Mathematical Research of Quantum Information and Quantum Computing

量子信息与量子计算的数学研究

• 批准号: 14340028
• 负责人: OZAWA Masanao
• 金额: $ 7.3万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340028/
• 关键词: uncertainty principle; quantum measurement; quantum computing; quantum information; conservation law; completely positive maps; quantum gates; fault-tolerant quantum computing; 量子完全相関; 量子回路; 量子Turing機械; 量子推定理論; 量子Fisher情報量; 測定誤差; 擾乱; Wigner

1.Heisenberg's uncertainty principle, which gives the trade-off between measurement noise and disturbance, has been reconsidered. Its limitation of the validity has been clarified and the universally valid generalization (universal uncertainty principle) has been established. If any measuring apparatus measures an observable A with noise ε(A) and disturbance η(B) on B, we have the relation ε(A)η(B)+ε(A)σ(B)+σ(A)η(B)≧(1/2)|<[A,B]>|, where σ stands for the standard deviation in the state just before the measurement.2.It has been shown that, if the noise ε(A) and the disturbance η(B) are statistically independent from the initial state of the measured object, we have the original Heisenberg relation ε(A)η(B)≧(1/2)|<[A,B]>|.3.From the universal uncertainty principle, Wigner-Araki-Yanase (WAY), found in 1952 to set a limitation of measurements under conservation law, has been quantitatively generalized to a reciprocal relation between the mean-square-noise of the measurement and the variance of the conserved quantity.4.As a consequence from the universal uncertainty principle, it has been shown that the angular momentum conservation law sets a limitation to the accuracy of quantum gate operations, the accuracy of the CNOT gate controlled by an interaction with n qubit ancilla system is limited with error probability at least 1/(4n^2), and the CNOT controlled by the electro-magnetic field with average photon N is limited with error probability at least 1/(16N).

1.重新考虑了海森堡不确定性原理，它给出了测量噪声和干扰之间的权衡，其有效性的局限性已经得到澄清，并且已经建立了普遍有效的概括（通用不确定性原理）。可观测 A，B 上带有噪声 ε(A) 和扰动 η(B)，我们有以下关系ε(A)η(B)+ε(A)σ(B)+σ(A)η(B)≧(1/2)|<[A,B]>|，其中 σ 代表标准差2.已经证明，如果噪声ε(A)和扰动η(B)与被测物体的初始状态完全无关，则有原始的海森堡关系ε(A)η(B)≧(1/2)|<[A,B]>|.3.根据普遍不确定性原理，Wigner-Araki-Yanase (WAY)，于 1952 年发现，设置测量限制根据守恒定律，已被定量地推广为测量的均方噪声与守恒量的方差之间的倒数关系。4.根据普遍不确定性原理，已经证明角动量守恒定律对量子门操作的精度设置了限制，由与 n 量子位辅助系统相互作用控制的 CNOT 门的精度受到限制，误差概率至少为 1/(4n^2)，并且由平均光子 N 的电磁场受到限制，误差概率至少为 1/(16N)。

项目成果

UNCERTAINTY PRINCIPLE FOR QUANTUM INSTRUMENTS AND COMPUTING

• DOI: 10.1142/s0219749903000437
• 发表时间: 2003-10
• 期刊: International Journal of Quantum Information
• 影响因子: 1.2
• 作者: M. Ozawa

An interacting Fock space with periodic Jacobi parameter obtained from regular graphs in large scale limit

具有大尺度极限正则图周期性雅可比参数的交互Fock空间

• 发表时间: 2006
• 期刊: Quantum information V(World Sci. Publ.)
• 作者: Hora;Akihito; obata;Nobuaki
• 通讯作者: Nobuaki

Computation of Iwasawa invariants

Iwasawa 不变量的计算

• 发表时间: 2002
• 期刊: Transactions of Japan Society for Industrial and Applied Mathematics 12
• 作者: H.Taya;T.Fukuda
• 通讯作者: T.Fukuda

Universal uncertainty principle and quantum state control under

普遍不确定性原理与量子态控制

• 发表时间: 2004
• 期刊: AIP Conference Proceedings : Quantum Communication, Measurement and Computing 734
• 作者: Mitsuhiro T.Nakao;K.Nagatou;K.Hashimoto;M.Ozawa
• 通讯作者: M.Ozawa

Universal uncertainty principle and quantum state control under conservation laws

守恒定律下的普遍不确定性原理和量子态控制

• 发表时间: 2004
• 期刊: AIP Conference Proceedings : Quantum Communication, Measurement and Computing 734
• 作者: M.Hotta;T.Karasawa;M.Ozawa;M.Ozawa
• 通讯作者: M.Ozawa