Related papers: Classical simulability and the significance of mod…
We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer…
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994,…
The quantum Fourier transform (QFT) is sometimes said to be the source of various exponential quantum speed-ups. In this paper we introduce a class of quantum circuits which cannot outperform classical computers even though the QFT…
We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…
A major obstacle to implementing Shor's quantum number-factoring algorithm is the large size of modular-exponentiation circuits. We reduce this bottleneck by customizing reversible circuits for modular multiplication to individual runs of…
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by…
The conventional Quantum Fourier Transform, with exponential speedup compared to the classical Fast Fourier Transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, the…
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…
We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in…
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…
It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of $log_2 N$ qubits in an…
Grover's algorithm relies on the superposition and interference of quantum mechanics, which is more efficient than classical computing in specific tasks such as searching an unsorted database. Due to the high complexity of quantum…
Simulating quantum systems constructively furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this letter, we directly simulate and explore the entanglement structure present in a…
The Quantum Fourier Transform (QFT) is a fundamental component of many quantum computing algorithms. In this paper, we present an alternative method for factoring this transformation. Inspired by this approach, we introduce a new quantum…
We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The…
We demonstrate that a tensor product structure and optical analogy of quantum entanglement can be obtained by introducing pseudorandom phase sequences into classical fields with two orthogonal modes. Using the classical analogy, we discuss…
We report the realization of a nuclear magnetic resonance (NMR) quantum computer which combines the quantum Fourier transform (QFT) with exponentiated permutations, demonstrating a quantum algorithm for order-finding. This algorithm has the…
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…
Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…
With the advancement of quantum technologies, there is a potential threat to traditional encryption systems based on integer factorization. Therefore, developing techniques for accurately measuring the performance of associated quantum…