noncliff: Documentation for overview of Generalized Stabilizer Representation

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"

docs/src/genstab.md

@@ -0,0 +1,98 @@
+# [Generalized Stabilizer Representation](@id Generalized-Stabilizer-Overview)
+
+Gottesman's introduction of stabilizer formalism in 1997 greatly impacted quantum complexity and coding
+theory. The key insight of the Gottesman-Knill theorem lies in utilizing the Heisenberg representation[^1] for
+quantum states, allowing classical simulations to work with only `n` Pauli operators, rather than processing
+an exponentially large complex vector with approximately `2ⁿ` entries for an `n`-qubit state. 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] introduced the 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 introduces stabilizer formalism and the Gottesman-Knill theorem.
+    2002 : Bartlett et al. expand to continuous variable quantum computation.
+    2004 : Aaronson and Gottesman improve measurement time complexity to 𝒪(n²).
+    2006 : Anders and Briegel achieve 𝒪(n log n) speedup in time complexity with graph states.
+    2012 : Bermejo-Vega and Van den Nest generalize to any finite Abelian group from n-qubits ℤ₂ⁿ.
+    2012 : Yoder develops 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.
+
+# Strong Simulation vs Weak Simulation
+
+Yoder[^2] investigated the classical simulation of quantum circuits in the context of `strong simulation`, which
+focuses on calculating exact probabilities for specific measurement outcomes. A notable gap exists between weak
+and strong simulation problems, with some circuits being `#P`-complete for strong simulation while weak simulation
+is in `BPP`.The key differences between these two approaches are as follows:
+
+```@raw html
+<div class="mermaid">
+mindmap
+  root((Classical Simulation of Quantum Circuits))
+    Strong Simulation
+      Goal: Exact probability of specific outcome
+      Example: Calculate probability of outcome 01
+      Precision: High, requires precise probabilities
+      Complexity: Can be #P-complete
+    Weak Simulation
+      Goal: Sample outcome close to quantum distribution
+      Example: Sample outcomes like 00, 01, 10, 11
+      Precision: Lower, approximate sampling
+      Complexity: Generally in BPP
+</div>
+```
+
+# 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)

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,