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arXiv:1908.10869 · Physical Review Research

Harnessing symmetry-protected topological order for quantum memories

M. Goihl, Nathan Walk, J. Eisert, N. Tarantino

Published Feb 4, 2020• 8 citations
arXivHTMLPDF

Abstract

By using figures of merit taken from quantum information theory, the authors characterize the information storage capabilities of a disordered and interacting topological spin chain. The performance of the chain depends on the method of encoding and disorder levels in a surprising way.
DOI
10.1103/physrevresearch.2.013120

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Spin chains with symmetry-protected edge modes are promising candidates to realize intrinsically robust physical qubits that can be used for the storage and processing of quantum information. In any experimental realization of such physical systems, weak perturbations in the form of induced interactions and disorder are unavoidable and can be detrimental to the stored information. At the same time, the latter may in fact be beneficial; for instance, by deliberately inducing disorder which causes the system to localize. We explore the potential of using an XZX cluster Hamiltonian to encode quantum information into the local edge modes and comprehensively investigate the influence of both many-body interactions and disorder on their stability over time, adding substance to the narrative that many-body localization may stabilize quantum information. We recover the edge state at each time step, allowing us to reconstruct the quantum channel that captures the locally constrained out-of-equilibrium time evolution. With this representation in hand, we analyze how well classical and quantum information are preserved over time as a function of disorder and interactions. We find that the performance of the edge qubits varies dramatically between disorder realizations. Whereas some show a smooth decoherence over time, a sizable fraction is rapidly rendered unusable as memories. We also find that the stability of the classical information-a precursor for the usefulness of the chain as a quantum memory-depends strongly on the direction in which the bit is encoded. When employing the chain as a genuine quantum memory, encoded qubits are most faithfully recovered for low interaction and high disorder.
Spin chains with symmetry-protected edge zero modes can be seen as prototypical systems for exploring topological signatures in quantum systems. These are useful for robustly encoding quantum information. However in an experimental realization of such a system, spurious interactions may cause the edge zero modes to delocalize. To stabilize against this influence beyond simply increasing the bulk gap, it has been proposed to harness suitable notions of disorder. Equipped with numerical tools for constructing locally conserved operators that we introduce, we comprehensively explore the interplay of local interactions and disorder on localized edge modes in the XZX cluster Hamiltonian. This puts us in a position to challenge the narrative that disorder necessarily stabilizes topological order. Contrary to heuristic reasoning, we find that disorder has no effect on the edge modes in the Anderson localized regime. Moreover, disorder helps localize only a subset of edge modes in the many-body interacting regime. We identify one edge mode operator that behaves as if subjected to a non-interacting perturbation, i.e., shows no disorder dependence. This implies that in finite systems, edge mode operators effectively delocalize at distinct interaction strengths. In essence, our findings suggest that the ability to identify and control the best localized edge mode trumps any gains from introducing disorder.
In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in S=1S=1S=1 spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal TTT symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with D2+T{D}_{2}+TD2​+T symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in S=1S=1S=1 spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of D2+T{D}_{2}+TD2​+T are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''
A Z2{{\mathbb {Z}}}_2Z2​-Index of Symmetry Protected Topological Phases with Time Reversal Symmetry for Quantum Spin Chains