jimmycricket said:I'm currently working through Nielsen & Chuang's section on the circuit design for implementing the QFT. I'm confused as to why swap gates are used in the model to swap the order of qubits. Heres what I'm looking at http://www.johnboccio.com/research/quantum/notes/QC10th.pdf page 247 figure 5.1
Can anybody help?
Strilanc said:Here's a blog post about the QFT with some puzzles in a circuit simulator.
The basic reason is that the QFT's output has the qubits in the reverse order of what you typically want. It switches you from big-endian bits to little-endian bits (or vice versa), so that the amplitude of the state 10010 corresponds to the 7/N frequency instead of the 18/N frequency (or vice-versa).
You don't have to swap them back into the right order, it just makes it easier to think about the QFT when used as a tool because you're keeping the endian-ness consistent. It makes the result match the usual definition of the discrete Fourier transform, where F(1) corresponds to the lowest non-zero frequency (instead of what F(N-1) corresponds to).
Do you mean the 9/N frequency not 7/N. Am I right in saying the qubits in your example reversed would read 01001=9?Strilanc said:Here's a blog post about the QFT with some puzzles in a circuit simulator.
The basic reason is that the QFT's output has the qubits in the reverse order of what you typically want. It switches you from big-endian bits to little-endian bits (or vice versa), so that the amplitude of the state 10010 corresponds to the 7/N frequency instead of the 18/N frequency (or vice-versa).
jimmycricket said:Do you mean the 9/N frequency not 7/N. Am I right in saying the qubits in your example reversed would read 01001=9?
The Quantum Fourier Transform is a mathematical operation used in quantum computing to transform a quantum state from the time domain to the frequency domain. It is the quantum equivalent of the classical Fourier Transform and is essential for many quantum algorithms.
The Quantum Fourier Transform works by applying a series of quantum gates to a quantum state, resulting in a new state that is represented in the frequency domain. The gates used in the QFT include the Hadamard gate, the phase gate, and the controlled phase gate.
The Quantum Fourier Transform has many applications in quantum computing, including in quantum error correction, quantum simulation, and quantum cryptography. It is also used in algorithms such as Shor's algorithm for factoring large numbers and Grover's algorithm for database search.
The Quantum Fourier Transform and the classical Fourier Transform are similar in that they both transform a signal from the time domain to the frequency domain. However, the QFT is performed on a quantum state, whereas the classical Fourier Transform is applied to a classical signal. Additionally, the QFT can provide exponential speedup compared to the classical Fourier Transform in certain applications.
One of the main challenges with the Quantum Fourier Transform is the implementation of the necessary quantum gates, which can be difficult to achieve in a physical quantum system. Additionally, the QFT can be susceptible to errors and noise in the quantum system, which can affect the accuracy of the transformed state. However, ongoing research and advancements in quantum computing are addressing these challenges and limitations.