[论文解读] Quantum speedup in stoquastic adiabatic quantum computation

Keisuke Fujii|arXiv (Cornell University)|Mar 27, 2018

Quantum Computing Algorithms and Architecture参考文献 1被引用 6

一句话总结

该论文表明，当结合非标准基的单量子比特测量时，具有多项式有界能隙的绝热量子计算（stoquastic adiabatic quantum computation, stoqAQC）可实现通用量子计算与量子加速。该研究构建了一个6-体stoquastic哈密顿量模型，可在多项式时间内模拟通用量子计算与Shor算法，即使在非自适应测量下，也基于复杂性理论的猜想证明了其具备量子优越性。

ABSTRACT

Quantum computation provides exponential speedup for solving certain mathematical problems against classical computers. Motivated by current rapid experimental progress on quantum computing devices, various models of quantum computation have been investigated to show quantum computational supremacy. At a commercial side, quantum annealing machine realizes the quantum Ising model with a transverse field and heuristically solves combinatorial optimization problems. The computational power of this machine is closely related to adiabatic quantum computation (AQC) with a restricted type of Hamiltonians, namely stoquastic Hamiltonians, and has been thought to be relatively less powerful compared to universal quantum computers. Little is known about computational quantum speedup nor advantage in AQC with stoquastic Hamiltonians. Here we characterize computational capability of AQC with stoquastic Hamiltonians, which we call stoqAQC. We construct a concrete stoqAQC model, whose lowest energy gap is lower bounded polynomially, and hence the final state can be obtained in polynomial time. Then we show that it can simulate universal quantum computation if adaptive single-qubit measurements in non-standard bases are allowed on the final state. Even if the measurements are restricted to non-adaptive measurements to respect the robustness of AQC, the proposed model exhibits quantum computational supremacy; classical simulation is impossible under complexity theoretical conjectures. Moreover, it is found that such a stoqAQC model can simulate Shor's algorithm and solve the factoring problem in polynomial time. We also propose how to overcome the measurement imperfections via quantum error correction within the stoqAQC model and also an experimentally feasible verification scheme to test whether or not stoqAQC is done faithfully.

研究动机与目标

• 表征受限于stoquastic哈密顿量（stoqAQC）的绝热量子计算的计算能力。
• 确定在缺乏非stoquastic项的情况下，stoqAQC是否能实现量子计算优越性或通用量子计算。
• 构建一个具有多项式有界能隙的显式stoqAQC模型，通过非标准基中的自适应或非自适应测量实现通用量子计算。
• 解决测量误差问题，并在stoqAQC框架内提出一种基于量子纠错的容错验证方案。

提出的方法

• 该论文基于Feynman-Kitaev哈密顿量构建了一个6-体stoquastic哈密顿量模型，确保绝热演化过程中具有多项式下界的能隙。
• 通过在非标准泡利基中采用自适应单量子比特测量，实现通用量子计算，利用测量量子计算实现任意量子门的能力。
• 该模型被证明可在多项式时间内模拟整数分解的Shor算法，展示了实际的量子加速。
• 通过引入CSS量子纠错码（如Steane码）编码最终态，利用换位单量子比特操作实现容错的逻辑测量。
• 论文提出一种基于稳定子形式的验证协议，用于检验绝热演化与测量结果的正确性。
• 讨论了通过微扰工具将6-体模型简化为2-体stoquastic哈密顿量的方法，在特定映射条件下保持计算能力。

实验结果

研究问题

• RQ1 在无非stoquastic哈密顿量的情况下，stoquastic绝热量子计算能否实现量子计算优越性？
• RQ2 当受限于非标准基中的单量子比特测量时，stoqAQC是否能实现通用量子计算？
• RQ3 所提出的stoqAQC模型能否在多项式时间内模拟Shor算法进行整数分解？
• RQ4 如何在stoqAQC中缓解测量误差？该模型能否实现容错量子计算？
• RQ5 量子相干性与非标准基测量在stoquastic系统中实现量子加速方面起什么作用？

主要发现

• 所提出的stoqAQC模型具有多项式下界的能隙，确保通过绝热演化可在多项式时间内达到最终基态。
• 当对最终态应用非标准基中的自适应单量子比特测量时，该模型可实现通用量子计算，证明了量子计算通用性。
• 即使采用非自适应测量，该模型仍表现出量子计算优越性，因为在标准复杂性理论猜想下，经典模拟是不可行的。
• 该模型可在多项式时间内模拟Shor算法进行整数分解，为stoquastic设置下的量子加速提供了具体实例。
• 使用CSS码可通过换位单量子比特操作实现容错的逻辑测量，增强了对测量误差的鲁棒性。
• 该构造对测量误差具有鲁棒性，并支持基于稳定子测量的验证方案，可实现对正确绝热演化的实验验证。