Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations
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⟧ LD...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
American Physical Society
2025-08-01
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| Series: | PRX Quantum |
| Online Access: | http://doi.org/10.1103/7x71-8j7k |
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| Summary: | 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≥clog(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=CNe^{−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. |
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| ISSN: | 2691-3399 |