Mathematical Research on Mathematical Models of Quantum Computing

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)的局域转移函数一般是刻画的，包括多带情况。(2)首次提出了均匀量子电路族的概念，发展了其复杂性理论，证明了蒙特卡罗计算中量子电路族与均匀量子电路族在计算上的等价性。(3)为了解决量子逻辑门的停止问题，证明了在改进的停止协议下，停止标志的测量不会干扰计算输出的概率分布。在量子逻辑门的物理实现上得到了以下结果：(1)守恒定律限制了基本量子逻辑门的物理实现的精度。(2)尽管互换门与守恒定律没有冲突，但受控非门，这是通用量子…之一更多的逻辑门，在错误概率1/16内不能通过任何2量子比特旋转不变的么正运算来实现。(3)如果计算基用自旋分量表示，并且物理实现遵守角动量守恒定律，则任何具有n个量子比特Ancilla的物理可实现量子逻辑门都不能在误差概率1/(4N^2)内实现受控非门。(4)对于由平均光子数定义大小的玻色子附件，也存在类似的关系。任何一组通用门都不可避免地服从误差概率为O(n^&lt；-2&gt；)的相关限制。(5)目前的理论要求每个量子门的门限误差概率为10^5~10^6。因此，单个受控非门实际上不会是2量子比特系统上的么正运算，而是至少有100个量子比特的系统上的么正运算。(6)本研究建议对现有的计算基的选择进行修改，使计算基与守恒量互换。较少

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