Flat-Space Cosmology & Gauged Tensor Networks

• Flat-space cosmology is a framework that employs gauged tensor networks to enforce local gauge invariance, enabling exact simulation of physical Hilbert spaces.
• Techniques like symmetric tensor PEPS, Gaussian fermionic methods, and quantum circuit protocols are used to construct gauge-invariant states and optimize bond dimensions.
• Applications include simulating lattice gauge theories, exploring topological phases, and investigating holographic dualities, thereby advancing nonperturbative quantum field analyses.

Gauged tensor networks are tensor network architectures in which local or global gauge symmetries are imposed at the level of the tensor components, ensuring that the resulting many-body quantum state or path integral is explicitly gauge-invariant. These constructions span methodologies for both exact gauge-invariant representation of physical Hilbert spaces (e.g., quantum states in lattice gauge theories, topological phases, or quantum gravity) and algorithmic frameworks for variational optimization, simulation, and RG analysis of gauge theories. Their formalism encompasses matrix product states (MPS), projected entangled pair states (PEPS), Gaussian fermionic tensor networks, and more abstract tensorial or categorical constructions, always with a mechanism to enforce or exploit gauge freedom.

1. Algebraic Foundations and Lattice Formulation

At the core of gauged tensor networks is the explicit encoding of the local gauge symmetry algebra in the virtual and physical spaces of tensor networks. Consider a lattice (graph) with matter degrees of freedom on sites and gauge field degrees of freedom on links. A generic many-body Hilbert space is

$$\mathcal{H}_\text{tot} = \bigotimes_{v\in V} \mathcal{H}_v \bigotimes_{e\in E} \mathcal{H}_e,$$

where $\mathcal{H}_e \cong L^2(G)$ for a compact or finite group $G$, carrying both left and right regular representations $L_e(g), R_e(g)$.

Gauge invariance is implemented by projectors onto the invariant subspace at each vertex,

$$P_v = \int_G dg\, U_v(g)\bigotimes_{e:\,v\in\partial e} L/R_e(g/\!g^{-1}),$$

with $U_v(g)$ acting on $\mathcal{H}_v$ (matter). The global projector

$GI = \prod_{v\in V} P_v$

selects the gauge-invariant sector.

In a PEPS/MPS or more general tensor network, gauge symmetry entails that each tensor $T^{ph; a_1\ldots a_z}$ at a vertex transforms as a group-invariant intertwiner: $$\sum_{ph', a_i'} D^{ph}_{ph',a_i,a_i'}(g) T^{ph',a_1'\ldots a_z'} = T^{ph,a_1\ldots a_z},\quad \forall g \in G.$$ For abelian or nonabelian $G$, virtual legs carry irreps, and fusion (intertwiner) tensors ensure that the outgoing charges fuse to trivial—i.e., satisfy local Gauss law.

In the continuum, the physical states lie in the kernel of the local Gauss operators $G^a(x) = Q^a(x) - (1/g) D_i E^a_i(x)$, and all gauge-invariant states can be written as the continuum limit of appropriately gauged tensor networks (e.g., gauged PEPS, MPS, or their continuous analogs) (Roose et al., 13 Nov 2025).

2. Construction of Gauge-Invariant Tensors: Techniques and Algorithms

Symmetric Tensor PEPS Approach: Each tensor is built from blocks labeled by irreps of $G$, and is a $G$-invariant map (intertwiner). For example, in a 2D square lattice,

• Virtual legs: Each carries a $G$-irrep label $\alpha$, $\beta$ etc.
• The site tensor $$T^{\alpha\beta\gamma\delta,a}_{m_\alpha m_\beta m_\gamma m_\delta,m_a}$$ is constructed by fusing the virtual irreps to the physical index via CG coefficients and normalization factors,

$$T^{\alpha\beta\gamma\delta,a}_{m_\alpha m_\beta m_\gamma m_\delta, m_a} = \sum_{q,m_q} C^{\alpha,\beta\rightarrow q}_{m_\alpha,m_\beta,m_q} C^{q,\gamma,\delta\rightarrow a}_{m_q, m_\gamma, m_\delta; m_a} (\Delta_q)^{-1/2}$$

where $C^{\cdot}$ are CG coefficients and $\Delta_q = \dim(q)$ (Canals et al., 22 Dec 2024, Tagliacozzo et al., 2014).

Circuit Protocols for Preparation: One can prepare the PEPS state by a finite-depth quantum circuit: initialize virtual ancillas in a maximally entangled state, apply controlled $U(g)$ gates from ancilla to physical legs, then perform a Fourier transform and measure, post-selecting to enforce the Gauss-law projector (Canals et al., 22 Dec 2024).

Gaussian Fermionic PEPS (GGPEPS): For fermionic lattice gauge theories with matter, start from a globally $G$-invariant Gaussian PEPS, then "gauge" it by introducing physical gauge fields on links and local unitaries $U_G$ that rotate the virtual fermions according to the group action, ensuring local Gauss law. The final state is a superposition over gauge configurations, with the fermionic part efficiently evaluated via covariance matrices and Schur complements (Kelman et al., 19 Apr 2024, Zohar et al., 2017).

Quantum Link Model/Reduced Hilbert Space: In QLM approaches, each link hosts a finite-dimensional quantum-spin representation ("rishons"), and local isometries project matter+rishons to the gauge-singlet subspace. Link projectors enforce constraints (e.g., fixed rishon number per link), often realized as MPOs/PEPOs (Silvi et al., 2014).

Gauge Network (QGN) Ansatz: Rather than a global wavefunction, the QGN formalism defines local wavefunctions on spatial "patches" and non-unitary gauge connections between them, reconstructing global observables via insertion rules. Local gauge freedom is manifest, update rules are gauge-covariant, and computational cost does not increase with spatial dimension (Slagle, 2022).

3. Gauge Fixing, Canonical Forms, and Bond Optimization

Within any tensor network architecture, independent of a physical gauge-group, networks admit a "gauge freedom" in their internal contractions (i.e., distribution of invertible matrices $X$, $X^{-1}$ on internal legs). Properly fixing this gauge is essential for:

• Optimal truncation of bond dimension,
• Well-conditioned variational optimization,
• Achieving canonical or quasi-canonical forms in loopy networks.

The weighted-trace gauge (WTG) and Vidal gauge are canonical forms where singular values (bond spectra) are aligned, and left/right boundary matrices are proportional to identity. Gauge-fixing is achieved by computing bond environments and appropriate eigenproblems; optimal index truncation proceeds via iterative full environment truncation for fidelity maximization (Evenbly, 2018). Recent algorithms employ belief propagation to estimate the bond environments efficiently, outperforming standard SVD/QR-based methods (Tindall et al., 2023).

4. Applications: Lattice Gauge Theories, Topological Order, and Gravity

Lattice Gauge Theories: Gauged tensor networks enable the exact representation and simulation of the physical (Gauss-law) sector of lattice gauge theories with arbitrary matter and gauge content, including $U(1)$, $SU(N)$, and discrete groups. Techniques include:

• Ground-state search (DMRG, PEPS-VUMPS, TTN),
• Real-time and thermal evolution (TEBD, TDVP, MPO purification),
• Computation of Wilson loops, correlators, excitation spectra.

Resource requirements scale polynomially with bond and truncation dimension; major numerical progress has been achieved for the Schwinger model, $Z_N$ and $SU(2)$ gauge theories, and higher-dimensional abelian/nonabelian models (Buyens et al., 2015, Bañuls et al., 2018, Magnifico et al., 3 Jul 2024).

Topologically Ordered Phases: The $d$-dimensional gauge symmetry embedded in the local structure of tensors allows the systematic computation of string operators, $d$-brane operators, ground-state degeneracy, and braiding statistics, directly delineating topological order parameters (e.g., $\mathbb{Z}_2$ toric code PEPS, quantum double models) (Swingle et al., 2010). Decorated tensor network RG schemes preserve gauge invariance and efficiently locate phase boundaries and topological fixed points (Dittrich et al., 2014).

Quantum Gravity and Holography: Recent results establish diffeomorphism-invariant tensor networks that prepare bona fide solutions to all constraints of Chern-Simons-based 3D quantum gravity (e.g., Lorentzian AdS$_3$ via $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$). Gauge constraints project onto flat connections (magnetic) and G-invariant intertwiners at vertices (electric), guaranteeing Wheeler-DeWitt and momentum-constraint solution. The network features an algebra of non-commuting area operators, mirroring overlapping Ryu-Takayanagi surfaces in AdS/CFT (Balasubramanian et al., 15 Oct 2025).

Random tensor network models with explicit bulk gauge symmetry analytically reproduce the quantum-corrected Ryu–Takayanagi formula, with emergent area operators in the center of the operator algebra, operator-valued area contributions, and $n$-dependent Rényi mutual information, matching cosmic-brane backreaction effects in holography (Dong et al., 2023).

5. Coarse-Graining, Continuum Limits, and Numerical Performance

Real-space renormalization and coarse-graining of gauged tensor networks require gauge-preserving truncation algorithms. Decorated tensor networks introduce representation labels as "decorations" retained at all scales, while central tensors encode short-range entanglement. In the continuum, one lifts the gauged construction (e.g., gauged PEPS/MPS) to continuous projected entangled-pair states (gCPEPS), producing path-integral states in the continuum physical Hilbert space, with bond functionals representing auxiliary fields (Roose et al., 13 Nov 2025).

Resource scaling, particularly in $d\geq2$, is challenging due to bond dimension ($\chi$) and local truncation ($d_g$) scaling. Hybrid quantum–classical approaches, local basis optimization, and block RG of gauge fields are key for scaling to QCD-relevant systems (Magnifico et al., 3 Jul 2024).

6. Significance and Outlook

Gauged tensor networks enable sign-problem-free simulation of lattice gauge theories, encode topological order and gauge constraints at the local tensor level, and are potent for both analytical and numerical exploration of quantum many-body systems with gauge symmetry. They facilitate variational and RG methods for nonperturbative quantum field theories, offer new analytic handles on holographic dualities and gravitational codes, and underpin a variety of emerging numerical algorithms for high-dimensional and real-time gauge field dynamics.

Ongoing research directions include:

• Establishing the completeness of continuum gauged CPEPS/MPS families for quantum gauge theories,
• Extending to fermionic, nonabelian, or topological matter,
• Integrating dynamical adaptation of local bond dimension and error control,
• Embedding gauged tensor network blocks into hybrid quantum architectures,
• Probing emergent gauge symmetry, topological entanglement, and gravitational dualities using tensorial and categorical tools.