{"id":2101,"date":"2024-03-14T06:11:50","date_gmt":"2024-03-14T10:11:50","guid":{"rendered":"https:\/\/blogs.mathworks.com\/matlab\/?p=2101"},"modified":"2024-03-14T06:12:58","modified_gmt":"2024-03-14T10:12:58","slug":"pi-day-using-ai-gpus-and-quantum-computing-to-compute-pi","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/matlab\/2024\/03\/14\/pi-day-using-ai-gpus-and-quantum-computing-to-compute-pi\/","title":{"rendered":"Pi day: Using AI, GPUs and Quantum computing to compute pi"},"content":{"rendered":"<div class = rtcContent><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>14th March is <\/span><a href = \"https:\/\/en.wikipedia.org\/wiki\/Pi_Day\"><span>Pi Day<\/span><\/a><span>, celebrated by geeks everywhere and a great excuse for technical computing bloggers to publish something tenuously related to pi and their favorite technology of choice. This is not the first pi day celebration on a MathWorks blog and it will not be the last. Here are some of our past takes on the subject<\/span><\/div><ul  style = 'margin: 10px 0px 20px; padding-left: 0px; font-family: Helvetica, Arial, sans-serif; font-size: 14px; '><li  style = 'margin-left: 56px; line-height: 21px; min-height: 0px; text-align: left; white-space: pre-wrap; '><span>2023 <\/span><a href = \"https:\/\/blogs.mathworks.com\/community\/2023\/03\/14\/celebrate-pi-day-with-a-pi-patch\/\"><span>Adam Danz celebrated pi day with a pi patch<\/span><\/a><span> <\/span><\/li><li  style = 'margin-left: 56px; line-height: 21px; min-height: 0px; text-align: left; white-space: pre-wrap; '><span>2018 Cleve Moler <\/span><a href = \"https:\/\/blogs.mathworks.com\/cleve\/2018\/03\/14\/happy-pi-day\/\"><span>Looked at the first 10,000 digits of pi<\/span><\/a><\/li><li  style = 'margin-left: 56px; line-height: 21px; min-height: 0px; text-align: left; white-space: pre-wrap; '><span>2015 Loren Shure asked <\/span><a href = \"https:\/\/blogs.mathworks.com\/loren\/2015\/03\/16\/the-ultimate-pi-day-is-over-now-what\/\"><span>The Ultimate Pi Day is Over...Now What?<\/span><\/a><\/li><\/ul><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>This year, over at The Simulink Blog, Guy has been using <\/span><a href = \"https:\/\/uk.mathworks.com\/products\/simscape-multibody.html\"><span>Simscape Multibody<\/span><\/a><span> to <\/span><a href = \"https:\/\/blogs.mathworks.com\/simulink\/2024\/03\/14\/computing-%CF%80-simscape-multibody-style\"><span>simulate one way our ancestors could have computed pi<\/span><\/a><span> -- by rolling a big stone cylinder down a hill.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><img class = \"imageNode\" src = \"http:\/\/blogs.mathworks.com\/matlab\/files\/2024\/03\/PiDay2024_1.gif\" width = \"724\" height = \"373\" alt = \"\" style = \"vertical-align: baseline; width: 724px; height: 373px;\"><\/img><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>I, on the other hand, will be at the other end of the time line by making use of Artificial Intelligence, GPUs and Quantum computing. If only I could have fitted in 'Blockchain', I would have won buzzword bingo! <\/span><\/div><h3  style = 'margin: 15px 10px 5px 4px; padding: 0px; line-height: 18px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 17px; font-weight: 700; text-align: left; '><span>Using AI to compute pi<\/span><\/h3><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>One method for computing pi uses a <\/span><a href = \"https:\/\/en.wikipedia.org\/wiki\/Monte_Carlo_method\"><span>Monte Carlo simulation<\/span><\/a><span>. Its so well known that it's actually used as a demonstration of what a Monte Carlo method is in the <\/span><a href = \"https:\/\/en.wikipedia.org\/wiki\/Monte_Carlo_method\"><span>wikipedia entry for Monte Carlo Method<\/span><\/a><span>! For the sake of brevity, I'll leave you to explore the mathematics for yourself. Today, I'm just going to go ahead and use <\/span><a href = \"https:\/\/uk.mathworks.com\/matlabcentral\/playground\/new\"><span>The MATLAB AI Chat Playground<\/span><\/a><span> and ask it how to code such a simulation in MATLAB.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><img class = \"imageNode\" src = \"http:\/\/blogs.mathworks.com\/matlab\/files\/2024\/03\/PiDay2024_2.png\" width = \"682\" height = \"88\" alt = \"\" style = \"vertical-align: baseline; width: 682px; height: 88px;\"><\/img><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>This is what it came up with, with a small modification by me to show more decimal points:<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">%Added by Mike to show more decimal points in the result<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >format <\/span><span style=\"color: #a709f5;\">long<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Number of random points to generate<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >numPoints = 1e6;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Generate random points in the range [0,1]<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >x = rand(numPoints, 1);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >y = rand(numPoints, 1);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Count the number of points inside the unit circle<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >insideCircle = (x.^2 + y.^2) &lt;= 1;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Estimate pi using the ratio of points inside the circle to total points<\/span><\/span><\/div><\/div><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >piEstimate = 4 * sum(insideCircle) \/ numPoints<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsVariableStringElement\" uid=\"07AA89D3\" prevent-scroll=\"true\" data-testid=\"output_0\" tabindex=\"-1\" style=\"width: 1146px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div class=\"textElement eoOutputContent\" tabindex=\"-1\" data-previous-available-width=\"1109\" data-previous-scroll-height=\"20\" data-hashorizontaloverflow=\"false\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"variable output \" style=\"max-height: 261px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div style=\"white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><span class=\"variableNameElement\" style=\"white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\">piEstimate = <\/span>   3.139064000000000<\/div><\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>One of the issues with Monte Carlo simulations is that they require a huge number of samples to get an accurate answer. Indeed, the 1 million samples suggested by the AI only gives us 1 or 2 correct decimal places for pi. For more accuracy, you need a <\/span><span style=' font-weight: bold;'>lot<\/span><span> more samples.  Not just a few million more but millions of millions more.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>On my machine, the above code completes in a fraction of a second: 0.014 seconds to be precise which works out at around 71.4 million samples a second.  Sounds pretty fast but it would take almost 4 hours to compute <\/span><span style=' font-family: monospace;'>10^12<\/span><span> samples and, as written, I'd run out of memory if I even tried it.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Starting from the above code and using various parallelisation and optimisation methods, I got close to 700 million samples a second on my 8 core machine and might even be able to squeeze out some more but I'll save that discussion for another time.  Today, I'll take another route.  <\/span><\/div><h3  style = 'margin: 15px 10px 5px 4px; padding: 0px; line-height: 18px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 17px; font-weight: 700; text-align: left; '><span>Computing pi on the GPU<\/span><\/h3><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Without GPUs, many of the machine learning techniques we use today would not be computationally feasible and technologies like the AI Chat Playground would not exist. I thought it would be fitting, therefore, to show how you'd accelerate the above AI-generated monte carlo code using the very devices that power modern AI. <\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>It often surprises me that many people don't know how extensive the GPU support is in MATLAB. Using Parallel Computing Toolbox, over 1,000 MATLAB functions 'just work' on a GPU when you supply them with a <\/span><a href = \"https:\/\/uk.mathworks.com\/help\/parallel-computing\/gpuarray.html\"><span style=' font-family: monospace;'>gpuArray<\/span><\/a><span style=' font-family: monospace;'>.<\/span><span> Not only that but MATLAB can automatically compile a subset of the language to run on the GPU using <\/span><a href = \"https:\/\/uk.mathworks.com\/help\/parallel-computing\/gpuarray.arrayfun.html\"><span style=' font-family: monospace;'>arrayfun<\/span><\/a><span>. You can even get MATLAB to write <\/span><a href = \"https:\/\/uk.mathworks.com\/products\/gpu-coder.html\"><span>CUDA code using GPU Coder<\/span><\/a><span>. <\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Today, however, let's keep it simple. Here's the code from AI Chat Playground, modified to run on my NVIDIA GPU and with 100 million samples instead of only 1 million. Other than changing the <\/span><span style=' font-family: monospace;'>numPoints<\/span><span> line, I only needed to modify the two calls to <\/span><span style=' font-family: monospace;'>rand<\/span><span> to request that the random numbers, <\/span><span style=' font-family: monospace;'>x<\/span><span> and <\/span><span style=' font-family: monospace;'>y<\/span><span> are generated on the GPU. All subsequent computations automatically work on the GPU. More details on how this works in my post <\/span><a href = \"https:\/\/blogs.mathworks.com\/matlab\/2022\/05\/19\/how-to-make-a-gpu-version-of-this-matlab-program-by-changing-two-lines\/\"><span>How to make a GPU version of this MATLAB program by changing two lines<\/span><\/a><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% This code needs an NVIDIA GPU and Parallel Computing Toolbox<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Number of random points to generate<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >numPoints = 1e8;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Generate random points in the range [0,1]<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >x = rand(numPoints, 1, <\/span><span style=\"color: #a709f5;\">\"gpuArray\"<\/span><span >);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >y = rand(numPoints, 1, <\/span><span style=\"color: #a709f5;\">\"gpuArray\"<\/span><span >);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Count the number of points inside the unit circle<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >insideCircle = (x.^2 + y.^2) &lt;= 1;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Estimate pi using the ratio of points inside the circle to total points<\/span><\/span><\/div><\/div><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >piEstimate = 4 * sum(insideCircle) \/ numPoints<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" uid=\"6C064781\" prevent-scroll=\"true\" data-testid=\"output_1\" tabindex=\"-1\" style=\"width: 1146px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div class=\" textElement eoOutputContent\" tabindex=\"-1\" data-previous-available-width=\"1109\" data-previous-scroll-height=\"32\" data-hashorizontaloverflow=\"false\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"text output \" style=\"max-height: 261px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\">piEstimate =\r\n   3.141578440000000<\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Timing this correctly (A simple <\/span><span style=' font-family: monospace;'>tic\/toc<\/span><span> isn't the way to go with GPU computations, see <\/span><a href = \"https:\/\/uk.mathworks.com\/help\/parallel-computing\/measure-and-improve-gpu-performance.html\"><span>Measure and Improve GPU Performance<\/span><\/a><span> for details) resulted in a speed of 0.06 seconds which gives us 1,666 million samples per second. That's a very nice speed up for very little work.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>I can't push the above code much further though since I quickly run out of memory on the GPU. One way to proceed is to split the simulation into chunks. I'm also going to use <\/span><span style=' font-family: monospace;'>single<\/span><span> precision since its faster on my GPU, uses less memory and the drop in precision isn't an issue here. Let's also put the code into a function<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">function <\/span><span >pi = piMonteCarloSplit(N,numChunks)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >count = 0;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">for <\/span><span >i =1:numChunks<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    x = rand(N\/numChunks, 1, <\/span><span style=\"color: #a709f5;\">\"single\"<\/span><span >, <\/span><span style=\"color: #a709f5;\">\"gpuArray\"<\/span><span >);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    y = rand(N\/numChunks, 1, <\/span><span style=\"color: #a709f5;\">\"single\"<\/span><span >, <\/span><span style=\"color: #a709f5;\">\"gpuArray\"<\/span><span >);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    count = count + sum((x.^2 + y.^2)&lt;1);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">end<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >pi = 4*count\/N;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px 0px 4px 4px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">end<\/span><\/span><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Time this using 100x more samples than before: <\/span><span style=' font-family: monospace;'>1e10<\/span><span> split into 100 chunks, using a timing technique described at <\/span><a href = \"https:\/\/uk.mathworks.com\/help\/parallel-computing\/measure-and-improve-gpu-performance.html\"><span>Measure and Improve GPU Performance<\/span><\/a><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >dev = gpuDevice();<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >wait(dev);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >tic<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >piEstimate = piMonteCarloSplit(1e10,100);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >wait(dev);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >toc<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" uid=\"5656675D\" prevent-scroll=\"true\" data-testid=\"output_2\" tabindex=\"-1\" style=\"width: 1146px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div class=\" textElement eoOutputContent\" tabindex=\"-1\" data-previous-available-width=\"1109\" data-previous-scroll-height=\"17\" data-hashorizontaloverflow=\"false\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"text output \" style=\"max-height: 261px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\">Elapsed time is 0.825407 seconds.<\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>10,000 million samples in 3.123 seconds gives 12,195 million samples per second. Hugely faster than where we started with barely any additional effort at all.  Here's the estimate for pi we get now<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 4px 4px 0px 0px; padding: 6px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >piEstimate<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsTextElement\" uid=\"98AF1AF2\" prevent-scroll=\"true\" data-testid=\"output_3\" tabindex=\"-1\" style=\"width: 1146px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div class=\" textElement eoOutputContent\" tabindex=\"-1\" data-previous-available-width=\"1109\" data-previous-scroll-height=\"32\" data-hashorizontaloverflow=\"false\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"text output \" style=\"max-height: 261px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\">piEstimate =\r\n   3.141574913600000<\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Even with 10,000 million samples, I only got 4 decimal places correct. There's a reason why this technique is not used if high precision for computing pi is what you need.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Using additional techniques, I eventually squeezed almost 17,000 million samples a second out of my GPU and there could be more performance on the table. I may write this up another time if there is sufficient interest. The point here is that it can be really easy to get started with porting a lot of MATLAB code to the GPU and the performance is pretty good! <\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>For reference, my GPU is an NVIDIA GeForce RTX 3070 and I'm also using it as the display driver. <\/span><\/div><h2  style = 'margin: 3px 10px 5px 4px; padding: 0px; line-height: 20px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 20px; font-weight: 700; text-align: left; '><span>Calculating pi using a quantum computer<\/span><\/h2><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Guy's post took us back to ancient times. I wanted to take a glimpse at the future and look at how we could calculate pi with a quantum computer. I had my first play with quantum computers last year in the post <\/span><a href = \"https:\/\/blogs.mathworks.com\/matlab\/2023\/10\/17\/quantum-computing-in-matlab-r2023b-on-the-desktop-and-in-the-cloud\/\"><span>Quantum computing in MATLAB R2023b: On the desktop and in the cloud<\/span><\/a><span> and so googled how to compute pi using one.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>IBM <\/span><a href = \"https:\/\/www.youtube.com\/watch?v=PBXb2LyFg3o\"><span>posted the algorithm over at YouTube with an implementation in Qiskit<\/span><\/a><span>. I reached out to one of our Quantum computing experts at MathWorks, <\/span><a href = \"https:\/\/www.linkedin.com\/in\/matt-bowring-463610149\/\"><span>Matt Bowring<\/span><\/a><span>, and asked him to write a version for MATLAB making use of the <\/span><a href = \"https:\/\/uk.mathworks.com\/matlabcentral\/fileexchange\/125425-matlab-support-package-for-quantum-computing\"><span>MATLAB Support Package for Quantum Computing<\/span><\/a><span> which you'll need to install if you are following along. Here's his function<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">function <\/span><span >[pie,circ] = estimatePi(N)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Return pi estimation pie using Quantum Phase Estimation with N qubits<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Inspired by the pi day code at https:\/\/github.com\/Qiskit\/textbook\/blob\/main\/notebooks\/ch-demos\/piday-code.ipynb<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Estimate phase angle theta for U|1&gt; = exp(2*pi*i*theta)|1&gt;, setting U as<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% the r1Gate with phase angle 1, so that QPE finds 2*pi*theta = 1.<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Increasing N improves the estimate precision since possible theta values<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% are discretized over the 2^N measurable bitstring states.<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >estQubits = 1:N;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >phaseQubit = N+1;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >gates = [<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    <\/span><span style=\"color: #008013;\">% Initialize bottom qubit to |1&gt;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    xGate(phaseQubit)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    <\/span><span style=\"color: #008013;\">% Apply QPE using r1Gate with angle 1 by controlling it on the<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    <\/span><span style=\"color: #008013;\">% estimation qubits<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    hGate(estQubits)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    cr1Gate(estQubits, phaseQubit, 2.^(N-1:-1:0))<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    inv(qftGate(estQubits))<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    ];<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >circ = quantumCircuit(gates);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >sv = simulate(circ);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Use most probable measurement of the estimation qubits to find theta<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >[bitstrings, probs] = querystates(sv, estQubits);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >[~, idx] = max(probs);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >theta = bin2dec(bitstrings(idx)) \/ 2^N;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #008013;\">% Estimate pi knowing that 1 = 2*pi*theta<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >pie = 1 \/ (2*theta);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px 0px 4px 4px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">end<\/span><\/span><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Simulating this on my PC using a circuit with 8 qubits gets the first 2 digits correct<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 4px 4px 0px 0px; padding: 6px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >qPi = estimatePi(8)<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsVariableStringElement\" uid=\"AA63E1FB\" prevent-scroll=\"true\" data-testid=\"output_4\" tabindex=\"-1\" style=\"width: 1146px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div class=\"textElement eoOutputContent\" tabindex=\"-1\" data-previous-available-width=\"1109\" data-previous-scroll-height=\"20\" data-hashorizontaloverflow=\"false\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"variable output \" style=\"max-height: 261px; white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><div style=\"white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\"><span class=\"variableNameElement\" style=\"white-space: pre; font-style: normal; color: rgb(33, 33, 33); font-size: 12px;\">qPi = <\/span>   3.121951219512195<\/div><\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>As we increase the number of qubits, we get a better approximation. <\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >qubits = 2:12;<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >pies = zeros(1,length(qubits));<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">for <\/span><span >n = 1:length(qubits)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >    pies(n) = estimatePi(qubits(n));<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span style=\"color: #0e00ff;\">end<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '>&nbsp;<\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >plot(qubits, pies, <\/span><span style=\"color: #a709f5;\">'r-o'<\/span><span >)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >hold <\/span><span style=\"color: #a709f5;\">on<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >yline(pi)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >legend(<\/span><span style=\"color: #a709f5;\">'estimate'<\/span><span >, <\/span><span style=\"color: #a709f5;\">'true'<\/span><span >)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 0px none rgb(33, 33, 33); border-radius: 0px; padding: 0px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >xlabel(<\/span><span style=\"color: #a709f5;\">'Number of qubits'<\/span><span >)<\/span><\/span><\/div><\/div><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >hold <\/span><span style=\"color: #a709f5;\">off<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsFigure\" uid=\"58FF21E0\" prevent-scroll=\"true\" data-testid=\"output_5\" tabindex=\"-1\" style=\"width: 1146px;\"><div class=\"figureElement eoOutputContent\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"figure output \"><img decoding=\"async\" class=\"figureImage figureContainingNode\" src=\"http:\/\/blogs.mathworks.com\/matlab\/files\/2024\/03\/PiDay2024_3.png\" style=\"width: 560px;\"><\/div><div class=\"outputLayer selectedOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer activeOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer scrollableOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer navigationFocusLayer doNotExport\" aria-hidden=\"true\" tabindex=\"-1\" role=\"application\" aria-label=\"figure output \"><\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>The function can also return the circuit used in each case which allows us to visualize it. Here's the 4 qubit case<\/span><\/div><div style=\"background-color: #F5F5F5; margin: 10px 15px 10px 0; display: inline-block\"><div class=\"inlineWrapper\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 1px solid rgb(217, 217, 217); border-bottom: 0px none rgb(33, 33, 33); border-radius: 4px 4px 0px 0px; padding: 6px 45px 0px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >[qPi,circ] = estimatePi(4);<\/span><\/span><\/div><\/div><div class=\"inlineWrapper outputs\"><div  style = 'border-left: 1px solid rgb(217, 217, 217); border-right: 1px solid rgb(217, 217, 217); border-top: 0px none rgb(33, 33, 33); border-bottom: 1px solid rgb(217, 217, 217); border-radius: 0px; padding: 0px 45px 4px 13px; line-height: 18.004px; min-height: 0px; white-space: nowrap; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; '><span style=\"white-space: pre\"><span >plot(circ)<\/span><\/span><\/div><div  style = 'color: rgb(33, 33, 33); padding: 10px 0px 6px 17px; background: rgb(255, 255, 255) none repeat scroll 0% 0% \/ auto padding-box border-box; font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace; font-size: 14px; overflow-x: hidden; line-height: 17.234px; '><div class=\"inlineElement eoOutputWrapper disableDefaultGestureHandling embeddedOutputsFigure\" uid=\"E17BD8EF\" prevent-scroll=\"true\" data-testid=\"output_6\" tabindex=\"-1\" style=\"width: 1146px;\"><div class=\"figureElement eoOutputContent\" role=\"article\" aria-roledescription=\"Use Browse Mode to explore \" aria-description=\"figure output \"><img decoding=\"async\" class=\"figureImage figureContainingNode\" src=\"http:\/\/blogs.mathworks.com\/matlab\/files\/2024\/03\/PiDay2024_4.png\" style=\"width: 560px;\"><\/div><div class=\"outputLayer selectedOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer activeOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer scrollableOutputDecorationLayer doNotExport\" aria-hidden=\"true\"><\/div><div class=\"outputLayer navigationFocusLayer doNotExport\" aria-hidden=\"true\" tabindex=\"-1\" role=\"application\" aria-label=\"figure output \"><\/div><\/div><\/div><\/div><\/div><div  style = 'margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>Today I only simulated these quantum systems but you can also try running them on real quantum hardware too. How to do this using IBM's quantum services is covered in the MATLAB documentation <\/span><a href = \"https:\/\/uk.mathworks.com\/help\/matlab\/math\/run-quantum-circuit-on-hardware-using-IBM.html\"><span>Run Quantum Circuit on Hardware Using IBM Qiskit Runtime Services<\/span><\/a><span>. My blog post, <\/span><a href = \"https:\/\/blogs.mathworks.com\/matlab\/2023\/10\/17\/quantum-computing-in-matlab-r2023b-on-the-desktop-and-in-the-cloud\/\"><span>Quantum computing in MATLAB R2023b: On the desktop and in the cloud<\/span><\/a><span> discusses how to run on quantum systems via AWS Braket.<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span>So that's it from me and Guy. Whether you calculate your pi's using state of the art computing hardware or a large stone cylinder, happy pi day!<\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span><\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><span><\/span><\/div><div  style = 'margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(33, 33, 33); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left; '><\/div>\r\n<\/div><script type=\"text\/javascript\">var css = '.eoOutputWrapper { width: calc(90vw - 10px) !important; } \/* Styling that is common to warnings and errors is in diagnosticOutput.css *\/.embeddedOutputsErrorElement {    min-height: 18px;    max-height: 550px;} .embeddedOutputsErrorElement .diagnosticMessage-errorType {    overflow: auto;} .embeddedOutputsErrorElement.activeOutput .eoOutputContent {    user-select: text;    -webkit-user-select: text;} .embeddedOutputsErrorElement.activeOutput .eoOutputContent button {    user-select: none;    -webkit-user-select: none;} .embeddedOutputsErrorElement .eoOutputContent ::selection {    color: var(--rtc-output-textSelection-color);    background: var(--rtc-output-textSelection-bg-color);} .embeddedOutputsErrorElement.inlineElement {} .embeddedOutputsErrorElement.rightPaneElement {} \/* Styling that is common to warnings and errors is in diagnosticOutput.css *\/.embeddedOutputsWarningElement {    min-height: 18px;    max-height: 550px;} .embeddedOutputsWarningElement .diagnosticMessage-warningType {    overflow: auto;} .embeddedOutputsWarningElement.activeOutput .eoOutputContent {    user-select: text;    -webkit-user-select: text;} .embeddedOutputsWarningElement .eoOutputContent ::selection {    color: var(--rtc-output-textSelection-color);    background: var(--rtc-output-textSelection-bg-color);} .embeddedOutputsWarningElement.inlineElement {} .embeddedOutputsWarningElement.rightPaneElement {} \/* Copyright 2015-2023 The MathWorks, Inc. *\/\/* In this file, styles are not scoped to rtcContainer since they could be in the Dojo Tooltip *\/.diagnosticMessage-wrapper {    font-family: Menlo, Monaco, Consolas, \"Courier New\", monospace;    font-size: 12px;} .diagnosticMessage-wrapper.diagnosticMessage-warningType {    \/*This fallback value will be used for appdesigner warnings*\/    color: var(--rtc-warning-output-color, var(--mw-color-matlabWarning));} .diagnosticMessage-wrapper.diagnosticMessage-warningType a {    \/*This fallback value will be used for appdesigner warnings*\/    color: var(--rtc-warning-output-color, var(--mw-color-matlabWarning));    text-decoration: underline;} .rtcThemeDefaultOverride .diagnosticMessage-wrapper.diagnosticMessage-warningType,.rtcThemeDefaultOverride .diagnosticMessage-wrapper.diagnosticMessage-warningType a {    color: var(--mw-color-matlabWarning) !important;} .diagnosticMessage-wrapper.diagnosticMessage-errorType {    \/*This fallback value will be used in appdesigner error tooltip text*\/    color: var(--rtc-error-output-color, var(--mw-color-matlabErrors));} .diagnosticMessage-wrapper.diagnosticMessage-errorType a {    \/*This fallback value will be used in appdesigner error tooltip text*\/    color: var(--rtc-error-output-color, var(--mw-color-matlabErrors));    text-decoration: underline;} .rtcThemeDefaultOverride .diagnosticMessage-wrapper.diagnosticMessage-errorType,.rtcThemeDefaultOverride .diagnosticMessage-wrapper.diagnosticMessage-errorType a {    color: var(--mw-color-matlabErrors) !important;} .diagnosticMessage-wrapper .diagnosticMessage-messagePart,.diagnosticMessage-wrapper .diagnosticMessage-causePart {    white-space: pre-wrap;} .diagnosticMessage-wrapper .diagnosticMessage-stackPart {    white-space: pre;} .embeddedOutputsTextElement,.embeddedOutputsVariableStringElement {    white-space: pre;    word-wrap:  initial;    min-height: 18px;    max-height: 550px;} .embeddedOutputsTextElement .textElement,.embeddedOutputsVariableStringElement .textElement {    overflow: auto;} .embeddedOutputsTextElement.activeOutput .eoOutputContent,.embeddedOutputsVariableStringElement.activeOutput .eoOutputContent {    user-select: text;    -webkit-user-select: text;} \/*embeddedOutputsTextElement has a different dom structure than embeddedOutputsVariableStringElement.Unlike variableString, the text output has both TEXT nodes and elements as children. 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