From 2d27879f0bfd292e8f607d51cc364423e59ed01d Mon Sep 17 00:00:00 2001 From: Purva Thakre Date: Wed, 2 Oct 2024 12:40:14 -0500 Subject: [PATCH] vincent + nate feedback --- docs/source/guide/lre-5-theory.md | 28 ++++++++++++++-------------- docs/source/guide/lre.md | 7 +++++-- 2 files changed, 19 insertions(+), 16 deletions(-) diff --git a/docs/source/guide/lre-5-theory.md b/docs/source/guide/lre-5-theory.md index 248d6df7c..dccd92a98 100644 --- a/docs/source/guide/lre-5-theory.md +++ b/docs/source/guide/lre-5-theory.md @@ -12,7 +12,7 @@ kernelspec: --- -```{admonition} Warning: +```{warning} The user guide for LRE in Mitiq is currently under construction. ``` @@ -21,15 +21,17 @@ The user guide for LRE in Mitiq is currently under construction. Layerwise Richardson Extrapolation (LRE), an error mitigation technique, introduced in {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. Similar to [ZNE](zne.md), this process works in two steps: +noisy circuit. -- **Step 1: Intentionally create multiple noise-scaled but logically equivalent circuits** through unitary folding. +Similar to [ZNE](zne.md), this process works in two steps: -- **Step 2: Extrapolate to the noiseless limit** using multivariate richardson extrapolation. +- **Step 1:** Intentionally create multiple noise-scaled but logically equivalent circuits through unitary folding. + +- **Step 2:** Extrapolate to the noiseless limit using multivariate richardson extrapolation. LRE leverages the flexible configurational 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 +the quantum circuit as an independent variable. ## Step 1: Intentionally create multiple noise-scaled but logically equivalent circuits @@ -40,9 +42,9 @@ 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. -Each layer can have a different scale factor and we can create $M$ such variations of the scaled circuit. Let $\{λ_1, λ_2, λ_3, ..., λ_M\}$ be the scale factors vectors used to create multiple variations of the noise-scaled circuits $\{C_{λ_1}, C_{λ_2}, C_{λ_3}, ……, C_{λ_M}\}$ such that each vector $λ_i$ defines the scale factors for the different layers in the input circuit $\{{λ^1}_i, {λ^2}_i, {λ^3}_i, ….., {λ^l}_i\}^T$. +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 $\{{λ^1}_i, {λ^2}_i, {λ^3}_i, \ldots, {λ^l}_i\}^T$. -If $d$ is the chosen degree of our multivariate polynomial, $M_j(λ_i, d)$ corresponds to the terms in the polynomial. In general, the monomial terms for a variable $l$ up to degree $d$ can be determined through the [Stars and Bars method](https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29). +If $d$ is the chosen degree of our multivariate polynomial, $M_j(λ_i, d)$ corresponds to the terms in the polynomial. In general, the monomial terms for a variable $l$ up to degree $d$ can be determined through the [stars and bars method](https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29). $$ \text{total number of terms in the monomial basis with max degree } d = \binom{d + l}{d} @@ -64,22 +66,20 @@ $$ \end{bmatrix} $$ -Each monomial term in the sample matrix is evaluated using the values in the scale factor vectors. We aim to define the zero-noise limit as a linear combination of the noisy expectation values. Finding the coefficients in the linear combination becomes a problem solvable through a system of linear equations $Ac = z$ where c is the coefficients vector, $z$ is the vector of expectation values and $\mathbf{A}$ is the sample matrix evaluated using the values in the scale factor vectors. +Each monomial term in the sample matrix is evaluated using the values in the scale factor vectors. We aim to define the zero-noise limit as a linear combination of the noisy expectation values. Finding the coefficients in the linear combination becomes a problem solvable through a system of linear equations $Ac = z$ where $c$ is the coefficients vector, $z$ is the vector of expectation values and $\mathbf{A}$ is the sample matrix evaluated using the values in the scale factor vectors. ## Step 2: Extrapolate to the noiseless limit -Each noise scaled circuit $C_{λ_i}$ has an expectation value associated with it $\langle O(λ_i) \rangle$ such that we can define a vector of the noisy expectation values $z = (\langle O(λ_1) \rangle, \langle O(λ_2) \rangle, \langle O(λ_3) \rangle, ..., \langle O(λ_M)\rangle)^T$. These have a coefficient of linear combination associated with them such that +Each noise scaled circuit $C_{λ_i}$ has an expectation value associated with it $\langle O(λ_i) \rangle$ such that we can define a vector of the noisy expectation values $z = (\langle O(λ_1) \rangle, \langle O(λ_2) \rangle, \langle O(λ_3) \rangle, \ldots, \langle O(λ_M)\rangle)^T$. These have a coefficient of linear combination associated with them such that $$ - -O_{LRE} = \sum_{i=1}^{M} \eta_i \langle O(\boldsymbol{\lambda}_i) \rangle - +O_{LRE} = \sum_{i=1}^{M} \eta_i \langle O(\boldsymbol{\lambda}_i) \rangle. $$ The system of linear equations is used to find the numerous $\eta_i$. As we only need to find the noiseless expectation value, we do not need to calculate the full vector of linear combination coefficients if we use the [Lagrange interpolation formula](https://files.eric.ed.gov/fulltext/EJ1231189.pdf). $$ -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)} +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)}. $$ -To get the matrix $\mathbf{M}_i(\mathbf{0})$, replace the $i$-th row of the sample matrix $\mathbf{A}$ by $\mathbf{e}_1=(1, 0, \ldots, 0)^t$. +To get the matrix $\mathbf{M}_i(\mathbf{0})$, replace the $i$-th row of the sample matrix $\mathbf{A}$ by $\mathbf{e}_1=(1, 0, \ldots, 0)^T$. diff --git a/docs/source/guide/lre.md b/docs/source/guide/lre.md index a301fdd9f..db6454f32 100644 --- a/docs/source/guide/lre.md +++ b/docs/source/guide/lre.md @@ -1,5 +1,5 @@ -```{admonition} Warning: +```{warning}: The user guide for LRE in Mitiq is currently under construction. ``` @@ -8,7 +8,10 @@ The user guide for LRE in Mitiq is currently under construction. Layerwise Richardson Extrapolation (LRE), an error mitigation technique, introduced in {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)). +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. + You can get started with LRE in Mitiq with the following sections of the user guide: