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Efficient factorization with a single pure qubit

Scott Parker, Martin Bodo Plenio

Published Jan 18, 2000• 4 citations
OpenAlex

Abstract

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

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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.

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The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) the precise sense in which a quantum computer employing Shor's algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer can. It is made apparent that factoring $N$ generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.

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The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (conventional) computers. In 1994 Peter Shor showed that for sufficiently large N, a quantum computer could perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the nonexpert, the RSA encryption protocol; the various quantum computer manipulations constituting the Shor algorithm; how the Shor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer. It is made apparent that factoring N generally requires many successive runs of the algorithm. Our analysis reveals that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.

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Enrique Martín-López, Anthony Laing, Thomas Lawson +3 more

Quantum algorithms are computational routines that exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum factoring algorithm is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the large number of resources required, to date, there have been only four small scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which then qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. Significantly, the algorithmic output exhibits structure that is distinguishable from noise, in contrast to previous demonstrations.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•May 1, 2013•7 citations
It is commonly assumed that the initial state in Shor's quantum algorithm for the efficient factorization of large numbers requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of qubits in an arbitrary mixed state is sufficient to implement Shor's factorization algorithm efficiently.
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.
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.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) the precise sense in which a quantum computer employing Shor's algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer can. It is made apparent that factoring NNN generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (conventional) computers. In 1994 Peter Shor showed that for sufficiently large N, a quantum computer could perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the nonexpert, the RSA encryption protocol; the various quantum computer manipulations constituting the Shor algorithm; how the Shor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer. It is made apparent that factoring N generally requires many successive runs of the algorithm. Our analysis reveals that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
Quantum algorithms are computational routines that exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum factoring algorithm is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the large number of resources required, to date, there have been only four small scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which then qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. Significantly, the algorithmic output exhibits structure that is distinguishable from noise, in contrast to previous demonstrations.