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:

$$\rho \rightarrow \sum_k A_k \rho A_k^\dagger$$

Where A_k are Krauss operators. These operators are not necessarily unitary and must satisfy the trace-preserving property:

$$\sum_k A_k^\dagger A_k = I$$

As a noisy channel, Krauss 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 Krauss 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:

$$\rho \rightarrow \sum_k p_k U_k \rho U_k^\dagger {\rm ~where~} \sum_k p_k =1 {\rm ~and~ U_k U_k^\dagger= I}$$

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.

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:

$$ \rho \rightarrow (1-p_x-p_y-p_z) \rho + p_x X \rho X + p_y Y \rho Y + p_z Z \rho Z $$

Here p_x is the probability that the X Pauli gate is applied and no other gate is applied, and similarly for p_y and p_z.

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:

$$ \begin{aligned} \rho \rightarrow& \sum_{k=0}^3 M_k \rho M_k \newline M_0 =& \sqrt{p} \begin{bmatrix} 1 & 0 \cr 0 & \sqrt{1 - \gamma} \end{bmatrix} \newline M_1 =& \sqrt{p} \begin{bmatrix} 0 & \sqrt{\gamma} \cr 0 & 0 \end{bmatrix} \newline M_2 =& \sqrt{1-p} \begin{bmatrix} \sqrt{1-\gamma} & 0 \cr 0 & 1 \\ \end{bmatrix} \newline M_3 =& \sqrt{1-p} \begin{bmatrix} 0 & 0 \cr \sqrt{\gamma} & 0 \end{bmatrix} \end{aligned} $$

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.