# cirq.Rz

A gate, with matrix e^{-i Z rads/2}, that rotates around the Z axis of the Bloch sphere.

### Used in the notebooks

Used in the tutorials

The unitary matrix of Rz(rads=t) is:

exp(-i Z t/2) = [ e^(-it/2) 0 ] [ 0 e^(it/2)]

The gate corresponds to the traditionally defined rotation matrices about the Pauli Z axis.

exponent The t in gate**t. Determines how much the eigenvalues of the gate are scaled by. For example, eigenvectors phased by -1 when gate**1 is applied will gain a relative phase of e^{i pi exponent} when gate**exponent is applied (relative to eigenvectors unaffected by gate**1).
global_shift Offsets the eigenvalues of the gate at exponent=1. In effect, this controls a global phase factor on the gate's unitary matrix. The factor is:

exp(i * pi * global_shift * exponent)


For example, cirq.X**t uses a global_shift of 0 but cirq.rx(t) uses a global_shift of -0.5, which is why cirq.unitary(cirq.rx(pi)) equals -iX instead of X.

ValueError If the supplied exponent is a complex number with an imaginary component.

exponent

global_shift

## Methods

### controlled

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Returns a controlled ZPowGate, using a CZPowGate where possible.

The controlled method of the Gate class, of which this class is a child, returns a ControlledGate. This method overrides this behavior to return a CZPowGate or a ControlledGate of a CZPowGate, when this is possible.

The conditions for the override to occur are:

* The global_shift of the ZPowGate is 0.
* The control_values and control_qid_shape are compatible with
the CZPowGate:
* The last value of control_qid_shape is a qubit.
* The last value of control_values corresponds to the
control being satisfied if that last qubit is 1 and
not satisfied if the last qubit is 0.


If these conditions are met, then the returned object is a CZPowGate or, in the case that there is more than one controlled qudit, a ControlledGate with the Gate being a CZPowGate. In the latter case the ControlledGate is controlled by one less qudit than specified in control_values and control_qid_shape (since one of these, the last qubit, is used as the control for the CZPowGate).

If the above conditions are not met, a ControlledGate of this gate will be returned.

### in_su2

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Returns an equal-up-global-phase gate from the group SU2.

### num_qubits

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The number of qubits this gate acts on.

### on

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Returns an application of this gate to the given qubits.

Args
*qubits The collection of qubits to potentially apply the gate to.

### on_each

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Returns a list of operations applying the gate to all targets.

Args
*targets The qubits to apply this gate to. For single-qubit gates this can be provided as varargs or a combination of nested iterables. For multi-qubit gates this must be provided as an Iterable[Sequence[Qid]], where each sequence has num_qubits qubits.

Returns
Operations applying this gate to the target qubits.

Raises
ValueError If targets are not instances of Qid or Iterable[Qid]. If the gate qubit number is incompatible.
TypeError If a single target is supplied and it is not iterable.

### validate_args

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Checks if this gate can be applied to the given qubits.

By default checks that:

• inputs are of type Qid
• len(qubits) == num_qubits()
• qubit_i.dimension == qid_shape[i] for all qubits

Child classes can override. The child implementation should call super().validate_args(qubits) then do custom checks.

Args
qubits The sequence of qubits to potentially apply the gate to.

#### Throws:

• ValueError: The gate can't be applied to the qubits.

### with_canonical_global_phase

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Returns an equal-up-global-phase standardized form of the gate.

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### __call__

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Call self as a function.

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### __truediv__

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