Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer
Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer
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DOI：
10.1103/physreva.105.032419
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发表时间：
2021-12
影响因子：
2.9
通讯作者：
K. Seki;S. Yunoki
中科院分区：
文献类型：
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作者：
K. Seki;S. Yunoki
我们提出了一个扩展版本的自适应变分量子本征解（VQE），并将其应用到一个二分格点上的两个组件的费米-哈伯德模型。在扩展的自适应VQE方法中，在适当选择的子空间中的哈密顿量和参数化量子态的Rayleigh商在子空间内最小化，并在量子电路上实现的变分参数之间进行优化，以获得变分的基态和基态能量。相应的能量导数相对于变分参数表示为Hellmann-Feynman型公式的广义本征值问题的子空间，从而使我们能够使用参数移位规则的评估。自然梯度下降法也推广到优化量子子空间展开方法中的变分参数。作为一个子空间来近似哈密顿量的基态，我们考虑一个Krylov子空间产生的哈密顿量和一个投影的变分态，因此近似的基态可以恢复的哈密顿对称性破坏的参数化变分态准备在量子电路。我们表明，空间对称操作的费米子在占领的基础上可以表示为产品的最近邻费米子交换操作的量子电路。我们还描述了如何自旋和电荷对称操作，即，旋转，可以在量子电路上实现。通过数值模拟，我们证明了空间，自旋和电荷对称性投影可以提高参数化变分态的精度，这可以通过扩展Krylov子空间而不增加变分参数的数量来进一步系统地提高。
We propose an extended version of the symmetry-adapted variational-quantum-eigensolver (VQE) and apply it to a two-component Fermi-Hubbard model on a bipartite lattice. In the extended symmetry-adapted VQE method, the Rayleigh quotient for the Hamiltonian and a parametrized quantum state in a properly chosen subspace is minimized within the subspace and is optimized among the variational parameters implemented on a quantum circuit to obtain variationally the ground state and the ground-state energy. The corresponding energy derivative with respect to a variational parameter is expressed as a Hellmann-Feynman-type formula of a generalized eigenvalue problem in the subspace, which thus allows us to use the parameter-shift rules for its evaluation. The natural-gradient-descent method is also generalized to optimize variational parameters in a quantum-subspace-expansion approach. As a subspace for approximating the ground state of the Hamiltonian, we consider a Krylov subspace generated by the Hamiltonian and a symmetry-projected variational state, and therefore the approximated ground state can restore the Hamiltonian symmetry that is broken in the parametrized variational state prepared on a quantum circuit. We show that spatial symmetry operations for fermions in an occupation basis can be expressed as a product of the nearest-neighbor fermionic swap operations on a quantum circuit. We also describe how the spin and charge symmetry operations, i.e., rotations, can be implemented on a quantum circuit. By numerical simulations, we demonstrate that the spatial, spin, and charge symmetry projections can improve the accuracy of the parametrized variational state, which can be further improved systematically by expanding the Krylov subspace without increasing the number of variational parameters.