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I have problems understanding the following step in Nielsen's and Chuang's Quantum Computation and Quantum Information (page 218, equations 5.9 and 5.10):

Nielsen and Chuang, 2016, p.2018

Could someone help me please to understand this step? I tried to do it on an example n=3 and j=5 but could not get it to work. Somehow e^(2 * pi * i * j_k/2^k) has to be 1 if k<(n+1-l).

I tried to work with the definitions of binary fraction and the binary representation given on the same page.

Thanks a lot, please let me know if some information is missing.

DianaPrince
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1 Answers1

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I think I finally found out how it works. The exponent terms are indeed 1 for all bits of j up to position (n-l). Thus for l=1 only j_n stays whereas for l=1 to l=n-1 it holds that e^(2 * pi * i * j_k * 2^(n-k-l))=1 since either j_k is 0 or if j_k is 1, n-k-1 is positive.

DianaPrince
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  • I post my answer on Math StackExchange https://math.stackexchange.com/questions/4574359/a-question-about-product-representation-of-quantum-fourier-transform/ – bfhaha Nov 12 '22 at 06:04