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Co-authored-by: nate stemen <[email protected]>

docs/source/guide/lre-5-theory.md

@@ -29,7 +29,7 @@ Similar to [ZNE](zne.md), this process works in two steps:
 
 - **Step 2:** Extrapolate to the noiseless limit using multivariate richardson extrapolation.
 
-LRE leverages the flexible configurational space of layerwise unitary folding,
+LRE leverages the flexible configuration space of layerwise unitary folding,
 allowing for a more nuanced mitigation of errors by treating the noise level of each layer of
 the quantum circuit as an independent variable.
 
@@ -40,7 +40,7 @@ a specific pattern as a result of unitary folding. This pattern is often describ
 generated by the fold multiplier and the chosen degree for multivariate Richardson extrapolation polynomial. For more information
 on unitary folding, go to [What is the theory behind ZNE?](zne-5-theory.md).
 
-Suppose we want to estimate the noiseless expectation value of some observable in an $n$-qubit circuit with $l$ layers.
+Suppose we're interested in the value of some observable in an $n$-qubit circuit with $l$ layers.
 
 Each layer can have a different scale factor and we can create $M$ such variations of the scaled circuit. Let $\{λ_1, λ_2, λ_3, \ldots, λ_M\}$ be the scale factors vectors used to create multiple variations of the noise-scaled circuits $\{C_{λ_1}, C_{λ_2}, C_{λ_3}, \ldots, C_{λ_M}\}$ such that each vector $λ_i$ defines the scale factors for the different layers in the input circuit $\{{λ^1}_i, {λ^2}_i, {λ^3}_i, \ldots, {λ^l}_i\}^T$.
 
@@ -70,10 +70,10 @@ Each monomial term in the sample matrix is evaluated using the values in the sca
 
 ## Step 2: Extrapolate to the noiseless limit
 
-Each noise scaled circuit $C_{λ_i}$ has an expectation value associated with it $\langle O(λ_i) \rangle$ such that we can define a vector of the noisy expectation values $z = (\langle O(λ_1) \rangle, \langle O(λ_2) \rangle, \langle O(λ_3) \rangle, \ldots, \langle O(λ_M)\rangle)^T$. These have a coefficient of linear combination associated with them such that 
+Each noise scaled circuit $C_{λ_i}$ has an expectation value associated with it $\langle O(λ_i) \rangle$ such that we can define a vector of the noisy expectation values $z = (\langle O(λ_1) \rangle, \langle O(λ_2) \rangle, \ldots, \langle O(λ_M)\rangle)^T$. These have a coefficient of linear combination associated with them such that 
 
 $$
-O_{LRE} = \sum_{i=1}^{M} \eta_i \langle O(\boldsymbol{\lambda}_i) \rangle.
+O_{\mathrm{LRE}} = \sum_{i=1}^{M} \eta_i \langle O(\boldsymbol{\lambda}_i) \rangle.
 $$
 
 The system of linear equations is used to find the numerous $\eta_i$. As we only need to find the noiseless expectation value, we do not need to calculate the full vector of linear combination coefficients if we use the [Lagrange interpolation formula](https://files.eric.ed.gov/fulltext/EJ1231189.pdf). 

docs/source/guide/lre.md

@@ -9,8 +9,8 @@ Layerwise Richardson Extrapolation (LRE), an error mitigation technique, introdu
 {cite}`Russo_2024_LRE` works by creating multiple noise-scaled variations of the input
 circuit such that the noiseless expectation value is extrapolated from the execution of each
 noisy circuit (see the section [What is the theory behind LRE?](lre-5-theory.md)). Compared to
-unitary folding, the technique treats the noise in each layer of the input circuit as an
-independent variable.
+Zero-Noise Extrapolation, this technique treats the noise in each layer of the circuit
+as an independent variable to be scaled and then extrapolated independently.
 
 
 You can get started with LRE in Mitiq with the following sections of the user guide: