---
title: "Shor's algorithm"
date: 2021-04-13
categories:
- Quantum information
- Cryptography
- Algorithms
layout: "concept"
---

**Shor's algorithm** was the first truly useful quantum algorithm.
It can solve important problems,
most notably integer factorization,
much more efficiently than any classical algorithm.
It weakens widely-used cryptographic schemes,
such as RSA and [Diffie-Hellman](/know/concept/diffie-hellman-key-exchange/).

In essence, Shor's algorithm's revolutionary achievement
is that it can efficiently find the periods $s_1, ..., s_A$
of a function $f(x_1, ..., x_A)$ on a discrete finite field, where:

$$\begin{aligned}
    f(x_1, ..., x_A)
    = f(x_1 + s_1, ..., x_A + s_A)
\end{aligned}$$

This is a so-called *hidden subgroup problem* for a *finite Abelian group*.
With minimal modifications,
Shor's algorithm can solve practically every such problem.


## Integer factorization

Originally, Shor's algorithm was designed to factorize an integer $N$,
in which case the goal is to find the period $s$ of
the modular exponentiation function $f$ (for reasons explained later):

$$\begin{aligned}
    f(x)
    = a^x \bmod N
\end{aligned}$$

For a given $a$ and $N$.
The period $s$ is the smallest integer satisfying $f(x) = f(x+s)$.
To do this, the following $2q$-qubit quantum circuit is used,
with $q$ chosen so that $N^2 \le 2^q < 2 N^2$:

<a href="shors-circuit.png">
<img src="shors-circuit.png" style="width:70%">
</a>

Here, $\mathrm{QFT}_q$ refers to the $q$-qubit
[quantum Fourier transform](/know/concept/quantum-fourier-transform/),
and the oracle $U_f$ calculates $f(x)$ for predetermined values of $a$ and $N$.
It is an XOR oracle, working as follows:

$$\begin{aligned}
    \Ket{x} \Ket{y}
    \quad \to \boxed{U_f(a, N)} \to \quad
    \Ket{x} \Ket{y \oplus f(x)}
\end{aligned}$$

Execution starts by applying the [Hadamard gate](/know/concept/quantum-gate/) $H$
to the first $q$ qubits, yielding:

$$\begin{aligned}
    \Ket{0}^{\otimes q} \Ket{0}^{\otimes q}
    \quad \to \boxed{H^{\otimes q}} \to \quad
    \Ket{+}^{\otimes q} \Ket{0}^{\otimes q}
    = \frac{1}{\sqrt{Q}} \sum_{x = 0}^{Q - 1} \Ket{x} \Ket{0}^{\otimes q}
\end{aligned}$$

Where $Q = 2^q$, and $\Ket{x}$ is the computational basis state $\Ket{x_1} \cdots \Ket{x_q}$.
Moving on to $U_f$:

$$\begin{aligned}
    \frac{1}{\sqrt{Q}} \sum_{x = 0}^{Q - 1} \Ket{x} \Ket{0}^{\otimes q}
    \quad \to \boxed{U_f(a, N)} \to \quad
    \frac{1}{\sqrt{Q}} \sum_{x = 0}^{Q - 1} \Ket{x} \Ket{f(x)}
\end{aligned}$$

Then we measure $f(x)$, causing it collapse as follows,
for an unknown arbitrary value of $x_0$:

$$\begin{aligned}
    f(x_0) = f(x_0 + s) = f(x_0 + 2s) = \cdots = f(x_0 + (L-1) s)
\end{aligned}$$

Due to [entanglement](/know/concept/quantum-entanglement/),
the unmeasured (top $q$) qubits change state into a superposition:

$$\begin{aligned}
    \frac{1}{\sqrt{L}} \sum_{\ell = 0}^{L - 1} \Ket{x_0 + \ell s}
\end{aligned}$$

Clearly, there is a periodic structure here,
but we cannot measure it directly,
because we do not know the value of $x_0$,
which, to make matters worse, changes every time we run the algorithm.
This is where the QFT comes in, which outputs the following state:

$$\begin{aligned}
    \frac{1}{\sqrt{QL}} \sum_{k = 0}^{Q - 1} \bigg( \sum_{\ell = 0}^{L - 1} \omega_Q^{(x_0 + \ell s) k} \bigg) \Ket{k}
\end{aligned}$$

Where $\omega_Q$ is a $Q$th root of unity.
Measuring this state yields a $\Ket{k}$, with a probability $P(k)$:

$$\begin{aligned}
    P(k)
    = \frac{1}{QL} \bigg| \sum_{\ell = 0}^{L - 1} \omega_Q^{(x_0 + \ell s) k} \bigg|^2
    = \frac{1}{QL} \bigg| \omega_Q^{x_0 k} \sum_{\ell = 0}^{L - 1} \omega_Q^{\ell s k} \bigg|^2
    = \frac{1}{QL} \bigg| \sum_{\ell = 0}^{L - 1} \omega_Q^{\ell s k} \bigg|^2
\end{aligned}$$

The last step holds because $|\omega_Q| = 1$.
Surprisingly, this implies that we did not need
to perform the measurement of $f(x)$ earlier!
This makes sense: the period $s$ does not depend on $x_0$,
so why would we need an implicit $x_0$ to determine $s$?

So, what does the above probability $P(k)$ work out to?
There are two cases:

$$\begin{alignedat}{2}
    &\mathrm{if} \: \omega_Q^{sk} = 1: \qquad
    &&P(k) = \frac{1}{QL} |L|^2 = \frac{L}{Q}
    \\
    &\mathrm{if} \: \omega_Q^{sk} \neq 1: \qquad
    &&P(k) = \frac{1}{QL} \Bigg| \frac{1 - \omega_Q^{sk L}}{1 - \omega_Q^{sk}} \Bigg|^2
\end{alignedat}$$

Where the latter case was evaluated as a geometric series.
The condition $\omega_Q^{sk}\!=\!1$ is equivalent to asking
if $sk$ is a multiple of $Q$, i.e. if $sk = cQ$, for an integer $c$.

Recall that $L$ is the number of times that $s$ fits in $Q$,
so $L\!=\!\lfloor Q / s \rfloor$.
Assuming $Q/s$ is an integer, then $L\!=\!Q/s$ and $Q\!=\!s L$,
which tells us that
$\omega_Q^{sk}\!=\!\omega_{s L}^{s k}\!=\!\omega_L^k$.
This implies that if $k$ is a multiple of $L$ (i.e. $k\!=\!c L$),
then $\omega_L^k\!=\!1$, so $P(k) = L / Q$,
which is exactly what we got earlier!

In other words, the condition $\omega_Q^{sk}\!=\!1$
is equivalent to $Q/s$ being an integer.
In that case, we have that $Q\!=\!sL$,
which we substitute into $P(k)$ from earlier:

$$\begin{aligned}
    \mathrm{if} \: (Q/s) \in \mathbb{N}: \qquad
    P(k)
    = \frac{L}{Q}
    = \frac{1}{s}
\end{aligned}$$

And because $k$ is a multiple of $L$,
and $L$ fits $s$ times in $Q$,
there must be exactly $s$ values of $k$ that satisfy $P(k) = 1/s$.
Therefore the probability of all other $k$-values is zero!
This becomes clearer when you look at the sum used to calculate $P(k)$:
if $Q\!=\!sL$, then it sums $\omega_L^{\ell k}$ over $\ell$,
leading to perfect destructive interference for the "bad" $k$-values,
leaving only the "good" ones.

**So, to summarize: if** $Q/s$ **is an integer**,
then measuring only yields $k$-values that are multiples of $L\!=\!Q/s$.
Running Shor's algorithm several times then gives
several $k$-values separated by $L$.
That tells us what $L$ is, and we already know $Q$,
so we *finally* find the period $s = Q/L$.

That begs the question: what if $Q/s$ is not an integer?
We cannot *check* this, since $s$ is unknown!
Instead, we rewrite the probability $P(k)$ as follows:

$$\begin{aligned}
    \mathrm{if} \: (Q/s) \not\in \mathbb{N}: \qquad
    P(k)
    = \frac{1}{QL} \Bigg| \frac{1 - \omega_Q^{sk L}}{1 - \omega_Q^{sk}} \Bigg|^2
    = \frac{1}{QL} \Bigg| \frac{\sin(\pi s k L / Q)}{\sin(\pi s k / Q)} \Bigg|^2
\end{aligned}$$

This function peaks if $s k$ is close to a multiple of $Q$, i.e. $s k \approx c Q$,
which we rearrange:

$$\begin{aligned}
    \frac{k}{Q} \approx \frac{c}{s}
\end{aligned}$$

We know the left-hand side,
and, from the definition of $f(x)$,
clearly $s \le N$.
We chose $Q \sim N^2$,
so $s$ is quite small,
and consequently $c$ is too, since $k < Q$.

In other words, $c/s$ is a "simple" fraction,
so our goal is to find a "simple" fraction
that is close to the "complicated" fraction $k/Q$.
For example, if $k/Q\!=\!0.332$,
then probably $c/s\!=\!1/3$.

This can be done rigorously using the **continued fractions algorithm**:
write $k/Q$ as a continued fraction,
until the non-integer part of the denominator becomes small enough.
This part is then neglected,
and we calculate whatever is left, to get an estimate of $c/s$.

Of course, $P(k)$ is a probability distribution,
so even though the odds are in our favour,
we might occasionally measure a misleading $k$-value.
Running Shor's algorithm several times "fixes" this.

**So, to summarize: if** $Q/s$ **is not an integer**,
the measured $k$-values are generally close to $c Q / s$ for an integer $c$.
By approximating $k/Q$ using the continued fraction algorithm,
we estimate $c/s$.
Repeating this procedure gives several values of $c/s$,
such that $s$ is easy to deduce
by taking the least common multiple of the denominators.

In any case, once we think we have $s$,
we can easily verify that $f(x)\!=\!f(x\!+\!s)$.
Whether $s$ is the *smallest* such integer depends on how lucky we are,
but fortunately, for most applications of this algorithm,
that does not actually matter,
and usually we find the smallest $s$ anyway.

You typically need to repeat the algorithm $\mathcal{O}(\log{q})$ times,
and the QFT is $\mathcal{O}(q^2)$.
The bottleneck is modular exponentiation $f$,
which is $\mathcal{O}(q^2 (\log{q}) \log{\log{q}})$
and therefore worse than the QFT,
yielding a total complexity of $\mathcal{O}(q^2 (\log{q})^2 \log{\log{q}})$.

OK, but what does $s$ have to do with factorizing integers?
Well, recall that $f$ is given by:

$$\begin{aligned}
    f(x)
    = a^x \bmod N
\end{aligned}$$

$N$ is the number to factorize, and $a$ is a random integer *coprime* to $N$,
meaning $\gcd(a, N) = 1$.
The fact that $s$ is the period of $f$ for a certain $a$-value, implies that:

$$\begin{aligned}
    a^x
    = a^{x + s} \bmod N
    \quad \implies \quad
    1
    = a^s \bmod N
\end{aligned}$$

Suppose that $s$ is even. In that case,
we can rewrite the above equation as follows:

$$\begin{aligned}
    (a^{s/2})^2 - 1
    = 0 \bmod N
\end{aligned}$$

In other words, $(a^{s/2})^2 \!-\! 1$ is a multiple of $N$.
We then use that $(a\!-\!b) (a\!+\!b) = a^2\!-\!b^2$:

$$\begin{aligned}
    \big( a^{s/2} - 1 \big) \big( a^{s/2} + 1 \big)
    = 0 \bmod N
\end{aligned}$$

Because $s$ is even by assumption, the two factors on the left are integers,
and as just mentioned, their product is a multiple of $N$.
Then we only need to calculate:

$$\begin{aligned}
    \gcd\!\big( a^{s/2}\!-\!1, N \big) > 1
    \quad\:\: \mathrm{and} \quad\:\:
    \gcd\!\big( a^{s/2}\!+\!1, N \big) > 1
\end{aligned}$$

And there we have the factors of $N$!
The $\gcd$ can be calculated efficiently in $\mathcal{O}(q^2)$ time.

But what if $s$ is odd?
No problem, then we just choose a new $a$ coprime to $N$,
and keep repeating Shor's algorithm until we do find an even $s$.
We do the same if $a^{s/2}\!\pm\!1$ is itself a multiple of $N$.



## References
1.  J.S. Neergaard-Nielsen,
    *Quantum information: lectures notes*,
    2021, unpublished.
2.  S. Aaronson,
    *Introduction to quantum information science: lecture notes*,
    2018, unpublished.