docs: write standard rb docs

doc/source/protocols/index.rst

@@ -30,4 +30,5 @@ In this section we introduce the basics of all protocols supported by ``qibocal`
     readout_mitigation_matrix
     avoided_crossing
     readout_optimization
+    standard_rb
     references

doc/source/protocols/standard_rb.rst

@@ -0,0 +1,85 @@
+Standard Randomize Benchmarking
+===============================
+
+An approach to obtain the average gate fidelity is to perform randomized 
+benchmarking :cite:p:`Emerson_2005`.
+The key idea is that if we average the error process over the uniform space of 
+unitaries the result is a depolarizing channel that maps any pure state to the 
+maximally mixed state. 
+Such uniform space of unitaries is known as *Haar measure*.
+It can be shown :cite:p:`Emerson_2005` that the average induced error is proportional
+to the depolarization probability.
+However, this approach is inefficient because we sample randomly from the Haar measure.
+A simplification was proposed in :cite:p:`Knill2008` by restricting the unitaries 
+to the Clifford group, which consists of unitary rotations mapping the group 
+of Pauli operators in itself.
+Among the advantages of such group are the fact of the number of Clifford 
+gates is finite given the Hilbert space and being a group we can easily found 
+the inverse within the group.
+The generic procedure to perform a randomized benchmarking is the following:
+
+1. initialize the system in ground state
+2. for each sequence length :math:`m` draw sequence of Clifford group elements
+3. calculate inverse gate
+4. measure sequence and inverse gate
+5. repeat the process for multiple sequence of same length and varying the length
+
+The previous approach works because it has been shown :cite:p:`Nielsen_2002` that 
+randomization with Clifford gates provides again a depolarized noise channel
+
+.. math::
+    :name: eq:1
+
+    \rho \rightarrow \frac{d}{2} I + ( 1 - d) \rho
+
+with depolarization probability :math:`d`.
+If we follow the previous procedure and we measure the survival probability, i.e. 
+the probability of measuring the qubit in :math:`\ket{0}`, for
+different sequence length :math:`m` we expect the following behavior
+
+.. math::
+    :name: eq:2
+
+    F(m) = A p^m + B
+
+where :math:`1-p` is the rate of depolarization while :math:`A` and :math:`B` 
+capture state preparation and measurement errors.
+Finally, we can extract the average error per Clifford as
+
+.. math::
+    :name: eq:3
+
+    \epsilon_\text{Clifford} = \frac{1 - p}{1 - 2^{-n}}
+
+where :math:`n` is the number of qubits. The error per gate can be derived by dividing
+the Clifford error by the physical gates per Clifford which usually is 1.875.
+One of the main feature of RB is the possibility to estimate the gate fidelity 
+alone without taking into account both state preparation and measurement errors
+which can be computed using the :math:`A` and :math:`B` terms in :ref:`Eq. 2 <eq:2>`.
+
+Parameters
+^^^^^^^^^^
+
+
+.. autoclass::
+	qibocal.protocols.randomized_benchmarking.standard_rb.StandardRBParameters
+	:noindex:
+
+
+Example
+^^^^^^^
+
+It follows a runcard where we execute a standard RB.
+
+.. code-block:: yaml
+
+    - id: standard rb
+      operation: standard_rb
+      parameters:
+        depths: [1,5,10,20,50,100]
+        niter: 20
+        nshots: 100
+
+The expected output is the following:
+
+.. image:: standard_rb.png

doc/source/refs.bib

@@ -150,3 +150,43 @@ @misc{pedicillo2023
       archivePrefix={arXiv},
       primaryClass={id='quant-ph' full_name='Quantum Physics' is_active=True alt_name=None in_archive='quant-ph' is_general=False description=None}
 }
+
+@article{Knill2008,
+   title={Randomized benchmarking of quantum gates},
+   volume={77},
+   ISSN={1094-1622},
+   url={http://dx.doi.org/10.1103/PhysRevA.77.012307},
+   DOI={10.1103/physreva.77.012307},
+   number={1},
+   journal={Physical Review A},
+   publisher={American Physical Society (APS)},
+   author={Knill, E. and Leibfried, D. and Reichle, R. and Britton, J. and Blakestad, R. B. and Jost, J. D. and Langer, C. and Ozeri, R. and Seidelin, S. and Wineland, D. J.},
+   year={2008},
+   month={jan},
+}
+
+@article{Emerson_2005,
+   title={Scalable noise estimation with random unitary operators},
+   volume={7},
+   ISSN={1741-3575},
+   url={http://dx.doi.org/10.1088/1464-4266/7/10/021},
+   DOI={10.1088/1464-4266/7/10/021},
+   number={10},
+   journal={Journal of Optics B: Quantum and Semiclassical Optics},
+   publisher={IOP Publishing},
+   author={Emerson, Joseph and Alicki, Robert and Życzkowski, Karol},
+   year={2005},
+   month=sep, pages={S347–S352} }
+
+@article{Nielsen_2002,
+   title={A simple formula for the average gate fidelity of a quantum dynamical operation},
+   volume={303},
+   ISSN={0375-9601},
+   url={http://dx.doi.org/10.1016/S0375-9601(02)01272-0},
+   DOI={10.1016/s0375-9601(02)01272-0},
+   number={4},
+   journal={Physics Letters A},
+   publisher={Elsevier BV},
+   author={Nielsen, Michael A},
+   year={2002},
+   month=oct, pages={249–252} }