Adaptive Quantum Matter: Variational Organization through Ising Agents

TL;DR

Adaptive Quantum Ising Agents (AQIA) introduce a geometry-free, self-referential framework in which finite transverse-field Ising patches interact through state-dependent, information-space couplings. The couplings $w_{ij}^{(oldsymbol{\alpha})}$ are updated from local summaries $oldsymbol{m}_i=(S_i,B_i,U_i)$ via Gaussian kernels, producing a Hermitian $H_{ ext{eff}}$ that evolves self-consistently through a mean-field energy functional $E_{ ext{MF}}[oldsymbol{m}]$ and iterative maps $oldsymbol{m} ooldsymbol{F}[oldsymbol{m}]$. Three emergent adaptive regimes—adaptive domain formation near a critical balance, adaptive glass with frustration, and community polarization—arise from this feedback and exhibit distinct order parameters and network topologies, with finite-size scaling consistent with 2D Ising universality. The framework demonstrates adaptive universality and offers a practical route to programmable quantum matter on current platforms (superconducting, trapped-ion, Rydberg), with experimental signatures including hysteresis, correlation topology, and adaptive kernel spectra, positioning AQIA as a quantum cybernetic paradigm for self-learning quantum materials.

Abstract

The study introduces the Adaptive Quantum Ising Agents (AQIA) framework, a Hamiltonian-based methodology that extends programmable quantum matter into an adaptive domain. Each agent operates as a finite transverse-field Ising subsystem, maintaining internal quantum coherence while interacting through state-dependent feedback channels characterised by reduced observables such as spin polarisation, bond correlation, and internal energy. These informational couplings enable the transformation of a static lattice into a feedback-reconfigurable medium. The effective Hamiltonian generated, which remains Hermitian at each iteration, is resolved self-consistently using a mean-field approximation, where the feedback fields are iteratively adjusted to minimise the total energy. Numerical investigations identify three distinct regimes: domain formation near the feedback--fluctuation critical point, glass-like frustration due to competing feedback channels, and modular polarisation sustained by structured interactions. These phenomena occur independently of geometric embedding, illustrating that informational similarity alone can induce coherent organisation. The AQIA framework is adaptable to implementation on superconducting, trapped-ion, or Rydberg platforms, offering a minimalistic model for exploring self-organisation and learning in adaptive programmable quantum matter.

Figures (14)

• Figure 1: Feedback-iteration progression of local observables. Columns correspond to distinct adaptive regimes: QPT (critical-balance), Glass (frustrated), and Community (modular). Each panel tracks the evolution of (a) spin polarization $S_i$, (b) bond correlation $B_i$, and (c) local energy $U_i$ across successive mean-field iterations. Circles denote individual agents; diamonds indicate population means. The convergence trajectories highlight distinct feedback responses: smooth monotone alignment in the QPT regime, slow disordered relaxation in the glassy case, and structured cluster polarization in the community regime.
• Figure 2: Adaptive network reorganization across feedback iterations. Node colors encode local spin polarization ($S_i$); edge opacities represent coupling strengths $w_{ij}$. Rows correspond to adaptive regimes (QPT, Glass, Community), and columns show successive variational updates. In the QPT case (top row), the network separates into two coherent ferromagnetic clusters. In the glassy regime (middle row), connectivity fluctuates without stable domain formation. In the community regime (bottom row), two polarized modules emerge and persist, illustrating self-organized modularity under variational feedback.
• Figure 3: Variational phase diagram around the adaptive-critical base configuration. Each panel shows equilibrium observables versus intra-agent coupling $J$ and transverse field $\Gamma$: (top left) mean spin magnitude $\langle |S| \rangle$, (top right) Edwards--Anderson order $q_{\mathrm{EA}}$, (bottom left) susceptibility $\chi$, and (bottom right) modularity $Q$. The star marks the simulated base point ($J=1$, $\Gamma=1$). High $\chi$ and partial order indicate a near-critical zone where small variations in feedback parameters trigger large reorganizations of network structure, signifying a self-organized quantum critical regime.
• Figure 4: Variational phase diagram in the adaptive-glass regime. Each panel displays equilibrium observables versus intra-agent coupling $J$ and transverse field $\Gamma$: (top left) mean spin magnitude $\langle |S| \rangle$, (top right) Edwards--Anderson order $q_{\mathrm{EA}}$, (bottom left) susceptibility $\chi$, and (bottom right) modularity $Q$. The absence of sharp ridges or coherent valleys indicates suppression of collective modes. Weak, irregular patterns reflect a rugged energy landscape with numerous shallow minima corresponding to metastable glassy equilibria.
• Figure 5: Variational phase diagram in the community-polarization regime. Each panel displays equilibrium observables versus intra-agent coupling $J$ and transverse field $\Gamma$: (top left) mean spin magnitude $\langle |S| \rangle$, (top right) Edwards--Anderson order $q_{\mathrm{EA}}$, (bottom left) susceptibility $\chi$, and (bottom right) modularity $Q$. The strong ridge in $Q$ identifies the stability window of two coherent communities connected by weak residual bonds.