From 1f166808db685d4cb419f2c2d3384166aeac602a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Tom=20Kr=C3=BCger?= Date: Fri, 10 Feb 2023 22:35:50 +0100 Subject: [PATCH] added missing reverse permutation --- content.tex | 6 +++--- main.pdf | Bin 355413 -> 355746 bytes 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/content.tex b/content.tex index 61b1b0d..965552d 100644 --- a/content.tex +++ b/content.tex @@ -497,7 +497,7 @@ A quantum gate is just a unitary operator on a select set of qubits. Because the \label{def:gate_nbit_extension} Let $\ket{\mathbf{x}} = \ket{x_1, \dots, x_n}$ be the $n$ qubits of a quantum system and let $U$ be a $k$-qubit quantum gate, with $k < n$. The $k$-tuple $I = (i_1, \dots, i_k)$ with $i_l, i_g \in [1,n]$ and $i_l \neq i_g$ for all $l,g$ maps $k$ qubits from $\ket{\mathbf{x}}$ to the inputs of $U$. Then, \begin{equation*} - G_n(U,I) \coloneqq V_{\pi_I} (U \otimes \idmat_{n-k}) + G_n(U,I) \coloneqq V_{\pi_I} (U \otimes \idmat_{n-k}) V_{\pi_I}^\dagger \end{equation*} is the $n$-qubit extension of $U$, with $V_{\pi_I}$ being a permutation matrix that satisfies \begin{equation*} @@ -509,7 +509,7 @@ A quantum gate is just a unitary operator on a select set of qubits. Because the \begin{remark} Two gates $U_1 \in \C^{2^{k_1}}$ and $U_2 \in \C^{2^{k_2}}$ acting on $I_1 = (i_1^1,\dots,i_{k_1}^1)$ and $I_2 = (i_1^2,\dots,i_{k_2}^2)$ respectively can be computed in parallel iff $I_1 \cap I_2 = \emptyset$ and thus \begin{equation*} - G_n(U_1,I_1) G_n(U_2,I_2) = G_n(U_1 \otimes U_2, I_1I_2) = V_{\pi_{I_1 I_2}}(U_1 \otimes U_2 \otimes \idmat_{n - (k_1 + k_2)}) + G_n(U_1,I_1) G_n(U_2,I_2) = G_n(U_1 \otimes U_2, I_1I_2) = V_{\pi_{I_1 I_2}}(U_1 \otimes U_2 \otimes \idmat_{n - (k_1 + k_2)}) V_{\pi_{I_1 I_2}}^\dagger \end{equation*} with $I_1I_2$ being the concatenation of $I_1$ and $I_2$ and \begin{equation*} @@ -525,4 +525,4 @@ A quantum gate is just a unitary operator on a select set of qubits. Because the The circuit $C$ is said to be maximally parallelized if all neighboring gate extensions $G_n(U_l,I_l)$ and $G_n(U_{l+1}, I_{l+1})$ with $I_l \cap I_{l+1} = \emptyset$ are reduced to $G_n(U_l \otimes U_{l+1}, I_l I_{l+1})$. \end{definition} - +\contentsketch{Circuit example Deutsch's algorithm} diff --git a/main.pdf b/main.pdf index 9028db4c8fc54b4ea797b968a0d45f5820dfcb9c..69dd453ca9564b424b1a695ca079aedd79f5b07f 100644 GIT binary patch delta 5810 zcmV;j7ES5Z*cGDN6|iB}12{D~lMyK>f8AVLliRixe$TH^pQ@c&2yO((q?t~VHmN(E z>9n(L9^yRE5_iQcMQwPg?f&|ng9Ba#O4_xZqo1D}#w!gZqzyfl zS+=Wv`q)nB-1tLu%*Izo$P)D6@Z^Zfte>(@}VN%EhQ6QL)e@;#1hoT5B7IZl%Kg@0<{6-GW3esQM`_>dnSt8@#a(;{zh z$h!~)$2*h-iU7HQPwC@hbxhr)d)z3{VXaxeY^WQiE%MBdeZ_??cX@g5Kcydu3xPW8 zm1T+LPMbv%vB*R#1?J+He=hwnGuHZ;#m1~QqhYT&?@3P zbRN!hEGA2${ubF=qK2eN&rT1BRMcI3s4@^ig?8nd%bnx$Mk>j8JmvK+t7)XtY7I<& zBm=7&$4_iT_3s8aZG_+YSo9VOuc@a&O=z$hmg$)hq|?BNxErzgf2n<)H~mgt7uf-g z((3i?-Y--)KulWo+fCo*ewWVl;CF2Ij%X1J#uG_Yd)yk?n&#Z`;J(CX;p`6EkWzfV 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