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I'm reading Programming Quantum Computers trying to understand Shor's algorithm. I learned there that we prepare a state $|x^i \bmod N\rangle$, then apply the QFT to this state. The QFT changes the amplitudes from a uniform superposition to large amplitudes evenly spaced out by the period of $x^i \bmod N$. For example, here's a graph of the amplitudes after applying the QFT with $N = 35$. That's on page 241.

enter image description here

The book says there are 12 spikes evenly spaced. I see a lot more than 12 spikes evenly spaced. Should I count just the highest ones and stop when I've counted 12? But isn't that subjective? How would I figure out that the number is really 12 from just looking at this graph without knowing the right answer? (In other words, how do I get 12 out of this?)

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  • $\begingroup$ @MariiaMykhailova, you can probably answer this. And --- thanks --- if you do! $\endgroup$ – user14021 Dec 5 '20 at 14:53
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    $\begingroup$ You're right to count the local maxima. And you're right that it's subjective, but the point is that if you run the modular exponentiation and the QFT, say, $k$ times, you'll get $x_1, x_2,\ldots x_k$, with each $x_i$ likely corresponding to one of the local maxima. You can feed these to the classical parts of the algorithm. $\endgroup$ – Mark S Dec 5 '20 at 16:13
  • $\begingroup$ That makes some sense, but it's not clear to me still. By the way, I know the continued fraction algorithm, but it's not clear how to use it to get the period here yet. I asked a question on this. Perhaps you could answer it. Thanks! $\endgroup$ – user14021 Dec 5 '20 at 16:53
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Looking at the graphs you've reproduced, the left graph shows the evaluation of $2^x\bmod 35$ for $x\in\{0,\dots 63\}$ while the right graph illustrates the amplitude of the discrete Fourier transform for $\hat{x}\in\{0,\dots 63\}$. The comment that there are "12 evenly spaced spikes" indicates that the local maxima of the right graph repeat every $64/12=5.33$ values.

You are correct, you do not have access to $\hat{x}$ in a manner that lets you observe this periodicity in $\hat{x}$ immediately; however, what you do have access to is a way to sample $\hat{x}_i$ for multiple $i$ in a manner that returns $\hat{x}_i$ with probability given by the (square of the) height of the respective $\hat{x}_i$.

For example, if you were to run the modular exponentiation (left graph) followed by the QFT (right graph), and sample the first register, you are likely to get a value such as $0$ with higher probability than $5$, with higher probability than $32$, with higher probability than $11$, with higher probability than $6$, etc.

From these respective samplings of $\hat{x}_i$, you can run the classical portions (the continued fraction portion) of Shor's algorithm to deduce that, indeed, there were 12 evenly spaced spikes in $\hat{x}$, giving you the period of $12$ in $2^x\bmod 35$. There are a lot of details that I'm forgetting but the point is that you use the samples from your QFT as inputs to this classical portion.

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