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arXiv:quant-ph/0612021 · New Journal of Physics

Efficient classical simulation of the quantum Fourier transform

Dan E. Browne

Published May 23, 2007• 67 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/9/5/146

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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.
In this note we describe a simple and intriguing observation: the quantum Fourier transform (QFT) over ZqZ_qZq​, which is considered the most ``quantum'' part of Shor's algorithm, can in fact be simulated efficiently by classical computers. More precisely, we observe that the QFT can be performed by a circuit of poly-logarithmic path-width, if the circuit is allowed to apply not only unitary gates but also general linear gates. Recalling the results of Markov and Shi [MaSh] and Jozsa [Jo] which provided classical simulations of such circuits in time exponential in the tree-width, this implies the result stated in the title. Classical simulations of the FFT are of course meaningless when applied to classical input strings on which their result is already known; Our observation might be interesting only in the context in which the QFT is used as a subroutine and applied to more interesting superpositions. We discuss the reasons why this idea seems to fail to provide an efficient classical simulation of the entire factoring algorithm. In the course of proving our observation, we provide two alternative proofs of the results of [MaSh,Jo] which we use. One proof is very similar in spirit to that of [MaSh] but is more visual, and is based on a graph parameter which we call the ``bubble width'', tightly related to the path- and tree-width. The other proof is based on connections to the Jones polynomial; It is very short, if one is willing to rely on several known results.
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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.
The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor's algorithm for factoring products of primes.Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically.While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT.The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits.In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively.Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions.We demonstrate this speedup in test calculations on some simple functions.For data vectors of length 10 6 -10 8 , the speedup can be a few orders of magnitude.
The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called \emph{normalizer circuits}: a normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. In our main result we prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for ddd-level qudits arise as normalizer circuits over the group Zdm{\mathbf Z}_d^mZdm​. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.