Polynomial representations of Boolean functions over various rings such as Z and Zm have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer m≥2, each Boolean function has a unique multilinear polynomial representation over ring Zm. The degree of such polynomial is called modulo-m degree, denoted as degm(⋅). In this paper, we investigate the lower bound of modulo-m degree of Boolean functions. When m=pk (k≥1) for some prime p, we give a tight lower bound that degm(f)≥k(p−1) for any non-degenerated function f:{0,1}n→{0,1}, provided that n is sufficient large. When m contains two different prime factors p and q, we give a nearly optimal lower bound for any symmetric function f:{0,1}n→{0,1} that degm(f)≥n2+1p−1+1q−1.

In this paper, we study the online quantum state learning problem which is recently proposed by Aaronson et al. (2018). In this problem, the learning algorithm sequentially predicts quantum states based on observed measurements and losses and the goal is to minimize the regret. In the previous work, the existing algorithms may output mixed quantum states. However, in many scenarios, the prediction of a pure quantum state is required. In this paper, we first propose a Follow-the-Perturbed-Leader (FTPL) algorithm that can guarantee to predict pure quantum states. Theoretical analysis shows that our algorithm can achieve an O(√T) expected regret under some reasonable settings. In the case that the pure state prediction is not mandatory, we propose another deterministic learning algorithm which is simpler and more efficient. The algorithm is based on the online gradient descent (OGD) method and can also achieve an O(√T) regret bound. The main technical contribution of this result is an algorithm of projecting an arbitrary Hermitian matrix onto the set of density matrices with respect to the Frobenius norm. We think this subroutine is of independent interest and can be widely used in many other problems in the quantum computing area. In addition to the theoretical analysis, we evaluate the algorithms with a series of simulation experiments. The experimental results show that our FTPL method and OGD method outperform the existing RFTL approach proposed by Aaronson et al. (2018) in almost all settings. In the implementation of the RFTL approach, we give a closed-form solution to the algorithm. This provides an efficient, accurate, and completely executable solution to the RFTL method.

Let R be an invariant polynomial ring of a reductive group acting on a vector space, and let d be the minimum integer such that R is generated by those polynomials in R of degree no more than d. To upper bound such d is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups - the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups - the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if r-copies of them are local unitary equivalent for some r.

With the rapid progress made by industry and academia, quantum computers with dozens of qubits or even larger size are being realized. However, the fidelity of existing quantum computers often sharply decreases as the circuit depth increases. Thus, an ideal quantum circuit simulator on classical computers, especially on high-performance computers, is needed for benchmarking and validation. We design a large-scale simulator of universal random quantum circuits, often called “quantum supremacy circuits”, and implement it on Sunway TaihuLight. The simulator can be used to accomplish the following two tasks: 1) Computing a complete output state-vector; 2) Calculating one or a few amplitudes. We target the simulation of 49-qubit circuits. For task 1), we successfully simulate such a circuit of depth 39, and for task 2) we reach the 55-depth level. To the best of our knowledge, both of the simulation results reach the largest depth for 49-qubit quantum supremacy circuits.

Gaussian boson sampling is a promising model for demonstrating quantum computational supremacy, which eases the experimental challenge of the standard boson-sampling proposal. Here by analyzing the computational costs of classical simulation of Gaussian boson sampling,we establish a lower bound for achieving quantum computational supremacy for a class of Gaussian boson-sampling problems, where squeezed states are injected into every input mode. Specifically, we propose a method for simplifying the brute-force calculations for the transition probabilities in Gaussian boson sampling, leading to a significant reduction of the simulation costs. Particularly, our numerical results indicate that we can simulate 18 photons Gaussian boson sampling at the output subspace on a normal laptop, 20 photons on a commercial workstation with 256 cores, and suggest about 30 photons for supercomputers. These numbers are significantly smaller than those in standard boson sampling, suggesting Gaussian boson sampling may be more feasible for demonstrating quantum computational supremacy.