Learning how to teleport quantum states with PhysicsGraph

I had a one-week “staycation” at the beginning of August and decided to use it to check out PhysicsGraph, a new online service for learning physics.¹ PhysicsGraph is very much inspired by and modeled on the Math Academy service, which I previously reviewed at length. Since I was a math and physics major a long long time ago, I was interested in the service and decided to try it out.

The first (and at the time of writing, only) course offered on PhysicsGraph is Introduction to Quantum Computing. I decided to take advantage of a promotional offer (see below) and recently finished the entire course. And as I did with Math Academy, I decided to write a review.

PhysicsGraph compared to Math Academy

As noted above (and freely confirmed by Chris Sutherland, one of its two founders), PhysicsGraph is very much modeled on Math Academy. In particular, it shares the following features:

A finely-scaffolded knowledge graph. Physics, like mathematics, is a hierarchical field: you must know elementary topics by heart in order to understand more advanced ones. Both PhysicsGraph and Math Academy embody these dependencies in a “knowledge graph” for each course, a directed acyclic graph of individual topics (e.g., “quantum operations and quantum circuits”) showing which topics are prerequisites for other topics (e.g., “multi-qubit operations and quantum circuits”). As a student progresses through the knowledge graph, fulfillment of prerequisites (i.e., completing the corresponding lessons) unlocks lessons for more advanced topics.

To reduce cognitive load on students, ideally the topics in the knowledge graph should be “bite-sized,” each containing the minimum of material needed to advance to the next topics in the graph. This help ensure that students can fully master a given topic before going on to the next. (Justin Skycak of Math Academy refers to the ideal knowledge graph as being “finely scaffolded.”) For the most part this seems to be the case with PhysicsGraph, although there were some longer lessons (like quantum teleportation and superdense coding) where some might have preferred their being split in two.

As does Math Academy, PhysicsGraph publishes the knowledge graphs for topics in its course(s), so you can see the scaffolding for yourself. For example, see the knowledge graph for the Introduction to Quantum Computing course.

Experience Points (XP). Like Math Academy, PhysicsGraph assigns each lesson an XP value that presumably reflects the amount and difficulty of the material in that lesson. Students then earn XP for themselves by successfully answering questions, with XP bonuses handed out for successfully answering all questions in a lesson or review.

I found accumulating XP on PhysicsGraph to be easier than on Math Academy: I was able to do over a thousand XP in a week and well over a hundred XP per day. (I did 354 XP on one particularly productive day.) In contrast, I keep myself busy trying to maintain a pace of 40 XP per day on Math Academy. Part of the difference may be that I was already familiar with many topics in the Introduction to Quantum Computing course (e.g., matrix multiplication and vector spaces) from learning and practicing them on Math Academy.

Adaptive diagnostics. Like Math Academy, PhysicsGraph has students take a diagnostic exam before starting a course. The exam is used to judge the level of the student’s knowledge and which prerequisite topics they can safely skip. That frees the student from getting bored covering material that they already know.

I found there were only a couple of times where I felt PhysicsGraph assumed knowledge I wasn’t fully solid on, one being expressing complex numbers in polar form. (I had some problems with this in Math Academy too.)

Spaced repetition review. Like Math Academy, PhysicsGraph prompts students to periodically review previously-covered (and hopefully mastered) material, scheduled in such a way that a student reviews the material for a given topic at the point at which they’re in danger of forgetting it.

Math Academy has an elaborate algorithm for doing this (see my discussion of it), one that incorporates “implicit review” of material, i.e., reducing the overall number of reviews based on the fact that successfully reviewing a topic also implicitly reviews prerequisites for that topic. The PhysicsGraph website doesn’t contain detailed information about its SRS technology, but I wouldn’t be surprised to learn it has a similar scheme.

Im Math Academy the reviews tend to predominate as you reach the end of a course, so that you spend as much or more time reviewing previous material as you do learning the last few topics in the course. That wasn’t my experience with PhysicsGraph, which may be a good thing or bad thing: It’s possible that PhysicsGraph is not scheduling enough reviews to promote long-term retention.

Gamification. Like Math Academy, PhysicsGraph allows students to compare their progress to others by looking at a leaderboard displaying XP totals for the highest-ranking students. However, unlike Math Academy, participating in the leaderboard is not optional, and there is only a single leaderboard instead of the multiple “leagues” of Math Academy. I think participation in gamification schemes should be optional, since some students don’t want to compare themselves to others and don’t need the motivation provided by such comparisons. As for having multiple leagues, I don’t think that makes sense until the number of active PhysicsGraph students is one or two orders of magnitude larger than it is now.

Community forum. Like Math Academy, PhysicsGraph has a Discord server in which students can (at least in theory) share experiences, help each other with problems, provide feedback to the service’s founders, and so on. At present the PhysicsGraph server is relatively inactive, presumably because the number of students is still relatively small. (Even with Math Academy, which presumably has hundreds if not thousands of active students, only a relatively few people post to the Discord server; this is typical of social media in general, and I wouldn’t expect PhysicsGraph to be any different.)

Target market. PhysicsGraph will likely appeal to the same types of students that Math Academy does, namely homeschoolers, advanced K-12 students, university students, and adults of any age learning new fields for personal enjoyment (like me) or professional enrichment. In particular, based on discussions on the Math Academy discord server and elsewhere, taking a Math Academy course appears to be a popular way to prepare for a formal instructor-led course (e.g., at a college or university) or to supplement such a course while taking it. I expect PhysicsGraph to have a similar appeal, and in fact their proposed next courses (“Physics I” and “Physics II”) appear to directly address that use case.

No “freemium” option. Like Math Academy, PhysicsGraph is a paid service with no free offering (although it does offer a free trial with a money-back guarantee). In my opinion this is exactly the way it should be: These services offer real value to students, and students should be willing to pay for that value.

PhysicsGraph’s subscription pricing is currently $29 US per month, compared to $49 US per month for Math Academy. I think this is a fair price given the paucity of courses at present compared to what Math Academy offers; however I think there’s room for raising the price in future if PhysicsGraph can built out a complete set of undergraduate physics courses and even extend into adjacent areas like chemistry. (I should also add that PhysicsGraph’s current annual price of $199 US is an absolute bargain, representing a 43% discount off paying per month; with Math Academy, paying by the year gets you only a 15% discount.)

The lack of a free offering also means that PhysicsGraph, like Math Academy, has an easier path to becoming a sustainable business by bringing in revenue from day one. The long-term financial health of a service like PhysicsGraph is presumably important for its founders and employees. But it’s also important for those of its customers who are lifelong learners, who want to take more courses over time and continue to get reviews for those previously taken. And unlike a typical VC-funded “freemium” educational offering, a service that can sustain itself solely on revenue from students will also be resistant to adding intrusive ads, employing deceptive marketing techniques, and generally implementing other forms of “enshittification.”

There are also areas where PhysicsGraph differs from Math Academy, for better or worse:

No timed quizzes. Math Academy periodically tests students’ mastery of previously-covered material using timed quizzes, requiring the student to answer a series of questions within a 15-minute period. The results of the quizzes are then used to schedule additional reviews and to modify the difficulty of subsequent quizzes. Despite finding these quizzes to be stressful, I think they’re useful in exposing gaps in learning and determining how best to correct them. I hope PhysicsGraph sees fit to offer a similar feature in future.

No “free response” questions. Math Academy lessons, reviews, and quizzes typically use multiple-choice questions with five possible answers. However some questions require responses where the student types in the answer. (Math Academy provides some UI widgets to make this easier when entering formulas.) As with quizzes, I find these types of questions more stressful to do, because I can’t just test my tentative answer by comparing it with the presented list of choices; instead I have to double- or even triple-check my work. But, also as with quizzes, I think free-response questions help in learning the material. This is another feature that PhysicsGraph might consider for future implementation in cases where it makes sense.

No question answering when revisiting lessons. This is a subtle point, but one that I think is important: In both Math Academy and PhysicsGraph you can go back to previous lessons and go through the material again. In Math Academy you can also answer the questions again in the same manner as when taking the lesson originally, except that you do not receive any further XP. However, this is not possible in PhysicsGraph; when revisiting a lesson you see all the questions presented along with your previous answers (correct or incorrect) and the accompanying explanations. I think this is a mistake: a major reason to revisit a lesson is to solidify your understanding of that lesson’s topics, and working through the exercises again helps contribute to that. I think PhysicsGraph should consider implementing Math Academy’s approach.

No “AI-powered” marketing. Math Academy advertises itself as “AI-Powered,” and its website has a “How our AI works” web page. As I discussed in my review, I am not a fan of this strategy, which presumably seeks to exploit the current hype around ChatGPT and other LLM-based services to help market Math Academy. PhysicsGraph is blessedly free of this conflation of LLMs with the machine learning-based analytics underlying Math Academy and (presumably) PhysicsGraph. It’s not trying to advertise itself as A Young Lady’s Illustrated Feynman Lectures, and thank goodness for that.

(To be clear, I think LLMs may have a place in a service like PhysicsGraph or Math Academy. However I think they’re an unneeded distraction from the core functioning of the service, and figuring out when and where their use does make sense will require some thought and experimentation.)

Per-class pricing. For both Math Academy and PhysicsGraph the main offering is via a monthly or yearly subscription; I think this is the right choice for both services, since it provides ongoing revenue with which to improve the service and add more courses. However, as I previously noted with Math Academy, I think it’s useful to also offer students the option to pay a one-time fee for an individual course, including the ability to review material for that course after completing it.

PhysicsGraph has done this with the Introduction to Quantum Computing course, offering lifetime access for $99 US, discounted from $199 US as an early-bird promotion. The promotional price thus equates to over three months of the subscription pricing, with the regular pricing (when it goes into effect) amounting to over six months of a subscription.

I could have opted for a $29 per month subscription, knocked out the Introduction to Quantum Computing course in less than a month, and then cancelled the service. However, I went for the per-class price because I wanted to be able to review topics in the course on an ongoing basis without having to maintain a subscription.

PhysicsGraph compared to Quantum Country

PhysicsGraph is not the first attempt to teach the basics of quantum computing and leverage spaced repetition to help the reader remember them. A couple of years ago Andy Matuschak and Michael Nielsen created the Quantum Country website, billed as “a free introduction to quantum computing and quantum mechanics . . . presented in a new mnemonic medium which makes it almost effortless to remember what you read.” It covers much of the same material as the PhysicsGraph Introduction to Quantum Computing course, including a discussion of quantum teleportation.

How does Quantum Country compare to the PhysicsGraph quantum computing course? My snap judgment is that Quantum Country is superior as a scientific essay for a knowledgeable audience, and makes an honest attempt to sweat the details (as opposed to hand-waving over them). However, I think it’s inferior as a way to actually learn quantum computing basics.

First, and most trivially, Quantum Country assumes a basic level of knowledge that some readers won’t have, to be “comfortable with complex numbers and with linear algebra—vectors, matrices, and so on . . . [and] with the logic gates used in conventional computers—gates such as AND, OR, NOT, and so on.” That’s not a problem with a scientific essay, and as the authors note there are other resources to learn those details. But it is a problem for readers coming into the topic relatively cold.

Quantum Country also spends some time on the history of the ideas behind quantum computing and the context for its development, and includes a fair amount of material on the principles and benefits of spaced repetition. If you subscribe to Justin Skycak’s theory of skill development then a lot of that is arguably superfluous—like teaching someone how to play basketball by starting out with a discussion of the high-level principles that make it a great game.² PhysicsGraph’s (and Math Academy’s) approach is more like “let’s have you first learn how to dribble.”

Next, Quantum Country material is not divided up and scaffolded as it would be in PhysicsGraph or Math Academy; it’s more like an essay where the reader is continually led along. This may be a small point, but I think it’s important for a student to have a small sense of accomplishment after finishing one lesson and before beginning the rest.

I think it’s also important for the system to gate access to further lessons on successfully completing the current one; with Quantum Country you can completely skip (or wrongly answer) all the spaced repetition questions and just go on to read further without necessarily understanding what you’ve read already.

Finally, in various places Quantum Country mimics the “this exercise is left to the reader” practice typical of traditional textbooks. This is contrary to the instructional philosophy behind both Math Academy and PhysicsGraph, in which all topics should be fully explained (even over-explained), with the role of the student then being to practice on their own applying the explained techniques to problems that are very similar (sometimes only trivially different) to those covered in the instructional material.

However, Quantum Country does have one feature that I think might be profitably considered for PhysicsGraph (and Math Academy): it sends email messages to remind readers when reviews are due. Math Academy assumes a model in which students are accessing the system every day, will see offered reviews in the normal course of events, and do not need additional reminders.

PhysicsGraph also assumes that students are motivated to check the system on a regular basis, in part because they’re paying for a continuing subscription. But what about students (like me) who pay for lifetime access to a single PhysicsGraph course? There may come a time when reviews for that course are due only every few days, or even every few weeks or months. In that case it would be useful to get email reminders when reviews are due.

Learning to teleport quantum states

I think there are multiple areas where PhysicsGraph could be improved. In particular, after I completed the course and started to get reviews for various topics, I had real difficulty completing the reviews for the last topics in the course, superdense coding and quantum teleportation. Above and beyond any intellectual deficiencies on my part, I think this may be due to at least two factors:

First, and most important, I think not having periodic timed quizzes is a real lack. There have been multiple occasions in Math Academy where I didn’t do well on quizzes and had to go back and do additional reviews and even revisit lessons. I think this really helped me better understand and retain the material. This is missing in PhysicsGraph, and as a result I wasn’t doing as much review as I should have been.

I also think some of the topics in the Introduction to Quantum Computing course could be even more finely scaffolded than they are. As one example, I think the lesson on separable and entangled states and Bell states might be better split into two. I’ve had major problems remembering the symbols for the Bell states and their associated quantum states; that might have been at least partially remedied by having a separate lesson (along with associated questions and reviews) dedicated to learning about those states.

But even in its current somewhat embryonic state, PhysicsGraph is a fun and effective way to learn physics. The choice of quantum computing for the first course is an excellent one: The material is understandable with only a minimum of mathematics knowledge, and the course itself teaches any additional mathematics needed.

The topic is also intrinsically interesting. Quantum mechanics, on which quantum computing depends, is a shining example of our ability to understand the world and manipulate it, the theory enabling everything from your cat’s laser pointer to the laptop on which I’m writing this article. Quantum entanglement is a phenomenon that Einstein found baffling (he called it “spukhafte Fernwirkung,” “spooky action at a distance”) and the ability to teleport quantum states is one of its most surprising consequences.

Yet at the end of the Introduction to Quantum Computing course, after all the preliminary lessons about Kronecker products and Bell states, PhysicsGraph will take you to a point where you can not only understand how quantum teleportation works, but why it must work exactly that way. Doing it in practice is far from trivial—real magic is hard—but you will have unlocked the spell that underlies the trick.

I myself am not planning to subscribe to PhysicsGraph on an ongoing basis. My personal interest is in learning more about data science, and to that end I’m working through the relevant Math Academy courses to learn the needed background material. My interest in physics these days is a relatively casual one that can easily be satisfied through other means.

However, if you are interested in really learning physics, and not just reading popular articles or watching videos about it, I strongly recommend that you check out PhysicsGraph. It’s a worthy attempt to apply to learning physics the proven educational techniques—direct instruction, spaced repetition, and so on— implemented in the Math Academy service. PhysicsGraph is still under construction, but I have confidence that the people behind it will end up building something great.