Related papers: Group-theoretical analysis of quantum complexity: …
In Nielsen's geometric approach to quantum complexity, the introduction of a suitable geometrical space, based on the Lie group formed by fundamental operators, facilitates the identification of complexity through geodesic distance in the…
According to the pioneering work of Nielsen and collaborators, the length of the minimal geodesic in a geometric realization of a suitable operator space provides a measure of the quantum complexity of an operation. Compared with the…
Nielsen's geometric approach offers a powerful framework for quantifying the complexity of unitary transformations. In this formulation, complexity is defined as the length of the minimal geodesic in a suitably constructed geometric space…
The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion…
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
Recently it has been shown that the complexity of SU($n$) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one…
It is common practice to describe elementary particles by irreducible unitary representations of the Poincar\'e group. In the same way, multi-particle systems can be described by irreducible unitary representations of the Poincar\'e group.…
In this review, we have reached from the most basic definitions in the theory of groups, group structures, etc. to representation theory and irreducible representations of the Poincar'e group. Also, we tried to get a more comprehensible…
A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the…
Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it continuous} systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the…
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite…
Understanding the complexity of quantum states and circuits is a central challenge in quantum information science, with broad implications in many-body physics, high-energy physics and quantum learning theory. A common way to model the…
A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum…
The problem of how to obtain quasi-classical states for quantum groups is examined. A measure of quantum indeterminacy is proposed, which involves expectation values of some natural quantum group operators. It is shown that within any…
It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The…
Mathematical core of quantum mechanics is the theory of unitary representations of symmetries of physical systems. We argue that quantum behavior is a natural result of extraction of "observable" information about systems containing…
The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the…
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this…
Gaussian unitaries, generated by quadratic Hamiltonians, are fundamental in quantum optics and continuous-variable computing. Their structures correspond to symplectic (bosons) and orthogonal (fermions) groups, but physical realizations…
With the aim to construct a dynamical model with quantum group symmetry, the $q$-deformed Schr\"odinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space is investigated. After reviewing the differential…