Finish draft of OBP intro

docs/guides/qiskit-addon-obp.mdx

@@ -1,43 +0,0 @@
-# Operator backpropagation (OBP)
-
-Operator backpropagation (OBP) is a technique to reduce circuit depth by trimming operations from its end at the cost of more operator measurements. There are a number of ways in which operator backpropagation can be performed, and this package uses a method based on Clifford perturbation theory.
-
-As one propagates an operator further through a circuit, the size of the additional obervable needed for measurment grows exponentially. This results in both a classical and quantum resource overhead. However, for some circuits, the resulting distribution of additional Pauli observables is more concentrated than the worst-case exponential scaling. This implies that some terms in an observable which small coefficients can be truncated to reduce the quantum overhead. The error incurred by doing so can be controlled to find a suitable tradeoff between precision and efficiency.
-
-## Installation
-
-There are two ways to install the OBP package. Via PyPI and building from source. We also recommend installing these packages in a [virtual environment](https://docs.python.org/3.10/tutorial/venv.html) to ensure separation between package dependencies.
-
-### Install from PyPI
-
-The most straightforward way to install the `qiskit-addon-obp` package is via PyPI.
-
-```bash
-pip install qiskit-addon-obp
-```
-
-### Build from source
-
-Users who wish to contribute to this package or who want to install it manually may do so by first cloning the repository
-
-```bash
-git clone git@github.com:Qiskit/qiskit-addon-obp.git
-```
-
-and install the package via `pip`. The repository also contains example notebooks which can be run. And if you plan on developing in the repository, you may want to also install the `dev` dependencies.
-
-Adjust the options to suit your needs
-
-```bash
-pip install tox notebook -e '.[notebook-dependencies, dev]'
-```
-
-## Theoretical background
-
-When using the Estimator primitive, the output of a quantum workload is the estimation of one or more expectation values $\langle O \rangle$ with respect to some state prepared using a QPU. Mathematically this can be written as
-
-$$ \langle O \rangle_{U|\psi\rangle} = \langle\psi | U^\dagger O U |\psi \rangle $$
-
-where $\langle O \rangle$ is the expectation value to estimate, $U$ is the circuit to execute, and \psi is some initial state.
-
-To distribute this problem across both classical and quantum resources, we can split the circuit $U$ into two subcircuits, $U_C$ and $U_Q$. The subcircuit $U_C$ is selected

docs/guides/qiskit-addons-obp.mdx

@@ -0,0 +1,90 @@
+# Operator backpropagation (OBP)
+
+Operator backpropagation (OBP) is a technique to reduce circuit depth by trimming operations from its end at the cost of more operator measurements. There are a number of ways in which operator backpropagation can be performed, and this package uses a method based on Clifford perturbation theory.
+
+As one propagates an operator further through a circuit, the size of the additional obervable needed for measurment grows exponentially. This results in both a classical and quantum resource overhead. However, for some circuits, the resulting distribution of additional Pauli observables is more concentrated than the worst-case exponential scaling. This implies that some terms in an observable which small coefficients can be truncated to reduce the quantum overhead. The error incurred by doing so can be controlled to find a suitable tradeoff between precision and efficiency.
+
+## Installation
+
+There are two ways to install the OBP package. Via PyPI and building from source. We also recommend installing these packages in a [virtual environment](https://docs.python.org/3.10/tutorial/venv.html) to ensure separation between package dependencies.
+
+### Install from PyPI
+
+The most straightforward way to install the `qiskit-addon-obp` package is via PyPI.
+
+```bash
+pip install qiskit-addon-obp
+```
+
+### Build from source
+
+Users who wish to contribute to this package or who want to install it manually may do so by first cloning the repository
+
+```bash
+git clone git@github.com:Qiskit/qiskit-addon-obp.git
+```_
+
+and install the package via `pip`. The repository also contains example notebooks which can be run. And if you plan on developing in the repository, you may want to also install the `dev` dependencies.
+
+Adjust the options to suit your needs
+
+```bash
+pip install tox notebook -e '.[notebook-dependencies, dev]'
+```
+
+## Theoretical background
+
+When using the Estimator primitive, the output of a quantum workload is the estimation of one or more expectation values $\langle O \rangle$ with respect to some state prepared using a QPU. In this section we will summarize the procedure.
+
+We first start by writing the expectation value measurement of an observable $O$ in terms of some initial state $|\psi\rangle$ and a quantum circuit $U_Q$:
+
+$$ \langle O \rangle_{U|\psi\rangle} = \langle\psi | U^\dagger O U |\psi \rangle. $$
+
+To distribute this problem across both classical and resources, we can split the circuit $U$ into two subcircuits, $U_C$ and $U_Q$, classically simulate the circuit $U_C$, then execute the circuit $U_Q$ on quantum hardware and use the results of the classical simulation to reconstruct the measurement of the obervable $O$.
+
+![OBP diagram depicting splitting a circuit into two subcircuits, classically computing one of the subcircuits, then measuring the other circuit using quantum hardware](/images/guides/qiskit-addons/obp-diagram.png)
+
+
+The subcircuit $U_C$ should be selected to be classically simulable and will compute the expectation value
+
+$$ \langle O' \rangle \equiv U_C^\dagger O U_C $$
+
+which is the version of the initial operator, $O$ evolved through the circuit $U_C$. Once $O'$ has been determined, the quantum workload is prepared wherein the state $|\psi\rangle$ is initiated, has the circuit $U_Q$ applied to it, and then measures the expectation value $O'$. We can show that this is equivalent to measuring $\langle O \rangle$ by writing:
+
+$$ \langle \psi | U_Q^\dagger O' U_Q \psi \rangle = \langle \psi | U_Q^\dagger U_C^\dagger O U_CU_Q \psi \rangle = \langle\psi | U^\dagger O U |\psi \rangle = \langle O \rangle_{U|\psi\rangle}$$
+
+
+Lastly, in order to measure the expectation value $\langle O' \rangle$, we must require it to be decomposable into a sum of Pauli strings
+
+$$ O' = \sum_P c_P P $$
+
+where $c_P$ are real coefficients of the decomposition and $P$ is some Pauli string composed of $I$, $X$, $Y$, and $Z$ operators. This ensures that we can reconstruct the expectation value of $O$ by
+
+$$ \langle \psi | U_Q^\dagger O' |\psi \rangle = \sum_P c_P \langle \psi | U_Q^\dagger P U_Q | \psi \rangle. $$
+
+
+### Truncating terms
+
+This scheme offers a tradeoff between the required circuit depth of $U_Q$, the number of circuit executions on quantum hardware, and the amount of classical computing resources needed to compute $O'$. In general, as you choose to backpropagate further through a circuit, the number of Pauli strings to measure as well as the error-mitigation overhead both grow exponentially (alongside the classical resources needed to simulate $U_C$).
+
+Fortunately, the decomposition of $O'$ can often contain coefficients which are quite small and can be truncated from the final measurements used to reconstruct $O$ without incurring much error. The `qiskit-addon-obp` package possesses functionality to specify an error budget which can be used to automatically search for terms which can be truncated, to within some error tolerance.
+
+
+### Clifford perturbation theory
+
+Lastly, the `qiskit-addon-obp` package approaches operator backpropagation based on Clifford perturbation theory. This method has the benefit that the overhead incurred by backpropagating various gates is determined by the non-Cliffordness of that gate (i.e. how much of a particular portion of $U_C$ which contains non-Clifford instructions).
+
+This approach to OBP begins by splitting the simulated circuit, $U_C$, into *slices*
+
+$$ U_C = \prod_{s=1}^S \mathcal{U}_s = \mathcal{U}_S...\mathcal{U}_2\mathcal{U}_1 $$
+
+where $S$ represents the total number of slices and $\mathcal{U}_s$ denotes a single slice of the circuit $U_C$. Each of these slices are then analytically applied in sequence to measure the back propogated operator $O'$ and may or may not contribute to the overall size of the sum, depending on if the slice is a Clifford vs non-Clifford operation.
+
+
+
+## Next steps
+
+<Admonition type="tip" title="Recommendations">
+    - Get started with OBP
+    - Become familiar with the error mitigation techniques available in Qiskit Runtime
+</Admonition>