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.

翻译：量子计算与现代基于张量的计算之间存在紧密联系，这种联系尤其体现在利用张量网络模拟量子计算的过程中。相反方向的研究则较少受到关注：量子计算鲜少被应用于张量相关问题。针对张量分解领域，我们聚焦于发现实用矩阵乘法算法，并开发了两种在量子计算机上计算分解的算法。这些算法被表述为高阶无约束二进制优化问题，进而转化为二次无约束二进制优化问题。我们的首个算法采用分解式策略，以保持优化问题在当前量子设备上的可行性。该算法从合适的初始点出发，利用当前量子退火器发现了对应于著名Strassen矩阵乘法算法的张量分解。由于分解式算法无法保证所得张量分解具有最小长度，我们进一步开发了能够发现固定长度分解的整体式算法。理论上，通过设定比已知最优分解更短的固定长度，我们可以确保整体优化问题的解将产生更快的矩阵乘法算法。