Choi representation #4132
Choi representation #4132
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It seems like we may be able to use this to speed up density matrix simulations, since (if I'm reading this correctly) the current implementation must apply each Kraus operator to the state independently.
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@95-martin-orion Each way of representing a channel has its own way of applying it to a density matrix and it's quite likely their costs are different. Also, the cost depends on the channel, e.g. applying unitary/isometric channels using a Kraus representation is probably fairly fast. Applying a channel using its Choi matrix involves transposing the density matrix, tensoring it with identity, multiplying by the Choi and tracing out the input space. It's not immediately clear that this is cheaper. However, a follow-up PR to this is a matrix representation of a channel (aka "the natural representation" in John Watrous' book). In this representation, applying a channel to a density matrix is just matrix multiplication (though the matrices involved are larger). It's quite possible that this would be the fastest way in the worst case. |
cirq-core/cirq/qis/channels.py
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| def choi(operation: 'protocols.SupportsChannel') -> np.ndarray: | ||
| """Returns the unique Choi matrix associated with a superoperator.""" | ||
| ks = protocols.channel(operation) | ||
| d = np.prod(ks[0].shape) |
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Should we extract a kraus_to_choi here?
cirq-core/cirq/__init__.py
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| @@ -330,6 +330,7 @@ | |||
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| from cirq.qis import ( | |||
| bloch_vector_from_state_vector, | |||
| choi, | |||
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Should this be a more specific name here to avoid future naming conflict? operation_to_choi or something?
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Good point, it does appear we might have a protocol named choi in near future.
This implements a simple utility for computing the Choi matrix of a channel from its Kraus representation.
Choi representation is more compact than Kraus representations and - unlike the Kraus representation - it is unique for every channel, so we can use this to e.g. compare channels.
The PR is a small step towards #3248. In particular, it does not introduce any new protocols. As discussed on the issue, we might at some point want a protocol to enable channel classes to implement something like
__choi__if there are cases where folks would prefer to specify channels using their Choi matrix rather than the Kraus operators. If we end up doing this, I suppose the protocol will use the utility in this PR to handle channels that provide Kraus operators.