# Noise¶

For simulation, it is useful to have `Gate`

objects that enact noisy quantum evolution.

Cirq supports modeling noise via *operator sum* representations of noise (these evolutions are also known as quantum operations, quantum dynamical maps, or superoperators).

This formalism models evolution of the density matrix via:

Where Ak are *Kraus* operators. These operators are not necessarily unitary and must satisfy the trace-preserving property:

As a noisy channel, Kraus operators are not unique. For more details of these operators, see John Preskill’s notes.

## Magic methods¶

A `Gate`

can represent an operator sum representation by supporting the `channel`

protocol. Alternatively, for channels that represent probabilistic mixtures of unitaries, one can implement the `mixture`

protocol.

### cirq.channel and def *channel*¶

To represent an operator sum evolution, a `Gate`

should implement the `SupportsChannel`

protocol. To do this, the `Gate`

should implement the `_channel_(self) -> Sequence[np.ndarray]:`

method.

This method should return the sequence of `numpy`

matrices corresponding to the Kraus operators. The basis in which this matrix is expressed is always implicit with respect to the object being called.

For example, in `GateOperations`

, these matrices must be ordered with respect to the list of qubits that the channel is applied to. The qubit-to-amplitude order mapping matches the ordering of `numpy.kron(A, B)`

, where `A`

is a qubit earlier in the list than the qubit `B`

.

If one has defined `_channel_`

, then that `Gate`

and any `GateOperation`

that uses that gate can be used as an argument to `cirq.channel`

and `cirq.channel`

will return this sequence of matrices.

Besides objects that support `_channel_`

, `cirq.channel`

will also fall back to other objects that can be interpreted as channels. For example, if a channel is a probabilistic mixture of unitary gates (see below), then `cirq.channel`

will fall back to seeing if the object supports `_mixture_`

. If `_mixture_`

is not supported, then `cirq.channel`

checks to see if `_unitary_`

is supported.

In addition to supporting `_channel_`

, objects that are channels should also implement `_has_channel_(self) -> bool`

to return `True`

. This method is used to determine whether an object has a `_channel_`

or not without having to do the potentially expensive creation of the matrices for the channel.

### cirq.mixture and def *mixture*¶

Some channels can be interpreted as probabilistically selecting between different unitary evolutions:

In this case, it is possible to perform **Monte Carlo simulations** of these gates using a wave function based simulator (and not a density matrix based simulator).

Instead of implementing the `SupportsChannel`

protocol, one should implement the `SupportsMixture`

protocol. To do this, one should implement the `_mixture_(self) -> Sequence[Tuple[float, np.ndarray]]`

protocol. This returns a sequence of tuples.

The first element of each tuple is the probability of the unitary, and the second element is the unitary. Like the `_channel_`

method described above, the basis for these matrices is implicit with respect to the object being called. One should also make `_has_mixture_`

return `True`

to indicate to callers that the object supports the mixture protocol.

If one wants to get the mixture channel directly, one can call `cirq.mixture`

.

## Common Channels¶

Cirq supports many commonly used quantum channels out of the box, see ``ops/common_channels.py`

<https://github.com/quantumlib/Cirq/blob/master/cirq/ops/common_channels.py>`__.

### AsymmetricDepolarizingChannel, DepolarizingChannel, BitFlipChannel, and PhaseFlipChannel¶

The asymmetric depolarizing channel represents probabilistically selecting one of three Pauli gates to apply or doing nothing to the state. This is implemented via a `_mixture_`

method so that a Monte Carlo simulation with a wave function simulator can be used.

This channel implements the evolution:

Here px is the probability that the X Pauli gate is applied and no other gate is applied, and similarly for py and pz.

A particular case of the asymmetric depolarizing channel is the case when each of the different Paulis occur with the same probability. This is encapsulated in the `DepolarizingChannel`

gate, which takes a probability `p`

such that each Pauli gate occurs with probability `p/3`

.

To construct channels, useful helpers are provided `cirq.asymmetric_depolarize`

and `cirq.depolarize`

.

Another common case is when only a Pauli X (bit flip) can occur, or when only a Pauli Y (phase flip) can occur. These correspond to `BitFlipChannel`

and `PhaseFlipChannel`

with helpers `cirq.bit_flip`

and `cirq.phase_flip`

.

### GeneralizedAmplitudeDampingChannel and AmplitudeDampingChannel¶

The generalized amplitude damping channel models the effect of energy dissipation to a surrounding environment as well as dephasing that does not exchange energy. The amplitude damping channel only models dissipation of energy to a surrounding environment.

Cirq has implementations of both of these channels. The generalized amplitude damping channel corresponds to:

Where γ is the probability of the interaction being dissipative, and `p`

is the probability that the qubit and environment exchange energy. The amplitude damping channel corresponds to `p=1`

.

Cirq provides the helpers `cirq.generalized_amplitude_damp`

and `cirq.amplitude_damp`

to construct these noisy gates.