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arXiv:1209.5932

Efficient quantum algorithm to construct arbitrary Dicke states

Kaushik Chakraborty, Byung-Soo Choi, Arpita Maitra, Subhamoy Maitra

Published Sep 26, 2012• 1 citation
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Abstract

In this paper, we study efficient algorithms towards the construction of any arbitrary Dicke state. Our contribution is to use proper symmetric Boolean functions that involve manipulations with Krawtchouk polynomials. Deutsch-Jozsa algorithm, Grover algorithm and the parity measurement technique are stitched together to devise the complete algorithm. Further, motivated by the work of Childs et al (2002), we explore how one can plug the biased Hadamard transformation in our strategy. Our work compares fairly with the results of Childs et al (2002).

DOI
10.48550/arxiv.1209.5932

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In this paper, we study efficient algorithms towards the construction of any arbitrary Dicke state. Our contribution is to use proper symmetric Boolean functions that involve manipulations with Krawtchouk polynomials. Deutsch-Jozsa algorithm, Grover algorithm and the parity measurement technique are stitched together to devise the complete algorithm. Further, motivated by the work of Childs et al (2002), we explore how one can plug the biased Hadamard transformation in our strategy. Our work compares fairly with the results of Childs et al (2002).
Quantum [Formula: see text]-qubit states that are totally symmetric under the permutation of qubits are essential ingredients of important algorithms and applications in quantum information. Consequently, there is significant interest in developing methods to prepare and manipulate Dicke states, which form a basis for the subspace of fully symmetric states. Two simple protocols for transforming Dicke states are considered. An algebraic characterization of the operations that these protocols induce is obtained in terms of the Weyl algebra [Formula: see text] and [Formula: see text]. Fixed points under the application of the combination of both protocols are explicitly determined. Connections with the binary Hamming scheme, the Hadamard transform and Krawtchouk polynomials are highlighted.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
This paper proposes using Dicke states to improve Grover’s quantum search algorithm for solving the k-vertex cover problem in a given graph with n vertices, a well-known NP-complete problem. A Dicke state $\left| {D_k^n} \right\rangle $ is a special entangled quantum state representing the symmetric superposition of all n-qubit states where exactly k qubits are in |1⟩. It is used to prepare the initial quantum state to reduce the quantum search space from 2nto C(n,k). Consequently, the operations required by Grover’s algorithm are reduced from O(2n){\mathbf{O}}\left({\sqrt {{2^n}} }\right)O(2n​) to O(C(n,k)){\mathbf{O}}(\sqrt {C(n,k)} )O(C(n,k)​), which is O(nk){\mathbf{O}}\left({\sqrt {{n^k}} }\right)O(nk​) for 1 ≤ k ≪ n/2 or n − k ≪ n/2.
We provide a framework wherein one can simulate the Deutsch-Jozsa quantum algorithm on a regular computer within polynomial time, and with linear memory consumption. Under certain reasonable assumptions the simulation solves the problem with a bounded error of probability with only one function evaluation, which is comparable with the efficiency of the quantum algorithm. The provided framework lies within a slight extension of the toy model purposed by Robert W. Spekkens Phys. Rev. A 75 (2007), and consists of transformations that are reminiscent of transformations in quantum mechanics.
Deutsch's algorithm and Grover's quantum search are theoretically interesting, but it is not obvious that these two algorithms are actually useful for anything important. We next consider a selection of more advanced algorithms, several of which may have real-life applications. Some of these will be too complicated to explain fully, and their properties will only be sketched briefly. The Deutsch–Jozsa algorithm Deutsch's algorithm is simple, but important, as it shows that a quantum algorithm can find a property of an unknown function (its parity) with a smaller number of queries than any possible classical algorithm (one rather than two). For this reason we can say that quantum computing is more efficient than classical computing within the oracle model of function evaluation. (It is widely believed that quantum computing is more efficient than classical computing in general, but this is a surprisingly hard thing to prove.) The simplicity of the algorithm is also an advantage, as it can be implemented on very primitive quantum computers. Beyond this, however, Deutsch's algorithm is also important as the simplest member of a large family of quantum algorithms, including most notably Shor's quantum factoring algorithm. The second simplest algorithm in the family is the Deutsch–Jozsa algorithm, which solves a very closely related problem. Consider an unknown binary function with n input bits, giving N = 2 n possible inputs, and a single output bit.