Shor's algorithm is mathematically quite difficult, so many books on quantum computing often only briefly mention it. On the other hand, when an explanation is given, it is difficult to understand because it is a list of many mathematical formulas. In this context, the book by Prof. Barry Burd [1] is surprisingly easy to understand and explains the basics. Chapter 9 of this book takes 40 pages to thoroughly explain the essence of Shor's algorithm. Using a concrete example, he explains that if you can find the period (frequency) in a coprime powers sequence, you can factorize the public key number. He then demonstrated quantum Fourier transform (QFT) to find that frequency, expressed it in Qiskit code, and ran it on IBM Quantum Lab. This is fantastic! With this, I was able to grasp the heart of Shor's algorithm!
I have so far developed my own quantum circuit simulator for 3-qubit as a mobile phone app with MIT App Inventor. The outline is shown in Fig.1.
This time, I was able to use my simulator to identify frequencies using quantum Fourier transform, following the instructions in this book. The results matched those from IBM Quantum Lab. In other words, using just a mobile phone, they were able to perform a quantum Fourier transform and obtain the results, just like a scientific calculator. The situation is shown in Fig.2 and Fig.3.
(Notes)
As of 02/23/2024, the execution environment of Qiskit (IBM Quantum Lab) has changed, so it is necessary to modify the original Python code (for example, Chapter09.ipynb) to run it.
●Here is how to fix it:
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(A)change libraries:
from qiskit import QuantumCircuit, Aer, execute
from qiskit.tools.visualization import plot_histogram
↓
from qiskit.primitives import Sampler
from qiskit.visualization import plot_histogram
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(B)use 'Sampler' instead of 'Aer' as follows:
sampler = Sampler()
result = sampler.run(circ, shots=1000).result()
print("result: ", result)
quasi_dists = result.quasi_dists
print("quasi_dists: ", quasi_dists)
display(plot_histogram(quasi_dists))
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●More details here:
Migration examples
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Reference
[1] Barry Burd, Quantum Computing Algorithms -Discover how a little math goes a long way, Packt Publishing, 2023.
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