Quantum Epidemic Hamiltonians Overview

• Quantum epidemic Hamiltonians are a framework that maps classical epidemic spreading processes to quantum many-body models, providing insight into coherent transitions and emergent phenomena.
• The methodology involves transforming Markovian epidemic models into quantum tight-binding and spin Hamiltonians using square-root variable substitutions and classical-to-quantum mappings.
• Practical implications include simulating non-equilibrium phase transitions and quantum-like behaviors in open systems, with experimental realizability in platforms like Rydberg atom arrays.

Quantum epidemic Hamiltonians constitute a framework in which epidemic-like spreading processes are represented within quantum many-body systems, and conversely, where quantum mechanical dynamics can be mapped to generalized classical epidemic models. This formalism establishes precise mathematical analogies and, in certain cases, equivalence between classical epidemic dynamics and quantum tight-binding or spin model Hamiltonians, facilitating the study of non-equilibrium transitions and emergent universal behaviors in both quantum and classical domains. Developments in this area have been motivated by, and enable, the simulation and potential realization of "quantum-like" information processing in physical and engineered systems (Pomorski, 2020, Pérez-Espigares et al., 2017).

1. Formulation of Quantum Epidemic Hamiltonians

Quantum epidemic Hamiltonians originate from the mapping between classical Markovian epidemic models and quantum tight-binding systems. In its simplest instantiation, the two-state classical epidemic model with populations $p_1(t)$, $p_2(t)$ (corresponding to, for example, susceptible and infected states) evolves according to

$$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$

where the $s_{jk}$ encode Markov transition (infection/recovery) rates. By introducing "square-root" variables $\psi_j = \sqrt{p_j}$ and performing a change of variables, one obtains a classical Hamiltonian acting in an analogous fashion to a quantum Schrödinger equation in imaginary time, with the correspondence

$$H_\text{cl}(\tau)\vec{\psi}(\tau) = i\hbar\,\frac{d\vec{\psi}}{d\tau}$$

where $\tau = it/\hbar$. This structure can be extended to real-time (Hermitian limit), yielding a tight-binding form with off-diagonal tunneling and diagonal onsite terms, and further generalized to include non-Hermitian dissipative effects via complex on-site energies (Pomorski, 2020).

In an open quantum spin system (for example, a 2D lattice of Rydberg atoms), the quantum epidemic Hamiltonian may take the form

$H = \Omega \sum_k \Pi_k \sigma^x_k$

with $\Pi_k$ projecting onto configurations allowing only specific (facilitated) transitions based on the local "infection" environment (e.g., exactly one neighboring site excited), and $\sigma^x_k$ mediating $p_2(t)$0 flips between healthy and infected states. This directly implements quantum analogues of classical population dynamics and allows for the direct study of quantum coherent and incoherent spreading phenomena (Pérez-Espigares et al., 2017).

2. Classical-to-Quantum Mapping and Density Matrix Evolution

The mapping between epidemic dynamics and quantum Hamiltonian evolution extends to the construction of classical analogues of the quantum density matrix. The classical pure-state density matrix is defined as

$p_2(t)$1

and evolves according to the anticommutator equation

$p_2(t)$2

contrasted with the quantum von Neumann equation

$p_2(t)$3

This mathematical structure allows the classical system to mimic various quantum effects when recast in this formalism, including coherent superpositions and entanglement between effective degrees of freedom (Pomorski, 2020).

In the context of open quantum systems, the full Lindblad master equation incorporates both coherent Hamiltonian evolution and dissipative processes such as dephasing and radiative decay (immunization). In the strong dephasing limit, adiabatic elimination leads to a purely classical master equation for the population probabilities (Pérez-Espigares et al., 2017).

3. Doubling of the State Space and S-matrix Formalism

By decomposing each amplitude $p_2(t)$4 into real and imaginary parts, the quantum system with $p_2(t)$5 levels is recast in terms of $p_2(t)$6 real dynamical variables. The corresponding epidemic evolution is then described by a real $p_2(t)$7 "S-matrix", which for $p_2(t)$8 (two-level system) explicitly encodes the dynamical behavior: $p_2(t)$9 For an $$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$0-site system or $$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$1-qubit system, the dimension of the classical S-matrix is always doubled compared to the quantum Hamiltonian. This construction is key in showing that classical embeddings can fully reproduce the unitary (and dissipative) dynamics of quantum tight-binding models and, by extension, phenomenology such as entanglement and superpositions for this class of models (Pomorski, 2020).

4. Emergence of Quantum Phenomena in Epidemic Models

Quantum epidemic Hamiltonians can realize core quantum phenomena using classical statistical models formulated in the proper amplitude-phase variables. Notably:

• Coherent superposition and quantum-like entanglement: Off-diagonal blocks in the S-matrix allow for the appearance of entanglement and superposition in purely classical epidemic dynamics.
• Rabi-like oscillations: For time-independent two-level epidemic models, the solutions exhibit sinusoidal population exchanges with a frequency matching the Rabi frequency of a quantum two-level system:

$$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$2

• Aharonov–Bohm phases: Magnetic/gauge flux analogues can be encoded via Peierls substitution, modulating infection transition amplitudes with synthetic gauge phases, mapping directly to the classical phase variables in the epidemic model (Pomorski, 2020).

5. Quantum Epidemic Dynamics in Open Spin Systems and Nonequilibrium Phase Transitions

Epidemic dynamics in open quantum spin systems, such as Rydberg atom arrays, demonstrate that the interplay of coherent (quantum) and incoherent (classical, dissipative) processes leads to fundamentally distinct types of nonequilibrium phase transitions. The main findings include:

• In the classical (strong dephasing) regime, epidemic spreading undergoes a continuous percolation-type transition akin to the General Epidemic Process (GEP), with a critical facilitation rate $$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$3 and critical exponents matching the 2D isotropic percolation universality class.
• In the quantum (zero dephasing) regime, the system exhibits a sequence of discontinuous jumps in the stationary population of immune sites, each associated with the onset of additional coherent outbreak wavefronts. This is not captured by classical epidemic theory and constitutes a novel form of first-order–like nonequilibrium phase transitions (Pérez-Espigares et al., 2017).

This behavior is accessible in current Rydberg experimental platforms, with tunable parameters such as the dephasing rate $$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$4, detuning $$\frac{d\vec{p}}{dt} = S(t)\vec{p}(t), \quad S(t) = \begin{pmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{pmatrix},$$5, and the effective blockade radius governing the interaction tails.

6. Practical Implications, Experimental Realizability, and Emergent Quantum Interpretation

The mapping from classical epidemic models to quantum Hamiltonian dynamics suggests that characteristic quantum phenomena traditionally thought to be signatures of irreducible quantum behavior—such as coherence, entanglement, and even Aharonov–Bohm interference—may arise naturally in suitably structured classical stochastic machines. For tight-binding models, all quantum dynamics can in principle be simulated by an underlying classical statistical model provided it is augmented with hidden variables encoding amplitudes and phases.

This recognition opens avenues for "quantum-like" information processing and communication using fully classical hardware—such as engineered CMOS devices with coupled rate equations emulating ideal quantum tight-binding behavior. It also raises foundational questions concerning the ontology of quantum mechanics, specifically, whether quantum mechanics is a fundamental description or only an effective, emergent theory for a richer underlying classical statistical physics (Pomorski, 2020).

Experimentally, quantum epidemic Hamiltonians and their predicted universality classes, transitions, and coherent phenomena are accessible in atomic and solid-state platforms, providing a versatile framework for exploring nonequilibrium critical phenomena, robust quantum simulations, and benchmarking classical-quantum analogies (Pérez-Espigares et al., 2017).

7. References to Key Literature

Reference Type	Title	arXiv ID
Equivalence & mapping	Equivalence between classical epidemic model and non-dissipative and dissipative quantum tight-binding model	(Pomorski, 2020)
Quantum Rydberg system	Epidemic dynamics in open quantum spin systems	(Pérez-Espigares et al., 2017)