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

Finding and building algebraic structures in finite-dimensional Hilbert spaces for quantum computation and quantum information

Lin, Robert Henry

Published Jan 1, 2023• 0 citations
OpenAlex

Abstract

In this dissertation, we investigate algebraic structures in finite-dimensional Hilbert spaces, as concerns quantum computation and quantum information, as well as these structures' applications to lattices. On the quantum computation side, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Subsequently, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many algebraic identities, including a novel algebraic proof of a Yang-Baxter equation. We also derive a new identity for the braid elements, which is key to our proofs. We then use the Yang-Baxter equation proof to resolve an open question of Cobanera and Ortiz. We demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. In addition, we show how to explicitly compute various vector states in an efficient manner using algebraic methods. On the quantum information side, we introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like operator decompositions (Kraus-like decompositions for short). An important motivation for this new decomposition is a general nonexistence result that we show for Kraus operator decompositions for quantum channels in this setting. We show that the notion of convex Kraus-like operator decompositions (in which the coefficients in the sum decomposition are nonnegative and satisfy a sum rule) that are induced by the irreducible characters of a finite group is equivalent to the notion of a conditionally negative-definite length when the length is a class function. For a general finite group G, we prove a stability condition which shows that if the semigroup associated with a length has a convex Kraus-like operator decomposition for all t>0 small enough, then it has a convex Kraus-like operator decomposition for all time t>0. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. By a result of Schoenberg, our result implies that in the group algebra setting, a semigroup P_t induced by a length function which is a class function is a quantum channel for all t greater than or equal to 0 if and only if it possesses a convex Kraus-like operator decomposition for all t>0. Finally, motivated by the importance of lattice problems in quantum cryptography, we extend the algebraic framework for multi-qudit representations of generalized Clifford algebras to d-dimensional lattices mod P. We show that under suitable number-theoretic conditions, the subalgebra induced by a lattice has trivial center. Under the trivial center constraint, we construct for pairs of lattice vectors satisfying an algebraic constraint a unitary operator based on the product of generalized Clifford algebra generators associated to each lattice vector.

Related papers

A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

Robert Lin

In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor ζ , which is an appropriate square root of a primitive root of unity.

arXiv:2103.16081•Quantum•Nov 17, 2025•0 citations

An Algebraic Framework for Multi-Qudit Computations with Generalized Clifford Algebras

Lin, Robert

In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of generalized Clifford algebras using purely algebraic methods, which is addressed in a follow-up paper.

arXiv:2103.15324•Mar 29, 2021•1 citation

The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information

Kagwe A. Muchane

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for $n$-qubit quantum computation based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$. In this setting, the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on the $J$-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the $n$-qubit computational basis state $|0\cdots 0\rangle$ is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for $n$ qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

quant-pharXiv:2512.07902•Dec 9, 2025•0 citations

Algebraic and Mathematical Foundations of Quantum Computation: Hilbert Spaces, Group Structures, and Quantum Algorithms

Bhavesh B. Prajapati

Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.

Communications on Applied Nonlinear Analysis•Jan 24, 2025•0 citations

Matrix Results And Techniques In Quantum Information Science And Related Topics

Diane Christine Pelejo

In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at $2^n\times 2^n$ unitary matrices, which describe operations on a closed $n$-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general $2^n\times 2^n$ unitary matrix to the product of no more than $2^{n-1}(2^n-1)$ single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element $\Phi$ among a given set of quantum channels $\mathcal{S}$ that will produce the optimal value of a scalar function $D(\rho_1,\Phi(\rho_2))$, on two fixed quantum states $\rho_1$ and $\rho_2$. Some of the functions we considered for $D(\cdot,\cdot)$ are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers $a_1\geq\ldots\geq a_{2n}\geq 0$, we want to determine if there exist a $2n\times 2n$ density matrix $\rho$ having eigenvalues $a_1,\ldots,a_{2n}$ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on $a_1,\ldots,a_{2n}$ for $n\leq 6$ and state some observations on general values of $n$. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that $\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}$ for given $\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_n$ and $\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_m$, (b) constructing a multipartite state $\rho$ having a prescribed set of reduced states $\rho_1,\ldots, \rho_r$ on $r$ of its subsystems, (c) constructing a multipartite state$\rho$ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix $A$ can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets $K_1$ and $K_2$ of $\IC$ given by $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that $K_1K_2$ is star-shaped when $K_1$ and $K_2$ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk.

•Jan 1, 2016•0 citations

Reductions in finite-dimensional quantum mechanics : from symmetries to operator algebras and beyond

Oleg Kabernik

The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.

arXiv:2103.08226•Jan 1, 2021•2 citations
In this dissertation, we investigate algebraic structures in finite-dimensional Hilbert spaces, as concerns quantum computation and quantum information, as well as these structures' applications to lattices. On the quantum computation side, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Subsequently, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many algebraic identities, including a novel algebraic proof of a Yang-Baxter equation. We also derive a new identity for the braid elements, which is key to our proofs. We then use the Yang-Baxter equation proof to resolve an open question of Cobanera and Ortiz. We demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. In addition, we show how to explicitly compute various vector states in an efficient manner using algebraic methods. On the quantum information side, we introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like operator decompositions (Kraus-like decompositions for short). An important motivation for this new decomposition is a general nonexistence result that we show for Kraus operator decompositions for quantum channels in this setting. We show that the notion of convex Kraus-like operator decompositions (in which the coefficients in the sum decomposition are nonnegative and satisfy a sum rule) that are induced by the irreducible characters of a finite group is equivalent to the notion of a conditionally negative-definite length when the length is a class function. For a general finite group G, we prove a stability condition which shows that if the semigroup associated with a length has a convex Kraus-like operator decomposition for all t>0 small enough, then it has a convex Kraus-like operator decomposition for all time t>0. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. By a result of Schoenberg, our result implies that in the group algebra setting, a semigroup P_t induced by a length function which is a class function is a quantum channel for all t greater than or equal to 0 if and only if it possesses a convex Kraus-like operator decomposition for all t>0. Finally, motivated by the importance of lattice problems in quantum cryptography, we extend the algebraic framework for multi-qudit representations of generalized Clifford algebras to d-dimensional lattices mod P. We show that under suitable number-theoretic conditions, the subalgebra induced by a lattice has trivial center. Under the trivial center constraint, we construct for pairs of lattice vectors satisfying an algebraic constraint a unitary operator based on the product of generalized Clifford algebra generators associated to each lattice vector.
In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor ζ , which is an appropriate square root of a primitive root of unity.
In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of generalized Clifford algebras using purely algebraic methods, which is addressed in a follow-up paper.
We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for nnn-qubit quantum computation based on the tensor product Cℓ2,0(R)⊗nC\ell_{2,0}(\mathbb{R})^{\otimes n}Cℓ2,0​(R)⊗n. In this setting, the bivector J=e12J = e_{12}J=e12​ satisfies J2=−1J^{2} = -1J2=−1 and supplies the complex structure on the JJJ-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the nnn-qubit computational basis state ∣0⋯0⟩|0\cdots 0\rangle∣0⋯0⟩ is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for nnn qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.