LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations

• URL: http://arxiv.org/abs/2411.02384v1
• Date: Mon, 04 Nov 2024 18:52:44 GMT
• Title: LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations
• Authors: Wojciech De Roeck, Vedika Khemani, Yaodong Li, Nicholas O'Dea, Tibor Rakovszky,
• Abstract summary: We generalize the proof of stability of topological order, due to Bravyi, Hastings and Michalakis, to Hamiltonians corresponding to low-density parity check codes. We show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting.
• Score: 0.025206105035672277
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• 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 $d \geq c \log(N)$, 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 quasi-local on the interaction graph defined by $H_0$, and which 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 $\delta = C N e^{-\Theta(d)}$, 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 quasi-local 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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