The ferromagnetic state of an Ising chain can represent a twofold degenerate subspace or equivalently a logical qubit that is protected from excitations by an energy gap. Here, we study a braiding-like exchange operation through the movement of the ferromagnetic state in the qubit subspace, which resembles that of the localized edge modes in a Kitaev chain. The system consists of two Ising chains in a one-dimensional geometry where the operation is simulated through the adiabatic time evolution of the ground state. The time evolution is implemented via the Suzuki-Trotter expansion on basic single- and two-qubit quantum gates using IBM's Aer QASM simulator. The fidelity of the system is investigated as a function of the evolution and system parameters to obtain optimum efficiency and accuracy for different system sizes. Various aspects of the implementation, including the circuit depth, Trotterization error, and quantum gate errors pertaining to noisy intermediate-scale quantum (NISQ) hardware, are discussed as well. We demonstrate that the quantum gate errors, i.e., bit-flip, phase errors, are the dominating factor in determining the fidelity of the system as the Trotter error and the adiabatic condition are less restrictive even for modest values of Trotter time steps. We reach an optimum fidelity $>99\%$ on systems of up to 11 sites per Ising chain. We show that running such simulations is far beyond the reach of current NISQ hardware with the most efficient implementation of a single braiding-like operation for a fidelity above $90\%$ requiring a circuit depth of the order of $\sim {10}^3$ restricting the individual gate errors to be less than $\sim {10}^{-6}$.

中文翻译：

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在数字量子计算机上研究 Ising 链的交换

Ising 链的铁磁态可以代表一个双重简并子空间，或者等效地代表一个逻辑量子位，它被能隙保护免受激发。在这里，我们通过量子位子空间中铁磁态的运动来研究类似编织的交换操作，这类似于 Kitaev 链中的局部边缘模式。该系统由一维几何中的两个伊辛链组成，其中通过基态的绝热时间演化来模拟操作。时间演化是通过 Suzuki-Trotter 扩展使用 IBM 的 Aer QASM 模拟器在基本的单和双量子位量子门上实现的。系统的保真度作为演化和系统参数的函数进行研究，以获得不同系统规模的最佳效率和准确性。还讨论了实现的各个方面，包括电路深度、Trotterization 误差和与噪声中等规模量子 (NISQ) 硬件有关的量子门误差。我们证明了量子门误差，即位翻转、相位误差是确定系统保真度的主要因素，因为即使对于 Trotter 时间步长的适中值，Trotter 误差和绝热条件的限制也较少。我们达到最佳保真度 是确定系统保真度的主要因素，因为即使对于 Trotter 时间步长的适中值，Trotter 误差和绝热条件的限制也较少。我们达到最佳保真度 是确定系统保真度的主要因素，因为即使对于 Trotter 时间步长的适中值，Trotter 误差和绝热条件的限制也较少。我们达到最佳保真度$>99\%$在每个 Ising 链最多 11 个站点的系统上。我们表明，运行此类模拟远远超出了当前 NISQ 硬件的能力范围，最有效地实现了单一编织类操作以实现上述保真度$90\%$ 需要的电路深度为 $～{10}^3$ 限制单个门错误小于 $～{10}^{-6}$.