Demonstrate local simulation in README

README.md

@@ -14,7 +14,7 @@ pip install qiskit-alice-bob-provider
 `pip` will handle installing all the python dependencies automatically and you
 will always install the  latest (and well-tested) version.
 
-## Use your Alice & Bob API key
+## Remote execution on Alice & Bob QPUs: use your API key
 
 To obtain an API key, please [contact Alice & Bob](https://alice-bob.com/contact/).
 
@@ -29,7 +29,7 @@ Where `MY_API_KEY` is your API key to the Alice & Bob API.
 
 ```python
 print(ab.backends())
-backend = ab.get_backend('SINGLE_CAT_SIMULATOR')
+backend = ab.get_backend('SINGLE_CAT')
 ```
 
 The backend can then be used like a regular Qiskit backend:
@@ -45,3 +45,61 @@ job = execute(c, backend)
 res = job.result()
 print(res.get_counts())
 ```
+
+## Local simulation of cat quit processors
+
+This project contains multiple simulators of multi cat qubit processors.
+
+```python
+from qiskit_alice_bob_provider import AliceBobLocalProvider
+from qiskit import QuantumCircuit, execute, transpile
+
+provider = AliceBobLocalProvider()
+print(provider.backends())
+# PHYSICAL_CATS_6, PHYSICAL_CATS_40, LESCANNE_2020
+```
+
+The `PHYSICAL_CATS` backends are theoretical models of quantum processors made
+up of physical cat qubits.
+They can be used to study the properties of error correction codes implemented
+with physical cat qubits, for different hardware performance levels
+(see the parameters of class `PhysicalCatProcessor`).
+
+The `LESCANNE_2020` backend is an interpolated model simulating the processor
+used in the [seminal paper](https://arxiv.org/pdf/1907.11729.pdf) by Raphaël
+Lescanne in 2020.
+This interpolated model is configured to act as a digital twin of the cat qubit
+used in this paper.
+It does not represent the current performance of Alice & Bob's cat qubits.
+
+The example below schedules and simulates a Bell state preparation circuit on
+a `PHYSICAL_CATS_6` processor, for different values of parameters `alpha` and
+`kappa_2`.
+
+```python
+from qiskit_alice_bob_provider import AliceBobLocalProvider
+from qiskit import QuantumCircuit, execute, transpile
+
+provider = AliceBobLocalProvider()
+
+circ = QuantumCircuit(2, 2)
+circ.initialize('0+')
+circ.cx(0, 1)
+circ.measure(0, 0)
+circ.measure(1, 1)
+
+# Default 6-qubit QPU with the ratio of memory dissipation rates set to
+# k1/k2=1e-5 and cat amplitude alpha set to 4.
+backend = provider.get_backend('PHYSICAL_CATS_6')
+
+print(transpile(circ, backend).draw())
+# *Displays a timed and scheduled circuit*
+
+print(execute(circ, backend, shots=100000).result().get_counts())
+# {'11': 49823, '00': 50177}
+
+# Changing the cat amplitude from 4 (default) to 2 and k1/k2 to 1e-2.
+backend = provider.get_backend('PHYSICAL_CATS_6', alpha=2, kappa_2=1e4)
+print(execute(circ, backend, shots=100000).result().get_counts())
+# {'01': 557, '11': 49422, '10': 596, '00': 49425}
+```