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Documentation for Generalized Stabilizer Representation
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# [Generalized Stabilizer Representation](@id Generalized-Stabilizer-Overview) | ||
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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. | ||
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# Advances in Stabilizer Formalism | ||
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Since its inception, the stabilizer formalism has undergone several improvements. Notable enhancements include: | ||
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```@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> | ||
``` | ||
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# Generalized Stabilizer Representation | ||
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The generalized stabilizer representation provides a flexible framework for simulating quantum circuits by: | ||
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- Enabling the representation of any quantum state, pure or mixed. | ||
- Allowing simulations of arbitrary quantum circuits, including unitary operations, measurements, and | ||
quantum channels. | ||
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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. | ||
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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. | ||
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# Simulation of Quantum Channels | ||
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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. | ||
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# Advantages of the Generalized Stabilizer | ||
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The proposed representation combines the rapid update capabilities of stabilizer states with the generality of | ||
density matrices. Key features include: | ||
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- 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`. | ||
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- 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. | ||
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# Implications for Classical and Quantum Computation | ||
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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. | ||
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[^1]: [gottesman1998heisenberg](@cite) | ||
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[^2]: [yoder2012generalization](@cite) | ||
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[^3]: [gottesman1997stabilizer](@cite) |
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