Quantum walk speedup of backtracking algorithms

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.Comment: 23 pages; v2: minor changes to presentatio

Transition Probabilities in Generalized Quantum Search Hamiltonian Evolutions

A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schr\"odinger's quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multi-parameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi-Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.Comment: 17 pages, 6 figures, 3 tables. Online ready in Int. J. Geometric Methods in Modern Physics (2020

Quantum Algorithms for Matrix Problems and Machine Learning

This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.Science and Engineering Research Board (Department of Science and Technology), Government of Indi

Fault-ignorant Quantum Search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Grover's algorithm) still outperform the best noiseless classical search algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published version (Introduction section significantly expanded, presentation clarified, results and order unchanged

Improved Quantum Query Complexity on Easier Inputs

Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].Comment: 35 pages, 2 figures. This article supersedes arXiv/2012.01276 (expanded author list, new application, improved algorithm