# IBM Q – The Moonshot We Might Be Waiting For

Summary

IBM Q is leading IBM’s research and commercialization efforts around quantum computing.

Practical quantum computers would herald a new revolution in computing.

IBM appears to be one of the leaders in the field, and long-term investors could be rewarded with a significant, new revenue stream.

1.0 Executive Summary

As opposed to an article with a specific investment thesis, this report is informational in nature. But I hope it also has an “enthusiastic” quality since my aim is to interest readers in what appears to be a pending revolution in computing. A revolution that IBM (IBM) is very much hoping to be a part of.

The revolution I’m talking about is quantum computing which may (literally) take us (humans) to places we have never been. It may herald the advent of applications and computational systems that can’t even be imagined right now.

IBM is not alone in the pursuit of this breakthrough technology. Intel (INTC), Microsoft (MSFT), and Google (NASDAQ:GOOG) (NASDAQ:GOOGL), among others, all have their own quantum computing programs. There are also private entities, such as Canada’s D-Wave and Berkeley, California’s Rigetti Computing; and government agencies, such as the NSA, that are aggressively pursuing research in this area. With that in mind, this report is specifically focused on IBM’s efforts and its IBM Q division.

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I want to reiterate there is no specific investment thesis contained herein. I wouldn’t go long/short or buy/sell IBM based on anything said here.

Perhaps because quantum computing is still largely research-centric and not broadly commercialized at this point, I’ve found that it is somewhat difficult to find papers and articles that explain the underlying concepts in easy-to-understand, layman’s terms. I have attempted to do that here, toward the end of the article in Section 5.0, for those readers/investors that may find the science interesting and compelling. My goal in that section is not to go into a great deal of scientific depth but rather to “paint in broad strokes” that readers will, with any luck, find exciting. Readers who do not care for the science may, of course, wish to skip the section entirely.

Let’s talk about quantum computing generally first and then get into IBM’s efforts and the market opportunity. As mentioned above, the last section will describe the science at a high level for those readers that are interested. I briefly discuss some of the ethical concerns and potential “arms race” associated with quantum computing in the Conclusion.

2.0 Quantum Computing: It’s Completely Different

Quantum computing is an entirely new approach to computing versus classical computing (i.e. the computers that we all use today). This is an important distinction. Here, I’m talking about how computational operations are performed, and not necessarily talking about how computers are built (although the physical construction of quantum computers is very different also). In regard to the last point, transistor-based computers today are obviously a heck of a lot faster (and smaller) than computers of the past which used vacuum tubes. So, modern computers are built differently; but the way they do things – in an operational sense, or perhaps I should say, in an algorithmic sense – is basically the same as how older computers worked. It’s just that operations on today’s computers are done much, much faster because the technology is better.

Quantum computers do not operate the same way as their classical counterparts. Whereas classical computers perform operations one-at-a-time (ignore computing architectures such as parallel CPUs for the moment), quantum computers can perform multiple operations simultaneously through the “weirdness” of quantum mechanics (the branch of physics which governs what happens in very small dimensions – think atomic). It is this “at-the-same-time” quality which allows quantum computers to solve certain kinds of problems with incredible, almost unimaginable speed, over a classical computer. An example, which I discuss further in Section 5.0, is the prime factorization of extremely large numbers. This kind of problem can, in theory, be solved on classical computers. But it takes classical computers so long, perhaps hundreds or thousands of years depending on the number and the computer, that the problem is effectively unsolvable. Modern day communication, which uses the product of large prime numbers to establish secure channels, relies on the fact that classical computers essentially can’t factor large numbers (i.e. they can’t do it in a reasonable amount of time). A quantum computer, on the other hand, could factor extremely large numbers into its primes in a fraction of the time by performing certain parts of the factorization algorithm simultaneously. If you’ve heard security experts talk about quantum computers rendering current security and encryption useless, it is this ability to perform a factorization computation with incredible speed that they’re talking about. The NSA is particularly worried about this. (Incidentally, quantum computing may actually give rise to a new kind of, possibly unbreakable, communication that exploits another “weird” property of quantum mechanics; I talk about it in Section 5.0).

Quantum computers are also probabilistic, whereas classical computers are deterministic. In simplistic terms, if you “ask” a classical computer a question (i.e. you give it some particular input), it gives you a specific answer back (i.e. it provides a particular output). Quantum computers don’t work this way. They provide a slew of answers to a “question” and provide a probability that a given answer is the correct one.

So, to drill the point home, quantum computers are unlike anything that exists today in the world of classical computing. It’s not just that they’re faster (in certain cases), it’s that they operate in a completely different way. (If you want to delve into the science a bit more first, jump to Section 5.0 and then come back here for the IBM and market-centric sections.)

3.0 IBM Q

IBM Q is the division of IBM tasked with leading research into quantum computing and developing systems which can ultimately be commercialized. As per the FY ’17 Annual Report, the company notes “IBM is the clear leader in quantum computing. The world’s first (and only) prototype 50-qubit system, announced in 2017, was a major step toward systems that can tackle problems beyond the scope of classical computation.” For readers unfamiliar with qubits, they are the information storage analog in quantum computers as compared with classical computers which use “regular” bits. Qubits are, shall we say, a bit (no pun intended) special; I discuss them in Section 5.0. But, for the purposes of this section, just understand they are how quantum computers store and manipulate information. The “power” of a quantum computer is effectively determined by the number of qubits that it operates with. As IBM’s quote implies, their development of a 50-qubit system is a significant advancement over previous systems. Simply, the physics – the actual production – of large quantum systems is very difficult right now. So, 50 qubits may not seem like a lot; but it is a big step forward. Some researchers have hypothesized that a practical quantum computer would need 10,000-100,000 qubits.

I should note here that D-Wave has a commercially available quantum computer system which they claim operates with 2000 qubits (Quantum Computing | D-Wave Systems). If true, that is a remarkable achievement. But I have also read articles online with some scientists questioning if D-Wave’s computer is a “true” quantum system. Personally, I have no idea; but, it is worth noting that D-Wave counts Lockheed-Martin and Google among its customers.

Companies love to proclaim they are the leader in this market, or that market; which usually means some skepticism is in order when a company makes a pronouncement like IBM’s. And it’s hard to know who is a leader in this field, because it is still very much in the research phase and much of the research is secret. However, there are some good reasons to think that IBM is in fact leading the pack for quantum computing. For example, Dr. Charles Bennett, who joined IBM Research in 1972, has pioneered work in that capacity “…applying quantum physics to the problems surrounding information exchange… [and] he has played a major role in elucidating the interconnections between physics and information, particularly in the realm of quantum computation” (Charles H. Bennett (computer scientist) – Wikipedia). In other words, at a (general) time when classical computing was still gaining its footing and companies like Microsoft (MSFT) were just getting started, IBM had researchers already thinking about “what could be” in a quantum computing world.

So, maybe IBM deserves the benefit of the doubt here, given their legacy in computing research. With that in mind, in some respects, the current state of quantum computing is perhaps a bit like the 1940s and 1950s when computers were masses of wires and large electronic components that filled entire rooms or even buildings. Here’s a picture of the IBM 50-qubit quantum computer referenced in the quote above:

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Source: Engadget This is what a 50-qubit quantum computer looks like

Granted, you probably won’t be putting one of these on the desk in your family room. Part of the reason that current quantum computers are so big is that quantum systems are very sensitive to “noise” which can essentially be thought of as environmental influences. Heat, for example, affects a quantum system. Noise introduces errors into the quantum system, and too much noise would essentially destroy any computational ability. As such, these systems must use heavy shielding to isolate the qubits from the external environment. (Check out the Engadget article for a more detailed description of IBM’s hardware.)

It would seem we (i.e. everyone in the world at this moment) are catching quantum computing at a time that is comparable to the early days of classical computing – the machines today are large and unwieldy because the technology is really still in its infancy in some ways. But, with continued advancements, and perhaps some breakthroughs analogous to the discovery of the transistor, we may in our lifetimes see quantum computing devices that become as common as laptops and mobile phones are today.

In the near future, we are more likely to see (physically) large, very expensive quantum computers – again, somewhat like the first commercially available computers – performing highly specialized tasks, which I suppose is also very much like the first commercially available computers. What would these “first” quantum computers be used for? Some of the potential applications include:

Simulation of molecules and chemical reactions, which could lead to discoveries of new drugs and materials.

Financial modeling and optimization.

New forms of machine learning and pattern recognition.

As IBM notes in its FY ’17 Annual Report, the company is working with “…a dozen clients, including partners JPMorgan Chase, Daimler AG, Samsung and JSR, [who] are now exploring practical applications on [IBM’s] latest commercial [quantum] systems”. It is also likely that these first quantum systems will work in tandem with classical systems to solve particular problems. One aspect of quantum computing is that these systems provide clearly superior performance for certain problems over their classical counterparts, but not all problems. You won’t be running Microsoft Word on a quantum computer because that is not a “problem” that a quantum computer would be good at solving. So, quantum computers are not going to replace classical computers; rather they will augment our overall computational capabilities, particularly around optimization problems.

So, how long do we have to wait for the first commercially available IBM computers? IBM says 5 years on the IBM Q website. To reiterate, those first computers are likely to be very expensive, perhaps only within the reach of governments and large organizations. From an investment standpoint, however, this could represent a significant new revenue stream for the company. And given that programming a quantum computer – at least for the moment – is nothing like programming a classical computer, there could be other, major pull-through revenue streams to help customers maximize the value from those systems (e.g. consulting). How much would these first systems cost? That also is a bit hard to say without proprietary knowledge from IBM which I don’t have. However, as a reference point, D-Wave’s commercial system mentioned earlier is around $15 million.

4.0 Market Opportunity

I suppose the question on a lot of readers’ minds is “how big is the market for quantum computing”? As you might guess, predictions are quite varied. Here are a few estimates, sorted in increasing order:

Analyst/Research Firm

Estimate

Markets & Markets

$495.3 million by 2023

Communications Industry Researchers

$1.9 billion by 2023 ($8.0 billion by 2027)

Homeland Security Research

$10.7 billion by 2024

Notably, IBM doesn’t mention a market size in its FY ’17 Annual Report for quantum computing. With that understanding, and the wide range of estimates above, it’s probably fair to say that no one really knows with any precision what the opportunity is right now. On the one hand then, revenues from this emerging business segment may be quite low for a long time and have little impact on IBM’s financial performance over the next few years. But the opposite is also a possibility where quantum computing systems become a “must-have” technology for the early-adopter organizations and governments that can afford it; and IBM realizes exponential growth in this business much like a start-up company.

Obviously, these 2 potential extremes don’t help investors too much from an investment-decision standpoint. Like I said in the Executive Summary, the point of this article is not to offer a particular investment thesis.

5.0 A Gentle Introduction to Quantum Computing

This section aims to provide a slightly more detailed, but still high-level discussion of the concepts and science that underpin quantum computing. Those readers who are not interested in the science may wish to skip this section and jump to the conclusion in Section 6.0. As I have seen quantum computing described as the intersection of math, physics, and computer science (and electrical engineering if one considers the hardware also), it is, of course, impossible to provide precise, granular explanations for “everything” that touches the subject in an article like this; not to mention that quantum computing is in essence the product of the greatest minds in human history. I encourage readers to therefore do their own research in specific areas where they might seek additional depth and clarity.

5.1 Quantum Mechanics: The Strange World We Live In

Perhaps most readers know or have heard that quantum mechanics is the realm of physics that governs what “happens” in our world when things get really small. By contrast, Newtonian physics are the laws that we “see” working in our day-to-day world of large things (i.e. the apple dropping on Sir Isaac Newton’s head). In any event, quantum mechanical properties are what allow quantum computers to operate the way they do.

When do the laws of physics switch from Newtonian to quantum? Or, to put it another way, what qualifies as small? There’s really no clear delineation of when quantum mechanics starts to kick-in and leave Newtonian physics behind, but I’ve read some papers where authors use distances below 1 nanometer as a rough point of demarcation. For reference, IBM Research is working on cutting-edge 5nm computer chips at the moment. Anyway, let’s just go with 1 nanometer and recognize that the realm of quantum mechanics involves really, really small dimensions.

(If you’re wondering if quantum mechanics actually works at larger scales, but perhaps the effects are too small to be noticeable, that may be so. But, physicists don’t seem to have a clear answer on this question at the moment. That is to say, there’s seems to be no clear consensus on why quantum mechanical effects are so dominant on smaller scales, and seem to “disappear” at larger scales.)

One of the weirdest things about quantum mechanics is that it is probabilistic, which flies in contrast to the deterministic world we live in. In regard to the latter, if I see a car traveling in a straight line down the street at a certain speed, I can calculate where it will be – say, in a few seconds. It’s deterministic. However, what if I try to determine the position of an electron moving around the nucleus of an atom? Quantum mechanics actually tells us (the Heisenberg Uncertainty principle) that if we know the momentum of an electron, we actually can’t know its precise position at a given time. The electron, in a sense, exists everywhere around the nucleus, and there is only a probability that it is in some particular location or another – until it is actually observed, at which point its position “collapses” to some particular place. So, quantum mechanics is probabilistic, not deterministic. Considering that the “normal” world we live in is not like this, a probabilistic reality is a bit mind-bending. But, this just seems to be the way it is.

(Actually, there are some physicists who think there may be a theory beyond quantum theory. Quantum theory may, in fact, just be an approximation of something more fundamental that describes the bizarre reality of the world/universe we live in.)

Quantum computers are probabilistic because quantum mechanics (and nature itself) is probabilistic.

Continuing with the example above, this “ability” for the electron to be in multiple states at once prior to being measured is known as superposition; and it is this property of quantum mechanics that quantum systems exploit to perform operations simultaneously.

Another property of quantum mechanics which affords us the unique possibilities of quantum computing is entanglement. In very simple terms, entanglement refers to a condition where pairs or groups of particles (again, let’s just say electrons for this example) are “connected” such that a change in state of one particle affects one or more other particles instantaneously, regardless of the distance between them. So, if 2 particles are entangled, a change to 1 would affect the other immediately, even if they were on opposite ends of the universe. This strange property of quantum mechanics troubled Einstein so deeply that he referred to it as “spooky action at a distance”. As we’ll discuss in sub-section 5.5, entanglement may give rise to a new kind of, perhaps unbreakable, communication.

5.2 Quantum Computing Gets Its Start

A lot of academics seem to feel that a seminal paper written by Dr. Richard Feynman in 1981 and published in 1982 provided the first description of what a quantum computer “architecture” would be. In that paper (which I include in the Appendix), Dr. Feynman more or less asked a question which seems rather trivial on the surface, but has great depth. I summarize the question in an overly simplistic (perhaps somewhat imprecise) way as follows:

Can you produce an exact simulation of nature on a classical computer?

The key word above is “exact”. We noted that classical computers are deterministic; if we provide it with some particular input, we get a particular answer as an output. But we also discussed earlier that nature at very small scales is probabilistic. So, you could never exactly simulate a probabilistic natural system on a deterministic computer. It’s just not possible; you can only approximate it. But you could exactly simulate a probabilistic natural system on a computer that is in-and-of-itself probabilistic.

Enter the idea of quantum computing…where the computer itself operates on probabilities.

5.3 Actually Computing With Quantum Computing

One might wonder how you could actually compute something useful with a computer that operates on probabilities. I mean, what’s the point of giving a computer some given input only to know that it will not give you a specific answer but rather a whole slew of answers, with each answer having some particular probability of being the “right” answer?

As it turns out, this “form” of computing happens to be incredibly useful for certain kinds of problems; problems which can’t be efficiently solved by today’s classical computers. Just as the key word a little while ago was “exact”, the key word in the sentence above is “efficiently”, so let’s provide some context around “efficiently” with an example.

Modern, secure communication (particularly public-key encryption) relies upon numbers that are the products of 2 unknown prime numbers. If those numbers could be factored into their primes, the security of many communication channels would be broken. For example, 15 is the product of 3 and 5, which are both prime. Of course, 15 is a pretty easy number to factor. But, what about 9999999942014077477? Its prime factors are 3162277633 and 3162277669. Clearly, that number would be much harder to factor. What if the number had over 200 digits; how long would it take you to factor? It took scientists “…two years to factor [a 232-digit number] using hundreds of classical computers operating in parallel” (The beginning of the end for encryption schemes?).

(The factorization of the 232-digit number mentioned above was a successful attempt at cracking one number among a series that were published as part of a contest by the encryption company RSA in March 1991. In fact, the largest number in the contest had 617 digits; that number, as well as several others, remain unbroken. As the referenced article notes, it took so long for the 232-digit number to be cracked, that the effort did not qualify for the contest prize money.)

Secure communication basically works by using numbers that are so large, it would take a classical computer too long to determine its prime factors. In other words, classical computers cannot factor these large numbers efficiently.

However, in 1997, Dr. Peter Shor discovered that a quantum computer could efficiently factor large prime numbers, rendering modern secure communication useless – if a suitable quantum computer could actually be built. Suffice it to say for the purposes of this article that a prime factorization computation for some large number, that might take a classical computer hundreds of years, would take a fraction of the time on a quantum computer – perhaps just a few hours.

(For mathematically inclined readers, as well as those with a background in formal computer science, the best known classical computer algorithm for factoring a number into its primes, the general number field sieve, has a sub-exponential running time, meaning it takes a lot longer to run than an algorithm running in polynomial time. The quantum algorithm developed by Dr. Shor runs in polynomial time, and is therefore much more efficient than the general number field sieve.)

Dr. Shor’s breakthrough was the first practical use identified for a quantum computer – if a suitable quantum computer could be built. The latter statement should assure you that your communications are quite safe at the moment since, for example, it was not too long ago that researchers were heralding the development of a hard-wired quantum computer that successfully factored the number 15 into 3 and 5 (Will this quantum computer take down internet banking?).

But the point is that it has been proven scientifically that quantum computers can do things that classical computers effectively can’t. The speed-up in computation for certain problems like prime factorization comes from the ability of the quantum computer to exploit the property of superposition mentioned earlier.

5.4 Everything All At Once: Superposition

You know that classical computers use bits, 0s and 1s, to store and manipulate information. As a physical analogy, we can think of modern computer chips with billions of incredibly tiny light switches. If the light switch is on, it represents a “1” and if the light switch is off, it represents a “0”. But these light switches can only be on or off; we can only have a 1 or 0 at a given time.

Quantum computers work with “qubits”, which also take on the values of 0 and 1. But here’s where one of the key differences with classical computing lies – and where we leave our “normal” world conventions behind: a qubit can be 0 and 1 at the same time. Using our light switch analogy from above, it’s as if the light switch can be on and off at the same time. But the laws of quantum mechanics – bizarre as they are – actually allow this. This ability for qubits to take on multiple values at the same time is a manifestation of superposition.

How is superposition useful? Think about it this way: let’s say we wanted to represent all the possible combinations of 3 bits so we could do some operation on them. First, let’s list out all the possible combinations (of which there are 23 = 8):

000 100 010 001 110 011 101 111

On a classical computer we’d actually need 24 bits (8 combinations x 3 bits per combination) to represent this information.

But, on a quantum computer, we’d only need 3 qubits. Why? Because, as we stated above, qubits can represent 0 and 1 at the same time. So, 3 qubits can exist in a superposition of the 8 combinations above at the same time!

This means a quantum computer can represent information far more efficiently than a classical computer. Instead of 3 bits, what if we wanted to represent all the combinations of 1,000,000 bits? On a classical computer, we’d need 21,000,000 x 1,000,000 bits of storage which is a crazy-ridiculously-large number that is (literally) physically impossible to print in this article. On, a quantum computer, we’d “only” need 1,000,000 qubits. So, in effect, it’s possible to store and manipulate information that is not efficient (or effectively possible) on classical computers. (For the record, no one is even close at the moment to building a quantum computer with 1,000,000 qubits.)

So, how does this help us with something like factoring large numbers into its primes? In a simple (not precisely correct, but good enough here) view, a classical computer has to try all possibilities one-by-one to figure out what the prime factors are for a given number. That’s why it takes so long for very large numbers. But, with Dr. Shor’s algorithm, a quantum computer can “try” certain steps involved in the factoring process all at once, providing the incredible speedup discussed earlier.

5.5 Use Cases for Quantum Computers

Beyond factoring large numbers, there are many other interesting use cases for quantum computers. Broadly, these use cases fall under the category of optimization problems. We mentioned a few earlier in Section 3.0, such as financial modeling and simulation of molecules. D-Wave (D-Wave Systems) mentions some additional examples on their website. I thought to include 2 other use cases and elaborate on them to help elucidate the power of quantum computing:

Searching for a specific item in an unsorted large database.

Unbreakable, secure communication.

In regard to the first case, some quantification is helpful. In a simple scenario, a classical computer would search the database one item at a time and check if it’s found the right entry (the key for this problem is that the database is unsorted, so our classical computer doesn’t have an obvious shortcut to find the target). This means the search time is dependent on the size of the database itself in terms of the items that it holds. Let’s say the database has N items where N = 1015; and it just so happens that our target item is the last entry in the database – so clearly, our classical computer will have to check each item until it gets to the very last one. Also, let’s assume that it takes 1 millisecond to check each item. On our classical computer, the approximate running time to scan the database would be:

1015 x (1 ms/item) x (1 hr/3,600,000 ms) x (1 yr/8,760 hr) =

31,710 years

You’d be waiting around a long time on our classical computer. As it turns out, a quantum computer could do the same thing with a total running time based on √N, not N as before. This, again, is due to the quantum computer being able to use the property of superposition to – in a physical sense – check multiple items at the same time. So, the approximate running time on our quantum computer would be:

√1015 x (1 ms/item) x (1 hr/3,600,000 ms) =

8.8 hours

That’s quite difference. If you’re interested in the math and science behind this use case, do a search on “Grover’s algorithm”.

The second case is also quite interesting and exploits the property of entanglement mentioned previously.

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Source: “Quantum Information and Computation for Dummies”, Peter Samuelsson

The diagram above shows how classical communication works via the “squiggly” line between Alice and Bob. Basically, classical communication is based on sending information via a physical connection between parties such as electrons traveling through a wire or photons traveling through a fiber optic cable. Even the radio waves for your wireless router represent a physical connection because the radio waves themselves are something tangible. However, quantum mechanics gives us a revolutionary approach to communication where no physical connection exists between communicating parties. In the same diagram above, we see the purple box which is used for quantum communication. As you can see, the box emits a pair of entangled particles (the purple ones) – with one entangled particle going to Alice and the other to Bob. (To reiterate, there is no physical connection between these entangled particles.) You can also see a pink particle on the left, representing information that Alice wants to transmit to Bob. In very, very, very high-level terms, Alice’s pink particle “interacts” with her entangled purple particle. Since her purple particle is entangled with Bob’s purple particle, the information from Alice’s pink particle is instantaneously transferred to Bob’s purple particle, even though no physical connection exists! In the diagram, the pink particle “leaving” Bob, represents how his (originally) purple particle has “changed” into the information that Alice communicated to him.

This process is known as quantum teleportation, because the information from Alice is teleported to Bob, not “sent” to him in a classical sense. In our modern, classical communication schemes, information can be intercepted because we are using some physical medium to communicate. If an intelligence service wants to spy on someone (e.g. a whistle-blower), it will attempt to physically intercept an individual’s communications (cellphone signals, electronic transmission of data, etc.) In a quantum communication scheme, interception is impossible because there is nothing to intercept; there is no physical connection. The information is not transmitted, it is teleported.

Before you start thinking a quantum cellphone might be in your pocket in the near future, bear in mind there are practical, implementation challenges with the scheme described above. For example, we talked about “noise” as a problem with quantum systems; and noise is a challenge with quantum communication in terms of information accuracy. However, Chinese researchers recently made a breakthrough in quantum communication, teleporting information 870 miles from Tibet to a satellite in orbit (Chinese Scientists Just Set the Record for the Farthest Quantum Teleportation).

6.0 Conclusion

While the advent of quantum computing holds the promise of many positive advancements, such as revolutionizing how new medicines are discovered, it may also carry with it significant risks. For example, it is theoretically possible that “bad actors” may seek to capture encrypted information now with the intention of decrypting it “later” when a sufficiently powerful quantum computer exists. (Although, some might argue the actual value of the encrypted information would be “long gone” by the time a feasible quantum computer is available.) Also, quantum communication, which could be used to protect highly sensitive information, could also be used to protect illicit activities. Many experts feel that the major countries are thus in something of an “arms-race” to develop quantum computing technologies (The U.S. and China “Quantum Computing Arms Race” Will Change Long-Held Dynamics in Commerce, Intelligence, Military Affairs and Strategic Balance of Power).

Only time will tell how the “story” of quantum computing plays out, with – hopefully – more positive benefits than negative ones. Perhaps though, in the next 5-10 years, when quantum computing is more broadly commercialized but still arguably in an early phase of its maturity cycle, investors in companies like IBM may be rewarded with a substantial new revenue stream, albeit one whose magnitude cannot be reliably forecast at this time.

Appendix

ibm_ar_fy2017.pdf

quantum_computing_market_size_cir.pdf

quantum_computing_market_size_marketwatch.pdf

quantum_computing_market_size_prnewswire.pdf

Feynman.pdf

quantum_information_and_computation_for_dummies.pdf

https://seekingalpha.com/article/4192580-ibm-q-moonshot-might-waiting