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Quantum Fourier Transform(QFT)
Discrete Fourier transform (DFT) is implemented in a quantum manner via the quantum Fourier transform (QFT). Many quantum algorithms, like Shor’s factoring of big numbers, depend on the QFT as a fundamental subroutine. The QFT converts a quantum state from the computational (Z) basis to the Fourier basis. This therefore alters the basis of the quantum state from one in which the basis states reflect the numbers 0 to N-1 to one in which the basis states reflect the frequencies of a signal.
Starting with a vector of N complex numbers, the conventional DFT generates a vector of N complex Fourier coefficients. By contrast, the QFT works on a quantum state—a superposition of basic states. Every basis state is an integer between 0 and N-1; the amplitude of every basis state shows the coefficient of that integer in the input vector. The QFT turns the amplitudes of the input state into those of the output state. The Fourier coefficients of the input vector show up as the output state’s amplitudes.
Using a Poisson number of quantum gates, the QFT may be performed on a quantum computer far quicker than the classical fast Fourier transform (FFT), which requires O(N log N). Whereas the FFT requires Θ(2nn), where n is the number of qubits, the QFT scales in Θ(n2.) Consequently, for big N the QFT is exponentially quicker than the FFT.
∣x⟩=∑j=0 N−1αj∣j⟩,
N is the dimension of the Hilbert space, the QFT transforms it into:
QFT(∣x⟩)= ∑ k=0 N−1βk∣k⟩,
Where the amplitudes βk are computed as
βk =1/√N∑j=0N−1 αje2πijk/N
Properties of Quantum Fourier Transform
A basic tool in quantum computing, the Quantum Fourier Transform (QFT) has numerous unique qualities.
linearity
Being a linear transformation, the QFT honors the superposition principle of quantum physics. Should the input state be a linear combination of basis states, the output is the linear combination of the transformed basis states:
QFT(a∣x⟩+b∣y⟩)=a⋅QFT(∣x⟩+)b⋅QFT(∣y⟩).
Reversibility
The QFT is unitary and thus reversible, much like other quantum processes. Computed effectively, its inverse, the Inverse Quantum Fourier Transform (IQFT), is This quality guarantees that the transformation preserves no information loss.
Efficient Implementation
For a n-qubit system, the QFT may be applied on a quantum computer using O(n2) quantum gates, far quicker than the conventional discrete Fourier transform, which depends on O(NlogN) operations for N=2n.
Periodicity
Extensive periodic properties of a quantum state are ideal for the QFT. For algorithms such as Shor’s method, which searches for hidden periodicities in functions, this quality is absolutely essential.
Compact Representation
The QFT enables the compact quantum form of encoding periodic information by mapping computational basis states to quantum states with certain phase connections. For instance,
∣j⟩↦1/√N∑k=0N−1 e2πijk/N ∣k⟩
Interference
Highly successful for issues requiring symmetry or periodicity, the QFT employs quantum interference to enhance or suppress specific outcomes in the quantum state.
Phase Encoding
The computational basis state is turned by the QFT into a superposition with phase factors dependent on the input value. Crucially important for frequency information extraction, these phases encode the Fourier coefficients.
Scalability
Since its operations easily extend to more qubits, the QFT is intrinsically scalable. Applications like factoring big numbers depend on this scalability practically.
Relationship to Classical DFT
Leveraging quantum parallelism, the QFT is similar to the conventional discrete Fourier transform only runs on the quantum amplitudes rather than deterministic values.
What is the Quantum Fourier transform used for?
Many quantum algorithms depend on the quantum Fourier transform (QFT), which lets quantum computers do some calculations exponentially faster than conventional computers. Shor’s method is among the most significant uses of the QFT as it finds the period of a function in order to factor a big number.
The QFT can also be used for:
- quantum phase estimation to learn about the features of a quantum system by estimating the eigenvalue of a unitary operator.
- resolving the abelian group hidden subgroup issue. This finds use in cryptanalysis and encryption.
- Constructing a Poisson solver for the method of vortex-in-cell.
The QFT transforms a quantum state from the computational basis, where basis states represent integers, and basis states represent frequencies. A quantum circuit constructed from a mix of Hadamard gates and conditional phase gates drives this change. Whereas the conditional phase gate functions on two qubits, phase shifting a target qubit depending on the state of the control qubit, the Hadamard gate acts on a single qubit, producing a superposition of states. This enables the QFT to be executed in polyn time, therefore accelerating it considerably quicker than the traditional fast Fourier transform (FFT).
A great technique enabling quantum computers to use quantum events to attain notable speedups over classical algorithms is the QFT. Expanding the possibilities of quantum computing depends on constant improvement of it.
How Quantum Fourier Transform work in Quantum Computing?
Input Quantum State
Quantum state represents a superposition of computational basis states. For an n-qubit system, the input state is:
∣x⟩=∑j=0 N−1αj∣j⟩,
where N=2n is the total number of states, and αj is the amplitudes of the basis states ∣j⟩.
Mathematical Transformation
The QFT maps the input state to a new state by applying the discrete Fourier transform on the amplitudes:
QFT(∣x⟩)= ∑ k=0 N−1βk∣k⟩,
Where the amplitudes βk are computed as
βk =1/√N∑j=0N−1 αje2πijk/N
This encodes the Fourier coefficients of the input amplitudes into the quantum state.
Circuit Construction
The QFT is using a quantum circuit consisting of the following components:
Hadamard Gate:Apply a Hadamard gate (H) to the most significant qubit (MSQ). The Hadamard gate creates a superposition of the basis states.
Controlled Phase Shift Gates:Apply a sequence of controlled phase shift gates (Rk) to entangle the MSQ with the remaining qubits. The phase shift introduces Fourier phase factors:
Repeat for Remaining Qubits Move to the next qubit and repeat the Hadamard and controlled phase shift operations, entangling it with the qubits to its right.
Inverse Order
The QFT circuit processes qubits in reverse order, starting from the MSQ and working toward the least significant qubit (LSQ). This reverse processing helps implement the phase entanglement correctly.
Swap Qubits
Finally, apply a series of SWAP gates to reverse the qubit order. This step is necessary because the QFT inherently transforms the qubits in reverse order relative to the computational basis.
Output Quantum State
The resulting quantum state after the QFT is:
∣x‘⟩=∑k=0 N−1∣k⟩,