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arXiv:2411.02384 · PRX Quantum

Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations

Wojciech De Roeck, Vedika Khemani, Yaodong Li, Nicholas O’Dea, Tibor Rakovszky

Published Jul 2, 2025• 10 citations
arXivHTMLPDF

Abstract

We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians H 0 defined by ⟦ N , K , d ⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance d ≥ c log  ( N ) , implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by H 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the O ( 1 ) smallest eigenvalues of H 0 . The band originating from the smallest eigenvalue has 2 K states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth δ = C N e − Θ ( d ) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.

DOI
10.1103/7x71-8j7k

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