标题： LDPC稳定器码作为有能隙的量子相：在图局部扰动下的稳定性 显示英文标题

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

摘要： 我们将Bravyi、Hastings和Michalakis关于拓扑序稳定性的证明推广到低密度奇偶校验（LDPC）码对应的稳定子哈密顿量，而不再局限于欧几里得空间中的几何局部性。我们考虑由$[[N,K,d]]$LDPC码定义的哈密顿量$H_0$，这些码满足某些拓扑量子序条件：(i) 码距$d \geq c \log(N)$，意味着基态的局部不可区分性，以及(ii) 基态的局部和全局相容性的轻微条件；这些包括好的量子LDPC码，以及其他如双曲格点上的表面码。我们考虑在由$H_0$定义的相互作用图上具有准局部性的弱扰动，并且可以表示为有界范数项的和。只要局部扰动强度小于一个有限常数，我们证明扰动后的哈密顿量具有源自$O(1)$的最小特征值的明确谱带的$H_0$。 从最小特征值起源的能带具有$2^K$个状态，它与谱的其余部分由有限的能量间隙分隔，并且具有指数级窄的带宽$\delta = C N e^{-\Theta(d)}$，这比欧几里得情况下的最佳已知界限还要紧。 我们还得出新的基态子空间通过准局部酉变换与初始码子空间相关，从而可以关联它们的物理性质。 我们的证明使用了一种迭代过程，通过连续旋转来消除哈密顿量中的非无矛盾项。 我们的结果适用于由经典LDPC码构建的量子哈密顿量，这些哈密顿量导致稳定的对称性破缺相。 这些结果表明，LDPC码非常普遍地定义了稳定的间隙量子相，即使在非欧几里得情况下也是如此，从而开启了对这类物质相的系统研究。 显示英文摘要

摘要： 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.