Quantum Algorithms for Identifying Hidden Strings with Applications to Matroid Problems

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of identifying a pair of $n$-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string $s, s'\in\{0, 1\}^n$ satisfying the above constraints, for any $x\in\{0, 1\}^n$ the max inner product oracle $O_{max}(x)$ returns the max value between $s\cdot x$ and $s'\cdot x$, and the sub-set oracle $O_{sub}(x)$ indicates whether the index set of the 1s in $x$ is a subset of that in $s$ or $s'$. We present a quantum algorithm consuming $O(1)$ queries to the max inner product oracle for identifying the pair $\{s, s'\}$, and prove that any classical algorithm requires $\Omega(n/\log_{2}n)$ queries. Also, we present a quantum algorithm consuming $\frac{n}{2}+O(\sqrt{n})$ queries to the subset oracle, and prove that any classical algorithm requires at least $n+\Omega(1)$ queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called $k$-bases if it has $k$ bases

Compressibility-Aware Quantum Algorithms on Strings

Sublinear time quantum algorithms have been established for many fundamental problems on strings. This work demonstrates that new, faster quantum algorithms can be designed when the string is highly compressible. We focus on two popular and theoretically significant compression algorithms -- the Lempel-Ziv77 algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT), and obtain the results below. We first provide a quantum algorithm running in $\tilde{O}(\sqrt{zn})$ time for finding the LZ77 factorization of an input string $T[1..n]$ with $z$ factors. Combined with multiple existing results, this yields an $\tilde{O}(\sqrt{rn})$ time quantum algorithm for finding the RL-BWT encoding with $r$ BWT runs. Note that $r = \tilde{\Theta}(z)$. We complement these results with lower bounds proving that our algorithms are optimal (up to polylog factors). Next, we study the problem of compressed indexing, where we provide a $\tilde{O}(\sqrt{rn})$ time quantum algorithm for constructing a recently designed $\tilde{O}(r)$ space structure with equivalent capabilities as the suffix tree. This data structure is then applied to numerous problems to obtain sublinear time quantum algorithms when the input is highly compressible. For example, we show that the longest common substring of two strings of total length $n$ can be computed in $\tilde{O}(\sqrt{zn})$ time, where $z$ is the number of factors in the LZ77 factorization of their concatenation. This beats the best known $\tilde{O}(n^\frac{2}{3})$ time quantum algorithm when $z$ is sufficiently small

High Performance Computing for DNA Sequence Alignment and Assembly

Recent advances in DNA sequencing technology have dramatically increased the scale and scope of DNA sequencing. These data are used for a wide variety of important biological analyzes, including genome sequencing, comparative genomics, transcriptome analysis, and personalized medicine but are complicated by the volume and complexity of the data involved. Given the massive size of these datasets, computational biology must draw on the advances of high performance computing. Two fundamental computations in computational biology are read alignment and genome assembly. Read alignment maps short DNA sequences to a reference genome to discover conserved and polymorphic regions of the genome. Genome assembly computes the sequence of a genome from many short DNA sequences. Both computations benefit from recent advances in high performance computing to efficiently process the huge datasets involved, including using highly parallel graphics processing units (GPUs) as high performance desktop processors, and using the MapReduce framework coupled with cloud computing to parallelize computation to large compute grids. This dissertation demonstrates how these technologies can be used to accelerate these computations by orders of magnitude, and have the potential to make otherwise infeasible computations practical

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Initial state preparation for quantum chemistry on quantum computers

Quantum algorithms for ground-state energy estimation of chemical systems require a high-quality initial state. However, initial state preparation is commonly either neglected entirely, or assumed to be solved by a simple product state like Hartree-Fock. Even if a nontrivial state is prepared, strong correlations render ground state overlap inadequate for quality assessment. In this work, we address the initial state preparation problem with an end-to-end algorithm that prepares and quantifies the quality of initial states, accomplishing the latter with a new metric -- the energy distribution. To be able to prepare more complicated initial states, we introduce an implementation technique for states in the form of a sum of Slater determinants that exhibits significantly better scaling than all prior approaches. We also propose low-precision quantum phase estimation (QPE) for further state quality refinement. The complete algorithm is capable of generating high-quality states for energy estimation, and is shown in select cases to lower the overall estimation cost by several orders of magnitude when compared with the best single product state ansatz. More broadly, the energy distribution picture suggests that the goal of QPE should be reinterpreted as generating improvements compared to the energy of the initial state and other classical estimates, which can still be achieved even if QPE does not project directly onto the ground state. Finally, we show how the energy distribution can help in identifying potential quantum advantage