Create `qml.PauliError`

Feature details

As requested in #971, we would like to add a channel for Pauli errors.

See the following references:

• https://arxiv.org/pdf/1907.12976.pdf
• https://arxiv.org/pdf/2101.02109.pdf

See the addition of a Reset error as an example.

Implementation

List of tasks for completion:

A.

1. Add a new qml.PauliError class to pennylane/ops/channel.py that inherits from Channel and defines the Kraus operators.
2. Add an entry to the doc/introduction/operations.rst file for the new operation
3. Add tests for the new channel in the tests/ops/test_channel_ops.py file.

B.

1. Add support for qml.PauliError in the default.mixed device by adding it to the list of supported operations.
2. Add tests for checking the Kraus operators of the channel and for applying the new channel using default.mixed.

How important would you say this feature is?

1: Not important. Would be nice to have.

Additional information

No response

[ingstra]

@MoritzWillmann sure! I’m not working on it.

[antalszava]

Hi @MoritzWillmann, awesome, I’ll have a go at addressing them. 🙂 Note that in general we have some flexibility here.

When you say “Pauli Error acting on an arbitrary number of qubits”, does that mean the Kraus Matrices “only” need to be upscaled to act on the according state vector?

What I had in mind there is that we could specify a Pauli string, the wires the error acts on and the corresponding probability , e.g., “XYZ”, wires=[0,1,2] and p=0.1 for example.

In such a case, with probability 0.1 we’ll apply the following

qml.PauliX(0)
qml.PauliY(1)
qml.PauliZ(2)

and with a probability of 0.9 the state is reserved.

One step further, we could define 3 lists:

1. a list of Pauli-strings;
2. a list of wires;
3. a list of probabilities.

For example, qml.PauliError(["XX","YY","ZZ"],[ [0,1], [1,2], [2,0]],[0.1, 0.2, 0.3]), such that

1. With probability 0.1 we’ll apply

qml.PauliX(0)
qml.PauliX(1)

with probability 0.2 we’ll apply

qml.PauliY(1)
qml.PauliY(2)

with probability 0.3 we’ll apply the following

qml.PauliZ(2)
qml.PauliZ(0)

and with a probability 0.4 the state is preserved.

Constructing the Kraus matrices will likely be computationally heavy when we have many wires.

We could then express your example as:

• qml.PauliError("X", [0], p) which has the same syntax as a bitflip essentially or;
• qml.PauliError("XI", [0,1], p), provided that we allow identities in the Pauli string.

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Should there be a possibility to choose which Pauli Error channel is applied or should all 4 Kraus Matrices be returned?

Not quite sure about this one, what would “choose which Pauli Error channel is applied” mean?