Phase estimation error analysis

Question

This question is about Lemma $7.1.2$ in Kaye, Laflamme, and Mosca's textbook:

Let $\omega = \frac{x}{2^n} = 0.x_1x_2\ldots x_n$ be some fixed number. The phase estimation algorithm applied to the input state $|\psi\rangle = \frac{1}{\sqrt{2^n}}\sum_{y=0}^{2^n-1}e^{2\pi i \omega y}|y\rangle$ outputs the integer $x$ with probability $$p(x) = \frac{1}{2^{2n}}\frac{\sin^2(\pi(2^n\omega-x))}{\sin^2(\pi(\omega-x/2^n))}$$

Is there a mistake here? I thought that the phase estimation algorithm outputs the integer $x$ with 100% certainty, since that's the whole point of the algorithm.

By definition, the quantum Fourier transform is the map:

$$\mathrm{QFT}_m^{-1} : \frac{1}{\sqrt{m}}\sum_{y=0}^{m-1}e^{2\pi i \frac{x}{m} y}|y\rangle \mapsto|x\rangle$$

Why is this not 100% successful?

Answer 1

Let me augment the discussion by adding some insight into the derivation of the estimate provided. This will give you a good understanding of when the result is an approximation and when it is precise. After the algorithm has run, we are left with the following state on the first register:

$$\frac{1}{2^{n}}\sum_{x=0}^{2^n - 1} \sum_{k=0}^{2^n - 1}e^{-\frac{2\pi i k}{2^n} \left ( i_{0} - 2^n \theta \right )} |x\rangle,$$

where $i_{0}$ is the integer closest to $2^{n}\omega$ such that $i_{0} < 2^{n}\omega$ ($2^{n}\omega = i_{0}-2^{n}\delta$). When measurement occurs, the axioms of quantum mechanics give you that the probability of measuring $i_{0}$ is $$\mathrm{prob}(i_{0})=\frac{1}{2^{2n}} \left | \sum_{k=0}^{2^n-1} e^{2 \pi i \delta k} \right |^2.$$

Notice that at the point you are asked to evaluate a finite geometrical series. We have the formula: $$\sum_{k=0}^{n-1} ar^k= a \left(\frac{1-r^{n}}{1-r}\right), (1)$$

valid only for $r\neq 1$ . This is a key point in the whole discussion as for $\delta = 0$ we simply end up adding $\frac{1}{2^{n}}$ $2^{n}$ times and so we get $\frac{(2^{n})^{2}}{2^{2n}}=1$ (we do not use the formula). So in the case of $2^{n}\omega$ being an integer, we have concluded that the probability of getting the exact value of the phase is 100%. If $\delta\neq 0$ we use the formula (1) and get $$\mathrm{prob}(i_{0})=\frac{1}{2^{2n}} \left | \frac{1- {e^{2 \pi i 2^n \delta}}}{1-{e^{2 \pi i \delta}}} \right |^2.(2)$$

Now notice that $$4\left | \sin(x)) \right |^{2}=4\left | \frac{e^{-ix}-e^{ix}}{2i}\right |^{2}=\left | e^{-ix}-e^{ix} \right |^{2}=\left | 1-e^{2ix} \right |^{2}.(3)$$

Using identity (3) we may rewrite (2) as $$\mathrm{prob}(i_{0})=\frac{1}{2^{2n}} \frac{\sin^{2}(\pi \delta 2^{n})}{\sin^{2}(\pi \delta)},$$

using the definition of $\delta$ we get (for $\delta \neq 0$) $$\mathrm{prob}(i_{0})=\frac{1}{2^{2n}} \frac{\sin^{2}(\pi (2^{n}\omega -i_{0}))}{\sin^{2}(\pi(\omega -\frac{i_{0}}{2^{n}}))},$$

which is exactly the result you were looking for. With a bit more effort it is possible to show that in the case where $\delta \neq 0$ the following holds $$\mathrm{prob}(i_{0})\geqslant \frac{4}{\pi ^{2}}.$$

So we may conclude with stating that for some special phases (such that $2^{n}\omega$ is an integer) we can extract the exact value of the phase. However, in the general case where $2^{n}\omega$ is not an integer, we get an approximation of the phase with a probability of at least $\frac{4}{\pi ^{2}}$.

Answer 2

First point: in most of the cases, the QPE algorithm cannot output the integer $x$. That is why $w$ is introduced in the algorithm: to represent the closest approximation of $x$ that can be returned by the QPE.

About your question, no the QPE is not always 100% successful (in this case successful means that the algorithm returns $w$, the closest approximation of $x$ possible on $n$ qubits).

The final state of the QPE before measurement is given by (according to Wikipedia):

$$ \frac{1}{2^{n}} \sum_{x=0}^{2^n - 1} \sum_{k=0}^{2^n - 1} e^{-\frac{2\pi i k}{2^n} \left ( x-w \right )} e^{2 \pi i \delta k} |x\rangle \otimes |\psi\rangle. $$ where:

1. $n$, $w$, and $x$ are defined as in your question.
2. $\delta = \left\vert \frac{x}{2^n} - w \right\vert = \left\vert \frac{x}{2^n} - \text{round}\left(\frac{x}{2^n}\right) \right\vert$ is the error due to the finite representation of $w$.

I will not rewrite all the calculations, but you can check on the intermediate results on the Wikipedia page.

The probability of getting the right result $w$ when measuring is then

$$ \Pr(w) = \left | \left \langle w \underbrace{\left | \frac{1}{2^{n}} \sum_{x=0}^{2^n-1} \sum_{k=0}^{2^n-1} e^{\frac{-2\pi i k}{2^n}(x-w)} e^{2 \pi i \delta k} \right |}_{\text{State of the first register}} x \right \rangle \right |^2 = \frac{1}{2^{2n}} \left | \sum_{k=0}^{2^n-1} e^{2 \pi i \delta k} \right |^2 = \begin{cases}1 & \delta = 0\\ & \\ \frac{1}{2^{2n}} \left | \frac{1- {e^{2 \pi i 2^n \delta}}}{1-{e^{2 \pi i \delta}}} \right|^2 & \delta \neq 0 \end{cases} $$

So:

• if the result $x$ can be represented exactly on $n$ qubits (i.e. $\delta$, the error, is $0$), the QPE succeed with a probability of $1$,
• but if $n$ bits are not enough to represent exactly $x$, the QPE might fail.

I tried quickly to derive the expression of your book from the one found on Wikipedia, without success but when the error $\delta$ tends to $0$, both formulas have the same behaviour: the probability of error is $ \sim \mathcal{O}\left( \frac{1}{2^{2n} \delta^2} \right)$.

We can simplify further this big-$\mathcal{O}$ notation by noticing that, by construction, $\delta \leqslant \frac{1}{2^{n+1}}$. The final probability of error is then $$ \Pr(\text{measurement not returning } \vert w \rangle) \sim \mathcal{O}\left( \frac{1}{2^{4n+1}} \right) $$

This means that, when you increase the number of qubits $n$ used to represent the solution by $1$, the probability of the QPE returning a bad result is divided by $16$^[1].

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^[1]: There are constants ignored by the big-$\mathcal{O}$ notation that change this value of $16 = 2^4$ to something like $\frac{2^5}{\pi^2} = \frac{32}{\pi^2}$. The additional factor of $2$ in the previous constant is needed to bound the expression $\left\vert 1- {e^{2 \pi i 2^n \delta}} \right\vert$ found for $\Pr(w)$ above but is not needed for the expression given in your book.

Answer 3

You are right. There is a slight typo. What the lemma is supposed to say is $\omega=0.x_1 x_2 \ldots x_n x_{n+1} \ldots$ or $\omega=0.x_1 x_2 \ldots x_n + a$ where $a \leq 2^{-(n+1)}$. In other words, $x=x_1, x_2, \ldots x_n$ are the the first $n$ digits of $\omega$.

Note that as written, the lemma doesn't make much sense since the denominator of $p(x)$ becomes $sin^2(\pi(\omega-x/2^n))=sin^2(0)=0$ when $a=0$. On the other hand, note that $$\lim_{a\rightarrow 0} p(x) = \lim_{b\rightarrow 0}\frac{1}{2^{2n}}\left(\frac{sin(2^n b)}{sin(b)}\right)^2=\lim_{b\rightarrow 0}\frac{1}{2^{2n}}\left(\frac{2^n b}{b}\right)^2 = 1$$ since $sin(x) \approx x$ for small $x$. So if you extend $f(a)=p(0.x_1\ldots x_n + a)$ continuously to $a=0$, then the lemma makes sense for the edge case as well.