# What happens when I use PEC?#

When applying PEC with Mitiq, the workflow of quantum and classical data is represented in the figure below.

• The user provides a QPROGRAM, (i.e. a quantum circuit defined via any of the supported frontends).

• Mitiq probabilistically generates a list of auxiliary circuits. This step depends on the quasi-probability representations provided by the user.

• The generated circuits are executed via a user-defined Executor.

• Mitiq infers an unbiased estimate of the ideal expectation value from a classical post-processing of the measured data.

• The error mitigated expectation value is returned to the user.

The general workflow of PEC is similar to the workflow of ZNE. The main difference is that in ZNE the auxiliary circuits are obtained by noise scaling, while in PEC they are probabilistically generated. As a consequence, the final inference step is different too. The inference step of PEC is based on the Monte Carlo estimation protocol discussed in What is the theory behind PEC?

As shown in the Section How do I use PEC?, the standard way of applying PEC in Mitiq is based on the function execute_with_pec(). However, one may be interested in applying PEC with a lower-level of abstraction for reasons of efficiency or for customization proposes. In particular, it could be useful to explicitly split the PEC process into the two main steps shown in the figure above:

• Step 1: Probabilistic generation of all the auxiliary circuits;

• Step 2: Inference of the ideal expectation value from the noisy execution of the auxiliary circuits.

## Step 1: Probabilistic generation of all the auxiliary circuits#

### Define the circuit of interest and the quasi-probability representations#

We define the circuit of interest as we did in How do I use PEC?

import mitiq
from mitiq import benchmarks

frontend = "qiskit"  # Supported: "cirq", "qiskit", "pyquil", "braket", "pennylane".

circuit = benchmarks.generate_rb_circuits(
n_qubits=1, num_cliffords=2, return_type = frontend,
)[0]
print(circuit)

   ┌──────────┐┌────┐┌─────────┐┌──────────┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└─────────┘└──────────┘└────┘└───┘└─────────┘


We assume local depolarizing noise and we define the list of OperationRepresentations (one for each gate) as we did in How do I use PEC?.

from mitiq.pec.representations.depolarizing import represent_operations_in_circuit_with_local_depolarizing_noise

noise_level = 0.01
reps = represent_operations_in_circuit_with_local_depolarizing_noise(circuit, noise_level)
print(f"{len(reps)} OperationRepresentation objects produced, assuming {noise_level :.2%} depolarizing noise.")

4 OperationRepresentation objects produced, assuming 1.00% depolarizing noise.


Each element of reps is an OperationRepresentation object that represents an ideal gate $$\mathcal G_i$$ of circuit as a linear combination of noisy operations $$\{\mathcal O_{i, \alpha} \}$$:

$\mathcal G_i = \sum_\alpha \eta_{i, \alpha} \mathcal O_{i, \alpha}, \quad \sum_\alpha \eta_{i, \alpha} = 1, \quad \sum_\alpha |\eta_{i,\alpha}|=\gamma_i,$

where $$\{\eta_{i, \alpha}\}$$ is a real quasi-probability distribution with one-norm $$\gamma_i$$.

For example, the first OperationRepresentation in reps is:

print(reps[0])

q_0: ───Ry(-0.5π)─── = 1.010*(q_0: ───Ry(-0.5π)───)-0.003*(q_0: ───Ry(-0.5π)───X───)-0.003*(q_0: ───Ry(-0.5π)───Y───)-0.003*(q_0: ───Ry(-0.5π)───Z───)


### Probabilistic generation of all the auxiliary circuits#

According to the theory of PEC, to probabilistically generate an auxiliary circuit one should:

• Define an empty sampled_circuit to be populated with probabilistic operations $$\mathcal O_{j,\alpha}$$;

• Define an empty gate_sign_list to be populated with the sign values $${\rm sgn}(\eta_{j,\alpha})$$;

• Start a loop over the ideal operations of circuit;

• For each gate of circuit, search for the corresponding OperationRepresentation in reps;

• Sample a noisy gate from the quasi-probability distribution of the ideal gate using rep[j].sample() as shown in What additional options are available in PEC?;

• Append the sampled gate to sampled_circuit and the corresponding sign to gate_sign_list;

• When the loop is completed, return sampled_circuit and the associated sampled_sign (which is the product of all the elements of gate_sign_list).

Instead of manually applying all the previous steps, one can call the Mitiq function sample_circuit() to probabilistically generate an integer number num_samples of auxiliary circuits.

from mitiq import pec

sampled_circuits, sampled_signs, one_norm = pec.sample_circuit(circuit, reps, num_samples=5, random_state=30)
for circuit in sampled_circuits:
print(circuit)
print("Signs:", sampled_signs)
print("One-norm:", one_norm)

   ┌──────────┐┌────┐┌─────────┐┌──────────┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└─────────┘└──────────┘└────┘└───┘└─────────┘
┌──────────┐┌────┐┌───┐┌─────────┐┌──────────┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└───┘└─────────┘└──────────┘└────┘└───┘└─────────┘
┌──────────┐┌────┐┌─────────┐┌──────────┐┌───┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ Y ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└─────────┘└──────────┘└───┘└────┘└───┘└─────────┘
┌──────────┐┌────┐┌─────────┐┌──────────┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└─────────┘└──────────┘└────┘└───┘└─────────┘
┌──────────┐┌────┐┌─────────┐┌──────────┐┌────┐┌───┐┌─────────┐
q: ┤ Ry(-π/2) ├┤ √X ├┤ Ry(π/2) ├┤ Ry(-π/2) ├┤ √X ├┤ X ├┤ Ry(π/2) ├
└──────────┘└────┘└─────────┘└──────────┘└────┘└───┘└─────────┘
Signs: [1, -1, -1, 1, 1]
One-norm: 1.1508179395702227


The above result contains:

1. The list of num_samples auxiliary circuits that have been probabilistically generated according to the quasi-probability distributions in reps.

2. the list of the sampled signs ($$\pm 1$$) associated to each auxiliary circuit.

3. The one-norm $$\gamma$$ of the global quasi-probability distribution of the full circuit.

Let us increase the value of num_samples and unpack the result in three different variables:

sampled_circuits, sampled_signs, one_norm = pec.sample_circuit(
circuit, reps, num_samples=200,
)


We are now ready for the second step of PEC, i.e., the inference of the ideal expectation value from the noisy execution of the sampled_circuits.

## Step 2: Inference of the ideal expectation value from the noisy execution of the auxiliary circuits#

### Noisy execution of all the auxiliary circuits#

To execute the auxiliary circuits we define the same execute function that we used in the Section “How do I use PEC?”. Actually, since we have a list of circuits to execute, we define a batched version of the function, i.e., a function that takes a list of circuits as input and returns a list of expectation values. More details on batched executors can be found in the “Executors” section.

from typing import List

from cirq import DensityMatrixSimulator, depolarize
from mitiq.interface import convert_to_mitiq

def batched_execute(circuits: List[mitiq.QPROGRAM], noise_level: float=0.01)->List[float]:
"""Returns [Tr[ρ_1 |0⟩⟨0|], Tr[ρ_2 |0⟩⟨0|]... ] where ρ_j is the state prepared by the
j_th circuit in the input argument "circuits".
"""
# Replace with code based on your frontend and backend, possibly using a batched execution.
expectation_values = []
for circuit in circuits:
mitiq_circuit, _ = convert_to_mitiq(circuit)
noisy_circuit = mitiq_circuit.with_noise(depolarize(p=noise_level))
rho = DensityMatrixSimulator().simulate(noisy_circuit).final_density_matrix
expectation_values.append(rho[0, 0].real)
return expectation_values


We also initialize a Mitiq Executor object. This step is optional but recommended to store/access executed circuits and results.

executor = mitiq.Executor(batched_execute, max_batch_size=100)


We execute all the auxiliary circuits generated in the previous section to obtain a list of noisy expectation values.

noisy_expecation_values = executor.evaluate(
sampled_circuits,
force_run_all=False,  # Set True if shot noise is present in quantum results.
)

# Unique noisy expectation values associated to unique circuits in sampled_circuits
executor.quantum_results

[0.9551589,
0.9431021,
0.05090983,
0.9431021,
0.05090987,
0.05090987,
0.05090983,
0.94909006,
0.9490901,
0.05090987,
0.05090987]


Since expectation values are evaluated from the simulation of an exact density matrix, we executed equal circuits only once by setting force_run_all=False.

Note: If shot noise is present, one should instead use force_run_all=True to ensure the statistical independence of the quantum results.

print(f"{len(noisy_expecation_values)} noisy expectation values efficiently evaluated by executing only {len(executor.quantum_results)} unique circuits.")

200 noisy expectation values efficiently evaluated by executing only 11 unique circuits.


Equivalently, but less efficiently, we could have done as follows:

noisy_expecation_values_direct = batched_execute(sampled_circuits)

assert noisy_expecation_values_direct == noisy_expecation_values


### Estimate the ideal expectation value from the noisy results#

According to the theory of PEC, the ideal expectation value can be estimated as an average of the noisy auxiliary expectation values, after scaling them by the corresponding sampled_signs and by the one_norm coefficient (obtained in the previous section).

# Scale noisy results by one-norm coefficient and by sampled signs
unbiased_samples = [
]

# Estimate ideal expectation value
pec_value = sum(unbiased_samples) / len(unbiased_samples)

unmitigated_value = executor.evaluate(circuit)[0]

print(f"Expectation value without error mitigation:    {unmitigated_value}")
print(f"Expectation value with error mitigation (PEC): {pec_value}")

Expectation value without error mitigation:    0.9551588892936707
Expectation value with error mitigation (PEC): 1.0198914969730744


To obtain a better PEC estimation (with reduced statistical fluctuations) one should increase the number of probabilistically generated circuits, i.e., one should increase the argument num_samples when calling the sample_circuit() function.