Parametrized controlled phase shift gate
| Identifier | Operator | Example statement |
|---|---|---|
| CRk | \(CRk(\theta)\) | CRk(2) q[0], q[1] |
Description
The parametrized controlled phase shift, or CRk, gate is a two-qubit gate. It is the parametrized version of the controlled phase shift gate, with angle \(\theta\), parametrized by an integer \(k\): \(\theta(k) = \frac{2\pi}{2^k}\).
The CRk gate is especially useful for calculating the Quantum Fourier Transform.
Special cases of the CRk gate include:
- \(I = CR_k(0)\),
- \(CZ = CR_k(1)\),
- \(CR(2\pi/2^k) = CR_k(k)\).
Representation
\[\begin{align}
CR_k(k) &= \left(\begin{matrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\frac{2\pi}{2^k}}
\end{matrix}\right)
\end{align}\]
which is equal to:
\[CR_k(k) = I \otimes |0\rangle\langle 0| + R_k(k) \otimes |1\rangle\langle 1|,\]
with
\[R_k(k) = \left(\begin{matrix}
1 & 0 \\
0 & e^{i\frac{2\pi}{2^k}}
\end{matrix}\right).\]
Operation examples
Standard basis
\[\begin{align}
CR_k(k)\,|00\rangle &= |00\rangle \\
\\
CR_k(k)\,|01\rangle &= |01\rangle \\
\\
CR_k(k)\,|10\rangle &= |10\rangle \\
\\
CR_k(k)\,|11\rangle &= e^{i\frac{2\pi}{2^k}}|11\rangle \\
\end{align}\]
Qubit state ordering
Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,
\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]