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Physical Review A

Quantum state transfer on square lattices with topology

Mei-Song Wei, Ming-Jie Liao, Yiqing Wang, Zijian Lin, Xingze Qiu, Ce Wang, Jingping Xu, Yaping Yang

Published Dec 1, 2023• 4 citations
Publisher

Abstract

The topological properties of a Heisenberg $XY$ spin model on the square lattice in the form of a two-dimensional Su-Schrieffer-Heeger model is investigated through the quench dynamics. Based on the topological edge states or the higher-order topological corner states of the square lattice, we propose a scheme to transmit quantum states with high quality, including single-excitation state and entangled state, on square lattices. For the transmission of the single-excitation state, when the square lattice is in the topological phase, the maximum fidelity at the corner sites of the lattice is much larger than that of bulk sites. In addition, it has been found that, due to the different symmetries of the spin ladder and the square lattice, the maximally entangled state can be generated between two qubits when a qubit at the edge of the spin ladder is excited initially, while it cannot be generated between two qubits at the corners of the square lattice. However, the maximally entangled state of two qubits at the edge of the spin ladder or the corners of the lattice can both be transmitted to the two qubits at the other side of the lattices with high quality.

DOI
10.1103/physreva.108.062401

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The topological properties of a Heisenberg XYXYXY spin model on the square lattice in the form of a two-dimensional Su-Schrieffer-Heeger model is investigated through the quench dynamics. Based on the topological edge states or the higher-order topological corner states of the square lattice, we propose a scheme to transmit quantum states with high quality, including single-excitation state and entangled state, on square lattices. For the transmission of the single-excitation state, when the square lattice is in the topological phase, the maximum fidelity at the corner sites of the lattice is much larger than that of bulk sites. In addition, it has been found that, due to the different symmetries of the spin ladder and the square lattice, the maximally entangled state can be generated between two qubits when a qubit at the edge of the spin ladder is excited initially, while it cannot be generated between two qubits at the corners of the square lattice. However, the maximally entangled state of two qubits at the edge of the spin ladder or the corners of the lattice can both be transmitted to the two qubits at the other side of the lattices with high quality.
The geometrical significance of winding number is used to study the topological characteristics of a single-excitation Heisenberg XY spin model represented by the Su-Schrieffer-Heeger (SSH) model. Due to the presence of topological edge states, the quantum state may be conveyed from the first site to the last site with a high degree of fidelity. Also examined is a Heisenberg XY model on the kagome lattice. In the topological phase of the kagome lattice, the three corner states are not affected by the Dzyaloshinsky-Moriya (DM) interactions. For the kagome lattice with and without the DM interactions, the state of a spin on one corner site can be transferred to the other two corner sites in various ways during the topological phase. The quantum state transfer (QST) for the XY model on the kagome lattice without the DM interactions is symmetrical and can only produce a three-body entangled state. In contrast, the QST for the XY model on the kagome lattice with the DM interactions is in an asymmetrical form, similar to one-way propagation, and it can produce both product states and three-body entangled states.
The recent experimental realization of a programmable two-dimensional square superconducting qubit array opens a new avenue for quantum simulation with qubits [Science 372, 948 (2021)]. Here, we present an experimentally feasible method to achieve two-dimensional superconducting qubit lattice with tunable coupling strengths. A configuration with higher-order topological phases is constructed, featuring topologically protected boundary states in lower dimension (corner states). We show that the quadrupole topological phases can be effectively characterized by the dynamics of the single-excitation quantum state. Moreover, we also explore the detection and dynamics of the corner modes in our qubit lattice system. Particularly, we propose an effective scheme to realize quantum information transfer via the corner states. Our work suggests that superconducting circuits systems are a fertile platform to study novel topological quantum phases and may shed light on the ongoing exploration of topologically protected quantum information processing.
Quantum technologies, holding the promise of exponentially superior performance in comparison to their classical counterparts for certain tasks, have consistently encountered challenges, including instability in quantum light sources, quantum decoherence, and vulnerability to losses that topological photonics happens to adeptly address. Here, we theoretically put forth a quantum Su-Schrieffer-Heeger-type chain designed to greatly enhance single-photon blockade (single PB) effect with topological protection. By designing the deliberate coupling strengths, the quantum level lattices take the form of a one-dimensional array with a topological edge state in single-excitation space and a two-dimensional square breathing lattice with topological corner states in two-excitation space, resulting in enhanced single-photon excitation and the suppression of two-photon transitions. Therefore, the second-order correlation function is diminished by up to 2 orders of magnitude at the cavity resonance frequency, accompanied by stronger brightness. Furthermore, the PB effect is robust to local perturbations in cavity-qubit coupling and qubit frequency, benefitting from topological protection.
High-order topological phases of matter refer to the systems of n-dimensional bulk with the topology of m-th order, exhibiting (n-m)-dimensional boundary modes and can be characterized by topological pumping. Here, we experimentally demonstrate two types of second-order topological pumps, forming four 0-dimensional corner localized states on a 4×4 square lattice array of 16 superconducting qubits. The initial ground state of the system at half-filling, as a product of four identical entangled 4-qubit states, is prepared using an adiabatic scheme. During the pumping procedure, we adiabatically modulate the superlattice Bose-Hubbard Hamiltonian by precisely controlling both the hopping strengths and on-site potentials. At the half pumping period, the system evolves to a corner-localized state in a quadrupole configuration. The robustness of the second-order topological pump is also investigated by introducing different on-site disorder. Our Letter studies the topological properties of high-order topological phases from the dynamical transport picture using superconducting qubits, which would inspire further research on high-order topological phases.
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.
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.