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Grover basis - generic algorithm
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19 September 2019 - A day with an attempt to generate Graver basis vectors | |
- https://ie.technion.ac.il/~onn/Nachdiplom/3.pdf (Followed this lecture notes, and replicated the example) | |
- https://ie.technion.ac.il/~onn/Book/NDO.pdf (Also followed the book) | |
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{ | |
"cells": [ | |
{ | |
"cell_type": "code", | |
"execution_count": 72, | |
"metadata": {}, | |
"outputs": [], | |
"source": [ | |
"import numpy as np\n", | |
"import math" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 285, | |
"metadata": {}, | |
"outputs": [], | |
"source": [ | |
"A = np.array([[1, 2, 1]])\n", | |
"x = np.array([[1], [2], [0]])\n", | |
"I = np.eye(3)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 286, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"array([[1],\n", | |
" [2],\n", | |
" [1]])" | |
] | |
}, | |
"execution_count": 286, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"A.T" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 287, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"(1, 3)" | |
] | |
}, | |
"execution_count": 287, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"A.shape" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 288, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"array([[1],\n", | |
" [2],\n", | |
" [0]])" | |
] | |
}, | |
"execution_count": 288, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"x" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 289, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"array([[5]])" | |
] | |
}, | |
"execution_count": 289, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"np.dot(A, x)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 290, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"(3, 3)" | |
] | |
}, | |
"execution_count": 290, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"I\n", | |
"I.shape" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 291, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"array([[1.],\n", | |
" [2.],\n", | |
" [0.]])" | |
] | |
}, | |
"execution_count": 291, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"np.dot(I, x)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 292, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"3" | |
] | |
}, | |
"execution_count": 292, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"np.linalg.matrix_rank(I)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 293, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"1" | |
] | |
}, | |
"execution_count": 293, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"np.linalg.matrix_rank(A)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 294, | |
"metadata": {}, | |
"outputs": [], | |
"source": [ | |
"def get_all_points(minimum, maximum, dimension):\n", | |
" x = np.arange(minimum, maximum, 1.0)\n", | |
" Z = np.meshgrid(*[x for _ in range(dimension)])\n", | |
" return np.array([z.flatten() for z in Z]).T\n", | |
"\n", | |
"def partially_ordered_2(x, y):\n", | |
" # this checks if two vectors are partially ordered with constraint 2\n", | |
" flag = True\n", | |
" for i in range(x.shape[0]):\n", | |
" if abs(x[i]) <= abs(y[i]):\n", | |
" flag = True\n", | |
" else:\n", | |
" flag = False\n", | |
" break\n", | |
" \n", | |
" return flag\n", | |
" \n", | |
"def distill_set(X):\n", | |
"# print(\"Distill\", X)\n", | |
" for x in X.copy():\n", | |
" for y in X.copy():\n", | |
" # print(x, y)\n", | |
" if x == y:\n", | |
" continue\n", | |
" if partially_ordered_2(np.array(x), np.array(y)):\n", | |
" X.remove(y)\n", | |
" " | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 295, | |
"metadata": {}, | |
"outputs": [], | |
"source": [ | |
"def generate_graver_basis(A):\n", | |
" # construct non-zero lattice of A, L(A)\n", | |
" L = set() \n", | |
" \n", | |
" # delta(A) upper bound\n", | |
" m, n = A.shape\n", | |
" delta_A = math.pow(math.sqrt(m)*np.max(A), m)\n", | |
" \n", | |
" # n=A.shape[0], m=1\n", | |
" bound = (n-np.linalg.matrix_rank(A))*delta_A\n", | |
" \n", | |
" print(delta_A)\n", | |
" print(bound)\n", | |
" # create all points in Z^n bounded by <bound\n", | |
" Z = np.arange(-bound, bound, 1.0) \n", | |
" grid = np.meshgrid(*[Z for i in range(n)])\n", | |
" x_points = np.array([g.flatten() for g in grid]).T\n", | |
" \n", | |
" # select x points that make A zero\n", | |
" for x in x_points:\n", | |
" if np.dot(A, x) == 0:\n", | |
" L.add(tuple(x))\n", | |
" \n", | |
" \n", | |
" # distill the vectors \n", | |
" L.remove(tuple(np.zeros(n)))\n", | |
" \n", | |
" # there are 2^n orthants in n-dimensional space, need to find and select partial minimal in each orthant\n", | |
" G = set() # graver basis set\n", | |
" groups = {}\n", | |
" for mask in (get_all_points(0, 2, n)*2 - 1):\n", | |
" groups[tuple(mask)] = set()\n", | |
" \n", | |
" for l in L:\n", | |
" for mask in groups:\n", | |
" flag = 0\n", | |
" for i in range(n):\n", | |
" if mask[i]*l[i] >= 0:\n", | |
" flag += 1\n", | |
" \n", | |
" \n", | |
" if flag == n:\n", | |
" groups[mask].add(l)\n", | |
" break\n", | |
" \n", | |
" # TODO: distill the basis by partial minimal \n", | |
" for mask, value in groups.items():\n", | |
" # print(mask)\n", | |
" if len(value) == 0:\n", | |
" continue\n", | |
" distill_set(value)\n", | |
"# print(\"Distilled:\", value)\n", | |
" for v in value:\n", | |
" G.add(v)\n", | |
" \n", | |
" return G\n", | |
" " | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 296, | |
"metadata": { | |
"scrolled": false | |
}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"2.0\n", | |
"4.0\n" | |
] | |
}, | |
{ | |
"data": { | |
"text/plain": [ | |
"{(-3.0, 1.0, 1.0),\n", | |
" (-2.0, 1.0, 0.0),\n", | |
" (-1.0, 0.0, 1.0),\n", | |
" (-1.0, 1.0, -1.0),\n", | |
" (0.0, -1.0, 2.0),\n", | |
" (0.0, 1.0, -2.0),\n", | |
" (1.0, -1.0, 1.0),\n", | |
" (1.0, 0.0, -1.0),\n", | |
" (1.0, 1.0, -3.0),\n", | |
" (2.0, -1.0, 0.0)}" | |
] | |
}, | |
"execution_count": 296, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"generate_graver_basis(A)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": {}, | |
"outputs": [], | |
"source": [] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": {}, | |
"outputs": [], | |
"source": [] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 297, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"[-1. -1.]\n", | |
"[ 1. -1.]\n", | |
"[-1. 1.]\n", | |
"[1. 1.]\n" | |
] | |
} | |
], | |
"source": [ | |
"for p in get_all_points(0, 2, 2)*2 - 1:\n", | |
" print(p)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 298, | |
"metadata": {}, | |
"outputs": [], | |
"source": [ | |
"x = np.matrix([[1, 1], [2, -3]])\n", | |
"xmax = x.flat[abs(x).argmax()]" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": {}, | |
"outputs": [], | |
"source": [] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 299, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"True" | |
] | |
}, | |
"execution_count": 299, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"x = np.array([-1.0, 0.0, 1.0])\n", | |
"y = np.array([-3.0, 0.0, 3.0])\n", | |
"\n", | |
"partially_ordered_2(x, y)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 300, | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"{(0.0, -1.0, 2.0), (-2.0, 0.0, 2.0), (-1.0, 0.0, 1.0), (-1.0, -1.0, 3.0), (-3.0, 0.0, 3.0)}\n", | |
"(-1.0, 0.0, 1.0) (-1.0, 0.0, 1.0)\n", | |
"(-1.0, 0.0, 1.0) (0.0, -1.0, 2.0)\n", | |
"(-1.0, 0.0, 1.0) (-2.0, 0.0, 2.0)\n", | |
"Remove (-2.0, 0.0, 2.0)\n", | |
"(-1.0, 0.0, 1.0) (-1.0, -1.0, 3.0)\n", | |
"Remove (-1.0, -1.0, 3.0)\n", | |
"(-1.0, 0.0, 1.0) (-3.0, 0.0, 3.0)\n", | |
"Remove (-3.0, 0.0, 3.0)\n", | |
"(0.0, -1.0, 2.0) (-1.0, 0.0, 1.0)\n", | |
"(0.0, -1.0, 2.0) (0.0, -1.0, 2.0)\n", | |
"(-2.0, 0.0, 2.0) (-1.0, 0.0, 1.0)\n", | |
"(-2.0, 0.0, 2.0) (0.0, -1.0, 2.0)\n", | |
"(-1.0, -1.0, 3.0) (-1.0, 0.0, 1.0)\n", | |
"(-1.0, -1.0, 3.0) (0.0, -1.0, 2.0)\n", | |
"(-3.0, 0.0, 3.0) (-1.0, 0.0, 1.0)\n", | |
"(-3.0, 0.0, 3.0) (0.0, -1.0, 2.0)\n", | |
"Final: {(0.0, -1.0, 2.0), (-1.0, 0.0, 1.0)}\n", | |
"Final: {(0.0, -1.0, 2.0), (-1.0, 0.0, 1.0)}\n" | |
] | |
} | |
], | |
"source": [ | |
"Xs = set()\n", | |
"Xs.add((0.0, -1.0, 2.0))\n", | |
"Xs.add((-2.0, 0.0, 2.0))\n", | |
"Xs.add((-1.0, 0.0, 1.0))\n", | |
"Xs.add((-1.0, -1.0, 3.0))\n", | |
"Xs.add((-3.0, 0.0, 3.0))\n", | |
"\n", | |
"print(Xs)\n", | |
"for x in Xs.copy():\n", | |
" for y in Xs.copy():\n", | |
" print(x, y)\n", | |
" if x == y:\n", | |
" continue\n", | |
" if partially_ordered_2(np.array(x), np.array(y)):\n", | |
" Xs.remove(y)\n", | |
" print(\"Remove\", y)\n", | |
"\n", | |
"print(\"Final:\", Xs)\n", | |
"\n", | |
"Xs = set()\n", | |
"Xs.add((0.0, -1.0, 2.0))\n", | |
"Xs.add((-2.0, 0.0, 2.0))\n", | |
"Xs.add((-1.0, 0.0, 1.0))\n", | |
"Xs.add((-1.0, -1.0, 3.0))\n", | |
"Xs.add((-3.0, 0.0, 3.0))\n", | |
"\n", | |
"distill_set(Xs)\n", | |
"print(\"Final:\", Xs)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": {}, | |
"outputs": [], | |
"source": [] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": {}, | |
"outputs": [], | |
"source": [] | |
} | |
], | |
"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": "3.6.7" | |
} | |
}, | |
"nbformat": 4, | |
"nbformat_minor": 2 | |
} |
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