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arXiv 1609.08571quant-ph
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修正的回路-哈密顿构造的分析与局限性

Analysis and limitations of modified circuit-to-Hamiltonian constructions

Johannes Bausch, Elizabeth Crosson

备注：
36 pages, 2 figures. Significantly extends v1: sec. 4, 5 and 6 (apart from 6.1) are new content
发表信息：
Quantum 2, 94 (2018)
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摘要：

费曼的电路到哈密顿构造将量子计算和多体量子系统的基态联系起来。Kitaev用这个构造来证明局部哈密顿问题的QMA完备性，Aharanov等人用它来证明绝热计算和量子线路模型的等价性。在这项工作中，我们分析了一类修改的电路哈密顿量，其中包括功能，如复杂的重量和分支转移的低能量属性。对于具有线性时钟和复杂权重的历史状态，我们开发了一种用于修改电路传播哈密顿量的方法，以实现在无挫折基态中电路的时间步长上的任何期望的分布，并且表明这可以用于获得用于通用绝热计算的恒定输出概率，同时保留谱隙的$Ω(T^{-2})$缩放，并且在量子位的数量方面没有任何额外的开销。此外，我们建立限制地面能量的增加，由于输入和输出惩罚项修改的三对角时钟的时间步长上的非均匀分布，通过证明一个紧的$O(T^{-2})$上限的产品的频谱间隙和基态重叠的端点的计算。使用变分技术，超越了$Ω(T^{-3})$标度，遵循通常的几何引理，我们证明，标准的费曼-Kitaev哈密顿量已经饱和这个界限。我们审查的形式主义的酉标记图，取代通常的线性时钟的图形，允许分支和循环，我们扩展的$O(T^{-2})$绑定从线性时钟到这个更一般的设置。为了实现这一点，我们应用切比雪夫多项式的范围内的任意厄米矩阵的图直径的谱间隙上的上限。

英文摘要：

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the $Ω(T^{-2})$ scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits. Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight $O(T^{-2})$ upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the $Ω(T^{-3})$ scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the $O(T^{-2})$ bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.

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