Related papers: Lie groups for quantum complexity and barren plate…
Variational quantum computing offers a powerful framework with applications across diverse fields such as quantum chemistry, machine learning, and optimization. However, its scalability is hindered by the exponential concentration of the…
Quantum algebras (also called quantum groups) are deformed versions of the usual Lie algebras, to which they reduce when the deformation parameter q is set equal to unity. From the mathematical point of view they are Hopf algebras. Their…
The appearance of quantum groups in conformal field theories is traced back to the Poisson-Lie symmetries of the classical chiral theory. A geometric quantization of the classical theory deforms the Poisson-Lie symmetries to the quantum…
The Quantum Approximate Optimization Algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ans\"atze for…
The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups…
Variational quantum algorithms (VQAs) have emerged as a leading paradigm in near-term quantum computing, yet their performance can be hindered by the so-called barren plateau problem, where gradients vanish exponentially with system size or…
Quantum neural networks (QNNs) leverage quantum entanglement and superposition to enable large-scale parallel linear computation, offering a potential solution to the scalability limits of classical deep learning. However, their practical…
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential…
We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic…
There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such…
The performance of the Quantum Approximate Optimization Algorithm (QAOA) is closely tied to the structure of the dynamical Lie algebra (DLA) generated by its Hamiltonians, which determines both its expressivity and trainability. In this…
Lie Algebras are powerful mathematical structures used in physics to describe sets of operators and associated combinations. A central task is to identify a minimal set of generators from which the algebra can be constructed. The classical…
In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on…
In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…
We introduce several probabilistic quantum algorithms that overcome the normal unitary restrictions in quantum machine learning by leveraging the Linear Combination of Unitaries (LCU) method. Among our investigations are quantum native…
We introduce a generalization of Lie algebras within the theory of nonhomogeneous quadratic algebras and point out its relevance in the theory of quantum groups. In particular the relation between the differential calculus on quantum group…
Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our geometric (Clifford)…
This survey is about the fundamentals of the theory of finite dimensional Lie groups over the field of real numbers. The notion of the tangent space of a manifold at a point is considered to be defined via the well known chart and vector…
The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a quantum computational framework for solving quantum or classical optimization tasks. Here, we explore using QAOA for Binary Linear Least Squares (BLLS); a problem…
Combining classical optimization with parameterized quantum circuit evaluation, variational quantum algorithms (VQAs) are among the most promising algorithms in near-term quantum computing. Similar to neural networks (NNs), VQAs iteratively…