Path-Integral and Tensor-Network Complexity

Path-integral optimization assigns a cost to a family of Euclidean preparations, while tensor-network complexity counts or weights discrete tensors or gates. Both can produce hyperbolic-looking geometry, but their cost functionals, regulators, equivalence moves, and operational tasks differ from each other and from CV or CA. A geometric resemblance is evidence for a useful model, not an equality of complexities.

Required background. Path-Integral and Euclidean Preparation Complexity defines the preparation task. Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets fixes the comparison contract.

Helpful background. Entanglement Structure and Tensor-Network Ansätze supplies network language. Regulator Dependence and Continuum Complexity supplies cutoff discipline.

Liouville optimization in a two-dimensional CFT

Prepare the vacuum wavefunctional by a Euclidean path integral on $z\geq\epsilon$ with metric

$$\mathrm ds^2=e^{2\phi(z,x)}(\mathrm dz^2+\mathrm dx^2).$$

For one proposed measure of Weyl-rescaling cost,

$$I_L[\phi] =\frac{c}{24\pi} \int\!\mathrm dz\,\mathrm dx \left[(\partial\phi)^2+\mu e^{2\phi}\right],$$

with a boundary condition that fixes the UV metric at $z=\epsilon$. Variation gives

$$\partial^2\phi=\mu e^{2\phi}.$$

The translation-invariant solution

$$e^{2\phi}=\frac{1}{\mu z^2}$$

has constant negative curvature and is a hyperbolic half-plane. This is the central path-integral-optimization result of Caputa et al. Caputa et al. 2017.

The derivation is exact for the chosen functional and boundary data. It does not show that $I_L$ equals a minimal circuit gate count, nor that its hyperbolic metric is the unique bulk spatial slice. Adding allowed higher-derivative Weyl costs changes the Euler–Lagrange equation and optimized geometry.

Tensor networks and discretization cost

A tensor network prepares a regulated state by contracting maps of bond dimension $\chi$. Possible costs include tensor count, weighted gate count, contraction complexity, or a variational action. Network identities can add tensors without changing the state; blocking changes the count while preserving long-distance data. A meaningful cost must quotient or penalize such moves.

Hyperbolic tilings and MERA reproduce scale layers and entanglement patterns Swingle 2012. The number of layers to resolve scale $z$ grows like $\log(z/\epsilon)$, paralleling radial depth, but the coefficient depends on branching, bond dimension, and allowed tensors.

First application

Optimize the vacuum path integral with the Liouville functional and $e^{\phi(\epsilon)}=1/\epsilon$, taking $\mu=1$. Evaluate $I_L$ on $e^\phi=1/z$ with an infrared cutoff $z_{max}$. Then construct a binary scale network with lattice spacing $\epsilon$ and enough layers to reach $z_{max}$. Compare the continuum cost per logarithmic scale with a network cost

$$\mathcal C_{\mathrm{TN}} =\sum_{\text{tensors }v}w_v(\chi_v).$$

The comparison reports the regulator map, $w_v$, bond dimensions, and network equivalence moves. Matching scale dependence is a diagnostic; matching one coefficient after tuning $w_v$ is not an independent dictionary test.

Adversarial control

Add a Weyl-invariant or higher-derivative term to $I_L$, or refine every tensor into two inverse pairs that leave the prepared state invariant. If the cost or apparent geometry changes, the proposal needs an equivalence rule or scheme translation. Compare the same state using a different circuit gate set; no theorem forces the answers to agree.

Regime, evidence ceiling, and handoff

The CFT derivation is continuum but regulator dependent; the network is finite and model dependent. A holographic interpretation additionally needs large central charge, suitable sparsity, controlled $\alpha'$ and bulk loops, and a map between boundary cutoff and radial coordinate. None of these fixes the cost functional uniquely.

The evidence ceiling is a family of preparation models with geometric extrema and useful qualitative matches. Continue to Operational Meaning, Nonuniqueness, and Evidence Status for cross-proposal comparison and to Divergences, Counterterms, and Scheme Dependence for regulator dependence.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Caputa, P., Kundu, N., Miyaji, M., Takayanagi, T., and Watanabe, K. (2017), “Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories,” Physical Review Letters 119, 071602. DOI; arXiv:1703.00456.
• Swingle, B. (2012), “Entanglement Renormalization and Holography,” Physical Review D 86, 065007. DOI; arXiv:0905.1317.