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

arXiv:2003.09192

Representation of $\mathfrak{su}(8)$ in Pauli Basis

Chew, K. Y., Nurisya Mohd Shah, Chan, K. T.

Published Mar 20, 2020• 0 citations
arXivHTMLPDF

Abstract

Quantum computation started to become significant field of studies as it hold great promising towards the upgrade of our current computational power. Studying the evolution of quantum states serves as a good fundamental in understanding quantum information which lead to quantum computation. This was assisted with the respective mathematical tools such as Lie group and Lie algebra. In this study, the Lie algebra of $\mathfrak{su}(8)$ is represented in tensor product between three Pauli matrices. This is done by constructing the generalized Gell-Mann matrices and compared to the Pauli basis. This study will explicitly shows the one-to-one correlation of Gell-Mann matrices with the Pauli basis resembled change of coordinates. This is particularly useful when dealing with quantum circuit problems.

DOI
10.48550/arxiv.2003.09192

Related papers

Algebraic representation of Three Qubit Quantum Circuit Problems

Chew K. Y., Nurisya Mohd Shah, Chan K. T.

The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group.

Malaysian Journal of Mathematical Sciences•Sep 26, 2022•0 citations

From Pauli Strings to Quantum Dynamics: A Unified Characterization

Roberto Gargiulo, Paul Herringer, Robert Zeier

Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.

quant-pharXiv:2606.09773•Jun 9, 2026•0 citations

Matrix Elements of Fermionic Gaussian Operators in Arbitrary Pauli Bases: A Pfaffian Formula

M. A. Rajabpour, M. A. Seifi Mirjafarlou, Reyhaneh Khasseh

Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to $\mathfrak{so}(2L)$. This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.

arXiv:2506.02809•Jun 3, 2025•0 citations

Spin correlations in two-particle systems: a pedagogically motivated comparison of computational approaches

S. Martins-Filho

In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-$1/2$ particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form $\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle$, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of $2\times2$ complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.

physics.ed-phCross-listedarXiv:2606.02361•Revista Brasileira de Ensino de Física•Jun 2, 2026•0 citations

Full classification of Pauli Lie algebras

Aguilar, Gerard, Simon Cichy, Jens Eisert +1 more

Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class. Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits.

arXiv:2408.00081•Jul 31, 2024•1 citation

Efficiently manipulating Pauli strings with PauliArray

Maxime Dion, Tania Belabbas, Nolan Bastien

Pauli matrices and Pauli strings are widely used in quantum computing. These mathematical objects are useful to describe or manipulate the quantum state of qubits. They offer a convenient basis to express operators and observables used in different problem instances such as molecular simulation and combinatorial optimization. Therefore, it is important to have a well-rounded, versatile and efficient tool to handle a large number of Pauli strings and operators expressed in this basis. This is the objective behind the development of the PauliArray library presented in this work. This library introduces data structures to represent arrays of Pauli strings and operators as well as various methods to modify and combine them. Built using NumPy, PauliArray offers fast operations and the ability to use broadcasting to easily carry out otherwise cumbersome manipulations. Applications to the fermion-to-qubit mapping, to the estimation of expectation values and to the computation of commutators are considered to illustrate how PauliArray can simplify some relevant tasks and accomplish them faster than current libraries.

arXiv:2405.19287•May 29, 2024•0 citations
Representation of su(8)\mathfrak{su}(8)su(8) in Pauli Basis
Quantum computation started to become significant field of studies as it hold great promising towards the upgrade of our current computational power. Studying the evolution of quantum states serves as a good fundamental in understanding quantum information which lead to quantum computation. This was assisted with the respective mathematical tools such as Lie group and Lie algebra. In this study, the Lie algebra of su(8)\mathfrak{su}(8)su(8) is represented in tensor product between three Pauli matrices. This is done by constructing the generalized Gell-Mann matrices and compared to the Pauli basis. This study will explicitly shows the one-to-one correlation of Gell-Mann matrices with the Pauli basis resembled change of coordinates. This is particularly useful when dealing with quantum circuit problems.
The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group.
Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to so(2L)\mathfrak{so}(2L)so(2L). This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.
In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-1/21/21/2 particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form ⟨ψ∣Su^(1)Sv^(2)∣ψ⟩\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle⟨ψ∣Su^(1)​Sv^(2)​∣ψ⟩, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of 2×22\times22×2 complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.
Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class. Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits.
Pauli matrices and Pauli strings are widely used in quantum computing. These mathematical objects are useful to describe or manipulate the quantum state of qubits. They offer a convenient basis to express operators and observables used in different problem instances such as molecular simulation and combinatorial optimization. Therefore, it is important to have a well-rounded, versatile and efficient tool to handle a large number of Pauli strings and operators expressed in this basis. This is the objective behind the development of the PauliArray library presented in this work. This library introduces data structures to represent arrays of Pauli strings and operators as well as various methods to modify and combine them. Built using NumPy, PauliArray offers fast operations and the ability to use broadcasting to easily carry out otherwise cumbersome manipulations. Applications to the fermion-to-qubit mapping, to the estimation of expectation values and to the computation of commutators are considered to illustrate how PauliArray can simplify some relevant tasks and accomplish them faster than current libraries.