Related papers: Quantum Speedup and Categorical Distributivity
The number of steps any classical computer requires in order to find the prime factors of an $l$-digit integer $N$ increases exponentially with $l$, at least using algorithms known at present. Factoring large integers is therefore…
Quantum computing represents a significant advancement in computational capabilities. Of particular concern is its impact on asymmetric cryptography through, notably, Shor's algorithm and the more recently developed Regev's algorithm for…
We study some extensions of Grover's quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching…
Quantum computing devices are believed to be powerful in solving the prime factorization problem, which is at the heart of widely deployed public-key cryptographic tools. However, the implementation of Shor's quantum factorization algorithm…
We present a quantum algorithm solving the greatest common divisor (GCD) problem. This quantum algorithm possesses similar computational complexity with classical algorithms, such as the well-known Euclidean algorithm for GCD. This…
It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the…
We demonstrate that, in the case of Shor's algorithm for factoring, highly mixed states will allow efficient quantum computation, indeed factorization can be achieved efficiently with just one initial pure qubit and a supply of initally…
Solving the discrete logarithm problem (DLP) with quantum computers is a fundamental task with important implications. Beyond Shor's algorithm, many researchers have proposed alternative solutions in recent years. However, due to current…
The quantum computer algorithm by Peter Shor for factorization of integers is studied. The quantum nature of a QC makes its outcome random. The output probability distribution is investigated and the chances of a successful operation is…
We perform logical and physical resource estimation for computing binary elliptic curve discrete logarithms using Shor's algorithm on fault-tolerant quantum computers. We adopt a windowed approach to design our circuit implementation of the…
This paper studies the quantum computational complexity of the discrete logarithm (DL) and related group-theoretic problems in the context of generic algorithms -- that is, algorithms that do not exploit any properties of the group…
The quantum multicomputer consists of a large number of small nodes and a qubus interconnect for creating entangled state between the nodes. The primary metric chosen is the performance of such a system on Shor's algorithm for factoring…
This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the…
For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer…
In this work, we present a multi-layer quantum search method that generates an exponential speedup of the standard Grover's algorithm. As direct applications, any NP problems can be solved efficiently on a quantum circuit with only…
We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order $r$ in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the…
We consider a probabilistic quantum implementation of a variable of the Pocklington-Lehmer $N-1$ primality test using Shor's algorithm. O($\log^3 N \log\log N \log\log\log N$) elementary q-bit operations are required to determine the…
In this paper, we use the methods found in quant-ph/0201095 to create a continuous variable analogue of Shor's quantum factoring algorithm. By this we mean a quantum hidden subgroup algorithm that finds the period P of a function F:R-->R…
In this note we consider optimised circuits for implementing Shor's quantum factoring algorithm. First I give a circuit for which none of the about 2n qubits need to be initialised (though we still have to make the usual 2n measurements…
We apply majorization theory to study the quantum algorithms known so far and find that there is a majorization principle underlying the way they operate. Grover's algorithm is a neat instance of this principle where majorization works step…