The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary $R$ matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-$\frac{1}{2}$ XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on $4$ sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

中文翻译：

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代数贝特电路

代数 Bethe Ansatz (ABA) 是一种非常成功的分析方法，用于精确求解统计力学和凝聚态物理中的多个物理模型。在这里，我们将 ABA 引入单一形式，以便在量子计算机上直接实现。这是通过使用 QR 分解将构成 ABA 的非酉 $R$ 矩阵提炼成酉矩阵来实现的。我们的算法是确定性的，适用于 Bethe 方程的实根和复根。我们在 spin-$\frac{1}{2}$ XX 和 XXZ 模型上说明我们的方法。我们表明，使用这种方法，可以在量子计算机上有效地准备 XX 模型的本征态，其量子资源与以前的最先进方法相匹配。我们在 IBM 量子计算机上运行小规模的减少错误的实现，包括在 4 美元的网站上为 XX 和 XXZ 模型准备基态。最后，我们使用酉矩阵推导出新形式的杨-巴克斯特方程，并在量子计算机上对其进行验证。