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arXiv:quant-ph/0305039 · Physical Review A

Effects of dynamical phases in Shor’s factoring algorithm with operational delays

L. F. Wei, Xiao Li, Xuedong Hu, Franco Nori

Published Feb 22, 2005• 11 citations
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Abstract

Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent errors will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor's quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these coherent errors, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to avoid or suppress these coherent errors when using Shor's algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically and numerically for the factorization of several integers: 4, 15, 21, and 33.

DOI
10.1103/physreva.71.022317

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The intent of this thesis is to elucidate the quantum computing algorithm developed by Peter Shor called Shor's algorithm. We will provide a detailed description and simulation of the algorithm using MATLAB. Precursory information regarding quantum phenomena such as superposition, entanglement, and Dirac notation, will be described in great detail so that the reader may have a better understanding of the operations in Shor's algorithm. Quantum computers store and transport information quite differently than their classical counterparts. We will provide a quick overview of these differences to high- light the benefit of utilizing quantum phenomena in a computer in order to create massive parallel computations. Thus, reducing the complexity time for classical algorithms used to solve problems such as the prime factorization problem and the period finding problem. The Quantum Fourier Transform is a principle component in Shor's algorithm. We will explicitly define the Quantum Fourier Transform and show that it is a unitary transformation. We will also show how the Quantum Fourier Transform, as well as another unitary transformation called the Hadamard transform, functions in Shor's algorithm. One of the initial parameters in Shor's algorithm is to select a random variable. We will examine the erratic effects of this random variable as well as how it effects the probability of us successfully reducing an integer into a product of two primes. We will provide a thorough analysis of the randomness in Shor's algorithm. We will also show how measuring the state of our quantum system as well as selecting a suitable random variable impacts finding the period of the Quantum Fourier Transform which in turn will either give a high or low probability of obtaining a factor of some integer.

•Jan 1, 2016•0 citations
Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent errors will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor's quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these coherent errors, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to avoid or suppress these coherent errors when using Shor's algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically and numerically for the factorization of several integers: 4, 15, 21, and 33.
The unavoidable finite time intervals between the sequential operations needed for performing practical quantum computing can degrade the performance of quantum computers. During these delays, unwanted relative dynamical phases are produced due to the free evolution of the superposition wave-function of the qubits. In general, these coherent "errors" modify the desired quantum interferences and thus spoil the correct results, compared to the ideal standard quantum computing that does not consider the effects of delays between successive unitary operations. Here, we show that, in the framework of the quantum phase estimation algorithm, these coherent phase "errors", produced by the time delays between sequential operations, can be avoided by setting up the delay times to satisfy certain matching conditions.
The publication in 1994 of Shor's algorithm, which allows factorisation of composite number N in a time polynomial in its binary length L has been the primary catalyst for the race to construct a functional quantum computer. However, it seems clear that any practical system that may be developed will not be able to perform completely error free quantum gate operations or shield even idle qubits from inevitable error effects. Hence, the practicality of quantum algorithms needs to be investigated to not only determine limitations on such algorithms in a noisy quantum computer, but also to estimate what demands must be made of quantum error correction (QEC). Shor's algorithm is a combination of both classical pre and post-processing, and also a quantum period finding subroutine (QPF) which allows for the exponential speed up of this algorithm on a quantum device. This paper will look at the stability of this quantum subroutine under the effects of several error models. Direct simulation of the entire QPF subroutine required to factorise a given composite number N in the presence of errors shows that the circuit required to implement Shor's algorithm is very sensitive to a small number of errors within the entire calculation. Well designed and efficient error correction codes, quick gate times and very low gate error rates will be essential for any physical realisation of Shor's factoring algorithm.
Shor's algorithm (SA) is a quantum algorithm for factoring integers. Since SA has polynomial complexity while the best classical factoring algorithms are sub-exponential, SA is cited as evidence that quantum computers are more powerful than classical computers. SA is critically dependent on the Quantum Fourier Transform (QFT) and it is known that the QFT is sensitive to errors in the quantum state input to it. In this paper, we show that the polynomial scaling of SA is destroyed by input errors to the QFT part of the algorithm. We also show that Quantum Error Correcting Codes (QECC) are not capable of suppressing errors due to operator imprecision and that propagation of operator precision errors is sufficient to severely degrade the effectiveness of SA. Additionally we show that operator imprecision in the error correction circuit for the Calderbank-Shor-Steane QECC is mathematically equivalent to decoherence on every physical qubit in a register. We conclude that, because of the effect of operator precision errors, it is likely that physically realizable quantum computers will be capable of factoring integers no more efficiently than classical computers.
We study effects of imperfections induced by residual couplings between qubits on the accuracy of Shor's algorithm using numerical simulations of realistic quantum computations with up to 30 qubits. The factoring of numbers up to N=943N=943N=943 show that the width of peaks, which frequencies allow to determine the factors, grow exponentially with the number of qubits. However, the algorithm remains operational up to a critical coupling strength ϵc{ϵ}_{c}ϵc​ which drops only polynomially with log2N{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Nlog2​N. The numerical dependence of ϵc{ϵ}_{c}ϵc​ on log2N{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Nlog2​N is explained by analytical estimates that allows one to obtain the scaling for functionality of Shor's algorithm on realistic quantum computers with a large number of qubits.
Quantum computing promises revolutionary advances in cryptography, particularly through Shor's algorithm, which can factor large integers in polynomial time and threaten traditional RSA encryption. Through simulation experiments, our research compares Shor's algorithm implementation in ideal versus noisy quantum environments. Under ideal conditions, we factored semiprimes up to N=299 with reasonable resources, but with noise introduction, this ceiling dropped to approximately N=95, beyond which computational costs increased exponentially. The performance gap is striking: ideal simulations factored N=91 in 4 seconds, while noisy simulations required 2,600 seconds—a 650-fold increase. Our hybrid quantum-classical approach showed that noise simulation required modified circuit architectures, reducing gate count growth from 50 to 42.5 gates for comparable test numbers. Qubit requirements followed a step-function pattern in both environments, scaling from 12 qubits for N=15 to 21 qubits for N=91-95, while circuit depth increased from 11 to 17-19 units, with slightly shallower circuits in noisy conditions. In our paper, we propose a hybrid implementation strategy that addresses hardware limitations, present empirical findings on the impact of quantum noise on factoring efficiency, and evaluate the cryptographic security implications under realistic conditions. Our findings underscore the need for fault-tolerant quantum computing and error correction, especially as optimizations suggest factoring 2048-bit RSA keys would require approximately 20 million qubits running for 8 hours. This highlights the urgency in transitioning to post-quantum cryptographic algorithms like CRYSTALS-Dilithium and CRYSTALS-Kyber being standardized by NIST.
The intent of this thesis is to elucidate the quantum computing algorithm developed by Peter Shor called Shor's algorithm. We will provide a detailed description and simulation of the algorithm using MATLAB. Precursory information regarding quantum phenomena such as superposition, entanglement, and Dirac notation, will be described in great detail so that the reader may have a better understanding of the operations in Shor's algorithm. Quantum computers store and transport information quite differently than their classical counterparts. We will provide a quick overview of these differences to high- light the benefit of utilizing quantum phenomena in a computer in order to create massive parallel computations. Thus, reducing the complexity time for classical algorithms used to solve problems such as the prime factorization problem and the period finding problem. The Quantum Fourier Transform is a principle component in Shor's algorithm. We will explicitly define the Quantum Fourier Transform and show that it is a unitary transformation. We will also show how the Quantum Fourier Transform, as well as another unitary transformation called the Hadamard transform, functions in Shor's algorithm. One of the initial parameters in Shor's algorithm is to select a random variable. We will examine the erratic effects of this random variable as well as how it effects the probability of us successfully reducing an integer into a product of two primes. We will provide a thorough analysis of the randomness in Shor's algorithm. We will also show how measuring the state of our quantum system as well as selecting a suitable random variable impacts finding the period of the Quantum Fourier Transform which in turn will either give a high or low probability of obtaining a factor of some integer.