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Mathematical Research on Mathematical Models of Quantum Computing

量子计算数学模型的数学研究

基本信息

批准号：
11440028
负责人：
OZAWA Masanao
金额：
$ 6.72万
依托单位：
Tohoku University (2001)Nagoya University (1999-2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

The following results have been obtained on mathematical foundations on quantum Turing machine and quantumcircuits: (1) Local transition functions of quantum Turing machines (QTM) are generally characterized including multitape cases. (2) The notion of uniform quantum circuit families (UQCF) was introduced for the first time and developed their complexity theory nd proved the computational equivalence between QTMs and UQCF in Monte Caro type computations. (3) In order to solve the halting problem for QTMs, it has been proved that under a refined halting protocol measurements of halting flag do not disturb the probability distribution of the output of computations.The following results have been obtained on physical implementations of quantum logicgates: (1) Conservation laws limit theaccuracy of physical implementations of elementary quantum logic gates. (2) Although the SWAP gate has no conflict with the conservation law, the controlled-NOT gate, which is one of the universal quantum … More logic gates, cannot be implemented by any 2-qubit rotationally invariant unitary operation within error probability 1/16.. (3) If the computational basis is represented by a component of spin and physical implementations obey the angular momentum conservation law, any physically realizable quantum logicgates with n qubit ancilla cannot implement the controlled-NOT gate within the error probability 1/(4n^2). (4) An analogous relation holds for bosonic ancillae with the size defined through the average number of photons. Any set of universal gates inevitably obeys a related limitation with error probability O(n^<-2>). (5) The current theory demands the threshold error probability 10^5 10^6 for each quantum gate. Thus, a single controlled-NOT gate would not be in reality a unitary operation on a 2-qubitsystem but would be a unitary operation on a system with at least 100 qubits. (6) The present investigation suggests that the current choice of the computational basis should be modified so that the computational basis commutes with the conserved quantity. Less

在量子图灵机和量子电路的数学基础上得到了以下结果：(1)量子图灵机（QTM）的局部跃迁函数一般包括多带情况。(2)首次引入了均匀量子电路族的概念，发展了均匀量子电路族的复杂性理论，证明了均匀量子电路族与均匀量子电路族在Monte Caro型计算中的计算等价性。(3)为了解决QTMs的停机问题，证明了在一种改进的停机协议下，停机标志的测量不会干扰计算输出的概率分布。关于量子逻辑门的物理实现得到了以下结果：(1)守恒定律限制了基本量子逻辑门物理实现的准确性。(2)尽管SWAP门与守恒定律不冲突，但作为通用量子逻辑门之一的受控非门，在误差概率为1/16的范围内，无法通过任何2量子位旋转不变的幺正运算来实现。(3)如果计算基由自旋分量表示，且物理实现服从角动量守恒定律，则任何物理上可实现的带有n个量子比特辅助的量子逻辑门都不能在误差概率1/(4n^2)内实现受控非门。(4)用平均光子数来定义大小的玻色子辅助子也有类似的关系。任何一组通用门都不可避免地服从错误概率为O（n^<-2>）的相关限制。(5)目前理论要求每个量子门的阈值误差概率为10^5 10^6。因此，单个受控非门实际上不是2量子位系统上的幺正操作，而是至少具有100量子位的系统上的幺正操作。(6)本文的研究表明，当前计算基的选择应加以修改，使计算基与守恒量相适应。少