Tensor networks

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 introduces SVD truncation errors, we will illustrate that, at least in the shallow circuits, the error is almost negligible. We perform experiments using standard random circuits on two-dimensional lattices [22–24], which iteratively apply single-qubit gates and two-qubit controlled Z gates to the initial j0; 0;...; 0i state, and finally measure the amplitude of a specific qubit string. The generation protocol is described in detail in the Supplemental Material [18]. We evaluate the performance of our method against the recently developed state-of-the-art exact tensor contraction method [24], which has a precisely predictable space and time complexity. With depth d 1⁄4 8, our algorithm can handle circuits with at most 40 × 40 1⁄4 1600 qubits with SVD accumulated truncation error εSVD ≤ 10−12 on a workstation with 64 GB memory in an hour. (a) (b) (c) (d)