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.
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.
Shor's algorithm proved a significant milestone in quantum computing since it promises an exponential speedup over conventional algorithms for integer factorization, a problem critical to modern cryptography systems. This work covers implementation steps of Shor’s algorithm and analyzes Quantum Fourier Transform (QFT) usage for extracting periodicity from quantum superpositions. Firstly, theoretical foundations of algorithm are illustrated concentrating on QFT construction and operation within the setting of the quantum circuit. Thereafter, each implementation step such as qubit optimization, modular exponentiation, and circuit design is discussed. Our findings validate the theoretical effectiveness of the QFT in solving the period-finding subroutine, while also highlighting the practical difficulties and scaling constraints presented by existing quantum hardware. The paper ends with a summary of possible advancements and future paths for error mitigation strategies and quantum algorithm design.
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.
In 1994, Peter Shor presented an algorithm for integer factorization that used exponentiallyless operations than the most efficient known algorithm. His algorithm requiresthe use of a quantum computer, a theoretical computational device using quantum mechanicaleffects not utilized in contemporary computers. In this paper we have analysedthe mathematics behind Shor’s algorithm and the quantum circuits on which it operates.We have also studied the Quantum Fourier Transform, a central component of Shor’sAlgorithm. Furthermore we have written a program in C++ to simulate a quantum circuitperforming Shor’s algorithm and the quantum Fourier transform. We were able tounderstand the critical parts of Shor’s algorithm that contribute with the great increasein efficiency compared to classical algorithms.
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.
We provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected. Quanta 2023; 12: 41–130.
RSA cryptography is a principle part in today's cyber-security frameworks, which intensely depends on the diffi-culty of factorizing large integers. Finding a factor of large prime number is not a small feature compared to the time complexity for computation on a classical computer is far beyond comprehension. Back in 1994, Peter Shor proposed a quantum algorithm to factorize integers much more efficiently than a classical computer. This can effectively break asymmetric cryptosystems such as RSA or Diffie-Hellman, depicted as the most famous quantum algorithm. It leverages the efficacy proclaimed by quantum bits and clearly shows how quantum computers can outperform classical ones in specific tasks. In this paper, we show how Shor's algorithm factorizes an integer into its two prime factors. Shor uses both classical and quantum computation to perform this task in polynomial time, using Euclidean algorithm and find the greatest common divisor (gcd) of two numbers in the classical part. Thereafter, the power of Quantum Fourier Transform (QFT) and Quantum Phase Estimation (QPE) is utilised for period-finding using quantum bits. We have implemented this algorithm on IBM's Quantum Lab and successfully factorized the number 15 into its prime factors as 5 and 3, by using 5 qubits. This paper provides an overview of the implementation of Shor's algorithm involving QPE, QFT, Modular Exponentiation and the corresponding individual implementations, the only prerequisite being, to know how a qubit works.