New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

Data Structures in Classical and Quantum Computing

This survey summarizes several results about quantum computing related to (mostly static) data structures. First, we describe classical data structures for the set membership and the predecessor search problems: Perfect Hash tables for set membership by Fredman, Koml\'{o}s and Szemer\'{e}di and a data structure by Beame and Fich for predecessor search. We also prove results about their space complexity (how many bits are required) and time complexity (how many bits have to be read to answer a query). After that, we turn our attention to classical data structures with quantum access. In the quantum access model, data is stored in classical bits, but they can be accessed in a quantum way: We may read several bits in superposition for unit cost. We give proofs for lower bounds in this setting that show that the classical data structures from the first section are, in some sense, asymptotically optimal - even in the quantum model. In fact, these proofs are simpler and give stronger results than previous proofs for the classical model of computation. The lower bound for set membership was proved by Radhakrishnan, Sen and Venkatesh and the result for the predecessor problem by Sen and Venkatesh. Finally, we examine fully quantum data structures. Instead of encoding the data in classical bits, we now encode it in qubits. We allow any unitary operation or measurement in order to answer queries. We describe one data structure by de Wolf for the set membership problem and also a general framework using fully quantum data structures in quantum walks by Jeffery, Kothari and Magniez