Tensors: Asymptotic Geometry and Developments 2016–2018最新文献

Motivation, first definitions and properties 动机，第一个定义和性质

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/01

引用次数: 0

Detour into probability and information theory 绕道进入概率论和信息论

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/04

引用次数: 0

Tensor networks 张量网络

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/03

Shinji Takeda

{"title":"Tensor networks","authors":"Shinji Takeda","doi":"10.1090/cbms/132/03","DOIUrl":"https://doi.org/10.1090/cbms/132/03","url":null,"abstract":"graphs. Our results on error of free energies are compared against mean-field methods, including the naïve mean-field (NMF), Thouless-Anderson-Palmer equations (TAP), belief propagation, and the neural-network-based variational autoregressive networks (VAN). On the 2D lattice without the external field, the graph is planar, so there are exact solutions [30]. Whereas on the other graphs, we adopt the exact (carefully designed) exponential algorithms [31] (in a reasonable time) to compute exact free energy values for the evaluations. The results are shown in Fig. 4. We can see that, in all experiments, our method outperforms all mean-field methods and the neural-network-based methods, to a large margin. In regular random graphs, small world networks, and the Sherrington-Kirkpatrick model, our accuracy is only limited by the machine precisions (10−16). In the experiments, we choose D̂ 1⁄4 50 and χ̂ 1⁄4 500, and the computational time on each instance is of a few seconds. Empirically, our method is faster than the mean-field methods and the neural-network-based methods. More results about the dependence of the bond dimensions and the computational time can be found in the Supplemental Material [18]. Moreover, it is worth noting that combining with the autodifferential for tensor networks [32] immediately gives our method an ability to perform learning tasks using graphical models. In the Supplemental Material [18], we give an example of using our method to learn a generative model [33–43] on hand-written digits of the MNIST dataset [44]. Application to quantum circuit simulations.—The problem of computing free energy of graphical models is similar to the problem of computing single amplitude estimates of a superconducting quantum circuit [45], which can be treated as a graphical model with complex couplings. Classical simulation of quantum circuits is important for verifying and evaluating the computational advances of quantum computers [20,22–24,46,47]. However, the nearterm noisy intermediate-scale quantum circuits (including Google’s recently announced “supremacy circuit” [48]) are not perfect: each operation of them contains a small error. Thus, an important open question is whether approximate simulations of quantum circuits could beat the noisy quantum device. Answering this question apparently requires advanced studies of approximate algorithms for simulating quantum circuits. Our method directly applies to approximate singleamplitude simulation of quantum circuits with any kind of connectivities, such as two-dimensional lattice [23,24] and random regular graphs, as considered in the quantum approximate optimization algorithm [49], after converting the initial state, the measurement qubit string, and the gates into tensors. The key difference between our method and existing methods for quantum circuit simulation is that, by detecting low-rank structures in the circuit, our method heavily reduces the computational complexity. Although this intr","PeriodicalId":226550,"journal":{"name":"Tensors: Asymptotic Geometry and Developments\n 2016–2018","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133793270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 33

Quantum information theory and the asymptotic geometry of tensors 量子信息理论与张量的渐近几何

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/07

引用次数: 0

Rank and border rank 军衔和边界军衔

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/02

引用次数: 0

Quantum mechanics for quantum information theory 量子力学中的量子信息理论

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/06

引用次数: 0

Strassen’s laser method and spectral theory 施特拉森的激光方法和光谱理论

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/05

引用次数: 0

Moment maps and moment polytopes 矩映射和矩多边形

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2019-07-05 DOI: 10.1090/cbms/132/08

引用次数: 0

Hints and answers to selected exercises 选定练习的提示和答案

Tensors: Asymptotic Geometry and Developments 2016–2018 Pub Date : 2018-05-01 DOI: 10.1017/9781316823200.011

Elizabeth Meckes, M. Meckes

引用次数: 0