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arXiv:quant-ph/0403048

Quantum implementation of elementary arithmetic operations

Giuseppe Florio, DOMENICO PICCA

Published Mar 5, 2004• 14 citations
arXivHTMLPDF

Abstract

Quantum computation has received great attention in recent years for its possible application to difficult problem in classical calculation. Despite the experimental problems of implementing quantum devices, theoretical physicists have tried to conceive some implementations for quantum algorithms. We present here some explicit schemes for executing elementary arithmetic operations.

DOI
10.48550/arxiv.quant-ph/0403048

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Quantum computation has received great attention in recent years for its possible application to difficult problem in classical calculation. Despite the experimental problems of implementing quantum devices, theoretical physicists have tried to conceive some implementations for quantum algorithms. We present here some explicit schemes for executing elementary arithmetic operations.
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Quantum circuits for elementary arithmetic operations are important not only for implementing Shor's factoring algorithm on a quantum computer but also for understanding the computational power of small quantum circuits, such as linear-size or logarithmic-depth quantum circuits. This paper surveys some recent approaches to constructing efficient quantum circuits for elementary arithmetic operations and their applications to Shor's factoring algorithm. It covers addition, comparison, and the quantum Fourier transform used for addition.
In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.
Nowadays, quantum computers are capturing the attention of researchers because of their potential capability to solve problems otherwise not solvable with classical computers. In parallel to the constant research aimed at solving problems related to the physical implementation of such computers, there is also another branch of research trying to develop quantum algorithms. Many of the existing algorithms today need to perform arithmetic operations. In this thesis, the development of a software library of quantum arithmetic circuits - capable of interfacing with the main languages related to the world of Quantum Computing, such as Qiskit, Cirq and T-ket - is presented. With the aim of implementing this software library, the basics of quantum computation were first studied and subsequently an extensive bibliographic research was carried out, aimed at identifying the best performing arithmetic circuits currently available. After determining the circuits to be implemented, the software development has begun using the Python language. During the development of the library, with the aim of making simpler future development, code's modularity - by organizing the general package in many subpackages - and readability - by respecting the guidelines proposed in the PEP8 style guide - were looked for. In addition to what has already been said, a detailed documentation has been produced using Sphinx. Once the development of the library was completed, the functionality of each circuit was tested with simulations on classical computers; moreover, some components were tested on real free-accessible quantum computers, in order to characterize their behavior. Even though the results obtained on real hardware proved that many of the circuits implemented in this library will be reliably executed by more sophisticated and performing quantum computers, this library can be currently employed in an effective way in the simulation of quantum circuits on classical computers.
We are now ready to assemble quantum subroutines into quantum algorithms. A reasonable first question is: Are there problems which can be solved efficiently on a quantum computer? In his analysis of a universal quantum computer [25], Deutsch illustrated quantum parallelism by constructing a procedure to compute the parity f (0) ⊕ f (1) of a function on one bit. In a generalization of this context, Deutsch and Jozsa [27] defined a quantum algorithm that could solve a problem more efficiently than is possible on a classical computer. This problem helped motivate the search for algorithms that could solve “real” problems, and Simon [67] provided an example that inspired Shor’s subsequent work on the factoring problem and on the discrete logarithm problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.