Polynomial-bond PEPS approximation of gapped grid ground states
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Does every unique ground state of a gapped nearest-neighbor Hamiltonian on an open L×L square grid admit a projected entangled-pair state approximation of polynomial bond dimension? Local dimension q, interaction bound J and spectral gap Δ>0 are fixed; the target normalized vector error is L⁻¹.
Why it matters and what counts as progress
Why it matters: Polynomial bond dimension would provide a compact tensor-network representation for a broad class of two-dimensional gapped ground states.
What counts as progress: Prove bond dimension at most CLᶜ with constants controlled by q,J,Δ, or identify an obstruction under these same gap assumptions.
How the problem developed
- 29 Jan 2015 · Partial result [1]
Norbert Schuch, Michael M. Wolf, Frank Verstraete and J. Ignacio Cirac Established efficient PEPS approximations for Gibbs states and ground states under a polynomial density-of-states assumption, not from a global gap alone.
- 24 Sept 2026 · Proof claim [2]
OpenAI Released a manuscript claiming polynomial bond dimension and inverse-linear normalized-vector error for unique ground states of bounded local Hamiltonians on open square grids with a fixed global gap.
AI contribution
Polynomial PEPS approximation of gapped square-grid ground states
For fixed local dimension q, interaction bound J and global spectral gap Δ>0, the manuscript claims a PEPS with bond dimension at most CLᶜ and normalized vector error at most L⁻¹ for the unique ground state of any on-site plus nearest-neighbour Hamiltonian on an open L×L square grid.
Sources
- Schuch, Wolf, Verstraete and Cirac, Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states
- OpenAI, polynomial PEPS approximation manuscript
- Polynomial PEPS approximation of gapped square-grid ground states
- OpenAI, companion two-dimensional area-law manuscript
- OpenAI Math release provenance
- Cirac and collaborators, related PEPS question
Website timeline
Publication dates and editorial changes.