Quantum computing and modern tensor-based computing have a strong connection, which is especially demonstrated by simulating quantum computations with tensor networks. The other direction is less studied: quantum computing is not often applied to tensor-based problems. Considering tensor decompositions, we focus on discovering practical matrix multiplication algorithms and develop two algorithms to compute decompositions on quantum computers. The algorithms are expressed as higher-order unconstrained binary optimization (HUBO) problems, which are translated into quadratic unconstrained binary optimization (QUBO) problems. Our first algorithm is decompositional to keep the optimization problem feasible for the current quantum devices. Starting from a suitable initial point, the algorithm discovers tensor decomposition corresponding to the famous Strassen matrix multiplication algorithm, utilizing the current quantum annealers. Since the decompositional algorithm does not guarantee minimal length for found tensor decompositions, we develop a holistic algorithm that can find fixed-length decompositions. Theoretically, by fixing a shorter length than the length for the best-known decomposition, we can ensure that the solution to the holistic optimization problem would yield faster matrix multiplication algorithms.
In this paper, we present a quantum singular value decomposition algorithm for third-order tensors inspired by the classical algorithm of tensor singular value decomposition (t-svd) and then extend it to order-
p tensors. It can be proved that the quantum version of the t-svd for a third-order tensor
A∈RN×N×N achieves the complexity of
O(Npolylog(N)), an exponential speedup compared with its classical counterpart. As an application, we propose a quantum algorithm for recommendation systems which incorporates the contextual situation of users to the personalized recommendation. We provide recommendations varying with contexts by measuring the output quantum state corresponding to an approximation of this user's preferences. This algorithm runs in expected time
O(Npolylog(N)poly(k)), if every frontal slice of the preference tensor has a good rank-
k approximation. At last, we provide a quantum algorithm for tensor completion based on a different truncation method which is tested to have a good performance in dynamic video completion.
Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify
K high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.
Tensors are a natural generalization of matrices, and tensor networks are a natural generalization of matrix products. Despite the simple definition of tensor networks, they are versatile enough to represent many different kinds of "products" that arise in various theoretical and practical problems. In particular, the powerful computational model of quantum computation can be defined almost entirely in terms of matrix products and tensor products, both of which are special cases of tensor networks. As such, (classical) algorithms for evaluating tensor networks have profound importance in the study of quantum computation. In this thesis, we design and implement a parallel algorithm for tensor network contraction. In addition to finding efficient contraction orders for a tensor network, we also dynamically slice it into multiple sub-tasks with lower space and time costs, in order to evaluate the tensor network in parallel. We refer to such an evaluation strategy as a contraction scheme for the tensor network. In addition, we introduce a local optimization procedure that improves the efficiency of the contraction schemes we find. We also investigate the applications of our parallel tensor network contraction algorithm in quantum computation. The most ready application is the simulation of random quantum supremacy circuits, where we benchmark our algorithm to demonstrate its advantage over other similar tensor network based simulators. Other applications we found include evaluating the energy function of a Quantum Approximate Optimization Algorithm (QAOA), and simulating surface codes under a realistic error model with crosstalk.