Quantum Fourier transform

In summary, when using the Quantum Fourier transform to find the period of a function f(x)\equiv a^x\mod N, the input register is 2n qubits in size and the output register is n qubits. This is because the overall period finding circuit may use ancilla bits, but the Quantum Fourier Transform itself does not require them. Therefore, the input size should not include ancilla bits as they are used and discarded in the process of finding the period.
  • #1
jimmycricket
116
2
When using the Quantum Fourier transform to find the period of the function [itex]f(x)\equiv a^x\mod N[/itex] why is it that the input register is 2n qubits in size and the output register is n qubits?
 
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  • #2
The overall period finding circuit probably uses ancilla bits, but turns them into measured garbage along the way. The Quantum Fourier Transform doesn't need ancilla bits, or measurement, so it's not the cause for adding them.

Personally, I wouldn't count ancilla bits against the input size. The circuit takes n input bits and n ancilla bits, burns the neg-entropy in the ancilla bits, and produces n outputs.
 

FAQ: Quantum Fourier transform

What is a Quantum Fourier transform?

A Quantum Fourier transform (QFT) is a mathematical operation in quantum computing that transforms a quantum state into a superposition of different frequency states. It is a generalization of the classical discrete Fourier transform (DFT) and is used for many applications such as quantum phase estimation and quantum simulation.

How does a Quantum Fourier transform work?

A Quantum Fourier transform works by applying a series of quantum gates to a quantum state. These gates manipulate the state in such a way that when measured, it reveals the frequency components of the original state. This is achieved by using quantum phase estimation, which uses the properties of quantum superposition and entanglement to perform the Fourier transform.

What are the advantages of using a Quantum Fourier transform?

One of the main advantages of using a Quantum Fourier transform is its ability to process information significantly faster than classical algorithms. It also allows for the efficient simulation of quantum systems, which can be used for solving complex problems in fields such as chemistry, physics, and cryptography.

What are some applications of Quantum Fourier transform?

The Quantum Fourier transform has a variety of applications in quantum computing, including quantum phase estimation, quantum simulation, and quantum error correction. It is also used in algorithms for finding the period of a function, which has applications in cryptography and prime factorization.

What are some challenges of implementing a Quantum Fourier transform?

One of the main challenges of implementing a Quantum Fourier transform is the need for high-quality quantum gates and accurate control of the quantum system. Any errors in the gates or measurements can lead to inaccuracies in the final result. Additionally, the complex nature of quantum states and operations makes it challenging to analyze and optimize QFT algorithms.

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