Related papers: Shor's Factoring Algorithm and Modular Exponentiat…
Integer factorization is a significant problem, with implications for the security of widely-used cryptographic schemes. No efficient classical algorithm for polynomial-time integer factorization has been found despite extensive research.…
We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…
Richard Feynman's observation that quantum mechanical effects could not be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used quantum effects. This speculation…
The operator in a parton shower algorithm that represents the imaginary part of virtual Feynman graphs has a non-trivial color structure and is large because it is proportional to a factor of $4\pi$. In order to improve the treatment of…
Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum…
A common starting point of traditional quantum algorithm design is the notion of a universal quantum computer with a scalable number of qubits. This convenient abstraction mirrors classical computations manipulating finite sets of symbols,…
We compute the $n_h$ terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals…
Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\"o}nhage and Strassen designed an algorithm that improved the multiplication time for…
Reasoning about quantum programs remains a fundamental challenge, regardless of the programming model or computational paradigm. Despite extensive research, existing verification techniques are insufficient -- even for quantum circuits, a…
Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed.…
The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their…
The Quantum Fourier Transform (QFT) is required by hidden subgroup problem (HSP) algorithms, including Shor's algorithm for factoring. The circuit depth of the QFT remains challenging for near-term hardware. To find shallower alternatives…
Order finding is the core subroutine of Shor's algorithm. On NISQ hardware, phase estimation output distributions are often distorted by noise, making correct order recovery difficult. We study recoverability in noisy order finding: given a…
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which…
The largest number factored on a quantum device reported until now was 143. That quantum computation, which used only 4 qubits at 300K, actually also factored much larger numbers such as 3599, 11663, and 56153, without the awareness of the…
We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of…
Theory of computer calculations strongly depends on the nature of elements the computer is made of. Quantum interference allows to formulate the Shor factorization algorithm turned out to be more effective than any one written for classical…
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…
In this note we describe a simple and intriguing observation: the quantum Fourier transform (QFT) over $Z_q$, which is considered the most ``quantum'' part of Shor's algorithm, can in fact be simulated efficiently by classical computers.…
We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a…