The Life of Pi to date!
Could you ever have imagined that such an unnoticeable constant could have such a rich history? It was initially discovered by Archimedes of Syracuse around 4000 years ago. For more than 1500 years after it was discovered, not a lot of digits were added to its decimal place until calculus and the method of calculating infinite series were invented. A significant breakthrough was when an English mathematician derived a formula for pi as the product of an endless series of ratios.
Pi is the ratio of the circumference of a circle to its diameter. It is an irrational number whose exact value has not been calculated up till the current date. Even though trillions of digits of pi have been known still due to no pattern emerging among those digits, it’s tough to predict the value of the irrational number so much so that you can continue to calculate the digits up to infinity.
Pi isn’t just unique because of the difficulty involved in calculating its digits but also because of its ubiquity in nature as well as mathematics. You can even observe it at places that have no connection with circles at all. For instance, if we take a sample of random whole numbers, then the probability that any two such numbers do not have a common factor is equal to 6/pi². In nature, you can find pi in the spiral of the DNA double helix, the disk of the sun, the pupil of the eye, the concentric rings that travel outward from the splashes in a pond. We can also observe pi in Physics too — such as ripples of sound and light. It has even not left the Heisenberg’s Uncertainty principle equation that defines how precisely can we know the state of the universe.
Most of us would very well know that March 14 is celebrated as Pi day globally. An Indian has the world record to recite the maximum digits of pi, see the evidence here. There are competitions held on this day that help to approximate the digits of pi, every year. This is so because as there are advances in computing, there are better methods to calculate the digits of pi. For more information on chronological advances in the computation of pi, you can view this. Below is a summary of the same.
After the invention of the calculator, in 1949, Ferguson and Wrench were able to calculate 1,120 digits using a desk calculator. The first attempt to compute it on ENIAC again in 1949 took 70 hours and computed 2037 decimal places. By 1967, around half a million digits were approximated, and in 2009 Takahashi calculated 2.5 trillion digits of Pi using a supercomputer. All the previously mentioned computations were really on huge devices. Still, on the last day of 2009, Fabrice Bellard used a home computer — running an Intel Core i7 CPU (similar to what you are using now) to end up calculating 2.7 trillion digits of pi. The most recent record happens to have derived more than 30 trillion digits of pi. Today, we can even compute thousands of digits of pi on a standard iPhone; the kind of calculation would have boggled mathematicians 2000 years ago.
How quantum computing can estimate the digits of pi?
What do you think is the most appropriate way of calculating the digits on pi? If you have studied just sufficient mathematics back at school, you would know of the set of infinite series that can represent log, e^x, trigonometric functions, and also pi in turn. We are going to be using that too.
Calculating the value of pi
The value for pi can be determined using the infinite series for the inverse tangent function, also known as the Gregory series.
If we substitute the value of x=1, then we get the following:
After we combine two terms at a time, we get the following:
Now we can use the quantum algorithms of multiplication, division, addition, and subtraction to calculate the value of pi.
Another method discovered by IBM talks of using the Quantum Phase Estimation for estimating the digits of pi. The goal is not to calculate the highest number of digits of pi but to demonstrate how quantum algorithms can be used for its estimation and also to determine the accuracy of the quantum computers at IBM. It is a central building block for many quantum algorithms.
Quantum Phase Estimation
A Phase Estimation Algorithm is used to estimate an eigenvalue given the corresponding eigenvector and a unitary matrix. In Quantum Phase Estimation we have a unitary operator U which acts on n qubits and an eigenvector |Φ> satisfying the following equation:
The goal is to find the eigenvalue λ. Because U is unitary, eigenvalue can be expressed as e² π ι θ, with phase 0 ≤ θ ≤ π. Thus, the problem is to find the phase θ. Thus, the Phase Estimation Algorithm.
As a matter of fact, |ϕ> can be any input state and not necessarily the eigenvector of U. Thus, the Phase Estimation’s output when measured collapses into one of the possible eigenvalues, from all the possibilities of eigenvalues. This happens because any input state can be expressed in terms of the eigenvectors of the unitary matrix U. Thus, it is written as a superposition of the eigenvectors as these vectors form a complete basis. Hence, due to this nature of the algorithm, it is also called Quantum Phase Estimation.
Conclusion
With the increasing accuracy of the quantum computers and using the Quantum Phase Algorithm, we might be able to estimate a more significant number of digits of π and exceed the current world record.
Liked the article? Click that 👏 button for 5 seconds and show your ❤️.
