From bea7fe5a2a24442764d6c57409d1d3053081ae2b Mon Sep 17 00:00:00 2001
From: Fe-r-oz <feroz.ahmad.email@gmail.com>
Date: Wed, 30 Oct 2024 21:15:41 +0500
Subject: [PATCH] Documentation for Generalized Stabilizer Representation

---
 docs/make.jl            |  3 ++
 docs/src/genstab.md     | 81 +++++++++++++++++++++++++++++++++++++++++
 docs/src/references.bib | 39 ++++++++++++++++++++
 3 files changed, 123 insertions(+)
 create mode 100644 docs/src/genstab.md

diff --git a/docs/make.jl b/docs/make.jl
index 699924cda..fa109ea37 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -41,6 +41,9 @@ pages = [
     "Circuit Operations" => "noisycircuits_ops.md",
     "API" => "noisycircuits_API.md"
 ],
+"Generalized Stabilizer" => [
+    "Overview" => "genstab.md",
+    ],
 "ECC compendium" => [
     "Evaluating codes and decoders" => "ECC_evaluating.md"
     "API" => "ECC_API.md"
diff --git a/docs/src/genstab.md b/docs/src/genstab.md
new file mode 100644
index 000000000..327c33196
--- /dev/null
+++ b/docs/src/genstab.md
@@ -0,0 +1,81 @@
+# [Generalized Stabilizer Representation](@id Generalized-Stabilizer-Overview)
+
+Gottesman's introduction of stabilizer formalism in 1997 greatly impacted quantum complexity and coding
+theory.The Heisenberg representation[^1], vital to the Gottesman-Knill theorem, enables classical simulations
+to use fewer Pauli operators, easing computational demands. However, this approach is limited to stabilizer
+circuits with Clifford gates and measurements. While effective, the theorem has a narrow scope, making it
+essential to generalize it for broader quantum circuit simulations. Theodore Yoder[^2] introduces a generalized
+stabilizer representation to address this challenge.
+
+## Advances in Stabilizer Formalism
+
+Since its inception, the stabilizer formalism has undergone several improvements. Notable enhancements include:
+
+```@raw html
+<div class="mermaid">
+timeline
+    title Related Work in Generalization of the Gottesman-Knill Theorem
+    1997 : Gottesman [3] introduces stabilizer formalism and the Gottesman-Knill theorem.
+    2002 : Bartlett et al. [4] expand to continuous variable quantum computation.
+    2004 : Aaronson and Gottesman [5] improve measurement time complexity to 𝒪(n²).
+    2006 : Anders and Briegel [6] achieve 𝒪(n log n) speedup in time complexity with graph states.
+    2012 : Bermejo-Vega and Van den Nest [7] generalize to any finite Abelian group from n-qubits ℤ₂ⁿ.
+    2012 : Yoder [2] presents the Generalized Stabilizer with a novel state representation.
+</div>
+```
+
+## Generalized Stabilizer Representation
+
+The generalized stabilizer representation provides a flexible framework for simulating quantum circuits by:
+
+- Enabling the representation of any quantum state, pure or mixed.
+- Allowing simulations of arbitrary quantum circuits, including unitary operations, measurements, and
+quantum channels.
+
+This representation expands on the stabilizer formalism by incorporating non-stabilizer states and circuits,
+enabling the simulation of non-Clifford gates and broader quantum channels for diverse quantum computations.
+
+Unlike previous methods that may use a superposition of stabilizer states to represent arbitrary states,
+this approach employs the tableau construction developed by Aaronson and Gottesman[^3]. This method implicitly
+represents a set of orthogonal stabilizer states, forming a stabilizer basis capable of representing arbitrary
+quantum states.Updating the tableau takes only twice as long as updating a single stabilizer, enabling efficient
+updates of the entire stabilizer basis with minimal computational overhead.
+
+## Simulation of Quantum Channels
+
+The generalized stabilizer representation enables the simulation of arbitrary quantum channels, beyond just
+unitary gates and measurements. It does this by decomposing the Kraus operators of a channel into Pauli operators
+from the state’s tableau, allowing for a broader range of quantum operations.
+
+## Advantages of the Generalized Stabilizer
+
+The proposed representation combines the rapid update capabilities of stabilizer states with the generality of
+density matrices. Key features include:
+
+- High update efficiency for unitary gates, measurements, and quantum channels, influenced by the sparsity of
+the density matrix, `Λ(χ)`, which indicates the count of non-zero elements in `χ`.
+
+-  Simulations maintain linear complexity with respect to the number of measurements, and the representation
+remains straightforward, reflecting the principle that measurements simplify quantum states through collapse.
+
+## Implications for Classical and Quantum Computation
+
+Investigating stabilizer circuits enhances our understanding of classical and quantum computation. Simulating these
+circuits is a complete problem in the classical complexity class `⊕L`, a subset of `P`, indicating that stabilizer
+circuits may not be universal in classical computation contexts. Surprisingly, adding just one non-Clifford gate to
+circuits with Clifford gates and measurements generally enables universal quantum computation—a contrast that highlights
+intriguing questions about the computational boundaries between classical and quantum systems.
+
+[^1]: [gottesman1998heisenberg](@cite)
+
+[^2]: [yoder2012generalization](@cite)
+
+[^3]: [gottesman1997stabilizer](@cite)
+
+[^4]: [bartlett2002efficient](@cite)
+
+[^5]: [aaronson2004improved](@cite)
+
+[^6]: [anders2006fast](@cite)
+
+[^7]: [bermejo2012classical](@cite)
diff --git a/docs/src/references.bib b/docs/src/references.bib
index 09820c204..78fcc7945 100644
--- a/docs/src/references.bib
+++ b/docs/src/references.bib
@@ -191,6 +191,45 @@ @article{nahum2017quantum
   year = {2017}
 }
 
+% Generalized Stabilizer
+
+@article{yoder2012generalization,
+  title={A generalization of the stabilizer formalism for simulating arbitrary quantum circuits},
+  author={Yoder, Theodore J},
+  journal={See http://www. scottaaronson. com/showcase2/report/ted-yoder. pdf},
+  year={2012},
+  publisher={Citeseer}
+}
+
+@article{bartlett2002efficient,
+  title={Efficient classical simulation of continuous variable quantum information processes},
+  author={Bartlett, Stephen D and Sanders, Barry C and Braunstein, Samuel L and Nemoto, Kae},
+  journal={Physical Review Letters},
+  volume={88},
+  number={9},
+  pages={097904},
+  year={2002},
+  publisher={APS}
+}
+
+@article{anders2006fast,
+  title={Fast simulation of stabilizer circuits using a graph-state representation},
+  author={Anders, Simon and Briegel, Hans J},
+  journal={Physical Review A?Atomic, Molecular, and Optical Physics},
+  volume={73},
+  number={2},
+  pages={022334},
+  year={2006},
+  publisher={APS}
+}
+
+@article{bermejo2012classical,
+  title={Classical simulations of Abelian-group normalizer circuits with intermediate measurements},
+  author={Bermejo-Vega, Juan and Nest, Maarten Van den},
+  journal={arXiv preprint arXiv:1210.3637},
+  year={2012}
+}
+
 % codes
 
 @article{mackay2004sparse,