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Implementing quantum algorithms using classical electrical circuits: Deutsch, Deutsch-Jozsa and Grover

Charles Dyson

Published Mar 1, 2011• 4 citations
OpenAlex

Abstract

We develop the means to implement Deutsch's algorithm, the Deutsch-Jozsa algorithm and Grover's algorithm using electronic circuits, and review previous attempts to implement quantum algorithms classically. We attempt to demonstrate that these are not fundamentally quantum algorithms, but merely algorithms that scale exceptionally well on quantum computers. Finally, we discuss a prototype design for an electrical Hadamard gate and report experimental results.

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We develop the means to implement Deutsch's algorithm, the Deutsch-Jozsa algorithm and Grover's algorithm using electronic circuits, and review previous attempts to implement quantum algorithms classically. We attempt to demonstrate that these are not fundamentally quantum algorithms, but merely algorithms that scale exceptionally well on quantum computers. Finally, we discuss a prototype design for an electrical Hadamard gate and report experimental results.
The Deutsch and Deutsch-Jozsa algorithms are foundational in the study of quantum computing, providing early and elegant demonstrations of quantum speedup over classical computation. This paper offers a pedagogical exposition of both algorithms, beginning with the necessary background in quantum computation and culminating in detailed, step-by-step derivations of the algorithms, complete with mathematical formulations and circuit diagrams. Simulations using quantum gates are also discussed to enhance understanding. The aim is to make these seminal algorithms accessible to readers with a basic background in linear algebra and quantum mechanics.
In this paper, we briefly review the basic concepts of quantum computation, entanglement, quantum cryptography and quantum fourier transform. Quantum algorithms like Deutsch Jozsa, Shor’s factorization and Grover’s data search are developed using fourier transform and quantum computation concepts to build quantum computers. Researchers are finding a way to build quantum computer that works more efficiently than classical computer. Among the standard well known algorithms in the field of quantum computation and communication we describe mathematically Deutsch Jozsa algorithm in detail for 2 and 3 qubits. Calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.
Quantum computing is an emerging computational paradigm that leverages fundamental principles of quantum mechanics, including superposition and entanglement, to perform computations beyond the capabilities of classical computers.Unlike classical bits, quantum bits (qubits) can exist in multiple states simultaneously, enabling exponential computational advantages for specific classes of problems.This paper presents the fundamental distinctions between classical and quantum computation, emphasizing qubit operational principles, quantum processing methods, and quantum gate operations.Key quantum algorithms, including Shor's algorithm, Deutsch-Jozsa algorithm, and Grover's search algorithm, are analyzed to demonstrate the computational superiority of quantum systems.Furthermore, major implementation challenges and diverse application domains of quantum computing are discussed
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.
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states from given initial states, and introduces circuits for building oracles. Chapter 3 presents foundational algorithms such as entanglement creation, quantum teleportation, Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm. Chapters 4 and 5 cover algorithms based on the quantum Fourier transform, including phase estimation, period finding, factoring, and logarithm computation. These chapters also include complexity analysis and detailed quantum circuits suitable for implementation in code. Chapter 6 introduces Grover's algorithm for quantum search and amplitude amplification, including its realization via Hamiltonian simulation and a method for derandomization. Chapter 7 discusses basic techniques for Hamiltonian simulation, such as Lie-Trotter decomposition, sparse Hamiltonians, and the linear combination of unitaries. It also provides example circuits for simulating Hamiltonians expressed as linear combinations of Pauli operators. Chapter 8 introduces variational quantum algorithms, and Chapter 9 presents an algorithm for simulating fermionic many-particle systems, with an emphasis on molecular Hamiltonians. It also outlines the key transformations needed to map a molecular Hamiltonian to a form suitable for simulation on a quantum computer.