Related papers: An explicit family of unitaries with exponentially…
We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We compute a possibly small…
The paper is devoted to the study of geodesic orbit Riemannian metrics on nilpotent Lie groups. The main result is the construction of continuous families of pairwise non-isomorphic connected and simply connected nilpotent Lie groups, every…
We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one…
In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class. Here we apply this viewpoint to geometry and initiate a program…
The Pauli matrices are a set of three 2x2 complex Hermitian, unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum…
In this paper, we design quantum circuits for the exponential of scaled $n$-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of…
We construct explicit non-isotrivial families of polynomials over $\mathbb{Q}$ satisfying uniform boundedness for their rational preperiodic points.
Let $A$ be an associative algebra with a superinvolution $*$ over a field of characteristic zero, and let $c_n^*(A)$, $n = 1, 2, \ldots$, denote its sequence of $*$-codimensions. It is well known that this sequence is either polynomially…
We realize the Pauli group $P$ as Galois group of polynomials over the rational numbers. It is shown by construction that each pure polynomial in the infinite family of the form $X^8+k^2$ for $k\neq \lambda^2, 2\lambda^2; k,\lambda \in…
In this paper we use Euler-Seidel matrices method to find out some properties of exponential and geometric polynomials and numbers. Some known results are reproved and some new results are obtained.
We present a general procedure for determining possible (nonuniform) magnetic fields such that the Pauli equation becomes quasi-exactly solvable (QES) with an underlying $sl(2)$ symmetry. This procedure makes full use of the close…
When the gate set has continuous parameters, synthesizing a unitary operator as a quantum circuit is always possible using exact methods, but finding minimal circuits efficiently remains a challenging problem. The landscape is very…
A variety of quantum algorithms employ Pauli operators as a convenient basis for studying the spectrum or evolution of Hamiltonians or measuring multi-body observables. One strategy to reduce circuit depth in such algorithms involves…
We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates $C_{\delta}(U)$ to approximate a unitary $U$ up to an error of $\delta$ in operator norm distance. More precisely we show two bounds…
We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…
In this work, we study distributions of unitaries generated by random quantum circuits containing only symmetry-respecting gates. We develop a unified approach applicable to all symmetry groups and obtain an equation that determines the…
We construct Euclidean lattices whose sets of minimal vectors support some large equiangular families of lines, using notably reduction modulo~$2$ of lattices. %as considered in \cite{Ma1} and \cite{Ma2}. We also consider some related…
Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…
Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group…
Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit…