# CRZGate¶

class CRZGate(theta, label=None, ctrl_state=None)[source]

Bases: qiskit.circuit.controlledgate.ControlledGate

Controlled-RZ gate.

This is a diagonal but non-symmetric gate that induces a phase on the state of the target qubit, depending on the control state.

Circuit symbol:

q_0: ────■────
┌───┴───┐
q_1: ┤ Rz(λ) ├
└───────┘


Matrix representation:

$\begin{split}CRZ(\lambda)\ q_0, q_1 = I \otimes |0\rangle\langle 0| + RZ(\lambda) \otimes |1\rangle\langle 1| = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & e^{-i\frac{\lambda}{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{i\frac{\lambda}{2}} \end{pmatrix}\end{split}$

Note

In Qiskit’s convention, higher qubit indices are more significant (little endian convention). In many textbooks, controlled gates are presented with the assumption of more significant qubits as control, which in our case would be q_1. Thus a textbook matrix for this gate will be:

     ┌───────┐
q_0: ┤ Rz(λ) ├
└───┬───┘
q_1: ────■────

$\begin{split}CRZ(\lambda)\ q_1, q_0 = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes RZ(\lambda) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{-i\frac{\lambda}{2}} & 0 \\ 0 & 0 & 0 & e^{i\frac{\lambda}{2}} \end{pmatrix}\end{split}$

CU1Gate: Due to the global phase difference in the matrix definitions of U1 and RZ, CU1 and CRZ are different gates with a relative phase difference.

Create new CRZ gate.

Methods Defined Here

 inverse Return inverse CRZ gate (i.e.

Attributes

ctrl_state

Return the control state of the gate as a decimal integer.

Return type

int

decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

definition

Return definition in terms of other basic gates. If the gate has open controls, as determined from self.ctrl_state, the returned definition is conjugated with X without changing the internal _definition.

Return type

List

duration

Get the duration.

label

Return instruction label

Return type

str

name

Get name of gate. If the gate has open controls the gate name will become:

<original_name_o<ctrl_state>

where <original_name> is the gate name for the default case of closed control qubits and <ctrl_state> is the integer value of the control state for the gate.

Return type

str

num_ctrl_qubits

Get number of control qubits.

Returns

The number of control qubits for the gate.

Return type

int

params

Get parameters from base_gate.

Returns

List of gate parameters.

Return type

list

Raises

CircuitError – Controlled gate does not define a base gate

unit

Get the time unit of duration.