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Comment on Demonstrations of Shor's Algorithm in the Past Decades.

Zhengjun Cao, Zhenfu Cao, Lihua Liu

Published Jan 1, 2015• 3 citations
OpenAlex

Abstract

Abstract. We remark that the experimental demonstrations of Shor’s algorithm in the past decades are falsely claimed and flawed, because they had used too less qubits in the first quantum register to accomplish the step of Continued Fraction Expansion in Shor’s algorithm. More worse, the amount of qubits used in some experiments are too less to represent all residues modulo n, which means that the number n cannot be truly involved in the related computations.

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arXiv:quant-ph/0102136•Journal of Modern Optics•Jul 1, 2002•25 citations
Abstract. We remark that the experimental demonstrations of Shor’s algorithm in the past decades are falsely claimed and flawed, because they had used too less qubits in the first quantum register to accomplish the step of Continued Fraction Expansion in Shor’s algorithm. More worse, the amount of qubits used in some experiments are too less to represent all residues modulo n, which means that the number n cannot be truly involved in the related computations.
An efficient quantum modular exponentiation method is indispensible for Shor's factoring algorithm. But we find that all descriptions presented by Shor, Nielsen and Chuang, Markov and Saeedi, et al., are flawed. We also remark that some experimental demonstrations of Shor's algorithm are misleading, because they violate the necessary condition that the selected number q=2sq=2^sq=2s, where sss is the number of qubits used in the first register, must satisfy n2≤q<2n2n^2 \leq q < 2n^2n2≤q<2n2, where nnn is the large number to be factored.
Although the fundamental principles of quantum computing have been known for decades, it is only within the past few years that practical quantum computers have become available. Since these systems are limited to a small number of qubits, they cannot demonstrate quantum advantage for many practical problems. Accordingly, there has been an effort to develop near term implementations of algorithms purely for research and education. In this paper, we discuss a near term implementation of Shor's Algorithm using the Qiskit language on an IBM Q System One quantum computer. We extend currently available documentation by providing a full circuit diagram for Shor's Algorithm including gates specific to the implementation using Qiskit. We present an implementation capable of factoring small two-digit prime numbers, and discuss the limitations of noise when using real quantum computers vs. simulations.
We demonstrate that, in the case of Shor's algorithm for factoring, highly mixed states will allow efficient quantum computation, indeed factorization can be achieved efficiently with just one initial pure qubit and a supply of initally maximally mixed qubits (S. Parker and M. B. Plenio, Phys. Rev. Lett. 85, 3049 (2000)) . This leads us to ask how this affects the entanglement in the algorithm. We thus investigate the behaviour of entanglement in Shor's algorithm for small numbers of qubits by classical computer simulation of the quantum computer at different stages of the algorithm. We find that entanglement is an intrinsic part of the algorithm and that the entanglement through the algorithm appears to be closely related the amount of mixing. Furthermore, if the computer is in a highly mixed state any attempt to remove entanglement by further mixing of the algorithm results in a significant decrease in it's efficiency.
I will describe our recent implementation of a five-qubit NMR quantum computer capable of executing a quantum algorithm for order-finding. This is the first demonstration of a quantum algorithm with the structure of Shor's algorithm for efficient prime factorization, a key step in the experimental realization of quantum computers because its structure appears to be common to all quantum algorithms that achieve an exponential speed-up over their classical counterparts. The experiment was made possible by the synthesis of a 5-spin molecule with remarkable NMR properties, and by the introduction of new techniques for state initialization and quantum control.
We demonstrate that, in the case of Shor's algorithm for factoring, highly mixed states will allow efficient quantum computation, indeed factorization can be achieved efficiently with just one initial pure qubit and a supply of initially maximally mixed qubits (Parker, S. and Plenio, M. B., 2000, Phys. Rev. Lett., 85, 3049). This leads us to ask how this affects the entanglement in the algorithm. We thus investigate the behaviour of entanglement in Shor's algorithm for small numbers of qubits by classical computer simulation of the quantum computer at different stages of the algorithm. We find that entanglement is an intrinsic part of the algorithm and that the entanglement through the algorithm appears to be closely related to the amount of mixing. Furthermore, if the computer is in a highly mixed state any attempt to remove entanglement by further mixing of the algorithm results in a significant decrease in its efficiency.