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Commit 933356d4 authored by Stefan Stump's avatar Stefan Stump
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Added shors algorithmus and minor fixes in bernstein-vazirani

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content/latex/bernstein_vazirani.tex
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......@@ -3,8 +3,8 @@
\begin{document}
\begin{quantikz}[slice all, remove end slices=1, slice titles=$\psi_\col$]
\lstick{$\ket{0}^n$} &\qwbundle{n} &\gate{H^{\otimes n}} &\gate[2]{U_f} &\gate{H^{\otimes n}} &\metercw{s} & \\
\begin{quantikz}
\lstick{$\ket{0^n}$} &\qwbundle{n} \slice{\ket{\psi_0}} &\gate{H^{\otimes n}} \slice{\ket{\psi_1}} &\gate[2]{U_f} \slice{\ket{\psi_2}} &\gate{H^{\otimes n}} \slice{\ket{\psi_3}} &\metercw{s} & \\
\lstick{\ket{1}} &\qw &\gate{H} &\qw &\qw &\qw &
\end{quantikz}
......
......
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\documentclass{standalone}
\usepackage{quantikz}
\begin{document}
\begin{quantikz}
\lstick{$\ket{j_1}$} &\gate{H} &\gate{R_2} &\gate{R_3} &\qw &\qw &\qw &\swap{2} & \rstick{$\psi_1$} \\
\lstick{$\ket{j_2}$} &\qw &\ctrl{-1} &\qw &\gate{H} &\gate{R_2} &\qw &\qw & \rstick{$\psi_2$} \\
\lstick{$\ket{j_3}$} &\qw &\qw &\ctrl{-2} &\qw &\ctrl{-1} &\gate{H} &\targX{} & \rstick{$\psi_3$}
\end{quantikz}
\end{document}
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\documentclass{standalone}
\usepackage{quantikz}
\begin{document}
% Wires: j_1, j_2, j_{n-1}, j_n
\begin{quantikz}
\lstick{$\ket{j_1}$} &\gate{H} &\gate{R_2} &\push{\:\:\dots\:\:} &\gate{R_{n-1}} &\gate{R_n} & &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\swap{4} & \rstick{$\psi_1$} \\
\lstick{$\ket{j_2}$} &\qw &\ctrl{-1} &\qw &\qw &\qw &\gate{H} &\push{\:\:\dots\:\:} &\gate{R_{n-2}} &\gate{R_{n-1}} &\qw &\qw &\qw &\qw &\targX{} & \rstick{$\psi_2$} \\
\setwiretype{n} &\push{\vdots} & &\push{\vdots} & & & &\push{\vdots} & & &\push{\vdots} & & &\push{\vdots} \\
\lstick{$\ket{j_{n-1}}$} &\qw &\qw &\qw &\ctrl{-3} &\qw &\qw &\qw &\ctrl{-2} &\qw &\push{\:\:\dots\:\:} &\gate{H} &\gate{R_2} &\qw &\targX{} & \rstick{$\psi_{n-1}$} \\
\lstick{$\ket{j_n}$} &\qw &\qw &\qw &\qw &\ctrl{-4} &\qw &\qw &\qw &\ctrl{-3} &\push{\:\:\dots\:\:} &\qw &\ctrl{-1} &\gate{H} &\targX{} & \rstick{$\psi_n$}
\end{quantikz}
% Wires: j_1, j_2, j_3, j_{n-1}, j_n
%\begin{quantikz}
% \lstick{$\ket{j_1}$} &\gate{H} &\gate{R_2} &\gate{R_3} &\push{\:\:\dots\:\:} &\gate{R_{n-1}} &\gate{R_n} & &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\swap{5} & \rstick{$\psi_1$} \\
% \lstick{$\ket{j_2}$} &\qw &\ctrl{-1} &\qw &\qw &\qw &\qw &\gate{H} &\gate{R_2} &\push{\:\:\dots\:\:} &\gate{R_{n-2}} &\gate{R_{n-1}} &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\qw &\targX{} & \rstick{$\psi_2$} \\
% \lstick{$\ket{j_3}$} &\qw &\qw &\ctrl{-2} &\qw &\qw &\qw &\qw &\ctrl{-1} &\qw &\qw &\qw &\gate{H} &\push{\:\:\dots\:\:} &\gate{R_{n-3}} &\gate{R_{n-2}} &\qw &\qw &\qw &\qw &\targX{} & \rstick{$\psi_3$} \\
% \setwiretype{n} &\push{\vdots} & & &\push{\vdots} & & & &\push{\vdots} & & & & &\push{\vdots} & & &\push{\vdots} & & &\push{\vdots} \\
% \lstick{$\ket{j_{n-1}}$} &\qw &\qw &\qw &\qw &\ctrl{-4} &\qw &\qw &\qw &\qw &\ctrl{-3} &\qw &\qw &\qw &\ctrl{-2} &\qw &\push{\:\:\dots\:\:} &\gate{H} &\gate{R_2} &\qw &\targX{} & \rstick{$\psi_{n-1}$} \\
% \lstick{$\ket{j_n}$} &\qw &\qw &\qw &\qw &\qw &\ctrl{-5} &\qw &\qw &\qw &\qw &\ctrl{-4} &\qw &\qw &\qw &\ctrl{-3} &\push{\:\:\dots\:\:} &\qw &\ctrl{-1} &\gate{H} &\targX{} & \rstick{$\psi_n$}
%\end{quantikz}
\end{document}
\ No newline at end of file
content/latex/shor.tex 0 → 100644
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\documentclass{standalone}
\usepackage{quantikz}
\begin{document}
\begin{quantikz}
\lstick{$\ket{0^n}$} &\qwbundle{n}\slice{\ket{\phi_1}} &\gate{H^{\otimes n}}\slice{\ket{\phi_2}} &\gate[2]{U_f}\slice{\ket{\phi_3}} &\qw\slice{\ket{\phi_4}} &\gate{DFT_{2n}}\slice{\ket{\phi_5}} &\metercw{c}\slice{\ket{\phi_6}} & \\
\lstick{$\ket{0^m}$} &\qwbundle{m} &\qw &\qw &\metercw{y} &\qw &\qw &\qw
\end{quantikz}
\end{document}
\ No newline at end of file
content/shorsAlgorithm.qmd
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......@@ -2,6 +2,7 @@
One of the best known quantum algorithm is Shor's algorithm for finding the prime factors of an integer. It was developed by Peter Shor in 1994.
## Discrete Fourier Transformation
One of the tools required for Shor's algorithm is the Discrete Fourier Transformation (DFT). Generally, a Fourier transformation is a mathematical technique that decomposes a function into its constituent frequencies. We use the DFT to find the period of a vector.
......@@ -13,14 +14,16 @@ The DFT is defined as follows:
## Discrete Fourier Transformation (DFT)
The discrete Fourier transform (DFT) is a linear transformation on $\mathbb{C}^M$ represented by the matrix
The discrete Fourier transform (DFT) is a linear transformation on $\mathbb{C}^N$ represented by the matrix
$$
\operatorname{DFT}_M = \frac{1}{\sqrt{M}} (\omega^{kl})_{kl} \in \mathbb{C}^{M\times M}
\operatorname{DFT}_N = \frac{1}{\sqrt{N}} (\omega^{kl})_{kl=0,\dots,N-1} \in \mathbb{C}^{N\times N}
$$
with $\omega = e^{2i\pi/M}$, which is the $M$-th root of unity.
with $\omega = e^{2i\pi/N}$, which is the $N$-th root of unity.
:::
:::
It applies $\omega^N=1$ and $\omega^M \neq 1$ for all $0 < M < N$.
This transformation is best imagined as a process, which takes a periodic vector as an input and outputs the period of that vector. The DFT has some important properties, which help us later on.
::: {.callout-note appearance="minimal" icon=false}
......@@ -30,26 +33,39 @@ This transformation is best imagined as a process, which takes a periodic vector
Here are some properties of the DFT which can be used without further proof.
1. The DFT is unitary.
2. $\omega^t = \omega^{t\mod M}$ for all $t \in \mathbb{Z}$.
3. Given a quantum state $\psi \in \mathbb{C}^M$ which is $r$-periodic and where $r\mid M$, $\operatorname{DFT}_M \psi$ will compute a quantum state $\phi \in \mathbb{C}^M$, which has non-zero values on the multiples of $\frac{M}{r}$. Note that $\frac{M}{r}$ intuitively represents the frequency of $\psi$. This means, that
1. The $\operatorname{DFT}_N$ is unitary.
2. $\omega^t = \omega^{t\mod N}$ for all $t \in \mathbb{Z}$.
3. Let $t \mid N$ and $s$ be fix variables and the quantum state $\psi \in \mathbb{C}^N$ g given by
$$
|\psi_i| =
\begin{cases}
\sqrt{\frac{t}{N}} & \text{if } i = at+s \text{ for some } a \\
0 & \text{else.}
\end{cases}
$$
|\phi_i| = \begin{cases} \frac{1}{\sqrt{r}}, & \text{if}\ \frac{M}{r}\mid i \\ 0, & \text{otherwise} \end{cases}
In other words $\psi$ is $t$-periodic. Then for $\phi \coloneqq \operatorname{DFT}_N$ it applies
$$
|\phi_i| =
\begin{cases}
\frac{1}{\sqrt{t}} & \text{if } \frac{N}{t} \mid i \\
0 & \text{else.}
\end{cases}
$$
The first peak of $\phi$ is at $\frac{N}{t}$.
:::
:::
## Reducing factoring to period finding
With the DFT, we have seen, that we can use a unitary to find the period of a quantum state. We now look into using period finding to factor integers. We first look at the definition of the two problems:
## Reducing factoring to period finding
With the DFT, we have seen, that we can use a unitary to find the period of a quantum state. We now look into using period finding to factor integers. We first look at the definition of some problems:
::: {.callout-note appearance="minimal" icon=false}
::: {.definition #def-shor-factoring}
## Factoring problem
Given integer $N$ with two prime factors $p,q$ such that $pq=N$ and $p \neq q$, find $p$ and $q$.
Given integer $N$ with two prime factors $p,q > 2$ such that $pq = N$ and $p \neq q$, find $p$ and $q$.
:::
:::
Note that this definition of the factoring problem is a simplified version of the factoring problem, where $N$ has only 2 prime factors.
......@@ -59,8 +75,9 @@ Note that this definition of the factoring problem is a simplified version of th
## Period finding problem
Given $f: \mathbb{Z} \to X$ with $f(x) = f(y)$ iff $x \equiv y \bmod r$ for some fixed secret $r$. $r$ is called the *period* of $f$.
Find $r$.
Given $f: \mathbb{Z} \to X$ with $f(x) = f(y)$ iff $x \equiv y \bmod r$ for some fixed secret $r$, find $r$.
We call $r$ the *period* of $f$.
:::
:::
......@@ -72,40 +89,44 @@ To start the reduction, we need a special case of the period finding problem cal
## Order finding problem
For known $a$ and $N$ which are relatively prime, find the period $r$ of $f(i) = a^i \bmod N$. We call $r$ the order of $a$ written $r = \text{ ord } a$. (This is similar to finding the smallest $i > 0$ with $f(i) = 1$).
For known $a$ and $N$ which are relatively prime, find the period $r$ of $f(i) = a^i \bmod N$. We call $r$ the order of $a$ written $r = \text{ ord } a$. (This is similar to finding the smallest $i > 0$ with $f(i) = a^i \bmod N = 1$).
:::
:::
Since the order finding problem is just the period finding problem for a specific $f(x)$, we know that if we can solve the period finding problem within reasonable runtime, we can also solve the order finding problem within reasonable runtime. We now reduce the factoring problem to the order finding problem:
We have a integer $N$ as an input for the factoring problem.
We have an integer $N$ as an input for the factoring problem.
1. Pick an $a \in \{1,\dots,N-1\}$ with $a$ relatively prime to $N$.
2. Compute the order of $a$, so that $r = \text{ ord } a$ (using the solver for the order finding problem).
3. If the order $r$ is odd, we abort.
1. Pick an $a \in \{2, \dots, N-1\}$ relatively prime to $N$.
2. Compute the order of $a$, so that $r \coloneqq \operatorname{ord} \operatorname{mod} a$ (using one solver for the order finding problem).
3. If the order $r$ is odd, restart at 1.
4. Calculate $x:= a^{\frac{r}{2}} + 1 \bmod N$ and $y:= a^{\frac{r}{2}} - 1 \bmod N$.
5. If $\gcd(x,N) \in \{1,N\}$, we abort.
6. We compute $p = \gcd(x,N)$ and $q = \gcd(y,N)$.
5. If $\gcd(x,N) \in \{1,N\}$, we restart at 1.
6. Return $p = \gcd(x,N)$ and $q = \frac{N}{\gcd(y,N)}$.
The output of the reduction are $p,q$, such that $pq = N$. This holds, since
$$
xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N
xy = (a^{\frac{r}{2}}+1) (a^{\frac{r}{2}}-1) = a^r - 1 \equiv 1-1 = 0 \pmod N.
$$
This means that $N \mid xy$ an therefore $p \mid xy$ and $q \mid xy$. From this it can be concluded that $p \mid x$, $q \mid y$, $p \mid y$ and $q \nmid x$ which leads to $\gcd(x,N) = p$. Or $p$ and $q$ swapped.
::: {.callout-note appearance="minimal" icon=false}
::: {.therorem #thm-shor-abort}
## Probability of an abort
If $N$ has at least two different prime factors and $N$ is odd, then the probability to abort is $\leq \frac{1}{2}$.
If $N$ has at least two different prime factors and $N$ is odd, then the probability to restart is $\leq \frac{1}{2}$.
:::
:::
All in all this reduction shows, that if we have an oracle which can solve the period finding problem within reasonable runtime, we can also solve the factoring problem within reasonable runtime (since all other operations are classically fast to compute).
## The quantum algorithm for period finding {#sec-shor-algo}
We now look into an quantum algorithm that solves the period finding problem within reasonable runtime. The quantum circuit for Shor's algorithm requires a $f:\mathbb{Z}\rightarrow X$ which is $r$-periodic. $X$ denotes an arbitrary set here. We choose a number $m$ which needs to be big enough to encode the values of $X$ and choose a number $n$ under the condition of $n\geq 2 \log_2(r)$ for the post processing to work. Note that when using this algorithm for factoring, we choose $n$ to be $n:=2\lvert N \rvert$, since $r \leq N$. $\lvert N\rvert$ denotes the number of bits needed to encode $N$ here.
We now look into an quantum algorithm that solves the period finding problem within reasonable runtime.
For the quantum circuit we need an $f: \{0,1\}^n \rightarrow \{0,1\}^m$ which is $r$-periodic with $r < 2^n$.
The quantum algorithm for period finding is shown in this figure:
......@@ -113,15 +134,15 @@ The quantum algorithm for period finding is shown in this figure:
The algorithm works as follows:
1. We start with a $\ket{0}$ entry on every wire.
2. We bring the top wire into the superposition over all entries. The quantum state is then $2^\frac{-n}{2}\sum_x \ket{x} \otimes \ket{0^m}$.
3. We apply $U_f$, which is the unitary of $f:\{0,1\}^n\rightarrow\{0,1\}^m$. This calculates the superposition over all possible values $f(x)$ on the bottom wire. The resulting quantum state is $2^\frac{-n}{2}\sum_x \ket{x,f(x)}$.
4. To understand the algorithm better, we measure the bottom wire at this point. This will give us one random value $f(x_0)$ for some $x_0$. The top wire will then contain a superposition over all values $x$ where $f(x) = f(x_0)$. Since $f$ is known to be $r$-periodic, we know, that $f(x) = f(x_0)$ iff $x \equiv x_0 \bmod r$. This means, that the resulting quantum state on the top wire is periodic and can be written as $\frac{\sqrt{r}}{\sqrt{2^n}} \sum_{x\equiv x_0 \bmod r} \ket{x} \otimes \ket{f(x_0)}$. For simplicity we assume, that $r \mid 2^n$ holds.
5. We apply the Discrete Fourier Transform on the top wire. This will "analyze" the top wire for the period and output a vector with entries at multiples of $\frac{2^n}{r}$ as seen in @thm-dft-properties. For simplicity we still assume, that $r \mid 2^n$ holds.
6. We measure the top wire and get one random multiple of $\frac{2^n}{r}$, which we can denote as $a\cdot\frac{2^n}{r}$
1. We start with $\ket{\psi_1} = \ket{0^n} \otimes \ket{0^m}$.
2. We bring the top wire into the superposition over all entries. The quantum state is then $\ket{\psi_2} = G \cdot \sum_{x \in \{0,\dots,2^n-1\}} \ket{x} \otimes \ket{0^m}$ with the constant $G \coloneqq 2^\frac{-n}{2}$.
3. We apply $U_f$, which is the unitary of $f: \{0,1\}^n \rightarrow \{0,1\}^m$. This calculates the superposition over all possible values $f(x)$ on the bottom wire. The resulting quantum state is $\ket{\psi_3} = G \cdot \sum_{x \in \{0,\dots,2^n-1\}} \ket{x,f(x)}$.
4. To understand the algorithm better, we measure the bottom wire at this point. This will give us one random value $y = f(x_0)$ for some $x_0$. The top wire will then contain a superposition over all values $x$ where $f(x) = f(x_0)$. Since $f$ is known to be $r$-periodic, we know, that $f(x) = f(x_0)$ iff $x \equiv x_0 \bmod r$. This means, that the resulting quantum state on the top wire is periodic. So the complete quantum state is $\ket{\psi_4} = C \cdot \sum_{x\equiv x_0 \bmod r} \ket{x} \otimes \ket{f(x_0)}$ with the constant $C = \frac{\sqrt{r}}{\sqrt{2^n}}$.
5. We apply the Discrete Fourier Transform on the top wire. This will "analyze" the top wire for the period and output a vector with entries at multiples of $\frac{2^n}{r}$ as seen in @thm-dft-properties.
6. We measure the top wire and get one random multiple of $\frac{2^n}{r}$, which we can denote as $c = a \cdot \frac{2^n}{r}$ for some $a$.
Since we get a multiple of $\frac{2^n}{r}$ on each run, we can simply run the algorithm multiple times to get different multiples and then compute $\frac{2^n}{r}$ by taking the gcd of those multiples. From that we compute $r$. Unfortunately this only works because we assumed $r \mid 2^n$. Since this does usually not hold, we only get approximate multiples of $\frac{2^n}{r}$ (which is not even an integer) and thus we need a post processing.
Since we get a multiple of $\frac{2^n}{r}$ on each run, we can simply run the algorithm multiple times to get different multiples and then compute $\frac{2^n}{r}$ by taking the gcd of those multiples. From that we compute $r$. Unfortunately this only works because we assumed $r \mid 2^n$. Since this does usually not hold, we only get approximate multiples of $\frac{2^n}{r}$ (which is not even an integer) and thus post processing is a bit more complex.
## Post processing
......@@ -131,7 +152,7 @@ So far we have seen the DFT to analyze the period of a quantum state, we have se
::: {.therorem #thm-shor-post-process}
##
Iff $f: \mathbb{Z} \rightarrow X$ is $r$-periodic, the following holds with probability $\Omega(1/\log\log r)$:
Iff $f: \{0,1\}^n \rightarrow \{0,1\}^m$ is $r$-periodic, the following holds with probability $\Omega(1/\log\log r)$:
$$
\frac{-r}{2} \leq rc\bmod 2^n \leq \frac{r}{2}
$$
......@@ -174,7 +195,7 @@ For a given continued expansion, a prefix $[a_0,\dots,a_i]$ is called a converge
::: {.callout-tip icon=false}
## Example: continued expansion of a fraction
## Example: Continued expansion of a fraction
The number $2.3$ can be written as
$$
......@@ -265,7 +286,7 @@ The next step is to construct the $\psi_2$ state on the middle wire. We again us
On the bottom wire, we can just do a Hadamad-gate to bring $\ket{j_3}$ into the superposition $\frac{1}{\sqrt{2}}(\ket{0} + e^{2 \pi i 0.j_3} \ket{1})$. We then have $\psi_1$ on the bottom wire. The full circuit is described in this figure:
![The DTF for three qubits](img/dft1){width=100%}
![The DTF for three qubits](img/dft_3_qubits){width=100%}
When applying this circuit, we get the state $\psi_3 \otimes \psi_2 \otimes \psi_1$ as a result. This very close to our desired state $\psi_1 \otimes \psi_2 \otimes \psi_3$, just the order of the wires is flipped. To solve this, we apply a $\operatorname{SWAP}$ onto all wires, which flips the order of the wires an delivers the correct output for $\operatorname{DFT}_{2^3}$.
:::
......@@ -282,4 +303,4 @@ Note: If the the output of the DFT circuit is measured right after applying it (
The more general layout of the quantum circuit for the $\operatorname{DFT_{2^n}}$ with the $\operatorname{SWAP}$ is shown in this figure.
![The DTF for $n$ qubits](img/dft2){width=100%}
![The DTF for $n$ qubits](img/dft_n_qubits){width=100%}
index.qmd
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......@@ -14,10 +14,3 @@ If you spot an error, please report it to [Gitlab](https://git.rwth-aachen.de/un
These lecture notes are released under the CC BY-NC 4.0 license, which can be found [here](https://creativecommons.org/licenses/by-nc/4.0/).
The Jupyter notebooks created during the lectures can be found in the [JupyterHub](https://jupyter.rwth-aachen.de/) in the course "[IQC] Introduction to Quantum Computing" or in Moodle. These will be added shortly after the lectures. If changes are made to the files, they can be easily reset with the usual git commands using "Git" and then "Open Git Repository in Terminal.
## Changelog {.unnumbered}
#### 16.05.2025
- started the lecture notes for the summer term 2025
- added chapters 1 to 9 (except 4.3)
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