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Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It…
Pascal's triangle is widely used as a pedagogical tool to explain the "first-order" multiplet patterns that arise in the spectra of $I_N S$ coupled spin-1/2 systems in magnetic resonance. Various other combinatorial structures, which may be…
We show in detail how the Jordan-Wigner transformation can be used to simulate any fermionic many-body Hamiltonian on a quantum computer. We develop an algorithm based on appropriate qubit gates that takes a general fermionic Hamiltonian,…
Physical systems, characterized by an ensemble of interacting elementary constituents, can be represented and studied by different algebras of observables or operators. For example, a fully polarized electronic system can be investigated by…
We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form for evaluation on quantum computers. Symmetries, such as point-group symmetries in molecules, are apparent in the…
We investigate the simulation of fermionic systems on a quantum computer. We show in detail how quantum computers avoid the dynamical sign problem present in classical simulations of these systems, therefore reducing a problem believed to…
We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive…
We propose models of quantum neural networks through Clifford algebras, which are capable of capturing geometric features of systems and to produce entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras…
Understanding and predicting the properties of solid-state materials from first-principles has been a great challenge for decades. Owing to the recent advances in quantum technologies, quantum computations offer a promising way to achieve…
Recently, we have used a projection operator to fix the number of particles in a second quantization approach in order to deal with the canonical ensemble. Having been applied earlier to handle various problems in nuclear physics that…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can results in significant overhead due to the exponentially growing size of some…
The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity…
Significant developments made in quantum hardware and error correction recently have been driving quantum computing towards practical utility. However, gaps remain between abstract quantum algorithmic development and practical applications…
The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
In these lectures we develop the projection operator method for quantum groups. Here the term "quantum groups" means q-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. Contains of the lectures…
Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are…
Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of…
This paper introduces a novel abstraction for programming quantum operations, specifically projective Cliffords, as functions over the qudit Pauli group. Generalizing the idea behind Pauli tableaux, we introduce a type system and lambda…