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

Inherits From: EigenGate, SingleQubitGate, SupportsOnEachGate, Gate

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

[[g·(c-i·s/sqrt(2)), -i·g·s/sqrt(2)],
[-i·g·s/sqrt(2)], g·(c+i·s/sqrt(2))]]


c = cos(π·t/2)
s = sin(π·t/2)
g = exp(i·π·t/2).

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

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 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.





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Returns a controlled version of this gate. If no arguments are specified, defaults to a single qubit control.

num_controls: Total number of control qubits. control_values: For which control qubit values to apply the sub gate. A sequence of length num_controls where each entry is an integer (or set of integers) corresponding to the qubit value (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.


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The number of qubits this gate acts on.


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Returns an application of this gate to the given qubits.

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


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

*targets The qubits to apply this gate to.

Operations applying this gate to the target qubits.

ValueError if targets are not instances of Qid or List[Qid]. ValueError if the gate operates on two or more Qids.


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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.

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


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


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Call self as a function.


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