Expand description
Applies the quantum Fourier transform (QFT) to a specific subset of qubits of the register \p qureg.
The order of \p qubits affects the ultimate unitary. The canonical full-state QFT (applyFullQFT()) is achieved by targeting every qubit in increasing order.
The effected unitary circuit (shown here for \p numQubits = 4) resembles \f[ \begin{tikzpicture}[scale=.5] \draw (-2, 5) – (23, 5); \node[draw=none] at (-4,5) {qubits[3]}; \draw (-2, 3) – (23, 3); \node[draw=none] at (-4,3) {qubits[2]}; \draw (-2, 1) – (23, 1); \node[draw=none] at (-4,1) {qubits[1]}; \draw (-2, -1) – (23, -1); \node[draw=none] at (-4,-1) {qubits[0]};
\draw[fill=white] (-1, 4) – (-1, 6) – (1, 6) – (1,4) – cycle; \node[draw=none] at (0, 5) {H};
\draw(2, 5) – (2, 3); \draw[fill=black] (2, 5) circle (.2); \draw[fill=black] (2, 3) circle (.2); \draw(4, 5) – (4, 1); \draw[fill=black] (4, 5) circle (.2); \draw[fill=black] (4, 1) circle (.2); \draw(6, 5) – (6, -1); \draw[fill=black] (6, 5) circle (.2); \draw[fill=black] (6, -1) circle (.2);
\draw[fill=white] (-1+8, 4-2) – (-1+8, 6-2) – (1+8, 6-2) – (1+8,4-2) – cycle; \node[draw=none] at (8, 5-2) {H};
\draw(10, 5-2) – (10, 3-2); \draw[fill=black] (10, 5-2) circle (.2); \draw[fill=black] (10, 3-2) circle (.2); \draw(12, 5-2) – (12, 3-4); \draw[fill=black] (12, 5-2) circle (.2); \draw[fill=black] (12, 3-4) circle (.2);
\draw[fill=white] (-1+8+6, 4-4) – (-1+8+6, 6-4) – (1+8+6, 6-4) – (1+8+6,4-4) – cycle; \node[draw=none] at (8+6, 5-4) {H};
\draw(16, 5-2-2) – (16, 3-4); \draw[fill=black] (16, 5-2-2) circle (.2); \draw[fill=black] (16, 3-4) circle (.2);
\draw[fill=white] (-1+8+6+4, 4-4-2) – (-1+8+6+4, 6-4-2) – (1+8+6+4, 6-4-2) – (1+8+6+4,4-4-2) – cycle; \node[draw=none] at (8+6+4, 5-4-2) {H};
\draw (20, 5) – (20, -1); \draw (20 - .35, 5 + .35) – (20 + .35, 5 - .35); \draw (20 - .35, 5 - .35) – (20 + .35, 5 + .35); \draw (20 - .35, -1 + .35) – (20 + .35, -1 - .35); \draw (20 - .35, -1 - .35) – (20 + .35, -1 + .35); \draw (22, 3) – (22, 1); \draw (22 - .35, 3 + .35) – (22 + .35, 3 - .35); \draw (22 - .35, 3 - .35) – (22 + .35, 3 + .35); \draw (22 - .35, 1 + .35) – (22 + .35, 1 - .35); \draw (22 - .35, 1 - .35) – (22 + .35, 1 + .35); \end{tikzpicture} \f] though is performed more efficiently.
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If \p qureg is a state-vector, the output amplitudes are a kronecker product of the discrete Fourier transform (DFT) acting upon the targeted amplitudes, and the remaining. \n Precisely,
- let \f$|x,r\rangle\f$ represent a computational basis state where \f$x\f$ is the binary value of the targeted qubits, and \f$r\f$ is the binary value of the remaining qubits.
- let \f$|x_j,r_j\rangle\f$ be the \f$j\text{th}\f$ such state.
let \f$n =\f$ \p numQubits, and \f$N =\f$
\frac{1}{\sqrt{2^n}} \sum\limits_{j=0}^{2^N-1} \alpha_j \left( \sum\limits_{y=0}^{2^n-1} e^{2 \pi , i , x_j , y / 2^n} ; |y,r_j \rangle \right) \f]qureg.numQubitsRepresented.\n Then, this function effects \f[ (\text{QFT}\otimes 1) , \left( \sum\limits_{j=0}^{2^N-1} \alpha_j , |x_j,r_j\rangle \right)
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If \p qureg is a density matrix \f$\rho\f$, it will be changed under the unitary action of the QFT. This can be imagined as each mixed state-vector undergoing the DFT on its amplitudes. This is true even if \p qureg is unnormalised. \f[ \rho ; \rightarrow ; \text{QFT} ; \rho ; \text{QFT}^{\dagger} \f]
This function merges contiguous controlled-phase gates into single invocations of applyNamedPhaseFunc(), and hence is significantly faster than performing the QFT circuit directly.
Furthermore, in distributed mode, this function requires only \f$\log_2(\text{#nodes})\f$ rounds of pair-wise communication, and hence is exponentially faster than directly performing the DFT on the amplitudes of \p qureg.
@see
- applyFullQFT() to apply the QFT to the entirety of \p qureg.
@ingroup operator @param[in,out] qureg a state-vector or density matrix to modify @param[in] qubits a list of the qubits to operate the QFT upon @param[in] numQubits the length of list \p qubits @throws invalidQuESTInputError()
- if any qubit in \p qubits is invalid, i.e. outside [0,
qureg.numQubitsRepresented) - if \p qubits contain any repetitions
- if \p numQubits < 1
- if \p numQubits >
qureg.numQubitsRepresented@throws segmentation-fault - if \p qubits contains fewer elements than \p numQubits @author Tyson Jones