绝热量子计算的理论与应用

In this project, theory and application of adiabatic quantum computation are studied, which belong to mathematical fundamental and subject frontier of quantum information, and they are significant to the theoretical research and application value for the quantum computing science. The main contents and aims are as follows, by using operator matrix, spectral analysis and tensor products, the adiabatic approximate theorem which is suitable for adiabatic quantum computation is established, the error function between the system state and the ground state of Hamiltonian is introduced and the estimation theorem for adiabatic approximation error is proposed. The proper initial Hamiltonian and the evolution path are constructed, as well as the effective adiabatic quantum algorithm is designed, then the feasible adiabatic quantum computing model is obtained. Based on these results, the time evolution governed by Schrödinger equation is revealed and the adiabatic quantum computing error is estimated. The applications of adiabatic quantum algorithm in solving unsorted database search problem, satisfiability problem and optimization problem are explored. The explicit features and innovations of this project lie in first presenting the adiabatic evolution theory which is suitable for adiabatic quantum computation, and estimating the error bound for the adiabatic quantum computation, further exploring new problems to adiabatic quantum computation in quantum information. The research results of this project will not only provide a theoretical basis for adiabatic quantum computation, but also establish a foundation for studying robustness, decoherence, quantum error-correction in adiabatic quantum computation.

本项目研究绝热量子计算的理论与应用问题，它属于量子信息学的数学基础与学科前沿问题，对量子计算科学的发展具有重要的理论意义与应用价值。研究内容与目标如下：应用算子矩阵、谱分析及张量积等理论与方法，建立能够适合绝热量子计算的绝热逼近定理，引入系统状态与哈密尔顿量基态之间的误差函数，提出绝热逼近误差的估计定理；构造合适的初始哈密尔顿量和演化路径，设计有效的绝热量子算法，得到切实可行的绝热量子计算模型；在此基础上，揭示由薛定谔方程决定的时间演化规律，给出绝热量子计算的误差估计；探索已建立的绝热量子算法在解决无序数据搜索问题、满足性问题及优化问题等方面的应用。创新之处在于：首次提出能够适合绝热量子计算的绝热演化理论，首次估计绝热量子计算的误差上界，揭示绝热量子计算在量子信息中的新问题。预期研究成果不仅为绝热量子计算提供理论依据，而且为研究绝热量子计算中的鲁棒性、退相干性及量子纠错等方面奠定基础。

量子信息学是信息科学、物理学、数学与计算机科学等学科相结合的产物，它充分地利用量子力学中的基本原理和基本概念来实现信息的处理，是经典信息论的革命性发展。量子信息学开拓了量子力学应用的新天地，深化了量子力学基本原理的内涵，并进一步验证了量子论的科学性。.本项目以算子理论为背景，综合运用分析、代数、几何和拓扑的思想方法、以矩阵分块技巧、算子谱分解和完全正映射等工具，系统研究了绝热量子计算的理论与应用问题，认真执行了研究计划，圆满完成了研究任务，实现了预期研究目标。给出了混合态的绝热演化定理和绝热逼近误差估计；得到了同一初态由不同哈密尔顿演化后末态之间的距离上界；建立了伪自伴量子系统的酉演化及绝热逼近定理；刻画了基于协方差矩阵的非马尔科夫量子演化；引入了先验模型的互文性度量并讨论其性质；设计了互文蒸馏协议；给出了多体量子态的纠缠鲁棒性刻画；揭示了量子运算对量子相干性的影响；提出了一种高效可行的秘密共享方案；得到了超算符的两种表示之间的对应关系。具体研究内容包括：量子绝热演化、量子纠缠、量子互文和量子相干问题的研究。.本项目共发表科研论文15篇，其中12篇发表在SCI期刊，2篇发表在国内权威期刊，1篇发表在国内核心期刊。所取得的研究成果属于信息与数学的交叉领域，为算子论与算子代数本身提供了许多新的研究课题，开发了算子理论与算子代数的应用前景。