利用对称性感知谱界评估量子相位估计的查询复杂度极限
Assessing the query complexity limits of quantum phase estimation using symmetry aware spectral bounds
AI总结：
本文提出对称性感知谱界层级，以更精确评估量子相位估计的查询复杂度，并通过轨道优化计算证明其优于传统谱界，揭示1-范数降低的改进空间与挑战。
摘要：
物理和化学量子算法的计算成本与哈密顿量的谱密切相关，这一性质体现在其本征值所需的重新缩放中。使用1-范数作为谱范数的上界来重新缩放哈密顿量的典型方法适用于有界厄米算符的最一般情况，但忽略了化学系统中常见的对称性的影响。在本工作中，我们引入了一个对称性感知谱界的层级结构，为使用块编码电子结构哈密顿量的量子相位估计算法的性能提供了统一的理解。我们提出了一种基于轨道优化的变分且数值上易处理的方法来计算这些界，以证明对于多种分子基准系统，计算得到的界小于传统的谱界。我们还强调了这些界在热力学极限和完备基组极限下独特的解析和数值缩放行为。我们的工作表明，在降低1-范数方面仍有改进空间，这一改进尚未通过双重分解和张量超收缩等方法实现，但同时也指出了仅通过1-范数降低技术来提升当前量子算法性能的潜在挑战，即可能仅限于小常数因子的改进。
英文摘要：
The computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian, a property that manifests in the necessary rescaling of its eigenvalues. The typical approach of using the 1-norm as an upper bound to the spectral norm to rescale the Hamiltonian suits the most general case of bounded Hermitian operators but neglects the influence of symmetries commonly found in chemical systems. In this work, we introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms using block-encoded electronic structure Hamiltonians. We present a variational and numerically tractable method for computing these bounds, based on orbital optimization, to demonstrate that the computed bounds are smaller than conventional spectral bounds for a variety of molecular benchmark systems. We also highlight the unique analytical and numerical scaling behavior of these bounds in the thermodynamic and complete basis set limits. Our work shows that there is room for improvement in reducing the 1-norm, not yet achieved through methods like double factorization and tensor hypercontraction, but highlights potential challenges in improving the performance of current quantum algorithms beyond small constant factors through 1-norm reduction techniques alone.
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