Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Graduate texts in physics

The Stabilizer Formalism and the Gottesman–Knill Theorem

János A. Bergou, Mark S. Hillery, Mark E. Saffman

Published Jan 1, 2021• 2 citations
Publisher
DOI
10.1007/978-3-030-75436-5_10

Related papers

Extensions and Limitations of the Noisy Stabilizer Formalism

Matti, Jasmin

The stabilizer formalism is a central framework in quantum computing and one of the key building blocks for fault-tolerant quantum computers. According to the Gottesman–Knill theorem, we can efficiently simulate any quantum system consisting of stabilizer states, Clifford gates, and Pauli measurements on a classical computer. A key challenge in developing robust quantum technologies is that physical quantum systems are unavoidably affected by noise, which degrades quantum information. Accurate noise modeling is therefore essential. For stabilizer states, the noisy stabilizer formalism has recently been introduced as a framework for efficiently describing the evolution of graph states subject to Pauli-diagonal noise under Clifford operations and Pauli measurements. In this thesis, we explore and extend this formalism in two directions. First, we generalize the method from graph states to arbitrary stabilizer states, thereby extending the Gottesman-Knill theorem to noisy scenarios. Second, we extend the formalism to handle noise models beyond Pauli-diagonal channels. Moreover, we study the applicability of the noisy stabilizer formalism to entanglement purification protocols.

•Jan 1, 2025•0 citations

The Stabilizer Formalism

WORLD SCIENTIFIC eBooks•Aug 27, 2026•0 citations

On the stabilizer formalism and its generalization

Éloi Descamps, Borivoje Dakić

Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.

arXiv:2309.09815•Journal of Physics A Mathematical and Theoretical•Oct 11, 2024•1 citation

Stabilizer Formalism and Its Applications

Keisuke Fujii

SpringerBriefs in mathematical physics•Jan 1, 2015•3 citations

Fast simulation of stabilizer circuits using a graph-state representation

Simon Anders, Hans J. Briegel

According to the Gottesman-Knill theorem, a class of quantum circuits---namely, the so-called stabilizer circuits---can be simulated efficiently on a classical computer. We introduce an algorithm for this task, which is based on the graph-state formalism. It shows significant improvement in comparison to an existing algorithm, given by Gottesman and Aaronson, in terms of speed and of the number of qubits the simulator can handle. We also present an implementation.

arXiv:quant-ph/0504117•Physical Review A•Feb 21, 2006•163 citations

Generalization of the Hellmann–Feynman theorem

J.G. Esteve, Fernando Falceto, Carlos A. García Canal

arXiv:0912.4153•Physics Letters A•Dec 7, 2009•33 citations
The stabilizer formalism is a central framework in quantum computing and one of the key building blocks for fault-tolerant quantum computers. According to the Gottesman–Knill theorem, we can efficiently simulate any quantum system consisting of stabilizer states, Clifford gates, and Pauli measurements on a classical computer. A key challenge in developing robust quantum technologies is that physical quantum systems are unavoidably affected by noise, which degrades quantum information. Accurate noise modeling is therefore essential. For stabilizer states, the noisy stabilizer formalism has recently been introduced as a framework for efficiently describing the evolution of graph states subject to Pauli-diagonal noise under Clifford operations and Pauli measurements. In this thesis, we explore and extend this formalism in two directions. First, we generalize the method from graph states to arbitrary stabilizer states, thereby extending the Gottesman-Knill theorem to noisy scenarios. Second, we extend the formalism to handle noise models beyond Pauli-diagonal channels. Moreover, we study the applicability of the noisy stabilizer formalism to entanglement purification protocols.
Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.