Chris Ferrie

New papers dance!

Two new papers were recently posted on the arXiv with my first two official PhD students since becoming a faculty member! The earlier paper is titled Efficient online quantum state estimation using a matrix-exponentiated gradient method by Akram Youssry and the more recent paper is Minimax quantum state estimation under Bregman divergence by Maria Quadeer. Both papers are co-authored by Marco Tomamichel and are on the topic of quantum tomography. If you want an expert’s summary of each, look no further than the abstracts. Here, I want to give a slightly more popular summary of the work.

Efficient online quantum state estimation using a matrix-exponentiated gradient method

This work is about a practical algorithm for online quantum tomography. Let’s unpack that. First, the term work. Akram did most of that. Algorithm can be understood to be synonymous with method or approach. It’s just a way, among many possibilities, to do a thing. The thing is called quantum tomography. It’s online because it works on-the-fly as opposed to after-the-fact.

Quantum tomography refers to the problem of assigning a description to physical system that is consistent with the laws of quantum physics. The context of the problem is one of data analysis. It is assumed that experiments on this to-be-determine physical system will be made and the results of measurements are all that will be available. From those measurement results, one needs to assign a mathematical object to the physical system, called the quantum state. So, another phrase for quantum tomography is quantum state estimation.

The laws of quantum physics are painfully abstract and tricky to deal with. Usually, then, quantum state estimation proceeds in two steps: first, get a crude idea of what’s going on, and then find something nearby which satisfies the quantum constraints. The new method we propose automatically satisfies the quantum constraints and is thus more efficient. Akram proved this and performed many simulations of the algorithm doing its thing.

Minimax quantum state estimation under Bregman divergence

This work is more theoretical. You might call it mathematical quantum statistics… quantum mathematical statistics? It doesn’t yet have a name. Anyway, it definitely has those three things in it. The topic is quantum tomography again, but the focus is different. Whereas for the above paper the problem was to devise an algorithm that works fast, the goal here was to understand what the best algorithm can achieve (independent of how fast it might be).

Work along these lines in the past considered a single figure of merit, the thing the defines what “best” means. In this work Maria looked at general figures of merit called Bregman divergences. She proved several theorems about the optimal algorithm and the optimal measurement strategy. For the smallest quantum system, a qubit, a complete answer was worked out in concrete detail.

Both Maria and Akram are presenting their work next week at AQIS 2018 in Nagoya, Japan.

One is the loneliest prime number

You can’t prove 1 is, or is not, prime. You have the freedom to choose whether to include 1 as a prime or not and this choice is either guided by convenience or credulity.

I occasionally get some cruel and bitter criticism from an odd source. I’m putting my response here for two reasons: (1) so I that I can simply refer them to it and not have to repeat myself or engage in the equally impersonal displeasure of internet arguments, and (2) I think there is something interesting to be learned about mathematics, logic, and knowledge more generally.

It all started when I wrote a very controversial book about an extremely taboo topic: mathematics. In my book ABCs of Mathematics, “P is for Prime”. The short, child-friendly description I gave for this was:

A prime number is only divisible by 1 and itself.

I thought I did a pretty good job of reducing the concept and syllables down to a level palatable by a young reader. Oh, boy, was I wrong. Enter: the angriest group of people I have met on the internet.

You see, by the given definition, I had to include 1 as a prime number since, as we should all agree, it is divisible only by 1 and itself.

Big mistake. Because, apparently, it has been drilled into people’s heads that this is a grave error, a misconception that can eventually lead young impressionable minds to a life of crime and possibly even death! It might even end up on a list of banned books!

By a vast majority, people love the book. I am generally happy with the reponse. The baby books I write are not for everyone—I get that. And I do try to take advice from all the feedback I receive on my books. There is always room for improvement. But the intense emotions some people have with the idea of 1 being a prime number is truly perplexing. Here are some examples:

I actually love the book, but there is a big mistake. The number 1 is not a prime number! The book should not be sold like this and needs to be reprinted.

and

1 IS NOT PRIME! How could a supposed math book have an error like this in it? I am disgusted!

Yikes. So what gives? Is 1 prime, or not? The answer is: that’s not a valid question.

Let me explain.

First, let’s look at a typical definition. Compare to, for example, Wikipedia’s entry on prime numbers:

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Much more precise—no denying that. It’s grammatically correct, but probably hard to parse. I wanted to avoid negative definitions as much as I could in my books. But that’s beside the point. The reason 1 is not a prime is that the definition of prime itself is contorted to exclude it!

OK, so why is that? Well, the answer is probably not as satisfying as you might like: convenience. By excluding 1 as prime, one can state other theorems more concisely. Take the Fundamental Theorem of Arithmetic, for example:

Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

Now, this statement would not be true if 1 were a prime since, for example, 6 = 2 × 3 but also 6 = 2 × 3 × 1 and also 6 = 2 × 3 × 1 × 1, etc. That is, if 1 were prime, the representation would not be unique and the theorem would be false.

However, if we do chose to include 1 as a prime number, all is not lost. Then the Fundamental Theorem of Arithmetic would still be true if it were stated as:

Every integer is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors and the number of 1’s.

Which version do you prefer? In either case, both the definition and theorem treat 1 as a special number. I’d argue that in this context, the number 1 is more of an annoyance that gets in the way of the deeper concept behind the theorem. But in mathematics you must be precise with your language. And so 1 must be dealt with as an awkward special case no matter which way you slice it.

So, is 1 prime, or not? Well, it depends on how you define it. But in the end it doesn’t really matter, so long as you are consistent. And understanding that is a much bigger lesson than memorizing some fact you were told in grade school.

The definition given in ABCs of Mathematics is not “wrong” any more than all of the other simplifications and analogies I have made are “wrong”. But, in case you were wondering, the second printing will be modified with the hope that everyone can enjoy the book. Even the angry people on the internet deserve to be happy.

Quantum computing worst case scenario: we are Lovelace and Babbage

As we approach the peak of the second hype cycle of quantum computing, I thought it might be useful to consider the possible analogies to other technological timelines of the past. Here are three.

Considered Realism

We look most like Lovelace and Babbage, historical figures before their time. That is, many conceptual, technological, and societal shifts need to happen before—hundreds of years from now—future scientists say “hey, they were on to something”.

Charles Babbage is often described as the “father of the computer” and he is credited with inventing the digital computer. You might be forgiven, then, if you thought he actually built one. The Analytical Engine, Babbage’s proposed general purpose computer, was never built. Ada Lovelace is credited with creating the first computer program. But, again, the computer didn’t exist. So the program is not what you are currently imagining—probably, like, Microsoft Excel, but with parchment?

By the time computing began in earnest, Lovelace and Babbage were essentially forgotten. Eventually, historians restored them to their former glory—and rightfully so as they were indeed visionaries. Lovelace anticipated many ideas in theoretical computer science. However, the academic atmosphere at the time lacked the language and understanding to appreciate it.

Perhaps the same is true of quantum computation? After all, we love to tout the mystery of it all. Does this point to a lack of understanding comparable to that in computing 200 years ago?

This I see as the worst case scenario for quantum computation. We are missing several conceptual—and possibly societal—ideas to articulate this thing which obviously has merit. Eventually, humanity will have a quantum computer. But, will that future civilisation look at us as their contemporaries or a bunch of idiots mostly interested in killing each other while a few of our social elite played with ideas of quantum information?

Cautious Optimism

We are on the cusp off a quantum AI winter. We’re in for a long calm before any storm.

This is probably where most academic quantum scientists sit. We’ve seen 10-year roadmaps, 20-year roadmaps, even 50-year roadmaps. The truth is that every “scalable” proposal for quantum technology contains a little magic. We really don’t know what secret sauce is going to suddenly scale us up to a quantum computer.

On the other hand, very very few scientists believe quantum computing to be impossible—it’s going to happen eventually. At the same time, most would also not bet their own money on it happening any time soon. And, if most scientists are correct, the hype doesn’t match reality and we’re headed for a crash—a crash in funding, a crash in interest, and—worst of all—a crash in trust.

Some would argue that there are too many basic science questions unanswered before we harness the full potential of this theory that even its practitioners continue to call strange, weird, and counterintuitive. The science will march on anyway, though. Memes with truth and merit have a habit of slow and steady longevity. The ideas will evolve and—much like AI—eventually become mainstream, probably in our lifetime.

Unabated Opportunism

We will follow the same steady forward march that digital computers did the past 50 years.

If you are involved with a start-up company with an awkwardly placed “Q” in its name, this is where you sit. You believe our current devices are “quantum ENIAC machines”. Following the historical trajectory of classical computers, we just need some competitive players making a steady stream of breakthroughs and—voila!—quantum iPads for your alchemy simulations in no time. Along the way, we will reap continuing benefits from the spin-offs of quantum tech.

This is the quantum tech party line: quantum supremacy (yep, that’s a term of art now) is near. We are on the precipice of a technological—no, societal—revolution. It’s a new space race with equally high stakes. Get your Series A while the gettin’s good.

Like it or not, this is the best case scenario for the field. Scientists like to argue about what the “true” resource for quantum computation is. Turns out, it was money all along. Perhaps the hype will create a self-fulfilling prophecy that draws the the hobbyists and tinkerers that fueled much of the digital revolution. Can we engineer such a situation? I think we better find that out sooner rather than later.

Estimation… with quantum technology… using machine learning… on the blockchain

A snarky academic joke which might actually be interesting (but still a snarky joke).

Abstract

A device verification protocol using quantum technology, machine learning, and blockchain is outlined. The self-learning protocol, SKYNET, uses quantum resources to adaptively come to know itself. The data integrity is guaranteed with blockchain technology using the FelixBlochChain.

Introduction

You may have a problem. Maybe you’re interested in leveraging the new economy to maximize your B2B ROI in the mission-critical logistic sector. Maybe, like some of the administration at an unnamed university, you like to annoy your faculty with bullshit about innovation mindshare in the enterprise market. Or, maybe like me, you’d like to solve the problem of verifying the operation of a physical device. Whatever your problem, you know about the new tech hype: quantum, machine learning, and blockchain. Could one of these solve your problem? Could you really impress your boss by suggesting the use of one of these buzzwords? Yes. Yes, you can.

Here I will solve my problem using all the hype. This is the ultimate evolution of disruptive tech. Synergy of quantum and machine learning is already a hot topic¹. But this is all in-the-box. Now maybe you thought I was going outside-the-box to quantum agent-based learning or quantum artificial intelligence—but, no! We go even deeper, looking into the box that was outside the box—the meta-box, as it were. This is where quantum self-learning sits. Self-learning is protocol wherein the quantum device itself comes to learn its own description. The protocol is called Self Knowing Yielding Nearly Extremal Targets (SKYNET). If that was hard to follow, it is depicted below.

hypebox — Inside the box is where the low hanging fruit lies—**pip install tensorflow** type stuff. Outside the box is true quantum learning, where a “quantum agent” lives. But even further outside-the-meta-box is this work, *quantum self-learning*—SKYNET.

Blockchain is the technology behind bitcoin² and many internet scams. The core protocol was quickly realised to be applicable beyond digital currency and has been suggested to solve problems in health, logistics, bananas, and more. Here I introduce FelixBlochChain—a data ledger which stores runs of experimental outcomes (transactions) in blocks. The data chain is an immutable database and can easily be delocalised. As a way to solve the data integrity problem, this could be one of the few legitimate, non-scammy uses of blockchain. So, if you want to give me money for that, consider this the whitepaper.

Problem

99probs — Above: the conceptual problem. Below: the problem cast in its purest form using the formalism of quantum mechanics.

The problem is succinctly described above. Naively, it seems we desire a description of an unknown process. A complete description of such a process using traditional means is known as quantum process tomography in the physics community³. However, by applying some higher-order thinking, the envelope can be pushed and a quantum solution can be sought. Quantum process tomography is data-intensive and not scalable afterall.

The solution proposed is shown below. The paradigm shift is a reverse-datafication which breaks through the clutter of the data-overloaded quantum process tomography.

fuckyeahquantum — The proposed quantum-centric approach, called *self-learning*, wherein the device itself learns to know itself. Whoa.

It might seem like performing a measurement of $\{|\psi\rangle\!\langle \psi|, \mathbb I - |\psi\rangle\!\langle \psi|\}$ is the correct choice since this would certainly produce a deterministic outcome when $V = U$ . However, there are many other unitaries which would do the same for a fixed choice of $|\psi\rangle$ . One solution is to turn to repeating the experiment many times with a complete set of input states. However, this gets us nearly back to quantum process tomography—killing any advantage that might have been had with our quantum resource.

Solution

quantumintensifies — Schematic of the self-learning protocol, SKYNET. Notice me, Senpai!

This is addressed by drawing inspiration from ancilla-assisted quantum process tomography⁴. This is depicted above. Now the naive looking measurement, $\{|\mathbb I\rangle\!\langle\mathbb I |, \mathbb I - |\mathbb I\rangle\!\langle \mathbb I|\}$ , is a viable choice as

Though $|\langle V | U\rangle|$ is not accessible directly, it can be approximated with the estimator $P(V) = \frac{n}{N}$ , where $N$ is the number of trials and $n$ is the number of successes. Clearly, $\mathbb E[P(V)] = |\langle V | U\rangle|^2.$

Thus, we are left with the following optimisation problem:
$\min_{V} \mathbb E[P(V)] \label{eq:opt},$

subject to $V^\dagger V= \mathbb I$ . This is exactly the type of problem suitable for the gradient-free cousin of stochastic gradient ascent (of deep learning fame), called simultaneous perturbation stochastic approximation⁶. I’ll skip to the conclusion and give you the protocol. Each epoch consists of two experiments and a update rule:

$V_{k+1} = V_{k} + \frac12\alpha_k \beta_k^{-1} (P(V+\beta_k \triangle_k) - P(V-\beta_k \triangle_k))\triangle_k.$

Here $V_0$ is some arbitrary starting unitary (I chose $\mathbb I$ ). The gain sequences $\alpha_k, \beta_k$ are chosen as prescribed by Spall⁶. The main advantage of this protocol is $\triangle_k$ , which is a random direction in unitary-space. Each epoch, a random direction is chosen which guarantees an unbiased estimation of the gradient and avoids all the measurements necessary to estimation the exact gradient. As applied to the estimation of quantum gates, this can be seen as a generalisation of Self-guided quantum tomography⁷ beyond pure quantum states.

To ensure integrity of the data—to make sure I’m not lying, fudging the data, p-hacking, or post-selecting—a blochchain-based solution is implemented. In analogy with the original bitcoin proposal, each experimental datum is a transaction. After a set number of epochs, a block is added to the datachain. Since this is not implemented in a peer-to-peer network, I have the datachain—called FelixBlochChain—tweet the block hashes at @FelixBlochChain. This provides a timestamp and validation that the data taken was that used to produce the final result.

Results

SKYNET finds a description of its own process. Each $N$ is a different number of bits per epoch. The shaded region is the interquartile range over 100 trials using a randomly selected “true” gate. The solid black lines are fits which suggest the expected $1/\sqrt{N}$ performance.

Speaking of final result, it seems SKYNET works quite well, as shown above. There is still much to do—but now that SKYNET is online, maybe that’s the least of our worries. In any case, go download the source⁸ and have fun!

Acknowledgements

The author thanks the quantum technology start-up community for inspiring this work. I probably shouldn’t say this was financially supported by ARC DE170100421.

V. Dunjko and H. J. Briegel, Machine learning and artificial intelligence in the quantum domain, arXiv:1709.02779 (2017). ↩
N. Satoshi, Bitcoin: A peer-to-peer electronic cash system, (2008), bitcoin.org. ↩
I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997). ↩
J. B. Altepeter, D. Branning, E. Jerey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomography, Phys. Rev. Lett. 90, 193601 (2003). ↩
B. Schumacher, Sending quantum entanglement through noisy channels, arXiv:quant-ph/9604023 (1996). ↩
J. C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE Transactions on Automatic Control 37, 332 (1992). ↩ ↩
C. Ferrie, Self-guided quantum tomography, Physical Review Letters 113, 190404 (2014). ↩
The source code for this work is available at https://gist.github.com/csferrie/1414515793de359744712c07584c6990. ↩

David Wolfe doesn’t want you to share these answers debunking quantum avocados

Everyone knows you need to microwave your avocados to release their quantum memory effects.

Recently, I joined Byrne and Wade on Scigasm Podcast to talk about misconceptions of quantum physics. Apparently, people are wrong about quantum physics on the internet! Now, since the vast majority of people don’t listen to Scigasm Podcast [burn emoji], I thought I’d expand a bit on dispelling some of the mysticism surrounding the quantum.

Would it be fair to say quantum physics is a new field in the applied sciences, though it has been around for a while in the theoretical world?

No. That couldn’t be further from the truth. There are two ways to answer this question.

The super pedantic way: all is quantum. And so all technology is based on quantum physics. Electricity is the flow of electrons. Electrons are fundamental quantum particles. However, you could rightfully say that knowledge of quantum physics was not necessary to develop the technology.

In reality, though, all the technology around us today would not exist without understanding quantum physics. Obvious examples are lasers, MRI and atomic clocks. Then there are technologies such as GPS, for example, that rely on the precision timing afforded by atomic clocks. Probably most importantly is the develop of the modern transistor, which required the understanding of semiconductors. Transistors exist, and are necessary, for the probably of electronic devices surrounding you right now.

However, all of that is based on an understanding of bulk quantum properties—lots of quantum systems behaving the same way. You could say this is quantum technology 1.0.

Today, we are developing quantum technology 2.0. This is built on the ability to control individual quantum systems and get them to interact with each other. Different properties emerge with this capability.

Does the human brain operate using properties of the quantum world?

There are two things this could mean. One is legit and other is not. There is a real field of study called quantum biology. This is basically material physics, where the material is biological. People want to know if we need more than classical physics to explain, say, energy transfer in ever more microscopic biochemical interactions.

The other thing is called quantum consciousness, or something equally grandiose. Now, some well-known physicists have written about this. I’ll note that this is usually long after tenure. These are mostly metaphysical musings, at best.

In either case, and this is true for anything scientific, it all depends on what you mean by properties of the quantum world. Of course, everything is quantum—we are all made of fundamental particles. So one has to be clear what is meant by the “true” quantum effects.

Then… there are the crackpots. There the flawed logic is as follows: consciousness is mysterious, quantum is mysterious, therefore consciousness is quantum. This is like saying: dogs have four legs, this chair has four legs, therefore this chair is a dog. It’s a logical fallacy.

Quantum healing is the idea that quantum phenomena are responsible for our health. Can we blame quantum mechanics for cancer? Or can we heal cancer with the power of thought alone?

Sure, you can blame physics for cancer. The universe wants to kill us after all. I mean, on the whole, it is pretty inhospitable to life. We are fighting it back. I guess scientists are like jujitsu masters—we use the universe against itself for our benefit.

But, there is a sense in which diseases are cured by thought. It is the collective thoughts and intentional actions of scientists which cure disease. The thoughts of an individual alone are useless without a community.

Is it true that subatomic particles such as electrons can be in multiple places at once?

If you think of the particles has tiny billiard balls, then no, almost by definition. A thing, that is defined by its singular location, cannot be two places at once. That’s like asking if you can make a square circle. The question doesn’t even make sense.

Metaphors and analogies always have their limitations. It is useful to think this way about particles sometimes. For example, think of a laser. You likely are not going too far astray if you think of the light in a laser as a huge number of little balls flying straight at the speed of light. I mean that is how we draw it for students. But a physicist could quickly drum up a situation under which that picture would lead to wrong conclusions even microscopically.

Does quantum mechanics only apply to the subatomic?

Not quite. If you believe that quantum mechanics applies to fundamental particles and that fundamental particles make up you and me, then quantum mechanics also applies to you and me.

This is mostly true, but building a description of each of my particles and the way they interact using the rules of quantum mechanics would be impossible. Besides, Newtonian mechanics works perfectly fine for large objects and is much simpler. So we don’t use quantum mechanics to describe large objects.

Not yet, anyway. The idea of quantum engineering is to carefully design and build a large arrangement of atoms that behaves in fundamentally new ways. There is nothing in the rules of quantum mechanics that forbids it, just like there was nothing in the rules of Newtonian mechanics that forbade going to the moon. It’s just a hard problem that will take a lot of hard work.

Do quantum computers really assess every possible outcome at once?

No. If it could, it would be able to solve every possible problem instantaneously. In fact, we have found only a few classes of problems that we think a quantum computer could speed up. These are problems that have a mathematical structure that looks similar to quantum mechanics. So, we exploit that similarity to come up with easier solutions. There is nothing magical going on.

Can we use entanglement to send information at speeds faster than the speed of light?

No. Using entanglement to send information faster than light is like a perpetual motion machine. Each proposal looks detailed and intricate. But some non-physical thing is always hidden under the rug.

Could I use tachyons to become The Flash? And if so, where do I get tachyons?

This is described in my books. Go buy them.

Why are there so many symbols in math?

“Mathematics is the language of the universe.” — every science popularizer ever

I am a mathematician. In fact, I have a PhD in mathematics. But, I am terrible at arithmetic. Confused? I certainly would have been if a self-proclaimed mathematician told me that 15 years ago.

The answer to this riddle is simple: math is not numbers. Whenever a glimpse of my research is seen by nearly anyone but another mathematician, they ask where the numbers are. It’s just a bunch of gibberish symbols, they say.

IMG_20171212_090428 — The whiteboard in my office on 12 December 2017.

Well, they are right. Without speaking the language, it is just gibberish. But why—why all these symbols?

The symbols are necessary because communicating the ideas requires it. A simple analogy is common human language.

Mandarin Chinese, for example, has many more like-sounding syllables than English. This has led to a great number of visual puns, which have become a large part of Chinese culture. For example, the phrase 福到了(“fortune has arrived”) sounds the same as 福倒了(“fortune is upside down”). Often you will see the character 福 (“fortune”, fú, which you pronounce as ‘foo’ with an ascending pitch) upside down. While, like most puns, this has no literal meaning, it denotes fortune has arrived.

Fudao, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=30725479

Not laughing? OK, well, not even English jokes are funny when they have to be explained, but you get the idea. This pun just doesn’t translate to English. (Amusingly, there is also no simple common word for pun in Chinese.)

The point here is that upside down 福, with its intended emotional response, is not something you can even convey in English. The same is true in mathematics. Ideas can be explained in long-winded and confusing English sentences, but it is much easier if symbols are used.

And, there really is a sense in which the symbols are necessary. Much like the example of 福, most mathematicians use symbols in a way that is just impossible to translate to English, or any other language, without losing most of the meaning.

Here is a small example. In the picture above you will see p(x|θ). First, why θ? (theta, the eighth letter of the Greek alphabet, by the way). That’s just convention—mathematicians love Greek letters. So, you could replace all the θ’s by another symbol and the meaning wouldn’t change. It’s like the difference between writing Chinese using characters or pinyin: 拼音 = pīnyīn.

You might think that it is weird to mix symbols, such as Roman and Greek, but it now very common in many languages, particularly in online conversations. For example, Chinese write 三Q to mean “thank you”, because 三 is 3 and, in English, 3Q sounds like ‘thank you”. In English, and probably all languages now, emojis are mixed with the usual characters to great effect. You could easily write, “Have a nice day. By the way, my mood is happy and I am trying to convey warmth while saying this.” But, “Have a nice day :)” is much easier, and actually better at conveying the message.

OK, so we are cool with Greek letters now, how about p(x|θ)? That turns out to be easy to translate—it means “the probability of x given θ.” Unfortunately, much like any statement, context is everything. In this case, not even a mathematician could tell you exactly what p(x|θ) means since they have not been told what x or θ means. It like saying “Bob went to place to get thing that she asked for.” An English speaker recognises this as a grammatically correct sentence, but who is “she”, what is the “thing”, and what is the “place”? No one can know without context.

What the English speaker knows is that (probably) a man, named Bob, went to store to purchase something for a woman, whose name we don’t know. The amazing thing is that many more sentences could follow this and an English speaker could easily understand without the context. Have you ever read or listened to a story in which the characters are never named or described? You probably filled in your own context to make the story understandable for you. Maybe that invented context is fluid and changes as you hear more of the story.

The important point is that such actions are not taught. They come from experience—from being immersed in the language and a culture built from it. The same is true in mathematics. A mathematician with experience in probability theory could follow most of what is written on that whiteboard, or at least get the gist of it, without knowing the context. This isn’t something innate or magical—it’s just experience.

5 Picture Books that Encourage Abstract Thinking

Wow. It doesn’t take long for a blog to get neglected, does it? Let’s make an easy transition back in and start with a listicle. Here I am going list 5 books that we like which are suited to the 3–6 age range and encourage abstract thinking. This won’t be your typical reading experience. There will be a lot of interruptions, questions, and dialogue. It’ll be fun.

In My Heart: A Book of Feelings

The kids seem to like the novelty of the heart cut-outs, but they also enjoy the imagery the author instills for each emotion experienced by the sole character. The melodic flow makes it easy and fun to read as well. The books ends with a question, “how does your heart feel?” and I always get an interesting answer.

This is Not a Book

book (bʊk), noun: a written or printed work consisting of pages glued or sewn together along one side and bound in covers.

So this is a book. But your kids might argue with you on that. I often catch them “reading” this one on their own, even the ones who can’t read. It’s hard to describe. Just get it.

Ask Me

This might be for a little bit older audience, but it works for the younger ones in small spurts. It is book full of personal, intriguing questions accompanied by a photography or drawing. For example, one page asks, “Have you ever been really alone?” and there is a picture of a child under a tree staring up at the sky. Sometimes they just describe the picture or say “I don’t know”. Occasionally you get a interesting answer and learn something you maybe didn’t know about your child.

The Curious Guide to Things That Aren’t

For each letter of the alphabet, an idea or non-material thing is first described with clues. The reader must guess before it is revealed and described with a scientifically accurate explanation. For example, D is for Darkness. It is first described cryptically by where you can find it (a cave) and what makes it go away (flashlight).

The Book With No Pictures

I don’t get it. I mean, I can see why the first 10 times are funny. But over and over again, really? And now you are reading these words because that’s how blogs work. BLORK.