Related papers: Quantum polylogarithms
We define formally decohered quantum computers (using density matrices), and present a simulation of them by a probabalistic classical Turing Machine. We study the slowdown of the simulation for two cases: (1) sequential quantum computers,…
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…
In this paper we describe a quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials. The algorithm is nondeterministic and its expected resource requirements are polylogarithmic in…
It is proposed that the ability for a quantum circuit to thermalize under time evolution is a valid way to compute linear algebra problems. The algorithm makes use of the eigenstate thermalization hypothesis and full ergodicity in quantum…
Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of…
We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…
A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space…
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial…
Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F_q. Any bounded-error classical algorithm for this task requires Omega(n^d) queries…
Littlewood-Richardson, Kronecker and plethysm coefficients are fundamental multiplicities of interest in Representation Theory and Algebraic Combinatorics. Determining a combinatorial interpretation for the Kronecker and plethysm…
Mixed Tate motives are central objects in the study of cohomology groups of algebraic varieties and their arithmetic invariants. They also play a crucial role in a wide variety of questions related to multiple zeta values and…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum…
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the…
Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We develop a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory…