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Journal of Applied Physics

Topological order detection and qubit encoding in Su–Schrieffer–Heeger type quantum dot arrays

Nikolaos Petropoulos, Robert Bogdan Staszewski, Dirk R. Leipold, Elena Blokhina

Published Feb 18, 2022• 9 citations
Publisher

Abstract

In this study, building on the 1D topological Su–Schrieffer–Heeger (SSH) model, we propose a model of quantum dot arrays with odd and even parity and variable on-site local potentials to examine topological edge states and a possible quantum information encoding, using these states. We first investigate the SSH model with alternating tunneling amplitudes t1 and t2. We study the model in a ring-like structure and then proceed to minimal open-end chains with even (N=4) and odd (N=5) number of dots. Furthermore, we depart from the basic SSH model by introducing local potentials μi, which offer additional control at the cost of breaking the chiral symmetry of the Hamiltonian and study the implications. Then, we propose an idealized “static” charge qubit design, based on encoding the topological invariant ν as qubit states, that exploits the topological nature of the edge states and their collective character. We introduce perturbing noise δtij(t) into the system and demonstrate the robustness of the states for some range of the ratio ξ=t1/t2. Moreover, we show a possible way to detect the presence of topological order in the system using equilibrium dynamics for both even and odd chains. We utilize the quantum informatic measure of bipartite mutual information I{b:e}(2)(t) as a measure of bulk-edge quantum correlations and a quantitative indicator for the manifestation of bulk-edge correspondence; here, we also propose a dynamical qubit encoding with ν for specific quantum chain parity. Finally, we offer a few remarks on potential future explorations.

DOI
10.1063/5.0082214

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In this study, building on the 1D topological Su–Schrieffer–Heeger (SSH) model, we propose a model of quantum dot arrays with odd and even parity and variable on-site local potentials to examine topological edge states and a possible quantum information encoding, using these states. We first investigate the SSH model with alternating tunneling amplitudes t1 and t2. We study the model in a ring-like structure and then proceed to minimal open-end chains with even (N=4) and odd (N=5) number of dots. Furthermore, we depart from the basic SSH model by introducing local potentials μi, which offer additional control at the cost of breaking the chiral symmetry of the Hamiltonian and study the implications. Then, we propose an idealized “static” charge qubit design, based on encoding the topological invariant ν as qubit states, that exploits the topological nature of the edge states and their collective character. We introduce perturbing noise δtij(t) into the system and demonstrate the robustness of the states for some range of the ratio ξ=t1/t2. Moreover, we show a possible way to detect the presence of topological order in the system using equilibrium dynamics for both even and odd chains. We utilize the quantum informatic measure of bipartite mutual information I{b:e}(2)(t) as a measure of bulk-edge quantum correlations and a quantitative indicator for the manifestation of bulk-edge correspondence; here, we also propose a dynamical qubit encoding with ν for specific quantum chain parity. Finally, we offer a few remarks on potential future explorations.
We investigate the emergence of charge waves and their temporal dynamics in one-dimensional Su-Schrieffer-Heeger (SSH) topological chains. Contrary to the conventional view that charge oscillations are suppressed in gapped topological systems with preserved chiral symmetry, we show that such oscillations can indeed occur. The general condition for an arbitrary oscillation period is analysed, and we find that the charge waves propagating along the chain do not depend on its topology, except at the edges, where both topological phases exhibit essential differences. In chains with inequivalent atoms within the SSH unit cell, we observe regular long-period sublattice oscillations that appear simultaneously with even-odd charge oscillations. Furthermore, we study the nonequilibrium dynamics in SSH chains. After a quench, the time evolution of the local density of states and charge occupancies exhibits clear dynamical fingerprints that distinguish topologically trivial and nontrivial phases. Our results establish that transient charge dynamics can distinguish topologically trivial and nontrivial phases in real time by detecting the presence of topologically-protected edge states.
Robust quantum state transfer (QST) is the foundation for information exchange among nodes in quantum networks. In this paper, we propose a robust QST protocol that utilizes topological edge modes in the qubit chains to encode (decode) quantum states (flying qubits). By employing qubits with tunable couplings, we construct Su-Schrieffer-Heeger (SSH) chains as the nodes of a quantum network. The end qubit of each SSH chain is dissipatively coupled to a chiral waveguide, and the dissipative strength is a constant. We refer to the SSH chain with a dissipation channel at the end qubit as the non-Hermitian SSH chain. Comparing the symmetry and energy spectra of the non-Hermitian SSH chain with those of the SSH chain, our analysis reveals that the dissipative dynamics of the topological edge state in the non-Hermitian SSH chain are governed by its imaginary spectra. The edge mode with the imaginary spectrum can be used to encode (decode) quantum states (flying qubits), thereby enabling robust QST between two remote mirrored non-Hermitian SSH chains. Our numerical simulations demonstrate that high-fidelity QST can be achieved even in the presence of coupling errors. Furthermore, we extend our analysis to consider QST in imperfect chiral waveguides, providing insights into the robustness of our protocol under realistic conditions. Our discussion is applicable to various quantum platforms and holds significant implications for constructing large-scale quantum networks.
We show that topological edge states can be realized in a simple superconducting circuit composed of an inductively shunted flux qubit. The Josephson potential of that qubit forms a honeycomb lattice similar to the case of graphene. The additional shunting element produces a quartic potential along one flux direction, which effectively creates a boundary in flux space. We map analytically the low-energy dynamics of the circuit to an extended Su--Schrieffer--Heeger (SSH) chain, in which a quartic potential is added to the usual SSH Hamiltonian. This mapping is confirmed by numerical simulations of the full superconducting circuit. We argue that chiral symmetry still holds for the edge states of the extended model around a chiral point in parameter space. Close to this point, we numerically find that edge states in our superconducting circuit are topologically protected against perturbations that obey chiral symmetry, as is the case for charge noise. Our results show that topological effects previously thought to occur in multi-qubit systems, such as protection against decoherence, can also be found in much simpler single superconducting qubits.
We analyze interacting ultracold bosonic atoms in a one-dimensional superlattice potential with alternating tunneling rates t1 and t2 and inversion symmetry, which is the bosonic analogue of the Su-Schrieffer-Heeger model. A Z2 topological order parameter is introduced which is quantized for the Mott insulating (MI) phases. Depending on the ratio t1/t2 the n=1/2 MI phase is topologically nontrivial, which results in many-body edge states at open boundaries. In contrast to the Su-Schrieffer-Heeger model the bosonic counterpart lacks chiral symmetry and the edge states are no longer midgap. This leads to a generalization of the bulk-edge correspondence, which we discuss in detail. The edge states can be observed in cold atom experiments by creating a step in the effective confining potential, e.g., by a second heavy atom species, which leads to an interface between two MI regions with filling n=1 and n=1/2. The shape and energy of the edge states as well as the conditions for their occupation are determined analytically in the strong coupling limit and in general by density-matrix renormalization group simulations.
Quantum state transfer is one of the basic tasks in quantum information processing. We here propose a theoretical approach to realize arbitrary entangled state transfer through a qubit chain, which is a class of extended Su-Schrieffer-Heeger models and accommodates multiple topological edge states separated from the bulk states. We show that an arbitrary entangled state, from two qubits to N\mathcal{N}N qubits, can be encoded in the corresponding edge states, and then adiabatically transferred from one end to the other of the chain. The dynamical phase differences resulting from the time evolutions of different edge states can be eliminated by properly choosing evolution time. Our approach is robust against both the qubit-qubit coupling disorder and the evolution time disorder. For concreteness of discussions, we assume that such a chain is constructed by an experimentally feasible superconducting qubit system, but our proposal can also be applied to other systems.
Topological insulators have inspired the study with various quantum simulators. Exploiting the tunability of the qubit frequency and qubit-qubit coupling, we show that a superconducting qubit chain can simulate various topological band models. When the system is restricted to the single-spin excitation subspace, the Su-Schrieffer-Heeger (SSH) model can be equivalently simulated by alternating the coupling strength between neighboring qubits. The existence of topological edge states in this qubit chain is demonstrated in the quench dynamics after the first qubit is excited. This excitation propagates along the chain where the qubit-qubit coupling is homogeneous. In contrast, in our qubit chain, the spin-up state localizes at the first qubit and the rest qubits remain in the spin-down state. We further show that the spin-up state can be transported along the chain by modulating the coupling strengths and the qubit frequencies. This demonstrates adiabatic pumping based on the Rice-Mele model. Moreover, we also discuss possible ways to construct other topological models with different topological phenomena within the current technology of superconducting qubits.