This research and educational poster presents a key part of the Shor's factorization, namely the quantum implementation of the modular exponentiation function. In contrast to other works on a similar topic, where Shor's factorizations of small numbers such as 15, 21 or 35 are presented, an implementation of a modular exponentiation function applicable to the Shor's factorization of the number 247 is presented here. This poster shows one of the simpler ways of constructing a quantum circuit of a modular exponentiation function that allows this factorization. At the bottom of the poster, the measured results are presented and the sought factors are calculated, thus confirming the correct implementation.
This research and educational poster presents a key part of the Shor's factorization, namely the quantum implementation of the modular exponentiation function. In contrast to other works on a similar topic, where Shor's factorizations of small numbers such as 15, 21 or 35 are presented, an implementation of a modular exponentiation function applicable to the Shor's factorization of the number 247 is presented here. This poster shows one of the simpler ways of constructing a quantum circuit of a modular exponentiation function that allows this factorization. At the bottom of the poster, the measured results are presented and the sought factors are calculated, thus confirming the correct implementation.
A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.
Abstract The content of this paper is a detailed analysis of possible ways how to quantum implement a key part of Shor’s factorization algorithm, the modular exponentiation function. This implementation is a bottleneck for performing quantum factorization with polynomial complexity, which would make it possible to factorize really large numbers in a reasonable amount of time. In this paper, not only the general theory is presented, but also the results of successful factorizations of the numbers 247 and 143 using Shor’s algorithm from a quantum computer simulator. An interesting fact is that no ancillary qubits were needed in these factorizations. Based on the content of the paper, the conclusion also suggests possible future work on the development of this modular exponentiation function implementation.
The content of this poster is a description of the implementation of two well-known quantum integer factorization methods and, in addition, a proposal of a new, third way along with some examples of its use. For all three methods, the measured results of their circuits will be shown.The first method is Shor's algorithm, where a quantum circuit is used only to find the period of a specially assembled modular exponentiating function. The rest of the algorithm is preprocessing and postprocessing that can be done on a classical computer, so one could say that this is a hybrid algorithm. New results of the successful factorization of the number 119, performed by this algorithm on a real 7-qubit quantum computer, will be presented. It is worth mentioning here that the highest number successfully factorized by this algorithm on a real quantum computer so far was 35.The second method is variational quantum factorization (VQF). In this method, preprocessing builds a system of equations based on the classical binary multiplication procedure of two numbers, resulting in a factorized number. This system of equations is then used to form a Hamiltonian in such a way that its ground state corresponds to the solution of this equation system. Only now the hybrid quantum optimization comes into play to iteratively find this ground state. The advantage over Shor's algorithm is that this method does not require a fault-tolerant quantum computer, but on the other hand it approaches the solution iteratively using a classical computer, which reduces the potential quantum advantage.The new, third method employs Grover's search algorithm in an unusual way to create a quantum factorization circuit. A key part of this approach is the quantum implementation of a multiplying oracle. The main advantage of this method over the previous ones is that it can be used to assemble a relatively simple universal factorization circuit. Moreover, there is almost no preprocessing and postprocessing required. This method is still under development, but this poster will present the first results.
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.
Shor's algorithm for the prime factorization of numbers provides an exponential speedup over the best known classical algorithms. However, nontrivial practical applications have remained out of reach due to experimental limitations. The bottleneck of the experimental realization of the algorithm is the modular exponentiation operation. In this paper, based on a relation between the modular multiplication operator and generalizations of discrete Fourier transforms, we propose a quantum circuit for modular exponentiation. A distinctive feature of our proposal is that our circuit can be entirely implemented in terms of the standard quantum circuit for the discrete Fourier transform and its variants. The gate-complexity of our proposal is O(L3) where L is the number of bits required to store the number being factorized. It is possible that such a proposal may provide easier avenues for near-term generic implementations of Shor's algorithm, in contrast to existing realizations which have often explicitly adapted the circuit to the number being factorized.
This paper discusses and demonstrates the construction of a quantum modular exponentiation circuit in the Qiskit simulator for use in Shor's Algorithm for integer factorization problem (IFP), which is deemed to be able to crack RSA cryptosystems when a large-qubit quantum computer exists. We base our implementation on Vedral, Barenco, and Ekert (VBE) proposal of quantum modular exponentiation, one of the firsts to explicitly provide the aforementioned circuit. Furthermore, we present an example simulation of how to construct a 7xmod 15 circuit in a step-by-step manner, giving clear and detailed information and consideration that currently not provided in the existing literature, and present the whole circuit for use in Shor's Algorithm. Our present simulation shows that the 4-bit VBE quantum modular exponentiation circuit can be constructed, simulated, and measured in Qiskit, while the Shor's Algorithm incorporating this VBE approach itself can be constructed but not yet simulated due to an overly large number of QASM instructions.