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Physical Review A

Streamlining Shor's algorithm for potential hardware savings

Yunseong Nam, R. Blümel

Published Jun 26, 2013• 6 citations
Publisher

Abstract

We constructed a virtual quantum computer by running a complete, scaling, quantum-gate--by--quantum-gate implementation of Shor's algorithm on a 128-core classical cluster computer. In mode A [quantum period finding (PF) only, supplied with classical results for the modular exponentiation (ME) part of Shor's algorithm], factoring semiprimes up to $N=557\phantom{\rule{0.16em}{0ex}}993$ with up to $n=39$ qubits, we confirm earlier, smaller-$n$ results concerning the performance scaling of Shor's algorithm equipped with a truncated (banded) quantum Fourier transform. Running our virtual quantum computer in mode B (full quantum implementation of ME and PF), we find that a large number of gates may be discarded in a scalable way in both the ME and PF parts of Shor's algorithm in exchange for only a small reduction in performance. We explicitly state the associated scaling laws. Implying significant savings in quantum gates, we suggest that these results are of importance for future experimental and technical large-$n$ implementations of quantum computers.

DOI
10.1103/physreva.87.060304

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We constructed a virtual quantum computer by running a complete, scaling, quantum-gate--by--quantum-gate implementation of Shor's algorithm on a 128-core classical cluster computer. In mode A [quantum period finding (PF) only, supplied with classical results for the modular exponentiation (ME) part of Shor's algorithm], factoring semiprimes up to N=557993N=557\phantom{\rule{0.16em}{0ex}}993N=557993 with up to n=39n=39n=39 qubits, we confirm earlier, smaller-nnn results concerning the performance scaling of Shor's algorithm equipped with a truncated (banded) quantum Fourier transform. Running our virtual quantum computer in mode B (full quantum implementation of ME and PF), we find that a large number of gates may be discarded in a scalable way in both the ME and PF parts of Shor's algorithm in exchange for only a small reduction in performance. We explicitly state the associated scaling laws. Implying significant savings in quantum gates, we suggest that these results are of importance for future experimental and technical large-nnn implementations of quantum computers.
In excellent agreement with our numerical simulations of Shor's algorithm, equipped with a truncated quantum Fourier transform of bandwidth bbb, we find that its performance scales $\ensuremath{\sim}$${2}^{\ensuremath{-}{\ensuremath{\xi}}_{b}n}$, where nnn is the number of qubits, ${\ensuremath{\xi}}_{b}=1.1\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}2b}$, and the bandwidth bbb is the number of quantum states coupled by the quantum Fourier transform. Nonexponential behavior is observed for small nnn and explained analytically. The large-nnn exponential scaling implies that b=7b=7b=7 is sufficient to operate a 1000-qubit quantum computer running Shor's algorithm on the 9595%95 performance level and implies hardware savings of the order of half a million rotation gates.
Competing computationally with experimental groups for the construction of scaling quantum computers, we simulate a complete quantum-gate by quantum-gate implementation of Shor's algorithm on a classical 128-core cluster computer. The resulting virtual quantum computer serves as a convenient quantum laboratory for the investigation of the effect of defects in the quantum circuitry. The class of defects studied here is the removal of all rotation gates with rotation angles $\ensuremath{\theta}<\ensuremath{\pi}/{2}^{b}$. Factoring semiprimes N=21,33,35,39,55,57N=21,33,35,39,55,57N=21,33,35,39,55,57, we find that the quantum computer still operates with acceptable performance (success probability of factoring) down to b=2b=2b=2. This is surprising since the deletion of rotation gates results in large errors in the arithmetic circuitry of the quantum computer. Extrapolating on the basis of these results we conclude that for quantum computers of practical interest more than 99% of rotation gates may be discarded with acceptable consequences in quantum computer performance. This result may be of interest to experimental physicists and quantum engineers currently embarked on designing efficient circuitry for scaling quantum computers.
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.
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.
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.
Factoring large numbers poses a significant chal-lenge in contemporary cryptography and security practices. Shor's algorithm presents a remarkable advancement over clas- sical methods by providing exponential acceleration in factoring large numbers, presenting a possible risk to traditional cryptography methods. The algorithm relies on the Quantum Fourier Transform (QFT) and modular exponentiation principles to efficiently ascertain the period (x) of a given large number (N). Upon determining the period using classical methods, the prime factors of the large number can be easily computed. To eval- uate the algorithm's efficiency and scalability, both simulation- based experiments and theoretical analyses are conducted. The algorithm is simulated utilizing the Qiskit back end to estimate resources, including Clifford gates, T gates, T depth, Full depth, and qubits, across a range of large numbers.