Topologically Protected Edge States in a Single Flux Qubit

Abstract

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.

Can a single superconducting-qubit circuit host edge modes without a chain of coupled qubits? This paper proposes a three-junction flux qubit shunted by two parallel Josephson-junction arrays. Its Josephson potential forms a honeycomb lattice in node-flux coordinates; the arrays add approximately quartic confinement along one direction, creating an effective boundary in this synthetic space. A tight-binding reduction maps the low-energy dynamics to an SSH chain with spatially varying on-site energies, with intercell hopping $w=t$ and intracell hopping $v=t(1+e^{i2\pi k_2})$, where $k_2$ is an offset charge. The unconfined SSH criterion is $1/3<|k_2|\leq1/2$. Although confinement breaks global chiral symmetry, perturbation theory identifies a tuning condition under which edge-state dispersion and wrong-sublattice weight vanish through order $|v|^2$, leaving corrections of order $|v|^4$. Numerical diagonalization of the circuit Hamiltonian uses $E_J/E_C=30$, $E'_J/E_J=10$, and arrays of 100 and 200 junctions. An effective seven-cell chain supports an in-gap excited doublet, levels 6 and 7, with edge localization and approximately single-sublattice support. Charge-noise dephasing and transition matrix elements are numerically suppressed near $k_2=\pm1/2$; the tight-binding doublet splitting starts at order $|v|^7$. The results are theoretical, with array internal modes eliminated in a Born–Oppenheimer approximation. Protection is approximate and parameter-dependent: the simulated circuit does not exactly meet the tuning condition, only one doublet shows robust signatures, finite-size hybridization lifts degeneracy, and flux noise remains unprotected. No experimental coherence times are reported.