时间非齐次二态量子游荡的演化过程分析

Abstract

Abstract:

In this paper, we establish a mathematical model for a time-inhomogeneous two-state quantum walk and give a calculation of the position probability distribution. By calculating spectral values and spectral vectors, we analyze the evolution of quantum states. Furthermore, we derive an Itô formula and get a matrix decomposition and its interpretation.

Key words: Quantum walk, Inhomogeneous, Probability distribution, Spectral value, Itô lemma

CLC Number:

O211.6

Yunguo Lin. The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk[J].Acta mathematica scientia,Series A, 2021, 41(4): 1097-1110.

Trendmd

References 21

1	Aharonov Y , Davidovich L , Zagury N . Quantum random walks. Physical Review A, 1993, 48 (2): 1687- 1690 doi: 10.1103/PhysRevA.48.1687
2	Shenvi N , Kempe J , Whaley K B . Quantum random-walk search algorithm. Physical Review A, 2003, 67 (5): 052307 doi: 10.1103/PhysRevA.67.052307
3	Ambainis A . Quantum walk algorithm for element distinctness. SIAM Journal on Computing, 2007, 37 (1): 210- 239 doi: 10.1137/S0097539705447311
4	Farhi E , Gutmann S . Quantum computation and decision trees. Physical Review A, 1998, 58 (2): 915- 928 doi: 10.1103/PhysRevA.58.915
5	Watrous J . Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer System Sciences, 2001, 62 (2): 376- 391 doi: 10.1006/jcss.2000.1732
6	Andraca V, Elías S. Quantum Walks for Computer Scientists. USA, Vermont: Morgan & Claypool, 2008
7	Venegasandraca S E . Quantum walks: a comprehensive review. Quantum Information Processing, 2012, 11 (5): 1015- 1106
8	Higuchi Y , Konno N , Sato I , et al. Periodicity of the discrete-time quantum walk on a finite graph. Interdisciplinary Information Sciences, 2017, 23, 75- 86 doi: 10.4036/iis.2017.A.10
9	Krovi H , Brun T A . Hitting time for quantum walks on the hypercube. Physical Review A, 2006, 73 (3): 501- 507
10	Konno N , Namiki T , Soshi T . Symmetry of distribution for the one-dimensional hadamard walk. Interdisciplinary Information Sciences, 2002, 10 (1): 11- 22
11	Konno N . A new type of limit theorems for the one-dimensional quantum random walk. Journal of The Mathematical Society of Japan, 2005, 57 (4): 1179- 1195
12	韩琦, 陈芷禾, 殷世德, 等. 基于hadamard算子的二维离散量子行走的概率测度估计. 应用数学学报, 2020, 43 (1): 49- 61
	Han Q , Chen Z H , Yin S D , et al. Estimation of probability measure for 2-D discrete quantum walk based on hadamard operator. Acta Mathematicae Applicatae Sinica, 2020, 43 (1): 49- 61
13	Linden N , Sharam J . Inhomogeneous quantum walks. Physical Review A, 2009, 80 (5): 052327 doi: 10.1103/PhysRevA.80.052327
14	Shikano Y , Katsura H . Localization and fractality in inhomogeneous quantum walks with self-duality. Physical Review E, 2010, 82 (3): 031122 doi: 10.1103/PhysRevE.82.031122
15	Rousseva J , Kovchegov Y . On alternating quantum walks. Physica A: Statistical Mechanics and its Applications, 2017, 470, 309- 320 doi: 10.1016/j.physa.2016.11.138
16	Konno N , Luczak T , Segawa E , et al. Limit measures of inhomogeneous discrete-time quantum walks in one dimension. Quantum Information Processing, 2013, 12 (1): 33- 53 doi: 10.1007/s11128-011-0353-8
17	韩琦, 郭婷, 殷世德, 陈芷禾. 直线上空间非齐次三态量子游荡的平稳测度. 数学物理学报, 2019, 39 (1): 135- 144
	Han Q , Guo T , Yin S D , Chen Z H . The stationary measure of a space-inhomogeneous three-state quantum walk on the line. Acta Math Sci, 2019, 39 (1): 135- 144
18	Machida T . Limit distribution for a time-inhomogeneous 2-state quantum walk. Journal of Computational and Theoretical Nanoscience, 2013, 10 (7): 1571- 1578 doi: 10.1166/jctn.2013.3090
19	Konno N . A note on Itô's formula for discrete-time quantum walk. Journal of Computational and Theoretical Nanoscience, 2013, 10 (7): 1579- 1582 doi: 10.1166/jctn.2013.3091
20	Kang Y , Wang C . Itô formula for one-dimensional continuous-time quantum random walk. Physica A: Statistical Mechanics and its Applications, 2014, 414, 154- 162 doi: 10.1016/j.physa.2014.06.086
21	康元宝. 多维连续时间量子随机游动的Itô公式. 数学物理学报, 2016, 36A (4): 771- 782
	Kang Y B . Itô's formula for multidimensional continuous-time quantum random walk. Acta Math Sci, 2016, 36A (4): 771- 782

[1]	Lv Dongting. Globally Asymptotic Stability of 2-Species Reaction-Diffusion Systems of Spatially Inhomogeneous Models [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1171-1183.
[2]	Cui Jianan,Chai Shugen. Indirect Boundary Stabilization of Strongly Coupled Variable Coefficient Wave Equations [J]. Acta mathematica scientia,Series A, 2025, 45(2): 389-407.
[3]	Jian Hui, Gong Min, Wang Li. On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1350-1372.
[4]	Niu Cuixia,Ma Heping. A Leap-Frog Crank-Nicolson Multidomain Legendre-Tau Collocation Spectral Method for 2D Nonlinear Maxwell's Equations in Inhomogeneous Media [J]. Acta mathematica scientia,Series A, 2023, 43(3): 808-828.
[5]	Li Deke, Wang Qingxuan. Limit Behavior of Mass Critical Inhomogeneous Schrödinger Equation at the Threshold [J]. Acta mathematica scientia,Series A, 2023, 43(1): 123-131.
[6]	Xiaojin Bai,Jianfeng Zhu. Boundary Schwarz Lemma for Solutions to a Class of Inhomogeneous Biharmonic Equations [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1633-1639.
[7]	Junfeng Liu,Lei Mao,Zhi Wang. Moment Bounds for the Fractional Stochastic Heat Equation with Spatially Inhomogeneous White Noise [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1186-1208.
[8]	Nan Fan,Caishi Wang,Hong Ji. Perturbations of Canonical Unitary Involutions Associated with Quantum Bernoulli Noises [J]. Acta mathematica scientia,Series A, 2022, 42(4): 969-977.
[9]	Zhihua Du,Yumei Li. An L^p Inhomogeneous Polyharmonic Neumann Problem on Lipschitz Domains in $\mathbb{R} ^{2}$ [J]. Acta mathematica scientia,Series A, 2020, 40(2): 271-287.
[10]	Qi Han,Ting Guo,Shide Yin,Zhihe Chen. The Stationary Measure of a Space-Inhomogeneous Three-State Quantum Walk on the Line [J]. Acta mathematica scientia,Series A, 2019, 39(1): 133-142.
[11]	Yang Wenbin, Wu Jianhua. Some Dynamics in Spatial Homogeneous and Inhomogeneous Activator-Inhibitor Model [J]. Acta mathematica scientia,Series A, 2017, 37(2): 390-400.
[12]	Wang Jixia, Xiao Qingxian. Local Estimation of the Diffusion Coefficient for Time-Inhomogeneous Diffusion Models [J]. Acta mathematica scientia,Series A, 2015, 35(2): 332-344.
[13]	Yang Fen. Decay Rate of Solutions to An Inhomogenerous Semilinear Biharmonic Equation in Entire Space [J]. Acta mathematica scientia,Series A, 2015, 35(2): 282-287.
[14]	Kuang Jie,Wang Zejun. The Large Time Behavior of Inhomogeneous Burgers Equation with Periodic Initial Data [J]. Acta mathematica scientia,Series A, 2015, 35(1): 1-14.
[15]	LIANG Zhan-Ping, SU Jia-Bao. Solutions to Inhomogeneous Quasilinear Elliptic Problems with Concave-Convex Type Nonlinearities [J]. Acta mathematica scientia,Series A, 2014, 34(2): 217-226.

The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 21

Related Articles 15

Metrics

Comments

Recommended 10