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Add the theory page of LRE user guide #2522

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Add the theory page of LRE user guide #2522

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Description

Fixes #2486


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  • I license this contribution under the terms of the GNU GPL, version 3 and grant Unitary Fund the right to provide additional permissions as described in section 7 of the GNU GPL, version 3.

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codecov bot commented Oct 2, 2024

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.72%. Comparing base (6f912f6) to head (558ef8e).
Report is 3 commits behind head on main.

Additional details and impacted files
@@           Coverage Diff           @@
##             main    #2522   +/-   ##
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  Coverage   98.71%   98.72%           
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  Files          89       90    +1     
  Lines        4131     4156   +25     
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+ Hits         4078     4103   +25     
  Misses         53       53           

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@natestemen natestemen left a comment

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Going to suggest @vprusso take the first pass at a review of the material. My comments are small and shouldn't overlap (I hope!).

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Great start Purva! I can tell you have a great understanding of this technique, but the this page still needs some work to get all the nuances from your head into the doc so others can understand as well.

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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).

$$
O_{\rm LRE} = \sum_{i=1}^M \langle O (\boldsymbol{\lambda}_i)\rangle \frac{\det \left(\mathbf{M}_i (\boldsymbol{0}) \right)}{\det \left(\mathbf{A}\right)}.
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I think we should avoid using $\mathbf{M}$ here since the matrix above was defined with $M$.

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This is supposed to be one of the terms in the monomial basis evaluated at $\lambda=0$

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What is the theory behind LRE
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