Another step toward quantum computers

Carlos Alberto Cadavid, Paulina Hoyos, Jay A. Jorgenson +2 more

In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most $O((\log N)^{2})$ deterministic steps and at most $O((\log N)^{2})$ diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most $O((\log N)^{2}\log (\log N) \log (\log \log N))$ quantum steps on a quantum computer. As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of $N=33$ and $N=1363$.

Anders Søndberg Sørensen

Varun Garg, Anupama Pande

Abstract—Quantum mechanical computers unlike classical computers can be in a superposition of states and carry out multiple operations at the same time. Grover has shown that a quantum computer can do a random database search in O ( √ N) steps. He has further proved that using the superposition of the states in quantum computing, a quantum computer should be able to do a search in a single step. He has stated that the query generated to do this search will require Ω(NlogN) steps. In this paper we aim at optimizing the Grover’s search algorithm. In our algorithm, we have repeated the inversion step a number of times instead of stopping after a single step. Measurement after a single step required a larger number of subsystems which impaired the effectiveness of the algorithm leading to Ω(NlogN) steps in the preparation and processing of the query. Repetition of the inversion step brought the the number of subsystems down drastically by increasing the amplitude of the desired state.We have proved that by doing so the overall effectiveness of Grover’s algorithm can be brought down to Ω ( 3 √ N). Keywords: Quantum mechanical computers, Grover’s search algorithm, inversion step, probability and amplitude.

Ranjani G. Sundaram, Himanshu Gupta, C. R. Ramakrishnan

Quantum computing hardware is improving in robustness, but individual computers still have small number of qubits (for storing quantum information). Computations needing a large number of qubits can only be performed by distributing them over a network of smaller quantum computers. In this paper, we consider the problem of distributing a quantum computation, represented as a quantum circuit, over a homogeneous network of quantum computers, minimizing the number of communication operations needed to complete every step of the computation. We propose a two-step solution: dividing the given circuit’s qubits among the computers in the network, and scheduling communication operations, called migrations, to share quantum information among the computers to ensure that every operation can be performed locally. While the first step is an intractable problem, we present a polynomial-time solution for the second step in a special setting, and a O(log n)-approximate solution in the general setting. We provide empirical results which show that our two-step solution outperforms existing heuristic for this problem by a significant margin (up to 90%, in some cases).