Archimedes, The Farmers Conundrum, and the Implications of Quantum Computing for Security
Archimedes was one of the first to develop an algorithm perfectly suited for quantum computing. There’s nothing more imperfect than the value of pi. Polygons conform to perfect math, as do triangles with their relationship to circles and cumulative degrees. Trigonometry so elegantly follows a repeatable sine wave and X2 + Y2 always equals Z2. However, the value of pi is this strange constant that seems to defy math. Measurements that use the value of pi are never accurate and always contain an error based upon the precision used. Prior to Archimedes, the value of pi was determined through statistics. Draw a perfect circle, measure the diameter, and trace the diameter around the circle a few thousand times. By recording the measurements, and determining an average value of pi with a standard deviation, a value could be determined for the accurate measurement of most applications at the time.
Superposition
Archimedes was the first to apply “quantum thinking” or superposition when determining the value of pi. Through trigonometry, he knew that a triangle could be perfectly inscribed within the circle.
Think of the inscribed triangle, not as perfect circle, but a poorly drawn circle using three lines. If I took the measurements l1 + l2 + l3, we could determine that pi would have to be greater that this number. For example, if each segment has a value of 1, pi would have to be greater than 3.
Think of the same circle with a circumscribed triangle. If we took the measurements of the outside triangle o1 + o2 + o3, we could determine that pi would have to be less than this number (another poorly drawn circle). For example, if each outside segment has a value of 1.5, pi would have to be less than 4.5.
Archimedes new that in the first iteration the value of pi was greater than l1 + l2 + l3 and less than o1 + o2 + o3. In essence, pi could simultaneously be any value between these two values and be true at the same time. This is “quantum thinking” or superposition.
If we continue to perfect the inscribed and circumscribed triangles into more perfect circles using trigonometry, each derivative (D) produces a more accurate value.
Pi is greater than D2l1 + D2l2 + D2l3 + D2l1 4 D2l5 + D2l6 AND…..
Pi is less than O2l1 + O2l2 + O2l3 + O2l1 4 O2l5 + O2l6
Each derivative requires the number of calculations to double (squared), but also refines the accuracy of pi. The calculation of the line segments are simple trigonometry and awareness of the derivative or iteration (i.e. trigonometry is perfect).
A dozen or so derivatives will get a very accurate measure of pi, but aren’t sufficient for large calculations like approximating the volume of the moon. In this case, the iteration may be sufficiently large that typical computers could not perform the calculations in a timely manner. A perfect use of quantum computing!
The Qubit Conundrum
But not so fast; the first problem we face with quantum computing is the ability to output the value for useful consumption. Humans like discrete answers; quantum computers live in an analog continuum. I like to say, the answer is in there somewhere, but finding it may be difficult! Pi would simultaneously contain all the possible vales (qubit) for a specific iteration simultaneously. When we try to convert a qubit for binary consumption, we get a random value that may or may not approximate the most accurate value Pi. Say what!
We cannot evaluate every possible value of Pi and there would not be enough time in the universe to do so. We get around this by providing “good enough” or probability, or “in this direction” type of thinking. Grover’s algorithm is a methodology for reducing the data to a more manageable set, but beyond the scope of this article. Let’s assume we took a random sample of the Pi every millisecond for 1 minute. We then calculated the mean and standard deviation of Pi for all the values. We would get a very accurate value for Pi and somewhat juxtaposed to Grover’s algorithm. This must sound very familiar doesn’t it? Is it that dissimilar to the mean of the measurements observed by the earlier derivations of Pi? The early observes were trying to get an approximation of Pi “just good enough” for the purposes to serve them. When we think of quantum computing, we have to think in probabilities, analogs, and “good enough” resultant sets. We have to think in horizontal terms and not vertical ones.
Entanglement
It wouldn’t be a valid article if I couldn’t bring farming into it. In the first example, I attempted to make a visible example of superposition. Let’s now talk about entanglement. It’s the state of a qubit where it cannot be experienced (described, notice we do not use the word calculated) independently of others. Explaining quantum computing always brings out the academic criticism as not being relevant or an accurate example and rightfully so. Making scalar examples of a non-scalar science is difficult at best.
Let’s suppose you have an unlimited number chickens that produce eggs and a limited amount of grain to feed the chickens. You know that some chickens require more grain than others to produce an egg. For example, Henrietta will produce an egg for every 1cup of grain, Alfalfa will produce an egg for every 2 cups, and Hilda will produce an egg for every three cups. The problem is that these chickens eat grain at different rates that do not correlate to the eggs produced. For example, Hilda requires more grain and eats grain faster. The faster Hilda eats the less available to other chickens. I have made discrete examples, but imagine more chickens and a greater amount of variables (15 different chicken types). I now want to know what is the best combination of randomly selected chickens (I have no idea of the value of a chicken when I buy it), to a very specific amount of grain. The cost effective ratio of one chicken is dependent on other chickens limited by food and ingestion rate.
“What is the most economical ratio of randomly selected chickens to the least amount of grain that maximizes my profits?”
For a very simple problem, the entanglement is so complex that conventional computers could not systematically step through the possible permutations to derive a useful answer. The problem becomes more unmanageable with the number and variations of the chickens in relation to the available grain.
Quantum computing is a perfect medium for approximating this ratio. Again, we have the issue of determining the results, but could use some of the same examples as above. We may never determine the perfect answer, but we will get a better or more accurate answer than traditional computing techniques (in a timely fashion).
The Case for Security
Quantum computing isn’t going to be a cloud application where we substitute traditional computing techniques for Quantum analysis. Very similar to Archimedes potential use of a Quantum computer, if he had one, would be to make his trigonometry more accurate for larger approximations of volume and area. It didn’t replace his mathematics; rather, it made his calculations better. The case for security is very similar. The first uses for quantum computing would be to enhance existing analytics with information gleaned from quantum analysis.
Graph representation of security events are overwhelmed with time-series relationships and entities. The value of understanding these relationships are clouded by the enormity of the data. Graph reduction is essential moving forward, especially when considering the relationships significantly change over short periods of time. It’s both a time and relationship problem. Quantum analytics could be helpful in developing algorithms for graph reduction. When and how do relationships become significant? To what extent do relationships become a deterrent to comprehension? Another opportunity may be deep learning on anomalous behavior through relationship analytics. That is, the seasonal change in related entities over time.
The biggest hurdle to applying quantum computing to security is changing the fundamental way we think about the computational model. Our mindset must change from liner to analog and from vertical (sequential) to horizontal. We need to think of experience versus outcomes. We may never know the exact right answer, but we have better conclusions than conventional computing mechanisms. MIT is now using IBM quantum computing in their classes. More education is needed before we will be able to take full advantage of this technology.
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