Recovering quantum information in a noisy quantum channel

不可逆性似乎是我们日常经验中物理过程的普遍性质。融化的冰淇淋不会自发冻结成原来的形状，鸡尾酒一旦混合就不会蒸馏成酒。即使在宇宙中最宏观的尺度上也可以观察到可逆性；由于广义相对论原理，落入黑洞的粒子永远不会被找回。另一方面，支配力学的物理定律，如牛顿定律和量子么正演化，似乎具有时间反转对称性，有时还伴随着其他物理对称性，如电荷和宇称。换句话说，单个粒子的时间倒转轨迹就像时间的正向轨迹一样自然。在量子力学中，封闭量子系统中的所有动力学都是可逆的，这是所有量子计算算法所假定的特征。然而，在现实世界中，门操作是不完美的，不可避免地会添加噪声来破坏量子相干和纠缠，最终破坏这种量子可逆性。这个项目提出了一个有趣的方向来解决这个问题，将物理过程视为量子通信通道，传输以量子态编码的有价值的信息。从这个角度来看，量子过程的可逆性可以量化为解码过程可以检索到的量子信息量。这项研究的主要目的是建立一个理论框架，通过描述量子信息恢复的基本限制和确定实现量子信息恢复的最小资源，来定量描述在噪声量子处理信道中的量子信息恢复。通过采用PETZ恢复映射作为量子恢复/译码的信道，我们将从多个方面研究从噪声环境中恢复量子信息的精确条件：寻找量子通信的量子容量的改进界限，发展量子计算中量子纠错的新方法，以及表征和量化量子存储器在量子反馈控制中的作用。我们的研究旨在减少量子计算中噪声和门操作缺陷的影响，以便我们能够进行模拟或算法来展示量子优势。这一研究将为理解量子计算中的噪声可逆性条件开辟新的范式。同时，从拟议的研究中获得的资源高效的恢复协议有可能对包括量子通信和量子计算在内的实际应用产生重大影响，在这些应用中，量子态的高保真恢复是必要的。

Irreversibility seems to be a universal nature of physical processes in our daily experience. Melted ice cream will not spontaneously freeze into its original shape, and a cocktail will not be distilled into liquors once it is mixed. The reversibility can be observed even in the most macroscopic scale in the universe; a particle fell into a black hole will never be recovered due to the principle of general relativity. On the other hand, the physical laws governing mechanics, such as Newtonian's law and quantum unitary evolutions, seem to have the time-reversal symmetry, sometimes accompanied with other physical symmetries, e.g., charge and parity. In other words, the time-reversed trajectory of a single particle seems as natural as the forward trajectory in time.In quantum mechanics, all the dynamics in a closed quantum system is reversible and this is the feature assumed in all the algorithms for quantum computation. However, in the real-world situation, gate operations are imperfect and noise is inevitably added to destroy quantum coherences and entanglement and ultimately this quantum reversibility. This project proposes an intriguing direction to approach this problem by regarding a physical process as a quantum communication channel, transmitting valuable information encoded in a quantum state. From this viewpoint, the reversibility of quantum processes can be quantified as the amount of quantum information that can be retrieved by the decoding process. The main objective of the research is to establish a theoretical framework towards a quantitative description of the recovery of quantum information in a noisy quantum processing channel, by characterising the fundamental limitation of recovering quantum information and identifying a minimal resource to achieve the recovery. By adopting the Petz recovery map as the recovery / decoding quantum channel, we will investigate a precise condition to recover quantum information from a noisy environment in various aspects: to find an improved bound of the quantum capacity for quantum communication, to develop a new method for quantum error correction in quantum computation, and to characterise and quantify a role of quantum memory in the quantum feedback control. Our research aims to reduce the impact of noise and imperfections of gate operations in quantum computing so that we can perform simulations or algorithms to demonstrate quantum advantages.This research will open a new paradigm to understand the noise-reversibility condition in quantum computing. At the same time, the resource-efficient recovery protocols obtained from the proposed research has potential to deliver a significant impact on practical applications, including quantum communications and quantum computations, where a high-fidelity recovery of a quantum state is necessary.