From c26f1b19a036e5bc078a82ede84264977f434d51 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Mon, 9 Dec 2019 17:45:43 +0100 Subject: [PATCH] some more work on stabilizers --- .../projector_after_commute_C_L.ipynb | 8 +- thesis/chapters/stabilizer.tex | 120 +++++++++++++++++- 2 files changed, 122 insertions(+), 6 deletions(-) diff --git a/computations/projector_after_commute_C_L.ipynb b/computations/projector_after_commute_C_L.ipynb index 1a3fda5..d657ee3 100644 --- a/computations/projector_after_commute_C_L.ipynb +++ b/computations/projector_after_commute_C_L.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 11, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -53,7 +53,7 @@ " ⎦⎦" ] }, - "execution_count": 11, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -108,7 +108,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -123,7 +123,7 @@ "⎣0 -1⎦" ] }, - "execution_count": 12, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index 1b84384..1727dc5 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -80,6 +80,66 @@ This is an important insight that is used for simulations\cite{gottesman_aaronso updating the $n$ stabilizers that are a tensor product of $n$ Pauli matrices scales with roughly $\mathcal{O}(n^2)$ instead of $\mathcal{O}(2^n)$ for the state vector approach. +Every qbit can be measured in the $X, Y$ or $Z$ basis which is a projection using + +\begin{equation} + \frac{I + (-1)^s g_a}{2} +\end{equation} + +Where $g_a \in \{Z_a, Y_a, X_a\}$ and $s \in \{0, 1\}$. How the stabilizers change when +measuring a qbit is given by the following lemma: + +\begin{lemma} + \label{lemma:stab_measurement} + Let $J := \{ S_i | [g_a, S_i] \neq 0\}$. Then + + \begin{enumerate} + \item{If $J = \{\}$, one value is measured with probability $1$ and the stabilizers are unchanged.} + \item{If $J \neq \{\}$, $1$ and $0$ are measured with probability $\frac{1}{2}$ and the new state + $\ket{\psi'}$ is stabilized by + \begin{equation} + \langle \{(-1)^s g_a\} \cup \{K_G^{(i)} K_G^{(j)} | j \in J, i \in J \setminus \{j\} \} \cup \{K_G^{(i)} | i \in J^c\}\rangle + \end{equation}} + \end{enumerate} +\end{lemma} + +\begin{proof} + \begin{enumerate} + \item{As $g_a$ commutes with all stabilizers, $\ket{\psi}$ is an eigenstate of $g_a$, + so the result of the measurement is deterministic and $\ket{\psi}$ is left unchanged.} + \item{As $g_a$ is a Pauli operator and $S^{(i)} \in J$ are multi-local Pauli operators, + $S_i$ and $g_a$ anticommute. Choose a $S^{(j)} \in J$. Then + + \begin{equation} + \begin{aligned} + P(s=+1) &= \hbox{Tr}\left(\frac{I + g_a}{2}\ket{\psi}\bra{\psi}\right) \\ + &= \hbox{Tr}\left(\frac{I + g_a}{2}S^{(j)} \ket{\psi}\bra{\psi}\right)\\ + &= \hbox{Tr}\left(S^{(j)}\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\ + &= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}S^{(j)}\right)\\ + &= \hbox{Tr}\left(\frac{I - g_a}{2}\ket{\psi}\bra{\psi}\right)\\ + &= P(s=-1) + \end{aligned} + \notag + \end{equation} + + With $P(s=+1) + P(s=-1) = 1$ follows $P(s=+1) = \frac{1}{2} = P(s=-1)$. + Further for $S^{(i)},S^{(j)} \in J$ + + \begin{equation} + \begin{aligned} + \frac{I + (-1)^sg_a}{2}\ket{\psi} &= \frac{I + (-1)^sg_a}{2}S^{(j)}S^{(i)} \ket{\psi} \\ + &= S^{(j)}\frac{I + (-1)^{s + 1}g_a}{2}S^{(i)} \ket{\psi} \\ + &= S^{(j)}S^{(i)}\frac{I + (-1)^{s + 2}g_a}{2}\ket{\psi} \\ + &= S^{(j)}S^{(i)}\frac{I + (-1)^sg_a}{2}\ket{\psi} + \end{aligned} + \notag + \end{equation} + + the state after measurement is stabilized by $S^{(j)}S^{(i)} i,j \in J$, and by + $S^{(i)} \in J^c\setminus\{a\}$. $g_a$ trivially stabilizes $\ket{\psi'}$. + } + \end{enumerate} +\end{proof} \subsection{The Vertex Operator-Free Graph States} @@ -98,7 +158,7 @@ were derived from the vertex operator-free graph states. for all $i \in V$ where for some operator $O$ $O_i$ indicates that it acts on the $i$-th qbit. - A state $\ket{\overline{G}}$ is a $+1$ eigenstate of all $n$ $K^{(i)}_G$. + $\ket{\overline{G}}$ is the $+1$ eigenstate of all $n$ $K^{(i)}_G$. \end{definition} \begin{corrolary} @@ -163,5 +223,61 @@ were derived from the vertex operator-free graph states. \end{proof} These insights can be used to understand how measurement works on the vop-free graph state \cite{nielsen_chuang_2010}: -Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a hermitian $g$ that is to be measured. +Consider a state $\ket{\psi}$ that is stabilized by $g_1, ... g_n$ and a Pauli operator $g \in \{Z_a, Y_a, X_a\}$ +that is to be measured. +Recalling lemma \ref{lemma:stab_measurement} the following relations follow immideately: + +\begin{enumerate} + \item{For $g = X_a$ if $a$ is an isolated qbit $+1$ is measured with probability $1$ + and the state $\ket{\bar{G}}$ is unchanged.} + \item{For $g = X_a$ and $a$ is non-isolated, choose $b \in n_a$ and the new stabilizers are + \begin{equation} + \langle \{(-1)^sX_a\} + \cup \{K_G^{(b)}K_G^{(i)} | i \in n_a \setminus \{b\}\} + \cup \{K_G^{(i)} | i \in V \setminus n_a \setminus \{a\}\} \rangle + \end{equation}} + \item{For $g = Z_a$ the new stabilizers are + \begin{equation} + \langle \{(-1)^sZ_a\} + \cup \{K_G^{(i)} | i \in V \setminus n_a\} \rangle + \end{equation}} + \item{For $g = Y_a$ the new stabilizers are + \begin{equation} + \langle \{(-1)^sY_a\} + \cup \{K_G^{(a)}K_G^{(i)} | i \in n_a\} + \cup \{K_G^{(i)} | i \in V \setminus n_a \setminus \{a\}\} \rangle + \end{equation}} +\end{enumerate} + +The states after the measurement are in general no vop-free graph states anymore, +the following discussion will allow to construct new vop-free graph states and +Clifford transformations from the vop-free graph state to the resulting state. +In the case $g = X_a$ and $n_a = \{\}$ the graph is obviously unchanged. + +\begin{lemma} + \begin{enumerate} + \item{For a result $+Z_a$ the new state is + $\ket{+_Z}_a \otimes \ket{\bar{G}'}$ + with $\ket{\bar{G}'}$ being stabilized by $K_G^{(i)}$ for + $i \notin n_a$ and $Z_aK_G^{(i)}$ for $i \in n_a$. + } + \item{For a result $-Z_a$ the new state is $U\ket{-_Z}_a \otimes \ket{\bar{G}'}$ + with $\ket{\bar{G}'}$ being stabilized by $K_G^{(i)}$ for $i \notin n_a$ and $Z_aK_G^{(i)}$ for $i \in n_a$ + and $U = \prod\limits_{i \in n_a}Z_i$.} + \end{enumerate} +\end{lemma} +\begin{proof} + \begin{enumerate} + \item{ + It is trivial that $Z_a$ and $K_G^{(i)} i \neq a$ stabilize $\ket{+_Z}_a \otimes \ket{\bar{G}'}$. + $Z_aK_G^{(i)}$ for $i \in n_a$ do not act on $a$, so $\ket{\bar{G}'}$ is well-defined. + } + \item{ + The state $\ket{-_Z}_a \otimes \ket{\bar{G}'}$ is the $-1$ eigenstate of $K_G^{(i)}$ for all + $i \in n_a$. This can be corrected by transforming $K_G^{(i)}$ to $-K_G^{(i)} = Z_i K_G^{(i)} Z_i^\dagger$ + which stabilizes the state $U\ket{-_Z}_a \otimes \ket{\bar{G}'}$. + } + \end{enumerate} + +\end{proof}