{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"# Boson Sampling with MPS\n",
"\n",
"In this notebook, we explain how to use the MPS (Matrix Product State) backend to simulate a linear circuit. MPS simulation is based on a type of tensor network simulation, which gives an approximation of the output states [[1]](#References) [[2]](#References). It does the computation on each component of the circuits one-by-one, and not on the whole unitary. The states are represented by tensors, which are then updated at each component. These tensors can be seen as a big set of matrices, and the approximation is done by chosing the dimension of these matrices, called the *bond dimension*. For this example, we simulate a simple boson sampling problem, with 6 modes and 3 photons [[3]](#References)."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import perceval as pcvl\n",
"import matplotlib.pyplot as plt"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Definition of the circuit\n",
"\n",
"Just like in the Boson Sampling notebook, we generate a Haar-random unitary and its decomposition in a circuit :"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"6 modes triangular Boson Sampler :\n"
]
},
{
"data": {
"image/svg+xml": [
"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
"<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"\n",
" width=\"3321.25\" height=\"406.25\" viewBox=\"-28.5 0 2657.0 325.0\">\n",
"<defs>\n",
"</defs>\n",
"<path d=\"M10,25 L25,25\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M10,75 L25,75\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M10,125 L25,125\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M10,175 L25,175\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M10,225 L25,225\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M10,275 L25,275\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M25,275 L75,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,290 L39,290 L53,260 L44,260 L30,290 L39,290 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"288\" font-size=\"7\" text-anchor=\"start\">Φ=1.265255</text>\n",
"<path d=\"M25,225 L75,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,240 L39,240 L53,210 L44,210 L30,240 L39,240 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"238\" font-size=\"7\" text-anchor=\"start\">Φ=2.858772</text>\n",
"<path d=\"M25,175 L75,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,190 L39,190 L53,160 L44,160 L30,190 L39,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=3.402583</text>\n",
"<path d=\"M25,125 L75,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,140 L39,140 L53,110 L44,110 L30,140 L39,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=1.851956</text>\n",
"<path d=\"M25,75 L75,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,90 L39,90 L53,60 L44,60 L30,90 L39,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=2.413235</text>\n",
"<path d=\"M25,25 L75,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M30,40 L39,40 L53,10 L44,10 L30,40 L39,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"47\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=0.39409</text>\n",
"<path d=\"M75,25 L103,25 L122,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M128,44 L147,25 L175,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M75,75 L103,75 L122,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M128,56 L147,75 L175,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M100,43 L150,43 L150,57 L100,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"125\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"125\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M100,43 L150,43 L150,47 L100,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M143,50 L153,50 L153,60 L143,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"148\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M175,25 L225,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M180,40 L189,40 L203,10 L194,10 L180,40 L189,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"197\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=2.758766</text>\n",
"<path d=\"M175,75 L225,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M225,25 L253,25 L272,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M278,44 L297,25 L325,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M225,75 L253,75 L272,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M278,56 L297,75 L325,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M250,43 L300,43 L300,57 L250,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"275\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"275\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M250,43 L300,43 L300,47 L250,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M293,50 L303,50 L303,60 L293,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"298\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M325,75 L375,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M330,90 L339,90 L353,60 L344,60 L330,90 L339,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"347\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=3.479159</text>\n",
"<path d=\"M75,125 L375,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M375,75 L403,75 L422,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M428,94 L447,75 L475,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M375,125 L403,125 L422,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M428,106 L447,125 L475,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M400,93 L450,93 L450,107 L400,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"425\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"425\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M400,93 L450,93 L450,97 L400,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M443,100 L453,100 L453,110 L443,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"448\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M475,75 L525,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M480,90 L489,90 L503,60 L494,60 L480,90 L489,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"497\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=1.947194</text>\n",
"<path d=\"M475,125 L525,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M525,75 L553,75 L572,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M578,94 L597,75 L625,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M525,125 L553,125 L572,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M578,106 L597,125 L625,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M550,93 L600,93 L600,107 L550,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"575\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"575\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M550,93 L600,93 L600,97 L550,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M593,100 L603,100 L603,110 L593,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"598\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M625,125 L675,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M630,140 L639,140 L653,110 L644,110 L630,140 L639,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"647\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=2.84937</text>\n",
"<path d=\"M325,25 L625,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M625,25 L653,25 L672,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M678,44 L697,25 L725,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M625,75 L653,75 L672,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M678,56 L697,75 L725,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M650,43 L700,43 L700,57 L650,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"675\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"675\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M650,43 L700,43 L700,47 L650,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M693,50 L703,50 L703,60 L693,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"698\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M725,25 L775,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M730,40 L739,40 L753,10 L744,10 L730,40 L739,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"747\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=1.169331</text>\n",
"<path d=\"M725,75 L775,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M775,25 L803,25 L822,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M828,44 L847,25 L875,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M775,75 L803,75 L822,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M828,56 L847,75 L875,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M800,43 L850,43 L850,57 L800,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"825\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"825\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M800,43 L850,43 L850,47 L800,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M843,50 L853,50 L853,60 L843,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"848\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M875,75 L925,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M880,90 L889,90 L903,60 L894,60 L880,90 L889,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"897\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=2.251856</text>\n",
"<path d=\"M75,175 L675,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M675,125 L703,125 L722,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M728,144 L747,125 L775,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M675,175 L703,175 L722,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M728,156 L747,175 L775,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M700,143 L750,143 L750,157 L700,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"725\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"725\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M700,143 L750,143 L750,147 L700,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M743,150 L753,150 L753,160 L743,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"748\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M775,125 L825,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M780,140 L789,140 L803,110 L794,110 L780,140 L789,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"797\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=1.194179</text>\n",
"<path d=\"M775,175 L825,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M825,125 L853,125 L872,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M878,144 L897,125 L925,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M825,175 L853,175 L872,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M878,156 L897,175 L925,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M850,143 L900,143 L900,157 L850,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"875\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"875\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M850,143 L900,143 L900,147 L850,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M893,150 L903,150 L903,160 L893,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"898\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M925,175 L975,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M930,190 L939,190 L953,160 L944,160 L930,190 L939,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"947\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=5.333985</text>\n",
"<path d=\"M925,75 L953,75 L972,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M978,94 L997,75 L1025,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M925,125 L953,125 L972,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M978,106 L997,125 L1025,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M950,93 L1000,93 L1000,107 L950,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"975\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"975\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M950,93 L1000,93 L1000,97 L950,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M993,100 L1003,100 L1003,110 L993,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"998\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1025,75 L1075,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1030,90 L1039,90 L1053,60 L1044,60 L1030,90 L1039,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1047\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=5.036737</text>\n",
"<path d=\"M1025,125 L1075,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1075,75 L1103,75 L1122,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1128,94 L1147,75 L1175,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1075,125 L1103,125 L1122,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1128,106 L1147,125 L1175,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1100,93 L1150,93 L1150,107 L1100,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1125\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1125\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1100,93 L1150,93 L1150,97 L1100,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1143,100 L1153,100 L1153,110 L1143,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1148\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1175,125 L1225,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1180,140 L1189,140 L1203,110 L1194,110 L1180,140 L1189,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1197\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=2.642193</text>\n",
"<path d=\"M875,25 L1175,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1175,25 L1203,25 L1222,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1228,44 L1247,25 L1275,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1175,75 L1203,75 L1222,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1228,56 L1247,75 L1275,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1200,43 L1250,43 L1250,57 L1200,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1225\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1225\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1200,43 L1250,43 L1250,47 L1200,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1243,50 L1253,50 L1253,60 L1243,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1248\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1275,25 L1325,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1280,40 L1289,40 L1303,10 L1294,10 L1280,40 L1289,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1297\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=2.172429</text>\n",
"<path d=\"M1275,75 L1325,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1325,25 L1353,25 L1372,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1378,44 L1397,25 L1425,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1325,75 L1353,75 L1372,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1378,56 L1397,75 L1425,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1350,43 L1400,43 L1400,57 L1350,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1375\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1375\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1350,43 L1400,43 L1400,47 L1350,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1393,50 L1403,50 L1403,60 L1393,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1398\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1425,75 L1475,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1430,90 L1439,90 L1453,60 L1444,60 L1430,90 L1439,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1447\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=2.095071</text>\n",
"<path d=\"M75,225 L975,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M975,175 L1003,175 L1022,194\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1028,194 L1047,175 L1075,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M975,225 L1003,225 L1022,206\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1028,206 L1047,225 L1075,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1000,193 L1050,193 L1050,207 L1000,207 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1025\" y=\"235\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1025\" y=\"176\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1000,193 L1050,193 L1050,197 L1000,197 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1043,200 L1053,200 L1053,210 L1043,210 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1048\" y=\"207\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1075,175 L1125,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1080,190 L1089,190 L1103,160 L1094,160 L1080,190 L1089,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1097\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=5.661671</text>\n",
"<path d=\"M1075,225 L1125,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1125,175 L1153,175 L1172,194\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1178,194 L1197,175 L1225,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1125,225 L1153,225 L1172,206\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1178,206 L1197,225 L1225,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1150,193 L1200,193 L1200,207 L1150,207 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1175\" y=\"235\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1175\" y=\"176\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1150,193 L1200,193 L1200,197 L1150,197 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1193,200 L1203,200 L1203,210 L1193,210 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1198\" y=\"207\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1225,225 L1275,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1230,240 L1239,240 L1253,210 L1244,210 L1230,240 L1239,240 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1247\" y=\"238\" font-size=\"7\" text-anchor=\"start\">Φ=0.738002</text>\n",
"<path d=\"M1225,125 L1253,125 L1272,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1278,144 L1297,125 L1325,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1225,175 L1253,175 L1272,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1278,156 L1297,175 L1325,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1250,143 L1300,143 L1300,157 L1250,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1275\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1275\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1250,143 L1300,143 L1300,147 L1250,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1293,150 L1303,150 L1303,160 L1293,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1298\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1325,125 L1375,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1330,140 L1339,140 L1353,110 L1344,110 L1330,140 L1339,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1347\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=5.273074</text>\n",
"<path d=\"M1325,175 L1375,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1375,125 L1403,125 L1422,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1428,144 L1447,125 L1475,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1375,175 L1403,175 L1422,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1428,156 L1447,175 L1475,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1400,143 L1450,143 L1450,157 L1400,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1425\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1425\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1400,143 L1450,143 L1450,147 L1400,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1443,150 L1453,150 L1453,160 L1443,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1448\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1475,175 L1525,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1480,190 L1489,190 L1503,160 L1494,160 L1480,190 L1489,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1497\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=3.543207</text>\n",
"<path d=\"M1475,75 L1503,75 L1522,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1528,94 L1547,75 L1575,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1475,125 L1503,125 L1522,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1528,106 L1547,125 L1575,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1500,93 L1550,93 L1550,107 L1500,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1525\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1525\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1500,93 L1550,93 L1550,97 L1500,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1543,100 L1553,100 L1553,110 L1543,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1548\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1575,75 L1625,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1580,90 L1589,90 L1603,60 L1594,60 L1580,90 L1589,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1597\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=4.449777</text>\n",
"<path d=\"M1575,125 L1625,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1625,75 L1653,75 L1672,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1678,94 L1697,75 L1725,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1625,125 L1653,125 L1672,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1678,106 L1697,125 L1725,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1650,93 L1700,93 L1700,107 L1650,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1675\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1675\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1650,93 L1700,93 L1700,97 L1650,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1693,100 L1703,100 L1703,110 L1693,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1698\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1725,125 L1775,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1730,140 L1739,140 L1753,110 L1744,110 L1730,140 L1739,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1747\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=3.302943</text>\n",
"<path d=\"M1425,25 L1725,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1725,25 L1753,25 L1772,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1778,44 L1797,25 L1825,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1725,75 L1753,75 L1772,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1778,56 L1797,75 L1825,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1750,43 L1800,43 L1800,57 L1750,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1775\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1775\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1750,43 L1800,43 L1800,47 L1750,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1793,50 L1803,50 L1803,60 L1793,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1798\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1825,25 L1875,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1830,40 L1839,40 L1853,10 L1844,10 L1830,40 L1839,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1847\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=4.02537</text>\n",
"<path d=\"M1825,75 L1875,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1875,25 L1903,25 L1922,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1928,44 L1947,25 L1975,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1875,75 L1903,75 L1922,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1928,56 L1947,75 L1975,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1900,43 L1950,43 L1950,57 L1900,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1925\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1925\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1900,43 L1950,43 L1950,47 L1900,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1943,50 L1953,50 L1953,60 L1943,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1948\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1975,75 L2025,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1980,90 L1989,90 L2003,60 L1994,60 L1980,90 L1989,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1997\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=0.249399</text>\n",
"<path d=\"M75,275 L1275,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1275,225 L1303,225 L1322,244\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1328,244 L1347,225 L1375,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1275,275 L1303,275 L1322,256\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1328,256 L1347,275 L1375,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1300,243 L1350,243 L1350,257 L1300,257 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1325\" y=\"285\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1325\" y=\"226\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1300,243 L1350,243 L1350,247 L1300,247 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1343,250 L1353,250 L1353,260 L1343,260 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1348\" y=\"257\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1375,225 L1425,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1380,240 L1389,240 L1403,210 L1394,210 L1380,240 L1389,240 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1397\" y=\"238\" font-size=\"7\" text-anchor=\"start\">Φ=6.238378</text>\n",
"<path d=\"M1375,275 L1425,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1425,225 L1453,225 L1472,244\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1478,244 L1497,225 L1525,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1425,275 L1453,275 L1472,256\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1478,256 L1497,275 L1525,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1450,243 L1500,243 L1500,257 L1450,257 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1475\" y=\"285\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1475\" y=\"226\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1450,243 L1500,243 L1500,247 L1450,247 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1493,250 L1503,250 L1503,260 L1493,260 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1498\" y=\"257\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1525,275 L1575,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1530,290 L1539,290 L1553,260 L1544,260 L1530,290 L1539,290 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1547\" y=\"288\" font-size=\"7\" text-anchor=\"start\">Φ=2.976814</text>\n",
"<path d=\"M1525,175 L1553,175 L1572,194\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1578,194 L1597,175 L1625,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1525,225 L1553,225 L1572,206\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1578,206 L1597,225 L1625,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1550,193 L1600,193 L1600,207 L1550,207 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1575\" y=\"235\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1575\" y=\"176\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1550,193 L1600,193 L1600,197 L1550,197 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1593,200 L1603,200 L1603,210 L1593,210 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1598\" y=\"207\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1625,175 L1675,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1630,190 L1639,190 L1653,160 L1644,160 L1630,190 L1639,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1647\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=5.71687</text>\n",
"<path d=\"M1625,225 L1675,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1675,175 L1703,175 L1722,194\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1728,194 L1747,175 L1775,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1675,225 L1703,225 L1722,206\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1728,206 L1747,225 L1775,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1700,193 L1750,193 L1750,207 L1700,207 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1725\" y=\"235\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1725\" y=\"176\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1700,193 L1750,193 L1750,197 L1700,197 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1743,200 L1753,200 L1753,210 L1743,210 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1748\" y=\"207\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1775,225 L1825,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1780,240 L1789,240 L1803,210 L1794,210 L1780,240 L1789,240 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1797\" y=\"238\" font-size=\"7\" text-anchor=\"start\">Φ=3.793155</text>\n",
"<path d=\"M1775,125 L1803,125 L1822,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1828,144 L1847,125 L1875,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1775,175 L1803,175 L1822,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1828,156 L1847,175 L1875,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1800,143 L1850,143 L1850,157 L1800,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1825\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1825\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1800,143 L1850,143 L1850,147 L1800,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1843,150 L1853,150 L1853,160 L1843,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1848\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M1875,125 L1925,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1880,140 L1889,140 L1903,110 L1894,110 L1880,140 L1889,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1897\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=0.998835</text>\n",
"<path d=\"M1875,175 L1925,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1925,125 L1953,125 L1972,144\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1978,144 L1997,125 L2025,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1925,175 L1953,175 L1972,156\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1978,156 L1997,175 L2025,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1950,143 L2000,143 L2000,157 L1950,157 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1975\" y=\"185\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"1975\" y=\"126\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M1950,143 L2000,143 L2000,147 L1950,147 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M1993,150 L2003,150 L2003,160 L1993,160 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"1998\" y=\"157\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M2025,175 L2075,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2030,190 L2039,190 L2053,160 L2044,160 L2030,190 L2039,190 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2047\" y=\"188\" font-size=\"7\" text-anchor=\"start\">Φ=0.329839</text>\n",
"<path d=\"M2025,75 L2053,75 L2072,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2078,94 L2097,75 L2125,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2025,125 L2053,125 L2072,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2078,106 L2097,125 L2125,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2050,93 L2100,93 L2100,107 L2050,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2075\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"2075\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M2050,93 L2100,93 L2100,97 L2050,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M2093,100 L2103,100 L2103,110 L2093,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2098\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M2125,75 L2175,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2130,90 L2139,90 L2153,60 L2144,60 L2130,90 L2139,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2147\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=0.382743</text>\n",
"<path d=\"M2125,125 L2175,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2175,75 L2203,75 L2222,94\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2228,94 L2247,75 L2275,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2175,125 L2203,125 L2222,106\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2228,106 L2247,125 L2275,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2200,93 L2250,93 L2250,107 L2200,107 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2225\" y=\"135\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"2225\" y=\"76\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M2200,93 L2250,93 L2250,97 L2200,97 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M2243,100 L2253,100 L2253,110 L2243,110 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2248\" y=\"107\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M2275,125 L2325,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2280,140 L2289,140 L2303,110 L2294,110 L2280,140 L2289,140 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2297\" y=\"138\" font-size=\"7\" text-anchor=\"start\">Φ=1.389795</text>\n",
"<path d=\"M1975,25 L2275,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2275,25 L2303,25 L2322,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2328,44 L2347,25 L2375,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2275,75 L2303,75 L2322,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2328,56 L2347,75 L2375,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2300,43 L2350,43 L2350,57 L2300,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2325\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"2325\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M2300,43 L2350,43 L2350,47 L2300,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M2343,50 L2353,50 L2353,60 L2343,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2348\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M2375,25 L2425,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2380,40 L2389,40 L2403,10 L2394,10 L2380,40 L2389,40 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2397\" y=\"38\" font-size=\"7\" text-anchor=\"start\">Φ=4.488344</text>\n",
"<path d=\"M2375,75 L2425,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2425,25 L2453,25 L2472,44\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2478,44 L2497,25 L2525,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2425,75 L2453,75 L2472,56\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2478,56 L2497,75 L2525,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2450,43 L2500,43 L2500,57 L2450,57 Z\" stroke=\"black\" fill=\"black\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2475\" y=\"85\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<text x=\"2475\" y=\"26\" font-size=\"7\" text-anchor=\"middle\"></text>\n",
"<path d=\"M2450,43 L2500,43 L2500,47 L2450,47 Z\" stroke=\"black\" fill=\"lightgray\" stroke-linejoin=\"miter\" />\n",
"<path d=\"M2493,50 L2503,50 L2503,60 L2493,60 Z\" stroke=\"black\" fill=\"thistle\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2498\" y=\"57\" font-size=\"6\" text-anchor=\"middle\">Rx</text>\n",
"<path d=\"M2525,75 L2575,75\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2530,90 L2539,90 L2553,60 L2544,60 L2530,90 L2539,90 Z\" stroke=\"black\" fill=\"gray\" stroke-linejoin=\"miter\" />\n",
"<text x=\"2547\" y=\"88\" font-size=\"7\" text-anchor=\"start\">Φ=3.753082</text>\n",
"<path d=\"M2525,25 L2575,25\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2325,125 L2575,125\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2075,175 L2575,175\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1825,225 L2575,225\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M1575,275 L2575,275\" stroke=\"darkred\" stroke-width=\"3\" fill=\"none\" />\n",
"<path d=\"M2575,25 L2590,25\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M2575,75 L2590,75\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M2575,125 L2590,125\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M2575,175 L2590,175\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M2575,225 L2590,225\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<path d=\"M2575,275 L2590,275\" stroke-width=\"3\" stroke=\"darkred\" stroke-linejoin=\"miter\" fill=\"none\" />\n",
"<text x=\"2600\" y=\"28\" font-size=\"7\" text-anchor=\"end\">0</text>\n",
"<text x=\"2600\" y=\"78\" font-size=\"7\" text-anchor=\"end\">1</text>\n",
"<text x=\"2600\" y=\"128\" font-size=\"7\" text-anchor=\"end\">2</text>\n",
"<text x=\"2600\" y=\"178\" font-size=\"7\" text-anchor=\"end\">3</text>\n",
"<text x=\"2600\" y=\"228\" font-size=\"7\" text-anchor=\"end\">4</text>\n",
"<text x=\"2600\" y=\"278\" font-size=\"7\" text-anchor=\"end\">5</text>\n",
"<text x=\"0\" y=\"28\" font-size=\"7\" text-anchor=\"start\">0</text>\n",
"<text x=\"0\" y=\"78\" font-size=\"7\" text-anchor=\"start\">1</text>\n",
"<text x=\"0\" y=\"128\" font-size=\"7\" text-anchor=\"start\">2</text>\n",
"<text x=\"0\" y=\"178\" font-size=\"7\" text-anchor=\"start\">3</text>\n",
"<text x=\"0\" y=\"228\" font-size=\"7\" text-anchor=\"start\">4</text>\n",
"<text x=\"0\" y=\"278\" font-size=\"7\" text-anchor=\"start\">5</text>\n",
"</svg>"
],
"text/plain": [
"<drawsvg.drawing.Drawing at 0x2317cfd6320>"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m = 6\n",
"unitary = pcvl.Matrix.random_unitary(m)\n",
"mzi = (pcvl.BS() // (0, pcvl.PS(phi=pcvl.Parameter(\"φ_a\")))\n",
" // pcvl.BS() // (1, pcvl.PS(phi=pcvl.Parameter(\"φ_b\"))))\n",
"linear_circuit = pcvl.Circuit.decomposition(unitary, mzi,\n",
" phase_shifter_fn=pcvl.PS,\n",
" shape=\"triangle\")\n",
"\n",
"print(m, \" modes triangular Boson Sampler :\")\n",
"pcvl.pdisplay(linear_circuit, compact = True)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## MPS simulation\n",
"\n",
"Let us now define the MPS simulation, using the MPS backend in Perceval."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"mps = pcvl.MPSBackend()\n",
"mps.set_circuit(linear_circuit)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We now choose the size of the matrices (the bond dimension) for our simulation."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"chi = 8\n",
"mps.set_cutoff(chi)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"And finally, we get the output probability distribution from a given input state with 3 photons."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table>\n",
"<thead>\n",
"<tr><th>state </th><th style=\"text-align: right;\"> probability</th></tr>\n",
"</thead>\n",
"<tbody>\n",
"<tr><td>|0,0,1,0,2,0></td><td style=\"text-align: right;\"> 0.089692</td></tr>\n",
"<tr><td>|1,1,1,0,0,0></td><td style=\"text-align: right;\"> 0.074157</td></tr>\n",
"<tr><td>|0,1,0,0,0,2></td><td style=\"text-align: right;\"> 0.066353</td></tr>\n",
"<tr><td>|0,0,1,0,1,1></td><td style=\"text-align: right;\"> 0.057677</td></tr>\n",
"<tr><td>|0,1,1,0,0,1></td><td style=\"text-align: right;\"> 0.057048</td></tr>\n",
"<tr><td>|0,0,0,0,1,2></td><td style=\"text-align: right;\"> 0.052347</td></tr>\n",
"<tr><td>|0,1,1,0,1,0></td><td style=\"text-align: right;\"> 0.048586</td></tr>\n",
"<tr><td>|0,1,2,0,0,0></td><td style=\"text-align: right;\"> 0.039534</td></tr>\n",
"<tr><td>|1,1,0,0,0,1></td><td style=\"text-align: right;\"> 0.034403</td></tr>\n",
"<tr><td>|2,0,1,0,0,0></td><td style=\"text-align: right;\"> 0.034118</td></tr>\n",
"<tr><td>|1,0,2,0,0,0></td><td style=\"text-align: right;\"> 0.029711</td></tr>\n",
"<tr><td>|0,0,0,1,2,0></td><td style=\"text-align: right;\"> 0.029622</td></tr>\n",
"<tr><td>|0,0,0,0,3,0></td><td style=\"text-align: right;\"> 0.027699</td></tr>\n",
"<tr><td>|0,0,2,0,1,0></td><td style=\"text-align: right;\"> 0.026586</td></tr>\n",
"<tr><td>|0,0,0,1,1,1></td><td style=\"text-align: right;\"> 0.022117</td></tr>\n",
"<tr><td>|2,0,0,0,0,1></td><td style=\"text-align: right;\"> 0.021964</td></tr>\n",
"<tr><td>|0,1,0,1,0,1></td><td style=\"text-align: right;\"> 0.020708</td></tr>\n",
"<tr><td>|1,1,0,1,0,0></td><td style=\"text-align: right;\"> 0.020591</td></tr>\n",
"<tr><td>|1,0,0,0,0,2></td><td style=\"text-align: right;\"> 0.018927</td></tr>\n",
"<tr><td>|0,0,2,0,0,1></td><td style=\"text-align: right;\"> 0.01776 </td></tr>\n",
"</tbody>\n",
"</table>"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"n = 3\n",
"input_state = pcvl.BasicState([1]*n + [0]*(m-n))\n",
"mps.set_input_state(input_state)\n",
"probs = mps.prob_distribution()\n",
"pcvl.pdisplay(probs, max_v=20)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Certification of the MPS method :\n",
"\n",
"As we make an approximation by chosing the bond dimension, we have to check when does this approximation becomes good enough. Unfortunately, there is no formula giving the minimal size for a given approximation error. What we can do though is to compute the *Total Variance Distance* (TVD) between an ideal simulation of Boson Sampling, and an approximated one. To compute the ideal one, we can for instance use the *SLOS* backend on Perceval :"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table>\n",
"<thead>\n",
"<tr><th>state </th><th style=\"text-align: right;\"> probability</th></tr>\n",
"</thead>\n",
"<tbody>\n",
"<tr><td>|0,0,1,0,2,0></td><td style=\"text-align: right;\"> 0.089692</td></tr>\n",
"<tr><td>|1,1,1,0,0,0></td><td style=\"text-align: right;\"> 0.074157</td></tr>\n",
"<tr><td>|0,1,0,0,0,2></td><td style=\"text-align: right;\"> 0.066353</td></tr>\n",
"<tr><td>|0,0,1,0,1,1></td><td style=\"text-align: right;\"> 0.057677</td></tr>\n",
"<tr><td>|0,1,1,0,0,1></td><td style=\"text-align: right;\"> 0.057048</td></tr>\n",
"<tr><td>|0,0,0,0,1,2></td><td style=\"text-align: right;\"> 0.052347</td></tr>\n",
"<tr><td>|0,1,1,0,1,0></td><td style=\"text-align: right;\"> 0.048586</td></tr>\n",
"<tr><td>|0,1,2,0,0,0></td><td style=\"text-align: right;\"> 0.039534</td></tr>\n",
"<tr><td>|1,1,0,0,0,1></td><td style=\"text-align: right;\"> 0.034403</td></tr>\n",
"<tr><td>|2,0,1,0,0,0></td><td style=\"text-align: right;\"> 0.034118</td></tr>\n",
"<tr><td>|1,0,2,0,0,0></td><td style=\"text-align: right;\"> 0.029711</td></tr>\n",
"<tr><td>|0,0,0,1,2,0></td><td style=\"text-align: right;\"> 0.029622</td></tr>\n",
"<tr><td>|0,0,0,0,3,0></td><td style=\"text-align: right;\"> 0.027699</td></tr>\n",
"<tr><td>|0,0,2,0,1,0></td><td style=\"text-align: right;\"> 0.026586</td></tr>\n",
"<tr><td>|0,0,0,1,1,1></td><td style=\"text-align: right;\"> 0.022117</td></tr>\n",
"<tr><td>|2,0,0,0,0,1></td><td style=\"text-align: right;\"> 0.021964</td></tr>\n",
"<tr><td>|0,1,0,1,0,1></td><td style=\"text-align: right;\"> 0.020708</td></tr>\n",
"<tr><td>|1,1,0,1,0,0></td><td style=\"text-align: right;\"> 0.020591</td></tr>\n",
"<tr><td>|1,0,0,0,0,2></td><td style=\"text-align: right;\"> 0.018927</td></tr>\n",
"<tr><td>|0,0,2,0,0,1></td><td style=\"text-align: right;\"> 0.01776 </td></tr>\n",
"</tbody>\n",
"</table>"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"slos = pcvl.SLOSBackend()\n",
"slos.set_circuit(pcvl.Unitary(unitary))\n",
"slos.set_input_state(input_state)\n",
"probs_slos = slos.prob_distribution()\n",
"pcvl.pdisplay(probs_slos, max_v=20)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We also have to define the TVD function."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def tvd(probs1, probs2):\n",
" tvd = 0\n",
" for state, prob in probs1.items():\n",
" tvd += abs(prob - probs2[state])\n",
" return tvd\n"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we compute the TVD between the two simulations for different bond dimensions, going from 1 to 10."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"TVD = []\n",
"for chi in range(1,11):\n",
" mps.set_cutoff(chi)\n",
" mps.set_input_state(input_state)\n",
" probs_mps = mps.prob_distribution()\n",
" TVD.append(tvd(probs_mps, probs_slos))\n",
"\n",
"plt.plot(TVD)\n",
"plt.xlabel('Bond dimension')\n",
"plt.ylabel('TVD')\n",
"plt.title('Evolution of the TVD between SLOS and MPS probability distributions');"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see that the TVD decreases as the size of the matrices increases, untill reaching 0 for a bond dimension of 7."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"\n",
"> [1] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. [Annals of Physics](https://doi.org/10.1016/j.aop.2010.09.012), 326(1):96–192, jan 2011.\n",
"\n",
"> [2] Changhun Oh, Kyungjoo Noh, Bill Fefferman, and Liang Jiang. Classical simulation of lossy boson sampling using matrix product operators. [Physical Review A](https://doi.org/10.1103/PhysRevA.104.022407), 104(2), aug 2021.\n",
"\n",
"> [3] Hui Wang, et al. Boson Sampling with 20 Input Photons and a 60-Mode Interferometer in a $10^{14}$-Dimensional Hilbert Space. [Physical Review Letters](https://link.aps.org/doi/10.1103/PhysRevLett.123.250503), 123(25):250503, December 2019. Publisher: American Physical Society."
]
}
],
"metadata": {
"language_info": {
"name": "python"
}
},
"nbformat": 4,
"nbformat_minor": 2
}