Related papers: Revisiting Shor's quantum algorithm for computing …
Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but these are only useful if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a…
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…
Shor's algorithm is one of the most prominent quantum algorithms, yet finding efficient implementations remains an active research challenge. While many approaches focus on low-level modular arithmetic optimizations, a broader perspective…
We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can…
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as…
Pollard's Rho is a method for solving the integer factorization problem. The strategy searches for a suitable pair of elements belonging to a sequence of natural numbers that given suitable conditions yields a nontrivial factor. In…
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…
It is currently known from the work of Shoup and Nechaev that a generic algorithm to solve the discrete logarithm problem in a group of prime order must have complexity at least $k\sqrt{N}$ where $N$ is the order of the group. In many…
Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum algorithm for fast number factoring is a key example and the prime motivator in the…
Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular…
In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His…
We show that a classical algorithm efficiently simulating the modular exponentiation circuit, for certain product state input and with measurements in a general product state basis at the output, can efficiently simulate Shor's factoring…
Shor's algorithm contains a classical post-processing part for which we aim to create an efficient, understandable method aside from continued fractions. Let r be an unknown positive integer. Assume that with some constant probability we…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
We report an experimental demonstration of a complied version of Shor's algorithm using four photonic qubits. We choose the simplest instance of this algorithm, that is, factorization of N=15 in the case that the period $r=2$ and exploit a…
Conditions on sure-success decidability of weights of Boolean functions are presented for a given number of generalized Grover iterations. It is shown that the decidability problem reduces to a system of algebraic equations of a single…
Shor's quantum factoring algorithm and a few other efficient quantum algorithms break many classical crypto-systems. In response, people proposed post-quantum cryptography based on computational problems that are believed hard even for…
Grover's search algorithm is one of the first quantum algorithms to exhibit a provable quantum advantage. It forms the backbone of numerous quantum applications and is widely used in benchmarking efforts. Here, we report…
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate…
The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S.…