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arXiv:1505.01116

Approaching unstructured search from function bilateral symmetry detection - A quantum algorithm

Dinesh Kumar, Srivastava, Pankaj

Published May 3, 2015• 0 citations
arXivHTMLPDF

Abstract

Detection of symmetry is vital to problem solving. Most of the problems of computer vision and computer graphics and machine intelligence in general, can be reduced to symmetry detection problem. Unstructured search problem can also be looked upon from symmetry detection point of view. Unstructured search can be thought as searching a binary string satisfying some search condition in an unsorted list of binary strings. In this paper unstructured search problem is reduced to function bilateral symmetry detection problem with polynomial overhead in terms of the size of the input. Keywords: Unstructured Search, Quantum algorithm, Function bilateral symmetry detection, Decision Problem, Quantum Black Box, Solving NP complete problems.

DOI
10.48550/arxiv.1505.01116

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Detection of symmetry is vital to problem solving. Most of the problems of computer vision and computer graphics and machine intelligence in general, can be reduced to symmetry detection problem. Unstructured search problem can also be looked upon from symmetry detection point of view. Unstructured search can be thought as searching a binary string satisfying some search condition in an unsorted list of binary strings. In this paper unstructured search problem is reduced to function bilateral symmetry detection problem with polynomial overhead in terms of the size of the input. Keywords: Unstructured Search, Quantum algorithm, Function bilateral symmetry detection, Decision Problem, Quantum Black Box, Solving NP complete problems.
We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, f(x)f(\mathbf{x})f(x), such as the objective functions in combinatorial and other discrete optimisation problems. The first step of the Quantum Search for Extrema and Roots Algorithm (QSERA) is to translate conditions of the form f(x∗)≃f∗f(\mathbf{x}_*) \simeq f_*f(x∗​)≃f∗​, where f∗f_*f∗​ is the extremum or zero, to an unstructured search problem for x∗\mathbf{x}_*x∗​. This is achieved by mapping f(x)f(\mathbf{x})f(x) to a function u(z)u(z)u(z) to create a quantum oracle, such that u(z∗)=1u(z_*) = 1u(z∗​)=1 and u(z≠z∗)=0u(z \neq z_*) = 0u(z=z∗​)=0. The next step is to employ Grover's algorithm to find z∗z_*z∗​, which offers a quadratic speed-up over classical algorithms. The number of operations needed to map f(x)f(\mathbf{x})f(x) to u(z)u(z)u(z) determines the accuracy of the result and the circuit depth. We describe the implementation of QSERA by assembling a quantum circuit for portfolio optimisation, which can be formulated as a combinatorial problem. QSERA can handle objective functions with higher order terms than the commonly-used Quadratic Unconstrained Binary Optimisation (QUBO) framework. Moreover, while QSERA requires some a priori knowledge of the extrema of f(x)f(\mathbf{x})f(x), it can still find approximate solutions even if the conditions are not exactly satisfied.
We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computational complexity of NP-Complete problems after the adaptations of both the classical as well as quantum [1] search algorithms remains of the exponential order as the exponent for quantum [1] algorithm changes only to one-half times the exponent for classical algorithm. But for our quantum algorithms the exponent falls substantially so that our new quantum algorithms for unstructured search are capable if reducing the computational complexity of NP-Complete problems to polynomial order!
Given an item and a list of values of size NNN. It is required to decide if such item exists in the list. Classical computer can search for the item in O(N). The best known quantum algorithm can do the job in O(N)O(\sqrt{N})O(N​). In this paper, a quantum algorithm will be proposed that can search an unstructured list in O(1) to get the YES/NO answer with certainty.
The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speedup over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.
Formulation symmetry in mixed-integer programming (MIP) can hinder solver performance by inducing redundant search, but detecting such symmetries is also a significant computational challenge. This paper explores the potential for quantum computing to handle symmetry detection. Quantum is a promising alternative to classical compute, but this emerging technology has limited hardware capacity in terms of input problem size. This paper explores the use of Quadratic Unconstrained Binary Optimization (QUBO) models for symmetry detection, as QUBO is the canonical format for quantum optimization platforms. To help address the input size bottleneck, we develop full, reduced, and decomposed QUBO as well as QUBO-Plus formulations for MIP symmetry detection. Computational experiments on the MIPLIB 2017 benchmark are used to estimate the quantum computing resources needed for practical problems.
Grover’s algorithm is a quantum search algorithm that makes a huge difference in finding an element of interest from unsorted and unstructured data. In the case of a classical search algorithm, testing O(N) elements within a dataset of size N is a significant number. Grover’s algorithm reduces this number to O(N)O(\sqrt N )O(N​) so the gain is obvious. This paper covers applied aspects of Grover’s algorithm, namely quantum circuit layout, efficiency of gates, fidelity of qubits, and the optimization of resources for searching unstructured data. Focus has been placed on quantum error correction: namely, planar surface code; resource costs involved have been quantified. The paper discusses the implications of quantum computing on data retrieval and how algorithms in quantum computing are crucial for solving issues presented by a changing environment of computation.