Controlled phase shift gate

Identifier	Operator	Example statement
CR	\(CR(\theta)\)	CR(pi) q[0], q[1]

Description

The Controlled phase shift, or CR, gate is a two-qubit gate. It is the controlled version of the phase shift gate, with angle \(\theta\) (radians).

The CR gate is a generalization of the CZ gate: \(CZ = CR(\pi)\)

Properties

• Ising gate.

Representation

\[\begin{align} CR(\theta) &= \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{i\theta} \end{matrix}\right) \end{align}\]

which is equal to:

\[CR(\theta) = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes R(\theta),\]

with

\[R(\theta) = \left(\begin{matrix} 1 & 0 \\ 0 & e^{i\theta} \end{matrix}\right).\]

Operation examples

Standard basis

\[\begin{align} CR(\theta)\,|00\rangle &= |00\rangle \\ \\ CR(\theta)\,|01\rangle &= |01\rangle \\ \\ CR(\theta)\,|10\rangle &= |10\rangle \\ \\ CR(\theta)\,|11\rangle &= e^{i\theta}|11\rangle \\ \end{align}\]

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 \(|q_0\rangle \otimes |q_1\rangle\).