QFT

Functions for applying the Quantum Fourier Transform. More...

Functions
void	applyFullQuantumFourierTransform (Qureg qureg, bool inverse)
void	applyQuantumFourierTransform (Qureg qureg, int *targets, int numTargets, bool inverse)

Detailed Description

Functions for applying the Quantum Fourier Transform.

Function Documentation

applyFullQuantumFourierTransform()

void applyFullQuantumFourierTransform	(	Qureg	qureg,
		bool	inverse )

Applies the Quantum Fourier Transform upon all qubits in qureg. Alternatively, applies the Inverse Quantum Fourier Transform according to inverse.

Formulae

The Quantum Fourier Transform maps each computational basis state \( \ket{j} \) in an \( N \) qubit qureg according to

\[ \ket{j} \rightarrow \frac{1}{\sqrt{2^N}} \sum_{k=0}^{2^N-1} e^{2 \pi i j k / 2^N} \ket{k}. \]

Similarly the Inverse Quantum Fourier Transform maps each basis state like

\[ \ket{j} \rightarrow \frac{1}{\sqrt{2^N}} \sum_{k=0}^{2^N-1} e^{-2 \pi i j k / 2^N} \ket{k}. \]

Equivalences

• This function wraps applyQuantumFourierTransform(), passing all qubits in the qureg as targets.

Parameters

[in,out]	qureg	the state to modify.
[in]	inverse	whether to apply the inverse QFT or forward QFT

Exceptions

error

if qureg is uninitialised.

See also

• applyQuantumFourierTransform()

Author

Vasco Ferreira

Definition at line 1793 of file operations.cpp.

                                                                 {
    validate_quregFields(qureg, __func__);
 
    auto targets = util_getRange(qureg.numQubits);
    applyQuantumFourierTransform(qureg, targets.data(), targets.size(), inverse);
}

applyQuantumFourierTransform()

void applyQuantumFourierTransform	(	Qureg	qureg,
		int *	targets,
		int	numTargets,
		bool	inverse )

Applies the Quantum Fourier Transform upon the specified targets of qureg. Alternatively, applies the Inverse Quantum Fourier Transform according to inverse.

Formulae

Letting \( N \) = numTargets, the \( N \) qubit Quantum Fourier Transform maps each computational basis state of the targeted qubits, \( \ket{j} \), according to

\[ \ket{j} \rightarrow \frac{1}{\sqrt{2^N}} \sum_{k=0}^{2^N-1} e^{2 \pi i j k / 2^N} \ket{k}. \]

Similarly the Inverse Quantum Fourier Transform maps each basis state like

\[ \ket{j} \rightarrow \frac{1}{\sqrt{2^N}} \sum_{k=0}^{2^N-1} e^{-2 \pi i j k / 2^N} \ket{k}. \]

Parameters

[in,out]	qureg	the state to modify.
[in]	targets	the indices of the target qubits.
[in]	numTargets	the length of list targets
[in]	inverse	whether to apply the inverse QFT or forward QFT

Exceptions

error

if qureg is uninitialised. if targets are invalid qubit indices. if targets are not unique. if numTargets < 1.

See also

• applyFullQuantumFourierTransform()

Author

Vasco Ferreira

Definition at line 1755 of file operations.cpp.

                                                                                           {
    validate_quregFields(qureg, __func__);
    validate_targets(qureg, targets, numTargets, __func__);
 
    /// @todo
    /// change this placeholder implementation to the bespoke, optimised routine,
    /// wherein each contiguous controlled-phase gate is merged
 
    int mid = numTargets/2; // floors
 
    // don't think it is worth overcomplicating this to avoid slight repetition
    if (inverse) {
        for (int n=0; n<mid; n++)
            applySwap(qureg, targets[n], targets[numTargets-1-n]);
 
        for (int n=0; n<numTargets; n++) {
            for (int m=0; m<n; m++) {
                qreal arg = - const_PI / powerOf2(m+1);
                applyTwoQubitPhaseShift(qureg, targets[n], targets[n-m-1], arg);
            }
            applyHadamard(qureg, targets[n]);
        }
 
    } else {
        for (int n=numTargets-1; n>=0; n--) {
            applyHadamard(qureg, targets[n]);
 
            for (int m=0; m<n; m++) {
                qreal arg = const_PI / powerOf2(m+1);
                applyTwoQubitPhaseShift(qureg, targets[n], targets[n-m-1], arg);
            }
        }
 
        for (int n=0; n<mid; n++)
            applySwap(qureg, targets[n], targets[numTargets-1-n]);
    }
}

Referenced by applyFullQuantumFourierTransform(), and applyQuantumFourierTransform().