qftGate

Quantum Fourier transform gate

Since R2023a

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

Syntax

cg = qftGate(targetQubits)

Description

cg = qftGate(targetQubits) applies the quantum Fourier transform (QFT) to the target qubits and returns a quantum.gate.CompositeGate object with a Name property of "qft".

Examples

Quantum Fourier Transform Gate Acting on Three Qubits

Create a quantum Fourier transform gate that acts on three qubits.

cg = qftGate(1:3)

cg = 

  CompositeGate with properties:

             Name: "qft"
    ControlQubits: [1×0 double]
     TargetQubits: [3×1 double]
            Gates: [7×1 quantum.gate.SimpleGate]

Get the matrix representation of the gate.

M = getMatrix(cg)

M =

  Columns 1 through 5

   0.3536 + 0.0000i   0.3536 + 0.0000i   0.3536 + 0.0000i   0.3536 + 0.0000i   0.3536 + 0.0000i
   0.3536 + 0.0000i   0.2500 + 0.2500i   0.0000 + 0.3536i  -0.2500 + 0.2500i  -0.3536 + 0.0000i
   0.3536 + 0.0000i   0.0000 + 0.3536i  -0.3536 + 0.0000i  -0.0000 - 0.3536i   0.3536 + 0.0000i
   0.3536 + 0.0000i  -0.2500 + 0.2500i  -0.0000 - 0.3536i   0.2500 + 0.2500i  -0.3536 + 0.0000i
   0.3536 + 0.0000i  -0.3536 + 0.0000i   0.3536 + 0.0000i  -0.3536 + 0.0000i   0.3536 + 0.0000i
   0.3536 + 0.0000i  -0.2500 - 0.2500i   0.0000 + 0.3536i   0.2500 - 0.2500i  -0.3536 + 0.0000i
   0.3536 + 0.0000i  -0.0000 - 0.3536i  -0.3536 + 0.0000i   0.0000 + 0.3536i   0.3536 + 0.0000i
   0.3536 + 0.0000i   0.2500 - 0.2500i  -0.0000 - 0.3536i  -0.2500 - 0.2500i  -0.3536 + 0.0000i

  Columns 6 through 8

   0.3536 + 0.0000i   0.3536 + 0.0000i   0.3536 + 0.0000i
  -0.2500 - 0.2500i  -0.0000 - 0.3536i   0.2500 - 0.2500i
   0.0000 + 0.3536i  -0.3536 + 0.0000i  -0.0000 - 0.3536i
   0.2500 - 0.2500i   0.0000 + 0.3536i  -0.2500 - 0.2500i
  -0.3536 + 0.0000i   0.3536 + 0.0000i  -0.3536 + 0.0000i
   0.2500 + 0.2500i  -0.0000 - 0.3536i  -0.2500 + 0.2500i
  -0.0000 - 0.3536i  -0.3536 + 0.0000i   0.0000 + 0.3536i
  -0.2500 + 0.2500i   0.0000 + 0.3536i   0.2500 + 0.2500i

Plot the returned QFT gate to show its internal gates.

plot(cg)

Quantum Circuit with Two Quantum Fourier Transform Gates

Create a quantum circuit that consists of two QFT gates. The first gate acts on qubits with indices 2 to 4. The second gate acts on qubits with indices 1, 3, and 5.

gates = [qftGate(2:4); qftGate([1 3 5])];
c = quantumCircuit(gates)

c = 

  quantumCircuit with properties:

    NumQubits: 5
        Gates: [2×1 quantum.gate.CompositeGate]
         Name: ""

Plot the quantum circuit.

plot(c)

The plotted circuit consists of five qubits with indices 1 to 5. The plot shows that qubits 2 to 4 of the outer circuit are mapped to qubits 1 to 3 of the inner circuit of internal gates that construct the first QFT gate. For the second QFT gate, qubit 1 of the outer circuit is mapped to qubit 1 of the inner circuit, qubit 3 of the outer circuit is mapped to qubit 2 of the inner circuit, and qubit 5 of the outer circuit is mapped to qubit 3 of the inner circuit.

Click either QFT gate in the plot. A new figure showing the internal gates of the composite gate appears.

Input Arguments

targetQubits — Target qubits to apply QFT

positive integer vector

Target qubits to apply QFT, specified as a positive integer vector of qubit indices.

More About

Quantum Fourier Transform

The quantum Fourier transform (QFT) acts on a quantum state $$|x\rangle ={\displaystyle \sum _{m=0}^{N-1}{x}_m}|m\rangle$$ and maps it to a quantum state $$|y\rangle ={\displaystyle \sum _{k=0}^{N-1}{y}_k}|k\rangle$$ according to the formula:

$$y_k=\frac{1}{\sqrt{N}}{\displaystyle \sum _{m=0}^{N-1}{x}_m}{\omega }_N^{mk}k=0,1,\dots ,N-1,$$

where $${\omega }_N=\exp \left(\frac{2\pi i}{N}\right)$$ and $${\omega }_N^k$$ is an Nth root of unity.

Here, $$N=2^n$$ is the number of basis states for the n qubits that are quantum Fourier transformed. The state $$|m\rangle$$ (or $$|k\rangle$$) represents the binary notation $$|m_1{m}_2\dots m_n\rangle$$ of the possible basis states, where the value of m is $$m=2^{n-1}{m}_1+2^{n-2}{m}_2+\dots +2^1{m}_{n-1}+2^0{m}_n$$. By this definition, the effect of the QFT on the state vector matches the inverse discrete Fourier transform, but with a difference in the scaling factor because QFT uses $$\text{1/}\sqrt{N}$$ instead of $$\text{1/}N$$.

In matrix representation, the QFT can also be expressed as

$$QFT=\frac{1}{\sqrt{N}}\left[\begin{array}{cccccc}1& 1& 1& 1& \dots & 1\\ 1& \omega & {\omega }^2& {\omega }^3& \dots & {\omega }^{N-1}\\ 1& {\omega }^2& {\omega }^4& {\omega }^6& \dots & {\omega }^{2\left(N-1\right)}\\ 1& {\omega }^3& {\omega }^6& {\omega }^9& \dots & {\omega }^{3\left(N-1\right)}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\omega }^{N-1}& {\omega }^{2\left(N-1\right)}& {\omega }^{3\left(N-1\right)}& \dots & {\omega }^{\left(N-1\right)\left(N-1\right)}\end{array}\right].$$

References

[1] Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information. 10th anniversary ed, Cambridge; New York: Cambridge University Press, 2010.

Version History

Introduced in R2023a