Differential Geometric Methods for Quantum Information Processing

量子信息处理（QIP）蕴藏着巨大的潜在科学和技术回报，其算法从根本上比经典算法更强大。实现QIP最有希望的方法之一涉及绝热过程，其中系统从初始最低能量输入状态缓慢调整到最终输出最低能量状态，其中包含计算难题的答案。这种方法被称为绝热量子计算（AQC）。AQC的速率限制步骤是通过一个到最近激发态的能隙最小的状态。在大系统尺寸的极限下，这一过程通常伴随着量子相变（QPT）。AQC可以理解为用于调整系统的控制参数的流形上的流。本项目旨在开发新的微分几何和变分微积分工具，用于分析和统一AQC和QPT。对于AQC，这将通过使用两种互补的方法来完成。一方面，将AQC视为时间最优控制问题是很自然的，其中尝试最小化根据绝热约束调整系统所需的时间。这导致了AQC作为变分优化问题的表述。另一方面，最优AQC可以看作是具有绝热度量张量的流形上的测地线。该项目旨在将这两种方法置于严格的基础上，并显示它们的等效性。通过求解相应的欧拉-拉格朗日方程和测地线方程，研究人员将找到新的最优AQC算法。对于qpt，本项目旨在基于算子保真度敏感性（OFS）的概念开发一种强大的新微分几何方法，OFS也是一个度量，即在射影量子空间上的度量回拉到控制参数流形上。随着控制参数的变化，量子系统处于随时间变化的状态，OFS可以检测到操作符之间保真度的变化。这种方法可以直接应用于当AQC算法通过一个临界区域时出现的qpt问题。以这种方式，研究者希望统一描述的AQC和qpt使用微分几何方法，并揭示新的光对两者。更广泛的影响：将微分几何引入到AQC的分析中，可能会为更大的量子信息科学界带来新的见解，也会使数学家更接近量子计算领域中出现的大量迷人问题。通过微分几何的AQC和qpt的统一也将使从事凝聚态物理中涉及强相关系统问题的数学物理学家受益。这项工作为所涉及的研究生和博士后提供了重要的培训和教育机会。研究生、博士后和主要研究人员将参与全院范围内的项目，以提高未被充分代表的少数民族在科学领域的作用。

Quantum information processing (QIP) harbors vast potential scientific and technological payoff, in the form of algorithms which are fundamentally more powerful than their classical counterparts. One of the most promising ways of achieving QIP involves adiabatic processes, wherein a system is slowly tuned from an initial lowest-energy input state to a final output lowest-energy state which contains the answer to a computationally difficult problem. This approach is known as adiabatic quantum computation (AQC). The rate limiting step for AQC is passage through a regime where the energy gap to the nearest excited state is smallest. In the limit of large system sizes this passage is usually accompanied by a quantum phase transition (QPT). AQC can be understood as a flow on the manifold of control parameters used to tune the system. This project aims to develop new differential geometry and variational calculus tools for an analysis and unification of AQC and QPT. For AQC this will be done by using two complementary approaches. On the one hand, it is natural to view AQC as a time-optimal control problem, wherein one attempts to minimize the time it takes to tune the system subject to the constraint of adiabaticity. This leads to the formulation of AQC as a variational optimization problem. On the other hand, optimal AQC can be seen as a geodesic on a manifold equipped with an adiabatic metric tensor. This project aims to place these two approaches on a rigorous footing, and to show their equivalence. The investigators will find new optimal AQC algorithms, by solving the corresponding Euler-Lagrange and geodesic equations. For QPTs, this project aims to develop a powerful new differential geometric approach based on the concept of operator fidelity susceptibility (OFS), which is also a metric, namely a pull-back of the metric on the projective quantum space onto the control parameter manifold. The OFS detects the change in fidelity between operators for a quantum system in a state that is evolving over time as control parameters are varied. This approach can be directly applied to the problem of QPTs that arise as an AQC algorithm is taken through a critical region. In this manner the investigators hope to unify the description of AQC and QPTs using differential geometric methods, and shed new light on both. Broader Impacts: The introduction of differential geometry into the analysis of AQC is likely to generate new insights for the larger quantum information science community, as well as bring mathematicians closer to the plethora of fascinating problems that have arisen in the field of quantum computation. The unification of AQC and QPTs via differential geometry will also benefit mathematical physicists working on problems involving strongly correlated systems in condensed matter physics. The work provides a significant training and educational opportunity to the graduate student and postdoc involved. The graduate student, postdoc, and principal investigators will participate in institute-wide programs to increase the role of underrepresented minorities in the sciences.