From 53dffcc6747b016bab75bc9fcf08fb69b309cb31 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Fri, 14 Feb 2020 11:03:27 +0100 Subject: [PATCH] some work on the paper --- thesis/chapters/stabilizer.tex | 108 ++++++++++++++++++++++++++++++--- thesis/cover.png | Bin 179386 -> 188148 bytes thesis/main.bib | 10 +++ 3 files changed, 108 insertions(+), 10 deletions(-) diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex index fafe142..4955f7c 100644 --- a/thesis/chapters/stabilizer.tex +++ b/thesis/chapters/stabilizer.tex @@ -371,6 +371,7 @@ This transformation toggles the neighbourhood of $a$ which is an operation that will be used later. \begin{lemma} + \label{lemma:M_a} When applying $M_a$ to a state $\ket{\bar{G}}$ the new state $\ket{\bar{G}'}$ is again a VOP-free graph state and the graph is updated according to @@ -454,6 +455,7 @@ can be constructed from $CZ$ and $C_L$ and in the following discussion it will b clear that both $C_L$ and $CZ$ can be applied to a general graph state. \subsubsection{Graph States and Vertex Operators} +\label{ref:g_states_vops} \begin{definition} \label{def:g_state} @@ -522,19 +524,105 @@ it is clear that just the vertex operators are changed and the new vertex operat \end{equation} The action of a $CZ$ gate on the state $(V, E, O)$ is in most cases less trivial. -Let $i \neq j$ be two qbits, now consider the action of $CZ_{i,j}$ on $(V, E, O)$. -The following discussion closely follows \cite{andersbriegel2005} and is analogous -to the implementation. +Let $i \neq j$ be two qbits, now consider the action of $CZ_{a,b}$ on $(V, E, O)$. +The cases given here follow the implementation of a $CZ$ application in \cite{pyqcs}, +the respective paragraphs from \cite{andersbriegel2005} are given in italic. -\textbf{Case 1:} -Both $o_i$ and $o_j$ commute with $CZ_{i,j}$. This is the case for exactly +\textbf{Case 1}(\textit{Case 1})\textbf{:}\\ +Both $o_a$ and $o_b$ commute with $CZ_{a,b}$. This is the case for exactly four vertex operators: $\mathcal{Z} = \left\{I, Z, S, S^\dagger\right\}$. In this case the CZ can be pulled past the vertex operators and just the edges -are changed to $E' = E \Delta \left\{\{i,j\}\right\}$. +are changed to $E' = E \Delta \left\{\{a,b\}\right\}$. -\textbf{Case 2:} -At least one vertex operator does not commute with $CZ_{i,j}$. +\textbf{Case 2}(\textit{Sub-Sub-Case 2.2.1})\textbf{:}\\ +The two qbits are isolated: From the definition of the graph state it is clear that +any isolated clique of the graph can be treated independently. Therefore the two isolated qbits +can be treated as an independent state and the set of two qbit stabilizer states is finite. An +upper bound to the number of two qbit stabilizer states is given by $2\cdot24^2$: with or without +an edge between the qbits and $24$ Clifford operators on each vertex. -\textbf{Sub-Case 2.1} -Both +All those states and the resulting state after a $CZ$ application can be computed and while doing so one +gets another interesting result that will be useful later: If one vertex has the vertex operator $I$ the +resulting state can be chosen such that at least one of the vertex operators is $I$ again and in particular +the identity on the vertex can be preserved under the application of a $CZ$. + +\textbf{Case 3}(\textit{Case 2, but one sub-case has been handled})\textbf{:}\\ +At least one vertex operator does not commute with $CZ$ and at least one vertex +has non-operand neighbours. In this case one can try to clear the vertex operators +which will succeed for at least one qbit. + +The transformation given in +Lemma \ref{lemma:M_a} is used to "clear" the vertex operators. Recalling that +the transformation $M_j$ toggles the neighbourhood of vertex $j$ gives substance +to the following theorem: + +\begin{theorem} + A graph state $\ket{G}$ associated with $(V, E, O)$ is invariant when + simultaneously toggling the neighbourhood of a non-isolated qbit $j$ + and right-multiplying $M_j^\dagger$ to the vertex operators in the sense + that $\sqrt{-iX}^\dagger = \sqrt{iX}$ is right-multiplied to $o_j$ and + $\sqrt{iZ}^\dagger = \sqrt{-iZ}$ is right-multiplied to $o_l$ for all + neighbours $l$ of $j$. + + Without proof. +\end{theorem} + +As stated in \ref{ref:g_states_vops} $C_L$ is also generated by $\sqrt{-iX}$ and $\sqrt{iZ}$. +This yields an algorithm to reduce the vertex operator of a non-isolated qbit $j$ to the identity. +The combined operation of toggling the neighbourhood of $j$ and right-multiplying +$M_j^\dagger$ is called $L_j$ transformation and it transforms $(V, E, O)$ into a so-called +local Clifford equivalent graphical representation. + +\begin{enumerate} + \item{Check whether $n_a \setminus \{b\} \neq \{\}$. If this condition does not hold + the vertex operator on $a$ cannot be cleared } + \item{Express the vertex operator $o_a$ as a product of $\sqrt{-iX}$ and $\sqrt{iZ}$.} + \item{Move from right to left through the product + \begin{enumerate} + \item{If the current matrix is $\sqrt{-iX}$ apply $L_a$} + \item{If the current matrix is $\sqrt{iZ}$ select a vertex + $j \in n_a \setminus \{b\}$ (which is possible as it has been checked before) + and apply $L_j$.} + \end{enumerate} + } +\end{enumerate} + +This algorithm has the important properties that if the algorithm succeeds +$o_a = I$ and $o_b$ has picked up powers of $\sqrt{iZ}^\dagger$. +If $b$ has non-operand neighbours after clearing the vertex operators on $a$ the vertex operators on $b$ +can be cleared using the same algorithm which gives $o_b = I$ and $o_a = (\sqrt{iZ}^\dagger)^l = S^l \in \mathcal{Z}$ +for some $l \in \{1, ..., 5\}$. Therefore Case 1 can now be applied. + +If $o_a$ could not be cleared $o_b$ can be cleared using the same procedure and after +clearing $o_b$ one can retry to clear $o_a$. + +In any case at least one vertex operator has been cleared. If both vertex operators have been +cleared Case 1 will be applied, if there is just one cleared vertex operator it is sure that it +is the vertex with non-operand neighbours. Using this one can still apply a $CZ$: Without loss of generality +assume that $a$ has non-operand neighbours and $b$ does not. Now the state $\ket{G}$ has the form\cite{andersbriegel2005} + +\begin{equation} + \begin{aligned} + \ket{G} &= \left(\prod\limits_{o_i \in O} o_i\right) \left(\prod\limits_{\{i,j\} \in E} CZ_{i,j}\right) \ket{+}_n \\ + &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) o_b (CZ_{a,b})^s \ket{+}_n \\ + &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(o_b (CZ_{a,b})^s \ket{+}_2\right) \\ + \end{aligned} +\end{equation} + +as $o_b$ commutes with all operators but $CZ_{a,b}$ and $s \in \{0, 1\}$ indicates whether there is an edge between $a$ and $b$. + +\begin{equation} + \begin{aligned} + CZ_{a,b}\ket{G} &= CZ_{a,b}\left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) o_b (CZ_{a,b})^s \ket{+} \\ + &= \left(\prod\limits_{o_i \in O \setminus \{o_b\}} o_i \right) \left(\prod\limits_{\{i,j\} \in E \setminus \{a,b\}} CZ_{i,j}\right) \ket{+}_{n-2} \otimes \left(CZ_{a,b} o_b (CZ_{a,b})^s \ket{+}_2\right) \\ + \end{aligned} +\end{equation} + +Using this one can re-use the method used in Case 2 to apply the $CZ$ while keeping the $o_a = I$. + +As both $CZ$ and $C_L$ can be applied to a graph state $\ket{G}$ this proofs constructively that +the graphical representation of a stabilizer state is indeed able to represent any stabilizer state. +If one wants to do computations using this formalism it is however also necessary to perform measurements. + +\subsubsection{Measurements on Graph States} diff --git a/thesis/cover.png b/thesis/cover.png index 224e64ba551da1898f53ea060de5559a23dbd986..566a8719098eafa2cc72cfe5d7141c85de0f615a 100644 GIT binary patch literal 188148 zcmeEug;&)1{`a7ZuDvRugxDx4-C!e9(hULv(hfM3gsY-RDoB^2(lB(xsFactGsFKF|A z$K8+r+y#HzA$0H){Ou3ts|uQb!e5?$-t&Xkdmh}q$F+E$- zll*C~rAJmR{jSr_1CLY`LeDC|JO9^TrR|yTj{f@v29vHzck(~4hF|Tt{GS(JPtiaA z&x=>r_G0M%{p#ck2F#9szqowxBIb{Ozj(~1j`{Q7FEA1xFn|5$h0`s%lmB_~-yeh? z#eW~&c0v5N82(!f+l7Gs@4v;#CeU|s)MeAP$Z?BBZ zRPOh^JLu@K-aMC5Q6cTpV-dO6C*Nm-W^y2&MfjMxo+;zdlKW&~m$sXCYl6%fh0rp^ zxbs%=&sE^z>6ZVjeo^n}Jr}z??XyMxp7y=QYo?Yt_wMH>7x?)@4v2Q2>@9ZWpkVn-zvN2LFUF+9C7W#%Ex(7(+)%(4|=)da_YxuA*;HkXd%`Q(~drCWcVQg zyT)1H^6{EfNe_4}^*>iJ&nk17?EQZ6`yZLQxu?As^Al^E#RmJnr`_2d!E3cN^?IVL7ga2ms1$YS)D&TB^O%IwJ*61oiwzA8r2G85yg`m$G=p4t zLp87EPaGdyX>n!A{K}_Vs_s?X=cMSlxiS_mJ35iy8mtm8$yhL`W8x)p!hQJlUUIC6 zEmNe4cd1!n;qq)FKPB2K;U&D5s0(6c=I~xN^;{iK$$dE9-dT^&Z;@=vHLve>D4#T| zXLk)eA>&aayEbu@>`3sEu{wykd@V-6{3d)^vM%uWQGDg1zEXm;1o{XbALBwpL%ZE5 zLFQ|d#UVy6J?9qtT?c>lm%G|kt&ARcC2_~ZvN?uP$JpJ%cM=uX`+|Z3I>8;cy5P~3 z%r;17rX_HV)W0&18)3x)4}eYL!R_vt_B6~ZAWTZW^NPI~R^ z{FtVZQpZ<5(f-{!GhESGgM+W^*XdfHpqhqoZc4oCz)LKp&TB&mJ5DourmCVJ1-jD3c#*=3POx91gNKIGhTE(yGIgMV_ zu{p00TJrnLKJ(4>m2lbYYcCFBp+Fc4rXNj}a&FE>5;VK3CX2{*j5>yD->X;x7`e4; 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} + +@online{ + pyqcs, + url={https://github.com/daknuett/pyqcs}, + urldate={13.01.2020}, + author={Daniel Knüttel}, + title={PyQCS}, + year=2020, + note={https://github.com/daknuett/pyqcs}, +}