Szegedy Walk Unitaries for Quantum Maps
Szegedy Walk Unitaries for Quantum Maps
复制标题
量子图的塞格迪步行酉式
DOI：
10.1007/s00220-023-04797-4
复制
发表时间：
2021
影响因子：
2.4
通讯作者：
K. Temme
中科院分区：
文献类型：
--
作者：
P. Wocjan;K. Temme
Szegedy开发了一种用于量化基于随机游走的经典算法的通用方法(Szegedy，in：第45届IEEE计算机科学基础研讨会，第32-41页，2004)。Https://doi.org/10.1109/FOCS.2004.53)。他的工作的一个主要贡献是为任何可逆的随机游动构造了一个游动么正。这种酉性具有两个重要性质：其本征位相为0的本征向量是随机游动极限分布的量子样本，且其本征相位间隙二次大于随机游动的谱间隙。是否可以将Szegedy的随机映射的量子化方法推广到量子映射，这是一个悬而未决的问题。我们通过给出详细平衡Lindbladians(量子马尔可夫半群的生成元)和详细平衡量子信道的Szegedy步么正的显式构造，肯定地回答了这个问题。我们证明了我们的Szegedy步酉化了Lindbladian的不动点作为本征位相为0的本征向量，并且它的本征相位差二次大于Lindbladian的谱间隙。为了构造步行么正，我们利用了详细的平衡Lindbladians的规范形式，表明它们在结构上与Davies生成器相关。我们还解释了如何将Lindbladians的量子化方法应用于量子通道。我们给出了一种有效的量子算法来量子化Davies生成元，这些生成元描述了开放量子系统的许多重要动力学，例如，耦合到浴池的量子系统的驰豫。我们的算法扩展了已知的在量子计算机上模拟量子系统动力学的技术。
Szegedy developed a generic method for quantizing classical algorithms based on random walks (Szegedy, in: 45th annual IEEE Symposium on Foundations of Computer Science, pp 32–41, 2004. https://doi.org/10.1109/FOCS.2004.53 ). A major contribution of his work was the construction of a walk unitary for any reversible random walk. Such unitary posses two crucial properties: its eigenvector with eigenphase 0 is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of the random walk. It was an open question if it is possible to generalize Szegedy’s quantization method for stochastic maps to quantum maps. We answer this in the affirmative by presenting an explicit construction of a Szegedy walk unitary for detailed balanced Lindbladians—generators of quantum Markov semigroups—and detailed balanced quantum channels. We prove that our Szegedy walk unitary has a purification of the fixed point of the Lindbladian as eigenvector with eigenphase 0 and that its eigenphase gap is quadratically larger than the spectral gap of the Lindbladian. To construct the walk unitary we leverage a canonical form for detailed balanced Lindbladians showing that they are structurally related to Davies generators. We also explain how the quantization method for Lindbladians can be applied to quantum channels. We give an efficient quantum algorithm for quantizing Davies generators that describe many important dynamics of open quantum systems, for instance, the relaxation of a quantum system coupled to a bath. Our algorithm extends known techniques for simulating dynamics of quantum systems on a quantum computer.
用于贝叶斯推理和估计配分函数的自适应量子模拟退火
DOI：
10.1137/1.9781611975994.12
发表时间：
2020
期刊：
Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms
影响因子：
--
作者：
Harrow, Aram W.;Wei, A
通讯作者：
Wei, A