Where Did the Universe Come From? and Other Cosmic Questions is a book I wrote with my academic colleague and friend Geraint Lewis. Geraint is a gifted scientific communicator and, if you aren’t already from Wales, you’ll love his Welsh accent. Geraint is also an astrophysicist, whereas I am a quantum physicist. In the world of academic physics, we couldn’t be further apart. We teach separate subjects and work in different parts of the university using different terminology and mathematics. When we visit our experimental colleagues, Geraint goes to observatories while I visit the labs in the basement. So, why did we come together to write this book?
I refer to the book as my love letter to quantum physics, and here’s why. When you look for popularizations of science, you tend to find the same perspective no matter who is offering it — the “cosmic perspective.” You should have read that in the voice of Niel deGrasse Tyson. Tyson, and his scientific hero Carl Sagan before him, were indeed astronomers. That means they may have shared beautiful astrophotography photos that they themselves had taken. But a vast array of (very gifted) scientific communicators study things completely disconnected from astronomy. Yet, when they get up on stage, they show you photos of stars and galaxies and artistic representations of black holes and other planets.
This makes sense, of course — we love the stars. I mean, you didn’t even need me to tell you that. But here’s a dirty little secret — everything we understand about the cosmos beyond what we can see through a 19th-century telescope is a consequence of quantum physics. Quantum physics has its fingerprints on everything, but it has been done a great disservice, being labeled as too complicated to understand even for the likes of Einstein. And here’s my dirty little secret — this book was my way of sneaking quantum physics into the public sphere through the cosmic perspective.
This book is about cosmic questions and how our best answers to them are given by quantum physics. Geraint is the expert on the questions and the real scientific data that prompts them. For example, where did the universe come from? sounds like a pretty innocent question. But, in fact, where that question comes from is just as interesting. As we discuss in the book, the astronomical data shows that all galaxies are moving away from each other — the universe is expanding. That means that all the galaxies were closer together in the past. And if we turn the clock back far enough (about 14 billion years), all the matter in the galaxies should have been in a single place. The entire cosmos was wrapped up in the tiniest fraction of the size of an atom. If we are going to have any attempt at explaining what happened then, we need the principles of quantum physics. That’s where I come in. In this case, we need to understand quantum uncertainty if we are to answer where the universe came from.
The book is a series of big questions about life, the universe, and everything. (No, 42 is not the answer every time.) With each question, Geraint and I go back and forth to connect the concepts in quantum physics to questions at the cosmic scale. It was insanely fun to write this book, and I hope you enjoy reading it as much as we did making it.
You can pick up a copy at your nearest local bookstore on 7 Sep 2021, or preorder online.
During a lecture for Introduction to Quantum Computing, the topic of the Quantum Zeno effect came up. We’ll get into the details of this, but essentially it says that — unlike a “watched pot” which eventually boils — a “watched quantum pot” never does boil. One of the students asked if this was similar to how Boo works in Super Mario Bros. The answer is almost, but no. However, we can use Boo to learn about quantum physics anyway!
Only 90’s kids
Nineties kids and younger millennial hipsters will have fond memories of Boo, the lovable ghost character introduced in Super Mario Bros. 3, the greatest video game of all time. If you are not familiar with Boo, you can see the game mechanics in this video:
Here’s the gist. If Boo touches Mario, Mario loses his power-up or dies (don’t worry, he comes back to life). Boo chases Mario if Mario is not facing Boo. If Mario is looking at Boo, Boo covers its face and doesn’t move. I made a simple animation summarising this.
How Boo works in Super Mario Bros. This is Classical Boo. Credit: own work.
Characters in video games obey rules. These rules are coded by the developers intentionally — if the game behaves in a way not intended, it is called a bug. If you looked at all the lines of code needed to make a video game, you would be able to see the rules. Sometimes these rules are called physics. In some games the “physics” is actually real physics. For example, in Angry Birds, the rules the birds obey are the actual equations of motions invented by Isaac Newton over 300 years ago to describe gravity. The “physics” of the game is the physics of real life. Mario, on the other hand, doesn’t obey the true laws of gravity when you make him jump. But, there was still a rule the character obeyed that we could call “physics”. In any case, all of the rules coded into nearly every video game ever made are derivatives of Newtonian physics. Newton’s laws are often called classical physics — the reason being that it has been superseded by modern physics, including quantum physics.
What if the rules Boo obeyed were those of quantum physics?
Quantum Mario Bros.
Disclaimer: if Boo obeyed quantum physics, we would not be able to see it do much of anything interesting on a screen because a screen displays classical information. The purpose here is to give you an intuition for some aspect of quantum physics. For a deeper understanding of quantum physics, you have to read this book.
With that in mind, we ask: what if instead of super Mario, we had quantum Mario? Actually, scratch that — let’s keep Mario classical. He will act as the scientist — or, as they are called in the quantum business, the observer. Instead, let’s make Boo behave quantumly.
The first difference between Classical Boo and Quantum Boo is that Quantum Boo doesn’t move along a smooth path from start to finish. Quantum Boo is never found in the space between the starting point and Mario. Quantum Boo can only be found at its starting point or on Mario. The only thing that changes for Quantum Boo is the probability of catching Mario. This probability can be visualised as transparency. The more transparent Boo is, the less likely it is to find it there.
Quantum Boo only has a probability of catching Mario or remaining put. Credit: own work.
Quantum Boo starts to disappear from its starting location and simultaneously reappears at Mario’s location. After some time, Boo’s probability of catching Mario is 100%. Before then, there is some chance that Boo has caught Mario and some chance that Boo hasn’t moved at all. At this midpoint, Quantum Boo is in a superposition of both locations.
Quantum Power-ups
Uh oh! Quantum Boo seems unstoppable. But wait! Mario has the power of observation — the power of quantum measurement. Whenever Mario looks toward Quantum Boo, Boo is only either found on top of Mario or at Boo’s starting position. If timed correctly, Mario can force Quantum Boo back to its starting position with a 0% chance of catching him. Of course, Mario could get unlucky and turn to find Boo on top of him as well — it all comes down to a quantum coin toss!
If Mario looks at Quantum Boo in superposition, he forces Quantum Boo to collapse. Note that this looping animation only shows one of the two possibilities! Credit: own work.
When Mario is not looking, Quantum Boo begins the transition. But this takes time. And, if Mario forces Quantum Boo to the starting position, the clock resets and Quantum Boo has to start the transition all over again. So, if Mario turns around often enough, he can force Quantum Boo to never move at all!
Well, that’s it. That is how a quantum Mario Bros. game would work. But now that your appetite is whetted, I know you want to dive deeper. So what are these principles of quantum physics on display in quantum Mario Bros.?
A ghostly state
In classical physics, the state of Boo is a list of all the important properties it possesses. Boo has a location, direction, speed, and whether or not it is hiding. Boo’s position can be anywhere, which means the possible states are continuous. Continuous means any small change is another allowed state. You can’t count the number of possible states in classical physics — it’s uncountably infinite. In quantum physics, the possible states Quantum Boo can be found in are discrete and finite — you can count them.
This is generally true in quantum physics, and the first identified departure from classical physics. In fact, a Nobel prize was awarded for this “quantum hypothesis” made in 1900. Now, if Quantum Boo can only be found in one of two locations, it must jump from one location to another without visiting the space between. Many of the early physicists studying the new quantum theory famously despised these “quantum jumps”.
Super Position Bros.
Yet, Quantum Boo does somehow move continuously between the two locations in an ephemeral way. The probability of being in one location or the other changes continuously. An instant after Mario turns his head away from Quantum Boo, Boo ceases to be in the starting position. Boo is also not at Mario’s position either at this point. In fact, Boo is in no definite position at all! This new state is called superposition. It’s not here or there, and it’s not both here and there. It is something entirely new.
A lot of words are written about quantum superposition that are just plain wrong. Here they might have said Quantum Boo is in two places at once. But this is wrong — Quantum Boo is at neither location, so it can’t be at both! When words are written that are correct about quantum superposition they are often contorted in unnecessary ways, as if they are skirting around some issue. The reason for this is the apparent need to always couch quantum physics in classical terminology. Well, here is a better analogy for quantum superposition. Imagine you have blue and yellow flowers. You’ve always had blue and yellow flowers and that is all you know. One day, you decide to cross-breed them. You end up with a new color of flower. What do you call it? Do you call it blue and yellow? No. Do you waste an entire blog post about explaining why it is neither blue nor yellow? No. You simply give it a new name, green. That’s quantum superposition. You can’t contort it into classical physics language — it’s just something new.
Phew. I didn’t mean to go off on that tangent. Now, where were we?
Quantum Boo gets a classical scare
Mario cannot “see” the quantum superposition state of Boo. Boo only occupies a superposition when Mario is not looking. There is no answer to what quantum superposition “looks like”. As soon as Mario looks, Boo is either found on top of him or way back at Boo’s starting location. Boo is never found in the space between. In quantum physics, Boo is said to have collapsed into the state it is found in.
This is the source of the so-called measurement problem in quantum physics and at the heart of all the meta-physics and philosophy under the name quantum foundations. In short, the problem is that there are two rules for how Quantum Boo behaves. When no one is looking, Boo seems to spread out potentially occupying every allowed state. But, when someone decides to look, Boo jumps instantly to one of those states. Some physicists say that this is a problem because laws of physics should apply independent of whether physicists, or Italian plumbers, decide to look. After more than 100 years there is still no consensus on this problem beyond the fact that quantum physics works impressively well when applied in the laboratory.
Intermission: who the hell is Zeno?
Zeno was the dictator of the galactic confederacy who brought billions of Teegeeack to Earth in a souped-up jumbo jet 75 million years ago only to kill them with hydrogen bombs releasing thetans which now stick to humans and cause spiritual harm. Wait. No. Wrong book. That was Xenu, not Zeno. My bad.
Zeno (of Elea) was an ancient Greek philosopher known especially for his “paradoxes”. Most of Zeno’s paradoxes are little arguments for the impossibility of motion. Of course, we move all the time — hence the paradox. The most relevant one is the “Arrow paradox”, which can be simplified as follows. At every instant an arrow is motionless (it is where it is). It takes time to move. But time is composed of instants. So it is always motionless. Therefore the arrow does not move.
There have been many arguments given about this paradox that I won’t repeat here. The simplest refutation is to deny that time is composed of instants of zero duration. Now back to our regularly scheduled programming.
Quantum Zeno
Would you believe me if I told you that now you have all the knowledge and intuition to explain the quantum Zeno effect to your friends? (Maybe don’t though unless quantum physics naturally comes up in conversation.)
Boo starts at 100% probability of being in the starting state and that number smoothly goes to 0%. After a few instants then, the probability of being in the starting state is still, say, 99%. If Mario turns around, there is a pretty good chance that Boo is found in the starting state. When Mario turns away, Boo moves again into slight superposition, slowing increasing the probability of landing on Mario. But, if Mario turns around again quickly, Boo will surely be found back in the starting state.
In other words, if Mario turns frequently enough, he can ensure that Quantum Boo never moves. It’s the quantum Zeno paradox, as taught you by quantum Mario.
During the Spring (southern hemisphere) of 2020 I’m teaching Introduction to Quantum Computing. In these Covid-19 times, the class is completely online. Hence a perfect time to flip the classroom.
The friendship paradox is an observed social phenomenon that most people have fewer friends than their friends have, on average. Sometimes it is stated more strongly that most people have fewer friends than *most* of their friends. It’s not clear from the popular articles about the topic whether the latter statement is generally true. Let’s investigate!
I would have thought this would be difficult to determine, and perhaps this is why it took until 1991 for someone to discover it. In Scott Feld’s original paper, Why your friends have more friends than you do, he suggested that this might be a source of feelings of inadequacy. But, I mean, it’s not like people keep tallies of each of their friend’s friends, let alone a list of their own friends, do they? This was probably true in 1991, but now in the age of Facebook we can do this with ease — and probably a lot of people do.
How many faces are in my book
In fact, why not, I’ll have a go. I have 374 friends on Facebook. I’m not sure why. Anyway, let’s take 10 random friends and count the number of friends they have. Here’s the tally:
The average number of friends these friends have is 1042. So I have fewer friends than my friends have, on average. Also, of these 10, I have fewer friends than all but one of them — poor friend number 8.
Okay, but this is not a paradox — especially if you know me. How about we look at someone else in my network, say Friend 2. Friend 2 has 451 friends, and because Facebook is such a secure website that naturally preserves the privacy of its users, we can click on profiles of the friends of Friend 2 to find out how many friends they have. The numbers of friends that 10 random friends of Friend 2 have are: 790, 928, 383, 73, 827, 1633, 202, 457, 860, and 121. On average, Friend 2’s friends have 627 friends, which is more than 451. Also, six of these friends have more than 451 friends and so Friend 2 has fewer friends than most of their friends as well. If we repeated this exercise for all of my friends, we would still find that most of them have fewer friends than their friends even though most already have more friends than me!
You probably have a gut feeling that this is paradoxical. My intuition for why it feels paradoxical comes from the following analogy. Consider your height. Perhaps you are shorter than average. Maybe you have some friends that are also shorter than average. But, you expect to find at least half the people out that are actually taller than average. That’s kind of the intuition of “average” after all. Why isn’t the same true for friends, then? That is, even though myself and Friend 2 have fewer friends than our friends, surely those people out there with lots of friends have more friends than their friends. This is true, but these popular people are extremely rare, and that is the difference between friend numbers and height, for example.
In a typical social network there are lots of people with few friends and few people with lots of friends. The very popular people are counted in many people’s friend circles. And, the unpopular people are not counted in many friend circles. Take my Facebook friendship as an example again. I have, apparently, few Facebook friends. So, it would seem that many people have more friends than me. But — and here is the most important point — those people are unlikely to count me and my low number of friends in their friend circles. This is evidenced by the fact that Friend 2 and I only have 2 friends in common. So all of those friends of Friend 2, when they go to count the number of friends their own friends have, aren’t going to have me and my low friend count skewing the distribution in their favor.
Is this why the “karate kid” was such a loner?
Let’s look at some smaller examples where we can count everyone and their friends in the network. The most famous social network (at least to network theorists) is Zachary’s karate club. It looks like this.
Zachary’s karate club. Node colors guide the eye and are marked based on the number of friends each person has.
(At this point I’d like to quickly mention that this post is duplicated in a Jupyter notebook which you can play with on Google Colab.)
This shows 34 people in a karate club (circa 1970) and who they interacted with outside class. Colors simply guide the eye to how many friends each member has. You can see one person only has 1 friend and another has a whopping 17! Already this graph is too big to count things by hand, but a computer can step through each node in the network and count the friends and the friends of friends. In Zachary’s karate club, then, we have,
The fraction of people with fewer friends than their friends have on average is 85.29%.The fraction of people with fewer friends than most of their friends is 70.59%.
Histogram of friends and friends of friends in Zachary’s karate club. The data is collected by looking at the friends of each friend for every person in the network. The distribution of those friends can be summarized by the mean and median. For each friend of each person, those numbers are tabulated and plotted in the lower row.
The above histograms show the data in detail. You can see the key feature again of a few people with lots of friends. Obviously those few people are going to have more friends than their friends have. But that’s the point — there are only a few such people! Once we look at the full distribution of “friends of friends” we find it flattens out significantly, and it is more likely to find a large number of friends. The lower two histograms show the mean and median number of friends of each person in the network. Note that in either case, each person has friends with roughly 9 friends. This much more than the vast majority of people in the network! Indeed, only 4 people out of 34 have more than 9 friends.
Go big or go… actually it doesn’t seem to matter how big we go
Now let’s go back to Facebook. This time we’ll use data collected from ten individuals, anonymized and made public. It includes roughly 4000 people connected to these 10 individuals in various social circles. The social network looks like this.
The Facebook Ego Graph, showing community structure and clustering around popular individuals.
We can do the counting of friends and friends of friends for this network as well. The details look like this.
Histogram of friends and friends of friends in the Facebook Ego Graph. The data is collected by looking at the friends of each friend for every person in the network. The distribution of those friends can be summarized by the mean and median. For each friend of each person, those numbers are tabulated and plotted in the lower row.
In summary, we find,
The fraction of people with fewer friends than their friends have on average is 87.47%.The fraction of people with fewer friends than most of their friends is 71.92%.
Very similar! But perhaps this is just a coincidence. It’s only two data points after all. How do we test the idea for any social network? Simulation!
Fake it ’til you make it
Simulation is a tool to understand something by way of studying scale models of it. A model airplane wing in a wind tunnel is a classic example. Today, many simulations are done entirely on computers. In the context of social networks there are many models, but out of sheer laziness we will choose the so-called Barabási–Albert model because it is already implement in NetworkX, the computer package I’m using. If we create a mock social network of 34 people (same number as the karate club), it will look something like this.
An instance of the Barabási–Albert model on 34 nodes. It is constructed node by node, connecting each new node to two previous nodes, with a preference for highly connected nodes. This is a example social network.
It doesn’t look all that much different from the karate club, does it? The numerical data are similar as well.
Histogram of friends and friends of friends in the example Barabási–Albert model graph above. The data is collected by looking at the friends of each friend for every person in the network. The distribution of those friends can be summarized by the mean and median. For each friend of each person, those numbers are tabulated and plotted in the lower row.
In this example, the numbers of interest are,
The fraction of people with fewer friends than their friends have on average is 79.41%.The fraction of people with fewer friends than most of their friends is 70.59%.
This is nice, but the real beauty of simulation is the ability to rapidly test multiple examples. The above is just one mock social network. To get full confidence in our conclusion (relative to the assumptions of the model of course), we need to perform many simulations over randomly generated social networks.
Simulate all the graphs!
If we repeat the above exercise over 10,000 random social networks with 34 individuals, we find,
The fraction of people with fewer friends than their friends have on average is 79.31%.The fraction of people with fewer friends than most of their friends is 65.31%.
So now we can say with some confidence that the friend paradox persists in any social network — at least one that has the same characteristics as the Barabási–Albert model with 34 people. Another thing we can easily do though is change the number of people in the network to see if the trend continues for large networks. Indeed, things get much worse if we increase the number of people. As the number of people in the social network increases, the fraction of people with fewer friends than their friends (in majority or on average) also increases.
The friendship paradox demonstrated over 100 randomly chosen social networks created with the Barabási–Albert model, for increasing network size.
Cool, cool. Anything less depressing to tell us?
It’s been shown that this paradox goes beyond friends as well. You are also probably deficient, when comparing yourself to your friends, in income, Twitter followers, and how happy you are — and this fact is probably not helping. But, okay, enough of this crap — we all feel terrible now! Surely there is something positive we can glean from all this, right? Yes!
Since your friends have more contacts than you, they will probably contract a virus — or anything else transmitted through communities — before you. Indeed, researchers showed that instead of tracking randomly chosen people to gauge the spread of a disease, it was much more efficient to ask those random people to name a friend, and track that friend instead! In the study, the group of friends got sick an average 2 weeks before the originally chosen people. There are probably lots of other applications out there waiting to be found for the friendship paradox.
Before we end, though, I want to ask one last question: does the friendship paradox have to happen in any social network? The answer to this is yes and no. For the statement of the paradox that uses averages the answer is yes and this can be proven mathematically. That is, the statement on average, people have less than or equal to the average number of friends of their friends is true for any social network you can conceive of. For the statement which uses majority (most of your friends have strictly more friends than you), the answer is no. The key ingredient is the existence — or non-existence — of popular individuals. We can even create completely random social networks for which the paradox does not hold. Consider the following network, again over 34 people.
This network (coming from the Erdős–Rényi model) is constructed by considering every person is a friend of every other person with some fixed probability (in this case 75%).
For this social network we have,
The fraction of people with fewer friends than their friends have on average is 44.12%.The fraction of people with fewer friends than most of their friends is 44.12%.
You can see from the detailed distribution of friends the difference more clearly. The number of friends is evenly spread around the average. This is similar to looking at the height of people. So, roughly half the people have fewer friends than their friends, another rough half has more friends than their friends, and the rest have exactly the same number as the average of their friends.
Histogram of friends and friends of friends in the example Erdős–Rényi model graph above. The data is collected by looking at the friends of each friend for every person in the network. The distribution of those friends can be summarized by the mean and median. For each friend of each person, those numbers are tabulated and plotted in the lower row.
Conclusion
Of course, it should be somewhat obvious that if everyone had exactly the same number of friends, then they would have the same number of friends as their friends! But this is also true when the distribution of friends is more evenly spread. What does this suggest? I guess it lends credence to claims of favorability of egalitarian communities. But it seems to be the case that more hierarchical networks (be they in friendships, corporations, Twitter followers, etc.) grow naturally as a way to accommodate the complexity arising from maintaining large numbers of connections. But that is the topic for another blog post. In the meantime,
Please don’t share this. Instead, tell a friend to share it.
So you are stuck at home, the children aren’t in school, but you still need to get some work done. The internet is now full of activities for you to, as they say, “keep the learning going”. As a parent of homeschooled children, and someone who was working from home a lot already, things have changed less in our home than they have for most parents of school-aged children. For that, we are grateful. And so I thought it might be useful to not give you yet another list of activities to do, but step back and discuss some more big-picture things as we struggle with the physical and emotional havoc the Covid-19 situation has caused. Here’s some advice.
Trust yourself
When you get your child’s report card back it all seems like a very well thought out and scientific evaluation process. Here’s a little secret: it’s not. But busy administrators need simple numbers to rank not only the students, but the teachers, principals, schools, and even countries! The irony is that “one size fits all” fits no one at all.
Testing can be useful if it is used and contextualized properly. But as a parent, you probably know your child better than they know themselves. It’s a problem in that you know them so well, you can hardly put into words what you know about them and how they will react to things. But this intuition is unique and yours alone. So you are the only one that can be trusted to know what is working for you and your child. Use this power to your advantage and don’t stress about what a particular day’s activities might mean for the far future.
Take it easy
Do you remember your time in school? Have you ever volunteered in or observed your child’s classroom? If so, you’ll know that the amount of formal learning — whether it’s a lesson or one-on-one — is quite limited in classrooms with upwards of 30 students.
There is a large variability among countries, but taking the average, students spend less than 1000 hours per year in the classroom. How many of these hours are effective? That’s impossible to say. But certainly less than half of them would involve direct teacher-to-student interaction. What does this all mean? It means, realistically, your children are getting — very roughly — 1.5 hours of formal learning per day (averaged over the year). The rest of their day is lost in thought, socialisation, and play. (These are arguably as important as formal learning for creating intelligent, healthy adults, and we’ll come back to that next.)
Since you are giving your child(ren) your mostly undivided attention, the amount of formal learning at home need only be a couple of hours at most. Some might breathe a sigh of relief. Ah ha! But what are you going to do with the rest of the day? Well, more learning, of course. The philosophy in our house is anything that is not mindless consumption of media is learning. This might involve playing board games, making a meal, playing hide-and-seek, drawing pictures, building with LEGOs, and so on. But make sure the child is choosing the activity — curing your own boredom is an essential skill many people are now realizing they don’t have! Remember: you can tell if mental or physical tools are being used and developed simply by observing.
Be proactive
In our house the order in which the activities play out can make a huge difference. We could do the same thing on two different days, in a different order, and one day could be great while the other terrible. Here are rules of thumb we play by.
Don’t start the day with media and distractions. Do the formal stuff first. If your children aren’t keen on breakfast, do this first. If they are hungry as soon as they wake up, do it immediately after breakfast.
Don’t end the day with media and distractions. If only for your own sanity, but also probably for a healthy sleep, end the day in bed with some calm reading rather than trying to tear a child away from their favorite movie.
Every activity ends in disaster if allowed to go on long enough. Whether it is copying out hand-writing exercises or playing an addictive app on the iPad, eventually a meltdown will occur. Don’t allow something to go on too long before a break happens. Try a walk, a stretch, or a snack to break up the day’s activities.
One activity must be completed and cleaned up before the next begins. This is not only to emphasize good organization and concentration, but also necessary if you are a parent working from home. The day simply cannot be a chaotic mess that requires your own constant attention.
Consistent with the above rules, the rest of the day is completely unstructured.
Be flexible
What works and doesn’t work isn’t something you are trying to find as if there were a fixed perfect schedule out there. Routine is important. But as we are all keenly aware now, those change. Hopefully they don’t have to change so abruptly often, but they will change. Adapting to change is generally something humans are good at, and successful people seem to be better at.
Your job is to find what works, while still working, knowing that what works will ebb and flow. Your children will be watching you now more than ever, learning how to react and deal with uncertainty, change, and boredom. It’s not going to be easy. And that’s why these might be the most important lessons your children receive.
When I started writing children’s books, they were for my own children. Since I never stop singing the praises of science, I wasn’t much concerned about how scientifically literate they would be. But how am I doing outside my own family? I don’t know! That’s where you come it 😁
I wrote a book a while back called Quantum Entanglement for Babies. But, now all those babies are grown into toddlers! I’ve been asked what is next on their journey to quantum enlightenment. Surely they have iPads now and know how to scroll, and so I give you Quantum Entanglement for Toddlers, the infographic!
Desiree Vogt-Lee maintains a list of quantum computing resources called Awesome Quantum Computing. It is indeed awesome and comprehensive. Here I am looking to answer the question where do I start with quantum computing? with a more concise list of my current favourite entry points.
But, before we get started, a general piece of advice if you want to study quantum computing (or anything else for that matter): learn more maths. More? Yes. More. It doesn’t matter how much you already know. In fact, I’m going to go learn some more maths after writing out this list. (I’m not joking — the next tab in my browser is Agent-based model – Wikipedia.)
Now — in order of some sense of difficulty — here are my favourite recommendations for starting points on learning quantum computing.
Undergraduate
The academically minded might be looking for a more traditional approach. Don’t worry. Got that covered by Quantum Computer Programming, a course lectured at Stanford University. Other standard lecture notes include those by David Mermin and John Preskill. The former is more computer-sciencey while the latter is more physicsy.
If you want to do some real quantum programming, The Quantum Katas by Microsoft Quantum is a set of tutorials on quantum programming using the Q# programming language. While it does start with the basics, there is a steep learning curve for those without a background in programming.
Quantum computing for the very curious by Andy Matuschak and Michael Nielsen is like an electronic textbook with exercises that use spaced repetition to assist in remembering key facts. This is an experimental learning tool, which at the time of writing, is still under construction.
Highschool
“Thinking Quantum”: Lectures on Quantum Theory by Barak Shoshany is a set of about 16 hours worth of lecture notes which was delivered to highschool students at an international summer school. Though it is more focused on quantum physics, the first half will give you all the basic tools needed to start analysing quantum algorithms. It is quite mathematical so the reader would have to be comfortable with some mathematical abstraction. However, much of the field of quantum computing comes from a physics background and the ideas and language of quantum physics are pervasive.
The Quantum Quest by members of the QuSoft team is a web class which contains videos, lecture notes, and a pared down version of Quirk. It starts with the basics of probability and linear algebra and quickly gets you up and running with quantum circuits and algorithms.
Quirk by Craig Gidney is a quantum circuit simulator. It is incredible expressive and provides many useful visualisations. This tools is simple enough for anyone to start creating quantum circuits. However, interpreting the output does require some guidance and knowledge of probability.
If you prefer the video playlist approach, Quantum computing for the determined by Michael Nielsen is a series of short YouTube videos going over the basics of quantum information. However, if you are not putting pen to paper yourself, you are not likely to absorb the necessary mathematics to understand quantum computing.
