Topological quantum order: stability under local perturbations (1001.0344v1)

Abstract: We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H_0 we prove that there exists a constant threshold \epsilon>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions the perturbed Hamiltonian H=H_0+\epsilon V has well-defined spectral bands originating from O(1) smallest eigenvalues of H_0. These bands are separated from the rest of the spectrum and from each other by a constant gap. The band originating from the smallest eigenvalue of H_0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb-Robinson bound.

• The paper proves that topological quantum order remains stable under weak local perturbations, ensuring distinct spectral bands in quantum spin systems.
• It employs discrete Hamiltonian flow equations and relatively bounded operator methods to rigorously control spectral deviations.
• The application of Lieb-Robinson bounds confirms localization of perturbation effects, reinforcing robust quantum error-correction in TQO.

Stability of Topological Quantum Order Under Local Perturbations

The paper by Bravyi et al. presents a comprehensive paper of the stability of topological quantum order (TQO) in the presence of weak, time-independent local perturbations. The research is grounded in the context of quantum spin Hamiltonians composed of geometrically local commuting projectors on a D-dimensional lattice, where certain conditions of topological order are met.

Key Contributions

The authors address the stability of topologically ordered phases at zero temperature, specifically considering interactions that are sums of short-range, bounded-norm perturbations. A central result is the proof that given an unperturbed Hamiltonian $H_0$, there exists a constant threshold $\epsilon > 0$ such that any perturbation $V$ satisfying the specified conditions ensures that the perturbed Hamiltonian $H = H_0 + \epsilon V$ maintains well-defined spectral bands. These bands originate from the $O(1)$ smallest eigenvalues of $H_0$ and are separated from the larger spectrum and each other by a constant gap. The band corresponding to the smallest eigenvalue of $H_0$ exhibits an exponentially diminishing width as a function of the lattice size.

Methodology and Proof Techniques

The proof methodology leverages several advanced theoretical tools:

1. Discrete Hamiltonian Flow Equations: The authors adapt these equations to transform and control the perturbations, aiming to maintain a block-diagonal form that respects the topological order.
2. Theory of Relatively Bounded Operators: This allows the researchers to mathematically handle and bound the spectral deviations introduced by perturbations.
3. Lieb-Robinson Bound: This bound is critical for establishing the locality of perturbation effects, ensuring that interactions do not propagate faster than an exponential decay rate.

Furthermore, they introduce conditions TQO-1 and TQO-2, which are vital for defining the stability of topological phases. TQO-1 demands macroscopic distance consistent quantum error-correcting conditions, while TQO-2 requires consistency between local and global ground subspaces.

Results and Implications

Numerically, the paper establishes stringent bounds showing that the perturbed Hamiltonian's spectrum remains robust against variations, provided the perturbation strength does not exceed specific limits tied to the topological model's properties. This finding is pivotal for supporting the notion that topologically ordered phases can withstand realistic experimental imperfections and remain faithful for quantum computational tasks.

The implications of these results are profound for both practical and theoretical developments in quantum information science. Practically, they suggest that complex quantum systems can be engineered to store and process information resiliently against local noise and perturbations, thereby supporting efforts towards realizing robust quantum memories and processors.

Future Directions

Future research may focus on extending these results to models with non-commuting local interactions or exploring dynamic perturbations. Furthermore, investigating how these stability results apply to various lattices and topologies could broaden the applicability of TQO in different quantum systems. Such endeavors would significantly advance our comprehension of fault-tolerant quantum computation and the scalability of quantum information processing devices.