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Documentation for Generalized Stabilizer Representation
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3 changes: 3 additions & 0 deletions docs/make.jl
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Expand Up @@ -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"
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74 changes: 74 additions & 0 deletions docs/src/genstab.md
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# [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 a 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.

# 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)
39 changes: 39 additions & 0 deletions docs/src/references.bib
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Expand Up @@ -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,
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