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arXiv:2302.03834 · Journal of Applied Physics

Simulation of Kitaev chain using one-dimensional chain of superconducting qubits and environmental effects on topological states

Yang Zhang, Yun-Qiu Ge, Yu-xi Liu

Published Aug 9, 2024• 6 citations
arXivHTMLPDF

Abstract

Kitaev chain is one of the important physical models for studying topological quantum states and quantum computing. We here propose an approach to simulate the one-dimensional Kitaev chain via a circuit of superconducting qubits. In our approach, all coupling parameters can be controlled independently, and a nontrivial gauge phase is constructed. We also study the environmental effects on the topological states of the Kitaev chain. In addition to the independent environment surrounding each qubit, we consider the common environment shared by neighboring qubits. Such an environment can generate effective non-Hermitian dissipative coupling between qubits. Through analysis and numerical calculation, we demonstrate that the common environment can significantly affect the topological properties of the qubit chain. Moreover, we observe that dissipative couplings at the edges of the Kitaev chain affect the topological states more strongly than those located elsewhere. Our work may provide a new way to explore topological phase transitions and environmental effects on topological physics using superconducting qubit circuits.

DOI
10.1063/5.0224271

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Kitaev chain is one of the important physical models for studying topological quantum states and quantum computing. We here propose an approach to simulate the one-dimensional Kitaev chain via a circuit of superconducting qubits. In our approach, all coupling parameters can be controlled independently, and a nontrivial gauge phase is constructed. We also study the environmental effects on the topological states of the Kitaev chain. In addition to the independent environment surrounding each qubit, we consider the common environment shared by neighboring qubits. Such an environment can generate effective non-Hermitian dissipative coupling between qubits. Through analysis and numerical calculation, we demonstrate that the common environment can significantly affect the topological properties of the qubit chain. Moreover, we observe that dissipative couplings at the edges of the Kitaev chain affect the topological states more strongly than those located elsewhere. Our work may provide a new way to explore topological phase transitions and environmental effects on topological physics using superconducting qubit circuits.
We report an experimental work on simulating the Kitaev chain of one-dimensional ppp-wave superconductivity with long-range coupling using superconducting quantum circuits. The effective Hamiltonian of this model in the momentum space is mapped onto that of a qubit driven by a tunable microwave control field. By monitoring the dynamics of the qubit, we generate intuitive yet precise visualization of topological characteristics of different quantum phases of the system. Topological invariants can be directly deduced from the experimental results, without relying on indirect extraction from measurements such as microwave spectroscopy and temporal averaging. Therefore a much enhanced efficiency of measurement can be realized compared to other methods that are often used to simulate topological phase transitions. As a consequence, a comprehensive phase diagram covering a wide range of the parameter space is obtained. Topological phase transitions and quantum multicritical points are clearly demonstrated. In particular, new topological phases emerge as a consequence of long-range coupling. The method used here can be readily generalized to simulate more complex systems on different experimental platforms.
Superconducting quantum circuits are a natural platform for quantum simulations of a wide variety of important lattice models describing topological phenomena, spanning condensed matter and high-energy physics. One such model is the bosonic analog of the well-known fermionic Kitaev chain, a 1D tight-binding model with both nearest-neighbor hopping and pairing terms. Despite being fully Hermitian, the bosonic Kitaev chain exhibits a number of striking features associated with non-Hermitian systems, including chiral transport and a dramatic sensitivity to boundary conditions known as the non-Hermitian skin effect. Here, using a multimode superconducting parametric cavity, we implement the bosonic Kitaev chain in synthetic dimensions. The lattice sites are mapped to frequency modes of the cavity, and the in situ tunable complex hopping and pairing terms are created by parametric pumping at the mode-difference and mode-sum frequencies, respectively. We experimentally demonstrate important precursors of nontrivial topology and the non-Hermitian skin effect in the bosonic Kitaev chain, including chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions. Our experiment is an important first step towards exploring genuine many-body non-Hermitian quantum dynamics.
Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits. We generate an arbitral number of segments in a Kitaev chain which are in the topological phase. A pair of Majorana-like edge states emerge at the edges of a topological sector, which is observed by impedance peaks. When the Kitaev chain contains NNN topological sectors, the degeneracy of the zero-admittance states is 2N2^{N}2N. A set of AC voltages along the chain plays a role of the wave function. An \textit{LC} resonator attached to each Majorana-like edge state registers the phase and the absolute value of the voltage. One qubit consists of the in-phase and opposite-phase states made of two \textit{LC} resonators installed in one topological sector. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. The present scheme may simulate topological quantum computations based on electric circuits.
Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays the role of the wave function. We generate an arbitral number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.
In this work, we study the topological phase transitions of a Kitaev chain generalized by the addition of nearest-neighbor Coulomb interaction. We show the presence of a robust topological phase as a function of the interaction strength and of the on-site energy with associated non-zero energy Majorana states localized at the chain edges. We provide an effective mean-field model that allows for the self-consistent computation of the mean value of the local particle number operator, and we also perform Density Matrix Renormalization Group numerical simulations based on a tensor network approach. We find that the two methods show a good agreement in reporting the phase transition between trivial and topological superconductivity. Temperature robustness within a physically relevant threshold has also been demonstrated. These findings shed light on an entire class of topological interacting one-dimensional systems in which the effects of residual Coulomb interactions play a relevant role.
The physics of the topological state of matter is the second revolution in quantum mechanics. We study the effect of interactions on the topological quantum phase transition and the quantum Berezinskii-Kosterlitz-Thouless (QBKT) transition in topological state of a quantum many-body condensed matter system. We predict a topological quantum phase transition from topological superconducting phase to an insulating phase for the interacting Kitaev chain. We observe interesting behaviour from the results of renormalization group study on the topological superconducting phase. We derive the renormalization group (RG) equation for QBKT through different routes with a few exact solutions along with the physical explanations, wherein we find the existence of two new important emergent phases apart from the two conventional phases of this model Hamiltonian. We also present results of a length-scale dependent study to predict asymptotic freedom like behaviour of the system. We do rigorous quantum field theoretical renormalization group calculations to solve this problem.