Related papers: Algebra, Logic and Qubits: Quantum Abacus
We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all…
We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $\gamma^a$ with the property $\{\gamma^a,\gamma^b\}_+ = 2 \eta^{ab}$, for representing quantum gates and quantum…
Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Important examples include QCD and other field theories at non-zero chemical potential, at non-zero vacuum angle, or with an odd number of flavors, as…
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical…
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra…
We present a new approach to scalable quantum computing--a ``qubus computer''--which realises qubit measurement and quantum gates through interacting qubits with a quantum communication bus mode. The qubits could be ``static'' matter qubits…
Quantum computation often suffers from artificial symmetry breaking. We should strive to suppress the artifact both by theoretical and technological improvements. The theoretical formalism of the lattice fermion with exact chiral symmetry…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
Case-Based Reasoning (CBR) is an artificial intelligence approach to problem-solving with a good record of success. This article proposes using Quantum Computing to improve some of the key processes of CBR, such that a Quantum Case-Based…
The states of the physical algebra, namely the algebra generated by the operators involved in encoding and processing qubits, are considered instead of those of the whole system-algebra. If the physical algebra commutes with the interaction…
We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each…
We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator…
The author introduces the notion of a quantum form of an algebraic torus. In the case of diagonal algebraic torus we get the algebra of Laurent twisted polynomials. Quantum algebraic torus can be characterized in terms of exact sequences.…
All classical Lie algebras can be realized \`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their $q$-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on…
Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A and B in the even and odd parity subspaces respectively, of two qubits. Using a Clifford algebra formalism we show that arbitrary uniform families of circuits of…
The physical interpretation of the main notions of the quantum group theory (coproduct, representations and corepresentations, action and coaction) is discussed using the simplest examples of $q$-deformed objects (quantum group…
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions…
{Although q-oscillators have been used extensively for realization of quantum universal enveloping algebras,such realization do not exist for quantum matrix algebras ( deformation of the algebra of functions on the group ). In this paper we…
There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way.…
Quantum contextuality, a fundamental feature distinguishing quantum theory from classical models, is investigated via algebraic and topological structures inherent in modular tensor categories. This work rigorously demonstrates that braid…