Shor's Factorization Algorithm Research Articles

A formally certified end-to-end implementation of Shor’s factorization algorithm

Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but it is useful only if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a less recognized obstacle to correctness is that of human programming errors-"bugs." Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics. To address this problem, we have been working to adapt formal methods to quantum programming. With such methods, a programmer writes a mathematical specification alongside the program and semiautomatically proves the program correct with respect to it. The proof's validity is automatically confirmed-certified-by a "proof assistant." Formal methods have successfully yielded high-assurance classical software artifacts, and the underlying technology has produced certified proofs of major mathematical theorems. As a demonstration of the feasibility of applying formal methods to quantum programming, we present a formally certified end-to-end implementation of Shor's prime factorization algorithm, developed as part of a framework for applying the certified approach to general applications. By leveraging our framework, one can significantly reduce the effects of human errors and obtain a high-assurance implementation of large-scale quantum applications in a principled way.

CAPACITIVE COMPLEXITY OF DETERMINING GCD IN THE SHOR S ALGORITHM

The article analyzes the results of finding the period r of the function y = axmodM (a is a random number) which is used in the Shor's factorization algorithm for quantum computers. The module M is the product of two primes p and q. The article analyzes the solutions r obtained for various a, for which the capacitive complexity H of finding the greatest common divisor GCD(ar/2 + 1, M) is the least. A digital quantum computer is a classic processor and its digital quantum coprocessor. A digital quantum coprocessor with hundreds and thousands of digital qubits can be implemented in one programmable logic integrated circuit FPGA. In the Shor’s algorithm, the factorization problem of the number M reduces to the problem of determining the period r of the function y. It is known that GCD(ar/2 + 1, M) can be a divisor of the number M The task of the quantum coprocessor in implementing the Shor’s algorithm is to find the period r. After that it is necessary to find the GCD. Since for random a the problem of finding the period r has many solutions, these solutions can be compared by the value of one of the arguments when finding the GCD - the number ar / 2 . In this case, H = (rlog2a)/2 is taken for analysis. It approximately represents the bit depth of binary codes that a classic computer will have to process when determining the GCD. H can vary over a wide range from tens to thousands of bits even for small values of M. In this research the period r, which ensures the least complexity of the subsequent task of finding the GCD, is most often a solution for a = 3 and a = 2, but it can also occur often with other values of a. To clarify the revealed patterns, especially for large M, it is necessary to conduct additional research.

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.

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.

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.

Period finding with adiabatic quantum computation

We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the “period”, or xor mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its theoretical and experimental implications for the existence of an adiabatic version of Shor's integer factorization algorithm that would have the same complexity as the original algorithm.

Quantum rotations

Work in quantum computer architecture has focused on communication, layout and fault tolerance, largely driven by Shor's factorization algorithm. For the first time, we study a larger range of benchmarks and find that another critical issue is the generation of code sequences for quantum rotation operations. Specifically, quantum algorithms require arbitrary rotation angles, while quantum technologies and error correction codes provide only for discrete angles and operators. A sequence of quantum machine instructions must be generated to approximate the arbitrary rotation to the required precision. While previous work has focused exclusively on static compilation, we find that some applications require dynamic code generation and explore the advantages and disadvantages of static and dynamic approaches. We find that static code generation can, in some cases, lead to a terabyte of machine code to support required rotations. We also find that some rotation angles are unknown until run time, requiring dynamic code generation. Dynamic code generation, however, exhibits significant trade-offs in terms of time overhead versus code size. Furthermore, dynamic code generation will be performed on classical (non-quantum) computing resources, which may or may not have a clock speed advantage over the target quantum technology. For example, operations on trapped ions run at kilohertz speeds, but superconducting qubits run at gigahertz speeds. We introduce a new method for compiling arbitrary rotations dynamically, designed to minimize compilation time. The new method reduces compilation time by up to five orders of magnitude while increasing code size by one order of magnitude. We explore the design space formed by these trade-offs of dynamic versus static code generation, code quality, and quantum technology. We introduce several techniques to provide smoother trade-offs for dynamic code generation and evaluate the viability of options in the design space.

Quantum Simulation of Tunneling in Small Systems

A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because of the number of gates and qubits necessary, however, digital quantum particle simulations remain untested. A contributing factor to the system size required is the number of ancillary qubits needed to implement matrix exponentials of the potential operator. Here, we show that a set of tunneling problems may be investigated with no ancillary qubits and a cost of one single-qubit operator per time step for the potential evolution, eliminating at least half of the quantum gates required for the algorithm and more than that in the general case. Such simulations are within reach of current quantum computer architectures.

A quantum-inspired genetic algorithm for dynamic continuous network design problem

The Network Design Problem (NDP), known to be NP-complete, arises in many application contexts. Transportation NDP involves optimizing the network performance under a budget constraint. Due to its non-convex and discontinuous solution space, a majority of the research efforts have been focused on tackling this problem using metaheuristics. Quantum computing is an active research area in which numerous advances have been made since the early 1990s. Quantum algorithms, such as Shor's factorization algorithm and Grover's database search algorithm, have opened up the possibility of solving NP-hard/NP-complete problems in polynomial time with quantum computers. Quantum-inspired genetic algorithms (QGA), as one of the novel meta-heuristics, seek to use insights gained through quantum computing to improve upon the performance of traditional genetic algorithms. By using the qubit as the basic unit of information in QGA, complex search space can be probabilistically characterized by superposition of states. With a suitable QGA updating mechanism, the qubit chromosome gradually converges to a single state as each qubit in the chromosome reaches the stable state of either 0 or 1. Therefore, without an exhaustive search of the qubit states combination, the solution spaces can be effectively explored and a better quality solution can be obtained with the same level of population, generation, and overall computational effort. In this research, we present a floating-point encoded quantum-inspired genetic algorithm to solve the dynamic transportation network design problem. The proposed algorithm is empirically applied to the Sioux-Falls and Monticello networks to test its effectiveness and applicability. Through numerical experiments, the QGA outperforms the conventional genetic algorithm, even with a smaller population size. Technical, computational and practical issues involved in developing and calibrating quantum-inspired genetic algorithm are discussed.

Quantum computaton from a quantum logical perspective

It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.

Efficient classical simulation of the quantum Fourier transform

A number of elegant approaches have been developed for the identification of quantum circuits which can be efficiently simulated on a classical computer. Recently, these methods have been employed to demonstrate the classical simulability of the quantum Fourier transform (QFT). Here we show that one can demonstrate a number of simulability results for QFT circuits in a straightforward manner using Griffiths and Niu's semi-classical QFT construction (Griffiths and Niu 1996 Phys. Rev. Lett. 76 3228). We use this to analyse the simulability properties of the QFT with a variety of classes of entangled input states. We then discuss the consequences of these results in the context of Shor's factorization algorithm.

Design of quantum VQ iteration and quantum VQ encoding algorithm taking O ( N 1/2 ) steps for data compression

Vector quantization (VQ) is an important data compression method. The key of the encoding of VQ is to find the closest vector among N vectors for a feature vector. Many classical linear search algorithms take O(N) steps of distance computing between two vectors. The quantum VQ iteration and corresponding quantum VQ encoding algorithm that takes O(N1/2) steps are presented in this paper. The unitary operation of distance computing can be performed on a number of vectors simultaneously because the quantum state exists in a superposition of states. The quantum VQ iteration comprises three oracles, by contrast many quantum algorithms have only one oracle, such as Shor's factorization algorithm and Grover's algorithm. Entanglement state is generated and used, by contrast the state in Grover's algorithm is not an entanglement state. The quantum VQ iteration is a rotation over subspace, by contrast the Grover iteration is a rotation over global space. The quantum VQ iteration extends the Grover iteration to the more complex search that requires more oracles. The method of the quantum VQ iteration is universal.

Quantitative measure of interference

We introduce an interference measure which allows to quantify the amount of interference present in any physical process that maps an initial density matrix to a final density matrix. In particular, the interference measure enables one to monitor the amount of interference generated in each step of a quantum algorithm. We show that a Hadamard gate acting on a single qubit is a basic building block for interference generation and realizes one bit of interference, an ``ibit.'' We use the interference measure to quantify interference for various examples, including Grover's search algorithm and Shor's factorization algorithm. We distinguish between ``potentially available'' and ``actually used'' interference, and show that for both algorithms the potentially available interference is exponentially large. However, the amount of interference actually used in Grover's algorithm is only about $3\phantom{\rule{0.3em}{0ex}}\mathrm{ibits}$ and asymptotically independent of the number of qubits, while Shor's algorithm indeed uses an exponential amount of interference.

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

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.

Implementing Shor’s algorithm on Josephson charge qubits

We investigate the physical implementation of Shor's factorization algorithm on a Josephson charge qubit register. While we pursue a universal method to factor a composite integer of any size, the scheme is demonstrated for the number 21. We consider both the physical and algorithmic requirements for an optimal implementation when only a small number of qubits are available. These aspects of quantum computation are usually the topics of separate research communities; we present a unifying discussion of both of these fundamental features bridging Shor's algorithm to its physical realization using Josephson junction qubits. In order to meet the stringent requirements set by a short decoherence time, we accelerate the algorithm by decomposing the quantum circuit into tailored two- and three-qubit gates and we find their physical realizations through numerical optimization.

Effects of Imperfect Gate Operations in Shor's Prime Factorization Algorithm

AbstractThe effects of imperfect gate operations in implementation of Shor's prime factorization algorithm are investigated. The gate imperfections may be classified into three categories: the systematic error, the random error, and the one with combined errors. It is found that Shor's algorithm is robust against the systematic errors but is vulnerable to the random errors. Error threshold is given to the algorithm for a given number N to be factorized.

Efficient Factorization with a Single Pure Qubit andlogNMixed Qubits

It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number N requires a pure initial state. Here we demonstrate that a single pure qubit, together with a collection of log 2N qubits in an arbitrary mixed state, is sufficient to implement Shor's factorization algorithm efficiently.

Decoherence and relevant universality in quantum algorithms via a dynamic theory for quantum measurement

It is well known that environment may decohere a quantum bit (qubit) system immersed in it, making a quantum computation invalid. But the quantitative features of the decoherence seem to depend on both the constitution of the environment and the details of its coupling with the qubit system. In this paper, based on the dynamic approach for quantum measurement developed from the Hepp-Coleman model [K. Hepp, Helv. Phys. Acta 45, 237 (1972)], we generally model the environment as a collection of a large number of subsystems and then consider to what extent and in which way the environment and its coupling with the qubit system may affect a quantum computation process. In the weak-coupling limit, we find that as far as decoherence time is concerned, there is no essential difference between an environment of two-level subsystems and an environment of harmonic oscillators. This implies that there exists some universality independent of specific constitutions of environments. However, it is also shown that this is not true at finite temperature or in the case of strong coupling. So only if the coupling is weak and the temperature low does there exist the possibility of developing a universal scheme of controlling a qubit system such that the decoherence is avoided. The possible effect of environment on the efficiency of a quantum algorithm is also explicitly illustrated through the example of Shor's prime factorization algorithm.