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We consider the problem of approximating arbitrary single-qubit z-rotations by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the…
Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work,…
Quantum algorithms to solve practical problems in quantum chemistry, materials science, and matrix inversion often involve a significant amount of arithmetic operations which act on a superposition of inputs. These have to be compiled to a…
This paper revisits the little-known Gibbs-Rodrigues representation of rotations in a three-dimensional space and demonstrates a set of algorithms for handling it. In this representation the rotation is itself represented as a…
In the pursuit of quantum advantage for financial engineering, researchers face a critical dilemma: analog rotation gates suffer from inherent 'sine-to-square' biases and error magnification, while digital arithmetic circuits (e.g.,…
The standard particle-in-cell algorithm suffers from grid heating. There exists a gridless alternative which bypasses the deposition step and calculates each Fourier mode of the charge density directly from the particle positions. We show…
We give three new algorithms for efficient in-place estimation, without using ancilla qubits, of average fidelity of a quantum logic gate acting on a d-dimensional system using much fewer random bits than what was known so far. Previous…
Recently, Armstrong, Guzm\'an, and Sing Long (2021), presented an optimal $O(n^2)$ time algorithm for strict circular seriation (called also the recognition of strict quasi-circular Robinson spaces). In this paper, we give a very simple…
Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to…
Active set method aims to find the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problem with bound constraints. To guarantee the finite step convergence, the existing active…
We describe an efficient quantum algorithm for the quantum Schur transform. The Schur transform is an operation on a quantum computer that maps the standard computational basis to a basis composed of irreducible representations of the…
We discuss efficient quantum logic circuits which perform two tasks: (i) implementing generic quantum computations and (ii) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the…
Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity…
In this study, we first use a three-qubit system as an example to demonstrate the construction of quantum circuits for the eight maximally entangled basis vectors, subsequently extending the approach to N-qubit systems. We employ a…
A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In spite of being an integral part of bundle adjustment and…
A new method to represent and approximate rotation matrices is introduced. The method represents approximations of a rotation matrix $Q$ with linearithmic complexity, i.e. with $\frac{1}{2}n\lg(n)$ rotations over pairs of coordinates,…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be…
We introduce a class of algorithms for constructing Fourier representations of Gaussian processes in $1$ dimension that are valid over ranges of hyperparameter values. The scaling and frequencies of the Fourier basis functions are evaluated…