From 52706fbcb6e5d129ad0a19bfd3d91a38502f841a Mon Sep 17 00:00:00 2001 From: Fe-r-oz Date: Wed, 30 Oct 2024 21:15:41 +0500 Subject: [PATCH] Documentation for Generalized Stabilizer Representation --- docs/make.jl | 3 ++ docs/src/genstab.md | 74 +++++++++++++++++++++++++++++++++++++++++ docs/src/references.bib | 39 ++++++++++++++++++++++ 3 files changed, 116 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..8a81412dc --- /dev/null +++ b/docs/src/genstab.md @@ -0,0 +1,74 @@ +# [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 +
+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. +
+``` + +# 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) 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,