Improvements in Quantum SDP-Solving with Applications
Improvements in Quantum SDP-Solving with Applications
复制标题
量子 SDP 求解的应用改进
DOI：
10.4230/lipics.icalp.2019.99
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
A. Gilyén
中科院分区：
文献类型：
--
作者：
Joran van Apeldoorn;A. Gilyén
自从Brand Ao和Svore在2016年发表了第一篇关于求解SDP的量子算法的论文后，量子优化算法取得了快速的发展。最近，Brand Ao等人。改进了所谓量子态输入模型中的量子SDP求解器，其中SDP的输入矩阵被给出为纯净的混合态。他们还给出了量子SDP的第一个非平凡应用-通过为阴影层析成像问题获得更高效的算法(由Aaronson在2017年提出)。在本文中，我们改进了所有以前的量子SDP-求解器。主要是我们构造了两个输入模型的更好的Gibbs采样器，这直接给出了SDP求解的更好界。对于包含$n\x n$矩阵的具有$m$约束的SDP，我们的改进在稀疏矩阵输入模型中得到了SDP求解的$宽倾斜{\数学O}左(\Sqrt{m}+\Sqrt{n}\伽马\右)$上界，在量子态输入模型中得到了$宽倾斜{\数学O}\左(\左(\SQrt{m}+B^{2.5}\伽马^{3.5}\右)B伽马^4\右)$上界.然后，我们将这些结果应用于阴影层析成像问题，以同时改进Aaronson和Brandao等人关于样本复杂性的已知上界。此外，我们还将我们的量子SDP求解器应用于量子态判别和E-最优设计问题。在这两种情况下，我们都在某些参数方面超过了经典的下界，但代价是严重依赖其他一些参数。最后，我们证明了用量子算法求解SDP的两个下界：(1)量子态输入模型中的$\tilde{\Omega}(\Sqrt{m}B/\Eps)$；(2)量子算符输入模型中的$\tilde{\Omega}(\Sqrt{m}\α/\Eps)$。这些下界表明，$Sqrt{m}$因子和对参数$B、α$和$1/EPS$的多项式依赖是必要的。
Following the first paper on quantum algorithms for SDP-solving by Brand\~ao and Svore in 2016, rapid developments has been made on quantum optimization algorithms. Recently Brand\~ao et al. improved the quantum SDP-solver in the so-called quantum state input model, where the input matrices of the SDP are given as purified mixed states. They also gave the first non-trivial application of quantum SDP-solving by obtaining a more efficient algorithm for the problem of shadow tomography (proposed by Aaronson in 2017). In this paper we improve on all previous quantum SDP-solvers. Mainly we construct better Gibbs-samplers for both input models, which directly gives better bounds for SDP-solving. For an SDP with $m$ constraints involving $n\times n$ matrices, our improvements yield an $\widetilde{\mathcal O}\left( \left( \sqrt{m} + \sqrt{n}\gamma \right)s \gamma^4\right)$ upper bound on SDP-solving in the sparse matrix input model and an $\widetilde{\mathcal O}\left( \left(\sqrt{m}+B^{2.5}\gamma^{3.5} \right)B\gamma^4 \right)$ upper bound in the quantum state input model. We then apply these results to the problem of shadow tomography to simultaneously improve the best known upper bounds on sample complexity due to Aaronson and complexity due Brandao et al. Furthermore, we apply our quantum SDP-solvers to the problems of quantum state discrimination and E-optimal design. In both cases we beat the classical lower bound in terms of some parameters, at the expense of heavy dependence on some other parameters. Finally we prove two lowers bounds for solving SDPs using quantum algorithms: (1) $\tilde{\Omega}(\sqrt{m}B/\eps)$ in the quantum state input model, and (2) $\tilde{\Omega}(\sqrt{m}\alpha/\eps)$ in the quantum operator input model. These lower bounds show that the $\sqrt{m}$ factor and the polynomial dependence on the parameters $B,\alpha$, and $1/\eps$ are necessary.
影响因子：
6.4
作者：
Chakrabarti, Shouvanik;Childs, Andrew M.;Li, Tongyang;Wu, Xiaodi
通讯作者：
Wu, Xiaodi