Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.

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Quantum networks (QNs) will play a key role in distributed quantum information processing. As the network size increases, network-level errors like random breakdown and intentional attack are inevitable; therefore, it is important to understand the robustness of large-scale QNs, similar to what has been done for the classical counterpart-the internet. For exponential networks such as Waxman networks, errors simply re-parameterize the network and lead to a linear decrease of the quantum capacity with the probability of error. The same linear decay happens for scale-free QNs under random breakdowns, despite the previously discovered robustness in terms of the connectivity. In presence of attack, however, the capacity of scale-free QNs shows a sharp exponential decay with the increasing attack fraction. Our results apply to quantum internet based on fibers for all kinds of quantum communications and provide implications for the future construction of QNs with regard to its robustness.

Entanglement is not only important for understanding the fundamental properties of many-body systems, but also the crucial resource enabling quantum advantages in practical information processing tasks. Although previous works on quantum networks focus on discrete-variable systems, light—as the only traveling carrier of quantum information in a network—is bosonic and thus requires a continuous-variable description. We extend the study to continuous-variable quantum networks. By mapping the ensemble-averaged entanglement dynamics on an arbitrary network to a random-walk process on a graph, we are able to exactly solve the entanglement dynamics. We identify squeezing as the source of entanglement generation, which triggers a diffusive spread of entanglement with a "parabolic light cone”. A surprising linear superposition law in the entanglement growth is predicted by the theory and numerically verified, despite the nonlinear nature of the entanglement dynamics. The equilibrium entanglement distribution (Page curves) is exactly solved and has various shapes depending on the average squeezing density and strength.

Variational quantum circuits (VQCs) have shown great potential in near-term applications. However, the discriminative power of a VQC, in connection to its circuit architecture and depth, is not understood. To unleash the genuine discriminative power of a VQC, we propose a VQC system with the optimal classical post-processing---maximum-likelihood estimation on measuring all VQC output qubits. Via extensive numerical simulations, we find that the error of VQC quantum data classification typically decay exponentially with the circuit depth, when the VQC architecture is extensive---the number of gates does not shrink with the circuit depth. This fast error suppression ends at the saturation towards the ultimate Helstrom limit of quantum state discrimination. On the other hand, non-extensive VQCs such as quantum convolutional neural networks are sub-optimal and fail to achieve the Helstrom limit, demonstrating a trade-off between ansatz complexity and classification performance in general. To achieve the best performance for a given VQC, the optimal classical post-processing is crucial even for a binary classification problem. To simplify VQCs for near-term implementations, we find that utilizing the symmetry of the input properly can improve the performance, while oversimplification can lead to degradation.