基于随机梯度下降算法实现对环上量子游走的动态完全控制

doi:10.13232/j.cnki.jnju.2022.02.005

南京大学学报(自然科学版) ›› 2022, Vol. 58 ›› Issue (2): 219–227.doi: 10.13232/j.cnki.jnju.2022.02.005

• • 上一篇

基于随机梯度下降算法实现对环上量子游走的动态完全控制

邵玉豪, 林嘉懿, 吴盛俊()

南京大学物理学院，南京，210093

收稿日期:2021-02-09 出版日期:2022-03-30 发布日期:2022-04-02
通讯作者: 吴盛俊 E-mail:sjwu@nju.edu.cn
作者简介:E⁃mail：sjwu@nju.edu.cn
基金资助:
国家自然科学基金(11475084)

Dynamic control of full quantum walk on a cycle based on stochastic gradient descent algorithm

Yuhao Shao, Jiayi Lin, Shengjun Wu()

Department of Physics, Nanjing University, Nanjing，210093, China

Received:2021-02-09 Online:2022-03-30 Published:2022-04-02
Contact: Shengjun Wu E-mail:sjwu@nju.edu.cn

摘要/Abstract

摘要：

寻找如何实现幺正量子操作是量子计算领域的基本问题，主要研究通过环上的离散时间量子游走实现任意幺正量子操作的可能.首先推广引入了特殊的环上的离散时间量子游走模型，并对模型实现任意量子操作的有效性进行了探讨.对于两量子比特的量子系统，给出了通用量子门集合与量子傅里叶变换的构造解.由于高维情况构造解较难精确给出，引入机器学习中常用的随机梯度下降算法，得以在高维系统近似实现所需要的幺正量子操作.此外，如对算法进行进一步微调，可以在位置空间上的实现任意的幺正量子操作以及两结果半正定算子测量.在高维情况下，这意味着通过控制两能级的硬币系统即可控制位置空间上大型系统，从而实现小系统对大系统的间接完全控制.这些任务的完成表明，基于随机梯度下降算法可以实现对整个环上量子游走过程的动态完全控制.

关键词: 环上的量子游走, 随机梯度下降, 幺正操作, 半正定算子测量, 动态控制

Abstract:

Finding implementations of unitary operations is a fundamental task in the field of quantum computation. This letter mainly investigates the ability of the discrete time quantum walk on a cycle for this task. First,we introduce a special discrete time quantum walk on a cycle model,and then discuss the effectiveness and feasibility of the model in achieving arbitrary unitary quantum operations. For two qubits system,the empirical solutions of the universal quantum gate set and quantum fourier transform are presented. Due to the difficulty of exactly giving the empirical solutions in high?dimensional cases,we would introduce the stochastic gradient descent algorithm which is commonly used in machine learning to find approximations to arbitrary desired unitary operations. Then,we would manage to modify the stochastic gradient descent algorithm in order to realize arbitrary unitary quantum operations and 2?outcome POVMs on the position space effeciently. In high?dimensional cases,it means one can control a large system via the discrete time quantum walk on a cycle indirectly on the basis of controlling the 2?level coin system. The completion of these tasks shows that we can achieve dynamic control of full discrete time quantum walk on a cycle based on the stochastic gradient descent algorithm.

Key words: quantum walk on a cycle, stochastic gradient descent algorithm, unitary operations, POVM, dynamic control

中图分类号:

O413.1

邵玉豪, 林嘉懿, 吴盛俊. 基于随机梯度下降算法实现对环上量子游走的动态完全控制[J]. 南京大学学报(自然科学版), 2022, 58(2): 219–227.

Yuhao Shao, Jiayi Lin, Shengjun Wu. Dynamic control of full quantum walk on a cycle based on stochastic gradient descent algorithm[J]. Journal of Nanjing University(Natural Sciences), 2022, 58(2): 219–227.

图/表 11

图1

表1

一步量子游走实现CNOT门的构造解"

I ? c

I ? c

表1

表2

两步量子游走实现V?=U?c?I?p的构造解"

U ? c

I ? c

U ? c

I ? c

表2

表3

四步量子游走实现V?=I?c?U?p的构造解"

I ? c

u 00 u 01 u 01 ? - u 00 ?

I ? c

I ? c

X ? c

u 00 ? - u 01 ? u 01 u 00

(u 00 u 11 - u 01 u 10) 0 - 1 10

I ? c

表3

表4

当单粒子量子门为对角阵时，仅需两步量子游走即可实现V?=I?c?U?p"

u 00 I ? c

I ? c

u 11 I ? c

I ? c

表4

表5

两步量子游走实现相位门的构造解"

$V ? = U ? c ? I ? p$	t=0	t=1	$V ? = I ? c ? U ? p$	t=0	t=1
x=0	$P h a s e$	$I ? c$	x=0	$I ? c$	$I ? c$
x=1	$P h a s e$	$I ? c$	x=1	$i ? I ? c$	$I ? c$

表5

图2

表6

三步量子游走实现量子傅里叶变换的构造解"

H ? c

H ? c

I ? c

12 1 - 1 11

12 i 1 - i 1

I ? c

表6

图3

图4

图5

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基于随机梯度下降算法实现对环上量子游走的动态完全控制

Dynamic control of full quantum walk on a cycle based on stochastic gradient descent algorithm

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