# Parity problem in Bernstein Vazirani Algorithm

• Ananthan9470
In summary, the output f(x) in the given quantum circuit is determined by multiplying the coefficients of each term in x with the corresponding coefficient in a and then taking the sum of the products (mod 2).
Ananthan9470
I don't understand the following aspect of the parity problem and if someone could please explain it to me, I would be grateful.
In the given quantum circuit, the output f(x) is defined to be x.a = x1a1+x2a2+x3a3(mod 2), where a is a fixed |a1a2a3>. For example, if x=|101> and a=|100>, x.a = 1+0+0(mod 2) = 1.

I hope I got till this much correct. My question is, what if x is something like α|000>+β|101>+γ|111>? Then how is f(x) defined? Any help will be appreciated. Thanks!

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If x is α|000>+β|101>+γ|111>, then f(x) is defined as f(x) = αa1+βa2+γa3 (mod 2). In other words, the value of f(x) is determined by multiplying the coefficients of each term in x with the corresponding coefficient in a and then taking the sum of the products (mod 2). For example, if x = α|000>+β|101>+γ|111> and a = |100>, then f(x) = α(1)+β(0)+γ(0) (mod 2) = α (mod 2).

## 1. What is the parity problem in Bernstein Vazirani Algorithm?

The parity problem in Bernstein Vazirani Algorithm is a mathematical problem in which a binary string of 0s and 1s is given and the goal is to determine the hidden value or pattern within the string. This problem is important in quantum computing as it is used as a benchmark to test the efficiency of various quantum algorithms.

## 2. How does Bernstein Vazirani Algorithm solve the parity problem?

Bernstein Vazirani Algorithm uses a quantum circuit to solve the parity problem. It applies a series of quantum gates to the input string, and then measures the output to reveal the hidden value or pattern. The algorithm is able to achieve this in only one iteration, making it more efficient than classical algorithms.

## 3. What makes the Bernstein Vazirani Algorithm efficient for solving the parity problem?

The efficiency of Bernstein Vazirani Algorithm lies in its use of a quantum oracle, which enables it to determine the hidden value or pattern in a single iteration. This is because quantum computers can process multiple inputs simultaneously, whereas classical computers can only process one input at a time.

## 4. Are there any limitations to the Bernstein Vazirani Algorithm?

One limitation of the Bernstein Vazirani Algorithm is that it can only be applied to binary strings with an odd number of 1s. This is because the algorithm relies on the parity of the input string, which is only meaningful for odd numbers of 1s. Additionally, the algorithm can only determine a single hidden value or pattern and cannot be used for more complex problems.

## 5. How is the Bernstein Vazirani Algorithm used in real-world applications?

Bernstein Vazirani Algorithm is mainly used in quantum computing research and development, as it is a fundamental algorithm for benchmarking quantum computers. However, it has potential applications in cryptography, where it could be used to solve certain types of encryption problems more efficiently than classical algorithms.

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