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change wording of Bi matrix
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purva-thakre committed Oct 11, 2024
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Finding the coefficients in the linear combination becomes a problem solvable through a system of linear equations $\mathbf{A} c = z$ where $c$ is the coefficients vector $(\eta_1, \eta_2, \ldots, \eta_N)^T$, $z$ is the vector of the noisy expectation values and $\mathbf{A}$ is the sample matrix evaluated using the values in the scale factor vectors.

The [general multivariate Lagrange interpolation formula](https://www.siam.org/media/wkvnvame/a_simple_expression_for_multivariate.pdf) is defined by replacing the $i$-th row of the sample matrix $\mathbf{A}$ with monomial terms evaluated using the generic variable λ. This matrix $\mathbf{B}_i$ defines an interpolating polynomial in variable λ of degree $d$. As we only need to find the noiseless expectation value, we can skip calculating the full vector of linear combination coefficients if we use the [Lagrange interpolation formula](https://files.eric.ed.gov/fulltext/EJ1231189.pdf) evaluated at $λ = 0$ i.e. the zero-noise limit.
The [general multivariate Lagrange interpolation polynomial](https://www.siam.org/media/wkvnvame/a_simple_expression_for_multivariate.pdf) is defined by a new matrix $\mathbf{B}_i$ obtained by replacing the $i$-th row of the sample matrix $\mathbf{A}$ with monomial terms evaluated using the generic variable λ. Thus, matrix $\mathbf{B}_i$ represents an interpolating polynomial in variable λ of degree $d$. As we only need to find the noiseless expectation value, we can skip calculating the full vector of linear combination coefficients if we use the [Lagrange interpolation formula](https://files.eric.ed.gov/fulltext/EJ1231189.pdf) evaluated at $λ = 0$ i.e. the zero-noise limit.

To get the matrix $\mathbf{B}_i(\mathbf{0})$, replace the $i$-th row of the sample matrix $\mathbf{A}$ by $\mathbf{e}_i=(1, 0, \ldots, 0)$ where except $M_1(0, d) = 1$ all the other monomial terms are zero when $λ=0$.

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