Quantum computers may solve problems that are hard for classical computers, such as prime factorization and discrete logarithms, by leveraging quantum mechanics.

Church's Thesis suggests that all computing devices can be simulated by a Turing machine, but this may not hold when considering quantum mechanics.

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Quantum computers could outperform classical computers for certain problems, challenging the "Strong Church’s Thesis" that all physical devices are polynomially equivalent.

Shor's algorithms efficiently solve integer factorization and discrete logarithms on quantum computers by using a series of quantum gates and transformations.

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The order-finding algorithm in Shor's method measures states to determine the order of elements in modular arithmetic, a crucial step for factoring.

Reversible computation is essential for quantum computing, using gates like Toffoli and Fredkin to preserve information and implement logic units.

Quantum mechanical systems require precise state manipulations to avoid errors, necessitating high precision and error correction strategies.

Shor's algorithm includes a method for solving discrete logarithms on quantum computers through efficient transformations and observations.

Building functional quantum computers faces challenges like decoherence and precise quantum state manipulation, impacting future development.