[论文解读] Increasing the quantum UNSAT penalty of the circuit-to-Hamiltonian construction

Johannes Bausch, Elizabeth Crosson|arXiv (Cornell University)|Sep 27, 2016

Quantum Computing Algorithms and Architecture参考文献 14被引用 7

一句话总结

该论文通过在改进的费曼-基塔耶夫哈密顿量中引入非均匀历史态，将电路到哈密顿量构造中的量子UNSAT惩罚从 Ω((1−√ε)T⁻³) 提升至 Ω((1−√ε)T⁻²)，在不增加局部项数量或算符范数的前提下实现了改进。该改进在三对角时钟哈密顿量框架内被证明是紧致的，通过在谱间隙与端点处基态重叠乘积上建立紧致的 O(T⁻²) 上限得以证实。

ABSTRACT

The Feynman-Kitaev Hamiltonian used in the proof of QMA-completeness of the local Hamiltonian problem has a ground state energy which scales as $\Omega((1-\sqrt{\epsilon}) T^{-3})$ when it is applied to a circuit of size $T$ and maximum acceptance probability $\epsilon$. We refer to this quantity as the quantum UNSAT penalty, and using a modified form of the Feynman Hamiltonian with a non-uniform history state as its ground state we improve its scaling to $\Omega((1-\sqrt{\epsilon})T^{-2})$, without increasing the number of local terms or their operator norms. As part of the proof we show how to construct a circuit Hamiltonian for any desired probability distribution on the time steps of the quantum circuit (which, for example, can be used to increase the probability of measuring a history state in the final step of the computation). Next we show a tight $\mathcal{O}(T^{-2})$ upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation for any clock Hamiltonian that is tridiagonal in the time register basis, which shows that the scaling of the quantum UNSAT penalty achieved by our construction cannot be further improved within this framework. Our proof of the upper bound applies a quantum-to-classical mapping for arbitrary tridiagonal Hermitian matrices combined with a sharp bound on the spectral gap of birth-and-death Markov chains. In the context of universal adiabatic computation we show how to reduce the number of qubits required to represent the clock by a constant factor over the standard construction, but show that it is otherwise already optimal in the sense we consider and cannot be further improved with tridiagonal clock Hamiltonians, which agrees with a similar upper bound from a previous study.

研究动机与目标

• 提升用于QMA完全性证明的电路到哈密顿量构造中的量子UNSAT惩罚。
• 在不增加电路复杂度或算符范数的前提下，降低费曼-基塔耶夫哈密顿量中的基态能量标度。
• 为三对角时钟哈密顿量建立谱间隙与时间寄存器端点处基态重叠乘积的紧致上界。
• 通过最小化时钟寄存器大小，优化通用绝热量子计算中的量子比特使用。

提出的方法

• 通过在非均匀历史态下构造改进的费曼-基塔耶夫哈密顿量，以提升量子UNSAT惩罚的标度。
• 设计一种电路哈密顿量，实现在量子计算时间步上的任意概率分布。
• 对三对角厄米矩阵应用量子到经典映射，以分析时钟哈密顿量的谱性质。
• 推导出生灭马尔可夫链谱间隙的精确界限，以建立谱间隙与基态重叠乘积的上界。
• 证明在三对角时钟哈密顿量框架内，改进后的 T⁻² 标度为最优。
• 展示该构造相比标准构造，可将时钟寄存器所需的量子比特数减少一个常数因子。

实验结果

研究问题

• RQ1 在不增加电路复杂度或算符范数的前提下，能否将电路到哈密顿量构造中的量子UNSAT惩罚提升至超过 Ω((1−√ε)T⁻³)？
• RQ2 T⁻² 标度的量子UNSAT惩罚是否为三对角时钟哈密顿量的最优标度？
• RQ3 是否可利用非均匀历史态在费曼-基塔耶夫构造中实现更优的基态能量标度？
• RQ4 三对角时钟哈密顿量中，谱间隙与基态重叠乘积的最紧可能上界是什么？
• RQ5 通过该改进构造，是否可将通用绝热量子计算中的量子比特数量减少一个常数因子？

主要发现

• 通过在改进的费曼-基塔耶夫哈密顿量中引入非均匀历史态，将量子UNSAT惩罚从 Ω((1−√ε)T⁻³) 提升至 Ω((1−√ε)T⁻²)。
• 证明了改进后的 T⁻² 标度为最优，因为建立了谱间隙与端点处基态重叠乘积的紧致 O(T⁻²) 上限。
• 该上界适用于所有三对角时钟哈密顿量，其推导基于量子到经典映射以及对生灭马尔可夫链谱间隙的精确界限。
• 构造了一种电路哈密顿量，可实现时间步上的任意概率分布，从而实现对历史态中测量结果的控制。
• 该构造相比标准构造，将表示时钟寄存器所需的量子比特数减少了常数因子。
• 结果确认，该改进构造在三对角框架内为最优，与独立研究中先前的上界一致。