From 01fa59c4b851296476c998246b8069bffe030a50 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= Date: Fri, 25 Oct 2019 10:27:05 +0200 Subject: [PATCH] initial version of the LC group elements --- .../local_clifford_group_elements_raw.ipynb | 876 ++++++++++++++++++ 1 file changed, 876 insertions(+) create mode 100644 computations/local_clifford_group_elements_raw.ipynb diff --git a/computations/local_clifford_group_elements_raw.ipynb b/computations/local_clifford_group_elements_raw.ipynb new file mode 100644 index 0000000..2008957 --- /dev/null +++ b/computations/local_clifford_group_elements_raw.ipynb @@ -0,0 +1,876 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from itertools import product\n", + "import sympy\n", + "i = sympy.I\n", + "exp = sympy.exp\n", + "M = sympy.Matrix\n", + "simplify = sympy.simplify\n", + "from sympy.physics.quantum.dagger import Dagger\n", + "\n", + "sympy.init_printing()" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "I = M([[1, 0], [0, 1]])\n", + "X = M([[0, 1], [1, 0]])\n", + "Y = M([[0, -i], [i, 0]])\n", + "Z = M([[1, 0], [0, -1]])\n", + "O = M([[0, 0], [0, 0]])" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "H = M([[1, 1], [1, -1]]) / sympy.sqrt(2)\n", + "S = M([[1, 0], [0, i]])\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2 √2 ⎤\n", + "⎢── ── ⎥\n", + "⎢2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2 -√2 ⎥\n", + "⎢── ────⎥\n", + "⎣2 2 ⎦" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & i\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 0⎤\n", + "⎢ ⎥\n", + "⎣0 ⅈ⎦" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "S" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "s2m = {\"H\": H, \"S\": S}" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "def evalstr(s):\n", + " m = I\n", + " \n", + " for c in s:\n", + " m *= s2m[c]\n", + " \n", + " m.simplify()\n", + " return m" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2 √2⋅ⅈ ⎤\n", + "⎢── ──── ⎥\n", + "⎢2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2 -√2⋅ⅈ ⎥\n", + "⎢── ──────⎥\n", + "⎣2 2 ⎦" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 √2 ⎤\n", + "⎢ ── ── ⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2⋅ⅈ -√2⋅ⅈ ⎥\n", + "⎢──── ──────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2 -√2 ⎤\n", + "⎢── ────⎥\n", + "⎢2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2 √2 ⎥\n", + "⎢── ── ⎥\n", + "⎣2 2 ⎦" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 √2⎤\n", + "⎢ ── ──⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2 √2⎥\n", + "⎢──── ──⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & - i\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 0 ⎤\n", + "⎢ ⎥\n", + "⎣0 -ⅈ⎦" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2 -√2⋅ⅈ ⎤\n", + "⎢── ──────⎥\n", + "⎢2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2 √2⋅ⅈ ⎥\n", + "⎢── ──── ⎥\n", + "⎣2 2 ⎦" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 √2 ⎤\n", + "⎢ ── ── ⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2⋅ⅈ √2⋅ⅈ⎥\n", + "⎢────── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 √2⋅ⅈ⎤\n", + "⎢ ── ────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2⋅ⅈ √2 ⎥\n", + "⎢──── ── ⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 -√2 ⎤\n", + "⎢ ── ────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢√2⋅ⅈ √2⋅ⅈ⎥\n", + "⎢──── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 √2⋅ⅈ⎤\n", + "⎢ ── ────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2 √2⋅ⅈ⎥\n", + "⎢──── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSHS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 -√2 ⎤\n", + "⎢ ── ────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2 -√2 ⎥\n", + "⎢──── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSHSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 -√2⋅ⅈ ⎤\n", + "⎢ ── ──────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2 -√2⋅ⅈ ⎥\n", + "⎢──── ──────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSHSSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2 -√2⋅ⅈ ⎤\n", + "⎢ ── ──────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2⋅ⅈ √2 ⎥\n", + "⎢────── ── ⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSSHSSS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{i}{2} & \\frac{1}{2} - \\frac{i}{2}\\\\\\frac{1}{2} - \\frac{i}{2} & \\frac{1}{2} + \\frac{i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 ⅈ 1 ⅈ⎤\n", + "⎢─ + ─ ─ - ─⎥\n", + "⎢2 2 2 2⎥\n", + "⎢ ⎥\n", + "⎢1 ⅈ 1 ⅈ⎥\n", + "⎢─ - ─ ─ + ─⎥\n", + "⎣2 2 2 2⎦" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSH\")# = exp(i*pi/4) SSSHSSS" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 1⎤\n", + "⎢ ⎥\n", + "⎣1 0⎦" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\i & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 1⎤\n", + "⎢ ⎥\n", + "⎣ⅈ 0⎦" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & i\\\\1 & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 ⅈ⎤\n", + "⎢ ⎥\n", + "⎣1 0⎦" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSSHS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}0 & i\\\\i & 0\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡0 ⅈ⎤\n", + "⎢ ⎥\n", + "⎣ⅈ 0⎦" + ] + }, + "execution_count": 25, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSSHS\") # = exp(i*pi/2) HSSH" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2⋅ⅈ -√2⋅ⅈ ⎤\n", + "⎢──── ──────⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ √2 √2 ⎥\n", + "⎢ ── ── ⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSSHSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ √2⋅ⅈ √2⎤\n", + "⎢ ──── ──⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢-√2⋅ⅈ √2⎥\n", + "⎢────── ──⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSHSSH\") # = exp(-i*pi/2) SSHSSS" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡√2⋅ⅈ √2 ⎤\n", + "⎢──── ── ⎥\n", + "⎢ 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ √2 √2⋅ⅈ⎥\n", + "⎢ ── ────⎥\n", + "⎣ 2 2 ⎦" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"HSSHSHS\") # = exp(-i*pi/2) SSSHSSS" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{i}{2} & \\frac{1}{2} - \\frac{i}{2}\\\\\\frac{1}{2} + \\frac{i}{2} & - \\frac{1}{2} + \\frac{i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 ⅈ 1 ⅈ ⎤\n", + "⎢─ + ─ ─ - ─ ⎥\n", + "⎢2 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢1 ⅈ 1 ⅈ⎥\n", + "⎢─ + ─ - ─ + ─⎥\n", + "⎣2 2 2 2⎦" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSH\")" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{i}{2} & \\frac{1}{2} + \\frac{i}{2}\\\\\\frac{1}{2} + \\frac{i}{2} & \\frac{i \\left(-1 + i\\right)}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 ⅈ 1 ⅈ ⎤\n", + "⎢─ + ─ ─ + ─ ⎥\n", + "⎢2 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢1 ⅈ ⅈ⋅(-1 + ⅈ)⎥\n", + "⎢─ + ─ ──────────⎥\n", + "⎣2 2 2 ⎦" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSHS\")" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{i}{2} & - \\frac{1}{2} + \\frac{i}{2}\\\\\\frac{1}{2} + \\frac{i}{2} & \\frac{1}{2} - \\frac{i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡1 ⅈ 1 ⅈ⎤\n", + "⎢─ + ─ - ─ + ─⎥\n", + "⎢2 2 2 2⎥\n", + "⎢ ⎥\n", + "⎢1 ⅈ 1 ⅈ ⎥\n", + "⎢─ + ─ ─ - ─ ⎥\n", + "⎣2 2 2 2 ⎦" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SHSHSS\") # = SHSH" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{i}{2} & \\frac{1}{2} - \\frac{i}{2}\\\\- \\frac{1}{2} + \\frac{i}{2} & - \\frac{1}{2} - \\frac{i}{2}\\end{matrix}\\right]$" + ], + "text/plain": [ + "⎡ 1 ⅈ 1 ⅈ ⎤\n", + "⎢ ─ + ─ ─ - ─ ⎥\n", + "⎢ 2 2 2 2 ⎥\n", + "⎢ ⎥\n", + "⎢ 1 ⅈ 1 ⅈ⎥\n", + "⎢- ─ + ─ - ─ - ─⎥\n", + "⎣ 2 2 2 2⎦" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evalstr(\"SSHSH\")" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}