"《量子计算与量子信息》课后答案：Nielsen和Chuang的解决方案"

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This is an unofficial solution manual for the book "Quantum Computation and Quantum Information" by Nielsen and Chuang. The solutions provided are for the 10th Anniversary Edition of the book. The work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Readers can access the latest version and the source LATEX code of the solutions on the GitHub repository at https://github.com/goropikari/SolutionForQuantumComputationAndQuantumInformation. The solutions in this manual aim to help readers better understand the concepts and problems presented in the original book. By providing step-by-step explanations and solutions to the exercises, readers can deepen their knowledge and enhance their understanding of quantum computation and quantum information. Whether you are a student, researcher, or enthusiast in the field of quantum computing, this solution manual can serve as a valuable resource to test your understanding and improve your problem-solving skills. Dive into the world of quantum computation and quantum information with confidence, knowing that there is a helpful guide available to assist you along the way.

CHAPTER 2. INTRODUCTION TO QUANTUM MECHANICS 4

hw|vi =



1 1





−1



= 1 − 1 = 0

|wi

k|wik

|wi

hw|wi

√





|vi

k|vik

|vi

hv|vi

√



−1



2.8

If k = 1,

i =

i − hv

i|v

k|w

i − hv

i|v

i = hv



i − hv

i|v

k|w

i − hv

i|v



i − hv

ihv

k|w

i − hv

i|v

= 0.

Suppose {v

, ···v

} (n ≤ d − 1) is a orthonormal basis. Then

n+1

i = hv



n+1

i −

i=1

n+1

i|v

k|w

n+1

i −

i=1

n+1

i|v



(j ≤ n)

n+1

i −

i=1

n+1

ihv

k|w

n+1

i −

i=1

n+1

i|v

n+1

i −

i=1

n+1

iδ

k|w

n+1

i −

i=1

n+1

i|v

n+1

i − hv

n+1

k|w

n+1

i −

i=1

n+1

i|v

= 0.

Thus Gram-Schmidt procedure produces an orthonormal basis.

2.9

= I = |0ih0| + |1ih1|

= X = |0ih1| + |1ih0|

= Y = −i |0ih1| + i |1ih0|

= Z = |0ih0| − |1ih1|

2.10

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