Your understanding of quantum computing algorithms exploring multiple possibilities simultaneously is correct. One of the fundamental properties of quantum computing that enables efficient prime factorization is called quantum superposition. In classical computing, you would indeed need to perform divisions sequentially by testing each potential factor. However, quantum computing algorithms, such as Shor's algorithm, leverage superposition and a phenomenon called quantum parallelism to perform computations on multiple inputs simultaneously.

Shor's algorithm, specifically designed for prime factorization, utilizes quantum parallelism to evaluate the function f(x) = a^x mod N, where N is the number to be factored and a is a randomly chosen base. By applying quantum techniques, the algorithm can evaluate this function for a range of values simultaneously, exploring multiple potential factors in parallel.

Moreover, quantum algorithms can also exploit another crucial quantum property known as entanglement. This property allows qubits (quantum bits) to become correlated in such a way that the state of one qubit depends on the state of another, even if they are physically separated. Shor's algorithm employs entanglement to extract information about the periodicity of the function f(x), which leads to the discovery of the prime factors.

Therefore, through the combined use of superposition and entanglement, quantum algorithms like Shor's can efficiently determine the prime factors of large composite numbers. This capability poses a significant challenge to certain cryptographic systems that rely on the computational difficulty of factorization for their security.

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Robert Collins

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Original Message:

Sent: Fri March 31, 2023 03:25 AM

From: Ken Iida

Subject: About Prime Factorization by Quantum Computer

I often hear that quantum computing can be used to prime factorize large composite numbers, making it easier to break cryptography that was considered secure . What properties of quantum computing algorithms make them suitable for prime factorization of large composite numbers?

I have an image that quantum computing algorithms explore multiple possibilities at the same time and gradually bring the correct answer to the surface.

If this image I have is correct, I can understand that quantum computers are suitable for problem such as the traveling salesman problem , searching mazes Etc... Why do I think so? It is because that these problems can be reduced to the problem of searching a huge binary tree. So we can take advantage of the quantum computer's characteristic to search multiple branches at the same time in the process of finding the correct leaf while tracing the branches.

In prime factorization of large numbers, does the algorithm perform something like dividing a given integer a by integers from 2 to √a in parallel?

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Ken Iida

Ken.Iida@kyndryl.com

NagoyaJapan

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