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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:

  • \(I = CR_k(0)\),
  • \(CZ = CR_k(1)\),
  • \(CR(2\pi/2^k) = CR_k(k)\).

Properties

  • Ising gate.

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:

\[CR_k(k) = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes R_k(k),\]

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 convention and matrix representation

Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,

\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]

Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).