# cirq.HPowGate

A Gate that performs a rotation around the X+Z axis of the Bloch sphere.

Inherits From: EigenGate, Gate

The unitary matrix of cirq.HPowGate(exponent=t) is:

$\begin{bmatrix} e^{i\pi t/2} \left(\cos(\pi t/2) - i \frac{\sin (\pi t /2)}{\sqrt{2} }\right) && -i e^{i\pi t/2} \frac{\sin(\pi t /2)}{\sqrt{2} } \\ -i e^{i\pi t/2} \frac{\sin(\pi t /2)}{\sqrt{2} } && e^{i\pi t/2} \left(\cos(\pi t/2) + i \frac{\sin (\pi t /2)}{\sqrt{2} }\right) \end{bmatrix}$

Note in particular that for $$t=1$$, this gives the Hadamard matrix

$\begin{bmatrix} \frac{1}{\sqrt{2} } & \frac{1}{\sqrt{2} } \\ \frac{1}{\sqrt{2} } & -\frac{1}{\sqrt{2} } \end{bmatrix}$

cirq.H, the Hadamard gate, is an instance of this gate at exponent=1.

exponent The t in gate**t. Determines how much the eigenvalues of the gate are phased 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 version of this gate. If no arguments are specified, defaults to a single qubit control.

Args
num_controls Total number of control qubits.
control_values Which control computational basis state to apply the sub gate. A sequence of length num_controls where each entry is an integer (or set of integers) corresponding to the computational basis state (or set of possible values) where that control is enabled. When all controls are enabled, the sub gate is applied. If unspecified, control values default to 1.
control_qid_shape The qid shape of the controls. A tuple of the expected dimension of each control qid. Defaults to (2,) * num_controls. Specify this argument when using qudits.

Returns
A cirq.Gate representing self controlled by the given control values and qubits. This is a cirq.ControlledGate in the base implementation, but subclasses may return a different gate type.

### 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.

Returns: a cirq.Operation which is this gate applied to the given qubits.

### 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.

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

### with_probability

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Creates a probabalistic channel with this gate.

Args
probability floating point value between 0 and 1, giving the probability this gate is applied.

Returns
cirq.RandomGateChannel that applies self with probability probability and the identity with probability 1-p.

### wrap_in_linear_combination

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Returns a LinearCombinationOfGates with this gate.

Args
coefficient number coefficient to use in the resulting cirq.LinearCombinationOfGates object.

Returns
cirq.LinearCombinationOfGates containing self with a coefficient of coefficient.

View source

### __call__

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Call self as a function.

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### __truediv__

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