The accurate computation of Hamiltonian ground, excited and thermal states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. In this regard, existing techniques based on phase estimation or variational algorithms display potential disadvantages; phase estimation requires deep circuits with ancillae, that are hard to execute reliably without error correction, while variational algorithms, while flexible with respect to circuit depth, entail additional high-dimensional classical optimization. Here, we introduce the quantum imaginary time evolution and quantum Lanczos algorithms, which are analogues of classical algorithms for finding ground and excited states. Compared with their classical counterparts, they require exponentially less space and time per iteration, and can be implemented without deep circuits and ancillae, or high-dimensional optimization. We furthermore discuss quantum imaginary time evolution as a subroutine to generate Gibbs averages through an analogue of minimally entangled typical thermal states. Finally, we demonstrate the potential of these algorithms via an implementation using exact classical emulation as well as through prototype circuits on the Rigetti quantum virtual machine and Aspen-1 quantum processing unit.

在量子计算机上准确计算哈密顿基态，激发态和热态势必会影响物理和计算机科学中的许多问题，从量子模拟到机器学习。鉴于构建大型量子计算机所面临的挑战，这些任务应以节省资源的方式执行。在这方面，基于相位估计或变分算法的现有技术显示出潜在的缺点。相位估计需要带有辅助电路的深层电路，如果没有纠错，这些电路很难可靠地执行，而变分算法虽然在电路深度方面很灵活，但需要进行额外的高维经典优化。在这里，我们介绍了量子虚时间演化和量子Lanczos算法，这是用于查找基态和激发态的经典算法的类似物。与传统的同类产品相比，它们每次迭代所需的空间和时间都成倍减少，并且可以在没有深层电路，辅助电路或高维优化的情况下实现。我们进一步讨论了量子虚数时间演化作为子例程的过程，该子例程通过最小纠缠的典型热态的模拟来生成吉布斯平均值。最后，我们通过使用精确经典仿真的实现以及Rigetti量子虚拟机和Aspen-1量子处理单元上的原型电路，展示了这些算法的潜力。无需深层次的电路和辅助功能，也无需进行高维度的优化即可实现。我们进一步讨论了量子虚数时间演化作为子例程的过程，该子例程通过最小纠缠的典型热态的模拟来生成吉布斯平均值。最后，我们通过使用精确经典仿真的实现以及Rigetti量子虚拟机和Aspen-1量子处理单元上的原型电路，展示了这些算法的潜力。无需深层次的电路和辅助功能，也无需进行高维度的优化即可实现。我们进一步讨论了量子虚数时间演化作为子例程的过程，该子例程通过最小纠缠的典型热态的模拟来生成吉布斯平均值。最后，我们通过使用精确经典仿真的实现以及Rigetti量子虚拟机和Aspen-1量子处理单元上的原型电路，展示了这些算法的潜力。