Our Philosophy

neuraLQX is built on a clear and ambitious vision: to lead the way toward variational loop quantum gravity (vLQG), a computational and conceptual framework in which the fundamental structures of LQG can be studied, approximated, and optimised using modern variational and machine learning techniques.

Our guiding philosophy is to bridge the gap between the highly formal world of canonical LQG and the practical tooling of variational quantum simulation. The idea is not to re-implement NetKet for a niche use case, but to provide a layer of physical meaning on top of it, one that speaks the language of graphs, holonomies, constraints, and gauge invariance.

The result is a toolkit that is simultaneously theoretical and practical, rigorous enough to respect the mathematical structure of LQG, but accessible enough to support exploration and experimentation in real numerical settings.

Why neuraLQX exists

At its heart, neuraLQX aims to answer a central question:

Can we represent and optimise candidate states of quantum geometry directly, using neural-network-based variational approaches, without giving up the core structure of LQG?

To do this, neuraLQX brings together three worlds:

  1. Loop Quantum Gravity, where states are defined on graphs with edges labelled by group representations and vertices by intertwiners.

  2. Variational Quantum Simulation, where trial states are parameterised by neural networks or other flexible ansätze.

  3. High-performance numerical backends, like JAX and NetKet, which provide automatic differentiation, Monte Carlo sampling, and stochastic optimisation.

In practice, this means we treat the LQG Hilbert space as a variational manifold, a space of representable states whose parameters are tuned to minimise constraint violations and capture low-energy or diffeomorphism-invariant structures.

Design principles

From its inception, neuraLQX is guided by several simple but strict principles:

1. Physics-first, abstractions-second.

All abstractions, from graphs to solvers, exist to express physics cleanly. The framework is designed so that LQG objects (graphs, holonomies, constraints, operators) are the native building blocks.

2. Compatibility over duplication.

NetKet has an amazing suite of capabilities. We reuse as much of NetKet’s mature infrastructure as possible, but extend it only where LQG’s needs go beyond it. This means users can mix NetKet-native components (optimisers, samplers) with neuraLQX ones without friction.

3. Incremental realism.

We start from the Abelian \(U(1)^N\) sector, the “Abelian playground”, not because it is simple, but because it allows for fast iteration. Every API choice here is made with a clear path to the full \(SU(2)\) case in mind.

4. Transparency and reproducibility.

Every object (graph, operator, solver) is inspectable, serialisable, and meant to be logged. A simulation should be fully reconstructible from its logs.

5. Scalability and modularity.

The codebase is structured so that the same high-level abstractions, graphs, Hilbert spaces, solvers, can be reused for different levels of theory, from toy models to near-full LQG simulations.

What “variational LQG” means

The term variational loop quantum gravity refers to a new computational viewpoint: treating the physical Hilbert space of LQG as the target of a variational optimisation.

Instead of working with fixed basis states or explicit spin network sums, neuraLQX allows you to:

  • define a graph and Hilbert space consistent with a chosen truncation or cutoff,

  • express the quantum constraints (Hamiltonian, Gauss, diffeomorphism) as operators,

  • define a parameterised neural-network quantum state,

  • and train that state to minimise constraint expectation values or other physically motivated losses.

Symbolically:

\[|\psi_\theta\rangle = \arg\min_{\psi_\theta} \big[ \langle \psi_\theta | \hat{C}^2 | \psi_\theta \rangle + \lambda_\mathrm{penalty} \, \Phi[\psi_\theta] \big]\]

where \(\hat{C}\) denotes a constraint operator and \(\Phi\) optional penalties or regularisers (e.g., orthogonality, gauge averaging, or symmetry enforcement).

This allows LQG research to be phrased as optimisation problems on neural manifolds rather than purely symbolic constraint equations. It opens a door to quantitative experiments with quantum geometry.

Where neuraLQX fits in

neuraLQX is neither a toy simulator nor a black-box ML model. It is a research platform for the next generation of loop quantum gravity simulations.

The package currently serves three distinct but related goals:

  1. Numerical experimentation: Provide a sandbox where researchers can implement and test new constraint forms, penalty terms, and ansätze.

  2. Conceptual unification: Offer a shared language between LQG theorists and computational physicists, one that abstracts away technical details (sampling, JAX tracing, MPI) so they can focus on ideas.

  3. Long-term infrastructure: Build the backbone for variational LQG, where future non-Abelian, graph-changing, and coupled models can live.

The path forward

Our philosophy does not end with the current release. We see neuraLQX as a living project that will evolve with its community and the needs of quantum gravity research. The road ahead includes:

  • moving from \(U(1)^N\) to full \(SU(2)\) support,

  • implementing dynamic (graph-changing) Hamiltonians and diffeomorphism constraints,

  • and enabling training over ensembles of graphs rather than a single one.

Ultimately, our goal is for neuraLQX to make variational LQG a mature and reproducible research programme, a space where neural methods are used not as black boxes, but as tools of understanding.