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arXiv:1307.6538 · Europhysics Letters (EPL)

Period finding with adiabatic quantum computation

Itay Hen

Published Mar 1, 2014• 26 citations
arXivHTMLPDF

Abstract

We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.

DOI
10.1209/0295-5075/105/50005

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We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.
We report an experimental demonstration of a one-way implementation of a quantum algorithm solving Simon's problem-a black-box period-finding problem that has an exponential gap between the classical and quantum runtime. Using an all-optical setup and modifying the bases of single-qubit measurements on a five-qubit cluster state, key representative functions of the logical two-qubit version's black box can be queried and solved. To the best of our knowledge, this work represents the first experimental realization of the quantum algorithm solving Simon's problem. The experimental results are in excellent agreement with the theoretical model, demonstrating the successful performance of the algorithm. With a view to scaling up to larger numbers of qubits, we analyze the resource requirements for an n-qubit version. This work helps highlight how one-way quantum computing provides a practical route to experimentally investigating the quantum-classical gap in the query complexity model.
The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.
The authors analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'. Adiabatic quantum computation is a novel paradigm for the design of quantum algorithms; it is truly quantum in the sense that it can be used to speed up searching by a quadratic factor over any classical algorithm. On the question of whether this new paradigm may be used to efficiently solve NP-complete problems on a quantum computer, we show that the usual query complexity arguments cannot be used to rule out a polynomial time solution. On the other hand, we argue that the adiabatic approach may be thought of as a kind of 'quantum local search'. We design a family of minimization problems that is hard for such local search heuristics, and establish an exponential lower bound for the adiabatic algorithm for these problems. This provides insights into the limitations of this approach. It remains an open question whether adiabatic quantum computation can establish an exponential speed-up over traditional computing or if there exists a classical algorithm that can simulate the quantum adiabatic process efficiently.
Abstract Peter Shor’s quantum factoring algorithm is one of the crowning achievements of the field of quantum computation and information. This chapter shows how quantum computing offers a super-polynomial (i.e., nearly exponential) advantage for the order-finding problem relative to the best-known classical counterpart, which can then be applied to the problem of factoring. Not only is this the most celebrated result in theoretical quantum computer science, but it also has the seismic implication that if and when quantum computing becomes a reality, the vast number of cryptosystems built on the the Rivest–Shamir–Adleman cryptosystem (RSA) protocol will no longer be secure.
The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.