# CRZGate#

class qiskit.circuit.library.CRZGate(theta, label=None, ctrl_state=None)[source]#

Bases: 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.

Can be applied to a QuantumCircuit with the crz() method.

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.

Attributes

condition_bits#

Get Clbits in condition.

ctrl_state#

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

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.

duration#

Get the duration.

label#

Return instruction label

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.

num_clbits#

Return the number of clbits.

num_ctrl_qubits#

Get number of control qubits.

Returns:

The number of control qubits for the gate.

Return type:

int

num_qubits#

Return the number of qubits.

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.

Methods

inverse()[source]#

Return inverse CRZ gate (i.e. with the negative rotation angle).