Classical circuits can simulate quantum aspects

TL;DR

The paper demonstrates that quantum dynamics in finite-dimensional Hilbert spaces can be emulated by classical electrical networks through a generalized similarity transformation $\pmb{\Omega}_{\pmb{\omega}}(\pmb{H})$ mapping Schrödinger evolution $i\,\partial_t|\psi(t)\rangle = H|\psi(t)\rangle$ to a network-based AB dynamics. It introduces a synthesis framework where $n$ dipole subnetworks couple via an $n$-port interaction network with admittance $Y(s)$; enforcing $Y(s)=\alpha+\frac{1}{s}\beta$ yields $\mathbf{A}=\mathbf{C}^{-1}\alpha$ and $\mathbf{B}=\mathbf{C}^{-1}\beta+\omega_0^2$, enabling a direct mapping from quantum Hamiltonians to circuit parameters. The work provides explicit constructions, including an Electrical Pauli representation for two-level systems and generalizations to $n$-level systems, and reinterprets Born's rule in terms of the envelope of port signals. This approach offers practical, coherent-free analog simulators and lays groundwork for extensions to infinite-dimensional settings using transmission lines or optical techniques.

Abstract

This study introduces a method for simulating quantum systems using electrical networks. Our approach leverages a generalized similarity transformation, which connects different Hamiltonians, enabling well-defined paths for quantum system simulation using classical circuits. By synthesizing interaction networks, we accurately simulate quantum systems of varying complexity, from $2-$state to $n-$state systems. Unlike quantum computers, classical approaches do not require stringent conditions, making them more accessible for practical implementation. Our reinterpretation of Born's rule in the context of electrical circuit simulations offers a perspective on quantum phenomena.

Figures (4)

• Figure 1: There are $n$ dipole networks, denoted by $\{\mathcal{N}_1,{\cdots},\mathcal{N}_k,{\cdots},\mathcal{N}_n\}$ interconnected through an interaction network $\mathcal{N}$.
• Figure 2: Alternatives topologies for each dipole subnetwork $\mathcal{N}_k$. Using one of them, the signal: voltage $V_k$ (up) or current $I_k$ (down) and its initial excitation, was useful as a classical coordinate $q_k$ of \ref{['A B']}. The above network is the dual of the one below, and vice versa.
• Figure 3: Under the hypothesis that the $n-$port network $\mathcal{N}$ has an admittance representation with admittance matrix $\pmb{Y}(s)$, for each $k{\in} \mathcal{J}_n$: $L_k\Vert C_k$ tandem dipole circuit of the Figure \ref{['nTopology']} is connected to its $k-$port.
• Figure 4: The admittance representation of the complete circuit whose port$-$voltages $(V_1,V_2)$ reproduce the time evolution \ref{['schr eqt v2']} of the quantum system given by the hamiltonian \ref{['two level Hamiltonian']}. In order to simplify the figure, the initial condition $V_1(0)$ and $V_2(0)$ has been suppressed.