All right, good morning, cyber traders. Happy Tuesday, everybody. How's everybody doing this morning? Good to see you all. Welcome back. Good morning, Gene. Good to see you all. All right, good, good. Norman, fabulous. Good to see you all, everybody. Hope everyone enjoyed their Columbus Day. I know I did. I had hopefully all my Canadian friends out there. I know a lot of people took off and enjoyed their Thanksgiving. Congratulations. Hope you had a good time. Hope you had a turkey. Art is a turkey shortage, is there really? I don't know, that's what I heard. Hopefully, we'll find out in November. Hopefully, you guys didn't have that problem, too. But I heard you guys had a little bit of good fun. There was a couple of stocks. The market took a little bit of a correction yesterday, which is not a bad thing. It's a funny part. It's not really going anywhere. Like I said, I don't really see anything going anywhere. I'll tell you, I read a big article in this morning in my paper. I wanna bring this up really quick. My local paper, Merck, okay? So everybody here would happen with Merck. The news on Merck, okay? This is new to me. I don't know if you guys know this, but this is new to me. Everybody heard the news on Merck? Everybody heard it? I don't know, YouTube, Facebook Live. I don't know if you guys heard what happened on Merck. Anyone? Nobody heard the news of what happened with Merck? Okay, so apparently Merck has the cure for COVID-19. And they got this pill out there that, and I guess that was that big rally right here that you can see right here in that Mark went from 75 up to 84.  So they're trying to go, yeah, there's a little red pill, right? Yeah, so basically what they're doing is they go with FDA approval and to make sure it gets approved. Now, you know, now it's a catch-22, right? Apparently it's saying the FDA is starting to be very political. So check this out. I just found this out after reading this article. Do you guys know how much it cost for you to get the shot, to get the COVID shot? Does anybody here know what that cost? The COVID shot? Just out of curiosity, just realistically, yeah, it costs us zero, but what does it actually cost that actual shot? Remember, someone's paying for it. It's the government's paying for it. How much does it cost that shot? Dino says 700. Allen says 900. John says $25. Carl says a lot. Wayne says the government's paying a lot. Anybody on YouTube? Anybody know on YouTube? How about this, $2,000. I couldn't believe it. $2,000, everybody, every shot that someone's getting is costing $2,000. That's what they're billing. You know, I know that's what I said. I'm like,Tube says $30,000 to $3,600. $2,000, because they were doing an art, they were saying like the Merck shot, the Merck pill, that pill is $700 a pill. Okay. If you're like, wow, it's a lot of money. Like, oh no, no, it's not. It's a lot cheaper than paying $2,000, you know. So anyway, the catch way too that I was reading in the article is that now it's gonna cause a big issue, which this is why I think Merck's gonna have an issue. If people know that there's a cure out there, then no one's gonna wanna take the shot and they're trying to get everybody vaccinated. So everybody's like, listen, if I'm already had COVID, I have the antibodies and if I get sick, I'll just take the pill. You know, if it's a cure, and they said it's an unbelievable cure of a pill out there of a, you know, that's out there. So we're gonna see what's gonna happen, what's gonna go on. So, because if this gets approved, that means the other one's gonna tank. Now, remember that as a trader, we look at, you know, regardless, you know, with COVID or if it's the financial crisis or if it's a war or whatever it is, us traders look for these things in the market. And that's why I'm finding out more and more people wanna get into the trading business. Listen, I mean, I know a lot of people don't like politics. I know a lot of you don't like to watch the news and it's depressing this and that. But listen, as a trader, you have to look at it and wake up and say, listen, what's actually going on in this world? They said there's a huge shortage of truck drivers.  It's brand new, right? We just bought it last year and it's not working. And I'm like, I don't think it takes a chip for it. You know what I mean? Sometimes it's cheaper to buy a brand new one than try to wait to get that thing. So it's really gonna be a major, you know, hurdle and, you know, oil, listen, look at oil, right? There are already, some people are projecting it's gonna hit $140, $140, $150 a barrel. Some people said it's gonna hit $250 a barrel, okay? Due to the demand, what's out there. So what is gonna happen with a lot of these stocks?  So there's a lot of great opportunities and the reason why I bring this up about the shortage of workers and inflation is that what's happening is that a lot of people stayed home for so long and they thought that, you know, whatever it may be, I don't know what, you know, we could all debate on what the purpose of that is, but, you know, maybe protect COVID, whatever it is, but people realize, maybe I don't wanna go back to that job, you know, maybe I wanna, maybe I wanna do something different, I don't wanna go work, I like staying at home. What can I do? And more and more people are getting into trading and that's what really what I'm noticing, like more and more people are coming. I mean, I just did an event last week, I had 600 people, before that I had 1,000 people. I mean, this is five to 10 times more people are saying, you know what, the hell would go into work and dealing with a boss and that, I'll just sit home and do it.  So only they could tell you is we're in the right business. So I'm just glad you guys are here. By the way, before we go through the watch list and talk about it, I gotta make one, I gotta say one thing. Everybody see this? I gotta thank Josh, I'm gonna tell you a story about this, okay? So anyway, these are my bullhorns and if anybody's from Texas, it's made from Texas. So anyway, so a long time ago, about five years ago, Josh came to my office by accident, he dropped it off my wall, okay? He dropped my thing and it cracked and I felt so bad, I was so upset. I brought it from Texas, I went there, I had a radio show in Texas and I used to go down there and I brought it, I carried and he felt so bad over the years and stuff like that and I was upset about it, but whatever, things happen. So it was always sitting in my office and the horn fell off. So for my birthday, he got me this horn so I really like to thank him for getting that, for getting my new bullhorns, you know? So really look at awesome, but they look awesome. I'm gonna put that right up there on top so you guys could see it right up there above my New York City picture right there. So thanks Josh for that, appreciate it. Great, good job. All right, let's get back to work everybody. Everybody excited? Cause I know I am. All right, so anyway, we got a couple of stocks that did move yesterday before we go through the watch list. I wanna bring some of them up that some of you missed some of them, but we had this any stock, I heard you guys did pretty well on it. It was a, wow, what a beautiful stock.  I mean, it's always a new day here. It's like Groundhog's Day here at Cybertree University, but you guys did pretty well with the stock any I heard. Wow, what a nice moving stock, right at nine o'clock. The thing ran from 670 all the way to about 820. That's amazing, beautiful trend. Look at that, even up until 1030, which we all know, a cyber clock right there, look at that, perfect timing. Went from 680 all the way to 730. That's not bad move right there. So you guys did well on that one. GOTU is another one that moved pretty nicely. And I like this stock. I'm gonna put this back on the watch list because she had a really nice trend. She's testing yesterday's highs. So this is how we find out of a day trade becomes into a swing trade.  Did a little bit of shake right here. Kind of boring after that. And after 11 o'clock, I mean, nice little trend from 370 all the way up to 430. Nice, cheap, inexpensive stock. Listen, for some of you out there that always feel like I need to trade an expensive stock and this and that. You don't, you shouldn't care. Less risk, more reward. And you don't have to worry about thinking like you can't afford to trade the stock. You can afford them, that fine. Another stock, LCID, you guys traded yesterday. Well, this looks like a great short right at the gate, beautiful short right there. Went from 23 all the way down to 22. Not really that much of a downtrend, but it is definitely a nice short. You can see how I'm breaking new higher highs. That was really nice right there. That was beautiful. So you could have got it for that little pop right there. I could have constantly shorted. That was also good. One last one I'm gonna throw in there is the PROG. That one also, nice move out of the gate 930. I mean, this is what we trade for. Dollar 30 all the way to about dollar 65. Listen, you could buy a thousand chairs. You could afford 10,000 chairs. You could put 100 chairs. Nice, slow trending moving stock. Had a nice push around 12 o'clock. Buck 60 all the way up to about buck 90. And then obviously the last hour of the close. For some of you and the different on West Coast or the other part of the world, maybe not early birds, but look, 180 all the way going to the close ran to about 235. Nice, cheap and expensive stock. Nice. All right. So anyway, that's the past. So we're on the future. And let's focus on what's going on today. All right. So a couple of stocks that came on the watch list. The biggest percentage gainer on the NASDAQ right now is NTRB. Now this is a very interesting stock because, you know, we covered this in class. Very, very spotty. Very, very spotty here. But all of a sudden, I mean, the stock is trading on uncharted territories, trading seven million shares. This is not normal for the stock to trade these amount of shares. So anyway, nice pop right at the gate ran from $8 in pre-market, had a couple of shakes right here. You could see it. Beautiful trend. Did a shake, boom. Did a shake, boom. We all know how those shakes work out. After around this time, it's kind of holding right there. This looks like we can have a really, really nice, nice push going into the close. All right. So that one looks pretty good. We have INBX, okay. 270,000 shares of $5, a little spread, a little big. I know this came up on the watch list. It's number eight on the list. But be careful that spread. You know that we always say this, the biggest failure rate in today's market is 50% of the people fail is because the first tee, it's not tradable. You gotta make sure the stock is tradable. And that's more a category four. Doesn't mean you can't make money. Doesn't mean you can't go back up to 40, but spread has a lot to do with it. All right. LCID, this is the stock that took a big hit yesterday. It's making a little bit of a comeback. It traded half a million shares. Spread looks a little dangerous. You know how the stock traded yesterday? I might scratch this one out. PROG, another one is the one from yesterday. It's got some support levels around two. If it breaks here around that 212, looks like it probably can go back and test that 225, 230-ish. But other than that, there's really not that much out there. Shockingly, there's really not that much. The Dow's not doing anything right now. I'm looking at it. It's kind of flat. It's not like we're up 300, 400 points or down 300, 400 points. But other than that, everything looks flat. I mean, I'm not really seeing it as an exciting pre-market. I did see the stock right here. GWH is an IPO. We know IPO is very, very dangerous. This one just came out. I mean, we could throw this one on the list. The RRD, it's got some good movement here. I didn't put it on, but it looks like it just started making a little bit of a move. Good volume, three million shares. Spread looks decent. This one's not bad. This one's not bad. Other than that, I don't really see that much out there. Everything else looks like, you know, kind of flat. But that doesn't stop us. We always find something that will pop out in the morning. You know, we'll pop up going into this afternoon when the market opens up. Some people hold their orders and then all of a sudden you'll see something pop up. So we'll see how things go. But as of right now, not a very big list, okay? All right. Now, anything else that we, I did not call out. Anything that I miss anything? Anybody else see something that just showed up? If anybody out there on YouTube, Facebook, anyone? CTO traders? All right. So anyway, let's get ready for the market open up. Two things that I want you guys keep in mind. Today is Traders Talk Day. So if you're in a cyber group room, it's one of the most popular cyber group workshops that we have with Josh, where we go over and we take some of the traders.  He's gonna pick one of you, he's gonna work with it, talk about it. Traders Talk is a great, great event. If you're on YouTube, Facebook Live, you can always join us by just liking us, friend us, do a trial. You can get in there. It's our very popular workshop. And then we're doing closing bell today at four o'clock. So for anyone that's new out there wants to learn about our style, what we do, how we trade, everything else every week here at four o'clock when the market closes here at CTO. So we'll be here also at that time. All right guys. So we'll see you those two things in the meantime. Be safe, enjoy. Thank you very much, Josh, for my new bull horns. All right. So let's have a nice little bull market and we'll see you back here at Traders Talk. Good luck everyone.


CMG is a complex protein composed of 36 subunits that assembles and moves along the DNA during replication. It was previously thought that CMG moved along the DNA via diffusion or by unidirectional translocation. However, recent research suggests that CMG moves along the DNA by both methods simultaneously. This new finding provides insight into how CMG interacts with the DNA and could help scientists better understand the process of DNA replication. This article was authored by Daniel Ramirez Montero, Humberto Sanchez, Eddo Van Veen and others.


Backroads is made possible by the Minnesota Arts and Cultural Heritage Fund, with money by the vote of the people November 4th, 2008. Walk down this road a million times, as wide as it all, like it's the first time, like it's the last time. Remember the days when I was young, we'd watch the clouds dance with the sun. My name is Anthony Lam. I'm a musician from Fargo, North Dakota. Originally from Minnesota, I grew up in Minnesota. My sound is very much a blend of kind of the music I was around when I grew up. It's got a little bit of like Johnny Cash meets kind of your high school choir, meets my own personal kind of folky singer-songwriter style. And when did you first start getting into music? I guess the first time I really got into music was when I was 13. My dad got me my first guitar, and I just kind of really wanted to be a guitar player. I think my singing ability is probably a bit better than my guitar playing, but I still would love to learn as much more about guitar as I possibly could. I play electric and acoustic, but when I was 13 I got my first guitar, and I kind of never looked back. You know what I mean? I've been writing songs for quite a time, so. How do you approach songwriting? Well, I guess every song is a little different, but a lot of my songs will start out with maybe like a guitar lick or a vocal line that I did have in mind before, or a lot of times it'll start with topics. So I'll say, alright, I want to write a song that kind of, you know, maybe brings awareness to like homeless people or drug abuse, things that aren't really the nicest things to talk about, but music kind of helps get over those bridges a little bit, kind of get some messages across. That's kind of how it starts, and then I just kind of play from there, and I really try to play, I mean it sounds pretty cliche, but it's like you kind of play what you feel at the time, you know what I mean? Some songs you can tell when they're sad, and you can tell when they're kind of inspiring or uplifting, and sometimes they write themselves.  That's pretty rare, but it does happen every once in a while. So many different influences. I try to find like three professional artists to kind of model myself after, but I mean I think a lot of my influences actually come from so many of the artists I play with. Just like getting to meet so many people when you're on the road and how they do things and how they kind of making the leap and being a kid that's playing in houses to playing venues and getting consistent shows. That's kind of what's influenced me the most, but music-wise it's definitely Johnny Cash was something that I grew up on a lot. My mom always listened to it. Frank Sinatra was actually a big one too. That's kind of why I do my vocal melodies the way I do. It's almost like borderline opera. I'm trying to tell stories, you know what I mean? Above all, there are stories behind the words and I guess those are probably two of my biggest influences. Other than that, I mean I take a lot of inspiration from local artists that I meet, you know, especially around Fargo and Bemidji. How would you describe the atmosphere and what it's like to be a musician in this area? People want you to be things sometimes, especially on shows, but I think especially this day and age it's actually great. There's a lot of places compared to just even four or five years ago. There's so many more places that are open to doing music and all different kinds of styles of music. So right now there's a lot of talent out there and it's just kind of fun to be around. It's a big reason why I lived in Bemidji in Fargo because the art is just, you know, there's something about the cities that are just kind of alive, you know, it's ever-changing and ever-evolving. And how would you describe the experience of performing in front of an audience? It's a lot more like I'm home versus when I first started, you know, I used to get pretty nervous sometimes, and I used to get nervous singing in choir, you know, in front of people, but now I'm at the point where it's kind of what I do and it's very fortunate to get to do it as often as I do so. But it's a blessing. It's probably the best way to connect with other people by far. That's why I love doing it. This next song is called I Tried. I saw you on the street last night Nothing to see just kept on walking by Like every other day Let me do this You have done the same I just live in a thinking I can be saved Well, I get into heaven Even if I do you right Well, Jesus, forgive me For telling all those lies Get into heaven Even if I lose my mind Well, Jesus, forgive me I swear to God I tried I can't say I blame you a bit I don't even know my name When I needed you most You turned the other way Let me ask you this Would you have done the same I just live in a thinking I can be saved Well, I get into heaven Even if I do you right Well, Jesus, forgive me For telling all those lies Get into heaven Even if I lose my mind Well, Jesus, forgive me I swear to God I tried And the thing especially around here is People don't really care what genre you are If you're good at what you do People will give you respect And what I try to do is if people are starting out I try to give them as much advice as I can From when I started Because when you're doing this You're on your own You're promoting yourself You're your own boss You're your own manager So you go as far as you want to go And whenever I see other musicians That are starting out I try to give them advice And just so they can, you know Hopefully not hit as many roblox as I did I said it before And I'll say it again


How does it feel when someone is trying to persuade you to do something? It feels like there is a you against them dynamic. That's what persuasion assumes. That the person isn't already happy to do whatever you also want them to do, right? So instead of persuasion, which by the way is what is typically taught in marketing and sales, which is what makes so many people dislike marketing sales, is this feeling that they're wanting us to do something that we don't really want to do and they kind of twist our arm with the right kinds of copywriting and then write incentives and maybe scaring us a little bit into, well, what if you don't do this? This is what's going to happen in your life. And then we finally buy and oftentimes when we buy out of someone having persuaded us rather than our own excitement, that came naturally because it's really what we wanted. But when we are persuaded to do it, we typically get buyers remorse afterwards. We regret because we've had time to sleep on it and really think about our decision and realizing I shouldn't have been persuaded to do that thing or to buy that thing. So instead of persuasion, I recommend alignment. So alignment is basically making sure that several things are aligned. The offering, your product or service is aligned with the person that they actually want it. They actually need it, okay? Which is aligned with the timing of when they want or need it, which is also aligned with the price that they are happy to pay. If you can align offering, person, timing and price, there is no persuasion needed. You simply need to whisper that alignment and you have a sale. You have a new customer, a new client. So how does alignment work? Two things. One is you need to understand the person that you're trying to align with. Are they really right for your offering? And by the way, does that mean your offering needs to change? If you want to serve a particular type of person but your offering isn't aligned with them, then guess what? Either your offering needs to change or you need to talk to different people. That's it, right? So you need to understand what kind of people you're trying to talk to and understand how do you align your offering with what they really want and need. And the other thing is to pay attention as you work with your customers and clients and what they really need and what they really want. So as you work with them, you make your offering more and more aligned to people like them. So one is of course conversations with prospective customers and clients, but two is just paying attention in the work that you already are doing with your customers and clients so that you keep aligning the offer and making the offer more and more effective for people just like them. So instead of persuasion, instead of learning all the persuasion tactics that you know don't feel great to you, instead of that, learn alignment. I hope this is helpful and always open to your questions.


Good afternoon. So I'm very pleased to be with you this afternoon, not only because my flight has been delayed by two hours and it was challenging to make it, but also thanks to the beautiful weather we are having today. So unfortunately I had not the opportunity to see other previous sessions and but I suspect what one of the main theme today was about data and how people are using data to do analysis and to try to do predictions or to create new models that will help to shape the future. So today I will I will talk a bit about smart cities but not only not from let's say high level perspective but more on a pragmatic way. So the purpose of a conversation today is to describe how we make that happen and one of the I was saying one of the main issue we are having today is the amount of data that everybody has to collect and to manage today. So if we look at what is happening today around smart cities so there are lots of things happening around let's say the urbanization that accelerate we are seeing more and more people coming to live in the cities and the city is becoming bigger and bigger and that creates lots of issues in terms of housing of transportation so you have congestion everywhere security as well so you know as the more people they have the more it's a problem there might arise and the main topic today and we are seeing that everywhere is around let's say reducing carbon footprint and sustainability so how do we make the cities more sustainable and able to continue to grow and to host more citizens. So there is lots of let's say technology behind that so I will not go deep into all of those technologies but just in fact smart cities it's not rocket science it's mostly around what we call you know more generic IoT so the Internet of Things so the capability to collect data everywhere in the city to analyze them and make smart decisions and we are not let's say making city smarter just for the sake of technology but at the end of the day the goal is to make the life of citizens easier in the city so to shorten transportation or things like that so I've just put some buzzwords there just to do some let's say I would say education but just to talk about those technologies so 5G we have heard about that it's just a way to accelerate the collection of data and to have more devices connected Lora is another let's say network dedicated to capture data from from devices and sound source and they are all let's say technologies that enable to collect data and to connect objects at the end of the day the IoT is how we connect the physical world with a digital world and around that you have things like in the smart cities we can benefit from these technologies by using the data that is going to be generating by the sunsores the people the cars that everything here are in the city digital twins so this is a term that has been coined for the industry 4.0 as we call it so it's the capability to to create a digital model of something physical and to to use it to try to predict how the physical model is going to react so we see that in factories and things like that but we also begin to see that in cities so for example if you look at Singapore so Singapore has done a partnership with Dassault system to create a digital twin of Singapore so Dassault system has been modellizing the complete city to to have a digital model all hold the city so Singapore authorities can make smarter decision in terms of creating accommodation new building works transportation and so on and then let's say how do we make that happen is thanks to technologies around the big data so we will talk about that AI so why AI is important as well is that given the amount of data we are going to collect so we are talking not on but terabytes but terapetabytes or even exabytes of data that we are collecting so it makes difficult for human beings to manage those data so this is when AI comes into play because if we are to take decision in a real time only let's say a computer is able to analyze the data crunch it and let's say provide useful information back to the devices of the people in a timely manner and in terms of technology so you have two kinds of technology so what we call any directional so just collecting the data and analyzing them and then but be directional when you send back data to the devices in the smart city and this is especially important for autonomous vehicles or let's say connected vehicles because the more autonomous they are the more let's say quick the decision is and has to be communicated to the car so when you have let's say non-critical devices having a response time in one second that's okay but when you have a self-driving car I think that people would prefer to have let's say a couple of milliseconds instead of long latency so all these technologies are just there to make cities smarter and to improve the quality of life of citizens so there are different challenges behind that so in order to enable this technology so we have some technical challenges so multiple components to integrate how we collect and store and manage this huge amount of data challenges around organization so because building these models and this technology requires lots of money and as you know the money from the city is coming from the taxes and not all the people are willing to pay more taxes just to have more technology so there has to be compelling use cases where people are satisfied by the service or there's a partnership between public and private in order to fund those projects and this is one of the main issue we are facing today in terms of part cities is so what is let's say the benefit for the citizen and who is going to fund the project and make it work in terms of integration you know we are talking about sensors and one other thing that it's basic but if you have a sensor for example with temperature that is giving you the temperature every minute the information itself is not that useful and so in order to build a decent model you have to know where this sensor is located in the city so you you do not only have to collect the data but also to add metadata to make the thing more interesting and to create models and have let's say algorithm that are going to provide good results to the smart city and the last ones are really around security and let's say ethic because smart city are let's say collecting so much information and not only about devices but also about people so cities today in cities you we have a capability to know where you are not only just with your phone and smart beacons because you can locate people with our phones and we know where they are but also if you have digital cameras everywhere in a city with technologies like facial recognition we are able to identify people where they are all there let's say over where they are traveling within the city even when who they are meeting in the city so that creates issues around security so all this data is available we have to ensure that it is used in a good way for the benefit on the people and not to threat their privacy and we also have to protect this data to ensure that only the authorized people are able to access it but also to protect the devices because the more connected other devices the more let's say temptation it is for hackers to try to go in and to and to use them maybe in a bad in a bad way so if for example you have a city with where the traffic lights are all connected and you try to optimize the traffic by using the data of the cars and everything if someone acts into their system and modify how the traffic lights behave that might become quite dangerous for everybody so how do we collect all this data and manage all this data so today you have let's say different parties of the networks the people who are actually connecting the devices to have infrastructure so all this data has to flow somewhere where you will you will collect these petabytes of data so you have to store them somewhere so you have lots of interesting features in the cloud providers like Azure IWS and so on but you might not be willing to put all your private data into those clouds so one of the things that you can think of is have a dedicated infrastructure that you control where you can store all your data manage it and own it and one of the benefit of interconnection platform that that we are managing today is that we give a capability to create very innovative models so for example you can have all your data stored in a dedicated environment but still you can use features in the cloud providers because they are connected through private interconnection with very high bandwidth and very low latency so you can have the benefit and the scalability of the cloud providers while you keep the control on all your data and it gives also flexibility so you can really create a kind of hybrid environment where all the you can let's say decide to do something in a house or use cloud providers or use partners or anybody else who is it's a close to where you are operating so you can recreate hybridization not only between you and the cloud but also maybe between different providers if you want to use different features and so on and what I was just explaining today is just something we are doing only for Dublin not in other cities but we do operate globally and we have recently done let's say interesting projects so not only in city but in the transportation area so for example in Germany we are working with Siemens and we are trying to do predictive maintenance on the trains in order for the trains to run let's say more efficiently and to have less outages so what we are doing with Siemens is that we are collecting all the data so every time a train stops at the railway station all the data is uploaded so we can analyze it use a kind of digital twin a model when you can modelize the behavior on the train and then potentially do predictive maintenance so the train can be let's say running more often in a more reliable manner to better serve the citizens so thank you very much for your attention today I don't know if there is any question coming it was an abrupt ending apologies thank you very much that was a fantastic talk yeah fascinating to hear about smart cities that's a term that's really only come into awareness in the past few years and how we're looking at cities and trying to see how we can incorporate technology as you as you share with the devices all the all our smartphones all the data that we have how we can use this I'm just wondering a quick question now what would you consider to be the smartest city is there someone that is there a location that's leading the way in this in implementing that in their in their city sorry the as in is there a leader in this is there a city you would consider to be leading the way no there is no real leader today so there are lots of let's say experimental things happening some cities are trialing things by themselves some are doing partnerships with people like for example GCDoco you know all the furniture within the city that's a good place to put sensors or to enable say wi-fi for everybody wi-fi for everybody but now there is no clear leader and even in the from the technology perspective all the system integrators our cloud providers are working on this area but there is no clear leader yet okay maybe Dublin thank you


Thank you to the organizer for offering me to speak about this topic. It's one of my favorite topics, and I don't have the opportunity to teach a lot at Microsoft, so I'm very glad to be here. And I'm not going to assume much about quantum error correction. My goal is to talk to those of you who don't know much about error correction. I'm going to start even with a little bit about classical error correction, and I'm happy if it's interactive. Feel free to jump in if something is not clear. Feel free to ask questions. There will be many examples throughout the lecture. I'm happy to go through only half of my slides today if you have questions. So feel free to jump in anytime. And my main goal for the week is to teach you how to build your own quantum memory based on quantum LDPC codes. And we want this quantum memory to be fault-tolerant. I'm going to explain all these words. And when I say that, I'm not a physicist, so you're not going to build much, but by that I mean on paper. And for the first lecture, I'm going to start with classical codes. Then I'm going to discuss stabilizer codes. And I'm going to show you that we can build good stabilizer codes. And this is something we wanted to do for a long time with quantum LDPC codes. And then I'm going to go through some examples of construction of quantum LDPC codes if we have time. So first, classical codes. How does it work? I'm going to start with a very basic example with the repetition code. We want to send some information through a channel and some bits are flipped when we send this information. So to correct a bit flip, we are going to repeat the information. Here we repeat three times. We encode the information by repeating each bit three times. When we send it through the channel, some bits are flipped. But even if some bits are flipped, we can still see what is a dominant value and we can decode to find the original code word, the original information. Now what's happening if there is two bit flip in the same code word? Then we're going to make a mistake. So we can improve this scheme. Do you know how we improve this scheme? More repetition, okay, great. So we can do more repetition, we can repeat five times. And with that, we can correct two bit flips. But if there is three bit flips, we make a mistake. There is a decoding error. And what we like to do in classical and quantum error correction is to check the performance of a scheme we propose. We do that by a Monte Carlo simulation. So we take a family of codes, the repetition code, where we repeat a bit D times. And when we keep increasing D, we can correct more and more bit flips. And we simulate this scheme. So we simulate those random bit flips a million times. And we report the performance. And the kind of plot you see in most error correction paper, and quantum error correction paper, is this kind of plot. Where you see the family of repetition code here. I increase the code distance, the number of repetition. And when I increase the code distance, the performance improves if the bit flip probability is low enough. So if there is 30% bit flip, when I move from three repetition to five, I decrease the decoding failure rate. So my logical bits become better and better. And we like to plot it in log scale where we see that if we're at a low noise rate, there is an exponential decay of the failure rate when we increase the code size, the code distance. So you have seen the exact same plots with surface codes, or with LDPC codes, if you have looked at some papers with LDPC codes. So it's a way to check the performance. And what we like to report as a number is the code threshold. This code can tolerate 50% noise. We can correct 50% of bit flip, up to 50%, which is a lot, it's great. And we can represent this code with a graph. I'm going to use a lot of graphs. And the graph that represents this code is called the tanner graph, where we have two types of vertices. We have bits that are circles. We put one bit on each circle, and we have checks that are squares. And a check is checking that the parity of the incident bits is zero. So if I sum the value of these two bits, I want to find a bit zero. And when we do that, we do get, we enforce a repetition code here. Because if this value is zero, the sum of the two neighboring bits must be zero. So this value must be zero. And it propagates through to force the bit string zero or the bit string one everywhere. Yes, so the bigger code get worse, that's true. So in this case, there is a symmetry that allows us even to use the code in this regime. So you're right, it's a great code. So I believe it's not possible to have a threshold as high as that in the quantum setting because of the no cloning theorem. So if you can have a threshold above 50%, you can just send in the information to two people. Each of them will get 50% of the information and recover a copy. So you will have a clone. So it's impossible, sorry. So I haven't defined it, but you can define it as a value P such that below P, the logical error rate goes down when you increase the code distance. It goes down, so if you are below 50%, when you increase the code length, the code distance, the logical error rate will go down. For any values, strictly below 50%. And here there is a very clean threshold because the code is so simple. It's also something we see in statistical physics in the easing model. It's the same notion of threshold, of phase transition. It's only defined for a family of codes, yeah. Okay, so we have our graph representation of the code. And now what's happening in this graph? If there is, if we receive a code word like that, where three bits were flipped out of seven, we can correct them. We have distance seven here. We have seven bits, three flip. We are going to look at the value of the checks and we see that some checks are telling us that there is an issue. Some of the bits have been flipped. So we are going to use this value called the syndrome to correct errors. And yeah, so the, the, the repetition code is great. But it has one issue is that it encodes very little information. There's only one logical bit. So if you see, if you need to send 10 bits or 50 bits or 1,000 bits each time you want to send one bit of data, it's going to be far too expensive. So the solution is to go beyond this graph. This graph is too simple. We're going to look at more interesting graphs. And we can look at any bipartite graph. And we can get two things simultaneously. We can encode many logical bits, more than one. And we can correct many bit flip in the same time. I'm not going to show it. I'm not going to prove it for linear codes, for this class of codes. But I'm going to give the same proof for stabilizer codes, for quantum codes later. So you, so you have to trust me for this claim. And this is an example of a more interesting graph. And it defines a hamming code. It's a code with seven bits and three checks. So there is four bits of information. It's seven minus three. Each check is killing one bit, is removing one logical bit. And with that, we can correct any single bit flip. So we can correct a bit less than with a seven bit repetition code. But we can encode four bits instead of one. It can be an interesting trade off. And when we have a bit string like that, we get a syndrome zero. If we compute all these checks, we get the value zero. Because this is a proper code world. So this is one of the bit strings we can send through our channel, and which is protected. And another case is, we start with any bit string, and there is a flip. One of the bit is flipped. So bit four is flipped. Then the incident checks are gonna detect that there was something wrong. Here, the only incident check is this check. The value of the check is flipped. We can use this information to correct errors. And the last component we need is an algorithm to tell us what was the bit flip given the syndrome. And I'm going through these three components because we're gonna build the same in the quantum setting. So we need these three components, the code, the syndrome, and the decoder. So the decoder is the algorithm that tells you, given the possible values for your syndrome, for your three checks, what should you flip back? What was flipped? And we can build a table. There is a small number of values here. We have only seven values, and we can check what value corresponds to what flip. We can store that, store those seven values, and this is an efficient decoder. It doesn't work if you have a larger code because you will have an exponential number of values here. But for a small code, you can decode like that. Okay. And the issue is that the codes are not gonna be small. So in general, this algorithm does not work. This table is gonna have an exponential size. Because of that, we need a family of codes with an efficient decoding algorithm. Better than just storing everything in memory in an exponential space. Yes. So that's one thing we are gonna get back to, but in the classical setting, yes. When you send some information through your channel, the receiver can have a very reliable computer. If you think about cell phone communication, your cell phone will communicate with a tower. You are far, some bits will be flipped, but when you receive the information on your cell phone, the machine you have in your cell phone, the computing capacity you have in your cell phone, is perfect, is pretty much perfect. So you have no more noise. And that's a good point that in the quantum setting, these checks are gonna lie to us. They will be noisy. And it's a big issue. It's a major issue. Thank you. So the last issue I want to point with classical code is that they don't have this decoding algorithm. And that's why we move to sparse graph. Because sparse is always easier for an algorithm. And this is what our LDPC codes. Low density parity check. So those checks have low densities. They are sparse. And that means we have a graph with low degree. If we have a graph with low degree, it looks like that. We have a very large code, many bits. By many, I mean thousands for LDPC codes. And we have many checks, but they act only on a small number of bits. So for instance, we have a bit string here, and we have a check that checks that the sum of three of the bits is zero. And because it's sparse, we have an efficient decoder. And the idea for this decoder is to say, if the sum of three bits is one, then you know that one of these three bits, at least, was flipped. So each check gives you a lot of information. You have only three choices for the bit flip here. So by using all the checks and comparing the information they give you, you can easily decode and you can do that locally in a sparse graph. And this is what makes them efficient. So now we want all of that in the quantum setting. Yes? So low in the classical case, it's not very well defined. It depends on your application. In the quantum case, we typically want very low. If you think that you have to measure a check acting on 10 qubits, a 10 qubit measurement is already a lot. It's challenging in hardware. It's going to be noisy. So we like to use surface code with wait for, or we can consider, I will discuss, of quantum LDPC codes with wait seven. I would like below 10 ideally. Okay, any more comments about the classical case? Okay, great. So what do we need? We need to encode many logical qubits. We need to correct many poly errors instead of bit flip. We need an efficient decoder. And we need the fault tolerance. As you were saying, the checks are noisy. That's what we mean by fault tolerance. We need to be able to implement our quantum error correction scheme with a hardware that doesn't work. And this introduces more noise. So we want to make sure that we don't introduce more noise than we correct by running error correction. And right now, all the quantum error correction codes that have been implemented do introduce more noise than they remove. We are not in this regime yet. But it's going to come soon, hopefully. And that's why we want quantum LDPC codes. They seem to satisfy all these features in theory. And it's a class of quantum stabilizer codes. I'm going to first introduce those codes, which are the quantum version of linear codes. So I'm going to go very slowly. So let me know if I'm too slow. Or if, but my goal is mainly to make sure that the notations are clear because I'm going to use these notations for the whole week. So by parli operators, I mean those strings of identity X, Y, Z with a phase. So I'm going to denote it like that. Or I will remove the tensor product sign, but it's still there. Or I will denote it like that with X acting on qubit one, X acting on qubit three with a sparse notation. And I will use PN for the N qubit parli group. Now, one important thing for error correction is the commutation relation between those parli operators. And there is two possibilities. If you take two parli operators, either they commute or they anti-commute. So when you write their commutator, you get plus or minus one, the identity. For instance, do you think that these two parli operators commute? Can anyone see immediately if they commute? No. So we can see it by comparing each qubit. X anti-commute with Z. The others all commute pairwise. So it anti-commute once, they anti-commute. These two, we can do the same thing. I compare X with Z, there is one anti-commutation. Y with X, two. Z with Y, three times anti-commute, they anti-commute. And those two, okay. They commute. How do you see it right away? Okay, so it's made with I's and X. And we are gonna use that. We are gonna build some code that use the fact that those guys always commute. And those two, they anti-commute four times, so they commute. And I'm gonna use a notation that is maybe non-standard for the commutator.  It's gonna be just a single bit. So I keep basically the phase plus or minus one of the commutator, and I store a bit. The reason I do that is because this looks like a symplectic inner product. And if you write these as symplectic vectors, it's exactly this symplectic inner product. So it's convenient in some cases. Okay, so now what is the stabilizer code? It's a code defined as a fixed subspace of a family of polyoperators, a set of polyoperators. And it's a subset of the n-cubit space, the n-cubit space, which is C2 tensor n times. So we have this subset. And what can we say about this set S? If we have this definition, this set S has a strong structure. It's not any set. It's a group. It's a group, it's commutative, and it does not contain minus one. And conversely, if you take a group with these three properties, a subgroup of PN, that is commutative and that does not contain minus one, then the set of fixed vectors is a subspace of the n-cubit space, and it's a stabilizer code. So it's just a subspace of the Hilbert space fixed by polyoperators. So you have a convenient way to describe your subspace. And I'm going to use the term stabilizer group for this group with the three properties, commutation, a subgroup commutation and minus one is not in it. The elements of S, the operators that fix the code space are going to be called stabilizers. And the stabilizer generators are the generators of this group. If they are given, we have stabilizer generators. And I'm going to let you do the proof.  Most of them are simple. I'm happy to chat any of them if it's not clear. And some of them that are not trivial will be done during the problem session. Okay, so now we can put all of that in a matrix. So we can put our stabilizer generators in a matrix. Where the rows are the generators. And we can, when we look at this matrix, we can check that the rows commute. The first and the second one commute. We can check that all of them commute. It's a commutative subgroup and it does not contain minus i, so it's a stabilizer group. It defines a code that we call the five qubit code because it encodes some information inside five qubits. There's five physical qubits. Okay, now how much information does it encode? How many logical qubits do we have in the five qubit code? And before I was telling you, each time you add a check, it removes one logical bit. Here it's the same. Each time you have a stabilizer generator, it's one extract on strain. It's going to kill one logical bit. So for this one, what will be your guess? How many logical qubits do we have? One, we have five physical qubits and we kill four of them. We fix four of them. And it's a general claim. If we have a set of independent poly operators, that means that there is no product of these operators that is the identity. Then, if you have r of them, the number of logical qubits is n minus r. So we can compute the dimension of the code and the number of logical qubits. And you are going to prove that later. And n is called the code length, like for classical codes. And we denote the parameters that has nk, where k is the number of logical qubits. And we use double brackets for quantum codes. Yeah, this is a stabilizer group. Yeah, so it's not written here, but s is the previous stabilizer group. I may omit some assumptions sometimes. But thanks for clarifying. Okay, so now what do you think about this code? How many logical qubits do you see? Sorry? The rows are not independent. Okay, that was a trap. So the last row is related to the others. And you can see that if you take the product of all of them, you have two x's and two z's everywhere. They cancel out. So the rows are not independent, but if we remove the last one, we get the five qubit code. So there is one logical qubit. And for this one, one, okay. How do you see it? I'm sorry. Okay, so because you know the answer, that's not a great proof, but that's true. So it's one. So x and z are independent. So we don't need to check the independence between the first three and the others. And between the first three, this is a standard form for a matrix, right? It's a echelon form. So the first row is independent from the second one and the third one because it's the only one with the x here on the first column. The second column is independent from the others for the same reason. And the third one as well. So they are independent, same in z. We have six independent stabilizers. And we call that a CSS code because it has only x checks, x stabilizers, or z stabilizers, stabilizer generators. Okay, now we can introduce a second object we need. We have a code, we need a syndrome. We need to detect some information about errors. So the syndrome, it's a bit string, like in the classical case. So I will denote it's sigma of e. It's a syndrome of a poly error. And sigma of e, it's a bit string where each bit is given by the commutation. If the error commutes with the stabilizer generator, then there is a trivial syndrome bit, trivial check. If it anti-commutes, we put a one. We have a non-trivial syndrome bit. This is our bit string, but it's not enough. This bit string, in the classical case, we can directly compute it. We can compute a parity. In the quantum case, we cannot compute a parity. Those bits, where are they? How do we get them? Does anyone know how we get them in a real quantum computer? We need to measure something. Yes, so we need to measure something. And what do we measure? Okay, so to get sigma i, we are gonna measure si. And it's also something we need to prove. And it's also something you are gonna do yourself. But what you can prove is that if you measure si, the outcome of the measurement is, it's an eigenvalue plus or minus one, and it's plus or minus one. It's minus one to the sigma i. So the syndrome bit can be detected by measuring this operator. And when you measure, it changes your state. So if you start with a quantum state that suffers from a polyurethane, it does nothing to your quantum state. So you can measure, you can get your syndrome bit without changing the state of the system. So it's exactly what we want. And you can prove those two by using the postulates of quantum mechanics. Sigma i of e, it's this sigma i. It's the syndrome bit. So how do you measure a syndrome bit? You measure the corresponding stabilizer generator. And it's a proper quantum measurement. I haven't told you how to implement it, but it exists mathematically. It's the first step. Okay, so now let's look at our favorite example. So what is the syndrome of those three errors? So first the first one. One. So the syndrome, so first the syndrome, it's a vector that lives in which space? How many bits do you have? So you have one for each row. So you have six bits. So I want an answer that contains six bits. Okay, zero, zero, zero, zero, zero, yeah. Why? How do you see that? It commutes with everything. Did you check the commutation with everything? Okay, yes, that's the first stabilizer. So if it's a stabilizer, it commutes with all the other stabilizers. So the syndrome of stabilizers is always zero. And that makes sense. Those stabilizers are doing nothing to our quantum state. So we don't want to detect them. We don't care about them. They are not harmful. Another second one. This one is more harmful. So I look at X on qubit three. It's commuting with this row. It's commuting with this row, with this row. Anti-commute here, here, and that's all. So it has two non-trivial syndrome bits on those two bits, on bit three and four. So it's this one. What about the other one, X1, X2? So it's the same. Now, can you find an error with trivial syndrome? I know there are one. Sorry? All Xs? You could do all Xs. What if I tell you that there is a hint in this slide? Sorry, I cannot hear you. X1, X2, X3. How do you know that? Okay, so we are going to take the product of those two. And when we take the product of those two, the syndrome is a linear map. Because it's a linear map, the syndrome of the product of the two is going to be this syndrome plus this syndrome. They can sell out. So it's going to be zero. And is it a stabilizer? It's an operator with syndrome zero. The first answer you should have given me is this one. We did it already. So our second example is the sum of these two, but it's different. The operator X1, X2, X3 is not a stabilizer. It cannot be written as a product of the first three rows. It's an X operator, so only the first three rows can be used. And those rows, you can check that. Oops, okay. You can check that they overlap only on an even number of positions and they have weight four. So any product will have even weight. You cannot build X1, X2, X3 with even weight. So it's not a stabilizer. It's another type of error. It's something that you cannot detect, but that is not a stabilizer. It's an issue, right? You should detect it, but you cannot.  So now a logical error for a stabilizer code. It's a parli error with trivial syndrome. And there is two types of them. Those that we care about, that we want to correct, but we cannot, and those that we don't care about. So the stabilizers, we are fine with them. They do nothing to our code space, but those that are not stabilizer, I will call them non-trivial logical error.  They change your state, but you cannot detect it. In that case, you just cannot correct. And when you have a stabilizer code, you can build those non-trivial logical errors. You can build all of them. You can build a family that generates them. And they look exactly like standard parli operators. They can be written as X1 bar, Z1 bar, up to Xk bar, Zk bar. So it's like logical parli operators. And they look the same in the sense that they satisfy the same commutation relation. So the X i bar and the Z i bar, anti-commute, but X i and Z j for a different value of j, commute. Just like if you remove the bar. So they behave the same. And when i and j are different, so of course the X i commutes and the Z i commutes. It's just a notation. They don't even have to be X type parli operator for X i bar, but it's a very convenient basis. And I'm gonna let you prove it using Gram-Schmidt. Okay, here is an example. We have our five qubit code and we can take X1 bar equal X, X, X, X and Z1 bar, which is a Z. They have syndrome zero and they anti-commute. And there is only one pair because there is one logical qubit. So it's fair to call them logical parli operators. Okay, and now we know that there is some error that we cannot detect. It's gonna give us a notion of distance. The maximum amount of parli errors, the maximum number of parli errors that we can detect is gonna define the minimum distance of a code. So it's the minimum weight of a non-trivial logical error. The minimum weight of an error that you cannot correct. And for instance, with our previous code, with the Sting code or the Hamming code, the parameters are seven, one, three where we add the distance when we know it because we saw that there is a weight three operator that is a logical operator. It has syndrome zero and we cannot detect it and it's not a stabilizer. So we saw that and it's a code with distance three. And the reason why we care about the distance because the number of parli errors you can correct is half the distance. So I'm gonna prove it quickly. So for that we define a decoder. By a decoder, I mean a mathematical decoder. Not an implementation yet, not an algorithm. It's a map that maps our syndrome, our bit string, onto a parli correction. So given a syndrome, we want to determine the correction to apply. And we say that the decoder works, the decoder corrects a parli error. If when you give it the syndrome of this error, it gives you back an error which is equivalent up to a stabilizer. It doesn't have to be the same but if it's the same up to a stabilizer, they have the same effect on the code because stabilizers are trivial. And we like to build minimum weight decoders. A minimum weight decoder, it's a decoder that returns a minimum weight correction. Why do we like that? It's because if you have a minimum weight decoder you can correct any parli error with weight up to D over two. So we can guarantee that. If we can find an algorithm that gives you a minimum weight correction for each syndrome, we are sure that we can correct any parli error acting on D over two qubits. This is the same thing as in the classical case. And I'm gonna skip the proof but we can discuss it if you want after the lecture. Okay, so how many errors can we correct with this code? How many parli errors? I think I heard the answer. One? How do you know? Okay. Okay, great. So it has this, oh, sorry. It has distance. I think the battery in this thing is dying. Oh, maybe it's not in here. It, okay. So it has distance three. So we can correct three over two. We can correct one error. We can correct any single qubit parli error with this code. So the decoder, it's something that identifies the parli error that occurred. After you find it, you need to apply it in some way to your hardware so you can apply the inverse of this correction. No, it does not, yeah. At least not in my definition. Okay, so we can correct one error. That's a great news. Yes, a minimum distance. So the minimum distance is the minimum weight of a non-trivial logical error. Ah, it's here, right? A minimum weight error. It's an error with syndrome sigma that has minimum weight in the set of error with syndrome sigma. So I look at all the possible errors with syndrome sigma. I know that the error which occurs as syndrome sigma, I'm gonna select the one that has minimum weight. And there may be multiple of them. That's why it's a minimum weight decoder. You may be, you have two solutions with weight one. In some cases, you pick one of them. Thank you. Okay, so we can correct one error. I'm gonna move from correcting one error to correcting many errors with many logical qubits. So we're gonna make a big step and this is what we call a good code. A good code, it's a code that, it's a family of code with parameters N, K, D and we want K and D to be good in the sense to be optimal. They are both linear in the length. That's the best we can do. So we can correct a linear number of logical qubits and we can correct a linear number of poly errors on these logical qubits, on all of them. So we can encode a lot of information and protect it well. It may seem surprising, but this code exists and I'm gonna show you that we can achieve a linear ratio with some constant here for K over N. So K is linear in N and this is the constant and D is linear in N and this is the constant delta. So I'm gonna give you a proof because I think it's a very pretty argument. It's a probabilistic method. We are not gonna construct explicitly these codes. We're gonna pick a random one and see that a random codes does achieve that. So it's gonna be a little bit of combinatorics. And the idea is to consider a random variable. We pick a random stabilizer code. So our random variable will be, when you pick a random code, it should be a Q, sorry. What is the number of poly errors that are not in the stabilizer group but that have syndrome zero and that have weight below our bound, delta N. So I'm counting the number of low-weight poly errors and my goal is to show that this number is zero. There exists a family of codes for which this number is zero. There is no low-weight logical error, non-trivial logical error. That means I have a large distance, right? And to do that, we do what we always do in the probabilistic method. We look at the average. We're gonna show that in average, a code can always give you, in average a code give you a very small number of Y. So we're gonna pick K over N which is small enough. I will tell you how small later. And we're gonna show that the number of code words in the ball with radius delta N goes to zero. The average number. And because the average number, because the average goes to zero, there is one family of codes for which this number is exactly zero. Do you know why? So if I tell you that the average value of something goes to zero, you cannot always conclude that this thing itself is zero. It could be very close to zero, but not exactly zero. It works here because this number we are counting is an integer. So if it were not zero, it will be at least one and then the average will not go to zero. So it's because there is a gap between zero and the smallest positive value, strictly positive value, that we can have this argument. And by definition, that means that the distance of the code is at least delta N. Okay, now how do we do that? We are gonna first start by counting stabilizer codes. So the question is, I take a stabilizer group with N minus K generators. So a stabilizer code with K logical qubits. And I want to know how many stabilizer codes there are. Is there any guess? You should not answer this question. Okay, I'm sure that was your guess. And we are gonna try to explain this number. So first I want to count the number of stabilizer codes. So I'm gonna count the number of choices for our independent generators, the SI. And I'm gonna divide by the number of possible choices of SI that give you the same group so that I don't over count. And this is this fraction. So the top is gonna be number of choices for N minus K generators. How do we choose our SI? So first, we know that when we have a stabilizer group, we can change the phase SI. We can multiply it by minus one. So for each of them, we have a plus or minus one freedom which give us two to the K possible choices. This is two to the N minus K. There is two choices for each of them. Then we are gonna choose the first generator, S1. What do we want for S1? We want S1 to be independent with itself. That means just non-zero. So a non-zero parli operator, the phase is already chosen. So it's two to the two N parli operators minus the zero, the identity. I will always confuse zero and identity. Sorry about that. Now let's choose the second one. We have chosen S1. It's fixed and we need a second generator such that they form a stabilizer group. So this second generator must commute with the first one. And the number of commuting operator is two to the two N minus one. There is half of them that commute, half of them that anti-commute. And we remove, we cannot take zero but we cannot take S1 either. So we remove two operators. Then we take the third one, it should be a three. And we want something that commute with the first two. There is only a quarter of the parli operator. So it's two to the two N minus four. And we remove any product of S1 and S2. There's four such products. And we keep going and taking the product of all these numbers, we get the numerator. So we get our first term and the numerator of this product. And then we do the same thing to compute the number of generating set for this operator. And this gives us the denominator. I will not do it, but it's a similar argument. Okay, now that we have that, we can do the same thing for counting a second set. We fix the parli error, a non-trivial parli error. And we want to count the number of stabilizer groups or stabilizer codes for which this error has a trivial syndrome. How many of the codes are detecting E and how many are not? And we can count that with the exact same argument, but adding the commutation with E. So we get a similar product. And now we are gonna use that right now. So let's get back to our proof. Our proof is let's count the number of non-trivial logical errors. So the number of non-trivial logical errors, we are gonna write it as a sum of random variable it's a standard technique where each XE takes the value zero or one. It takes the value one if the syndrome is zero because we want to count those errors with syndrome zero. So we are writing this number, let's say it's N. I'm gonna write N as one plus one plus one plus one N times. It seems very useless and trivial, but it's powerful when we take the expectation. By linearity of the expectation, now we can take the expectation of this thing which is zero or one. And the expectation of this thing is the probability that the syndrome is zero. And it happens that we just computed this thing. So the probability that the syndrome is zero is the number of codes for which the syndrome is zero divided by the total number of codes. We computed those two things. When I take the ratio of the two previous fractions, I get these products and if I remove the second term, minus two to the I on both side, I get exactly one half and a product N minus K times. So it's two to the minus N minus K. And when you think about it, it's what we expect. We expect that if we have an error and we select N minus K stabilizer generator, each of them has 50% chances to detect it. So the probability that we don't detect it is one over two to the number of stabilizer generators that we add. Each of them has 50% chances to detect it. Okay, now that we have that, we can get back to our expectation and we can plug in this bound on the probability to have a non-trivial syndrome. And this bound gives us some formula that comes from sterling formula from bounds on the binomial coefficients.  So if it's two to the minus N times some constant, if this constant is strictly positive, then this expectation goes to zero and we are done. So we just need to figure out when this constant is strictly positive. And this is happening when we take the ratio K over N to be small enough. So we take K over N to be one minus H of delta minus this log minus epsilon. So as soon as we have this number of logical qubits, we are sure that we're gonna have a linear distance. That proves that we can build good quantum codes, good stabilizer codes. Okay, now what we want is not good stabilizer codes, we want good LDPC codes because again, they are not sparse. They are hard to measure. They are defined by high weight measurements. We want to move to low weight measurements. So I have seven minutes so I'm gonna go through three examples of construction and later I'm gonna discuss the proof of the parameter for one of them. So I'm just gonna give you the construction and let you check that it seems to work. They commute. The stabilizer generators do commute. And you're gonna see them in the problem session in details. So the kit I have to record is defined by taking a solution of the torus. So it's a square grid on the torus. So the green side on top is the same as the bottom side in the bottom. If you identify them on both opposite side, the red side is identified to the red side, you get a torus.  The white circles are qubits. And you're gonna define a X-stabilizer generator for each vertex.  So each black vertex like this one give you a stabilizer generator in X on the incident qubits. And each Z generator is defined from a face. And when you have a face and a star, they do commute. They overlap in two positions. This is kit I have codes. And it's also what led to the surface code which is one of the most popular codes today. We can generalize it to 3D. In 3D you take a cubic grid of a torus. Again, opposite sides are identified. This time there is three identifications. And it works for any cellulation of a three manifold more generally.  The face is something like that. There is horizontal face, vertical faces. There is three types of faces. So you have many qubits. And then you place X generators and Z generators. Like before, you're gonna place them on the dimension below and above. So when you have faces for your qubits, the dimension below is an edge. So when you have an edge, you place an X stabilizer on the four incident faces. And on the dimension above, above a face it's a three dimensional cell, a cube. So for each cube, you place a Z on each of the faces of the cube. So you have a weight six operator. And when you do that, you can see that the cube does commute with this cross operator. They overlap in two faces if you shift the cube. So it does define a proper stabilizer code. It does. It means that because you use higher weight stabilizer generators, it's gonna be easier to detect X errors than Z errors. And we are gonna see it in distance. The distance in X is gonna be bigger than the distance in Z. So there is a trade-off in 3D. And we can do something different. We can place qubits on edges. And we do the same thing. X generator on vertices, Z generator on faces. So all these codes are topological codes. They are defined from many fold. But there is another construction that I will discuss more in two details because it's based on graphs and it gives you better codes, better than all the topological codes. So I'm just gonna give you the example. You take two bipartite graphs. So two tanner graphs with squares, with checks and bits.  And you're gonna place qubits and stabilizers on the Cartesian product. So the Cartesian product has four types of vertices. There is circle, circle. So you place a qubit on each circle, circle. There's nine qubits here. There's also square, square.  And then there is two other types on which we are gonna place X generators. So we're gonna do it, for instance, for each square, circle node here. We are gonna place the X check that is acting on the qubit connected vertically, given by the vertical tanner graph. So there is two incident qubits here. And horizontally, the only horizontal edge comes to this vertex. So we have a weight three stabilizer generator in X. And we can do the same thing in Z. But with the circle square nodes. When we have a circle square and we select one of them, it defines a stabilizer generator corresponding to the same thing, horizontal edges. Each horizontal edge connected to this square defines a qubit, give you a qubit. And vertical edges connected to this qubit, to this bit, give you a qubit. And what you can check is that because you move horizontally and vertically in parallel ways, in the same way on each row and each column, they are gonna commute. Here you see that they overlap on two qubits exactly. They overlap on two qubits exactly. And this is because of this square structure in the graph. So I'm gonna stop here. What I'm gonna show you is the parameters of these codes. This is a theorem that I'm gonna prove next time. Partly prove. And if I give you two classical codes where I give you the number of bits and the number of checks.


Currently, steel making uses fossil fuel like coal. This is responsible for 10% of global emissions. To produce 1 ton of steel, we emit about 2 tons of carbon dioxide. Green steel is the solution. Green steel refers to the change in the steel making process. In this process, we replace the fossils like coal with an alternative fuel like hydrogen from solar and wind to produce steel. This is known as hydrogen-based steel making. In this process, the product steel is the same. What we are changing is the way we produce it. We produce it in a more climate-friendly manner where the by-product is no longer carbon dioxide emissions but the water.


This is the propeller for my zombie chopper the current one. I mean I've made several This one is really good Except I think it's a little bit too high pitch as in the propeller blades are tilted too far this way now just a quick little overview on pitch if your if your propeller blades were aimed directly this way and they were spinning around All they would do is mix the water spinning around right? But if they're turned this way all they would do is like slicing through the water Not really doing much of anything, but they'd be really easy to turn right and Now somewhere between this angle and that angle is like the optimum angle for your Motor your boat and everything and I think this is a little bit too steep. I Wanted to go back a little bit so It's rainy day today. Well the Sun's out now, but everything's soaking out. So I think I'm gonna make a new propeller New total propeller start with the mold This is the mold. It's just a bunch of blocks of wood glued together in a spiral shape Yeah, so this is the mold that I used to make these Propeller blades. You can actually see the outline of where I put the fiberglass on here So it kind of came off there, right? So I want to make another one of these And have each piece of wood each piece of wood thinner So that it ends up being more shallow slope up Oh back here.  I cut on my lumber mill and yes Well, that's definitely thinner What why is Lego over here? somebody's in trouble Put the comb back on this part as a piece of PVC pipe Inside the same thickness as this and that fits on the pipe on the boat So I basically want these to be wrapped around so I could You know put these against this When I make the fiberglass piece will be stuck right on Hopefully that's held steady enough. This is the closest I've got to the right size. Oh, yeah. Oh, that looks great Pretty good Definitely a more shallow angle than that one. I guess I'll do some kind of measurement like from this corner to here And make sure they're the same going on what actually I think I Want it to be a little steeper at the bottom And shallower angle as it goes up because it'll be accelerating the water as it goes, you know Supposedly I'll just change that a little bit So I'll start with maybe And all three centimeters here Add a millimeter on everyone That sounds great No, maybe four centimeters. There's maybe four centimeters down here and I'll add a Senate one millimeter To the angle of everyone Sounds good. Oh, wow time to get some more glue Think I can get enough out of there Okay, now that I'm about to start gluing things. I need to get real serious about these measurements Okay, this one There's only like three point two millimeters And my new pieces of wood are much thinner so I might be going too low pitch. So maybe I want to make this one like Three and a half Yeah, I think like three and a half And I want to go too low pitch because I'll defeat the point of doing I could do math to figure it out, but I think I have a pretty good idea of what Okay I'm gonna leave that for a few hours Maybe something else. Maybe this won't move Okay, someone's gonna ask why do you make this corkscrewy shape like this, you know thing Instead of just having a propeller blade that's like this, right? It's the flat thing Okay, it can't just be a flat thing because the distance the propeller blade travels on the outside is Much further than the distance that travels on the inside, right? So this is just only traveling this far and out here it's traveling that far and When this travels this far and that travels that far they have to travel the same distance Through the water Get it. So it's got to be the same thickness Obviously this can't be all steps so I'm gonna use a What's this called an aluminium flap wheel flap disc, whatever smooth it out But you guys back up this is gonna be loud. I also don't want to get dusty On the piece of 60 grit sandpaper on the ground the other day It's visiting somewhere. Oh, it's still got some use in it. That's great This is wax Nothing will stick to this Small can and a big can of wax. This is car wax. It's not as good This stuff is very good at not letting fiberglass stick It also makes the wood look so nice doesn't it? Lay it on a little thick to fill in some of the creases But it's not perfect Okay, you want to know how this thing works Okay, this is the piece. It's attaching to make sure don't get wax on it Let's do go like this And I'll put a piece of fiberglass in the shape of the propeller right there attached to this and then I'll rotate this 360 divided by three 120 degrees and Then make another one and then rotate again and then make another one Three three propellers. I think I'll make it pretty much like this one Oh, yeah, and then this will stick out the back have any scientific way that I'm Making this propeller or mathematical measurements or anything doing what looks about right and With a water propeller It's nice to have it go like you know kind of like sweep back So if grass or seaweed gets caught on it Ends up falling off the end whereas if you have like more like an airplane propeller where it kind of goes straight out Grass and stuff can get stuck on it, you know, so this is kind of what they call a weedless propeller Yeah, so like on here any grass would just kind of come right off the end But if the propeller was going this way forward it has like more of a flat front edge Grass could get you know just stuck on there and just want to make sure I get All three propeller blades Pretty much in this I'm doing it again Because I put this on then I realized I want to draw the propeller thing on there first so I sanded the Most of the wax off so I could draw on it and now I'm waxing it in Okay What I've got set up here is I've got a block of wood. This is screwed down to it. That's all clamped in place The block of the block of wood has a hole so this goes in the right height and then I've got a little notch filed out of here So I put this on top and line it up with a line right here Do the first blade and then I can rotate it a hundred and twenty degrees until well Doing a hundred and twenty degree angle wasn't as easy doing a sixty degree angle So I'm just gonna check the sixty degree angle on the back side Which you put the front side at a hundred eighty degree or a hundred and twenty degrees And I do that propeller blade and then the same thing One more line over here, that's 60 degrees from that one Okay, last thing I'll lean the crowbar against it just to make sure this is Pressed up against there, so it'll be the same every time All right I'm gonna cut out some pieces of fiberglass that are this shape was enough to go up onto this I Think I'll do three this shape and then three a little bit smaller So I can do two layers on each the pallet blade at the beginning Once I have one of these good I can Trace it for the rest You keep the stinky fumes going out the window Okay, no scrubs no scrubs All right, it's definitely more than I need I just need to let that cure After cures I should draw The line So they can trim it so they can be all exactly the same Hey, do you smell any resin? Do you know what it smells like? Smells like nail polish Do you smell any right now so you don't smell any right now do you? All right, you'd definitely be able to smell it if you smell it I kind of want to take this out into the Sun because Sun cures this stuff Faster But if I do that I'll have to disconnect it and screw up the alignment of everything so It's like it's pretty It's pretty solid Fanster Oh No before I move it Chantel this is showing up And I hope it is good wax job Jay Let's stick my head in the fan Previous one looks like it's not in the way The trick will be see how much of this gets in the way of the mold on the next one I might have to take some of that off whatever though Let's do the current step. Oh my line. Did you up good? This right two for two this one is probably gonna be trickier Trim that I just took the angle grinder to it It's the third one and mighty good so far Sure, it's not perfect, but that is pretty good looking Let's try the weaker one I can rotate this And a whole bunch of pieces cut to put on there And I just need to mix up some garbage and let's do it. I've got my fan going. Okay. Oh Yeah Little sandpaper and that'll look great Should be super strong now Now I have the conceptually easiest part just smooth it out, right? Oh sanding I can do some of it with an angle grinder But there's gonna be some sanding. All right It's not bad All right, I think I want to put some paint on it. Yeah, definitely some Sanding I could still do but That's good enough give it a test Make sure everything's good And I just I just want to test it. I don't feel like sanding anymore. Okay.  There's a little pit Okay, it's tomorrow. I'm somehow not itchy which is amazing and I think I'm gonna tackle these little dents All right, I just decided to sand the whole The whole propeller blade I'll do the other side after And I'm gonna mix a little bit of resin and find something flat Not this but something better than this that I can use to just smear And get the little divots all the scratches get everything out of there. You know what I've got I've actually got the thing for this All right, fingers crossed. That's pretty good Let that fast cure in the sun I guess we'll spray it Oh, it looks pretty good, of course as soon as I spray paint and silver Any blemishes will really stick out. Where let's go Oh, that one looks great. Not bad Yeah, good enough Okay No, I better just wait. All right. I gotta go eat some food or something I'm gonna come back to this in like 15 minutes when it's definitely dry. Otherwise, I'm gonna get fingerprints on it If I measured this hole right, I should be able to put this on the boat Get a lined up with a hole that's on the color shaft Then drill through the other side and it'll be all lined up and then That's screwing. I don't want to drill through the other side Here because it won't be perfectly lined up Well, it looks pretty good Here's the rest of the boat Take it out. Oh, I guess I better get this one out of the way Yeah, this is the boat we're doing. All right, let's go test it out


What is up everybody here, Jason here from JaspysCaseBreaks.com. We just sold out our Topps Chrome Black Baseball Sealed Case Giveaway and as it says details inside here guys in the description. So again, how does this promo work guys? So basically we were selling pre-order spots for Topps Chrome Black before Wednesday. So if you did buy before Wednesday, we have your names jotted down as you got two times the entry. So basically the pre-order entry and then the entry that you get in the breaks you get two times. So if anything bought before 10.30 on Tuesday, you got two times the entry. Everybody else of course that bought after from Wednesday to today Saturday, if you bought a straight up spot in numbers one through ten, you got at least one entry. What we're going to do is when we sold out all the ten cases like we did now, we're going to randomize the whole list with the pre-order entries, the straight up entries and we'll randomize those list of names. It's going to be a long one, but still one person is going to win a sealed case of Topps Chrome Black Baseball. Now again, obviously we did have some fillers and R&Bs that we did for some Picker teams. Just remember if you did buy into those as I was saying them in the R&Bs, you will not get entries. Only the teams that were sold straight up did. So we switch on over here. As you can see we organized where it says pre-order list for Picker team number one, pre-order list for Picker team number two, three, four, five, six, seven, eight, nine, ten. Obviously some of this were smaller than others and then of course starts the full list. So customers that bought in straight up from Picker team number one, two, three, four, five, six, seven, eight, nine and ten right there.  I could put a couple of stars in a plus sign so we can differentiate. Maybe he wins. Maybe he doesn't. I don't know. But good luck to everybody guys. So what we're going to do here is I'm going to copy the whole list. It's going to be a lot of names guys, but everybody still gets a chance. And there we go. It's going to be four hundred and eight entries, four hundred and eight entries. Right there. So here we go. Boom. There's four hundred and eight. We're going to click dice roll. It's going to be a six and a two for eight times, eight times, eight times, one, two, three, four, five, six, seven and eight and final time guys. Boom. Didn't see. Didn't see. Eight times. Now let's scroll on up guys and let's see where we land. Logan, sorry. Ended up at three fifty six for that extra extra mojo, but I know you had multiple spots so it still could be in the top. Of course guys, obviously only give me one winner, so four hundred and five sad people. One happy person. We're in the one seventies, one sixties, one fifties, one forties, one thirties, one twenties, officially in the hundreds, eighties, seventies, sixties, fifties, forties, thirties. And here we go guys, starting to get into crunch time. We are now in the top ten guys. In the top ten. Try not to move the mouse so fast. Oh, Christopher at three and four. I can't see who's at two. Oliver at two and EA, wow, congratulations man. EA at number one and Oliver so close.  But EA, I think EA bought the Reds and almost all of our picker teams or at least it seemed like and I'd say it paid off because I don't know which actual spot that was actually, you know, randomized but there you go. So there you go EA, congratulations man, top scrims back, sealed case giveaway, so we'll ship that to you, sealed man. Most likely it won't go out until maybe Monday or Tuesday, I mean obviously we're closed tomorrow and the post office is closed but whenever we have our next shipping day I'm pretty sure we'll have that shipped out to you. So there you go man, congratulations, like I said, we'll get that out to you as soon as we can and just like that guys, there you go. So I appreciate everybody that got in guys, can't say there won't be any more tops chrome because obviously there is no crazy releases for baseball in the next few weeks until Bowman Baseball on the 8th I believe it may. So we might see some more tops chrome black at some point this week. Obviously we have some other breaks we already posted up yesterday and today so you know NT there at 2017 left in 5 stars so we'll work on those. You know the NFL Draft Mixer we just posted it today I'm sure you'll get more advertising on that tomorrow and Monday and of course the last inter case there of black hockey tops chrome, I mean sorry, black diamond hockey I should say. So yeah, I appreciate it guys, thank you guys so much and like I said we'll have another case giveaway soon probably, thank you.


Hi and welcome to the session. Let us discuss the following question. Question says determine order and degree of following differential equations. Given differential equation is square of d square y upon dx square plus cos dy upon dx is equal to 0. Let us now start with the solution. First of all, let us rewrite the differential equation given in the question. It is square of d square y upon dx square plus cos dy upon dx is equal to 0. Now clearly we can see highest order derivative present in the given differential equation is d square y upon dx square. So its order is 2. So we can write the highest order derivative present in given differential equation is d square y upon dx square. So its order is 2. Now let us discuss about degree of the given differential equation. Degree of a differential equation is defined if it is a polynomial equation in its derivatives. Now clearly we can see this given differential equation is not a polynomial equation in its derivative dy upon dx. So we can write the given differential equation is not a polynomial equation in its derivatives and so its degree is not defined. So we get order of this differential equation is 2 and degree of this differential equation is not defined. So this is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.


Hi, my name's Darren Smith. I'm the officer in charge of the Logan District Road Policing Unit for the Queensland Police Service. Here's some helpful hints and tips that will help you have a safe and enjoyable time riding your motorcycle on the roadway. So it's also important to check all the lights on our motorcycle. You want to be highly visible, you want to make sure that everything's working properly and you want to know that other motorists can see what your intentions are. So make sure that you check your headlight, your park light, your indicators, your rear tail light and your brake light. Make sure they're all working before you hit the roadway. As police, we want you to be out there to have fun on the road. We want you to enjoy your ride, have a safe time out there and make sure that you get home to your families safe and sound.