What is the minimum number of qubits required for a quantum system to represent each password simultaneously, given an 8-character-long non-case-sensitive alphanumeric password?

Understand the Problem

The question asks for the minimum number of qubits needed in a quantum system to represent all possible 8-character alphanumeric passwords simultaneously, where the passwords are not case-sensitive.

Answer

42
Answer for screen readers

42

Steps to Solve

  1. Determine the number of possible characters

Since the passwords are alphanumeric and not case-sensitive, we have 26 letters (A-Z) and 10 digits (0-9), making a total of 36 possible characters.

  1. Calculate the total number of possible passwords

Each character in the password can be any of the 36 characters. Since the password is 8 characters long, the total number of possible passwords is $36^8$.

  1. Calculate the number of qubits needed

To represent $36^8$ different states, we need to find the smallest integer $n$ such that $2^n \geq 36^8$. We can use logarithms to solve for $n$: $2^n \geq 36^8$ Take the logarithm base 2 of both sides: $log_2(2^n) \geq log_2(36^8)$ $n \geq 8 * log_2(36)$

  1. Approximate $log_2(36)$

We know that $2^5 = 32$ and $2^6 = 64$. Since 36 is between 32 and 64, $log_2(36)$ is between 5 and 6. We can approximate $log_2(36)$ as 5.17.

  1. Calculate n

$n \geq 8 * 5.17$ $n \geq 41.36$

  1. Round up to the nearest integer

Since $n$ must be an integer, we round up to the nearest whole number: $n = 42$

Therefore, we need a minimum of 42 qubits.

42

More Information

Each qubit can represent two states, 0 or 1. Two qubits can represent four states (00, 01, 10, 11), and so on. In general, $n$ qubits can represent $2^n$ states. In this case, we found that we needed to represent $36^8 = 2,821,109,907,456$ different password states. This requires at least 42 qubits since $2^{41}=2,199,023,255,552 < 36^8$ and $2^{42} = 4,398,046,511,104 > 36^8$.

Tips

A common mistake is to calculate the number of bits needed to represent a single alphanumeric character (which is between 5 and 6 bits, so you would need 6 bits). Then multiply that number by the number of characters in the password (8). This would give you $6*8 = 48$ bits. However, you can achieve a more efficient representation by encoding all possible passwords together by calculating the total number of possible passwords, then finding the number of bits or qubits needed to represent that many states.

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