Shor's Quantum Research Articles

An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory.

It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.

Design Space Exploration for Efficient Quantum Most-Significant Digit-First Arithmetic

Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.

Quantum Cryptanalysis on a Multivariate Cryptosystem Based on Clipped Hopfield Neural Network.

Shor's quantum algorithm and other efficient quantum algorithms can break many public-key cryptographic schemes in polynomial time on a quantum computer. In response, researchers proposed postquantum cryptography to resist quantum computers. The multivariate cryptosystem (MVC) is one of a few options of postquantum cryptography. It is based on the NP-hardness of the computational problem to solve nonlinear equations over a finite field. Recently, Wang et al. (2018) proposed a MVC based on extended clipped hopfield neural networks (eCHNN). Its main security assumption is backed by the discrete logarithm (DL) problem over Matrics. In this brief, we present quantum cryptanalysis of Wang et al. 's eCHNN-based MVC. We first show that Shor's quantum algorithm can be modified to solve the DL problem over Matrics. Then we show that Wang et al. 's construction of eCHNN-based MVC is not secure against quantum computers; this against the original intention of that multivariate cryptography is one of a few options of postquantum cryptography.

Tunable Anisotropic Quantum Rabi Model via a Magnon–Spin-Qubit Ensemble

The ongoing rapid progress towards quantum technologies relies on new hybrid platforms optimized for specific quantum computation and communication tasks, and researchers are striving to achieve such platforms. We study theoretically a spin qubit exchange-coupled to an anisotropic ferromagnet that hosts magnons with a controllable degree of intrinsic squeezing. We find this system to physically realize the quantum Rabi model from the isotropic to the Jaynes-Cummings limit with coupling strengths that can reach the deep-strong regime. We demonstrate that the composite nature of the squeezed magnon enables concurrent excitation of three spin qubits coupled to the same magnet. Thus, three-qubit Greenberger-Horne-Zeilinger and related states needed for implementing Shor's quantum error-correction code can be robustly generated. Our analysis highlights some unique advantages offered by this hybrid platform, and we hope that it will motivate corresponding experimental efforts.

Integer Factorization – Cryptology Meets Number Theory

Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about be­haviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient fac­toring algorithm is equivalent to breaking RSA. The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern fac­toring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathemati­cians and cryptographers. Shor's quantum factorization algorithm with poly­nomial complexity is described and the impact on public-key encryption is discussed.

Optimising Matrix Product State Simulations of Shor's Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on r, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.

Factoring using 2n + 2 qubits with Toffoli based modular multiplication

We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli ba...

Quantum polynomial-time fixed-point attack for RSA

Security analysis of public-key cryptosystems is of fundamental significance for both theoretical research and applications in cryptography. In particular, the security of widely used public-key cryptosystems merits deep research to protect against new types of attacks. It is therefore highly meaningful to research cryptanalysis in the quantum computing environment. Shor proposed a well-known factoring algorithm by finding the prime factors of a number n = pq, which is exponentially faster than the best known classical algorithm. The idea behind Shor's quantum factoring algorithm is a straightforward programming consequence of the following proposition: to factor n, it suffices to find the order r; once such an r is found, one can compute gcd(a <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r/2</sup> ± 1, n) = p or q. For odd values of r it is assumed that the factors of n cannot be found (since a <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r/2</sup> is not generally an integer). That is, the order r must be even. This restriction can be removed, however, by working from another angle. Based on the quantum inverse Fourier transform and phase estimation, this paper presents a new polynomial-time quantum algorithm for breaking RSA, without explicitly factoring the modulus n. The probability of success of the new algorithm is greater than 4φ(r)/π <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> r, exceeding that of the existing quantum algorithm for attacking RSA based on factorization. In constrast to the existing quantum algorithm for attacking RSA, the order r of the fixed point C for RSA does not need to be even. It changed the practices that cryptanalysts try to recover the private-key, directly from recovering the plaintext M to start, a ciphertext-only attack attacking RSA is proposed.

Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip.

Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor's factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, nonfault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a silicon quantum photonic device. The approach is verified to be well suited for prethreshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.

An n-bit general implementation of Shor's quantum period-finding algorithm

The goal of this paper is to outline a general-purpose scalable implementation of Shor’s period-finding algorithm using fundamental quantum gates, and to act as a blueprint for linear optical implementations of Shor’s algorithm for both general and specific values of N. This offers a broader view of a problem often overlooked in favour of compiled versions of the algorithm.

Undergraduate computational physics projects on quantum computing

Computational projects on quantum computing suitable for students in a junior-level quantum mechanics course are described. In these projects students write their own programs to simulate quantum computers. Knowledge is assumed of introductory quantum mechanics through the properties of spin 1/2. Initial, more easily programmed projects treat the basics of quantum computation, quantum gates, and Grover's quantum search algorithm. These are followed by more advanced projects to increase the number of qubits and implement Shor's quantum factoring algorithm. The projects can be run on a typical laptop or desktop computer, using most programming languages. Supplementing resources available elsewhere, the projects are presented here in a self-contained format especially suitable for a short computational module for physics students.

Quantum Error Correction with magnetic molecules

Quantum algorithms often assume independent spin qubits to produce trivial , mappings. This can be unrealistic in many solid-state implementations with sizeable magnetic interactions. Here we show that the lower part of the spectrum of a molecule containing three exchange-coupled metal ions with and is equivalent to nine electron-nuclear qubits. We derive the relation between spin states and qubit states in reasonable parameter ranges for the rare earth Tb3+ and for the transition metal Cu2+, and study the possibility to implement Shor's Quantum Error Correction code on such a molecule. We also discuss recently developed molecular systems that could be adequate from an experimental point of view.

Bit-string oblivious transfer based on quantum state computational distinguishability

Oblivious transfer protocol is a basic building block in cryptography and is used to transfer information from a sender to a receiver in such a way that, at the end of the protocol, the sender does not know if the receiver got the message or not. Since Shor's quantum algorithm appeared, the security of most of classical cryptographic schemes has been compromised, as they rely on the fact that factoring is unfeasible. To overcome this, quantum mechanics has been used intensively in the past decades, and alternatives resistant to quantum attacks have been developed in order to fulfill the (potential) lack of security of a significant number of classical schemes. In this paper, we present a quantum computationally secure protocol for oblivious transfer between two parties, under the assumption of quantum hardness of state distinguishability. The protocol is feasible, in the sense that it is implementable in polynomial time.

Teaching Quantum Computing with the QuIDE Simulator

Recently, the idea of quantum computation is becoming more and more popular and there are many attempts to build quantum computers. Therefore, there is a need to introduce this topic to regular students of computer science and engineering. In this paper we present a concept of a course powered by the Quantum Integrated Development Environment (QuIDE), the new quantum computer simulator that joins features of GUI based simulators with inter- preters and simulation library approach. The idea of the course is to put together theoretical aspects with practical assignments realized on the QuIDE simulator. Such an approach enables studying a variety of topics in a way understandable for this category of students. The topics of the course included understanding the concept of quantum gates, registers and a series of algorithms: Deutsch and Bernstein-Vazirani Problems, Grover's Fast Database Search, Shor's Prime Factorization, Quantum Teleportation and Quantum Dense Coding. We describe results of QuIDE assessment during the course; our solution scored more points in System Usability Scale survey then the other tool previously used for that purpose. We also show that the most useful features of such a tool indicated by students are similar to the assumptions made on the simulator functionality.

Cofactor Sharing for Reversible Logic Synthesis

Improving circuit realization of known quantum algorithms by CAD techniques has benefits for quantum experimentalists. In this article, the problem of synthesizing a given function on a set of ancillea is addressed. The proposed approach benefits from extensive sharing of cofactors among cubes that appear on function outputs. Accordingly, it can be considered a multilevel logic optimization technique for reversible circuits. In particular, the suggested approach can efficiently implement any n -input, m -output lookup table (LUT) by a reversible circuit. This problem has interesting applications in the Shor's number-factoring algorithm and in quantum walk on sparse graphs. Simulation results reveal that the proposed cofactor-sharing synthesis algorithm has a significant impact on reducing the size of modular exponentiation circuits for Shor's quantum factoring algorithm, oracle circuits in quantum walk on sparse graphs, and the well-known MCNC benchmarks.

Fast quantum modular exponentiation architecture for Shor's factoring alogrithm

We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

APLICAÇÃO DE PROTOCOLOS QUÂNTICOS E ALGORITMO DE SHOR PARA A SEGURANÇA DA INFORMAÇÃO

The purpose of this article is to show, in an objective way, especially to Information Security students, some practical ways of operating with Quantum Cryptography. The whole approach was carried out on the tripod of Quantum Computing, which is Mathematics, Quantum Physics and Computation, with a view to current and tangible issues related to information security. The first approach is related to the generation of quantum keys and the use of quantum protocols (BB84 and B92). Regarding the use of algorithms, the Shor Quantum Algorithm is implemented in the JavaScript programming language, with its code partially commented to illustrate the operation of the algorithm.

Oversimplifying quantum factoring

Shor's quantum factoring algorithm exponentially outperforms known classical methods. Previous experimental implementations have used simplifications dependent on knowing the factors in advance. However, as we show here, all composite numbers admit simplification of the algorithm to a circuit equivalent to flipping coins. The difficulty of a particular experiment therefore depends on the level of simplification chosen, not the size of the number factored. Valid implementations should not make use of the answer sought.

New public key cryptosystems based on non‐Abelian factorization problems

ABSTRACTTwo novel public key encryption schemes based on the non‐Abelian factorization problems were proposed. Both of them are proved to be indistinguishable against adaptively chosen ciphertext attack (IND‐CCA2) in the random oracle models. These constructions have the potential to resist Shor's quantum algorithm attack proposed in 1994 and give affirmative answers for the open question announced by Myasnikov, Shpilrain and Ushakov in 2011. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Faster quantum number factoring via circuit synthesis

A major obstacle to implementing Shor's quantum number-factoring algorithm is the large size of modular-exponentiation circuits. We reduce this bottleneck by customizing reversible circuits for modular multiplication to individual runs of Shor's algorithm. Our circuit-synthesis procedure exploits spectral properties of multiplication operators and constructs optimized circuits from the traces of the execution of an appropriate greatest-common-divisor algorithm. Empirically, gate counts are reduced by four to five times, and circuit latency is reduced by larger factors.