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arXiv:quant-ph/0001066 · Physical Review Letters

Efficient Factorization with a Single Pure Qubit and logN Mixed Qubits

S. Parker, Martin Bodo Plenio

Published Oct 2, 2000• 130 citations
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Abstract

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.
DOI
10.1103/physrevlett.85.3049

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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.
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.
In this note we consider optimised circuits for implementing Shor's quantum factoring algorithm. First I give a circuit for which none of the about 2n qubits need to be initialised (though we still have to make the usual 2n measurements later on). Then I show how the modular additions in the algorithm can be carried out with a superposition of an arithmetic sequence. This makes parallelisation of Shor's algorithm easier. Finally I show how one can factor with only about 1.5n qubits, and maybe even fewer.