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

Journal Article · Tue Aug 19 00:00:00 EDT 2025 · PRX Quantum

DOI:https://doi.org/10.1103/7x71-8j7k· OSTI ID:2999739

De Roeck, Wojciech ^[1]; Khemani, Vedika ^[2]; Li, Yaodong ^[2]; O’Dea, Nicholas ^[2]; Rakovszky, Tibor ^[3]

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1. Katholieke Univ. Leuven (Belgium)
2. Stanford Univ., CA (United States)
3. Budapest Univ. of Technology and Economics (Hungary)

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 𝐻₀ 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 𝐻₀ 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 𝐻₀. 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.

• Research Organization: University of California, Berkeley, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP)
• Grant/Contract Number: SC0019380
• OSTI ID: 2999739
• Journal Information: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 3 Vol. 6; ISSN 2691-3399
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

mathematical physics
phase diagrams
quantum error correction
quantum memories