Related papers: Algorithm for computing canonical bases and foldin…
Mappings of classical computation onto statistical mechanics models have led to remarkable successes in addressing some complex computational problems. However, such mappings display thermodynamic phase transitions that may prevent reaching…
For $\imath$quantum covering groups $(\mathbf{U}, \mathbf{U}^\imath)$ of super Kac-Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as…
The recent advent of commercially available quantum annealing hardware (QAH) has expanded opportunities for research into quantum annealing-based algorithms. In the domain of power systems, this advancement has driven increased interest in…
Quantum advantage in computation refers to the existence of computational tasks that can be performed efficiently on a quantum computer but cannot be efficiently simulated on any classical computer. Identifying the precise boundary of…
The $\imath$quantum groups admit two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Based on these two realizations, we…
Within the framework of the individuality interpretation of quantum theory (QT), the basic equations of QT cannot be derived from the basic equations of classical mechanics (CM). The unbridgeable gap between CM and QT is given by the fact…
The matrix functions can be defined by Cauchy's integral formula and can be approximated by the linear combination of inverses of shifted matrices using a quadrature formula. In this paper, we show a concrete construction of a framework to…
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending…
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in…
We explore a way of universal quantum computation with particles which cannot occupy the same position simultaneously and are symmetric under exchange of particle labels. Therefore the associated creation and annihilation operators are…
We present elementary mappings between classical lattice models and quantum circuits. These mappings provide a general framework to obtain efficiently simulable quantum gate sets from exactly solvable classical models. For example, we…
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…
Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke…
For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie…
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…
It is NP-complete to find non-negative factors $W$ and $H$ with fixed rank $r$ from a non-negative matrix $X$ by minimizing $\|X-WH^\top\|_F^2$. Although the separability assumption (all data points are in the conical hull of the extreme…
Error correction will add so much overhead to large quantum computations that we suspect the most efficient algorithms will use a classical co-processor to do as much work as possible. We present a method to offload portions of a quantum…
It is shown how nonlinear versions of quantum mechanics can be refolmulated in terms of a (linear) C*-algebraic theory. Then also their symmetries are described as automorphisms of the correspondong C*-algebra. The requirement of…
We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any…
This article gives conjecturally correct algorithms to construct canonical bases of the irreducible polynomial representations and the matrix coordinate rings of the nonstandard quantum groups in GCT4 and GCT7, and canonical bases of the…