相关论文: Quantum Algorithms and the Fourier Transform

Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of…

The Quantum Fourier Transform (QFT) is a key component of many important quantum algorithms, most famously as being the essential ingredient in Shor's algorithm for factoring products of primes. Given its remarkable capability, one would…

A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum…

Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups…

Shor's quantum factoring algorithm finds the prime factors of a large number exponentially faster than any other known method a task that lies at the heart of modern information security, particularly on the internet. This algorithm…

We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast…

We report on the current state of factoring integers on both digital and analog quantum computers. For digital quantum computers, we study the effect of errors for which one can formally prove that Shor's factoring algorithm fails. For…

The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely…

The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…

Besides the superior efficiency compared to their classical counterparts, quantum algorithms known so far are basically task-dependent, and scarcely any common features are shared between them. In this work, however, we show that the…

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…

Simon's problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor's algorithm. The generalized Simon's problem is a natural extension of Simon's problem, and…

The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the…

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of…

This work presents a precise connection between Clifford circuits, Shor's factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems. We show that all…

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform…

In recent decades, the field of quantum computing has experienced remarkable progress. This progress is marked by the superior performance of many quantum algorithms compared to their classical counterparts, with Shor's algorithm serving as…

These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states…

Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any…