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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，得到了如下结果：（1）量子图灵机（QTM）的局部转移函数的一般特征，包括多带情形。(2)首次提出了均匀量子电路族（UQCF）的概念，发展了它们的复杂性理论，证明了QTM和UQCF在Monte Caro计算中的计算等价性。(3)为了解决QTM的停机问题，我们证明了在一个改进的停机协议下，停机标志的测量不会干扰计算输出的概率分布，并在量子逻辑门的物理实现上得到了以下结果：（1）守恒律限制了基本量子逻辑门的物理实现的精度。(2)虽然SWAP门与守恒定律没有冲突，但作为量子力学中普遍存在的一种， ...更多信息逻辑门，不能由错误概率1/16内的任何2量子比特旋转不变幺正操作来实现。(3)如果计算基础由自旋分量表示，并且物理实现遵循角动量守恒定律，则任何具有n量子位辅助的物理可实现的量子逻辑门都不能在错误概率1/（4n^2）内实现受控非门。(4)一个类似的关系也适用于玻色子的附肢，其大小是通过光子的平均数来定义的。任何一组通用门都不可避免地服从一个相关的限制，错误概率为O（n^<-2>）。(5)目前的理论要求每个量子门的阈值错误概率为10^5 10^6。因此，单个受控非门实际上不会是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)目前的调查表明，目前的选择的计算基础应加以修改，使计算基础与守恒量互换。少