Systematic Analysis of Majorization in Quantum Algorithms

Motivated by the need to uncover some underlying mathematical structure of optimal quantum computation, we carry out a systematic analysis of a wide variety of quantum algorithms from the majorization theory point of view. We conclude that step-by-step majorization is found in the known instances of fast and efficient algorithms, namely in the quantum Fourier transform, in Grover's algorithm, in the hidden affine function problem, in searching by quantum adiabatic evolution and in deterministic quantum walks in continuous time solving a classically hard problem. On the other hand, the optimal quantum algorithm for parity determination, which does not provide any computational speed-up, does not show step-by-step majorization. Lack of both speed-up and step-by-step majorization is also a feature of the adiabatic quantum algorithm solving the 2-SAT ``ring of agrees'' problem. Furthermore, the quantum algorithm for the hidden affine function problem does not make use of any entanglement while it does obey majorization. All the above results give support to a step-by-step Majorization Principle necessary for optimal quantum computation.Comment: 15 pages, 14 figures, final versio

Ising formulations of many NP problems

We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more references; v3: substantial revision and extension, to-be-published versio

A hybrid algorithm framework for small quantum computers with application to finding Hamiltonian cycles

Recent works have shown that quantum computers can polynomially speed up certain SAT-solving algorithms even when the number of available qubits is significantly smaller than the number of variables. Here we generalise this approach. We present a framework for hybrid quantum-classical algorithms which utilise quantum computers significantly smaller than the problem size. Given an arbitrarily small ratio of the quantum computer to the instance size, we achieve polynomial speedups for classical divide-and-conquer algorithms, provided that certain criteria on the time- and space-efficiency are met. We demonstrate how this approach can be used to enhance Eppstein's algorithm for the cubic Hamiltonian cycle problem, and achieve a polynomial speedup for any ratio of the number of qubits to the size of the graph.Comment: 20+2 page

QuASeR -- Quantum Accelerated De Novo DNA Sequence Reconstruction

In this article, we present QuASeR, a reference-free DNA sequence reconstruction implementation via de novo assembly on both gate-based and quantum annealing platforms. Each one of the four steps of the implementation (TSP, QUBO, Hamiltonians and QAOA) is explained with simple proof-of-concept examples to target both the genomics research community and quantum application developers in a self-contained manner. The details of the implementation are discussed for the various layers of the quantum full-stack accelerator design. We also highlight the limitations of current classical simulation and available quantum hardware systems. The implementation is open-source and can be found on https://github.com/prince-ph0en1x/QuASeR.Comment: 24 page

Exponential algorithmic speedup by quantum walk

We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification

Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm