# cirq.HPowGate¶

class cirq.HPowGate(*, exponent: Union[float, sympy.core.basic.Basic] = 1.0, global_shift: float = 0.0)[source]

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

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))]]


where

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.

__init__(*, exponent: Union[float, sympy.core.basic.Basic] = 1.0, global_shift: float = 0.0) → None

Initializes the parameters used to compute the gate’s matrix.

The eigenvalue of each eigenspace of a gate is computed by

1. Starting with an angle in half turns as returned by the gate’s
_eigen_components method:
θ

2. Shifting the angle by global_shift:

θ + s

3. Scaling the angle by exponent:

(θ + s) * e

4. Converting from half turns to a complex number on the unit circle:

exp(i * pi * (θ + s) * e)

Parameters
• 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.