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arXiv:2306.08473 · Physical Review A

Quantum multiplication algorithm based on the convolution theorem

Mehdi Ramezani, Morteza Nikaeen, Farnaz Farman, Seyed Mahmoud Ashrafi, Alireza Bahrampour

Published Nov 6, 2023• 8 citations
arXivHTMLPDF

Abstract

The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their theoretical limit and offer little room for further enhancement. However, with the advent of quantum computers and the need for quantum algorithms that can perform multiplication on quantum hardware, a new paradigm emerges. In this paper, inspired by the convolution theorem and quantum amplitude amplification paradigm we propose a quantum algorithm for integer multiplication with time complexity of $O(\sqrt{n}{log}^{2}n)$ which outperforms the best-known classical algorithm, the Harvey algorithm with time complexity of $O(nlogn)$. Unlike the Harvey algorithm, our algorithm does not have the restriction of being applicable solely to extremely large numbers, making it a versatile choice for a wide range of integer multiplication tasks. The paper also reviews the history and development of classical multiplication algorithms and motivates us to explore how quantum resources can provide new perspectives and possibilities for this fundamental problem.

DOI
10.1103/physreva.108.052405

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The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their theoretical limit and offer little room for further enhancement. However, with the advent of quantum computers and the need for quantum algorithms that can perform multiplication on quantum hardware, a new paradigm emerges. In this paper, inspired by the convolution theorem and quantum amplitude amplification paradigm we propose a quantum algorithm for integer multiplication with time complexity of O(nlog2n)O(\sqrt{n}{log}^{2}n)O(n​log2n) which outperforms the best-known classical algorithm, the Harvey algorithm with time complexity of O(nlogn)O(nlogn)O(nlogn). Unlike the Harvey algorithm, our algorithm does not have the restriction of being applicable solely to extremely large numbers, making it a versatile choice for a wide range of integer multiplication tasks. The paper also reviews the history and development of classical multiplication algorithms and motivates us to explore how quantum resources can provide new perspectives and possibilities for this fundamental problem.
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
Quantum multiplication is a fundamental operation in quantum computing. It is important to have a quantum multiplier with low complexity. In this paper, we propose the Quantum Multiplier Based on Exponent Adder (QMbead), a new approach that requires just log⁡2(n)\log_2(n)log2​(n) qubits to multiply two nnn-bit integer numbers, in addition to O(n)O(n)O(n) ancillary qubits used for quantum state preparation. The QMbead uses a so-called exponent encoding to respectively represent two multiplicands as two superposition states which are prepared by a quantum state preparation method, then employs a quantum adder to obtain the sum of these two superposition states, and subsequently measures the outputs of the quantum adder to calculate the product of the multiplicands. Different quantum adders can be used in the QMbead. The circuit depth and time complexity of the QMbead, using a logarithmic-depth quantum carry lookahead adder (QCLA) as adder, are O(log⁡n)O(\log n)O(logn) and O(nlog⁡n)O(n \log n)O(nlogn), respectively. The gate complexity of the QMbead is O(n)O(n)O(n). The circuit depth and gate complexity of the QMbead is better than existing quantum multipliers such as the quantum Karatsuba multiplier and the QFT based multiplier. The time complexity of the QMbead is identical to that of the fastest classical multiplication algorithm, Harvey-Hoeven algorithm. Interestingly, the QMbead maintains an advantage over the Harvey-Hoeven algorithm, given that the latter is only suitable for excessively large numbers, whereas the QMbead is valid for both small and large numbers. The multiplicand can be either an integer or a decimal number. The QMbead has been implemented on quantum simulators to compute products with a bit length of up to 273 bits using only 17 qubits, excluding the ancillary qubits used for quantum state preparation. This establishes QMbead as an efficient solution for multiplying large integer or decimal numbers with many bits.
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.
The advent of quantum computing has ushered in a new era of computational capabilities, promising to revolutionize the resolution of complex problems. Unlike classical computers, which rely on binary digits (bits) to process information, quantum computers utilize quantum bits (qubits) that can exist in multiple states simultaneously due to the principles of superposition and entanglement inherent in quantum mechanics. This fundamental difference allows quantum computers to perform certain calculations exponentially faster than their classical counterparts, offering profound implications for fields ranging from cryptography to material science. At the heart of this transformative potential lie quantum algorithms, specifically designed to harness the unique properties of quantum systems to tackle problems currently intractable for classical computers. Notable examples include Shor's algorithm for integer factorization, which threatens the security of widely used cryptographic schemes, and Grover's algorithm for unstructured search, which provides a quadratic speedup over classical approaches. Beyond these pioneering works, ongoing research continues to uncover new quantum algorithms capable of addressing a broad spectrum of applications. The proposed method leverages the principles of quantum mechanics to develop efficient algorithms that outperform classical approaches in solving complex computational problems. By harnessing the unique properties of quantum bits (qubits), such as superposition and entanglement, the method aims to achieve superior accuracy and efficiency in various applications. The performance of the proposed method was evaluated using several key metrics: an accuracy of 94.8%, a Root Mean Squared Error (RMSE) of 0.208, and a Mean Absolute Error (MAE) of 0.406. These metrics collectively underscore the robustness and reliability of the proposed quantum algorithms, highlighting their potential to significantly enhance computational efficiency and accuracy across various domains. This study delves into the development of quantum algorithms, exploring their theoretical foundations, practical implementations, and the challenges faced in their realization. By leveraging the principles of quantum mechanics, these algorithms hold the promise of not only solving existing problems more efficiently but also unlocking new computational paradigms previously deemed unattainable. Through a comprehensive analysis, this research aims to elucidate the current state of quantum algorithm research, highlight the advancements made, and identify the pathways forward in this rapidly evolving field.
The multiplication of superpositions of numbers is a core operation in many quantum algorithms. The standard method for multiplication (both classical and quantum) has a runtime quadratic in the size of the inputs. Quantum circuits with asymptotically fewer gates have been developed, but generally exhibit large overheads, especially in the number of ancilla qubits. In this work, we introduce a new paradigm for sub-quadratic-time quantum multiplication with zero ancilla qubits -- the only qubits involved are the input and output registers themselves. Our algorithm achieves an asymptotic gate count of O(n1+ε)\mathcal{O}(n^{1+ε})O(n1+ε) for any ε>0ε> 0ε>0; with practical choices of parameters, we expect scalings as low as O(n1.3)\mathcal{O}(n^{1.3})O(n1.3). Used as a subroutine in Shor's algorithm, our technique immediately yields a factoring circuit with O(n2+ε)\mathcal{O}(n^{2+ε})O(n2+ε) gates and only 2n+O(log⁡n)2n + \mathcal{O}(\log n)2n+O(logn) qubits; to our knowledge, this is by far the best qubit count of any factoring circuit with a sub-cubic number of gates. Used in Regev's recent factoring algorithm, the gate count is O(n1.5+ε)\mathcal{O}(n^{1.5+ε})O(n1.5+ε). Finally, we demonstrate that our algorithm has the potential to outperform previous proposals at problem sizes relevant in practice, including yielding the smallest circuits we know of for classically-verifiable quantum advantage.