has yet demonstrated the mapping of a density matrix system to classical analog circuit components. We review the procedure for emulating the dynamics of a finite state vector with four essential analog circuit components and extend this procedure to emulate density matrix dynamics. We then simulate these systems as analog circuits in the presence of noise. This protocol opens up exciting possibilities for further research and development in noisy quantum emulation and simulation using analog circuits for arbitrarily large or small systems.