EAGER: QSA: Eigenstate Thermalization and the Quantum Metropolis Algorithm

强相互作用的量子粒子系统能够实现量子物质的新相，它可以表现出惊人的宏观特性，如超流动性、超导性和量子信息的长期存储。尽管在使用传统超级计算机模拟这些系统方面做出了巨大的努力，但计算复杂性的基本障碍限制了进展，这一障碍掩盖了一个未被探索的物理学前沿，即纠缠前沿。量子计算机自然能够探索这一前沿，因为它们在基本操作中利用纠缠，因此成熟量子计算技术的最终发展将极大地促进我们对量子相变的理解，并使发现具有潜在变革技术和社会特性的量子物质的新阶段成为可能。所有这些都依赖于模拟量子系统热力学性质的有效量子算法的发展。该项目研究了量子Metropolis算法，这是一种量子算法，它直接推广了20世纪最成功的模拟经典统计物理的范式之一。通过在相应的经典算法中发生的并行发展，该项目试图确定暗示量子计算机可以有效地模拟量子热特性的一般条件。该项目中使用的主要新观点是基于物理热化（物理量子系统与其环境达到热平衡的过程）与量子Metropolis算法的行为和收敛之间的联系。本征态热化假设（ETH）是物理量子系统如何接近热平衡的一个重要解释，尽管它们的动力学行为原则上是时间可逆的。该项目将ETH背景下发展的数学不等式直接与控制量子Metropolis算法行为的跃迁概率联系起来。这种跃迁概率的定量表征为建立严格有效（多项式时间）量子算法的第一个非平凡案例打开了大门，用于模拟某些量子系统的热态，这些系统被强烈怀疑超出了有效经典模拟的范围。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Systems of strongly interacting quantum particles are capable of realizing novel phases of quantum matter, which can exhibit astonishing emergent macroscopic properties like superfluidity, superconductivity, and long-termstorage of quantum information. Despite significant effort in simulating these systems using conventional supercomputers, progress has been limited by a fundamental barrier of computational complexity which veils an unexplored frontier of physics called the entanglement frontier. Quantum computers are naturally able to probe this frontier because they harness entanglement in their elementary operations, and so the eventual development of mature quantum computing technology will greatly advance our understanding of quantum phase transitions and enable the discovery of new phases of quantum matter with potentially transformative properties for technology and society. All of this depends on the development of efficient quantum algorithms for simulating the thermodynamic properties of quantum systems. This project investigates the quantum Metropolis algorithm, which is a quantum algorithm that directly generalizes one of the most successful 20th century paradigms for simulating classical statistical physics. By paralleling developments that occurred in the corresponding classical algorithm, this project seeks to determine general conditions which imply that a quantum computer can efficiently simulate quantum thermal properties.The main new perspective used in this project is based on a connection between physical thermalization - the process by which a physical quantum system comes into thermal equilibrium with its environment - and the behavior and convergence of the quantum Metropolis algorithm. The eigenstate thermalization hypothesis (ETH) is a prominent explanation for how physical quantum systems approach thermal equilibrium, despite their dynamical behavior being time-reversible in principle. This project connects mathematical inequalities developed in the context of the ETH directly to the transition probabilities that govern the behavior of the quantum Metropolis algorithm. This quantitative characterization of transition probabilities opens the door to establishing the first non-trivial cases of rigorously efficient (polynomial-time) quantum algorithms for simulating thermal states of some class of quantum systems that is strongly suspected to be beyond the range of efficient classical simulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.