Question about running the find_optimal_representation method in the PEC module

[eshaspark]

I have a question about using the PEC module of mitiq:

In what form does one input a basis of noisy operations for the Quasiprobability decomposition?

I'm looking to run the find_optimal_representation method which decomposes ideal operations to a basis of noisy operations. The basis I'd like consists of a bunch of unitaries and post selection operations. The documentation mentions the function:

mitiq.pec.representations.optimal.find_optimal_representation(
    ideal_operation,
    noisy_operations,
    tol=1e-08,
    initial_guess=None,
    is_qubit_dependent=True,
)

Do the noisy_operations have to be initialized as quantum circuits? (I'm using qiskit frontend)

I tried that and it doesn't seem to be working. Also, I'd like to use post-selection operations so numerical matrices are preferred, for the noisy operation parameter. I can see there's an option to input a superoperator but it isn't clear to me whether that's sufficient.

[andreamari]

Hi @eshaspark,
to use the function find_optimal_representation you need to define, as a basis, a list of NoisyOperation objects. Each NoisyOperation object is defined by the underlying circuit (e.g. a Qiskit or Cirq circuit) and the associated channel_matrix (the superoprator as a numerical matrix).

You can find an example on how define noisy operations at this link.
You can find an example no how to use find_optimal_representation at this link.

In mitiq we assume that noisy operations are noisy versions of ideal unitary gates, so post-selection operations are not supported in principle. However, you may try to define noisy operations with post-selection gates, an see if you get an error in the process.

Alternatively, if you have the numerical super-operator matrices and you are just interested in solving the underlying optimization problem from a numerical perspective (i.e. no circuits, no NoisyOperations objects, etc..), you may have a look at the function minimize_one_norm . More info at this link.

[eshaspark]

Hi @andreamari,

Thanks for the explainer and the links. When defining the NoisyOperation object, I see one needs the underlying (Qiskit or cirq) circuit as well as numerical superoperator matrices. Defining post selections as superoperator matrices is straightforward ✅ .
It seems to implement PEC in mitiq (not just for solving the optimization problem from a numerical perspective, but to also run PEC), this superoperator matrix isn't enough and one needs the circuits for the noisy operations as well.

I want to understand how to include underlying basis circuits for post-selections. In Qiskit, the measurement + reset gate wouldn't achieve the same as it would reset all measurements to 0, regardless of the outcome. There aren't native post-selection gates in Qiskit as it technically isn't a gate, but rather the fact that certain measurement outcomes are discarded. Is there a way to include an operation like this to run PEC with mitiq?

[andreamari]

You are right. If you aim to implement PEC, the NoisyOperation objects must be applicable in a circuit as actual real gates on real qubits.
So, if the specific operations you are interested in are not implementable in Qiskit, I think there is no way to apply PEC with or without Mitiq.

About Qiskit conditional operations: you might be interested in this method. It allows to apply gates based on a previous measurement outcome. Likely this kind of operations won't be compatible with Mitiq due to conversion errors, but at least you may find a way to manually implement your noisy operations in a Qiskit circuit.

[eshaspark]

Ok, thanks for confirming that post-selections are currently not implementable on mitiq!

There is a second issue I encounter with find_optimal_representation method - when inputing circuits to the method. This issue is related to what you mean by 'real gates'. There are operations that are implementable on qiskit as gates in software, but are mitiq is unable to decompose these circuits when they are input to PEC. Pl refer to the screenshot below.

The only qiskit circuit inputs that are being accepted by mitiq without error involve native gates: h, cx, and so on. When I define an arbitary unitary and then create a Quantum Gate object from this unitary on qiskit, the input is not being accepted by mitiq. So it seems the PEC simulation can't be performed for arbitrary unitary circuits on qiskit (without circuit synthesis)? The numerical minimize_one_norm method works fine, but I'd like to be able to run a simulation after computing the coefficients.

[andreamari]

You are right. Custom gates such as those defined by numerical matrices are not supported in Mitiq. So one can apply PEC only if the circuit is already decomposed into native gates or at least standard gates such as H, S, T, CZ, etc... .