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Results in Physics

Non-classical computing problems: Toward novel type of quantum computing problems

Mohammed Zidan, Hichem Eleuch, Mahmoud Abdel‐Aty

Published Nov 2, 2020• 47 citations
Publisher

Abstract

Quantum teleportation draws our attention to propose a new type of problems which can not be solved using classical computers. In this paper, we propose one of these problems. Concretely, this paper extends the definition of Deutsch’s problem to decide whether a black box Uf applied on a given unknown qubit α|0〉+β|1〉, such that |α|>0,|β|>0,and|α|≠|β|, is constant or balanced Boolean function, besides, estimation of |α| and |β|. Although, this problem is very simple but it can not be solved using classical computers, because qubit can not be implemented physically using classical computers. A novel quantum algorithm based on principle of entanglement measure is proposed to solve this problem. IBMs 5-qubit quantum computer (ibmqx4) is used to realize the proposed algorithm experimentally.

DOI
10.1016/j.rinp.2020.103536

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Quantum teleportation draws our attention to propose a new type of problems which can not be solved using classical computers. In this paper, we propose one of these problems. Concretely, this paper extends the definition of Deutsch’s problem to decide whether a black box Uf applied on a given unknown qubit α|0〉+β|1〉, such that |α|>0,|β|>0,and|α|≠|β|, is constant or balanced Boolean function, besides, estimation of |α| and |β|. Although, this problem is very simple but it can not be solved using classical computers, because qubit can not be implemented physically using classical computers. A novel quantum algorithm based on principle of entanglement measure is proposed to solve this problem. IBMs 5-qubit quantum computer (ibmqx4) is used to realize the proposed algorithm experimentally.
Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called Deutsch's problem. Consider a Boolean function f: {0,1} → {0,1} and suppose that we have a (classical) black box to compute it. The problem asks whether f is constant [that is, f(0) = f(1)] or balanced [f(0) ≠ f(1)]. Classically, to solve the problem seems to require the computation of f(0) and f(1), and then the comparison of results. Is it possible to solve the problem with only one query on f? In a famous paper published in 1985, Deutsch posed the problem and obtained a "quantum" partial affirmative answer. In 1998, a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello and Mosca. Here we will show that the quantum solution can be de-quantized to a deterministic simpler solution which is as efficient as the quantum one. The use of "superposition," a key ingredient of quantum algorithm, is — in this specific case — classically available.
Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function f:{0,1}→{0,1}f: \{0,1\} \to \{0,1\}f:{0,1}→{0,1} and suppose that we have a (classical) black box to compute it. The problem asks whether fff is constant (that is, f(0)=f(1)f(0) = f(1)f(0)=f(1)) or balanced (f(0)≠f(1)f(0) \not= f(1)f(0)=f(1)). Classically, to solve the problem seems to require the computation of f(0)f(0)f(0) and $ f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on fff? In a famous paper published in 1985, Deutsch posed the problem and obtained a ``quantum'' {\em partial affirmative answer}. In 1998 a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello, and Mosca. Here we will show that the quantum solution can be {\it de-quantised} to a deterministic simpler solution which is as efficient as the quantum one. The use of ``superposition'', a key ingredient of quantum algorithm, is--in this specific case--classically available.
Distinguishing unknown non-orthogonal qubits is an essential requirement for addressing various challenges in quantum computation , including quantum machine learning, quantum communications , and quantum technologies . For instance, while quantum teleportation enables the transfer of unknown individual qubits between distant parties, an algorithm is necessary to define the associated state of a teleported qubit at the receiving end. In this paper, we propose a novel quantum algorithm designed to effectively determine the state of a given unknown qubit and distinguish a subset of non-orthogonal qubits. The proposed algorithm can efficiently identify the state of an unknown qubit in the form cos θ 2 | 0 〉 + sin θ 2 | 1 〉 using the M z operator. By estimating the angle θ through the measurement of entanglement degree, the proposed algorithm can identify the state of an unknown qubit. Experimental validation of the proposed algorithm is conducted using the IBM quantum computer simulator chip ibmqx2. Furthermore, a t-test is conducted to compare the proposed algorithm with the direct measurement approach. The results indicate a significant difference between the two methods, demonstrating the superior performance of the proposed algorithm.
Abstract Quantum computing is an inspiring technic on solving complicate problems, which shows the superiority over the classical computing. Contemporarily, the quantum-based algorithms were invented in many purposes on solving those problems. The thing is some of the algorithms were composed with oracles, which can be treated as a black box. It can be worked out/analyzed in theoretical mathematical expressions, but it can never fall in the ground without those actual implementations. Therefore, this paper will illustrate the idea and possible implementations on some quantum algorithm. Among various algorithms, this research will choose those relatively famous quantum algorithms, i.e., Deutsch-Jozsa Algorithm, Shor’s Algorithm, and Grover’s Algorithm. Different algorithms have the different intrinsic logics, which means there is no general description for all solution in this problem, but all of those can achieve the goal according to the same framework. Since those oracles are black boxes that can only be analyzed by its behavior with different cases, and then based on the behaviors and cases, and in IBM quantum platform, one can fill out the oracle by manipulate with different quantum operators. Lastly, the figures of some the possible implementation of those oracles with the three quantum algorithms will be shown in the IBM quantum platform, as well as the histograms of probability distributions that can justify the correctness of the implementations. These results shed light on guiding further exploration of the design and analysis on quantum algorithms.
Quantum computation is a part of quantum information theory. By using quantum mechanical phenomena it is possible to speed up the solving of some problems with quantum algorithms over classical probabilistic algorithms. In some cases quantum algorithms can utilise quantum parallelism in order to speed up problem solving. In this thesis experiments on IBM quantum computers are compared with ideal error free simulations. In the first part quantum teleportation is shown in practice, similarly the performance of Deutsch–Jozsa, Bernstein–Vazirani, Simon’s and Grover’s search algorithms are shown in action on IBM quantum computers and each is compared against an ideal simulation. The last experiment shows how the binary success probability of an unsharp measurement using Lüders instrument can be improved by adding sequential measurements by using an ideal simulation and a simulation using simple noise model.