Quantum algorithms have been developed for efficiently solving linear algebra tasks. However, they generally require deep circuits and hence universal fault-tolerant quantum computers. In this work, we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices. We show that the solutions of linear systems of equations and matrix–vector multiplications can be translated as the ground states of the constructed Hamiltonians. Based on the variational quantum algorithms, we introduce Hamiltonian morphing together with an adaptive ansätz for efficiently finding the ground state, and show the solution verification. Our algorithms are especially suitable for linear algebra problems with sparse matrices, and have wide applications in machine learning and optimisation problems. The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation. We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations. We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.

已经开发了量子算法来有效地解决线性代数任务。然而，它们通常需要深层电路，因此需要通用容错量子计算机。在这项工作中，我们提出了与嘈杂的中等规模量子设备兼容的线性代数任务的变分算法。我们表明，线性方程组和矩阵向量乘法的解可以转化为构造的哈密顿量的基态。基于变分量子算法，我们引入了哈密顿变体和自适应 ansätz 以有效地找到基态，并展示了解决方案验证。我们的算法特别适用于具有稀疏矩阵的线性代数问题，在机器学习和优化问题中具有广泛的应用。矩阵乘法算法也可用于哈密顿模拟和开放系统模拟。我们通过求解线性方程组的数值模拟来评估算法的成本和有效性。我们在 IBM 量子云设备上实现了该算法，解决方案保真度高达 99.95%。