# cirq.EigenGate

A gate with a known eigendecomposition.

Inherits From: `Gate`

EigenGate is particularly useful when one wishes for different parts of the same eigenspace to be extrapolated differently. For example, if a gate has a 2-dimensional eigenspace with eigenvalue -1, but one wishes for the square root of the gate to split this eigenspace into a part with eigenvalue i and a part with eigenvalue -i, then EigenGate allows this functionality to be unambiguously specified via the _eigen_components method.

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)
``````

`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 probabilistic 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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