Mathematical Research on Mathematical Models of Quantum Computing

量子计算数学模型的数学研究

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440028/
• 关键词: quantum computing; quantum Turing machines; quantum circuits; quantum complexity theory; quantum gates; quantum operation; fault-tolerant quantum computing; controlled NOT gate; 加法的保存量; 誤り訂正理論; 万能量子論理素子; 制御否定素子; 計算基底; 閾値定理; 量子オラクルTuring機械

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

在数学基础上获得了以下结果，并在量子量机上和量子循环中获得了以下结果：（1）量子图灵机（QTM）的局部过渡函数通常被表征在内，包括多层案例。 （2）首次引入了均匀量子电路家族（UQCF）的概念，并开发了其复杂性理论，提供了QTMS和UQCF之间的计算等效性，以蒙特卡罗类型计算。 （3）为了解决QTMS的停止问题，已证明，在制定的停止方案测量中，停止标志的测量结果不会使计算输出输出的概率分布分配。在物理实施量子逻辑门的物理实施方面已获得以下结果：（1）保护法律限制了基本量子量子的物理实现的局限性。 （2）尽管互换大门与保护法没有冲突，但受控的闸门是通用的量子……更多的逻辑大门之一，无法通过任何2 Q量的旋转旋转旋转不变的单一操作在误差概率中1/16中的统一操作。（3）无法在错误概率1/（4n^2）中实现受控的未门门。 （4）类似的关系适用于玻色词Ancillae，其大小通过平均照片数量定义。任何一套通用门都不可避免地会服从一个相关限制，而误差概率o（n^<-2>）。 （5）当前理论要求每个量子门的阈值误差概率10^5 10^6。这就是一个受控门不会实际上是在2 Qubitsystem上进行的单一操作，而是在具有至少100个配额的系统上的单一操作。 （6）本研究表明，应修改计算基础的当前选择，以便计算基础以配置的数量进行。较少的

项目成果

C.Donati-Martin, H.Hatsumoto, H.Yor: "On positive and negative moments of the integrals of geometric Brownian motions"Stat.Prob.Lett.. 49. 45-52 (2000)

C.Donati-Martin、H.Hatsumoto、H.Yor：“论几何布朗运动积分的正矩和负矩”Stat.Prob.Lett.. 49. 45-52 (2000)

Y.Matsubara: "Stronger ideals over P_kλ"Fundamanta Mathematicae. (印刷中).

Y. Matsubara：“比 P_kλ 更强大的理想”Fundamanta Mathematicae（正在出版）。

Y.Matsubara, S.Shelah: "Nowhere precipitousness of the non-stationary ideal over PκA"Journal of Mathematical Logic. (in press).

Y. Matsubara、S. Shelah：“PκA 上的非平稳理想无处陡峭”《数理逻辑杂志》（出版中）。

H.Ozawa: "Operations, disturbance, ands imultaneous measurability"Phys.Rev.. 63. 032109-1-032109-15 (2001)

H.Ozawa：“操作、干扰和同步可测量性”Phys.Rev.. 63. 032109-1-032109-15 (2001)

Y.Matsubara: "Stronger ideals over P_κλ"Fundamenta Mathematicae. (in press).

Y. Matsubara：“比 P_κλ 更强大的理想”数学基础（正在出版）。