407 lines
53 KiB
Plaintext
407 lines
53 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Computing the elements of the local clifford group"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import product\n",
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"import sympy\n",
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"i = sympy.I\n",
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"exp = sympy.exp\n",
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"M = sympy.Matrix\n",
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"simplify = sympy.simplify\n",
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"from sympy.physics.quantum.dagger import Dagger\n",
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"\n",
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"sympy.init_printing()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"I = M([[1, 0], [0, 1]])\n",
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"X = M([[0, 1], [1, 0]])\n",
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"Y = M([[0, -i], [i, 0]])\n",
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"Z = M([[1, 0], [0, -1]])\n",
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"O = M([[0, 0], [0, 0]])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"H = M([[1, 1], [1, -1]]) / sympy.sqrt(2)\n",
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"S = M([[1, 0], [0, i]])\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"C = [H, S]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"def is_math_the_same(v):\n",
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" return simplify(v[0] - v[1]) == O\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"from multiprocessing import Pool, cpu_count\n",
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"\n",
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"pool = Pool(cpu_count())"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"found Matrix([[1, 0], [0, 1]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)/2], [sqrt(2)*I/2, -sqrt(2)*I/2]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)*I/2], [sqrt(2)/2, -sqrt(2)*I/2]])\n",
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"found Matrix([[1, 0], [0, -1]])\n",
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"found Matrix([[1/2 + I/2, 1/2 - I/2], [1/2 - I/2, 1/2 + I/2]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)*I/2], [sqrt(2)*I/2, sqrt(2)/2]])\n",
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"found Matrix([[sqrt(2)/2, -sqrt(2)/2], [sqrt(2)/2, sqrt(2)/2]])\n",
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"found Matrix([[1, 0], [0, -I]])\n",
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"found Matrix([[1/2 + I/2, 1/2 - I/2], [1/2 + I/2, -1/2 + I/2]])\n",
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"found Matrix([[0, 1], [1, 0]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)/2], [-sqrt(2)*I/2, sqrt(2)*I/2]])\n",
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"found Matrix([[1/2 + I/2, 1/2 + I/2], [1/2 - I/2, -1/2 + I/2]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)*I/2], [-sqrt(2)/2, sqrt(2)*I/2]])\n",
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"found Matrix([[sqrt(2)/2, -sqrt(2)/2], [sqrt(2)*I/2, sqrt(2)*I/2]])\n",
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"found Matrix([[sqrt(2)/2, -sqrt(2)*I/2], [sqrt(2)/2, sqrt(2)*I/2]])\n",
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"found Matrix([[sqrt(2)*(1 + I)/2, 0], [0, sqrt(2)*(1 - I)/2]])\n",
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"found Matrix([[1/2 + I/2, 1/2 - I/2], [-1/2 + I/2, -1/2 - I/2]])\n",
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"found Matrix([[0, 1], [I, 0]])\n",
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"found Matrix([[1/2 - I/2, 1/2 + I/2], [1/2 + I/2, 1/2 - I/2]])\n",
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"found Matrix([[1/2 + I/2, 1/2 + I/2], [1/2 + I/2, I*(-1 + I)/2]])\n",
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"found Matrix([[0, I], [1, 0]])\n",
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"found Matrix([[sqrt(2)/2, sqrt(2)*I/2], [-sqrt(2)*I/2, -sqrt(2)/2]])\n",
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"found Matrix([[1/2 + I/2, -1/2 + I/2], [1/2 - I/2, I*(-1 + I)/2]])\n",
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"found Matrix([[sqrt(2)/2, -sqrt(2)/2], [-sqrt(2)/2, -sqrt(2)/2]])\n",
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"found Matrix([[sqrt(2)/2, -sqrt(2)*I/2], [sqrt(2)*I/2, -sqrt(2)/2]])\n"
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]
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}
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],
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"source": [
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"\n",
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"#def list_contains_matrix(l, m):\n",
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"# for elem in l:\n",
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"# test = simplify(elem - m)\n",
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"# if(test == 0):\n",
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"# return True\n",
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"# return False\n",
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"\n",
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"def list_contains_matrix(l, m):\n",
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" #print(l, m)\n",
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" for test in pool.map(is_math_the_same, [(e,m) for e in l]):\n",
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" if(test):\n",
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" return True\n",
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"\n",
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"\n",
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"#for i in range(10):\n",
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"while(len(C) < 24):\n",
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" new_ops = ([c * H for c in C]\n",
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" + [c * S for c in C])\n",
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" for new_op in new_ops:\n",
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" new_op.simplify()\n",
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" #if(new_op not in C): # and Dagger(new_op) not in C):\n",
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" if(not list_contains_matrix(C, new_op)):\n",
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" print(\"found\", new_op)\n",
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" C.append(new_op)\n",
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" #else:\n",
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" #print(new_op, \"is already known\")\n",
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" "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"for c in C:\n",
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" c.simplify()\n",
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"#C = [c for c in C if Dagger(c) not in C]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"scrolled": false
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},
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"outputs": [
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{
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"data": {
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"image/png": "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"text/latex": [
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"$\\displaystyle \\left[ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}1 & 0\\\\0 & i\\end{matrix}\\right], \\ \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right], \\ \\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], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}1 & 0\\\\0 & - i\\end{matrix}\\right], \\ \\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], \\ \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\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], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2} \\left(1 + i\\right)}{2} & 0\\\\0 & \\frac{\\sqrt{2} \\left(1 - i\\right)}{2}\\end{matrix}\\right], \\ \\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], \\ \\left[\\begin{matrix}0 & 1\\\\i & 0\\end{matrix}\\right], \\ \\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], \\ \\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], \\ \\left[\\begin{matrix}0 & i\\\\1 & 0\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\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], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], \\ \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]\\right]$"
|
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|
|
],
|
|||
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|
"text/plain": [
|
|||
|
|
"⎡⎡√2 √2 ⎤ ⎡ √2 √2 ⎤ ⎡√2 √2⋅ⅈ ⎤ ⎡1 ⅈ 1 \n",
|
|||
|
|
"⎢⎢── ── ⎥ ⎢ ── ── ⎥ ⎢── ──── ⎥ ⎢─ + ─ ─ \n",
|
|||
|
|
"⎢⎢2 2 ⎥ ⎡1 0⎤ ⎡1 0⎤ ⎢ 2 2 ⎥ ⎢2 2 ⎥ ⎡1 0 ⎤ ⎢2 2 2 \n",
|
|||
|
|
"⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ \n",
|
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|
|
"⎢⎢√2 -√2 ⎥ ⎣0 ⅈ⎦ ⎣0 1⎦ ⎢√2⋅ⅈ -√2⋅ⅈ ⎥ ⎢√2 -√2⋅ⅈ ⎥ ⎣0 -1⎦ ⎢1 ⅈ 1 \n",
|
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|
|
"⎢⎢── ────⎥ ⎢──── ──────⎥ ⎢── ──────⎥ ⎢─ - ─ ─ \n",
|
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|
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"⎣⎣2 2 ⎦ ⎣ 2 2 ⎦ ⎣2 2 ⎦ ⎣2 2 2 \n",
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"\n",
|
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|
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" ⅈ⎤ ⎡ √2 √2⎤ ⎡ √2 √2⋅ⅈ⎤ ⎡√2 -√2 ⎤ ⎡1 ⅈ 1 ⅈ ⎤ \n",
|
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"- ─⎥ ⎢ ── ──⎥ ⎢ ── ────⎥ ⎢── ────⎥ ⎢─ + ─ ─ - ─ ⎥ \n",
|
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|
|
" 2⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎢2 2 ⎥ ⎡1 0 ⎤ ⎢2 2 2 2 ⎥ ⎡0 1⎤ \n",
|
|||
|
|
" ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥,\n",
|
|||
|
|
" ⅈ⎥ ⎢-√2 √2⎥ ⎢√2⋅ⅈ √2 ⎥ ⎢√2 √2 ⎥ ⎣0 -ⅈ⎦ ⎢1 ⅈ 1 ⅈ⎥ ⎣1 0⎦ \n",
|
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|
|
"+ ─⎥ ⎢──── ──⎥ ⎢──── ── ⎥ ⎢── ── ⎥ ⎢─ + ─ - ─ + ─⎥ \n",
|
|||
|
|
" 2⎦ ⎣ 2 2 ⎦ ⎣ 2 2 ⎦ ⎣2 2 ⎦ ⎣2 2 2 2⎦ \n",
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"\n",
|
|||
|
|
" ⎡ √2 √2 ⎤ ⎡1 ⅈ 1 ⅈ ⎤ ⎡ √2 √2⋅ⅈ⎤ ⎡ √2 -√2 ⎤ ⎡√2 -√2⋅ⅈ ⎤ ⎡\n",
|
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|
|
" ⎢ ── ── ⎥ ⎢─ + ─ ─ + ─ ⎥ ⎢ ── ────⎥ ⎢ ── ────⎥ ⎢── ──────⎥ ⎢\n",
|
|||
|
|
" ⎢ 2 2 ⎥ ⎢2 2 2 2 ⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎢2 2 ⎥ ⎢\n",
|
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|
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" ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢\n",
|
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|
|
" ⎢-√2⋅ⅈ √2⋅ⅈ⎥ ⎢1 ⅈ 1 ⅈ⎥ ⎢-√2 √2⋅ⅈ⎥ ⎢√2⋅ⅈ √2⋅ⅈ⎥ ⎢√2 √2⋅ⅈ ⎥ ⎢\n",
|
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|
|
" ⎢────── ────⎥ ⎢─ - ─ - ─ + ─⎥ ⎢──── ────⎥ ⎢──── ────⎥ ⎢── ──── ⎥ ⎢\n",
|
|||
|
|
" ⎣ 2 2 ⎦ ⎣2 2 2 2⎦ ⎣ 2 2 ⎦ ⎣ 2 2 ⎦ ⎣2 2 ⎦ ⎣\n",
|
|||
|
|
"\n",
|
|||
|
|
"√2⋅(1 + ⅈ) ⎤ ⎡ 1 ⅈ 1 ⅈ ⎤ ⎡1 ⅈ 1 ⅈ⎤ ⎡1 ⅈ \n",
|
|||
|
|
"────────── 0 ⎥ ⎢ ─ + ─ ─ - ─ ⎥ ⎢─ - ─ ─ + ─⎥ ⎢─ + ─ \n",
|
|||
|
|
" 2 ⎥ ⎢ 2 2 2 2 ⎥ ⎡0 1⎤ ⎢2 2 2 2⎥ ⎢2 2 \n",
|
|||
|
|
" ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ \n",
|
|||
|
|
" √2⋅(1 - ⅈ)⎥ ⎢ 1 ⅈ 1 ⅈ⎥ ⎣ⅈ 0⎦ ⎢1 ⅈ 1 ⅈ⎥ ⎢1 ⅈ ⅈ\n",
|
|||
|
|
" 0 ──────────⎥ ⎢- ─ + ─ - ─ - ─⎥ ⎢─ + ─ ─ - ─⎥ ⎢─ + ─ ─\n",
|
|||
|
|
" 2 ⎦ ⎣ 2 2 2 2⎦ ⎣2 2 2 2⎦ ⎣2 2 \n",
|
|||
|
|
"\n",
|
|||
|
|
" 1 ⅈ ⎤ ⎡ √2 √2⋅ⅈ⎤ ⎡1 ⅈ 1 ⅈ ⎤ ⎡ √2 -√2 ⎤ ⎡ √2 \n",
|
|||
|
|
" ─ + ─ ⎥ ⎢ ── ────⎥ ⎢─ + ─ - ─ + ─ ⎥ ⎢ ── ────⎥ ⎢ ── \n",
|
|||
|
|
" 2 2 ⎥ ⎡0 ⅈ⎤ ⎢ 2 2 ⎥ ⎢2 2 2 2 ⎥ ⎢ 2 2 ⎥ ⎢ 2 \n",
|
|||
|
|
" ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ \n",
|
|||
|
|
"⋅(-1 + ⅈ)⎥ ⎣1 0⎦ ⎢-√2⋅ⅈ -√2 ⎥ ⎢1 ⅈ ⅈ⋅(-1 + ⅈ)⎥ ⎢-√2 -√2 ⎥ ⎢√2⋅ⅈ \n",
|
|||
|
|
"─────────⎥ ⎢────── ────⎥ ⎢─ - ─ ──────────⎥ ⎢──── ────⎥ ⎢──── \n",
|
|||
|
|
" 2 ⎦ ⎣ 2 2 ⎦ ⎣2 2 2 ⎦ ⎣ 2 2 ⎦ ⎣ 2 \n",
|
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|
|
"\n",
|
|||
|
|
"-√2⋅ⅈ ⎤⎤\n",
|
|||
|
|
"──────⎥⎥\n",
|
|||
|
|
" 2 ⎥⎥\n",
|
|||
|
|
" ⎥⎥\n",
|
|||
|
|
" -√2 ⎥⎥\n",
|
|||
|
|
" ──── ⎥⎥\n",
|
|||
|
|
" 2 ⎦⎦"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 10,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"\n",
|
|||
|
|
"C"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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\n",
|
|||
|
|
"text/latex": [
|
|||
|
|
"$\\displaystyle 28$"
|
|||
|
|
],
|
|||
|
|
"text/plain": [
|
|||
|
|
"28"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"len(C)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"This is bad. Does one have to ignore global phases? Note that the matrices 10 and 18 are the same, disregarding a phase of $pi/4$:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 13,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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\n",
|
|||
|
|
"text/latex": [
|
|||
|
|
"$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & - i\\end{matrix}\\right]$"
|
|||
|
|
],
|
|||
|
|
"text/plain": [
|
|||
|
|
"⎡1 0 ⎤\n",
|
|||
|
|
"⎢ ⎥\n",
|
|||
|
|
"⎣0 -ⅈ⎦"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 13,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"C[10]"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 20,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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\n",
|
|||
|
|
"text/latex": [
|
|||
|
|
"$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} \\left(1 + i\\right)}{2} & 0\\\\0 & \\frac{\\sqrt{2} \\left(1 - i\\right)}{2}\\end{matrix}\\right]$"
|
|||
|
|
],
|
|||
|
|
"text/plain": [
|
|||
|
|
"⎡√2⋅(1 + ⅈ) ⎤\n",
|
|||
|
|
"⎢────────── 0 ⎥\n",
|
|||
|
|
"⎢ 2 ⎥\n",
|
|||
|
|
"⎢ ⎥\n",
|
|||
|
|
"⎢ √2⋅(1 - ⅈ)⎥\n",
|
|||
|
|
"⎢ 0 ──────────⎥\n",
|
|||
|
|
"⎣ 2 ⎦"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 20,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"C[18]"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"XXX: continue here with computations"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 12,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"lookup_table = [ca * cb for ca, cb in product(C, C)]\n",
|
|||
|
|
"for l in lookup_table:\n",
|
|||
|
|
" l.simplify()"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 13,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"False"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 13,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"len(lookup_table) == 24*24"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 14,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"ltb = []\n",
|
|||
|
|
"row = []\n",
|
|||
|
|
"for i, result in enumerate(lookup_table):\n",
|
|||
|
|
" if(i % 24):\n",
|
|||
|
|
" ltb.append(row)\n",
|
|||
|
|
" row = []\n",
|
|||
|
|
" row.append(result)\n"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 15,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"#lookup_table = [[C.index(i) for i in row] for row in ltb]\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"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
|
|||
|
|
}
|